Nonlinear Industrial Control Systems: Optimal Polynomial Systems and State-Space Approach 1447174550, 9781447174554

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Michael J. Grimble Paweł Majecki

Nonlinear Industrial Control Systems Optimal Polynomial Systems and State-Space Approach

Nonlinear Industrial Control Systems

Michael J. Grimble Paweł Majecki •

Nonlinear Industrial Control Systems Optimal Polynomial Systems and State-Space Approach

123

Michael J. Grimble Department of Electronic and Electrical Engineering University of Strathclyde Glasgow, UK

Paweł Majecki Industrial Systems and Control Limited Glasgow, UK

ISBN 978-1-4471-7455-4 ISBN 978-1-4471-7457-8 https://doi.org/10.1007/978-1-4471-7457-8

(eBook)

MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 1 Apple Hill Drive, Natick, MA, 01760-2098, USA LabVIEW™ is a trademark of National Instruments. This book is an independent publication. National Instruments is not affiliated with the publisher or the author, and does not authorize, sponsor, endorse or approve this book. © Springer-Verlag London Ltd., part of Springer Nature 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer-Verlag London Ltd. part of Springer Nature. The registered company address is: The Campus, 4 Crinan Street, London, N1 9XW, United Kingdom

The fool doth think he is wise, but the wise man knows himself to be a fool. —William Shakespeare, (As You Like It, Act 5, Scene 1)

To my lovely family including wife Wendy, children Claire and Andrew, and grandchildren Callum and Emma. Michael J. Grimble To my Parents—Moim Rodzicom. And, of course, to Anka, for her patience during all these long hours (and years) spent on ‘the Book’. Paweł Majecki

Foreword

Control theory and its applications have made considerable progress by using model-based analysis of process dynamics to understand how controller design needs to meet the demands and compensate for the effects of the dynamical behaviour of that process on controller performance. It is difficult to conceive of any design techniques that avoid this basic need with any degree of success, particularly if the controller must produce significant performance improvements. A glance at the published texts and papers quickly reveals that much of the progress in our understanding assumes that the process has a linear dynamical model. This assumption has been vital and has opened up the possibility of techniques based on state-space modelling, Laplace transform, frequency-domain concepts and the related ideas of poles and zeros of the transfer-function (matrix) of the system. The stability of feedback systems has a formal representation in this language and is now familiar to many practicing control engineers. One branch of design theory is often seen as the task of manipulation of frequency-domain characteristics, guided by rules derived from simple low-order models that can be analysed on paper or using standard design software. Another uses optimization methodologies to ensure stability and guide performance outcomes by careful choice of a cost function to be minimized. Optimization of quadratic cost functions fits naturally with the linear process dynamics, as it tends to produce linear feedback laws. This situation is very attractive for design, commissioning and other issues that must be faced in applications. It has the advantage of being amenable to an analysis that often reveals useful computational tools and links process dynamics, the parameters in the cost function and the general form of performance that can be expected. It has a long history beginning over 60 years ago with the arrival of Pontriagin’s Maximum Principle and Dynamic Programming and then finding its feet in the engineering community when the seminal state-space analysis of Linear Quadratic Optimal Control Problems was published. It now has many variants; the most notable in the context of this monograph is the introduction of the effects of noise in Optimal Filtering Problems and the Linear Quadratic Gaussian Control Problem, where optimal control solutions in the presence of noise are characterized in a useful feedback form. ix

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The problem with nonlinear systems is revealingly expressed by saying that, whereas linearity is a property, nonlinearity is not a property—it is the absence of a property. Consequently, it is almost impossible to make any useful statement until more information is added. This explains why the well-known technique of linearization of process dynamics, about a known condition, plays a common role, if the nonlinearity is important but small enough in the region of operation to allow the successful use of an approximate linear model. If linearization is inadequate, progress is normally made by making assumptions about the form of the nonlinearity and it is here that control theory takes many paths, depending on the assumptions made. In these cases, any progress made is specific to the type of nonlinearity encountered and does not transfer to others. This text provides a helpful review of many of the linear optimization-based techniques with an emphasis on Minimum Variance (MV), Generalized Minimum Variance (GMV), Linear Quadratic Gaussian (LQG) optimal control and Predictive Control (MPC) paradigms. It contains elements of theory and design and adds an applications flavour that reflects the wide experience of the authors, using detailed design examples from applications including wind turbines, marine systems, steel making, robotics and aerospace. The central thrust of the development is to consider nonlinear systems for which so-called Nonlinear GMV (NGMV), Nonlinear Quadratic GMV (NQGMV) and the Nonlinear Predictive GMV (NPGMV) approaches can be successful. The motivation is that of the simplicity of design, commissioning and implementation in real-world applications. The core of the technical development is to build on known linear methods and to create approximation scenarios and design conditions within which an acceptable feedback solution to an optimization problem can be computed. It is an ambitious target that is approached using careful construction of design specifications, ensuring direct or indirect links to the linear case are retained, and emphasizing the structure of the controller implementation. Although the presentation is necessarily mathematical in its nature, the authors mix rigour with design rules and illustrate ideas using simple examples as the text progresses. Given the author’s considerable experience in the area of optimization-based control theory and applications, the text is an important contribution to the continuing efforts to improve and extend the arsenal of tools that are available for the systematic and successful design of nonlinear control systems. Many readers will benefit from a study of problem formulation, design discussions and examples and some will be inspired to build on these efforts to produce further useful refinements. Biographical Note Professor Owens is Principal Scientist at Zhengzhou University in the People’s Republic of China, an Emeritus Professor at the University of Sheffield and a Visiting Professor at the University of Southampton. Prof. David H. Owens FREng Zhengzhou University, Zhengzhou, China University of Sheffield 2017, Sheffield, UK

Preface

This text was developed mainly from the authors’ experience working for Industrial Systems and Control Limited (ISC) and for the University of Strathclyde’s Industrial Control Centre (ICC) in Glasgow. The former (ISC Limited) is a consulting company established by the University more than three decades ago to encourage technology transfer. The authors have also been inspired by the industrial training courses run under the umbrella of the Applied Control Technology Consortium (ACTC). This international training network has provided invaluable feedback from experienced engineers. Mike Grimble was instrumental in the establishment of the three activities that have provided most of the applications described. The text includes a mixture of fundamental concepts, design and applications experience. It covers quite a range of nonlinear-model-based design methods and focusses upon methods where there is a reasonable theoretical basis and some applications potential. There is a requirement for control laws that will satisfy an industrial need, and the main emphasis is, therefore, on control laws that are relatively simple to design and implement. Considering the complexity of nonlinear control problems, the methods are simple conceptually, even if the algebra is in places a little messy. It attempts to link the control design methods to existing industrial practice, where there is some confidence in the methods used. For example, links are established to PID control, to the Smith Predictor, to traditional model predictive control and to the internal model control principle. The synthesis and design of controllers and estimators for nonlinear industrial systems is the focus of the text, using various forms of system modelling. The system models include a combination of nonlinear operator and linear state-equation or polynomial matrix model forms. State-dependent, Linear Parameter-Varying (LPV) and Quasi-LPV model forms are also considered. Most of the results are presented in discrete-time systems form, based on the very reasonable expectation that most industrial systems will be implemented digitally. The applications chapters 14 and 15 provide a range of industrial control design studies based on experience gained from industrial or research projects undertaken.

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The book describes a range of modern linear and nonlinear control design methods, including Minimum Variance, Generalized Minimum Variance, Linear Quadratic Gaussian, H∞ Linear and Nonlinear Control, Generalized Predictive Control, Nonlinear Generalized Minimum Variance Control, Nonlinear Generalized Predictive Control and Nonlinear Predictive Generalized Minimum Variance Control methods. In addition, the text covers the use of time-varying Kalman filtering methods and introduces Nonlinear Minimum Variance estimation methods for condition monitoring and fault detection. The motivation for the text stems from the belief that much of the existing material on nonlinear control systems design is hard for engineers to use without a lot of background study in areas of mathematics that are unfamiliar. These methods are, therefore, some way from being applied in practice. However, nonlinear control is essential in certain applications, where nonlinearities limit performance or cause unpredictable and possibly unsafe behaviour. In many applications, where the boundaries of performance are reached, it is the nonlinear elements, which determine the limits of operation. For example, modern fighter aircraft can produce manoeuvres well outside the linear flight dynamics to achieve a tactical advantage. Control methods based upon local linearization are used but are expensive to validate and to certify in all regions of operation. There is, therefore, an opportunity for nonlinear controllers that can cover a wider range of flight conditions. Many of the nonlinear control algorithms described in this text are related to well known and popular linear control design methods. The book is concerned with nonlinear control synthesis methods that have some formal underlying theory, and it covers engineering design issues and a wide range of applications. The theory of control systems is normally aimed at generalizing the results as much as possible so that a comprehensive solution is obtained to a range of problems. However, in practice, the most practical solution to a real problem often depends upon a tailored solution that exploits the physical structure of the system. There is, therefore, a certain incompatibility between the development of a good general theory and the production of a simple and practical working solution. This gap has not been bridged here but there is an attempt throughout the design methods presented to be aware of the needs of the end user and to provide some of the tools needed for more bespoke solutions. The text is aimed at graduate-level researchers, engineers in industry, final-year degree students, and industrial taught course attendees. It has a balance of engineering design and basic theory. It is mostly concerned with optimal control methods for the design of nonlinear systems and includes numerous examples and industrial applications. It is accompanied by a MATLAB® software toolbox and builds on experience implementing the techniques in real applications. The toolbox includes all the control algorithms and the industrial application models, simulations and solutions. This toolbox was initially developed with the aid of United Kingdom Engineering and Physical Sciences Research Council grants on Nonlinear Industrial Control.

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Readers should find the toolbox similar to other MATLAB® toolboxes having documentation and demo examples. It should be relatively easy to use but it is not a commercial toolbox of course. The reader will be able to apply the control design methods proposed using the library of MATLAB® software provided. The toolbox was developed by co-author Dr. Paweł Majecki but it includes contributions from Dr. Dimos Fragopoulos, Dr. Shamsher Ali Naz and other colleagues in both the University and Company. We are grateful to MathWorks for access to MATLAB® under their book program. It was also valuable to have access to the Polynomial Toolbox by PolyX for use with the MIMO polynomial algorithms, and access to the Marine Systems toolbox due to Thor Fossen and colleagues. We are indebted to our colleagues within Industrial Systems and Control Limited that have provided us with a continuous source of inspiration and problems. These include Dr. Andy Clegg (Managing Director), Dr. Gerrit van der Molen (Engineering Director), together with Dr. Xiaohong Guan, Dr. Petros Savvidis, Dr. Arkadiusz Dutka (now with Diehl Controls), Dr. Megan McGookin and Dr. Luca Cavanini. We are particularly grateful for the application examples concerning wind turbine control and the aircraft electro-optical servos provided by our colleague Dr. Petros Savvidis in Chaps. 14 and 15, respectively. We miss the very kind and patient Chairman of our company, Mr. Andrew Buchanan, who sadly passed away. We should also like to acknowledge our debt to the members of the Industrial Control Centre of the University of Strathclyde that contributed so substantially to our research including current and past academic and research staff. We should particularly like to thank Prof. Michael A. Johnson for his many contributions to the development of the Industrial Control Centre. Mike Grimble would also like to acknowledge the inspiration provided by his experience on the Industrial Automation Group at Imperial College, and particularly the guidance of Prof. Greyham Bryant, and his friend and inspirational research advisor Dr. Martin Fuller. We would also like to thank and record the cooperation of our past and current academic staff colleagues including Prof. Bill Leithead, Dr. Reza Katebi, Prof. Andrzej Ordys, Dr. Hong Yue, Dr. Jacqueline Wilkie, Dr. Joe McGhee, Dr. Ian Henderson and Dr. Akis Petropoulakis. We are grateful to academic visitors including Prof. Ikuo Yamamoto (Nagasaki University), Prof. Pedro Albertos (Universidad Politécnica de Valencia), Prof. Jakob Stoustrup (Aalborg), Prof. Adrian Stoica (Bucharest) and Dr. Alkan Alkaya (Mersin University) for their contributions. The project collaboration with Profs. Basil Kouvaritakis and Mark Cannon of the University of Oxford, and with Dr. David Anderson, Prof. David Murray-Smith and Dr. Douglas Thomson, at the University of Glasgow, were appreciated. The many years of links and discussions with Prof. David Owens of the University of Sheffield were greatly valued. We should also like to thank Dr. Luisella Balbis for the introductory material on hybrid systems and the contribution from Mr. Ikponmwosa Kenneth Osagie on the Shell heavy oil fractionator example. Our fellow researchers in the Centre contributed in many ways to the text. These included: Prof. Yan Pang, Dr. Gerald Hearns, Dr. Arek Dutka, Dr. Leonardo Giovanini, Dr. Peter Martin, Dr. Shamsher

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Ali-Naz, Dr. Damien Uduehi, Dr. Jonas Balderud, Dr. Diyar Abdulkadar, Dr. Raad Hashim, Dr. Steven Forrest, Dr. Diane Biss, Dr. Susan Carr, Dr. Barry Connor, Dr. Mahmoud Akbar, Dr. Ilyas Eker, Dr. Ahmed Medhat, Dr. Alaa El-Dien Shawky Mohamedy, Dr. Gamal El-Sheikh, Dr. Massimo Calligaris, Dr. Santo Inzerillo, Dr. Marie O’Brien, Mr. Piotr Czechowski, Dr. Hao Xia, Dr. Sergio Enrique Pinto-Castillo, Dr. Gordon McNeilly, Dr. David Greenwood, Dr. Farid Benazzi, Dr. Luis Recalde, Mr. Domenico Gorni, Mr. Sultan Mubarak Q. Alotaibi and Mr. Cagatay Cebeci. We are also very grateful for discussions and contributions to the text from our friends in the industry including the following: • Dr. Chen-Fang Chang, Dr. Ibrahim Haskara, Dr. Yiran Hu and Dr. Hossein Javaherian of General Motors in Warren and Pontiac, Michigan. • Chief Engineer Craig Stephens, Dr. Eric Tseng, Dr. Scott Varnhagen at Ford, Dearborn, Michigan and Dr. Jeff Cook (Adjunct Professor at the University of Michigan). • Dr. Hussein Dourra, Michael J. Prucka, Ethan Bayer and Dr. Chen Gang at Fiat Chrysler Automobiles (FCA), Auburn Hills, Michigan. • Dr. Edmund P. Hodzen at Cummins Inc. in Columbus, Indiana. • Dr. Jeannie Falcon at National Instruments in Austin, Texas. • Dr. Ken Butts in Toyota at Ann Arbor, Michigan. • Dr. Matt Macdonald at SELEX Galileo in Edinburgh. The kind support in preparing the manuscript and his many contributions to the Centre’s development, by Andrew Smith (University of Strathclyde) was particularly valued and appreciated. Finally, the help of Sheila Campbell in producing papers and presentations for the group was appreciated. A Note on Permissions We are grateful to the following for permissions to reproduce copyright material: The Fig. 14.19 was adapted from the text “The Shell Process Control Workshop,” published by Butterworth, and by Prett, D. M. and Morari, M., page 356, 1987, with permission from Elsevier. The Figs. 15.37, 15.39 and 15.40 were based on a conference publication by Savvidis, P., Anderson, D., Grimble M. J., “Application of nonlinear generalised minimum variance to the nadir problem in 2-axis gimbal pointing and stabilization,” SPIE Conference on Defence, Security, and Sensing, Proceedings Volume 7696, Orlando, Florida, 5th– 9th April, 2010, doi:10.1117/12.849523, organised by the SPIE (the international society for optics and photonics) who kindly provided permission to publish. Michael J. Grimble Paweł Majecki Glasgow, UK

Contents

Part I 1

Introduction and Linear Systems

Introduction to Nonlinear Systems Modelling and Control . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 PID Control of Nonlinear Industrial Systems 1.1.2 Optimal Control of Nonlinear Industrial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Linear and Nonlinear Dynamic Systems . . . . 1.1.4 Nonlinearities in Physical Systems . . . . . . . . 1.1.5 Smooth and Non-smooth Nonlinearities . . . . 1.2 Types of Static Nonlinear Models . . . . . . . . . . . . . . . 1.2.1 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Stiction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Deadzone . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Coulomb Friction Characteristics . . . . . . . . . 1.3 Nonlinear Dynamic Systems Behaviour . . . . . . . . . . . 1.3.1 Nonlinear Systems Behaviour . . . . . . . . . . . 1.3.2 Stability Analysis and Concepts . . . . . . . . . . 1.4 Nonlinear Control Systems Analysis Methods . . . . . . . 1.4.1 Stability and Robustness Analysis Methods . 1.5 Nonlinear Control Design Methods . . . . . . . . . . . . . . 1.5.1 Describing Function Methods . . . . . . . . . . . 1.5.2 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . 1.5.3 Variable Structure and Sliding Mode Control 1.5.4 Lyapunov Control Design . . . . . . . . . . . . . . 1.5.5 Feedback Linearization . . . . . . . . . . . . . . . . 1.5.6 High-Gain Control . . . . . . . . . . . . . . . . . . . 1.5.7 Backstepping . . . . . . . . . . . . . . . . . . . . . . .

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1.5.8 Dynamic Inversion . . . . . . . . . . . . . . . . . . . . 1.5.9 Nonlinear Filtering . . . . . . . . . . . . . . . . . . . . 1.5.10 Adaptive Control . . . . . . . . . . . . . . . . . . . . . 1.6 Linearization, Piecewise-Affine Systems and Scheduling 1.6.1 Linearization Methods . . . . . . . . . . . . . . . . . . 1.6.2 Piecewise-Affine Dynamical Systems . . . . . . . 1.6.3 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . 1.7 LPV System Modelling and Control . . . . . . . . . . . . . . . 1.7.1 Modelling LPV Systems . . . . . . . . . . . . . . . . 1.7.2 Deriving an LPV Model by Jacobian Linearization . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Quasi-LPV and State-Dependent Models . . . . 1.7.4 Derivation of Quasi-LPV Models . . . . . . . . . . 1.7.5 Velocity-Based LPV Models . . . . . . . . . . . . . 1.7.6 Discrete-Time LPV State-Space Model . . . . . . 1.8 Introduction to Hybrid System Models and Control . . . . 1.8.1 Hybrid Phenomena and Definitions . . . . . . . . 1.8.2 Modelling Hybrid Systems . . . . . . . . . . . . . . 1.8.3 Optimization Methods and Hybrid Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 Hybrid Optimal Control and Optimization . . . 1.9 Nonlinear System Identification and Auto-tuning . . . . . 1.9.1 Nonlinear System Parameter Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Role of Nonlinear System Identification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.3 LPV or qLPV System Identification . . . . . . . . 1.9.4 Orthonormal Basis Functions . . . . . . . . . . . . . 1.9.5 Model Assessment . . . . . . . . . . . . . . . . . . . . 1.9.6 Relay Auto-tuning . . . . . . . . . . . . . . . . . . . . 1.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

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Review of Linear Optimal Control Laws . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stochastic System Polynomial Models and Control . . . . . 2.2.1 Minimum Variance Control Fundamentals . . . . 2.2.2 Minimum-Variance Control Law Derivation . . . 2.2.3 MV Control Law for Non-minimum-phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Stochastic Versus Deterministic Signal Models .

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Generalized Minimum Variance Control . . . . . . . . . . . . . 2.3.1 Scalar GMV Control Law Derivation . . . . . . . . 2.3.2 Remarks on the GMV Criterion . . . . . . . . . . . . 2.4 LQG, GLQG and GMV Control Laws . . . . . . . . . . . . . . 2.4.1 Multivariable Linear System Description . . . . . 2.4.2 Sensitivity Matrices and Signal Spectra . . . . . . 2.4.3 Solution of the Scalar LQG Cross-Product Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Scalar Linear Optimal Control Examples . . . . . 2.5 Multivariable LQG, GLQG, MV and H∞ Cost Problems . 2.5.1 Multivariable GLQG Cost-Function Problem . . 2.5.2 Solution of the Multivariable GLQG Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 LQG with Cross-Product Criterion and Links . . 2.5.4 Multivariable Minimum Variance Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Equivalence of the Multivariable GLQG and LQG Solutions . . . . . . . . . . . . . . . . . . . . . 2.5.6 H∞ Control Design and Relationship to GLQG 2.6 Restricted Structure Control Systems . . . . . . . . . . . . . . . 2.6.1 Controller Order Reduction . . . . . . . . . . . . . . . 2.6.2 LQG Restricted Structure Control for SISO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Parametric Optimization Problem . . . . . . . . . . . 2.6.4 RS Controllers for Benchmarking Performance . 2.6.5 LQG Restricted Structure Control for MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Solution of the Parametric Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.7 Control System Structures Assessment . . . . . . . 2.7 Nonlinear Systems and Uncertainty . . . . . . . . . . . . . . . . 2.7.1 Multiple-Model Control . . . . . . . . . . . . . . . . . 2.7.2 Multiple-Model Example . . . . . . . . . . . . . . . . . 2.7.3 Nonlinear Minimum Variance Control and Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.4 MV and GLQG Constrained or Saturating Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5 Switching Systems . . . . . . . . . . . . . . . . . . . . . 2.7.6 Linear Matrix Inequalities . . . . . . . . . . . . . . . . 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II 3

4

Polynomial Systems Nonlinear Control . . . . . . . . .

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Nonlinear GMV Feedback Optimal Control . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Strategy and Problem Formulation . . . . . . . . . . . 4.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Plant Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Nonlinear Generalized Minimum Variance Control . . . . . . 4.3.1 NGMV Optimal Feedback Control Solution . . . . 4.3.2 Non-square Systems . . . . . . . . . . . . . . . . . . . . . 4.3.3 Stability and Weighting Choice . . . . . . . . . . . . . 4.3.4 Stability of the System . . . . . . . . . . . . . . . . . . . 4.3.5 Asymptotic Conditions and Inverse Plant . . . . . . 4.3.6 Benchmarking Nonlinear Controllers . . . . . . . . . 4.4 Process Tank Level Nonlinear Control . . . . . . . . . . . . . . . 4.4.1 NGMV Feedback Control Design Results . . . . . . 4.5 Feedback, Tracking and Feedforward Control . . . . . . . . . . 4.5.1 Linear and Nonlinear Subsystem Models . . . . . . 4.5.2 Nonlinear GMV Control Problem with Tracking and Feedforward . . . . . . . . . . . . . . . . . . . . . . . .

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Open-Loop and Feedforward Nonlinear Control . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonlinear System Description . . . . . . . . . . . . . . . . . . . 3.2.1 Control Problem Options . . . . . . . . . . . . . . . . 3.2.2 Linear and Nonlinear System Structure . . . . . 3.2.3 Plant and Disturbance Subsystem Descriptions 3.2.4 Signals and Relationships . . . . . . . . . . . . . . . 3.3 NGMV Open-Loop and Feedforward Control . . . . . . . . 3.3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Solution of the Open-Loop/Feedforward Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Polynomial Models for System . . . . . . . . . . . 3.3.4 Diophantine Equations . . . . . . . . . . . . . . . . . 3.3.5 Optimization and Signals . . . . . . . . . . . . . . . 3.4 Design and Implementation Issues . . . . . . . . . . . . . . . . 3.4.1 Benchmarking Performance . . . . . . . . . . . . . . 3.4.2 Implementation Issues . . . . . . . . . . . . . . . . . . 3.4.3 Existence of the Nonlinear Operator Inverse . . 3.4.4 Weightings and Scheduling . . . . . . . . . . . . . . 3.4.5 Benefits of NGMV Open-Loop/Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Example of Feedforward and Tracking Control . . . . . . . 3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Solution for Feedback, Tracking and Feedforward Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Introduction of the Diophantine Equations . . . . . . 4.5.5 Optimization and Benchmarking . . . . . . . . . . . . . 4.5.6 Stability Conditions . . . . . . . . . . . . . . . . . . . . . . 4.6 Feedforward and Tracking Control Design Example . . . . . . 4.6.1 Results for Feedforward and Feedback Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Nonlinear Control Law Design and Implementation . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Desirable Characteristics of an Industrial Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Nonlinear Controllers and Design Methods . . . 5.1.3 Optimal Nonlinear Control . . . . . . . . . . . . . . . 5.1.4 Internal Model Control Principle . . . . . . . . . . . 5.2 Optimal Cost-Function Weighting Selection . . . . . . . . . . 5.2.1 NGMV Free Weighting Choice . . . . . . . . . . . . 5.2.2 NGMV PID Motivated Weightings . . . . . . . . . 5.2.3 NGMV a-PID Motivated Weightings . . . . . . . . 5.2.4 General Weighting Selection Issues . . . . . . . . . 5.2.5 Nonlinear GMV Control Design Example . . . . . 5.2.6 Scheduled Weightings . . . . . . . . . . . . . . . . . . . 5.3 NGMV Control of Plants with Input Saturation . . . . . . . . 5.3.1 Classical Anti-windup Protection . . . . . . . . . . . 5.3.2 NGMV Solution of the Anti-windup Problem . . 5.3.3 Example: Control of Plant with Saturation . . . . 5.3.4 Other Input Nonlinearities . . . . . . . . . . . . . . . . 5.4 Algebraic Loop and Implementation . . . . . . . . . . . . . . . . 5.4.1 Algebraic Loops in Implementing the Controller . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Example for Nonlinear State-Space Model . . . . 5.4.3 Use of Scaling in the Algebraic-Loop Solution . 5.4.4 Avoiding the Algebraic Loop and Scaling . . . . 5.4.5 Modified Weighting to Avoid Algebraic Loop . 5.5 Transport-Delay Compensation Structures . . . . . . . . . . . . 5.5.1 The Smith Predictor . . . . . . . . . . . . . . . . . . . . 5.5.2 Nonlinear Smith Predictor . . . . . . . . . . . . . . . . 5.5.3 Smith Predictor and PID-Motivated Weightings

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Robustness Analysis and Uncertainty . . . . . . . . . . . . . . 5.6.1 Control Sensitivity Functions and Uncertainty 5.6.2 Uncertainty and Fuzzy/Neural Networks . . . . 5.6.3 Quantitative Feedback Theory . . . . . . . . . . . . 5.6.4 Linear Matrix Inequalities . . . . . . . . . . . . . . . 5.6.5 Restricted Structure Nonlinear Controllers . . . 5.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Nonlinear Quadratic Gaussian and H∞ Robust Control . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Introduction to NQGMV Control Problem . . . . . . . . . . . . 6.2.1 Nonlinear System Description . . . . . . . . . . . . . 6.2.2 Signals in the Feedback System . . . . . . . . . . . . 6.2.3 Filter Spectral Factor . . . . . . . . . . . . . . . . . . . . 6.2.4 Control Spectral Factor . . . . . . . . . . . . . . . . . . 6.3 Nonlinear Quadratic Generalized Minimum Variance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Solution NQGMV Optimal Control Problem . . . 6.3.2 Diophantine Equations . . . . . . . . . . . . . . . . . . 6.3.3 Simplification of the Inferred Signal . . . . . . . . 6.3.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 NQGMV Design and Stability . . . . . . . . . . . . . . . . . . . . 6.4.1 Closed-Loop Stability . . . . . . . . . . . . . . . . . . . 6.4.2 Treatment of Explicit Transport Delay Terms . . 6.4.3 Example of NQGMV Control Design . . . . . . . . 6.5 Robust Control Design Philosophy . . . . . . . . . . . . . . . . . 6.5.1 H∞ Control Methods . . . . . . . . . . . . . . . . . . . 6.5.2 Problem Description . . . . . . . . . . . . . . . . . . . . 6.5.3 Uncertainty Descriptions . . . . . . . . . . . . . . . . . 6.5.4 Nonlinear Generalized Minimum Variance . . . . 6.6 NGH∞ Optimal Control Problem . . . . . . . . . . . . . . . . . . 6.6.1 Linking Lemma . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Robustness Weighting Function . . . . . . . . . . . . 6.6.3 Solution of the NGH∞ Control Problem . . . . . . 6.6.4 Super-Optimality and Eigenvalue Problem . . . . 6.6.5 Cost-Function Weighting Selection and Robustness . . . . . . . . . . . . . . . . . . . . . . . . 6.6.6 GH∞ Controller Example . . . . . . . . . . . . . . . . 6.6.7 Multivariable NGH∞ Control Example . . . . . . 6.7 Improving the Stability and Performance of H∞ Designs . 6.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7

Linear and Nonlinear Predictive Optimal Control . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Historical Perspective . . . . . . . . . . . . . . . . . . . . 7.1.2 Predictive Functional Control . . . . . . . . . . . . . . 7.1.3 Nonlinear Model Predictive Control . . . . . . . . . . 7.1.4 Implicit and Explicit MPC and Algorithms . . . . . 7.1.5 Simple Nonlinear Model Predictive Control . . . . 7.2 Nonlinear System Description . . . . . . . . . . . . . . . . . . . . . 7.2.1 Linear Subsystem Polynomial Matrix Models . . . 7.2.2 Optimal Linear Prediction Problem . . . . . . . . . . 7.2.3 Derivation of the Predictor . . . . . . . . . . . . . . . . 7.2.4 Matrix Representation of the Prediction Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Generalized Predictive Control for Linear Systems . . . . . . 7.3.1 GPC Optimal Control Solution . . . . . . . . . . . . . 7.3.2 Receding Horizon Principle . . . . . . . . . . . . . . . . 7.3.3 Choice of Prediction and Control Horizons . . . . 7.3.4 Equivalent GPC Cost Minimization Problem . . . 7.3.5 Modified Cost-Index Giving GPC Control . . . . . 7.4 NPGMV Control Problem . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Solution of the NPGMV Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Implementation of the NPGMV Optimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Properties of the Predictive Optimal Controller . . 7.5 Stability of the Closed-Loop System . . . . . . . . . . . . . . . . 7.5.1 Relationship to the Smith Predictor and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Design, Limits and Constraints . . . . . . . . . . . . . . . . . . . . 7.6.1 Limits, Constraints and Quadratic Programming . 7.6.2 Reference Governors . . . . . . . . . . . . . . . . . . . . . 7.7 Ship Roll Stabilization Using Fins and Rudder . . . . . . . . . 7.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.3

Nonlinear GMV Control . . . . . . . . . . . . . . . Solution for the NGMV Optimal Control . . Remarks on the NGMV Control Solution . . Structure of Controller in Terms of Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Avoiding Algebraic Loop Implementation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Stability of the Closed-Loop System . . . . . 8.4 Relationship to the Smith Predictor and Robustness . 8.4.1 Black-Box Nonlinearity and Robustness . . . 8.5 Multivariable Control Design Example . . . . . . . . . . . 8.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Optimal 8.3.1 8.3.2 8.3.3

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State-Space Nonlinear Predictive Optimal Control . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Solution Philosophy . . . . . . . . . . . . . . . . . . . . 9.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Linear State-Space Subsystem . . . . . . . . . . . . . 9.2.2 Linear State Prediction Model . . . . . . . . . . . . . 9.2.3 Output Prediction Model . . . . . . . . . . . . . . . . . 9.2.4 Kalman Estimator in Predictor Corrector Form . 9.3 Generalized Predictive Control . . . . . . . . . . . . . . . . . . . . 9.3.1 Vector Form of GPC Cost-Function . . . . . . . . . 9.3.2 Orthogonality and GPC Solution . . . . . . . . . . . 9.3.3 Equivalent GPC Cost Optimisation Problem . . . 9.3.4 Modified Cost-Function Generating GPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Nonlinear Predictive GMV Control Problem . . . . . . . . . . 9.4.1 Condition for Optimality . . . . . . . . . . . . . . . . . 9.4.2 NPGMV Optimal Control Solution . . . . . . . . . . 9.4.3 NPGMV Control Problem Results . . . . . . . . . . 9.4.4 Implementation of the Predictive Controller . . . 9.5 Systems Analysis and Stability . . . . . . . . . . . . . . . . . . . 9.5.1 System Stability . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Cost-Function Weightings and Stability . . . . . . 9.5.3 Relationship to the Smith Predictor . . . . . . . . . 9.6 Automotive Engine Control Example . . . . . . . . . . . . . . . 9.6.1 Gasoline Engine Control . . . . . . . . . . . . . . . . . 9.6.2 Air–Fuel Ratio Engine Predictive Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preview Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Preview Cost-Function and Solution . . . . . . 9.7.2 Merits of Preview and Possible Applications 9.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 LPV and State-Dependent Nonlinear Optimal Control . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 State-Dependent Riccati Equation Approach . . 10.2 Nonlinear Operator and qLPV Plant Model . . . . . . . . . . 10.2.1 Signal Definitions . . . . . . . . . . . . . . . . . . . . . 10.2.2 Quasi-LPV Model Dynamics . . . . . . . . . . . . . 10.2.3 Tracking Control and Reference Generation . . 10.2.4 Error and Observation Signals . . . . . . . . . . . . 10.2.5 Total Augmented System . . . . . . . . . . . . . . . 10.2.6 Definition of the Augmented System Matrices 10.3 Predicted Plant Outputs and States . . . . . . . . . . . . . . . . 10.3.1 Expression for Predicted Plant Outputs and States . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Kalman Predictor qLPV Model . . . . . . . . . . . 10.3.3 Kalman Predictor Corrector Form . . . . . . . . . 10.4 Nonlinear Generalized Minimum Variance Control . . . . 10.4.1 Minimization of NGMV Control Problem . . . . 10.4.2 Solution of the NGMV Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Advantages and Disadvantages of NGMV Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Controller Structure and Implementation . . . . . . . . . . . . 10.5.1 Kalman Filter-Based Controller Structure . . . . 10.5.2 Simplified Solution for a Special Case . . . . . . 10.6 Properties and Stability of the Closed-Loop . . . . . . . . . 10.6.1 Cost-Function Weighting Choice . . . . . . . . . . 10.6.2 Alternative Expression for Optimal Control . . 10.6.3 Use of Future Reference Information . . . . . . . 10.6.4 Enhanced NGMV Control . . . . . . . . . . . . . . . 10.7 Links to the Smith Predictor . . . . . . . . . . . . . . . . . . . . 10.8 SI Automotive Engine Multivariable Control . . . . . . . . 10.8.1 NARX Model Identification . . . . . . . . . . . . . . 10.8.2 Control Design and Simulation . . . . . . . . . . . 10.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 LPV/State-Dependent Nonlinear Predictive Optimal Control . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Nonlinear Operator and qLPV System Model . . . . . . . . . 11.2.1 Signal Definitions . . . . . . . . . . . . . . . . . . . . . . 11.2.2 qLPV Subsystem Models . . . . . . . . . . . . . . . . . 11.2.3 Augmented System . . . . . . . . . . . . . . . . . . . . . 11.2.4 Definition of Augmented System Matrices . . . . 11.3 qLPV Model Future State and Error Predictions . . . . . . . 11.3.1 State Estimates for qLPV Prediction Models . . . 11.3.2 Vector Matrix Notation . . . . . . . . . . . . . . . . . . 11.3.3 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Kalman Filter Predictor–Corrector Form . . . . . . 11.4 Nonlinear Generalized Predictive Control . . . . . . . . . . . . 11.4.1 Connection Matrix and Control Profile . . . . . . . 11.4.2 NGPC State-Dependent/qLPV Solution . . . . . . 11.4.3 NGPC Equivalent Cost Optimization Problem . 11.4.4 NGPC-Modified Cost-Function and Solution . . 11.5 Nonlinear Predictive GMV Optimal Control . . . . . . . . . . 11.5.1 NPGMV Cost-Index . . . . . . . . . . . . . . . . . . . . 11.5.2 NPGMV Optimal Control Solution . . . . . . . . . . 11.5.3 Nonlinear Predictive Control Summary . . . . . . 11.5.4 Including Anti-windup Protection in Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Stability of the Closed-Loop and Robust Design Issues . . 11.6.1 Cost-Function Weightings and Relationship to Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 MPC Robustness, Design and Guidelines . . . . . 11.6.3 Modelling the Parameter Variations . . . . . . . . . 11.6.4 Formal Robust Predictive Control Approaches . 11.7 NPGMV Simplifications for Implementation . . . . . . . . . . 11.7.1 NPGMV Special Weighting Case . . . . . . . . . . . 11.8 NPGMV in Terms of Finite Pulse Response . . . . . . . . . . 11.8.1 Simplified Controller Structure and Special Weighting Case . . . . . . . . . . . . . . . . . . . . . . . 11.9 Rotational Link Control Design . . . . . . . . . . . . . . . . . . . 11.10 Restricted Structure Generalized Predictive Control . . . . . 11.10.1 Restricted Structure GPC Criterion . . . . . . . . . 11.10.2 Parameterizing a Restricted Structure Controller 11.10.3 Parametrizing a General RS Controller . . . . . . . 11.10.4 Block Diagonal Parameterization Matrix . . . . . 11.10.5 Parameterizing the Vector of Future Controls . . 11.10.6 RS-GPC Theorem . . . . . . . . . . . . . . . . . . . . . . 11.10.7 Adaptive Behaviour . . . . . . . . . . . . . . . . . . . .

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11.11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 11.11.1 Pros and Cons of Some Model-Based Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Part IV

Estimation, Condition Monitoring and Fault Detection for Nonlinear Systems

12 Nonlinear Estimation Methods: Polynomial Systems Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Nonlinear Minimum Variance Options . . . . . . 12.1.2 Nonlinear Minimum Variance Estimation . . . . 12.2 Nonlinear Multichannel Estimation Problem . . . . . . . . . 12.2.1 Nonlinear Channel Models . . . . . . . . . . . . . . 12.2.2 Signals in the Signal Processing Problem . . . . 12.2.3 Spectral Factorization . . . . . . . . . . . . . . . . . . 12.3 Nonlinear Estimation Problem . . . . . . . . . . . . . . . . . . . 12.3.1 Solution of the Nonlinear Estimation Problem 12.3.2 Optimization and Solution . . . . . . . . . . . . . . . 12.3.3 Significance of Parallel Path Dynamics . . . . . 12.3.4 Design and Implementation Issues . . . . . . . . . 12.3.5 Nonlinear Channel Equalization Problem . . . . 12.4 Automotive Nonlinear Filtering Problem . . . . . . . . . . . 12.4.1 Design Problem and Simulation Results . . . . . 12.5 Introduction to the Wiener NMV Estimation Problem . . 12.5.1 Nonlinear Multichannel System Description . . 12.5.2 Signals in the Signal Processing System . . . . . 12.5.3 Design and Weighting Functions . . . . . . . . . . 12.6 Nonlinear Estimation Problem and Solution . . . . . . . . . 12.6.1 Solution of the WNMV Estimation Problem . . 12.6.2 Optimization to Compute the Estimator . . . . . 12.6.3 WNMV Optimal Estimator . . . . . . . . . . . . . . . 12.6.4 Limiting Features of WNMV and NMV Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.5 Parallel Path Dynamics and Uncertainty . . . . . 12.7 Example of Design Issues and Channel Equalization . . . 12.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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553 553 554 555 556 558 559 561 562 563 564 567 568 569 569 570 572 572 573 577 579 580 582 583

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585 586 586 593 594

13 Nonlinear Estimation and Condition Monitoring: State-Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 13.1.1 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

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Contents

13.1.2 Extended Kalman Filter . . . . . . . . . . . . . . . . 13.1.3 Particle Filters . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 Unscented Kalman Filter . . . . . . . . . . . . . . . . 13.1.5 Nonlinear Minimum Variance Filters . . . . . . . 13.2 Nonlinear Multi-channel Estimation Problem . . . . . . . . 13.2.1 Operator Forms of Nonlinear Channel Models 13.2.2 Signals in the Signal Processing System . . . . . 13.2.3 Signal Analysis and Noise Models . . . . . . . . . 13.2.4 Augmented State Model . . . . . . . . . . . . . . . . 13.3 Discrete-Time Kalman Filter . . . . . . . . . . . . . . . . . . . . 13.3.1 Return Difference and Spectral-Factorization . 13.4 Nonlinear Channel Estimation Problem . . . . . . . . . . . . 13.4.1 Solution of the Nonlinear Estimation Problem 13.4.2 Prediction Equation . . . . . . . . . . . . . . . . . . . . 13.4.3 Separation into Future and Past Terms . . . . . . 13.4.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.5 Benchmarking the Estimator . . . . . . . . . . . . . 13.5 Design and Implementation Issues . . . . . . . . . . . . . . . . 13.5.1 Parallel Path Dynamics and Uncertainty . . . . . 13.5.2 Implementation Issues . . . . . . . . . . . . . . . . . . 13.6 Automotive Nonlinear Filtering Problem . . . . . . . . . . . 13.7 Condition Monitoring and Fault Detection . . . . . . . . . . 13.7.1 NMV Estimator-Based Fault Detection . . . . . . 13.7.2 Fault Detection Example . . . . . . . . . . . . . . . . 13.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part V

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600 603 604 605 606 606 608 610 612 613 614 617 618 618 620 621 623 624 624 625 625 630 631 633 638 639

Industrial Applications

14 Nonlinear Industrial Process and Power Control Applications 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Wind Energy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Introduction to Individual Wind Turbine Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Controller Structure . . . . . . . . . . . . . . . . . . . . 14.2.3 Wind Turbine Model Description . . . . . . . . . . 14.2.4 LPV Wind Turbine Model for Control Design . 14.2.5 Wind Turbine Simulation and Performance . . . 14.3 Tension Control in Hot Strip Finishing Mills . . . . . . . . . 14.3.1 Hot Mill Control . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Looper Control Systems . . . . . . . . . . . . . . . . . 14.3.3 Looper Control System Design . . . . . . . . . . . . 14.4 Control of a Heavy Oil Fractionator . . . . . . . . . . . . . . . .

. . . . 645 . . . . 645 . . . . 646 . . . . . . . . . .

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648 649 650 654 655 659 661 662 664 667

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xxvii

14.4.1 14.4.2 14.4.3 Control 14.5.1 14.5.2

Modelling the Fractionator System . . . . . . . . . . Fractionator System Objectives and Constraints System Model and Controller Design . . . . . . . . 14.5 of a CSTR Process . . . . . . . . . . . . . . . . . . . . . . NGMV Control of a CSTR . . . . . . . . . . . . . . . . Using State-Dependent NGMV Control for the CSTR . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.3 Control Design and Simulation Results . . . . . . 14.6 Nonlinear Predictive Control of an Evaporator Process . . 14.6.1 Process Model and Evaporator Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 LPV Modelling of the Evaporator . . . . . . . . . . 14.6.3 Simulation Scenarios and Results . . . . . . . . . . . 14.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Nonlinear Automotive, Aerospace, Marine and Robotics Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Multivariable Control of a Two-Link Robot Manipulator . 15.2.1 State-Dependent Solution . . . . . . . . . . . . . . . . 15.2.2 LTI Output Block Solution . . . . . . . . . . . . . . . 15.2.3 Simplifying Implementation . . . . . . . . . . . . . . . 15.3 Ship Rudder and Fin Roll Stabilization . . . . . . . . . . . . . 15.3.1 Ship Rudder Roll-Stabilization Problem . . . . . . 15.3.2 NGMV Rudder Roll-Stabilization Controller . . . 15.3.3 Rudder Roll Simulation . . . . . . . . . . . . . . . . . . 15.4 Dynamic Ship Positioning Systems . . . . . . . . . . . . . . . . 15.4.1 Ship Motion Modelling . . . . . . . . . . . . . . . . . . 15.4.2 Dynamic Ship Positioning Control Design . . . . 15.4.3 Dynamic Positioning Predictive Control Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.4 Implementation Issues for NPGMV Ship Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Diesel Engine Modelling and Control . . . . . . . . . . . . . . . 15.5.1 Current Practice in the Control of Diesel Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 Model-Based Control Design Study . . . . . . . . . 15.5.3 Engine Model . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.4 LPV Modelling . . . . . . . . . . . . . . . . . . . . . . . . 15.5.5 Baseline Controller . . . . . . . . . . . . . . . . . . . . . 15.5.6 Design Approach and Weighting Options . . . . . 15.5.7 Simulation Results . . . . . . . . . . . . . . . . . . . . .

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668 669 670 677 680

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688 691 691 694 695

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699 699 701 703 707 707 712 713 715 719 721 722 725

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732 733 734 736 737 738 739

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15.6

Sightline Stabilization of Aircraft Electro-optical Servos 15.6.1 Nadir Problem . . . . . . . . . . . . . . . . . . . . . . . 15.6.2 Experimental Configuration . . . . . . . . . . . . . . 15.6.3 Model Equations . . . . . . . . . . . . . . . . . . . . . . 15.6.4 NGMV Control Design and Results . . . . . . . . 15.7 Design of Flight Controls . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Model Predictive Flight Control . . . . . . . . . . . 15.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 The Final Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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742 742 744 747 751 753 754 755 756 756

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

Abbreviations

APC ARE ARMA BFR CPC DC DCS DMC DOF EKF FIR GLQG GMV GPC IMC LMI LP LPV LQ LQG LQR LTI MBPC MIMO MIP MIQP MLD MPC MV NGMV

Advanced Process Control Algebraic Riccati Equation Autoregressive Moving Average Best Fit Rate Constrained Predictive Control Direct Current Distributed Control System Dynamic Matrix Control Degree of Freedom Extended Kalman Filter Finite Impulse Response Generalized Linear Quadratic Gaussian Generalized Minimum Variance Generalized Predictive Control Internal Model Control Linear Matrix Inequalities Linear Program Linear Parameter Varying Linear Quadratic Linear Quadratic Gaussian Linear Quadratic Regulator Linear Time Invariant Model-Based Predictive Control Multi-Input Multi-Output Mixed-Integer Programming Mixed-Integer Quadratic Program Mixed Logical Dynamical Model Predictive Control Minimum Variance Nonlinear Generalized Minimum Variance

xxix

xxx

NGPC NL NPGMV NQGMV OBF PFC PID PLC PWA QFT qLPV QP RDE RMPC RS RS-GPC SCADA SD SDP SDRE SID SISO UKF

Abbreviations

Nonlinear Generalized Predictive Control Nonlinear Nonlinear Predictive Generalized Minimum Variance Nonlinear Quadratic Generalized Minimum Variance Orthonormal Basis Function Predictive Functional Control Proportional–Integral–Derivative Programmable Logic Controller Piecewise-Affine Quantitative Feedback Theory Quasi-Linear Parameter Varying Quadratic Programming Riccati Difference Equation Robust Model Predictive Control Restricted Structure Restricted Structure-Generalized Predictive Control Supervisory Control And Data Acquisition State Dependent Semidefinite Program State-Dependent Riccati Equation System Identification Single-Input Single-Output Unscented Kalman Filter

Part I

Introduction and Linear Systems

Chapter 1

Introduction to Nonlinear Systems Modelling and Control

Abstract This chapter provides a very brief introduction to many areas in nonlinear systems theory and includes some of the basic mathematical modelling results that are needed later. Common static nonlinear functions are first described that are often used as part of a simple nonlinear system model. Dynamic nonlinear systems are introduced and methods of approximating nonlinear systems such as linearization methods, linear parameter-varying systems and state-dependent system models. The review of nonlinear control design methods is not meant to be exhaustive but it does provide a brief introduction to some of the most popular methods that are referred in later chapters. Hybrid systems and nonlinear system identification methods are topics that are covered briefly since they are of growing importance in applications and relate to some of the model structures employed later. This chapter can be skipped and just used for reference purposes for those wishing to move quickly to the more practical topics in control systems design.

1.1

Introduction

Control engineering plays a large part in providing safe and reliable operation of any process or industrial system. It is the actions of the controller that determine the actuator changes and thereby the stability of the system, the stress in components, and behaviour in the presence of disturbances, or even faults. A control system is part of an electromechanical system or chemical process, which cannot be seen, and it is usually a very small proportion of the plant costs, yet it has a disproportionate effect on the performance and reliability of the system. The design of control loops for industrial applications is, therefore, critically important, even if this is not obvious to engineers outside the controls community. For example, the plant may be built with oversized pumps that result in large changes in flow for small changes in valve position, and the layout/arrangement of plant vessels, piping and processes can be quite poor leading to a dynamical system that is very difficult to control. However, a control system can often make up for such deficiencies providing good performance in a bad situation. This is not true in reverse. If a control system design © Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_1

3

4

1 Introduction to Nonlinear Systems Modelling and Control

is poor, it is unlikely that changing the electromechanical devices or types of process tank will fix the problem. Despite the great virtues of feedback control, there remain challenges to the control loop designer, mainly due to nonlinearities, uncertainties and complexity. The focus here is on the former, since major advances have been made over the past two decades in providing practical solutions to model-based nonlinear control problems. Nonlinearities can cause very poor plant responses in certain operating conditions and severely limit the performance and quality of the product that is achievable. The time to tune a nonlinear system can be excessive, costing scarce engineering resources and sometimes a waste of high-value product. The nonlinearities can also result in unpredictable and on occasion’s unsafe behaviour. Early control design methods focussed on linear systems behaviour, and these methods are adequate for many industrial control loops that can be detuned when nonlinearities are troublesome. However, this text is mostly concerned with the design of industrial control loops where the nonlinearities dominate, and where advanced control is needed if good performance is to be achieved. This first chapter considers the nonlinear dynamics of a system and much of the discussion involves continuous-time models, since the industrial processes are normally represented by continuous-time differential equations. However, most of the rest of the text, which focusses on the controller design and potential applications, considers the discrete-time domain solution, which is how modern controllers are normally implemented.

1.1.1

PID Control of Nonlinear Industrial Systems

The most popular industrial control law is by far the Proportional–Integral– Derivative (PID) controller developed in the 1940s, and it is deployed widely in the process and many other industries. It uses a single degree of freedom feedback control structure, and tuning methods that are usually empirical and assume the plant is linear. The Ziegler–Nichols PID controller tuning method was published in 1942 and is still widely used. The limitations of PID control are often apparent in electrical machine control systems, where classical lead–lag control design is usually preferred. In practice, most processes are nonlinear with nonlinearities in actuators, plant and measurement systems, as illustrated in Fig. 1.1. Although PID control has been very successful for low-level single loop controls, it has limitations for systems that are multivariable and include significant delays. The tuning can be difficult for nonlinear systems, if they are required to control well over a wide range of operation. However, the ubiquitous PID controller often performs better than might be expected. One reason for its success is that it has a “low-order” structure. This may be thought of as a limitation, but if a low-order controller exists that provides reasonable performance, it often has good robustness to parameter variations and uncertainties. At the same time, it will limit the frequency-domain loop shaping that

1.1 Introduction

5 Disturbances

Desired output setpoint or reference + –

Controller Reference - measurement

Control signal

Actuators Actuators (nonlinear)

Plant dynamics including interactions

Output

(possibly very nonlinear)

Measured output

Sensor Measurements (nonlinear)

Fig. 1.1 Essential elements of a feedback control system including measurements, controller, plant actuators and dynamics

can be achieved. The result is that its performance may be limited, particularly for multivariable systems with significant high-order and difficult dynamics. All of the core elements of the feedback control system shown in Fig. 1.1 are nonlinear to a greater or lesser extent. Considering that the basic PID controller has no direct compensation for nonlinearities, it performs well in many situations where significant nonlinearities are present. Moreover, the PID controller is seldom employed in its basic form. It normally includes filters for noise attenuation, and techniques for avoiding integral-windup. It may also compensate for some nonlinear behaviour using scheduled gains, as discussed later. The main application area for PID control is in process control systems, where it is normally very reliable. However, even in this industry the benefits of improved control, using a more advanced controller, can be very significant. Possible benefits of using controllers that are more advanced in the process industries include the following: • • • • • • • •

Improving response times to setpoint and operating point changes. Reducing energy consumption. Improving product yield. Improving product quality and plant capacity. Improving product consistency. Reducing waste and the impact on the environment. Improving the safe and reliable operation of large complex systems. Reducing the loss of product due to poor control and avoiding the resulting costs of effluent processing.

With these possible improvements in mind, it is, therefore, not disloyal to say to our old friend PID that however useful, simple and reliable it may be, there are still times when improved control may be needed.

6

1.1.2

1 Introduction to Nonlinear Systems Modelling and Control

Optimal Control of Nonlinear Industrial Systems

The aim in the following is to derive controllers for industrial applications that take account of nonlinearities naturally by virtue of the model and control problem definition. There are inevitable compromises to be made, since controllers for industry must be simple to understand and implement. Most of the controllers discussed in the following will involve optimal solutions, where paradoxically optimality is not very important. An optimal control philosophy is utilized since it provides a formal framework for analysis and control design. The behaviour is then often predictable and tuning, even for complex systems, is relatively straightforward. Industry often needs formalized design procedures, which can incorporate the experience gained and simplify the design process for new engineers. Optimal design approaches provide such a framework, where an experienced engineer can set up a cookbook procedure to follow, that makes future designs faster and more reliable. This is more difficult to achieve using classical control design methods, particularly for multivariable and complex systems. Before turning our attention to the optimal control synthesis and design problems of interest, a brief overview will be provided of the various modelling, analysis and design methods often used for nonlinear systems.

1.1.3

Linear and Nonlinear Dynamic Systems

Linear systems can be described by a set of ordinary differential equations or difference equations and closed-form expressions for their solutions can be derived. If the system is linear time-invariant then a Laplace or Z-transform may be used for analysis that provides a deep insight into system behaviour. Sadly, none of this is true for nonlinear systems. A system is linear if it exhibits the properties of superposition and homogeneity (sometimes superposition is defined to include homogeneity). That is, if a system is linear its responses will satisfy the principle of superposition. If a linear system has two or more inputs then the time response due to the sum of the inputs will be the same as the time response due to the sum of the individual responses. To be more precise, if a system has two inputs u1 and u2, with input u1 giving an output y1 and input u2 giving an output y2, then if the system is linear the total output will be y1 þ y2 . Let an operator W represent a system, then an additive system satisfies Wðu1 þ u2 Þ ¼ Wðu1 Þ þ Wðu2 Þ. A linear system is clearly additive; however, a linear system also satisfies the homogeneity or scaling property. That is, if a is a scalar then Wðau1 Þ ¼ aWðu1 Þ. To summarize, a linear system is one that satisfies superposition and homogeneity, and for a linear system Wða1 u1 þ a2 u2 Þ ¼ a1 Wðu1 Þ þ a2 Wðu2 Þ. Any function that does not satisfy superposition and homogeneity is, therefore,

1.1 Introduction

7

nonlinear. In fact a nonlinear system can be defined as a system that is not linear. In practice, no system or component is actually linear, however, when the changes from a steady-state operating point are small, many systems can be analysed and designed very successfully using a linear model. A popular approach is to develop controllers based on a linearized model. This works well for the operation of the system around its operating point but can be problematic when large changes are needed because of product changes. Fortunately, there may not be a significant loss of accuracy in computing system responses, when a linear model is used, partly because of the beneficial effects of feedback action. However, linear analysis and design methods are not so suitable for systems that have discontinuities in their characteristics. The problems that arise become more serious as the nonlinearity becomes more severe [1]. Since the class of nonlinear problems is large, it is difficult to find methods that work well for all classes of nonlinear systems. Linearization is, of course, an approximation in the neighbourhood of an operating point and it can only predict local behaviour of the nonlinear system. It provides no guidance on the global behaviour of the system. The types of nonlinear control systems considered here can be described by nonlinear differential or difference equations, nonlinear state equations or operators [2, 3].

1.1.4

Nonlinearities in Physical Systems

The effect of nonlinearities can be very destabilizing and since the effective gains and time constants are operating point dependent they can also introduce mysterious behaviour compared with essentially linear systems [4]. For example, hysteresis (from the Greek, meaning late or lag in arrival) is found in many valves and magnetic systems. Hysteresis is not only destabilizing but limits tracking performance. Nonlinear systems also have more complex behaviour than linear systems including so-called chaotic behaviour. Phenomena like resonance jumps can occur, where a jump can arise in the amplitude and/or phase and/or frequency of a periodic output signal (likely to arise in systems with small stability margins). Friction is another common nonlinearity and is particularly important in servo-systems, where it is often assumed proportional to velocity. Coulomb or dry-friction is also observed in such systems where the friction effect is in this case independent of velocity. Dry-friction can arise when the rubbing parts have no substance such as oil or grease between them. It introduces a force whilst the system is in motion, with a constant absolute value and a direction that is opposite to the velocity. Nonlinearities in systems are also often accompanied by uncertainties, or they may introduce uncertainties. For example, an aircraft gas turbine fuel control system has many nonlinear effects changing with altitude and Mach number, including problems that arise through the presence of transport delays. At a single operating condition, like sea level, an engine responds more slowly at low power than at

8

1 Introduction to Nonlinear Systems Modelling and Control

higher power settings. An engine with a half-second time constant at full power may have a time constant of say 2 s for idle speed operation. Nonlinearities arise in physical systems in many ways. The differential equations describing the behaviour of systems, such as mechanical, electrical, thermal, fluidic or biological systems, are often nonlinear. Thermal systems may, for example, have nonlinear radiation and convective losses. Electrical systems, machines and transformers have nonlinearities in the form of saturation and nonlinear characteristics. Nonlinearities arise when ferromagnetic cores are present which is described by nonlinear magnetization curves and equations. Electrical systems also involve nonlinear dielectric and permeability effects, and current- and frequency-dependent input and output impedances. Mechanical systems may include significant static nonlinearities in cascade with system dynamics. These may include nonlinear friction, deadband, hysteresis and backlash. Many physical quantities, such as the actuator rate or length of travel have an upper bound. When that upper bound is reached, linearity is lost. Valve nonlinearities are often significant in process plants and this is exacerbated if the valve is the wrong size and is operating close to its limits. If, for example, the controller output is less than say 15%, or is greater than 85% during normal steady-state operation, then the valve may be classified as oversized or undersized, respectively. The valve characteristic may then be very nonlinear in the normal operating region. A relay characteristic is an example of an extreme nonlinearity. Relay characteristics are involved in many practical control systems for simple temperature or level control action.

1.1.5

Smooth and Non-smooth Nonlinearities

Nonlinear models may be classified into smooth and non-smooth cases. A smooth function has continuous derivatives. Non-smooth nonlinearities include backlash, hysteresis, deadband, saturation and dry-friction that arise naturally in devices. Non-smooth nonlinearities may also be introduced by hard limits such as on a power converter output current). Difficult non-smooth nonlinearities may require tailored compensation and control techniques, whereas conventional control may be adequate for systems with smooth nonlinearities. A bang–bang control system does, of course, involve an inherent non-smooth nonlinearity. The gain and phase characteristics of discontinuous nonlinearities are quite complex. For example, the effective gain of a saturation characteristic is reduced as the inputs increase, but there is no phase shift. The gain of a deadband, on the other hand, is increased as input amplitudes increase and again no phase shift arises. The effective gain of a hysteresis also increases with an increase in the input amplitude and in this case a phase shift is introduced that is smaller for larger inputs. Any nonlinearity like hysteresis, that includes memory and energy storage, is usually destabilizing, and when classical control loops are used they must often be detuned.

1.2 Types of Static Nonlinear Models

1.2

9

Types of Static Nonlinear Models

The lumped models of nonlinearities often involve a significant approximation. Nevertheless, a combination of lumped linear and static nonlinear models often provides a reasonable approximation to a complex nonlinear system. Static nonlinearities may be classified as single and multivalued as illustrated in Fig. 1.2 and Fig. 1.3, respectively. Real control systems may not be able to be separated into a convenient static nonlinearity and a linear dynamic system. However, such models do provide a starting point for designing a nonlinear control system. In fact, static characteristics can often provide a good approximation for the behaviour of nonlinear devices such as valves.

1.2.1

Saturation

Most of the nonlinearities, such as a saturation nonlinearity, can be classified as single-valued and time-invariant. The saturation function can be represented as in Fig. 1.4, with limits 1, and is defined as 8 > < uðtÞ for juðtÞj  1 /ðtÞ ¼ satðuðtÞÞ ¼ 1 for uðtÞ [ 1 > : 1 for uðtÞ\1

Fig. 1.2 Smooth and single-valued nonlinearities

y

y

y

x

Fig. 1.3 Multi-valued nonlinearity characteristics

x

x

y

y

x

x

sat(u(t))

Fig. 1.4 Saturation function

1

u(t) –1

10

1.2.2

1 Introduction to Nonlinear Systems Modelling and Control

Stiction

Control valves that are poorly maintained are sometimes thought to be suffering from valve stiction, even though the actual origin of problems may be due to a range of nonlinear behaviour. Stiction is an important model for friction and represents the resistance to the start of valve motion. For example, stiction exists when the static (starting) friction exceeds the dynamic (moving) friction inside a valve. The presence of stiction often results in cycling and product variability and can be more harmful than other valve problems. If a control valve sticks, the feedback control loop will not function correctly. The controller output u(t) may include a saw-tooth pattern and the output or process variable y(t) may then exhibit a square wave. Tuning will not normally remove the square wave and saw-tooth oscillation, and hence the valve must be changed. In many processes, 0.5% stiction is enough to cause process oscillations that are difficult to compensate.

1.2.3

Backlash

Backlash can occur in geared positional systems, where there is some significant play between the teeth of the gears. If backlash is present then a small input signal will produce no output, and this adds a phase delay to the system. The backlash in valves can lead to poor process control. For stem control valves with diaphragm actuators and digital positioning, the backlash is around 1–2%. For rotary valves that have been designed for on/off service, rather than throttling, the backlash can be much larger. It has less impact on a feedback control system if the loop gains are reduced, but this gain change can also result in sluggish behaviour.

1.2.4

Deadzone

A deadzone characteristic and a deadzone with saturation characteristic are shown in Fig. 1.5a and b, respectively. A spool-valve with these characteristics is shown in Fig. 1.5c.

1.2.5

Hysteresis

Hysteresis is important in electrical and mechanical systems, and is experienced in most control valves when there is a reversal of direction. The space between connections in mechanical parts leads to hysteresis. This adds dead time to a loop and makes the loop response slower near the setpoint when the loop is settling.

1.2 Types of Static Nonlinear Models

(a)

(b)

φ (t)

− xb

11 y

Q

(c)

yb

Nonlinear flow

− xa

xa

u(t)

xb

x

x

Deadband

− yb Saturation

Fig. 1.5 a Dead-zone function, b saturation characteristic with deadzone, c typical spool value characteristic (flow-rate versus valve lift characteristic)

(a)

(b)

φ (t)

u(t)

φ (t)

u(t)

Fig. 1.6 a Multi-valued static characteristic of hysteresis, b two-position relay with hysteresis

Hysteresis can induce cycling in feedback control loops and it is not always easy to understand whether oscillations are due to disturbances or to nonlinearities. Typical multivalued static nonlinear characteristics are illustrated in Fig. 1.6a, b.

1.2.6

Coulomb Friction Characteristics

Friction is important in high accuracy servo-system applications. Coulomb friction is an approximate model used to calculate the force of dry-friction and has the characteristic shown in Fig. 1.7 (due to Charles-Augustin de Coulomb). Recall that for two dry solid surfaces rubbing against one another, the magnitude of the kinetic friction is independent of the magnitude of the velocity, and the direction of the kinetic friction is opposite the direction of the velocity. Coulomb damping, which is often used in mechanical systems, provides a constant damping force. The energy is absorbed via the sliding friction. The movement of the throttle plate in an automotive electronic throttle encounters resistance in the form of Coulomb friction, which affects performance.

12

1 Introduction to Nonlinear Systems Modelling and Control

Fig. 1.7 Coulomb friction characteristic

Torque φ (t)

Velocity u(t)

1.3

Nonlinear Dynamic Systems Behaviour

The main problem with nonlinear systems lies in the wide variety of behaviour that may be exhibited, which makes the tuning of such systems both difficult and unpredictable. If the system is mildly nonlinear, and behaves like a linear system, all our usual intuition holds, but if not, an understanding of the nonlinear system dynamics is needed if good performance is to be achieved. State-space models are useful for linear and nonlinear systems. Consider a continuous-time dynamical system modelled by a finite number of coupled first-order ordinary differential equations: x_ ðtÞ ¼ f ðt; xðtÞ; uðtÞÞ

ð1:1Þ

where xðtÞ ¼ ½ x1 ðtÞ . . . xn ðtÞT denotes the state vector, uðtÞ ¼ ½ u1 ðtÞ . . . um ðtÞT denotes the input vector, and f ðÞ ¼ ½ f1 ðÞ . . . fn ðÞT represents a vector of nonlinear functions. The output equation, relating to the state-equation model (1.1), can be written in the form: yðtÞ ¼ hðt; xðtÞ; uðtÞÞ

ð1:2Þ

where yðtÞ ¼ ½ y1 ðtÞ . . . yr ðtÞT denotes the output vector. The two equations represent the state-equation nonlinear model. The difference between autonomous and non-autonomous systems may also be noted. An autonomous system: x_ ðtÞ ¼ f ðxðtÞÞ is one that is invariant to a shift in the origin of the time scale. That is, a change in the time t to s ¼ t  a will not change the function and the state response is only shifted in time. If a system is not autonomous it is called non-autonomous, or more commonly a time-varying system. A point x0 in the state-space is an equilibrium point of the linear autonomous system x_ ðtÞ ¼ AxðtÞ if when the state x reaches x0, it stays at x0 for all future time. Discrete-time nonlinear systems are of particular interest in the following chapters, and in this case, the difference equation model has the form: xðt þ 1Þ ¼ f ðt; xðtÞ; uðtÞÞ

ð1:3Þ

1.3 Nonlinear Dynamic Systems Behaviour

1.3.1

13

Nonlinear Systems Behaviour

Nonlinear systems can exhibit unusual behaviour and some of the conditions and properties that arise are now considered [4]. Finite escape time: The state of a nonlinear system can go to infinity in a finite time, whereas the state of a linear system can only go to infinity as time goes to  infinity. Consider the solution of the differential equation dðxðtÞÞ dt ¼ ðxðtÞÞ2 , where the initial condition xðt0 Þ ¼ 1. Note dð1=ð1  tÞÞ=dt ¼ 1=ð1  tÞ2 , and hence the solution is xðtÞ ¼ 1=ð1  tÞ, for 0  t\1, with a finite escape time tf ¼ 1. Clearly, the solution for the output of a nonlinear system may not exist over all time. Equilibrium points: An equilibrium point for a continuous-time autonomous system x_ ðtÞ ¼ f ðxðtÞÞ describes the points xe ðtÞ in the state-space, where all the derivatives are zero f ðxe ðtÞÞ ¼ 0. The system can stay forever in such a state. For a system with control input x_ ðtÞ ¼ f ðxðtÞ; uðtÞÞ, the pair ðxe ðtÞ; ue ðtÞÞ is an equilibrium point if f ðxe ðtÞ; ue ðtÞÞ ¼ 0. The equilibrium points of a system are the points in the state-space for which the time derivative of the state is null. A linear system can have only one equilibrium point, at the origin, which attracts the states irrespective of the initial state. Nonlinear systems may have multiple isolated equilibrium points, and the state will converge to one of them depending on the initial value of the state vector. Bifurcation: A nonlinear system may have sudden changes in its behaviour, referred to as bifurcation due to parameter variations or due to disturbances. This can occur when a small smooth change made to a parameter value causes a qualitative change in the system behaviour. A system is called structurally stable if small changes of a parameter do not cause the equilibria to change stability properties, and if additional equilibria are not created. The points in the space of system parameters, where the system is not structurally stable, are called the bifurcation points. Bifurcation problems can arise where the linearization has a large and possibly infinite-dimensional stable part and a small number of critical modes, which change from stable to unstable as the bifurcation parameters exceed a threshold [5]. Multiple modes of behaviour: A nonlinear system can exhibit multiple modes of behaviour based on the type of excitation: • An unforced system may have a limit cycle discussed below. • A periodic excitation signal may result in harmonic, subharmonic or chaotic behaviour based on the amplitude and the frequency of the input signal. • If the amplitude or frequency of the input is changed smoothly, the system may exhibit discontinuous jumps in the modes. Limit cycle behaviour: A limit cycle is a self-excited oscillation at a fixed amplitude and frequency that is common in poorly tuned nonlinear systems. Oscillations are unlikely to occur in systems that are dominantly linear, but in nonlinear systems limit cycles are not uncommon. A limit cycle is termed stable if

14

1 Introduction to Nonlinear Systems Modelling and Control

all the neighbouring trajectories approach the limit cycle as time approaches infinity. For the case of a linear system with a pair of eigenvalues on the imaginary axis, constant magnitude oscillations can arise with a magnitude determined by the initial conditions. In the case of nonlinear systems, oscillations can arise that have an amplitude and frequency that is not dependent on the initial conditions. A limit cycle in an electromechanical or process control system causes unnecessary wear and tear, which can easily lead to faults and ultimately failure. Sub-harmonic, harmonic and almost-periodic oscillations: A stable linear system with a periodic input provides an output at the same frequency but a nonlinear system under a periodic input can oscillate with submultiple or multiple frequencies of the input, or with so-called almost-periodic oscillations. The output signal of a nonlinear feedback system can, therefore, have very different frequency characteristics. The output can be a periodic signal that is independent or proportional to the input signal frequency or it may have the same frequency. The output can also have higher harmonic or sub-harmonic frequencies. Chaos: A chaotic system is a deterministic dynamical system that has irregular, seemingly random behaviour. A nonlinear system may have different steady-state behaviour, which is not an equilibrium point, periodic oscillation or almost-periodic oscillation. Chaotic systems are characterized by local instability and global boundedness of the trajectories. Chaotic motions appear random despite the deterministic nature of the system [6].

1.3.2

Stability Analysis and Concepts

Establishing the stability of a nonlinear system is a rather more complex problem than for a linear system. It is not possible to guarantee that a globally stabilizing feedback control can be found. As an example consider the case of an exponentially unstable linear system that has an input saturation characteristic, where it is not possible to stabilize the system globally. In this case, when the open-loop plant is unstable, it is only possible to guarantee local closed-loop stability. There follows a brief summary of some of the stability properties of nonlinear systems. Stability: may be considered about an equilibrium point, in the sense of Lyapunov or in an input–output sense. Lyapunov stability is a very mild requirement on equilibrium points. If all solutions of the dynamical system that start out near an equilibrium point xe ðtÞ stay near it over all future time, then xe ðtÞ is Lyapunov stable. It does not require that trajectories starting close to the origin tend to the origin asymptotically [7]. Asymptotic stability: A system is asymptotically stable at an equilibrium point xe ðt0 Þ ¼ 0 if this point is locally attractive (there exists d(t0) such that kxðt0 Þk\d ) limt!1 xðtÞ ¼ 0). This is a stronger requirement than Lyapunov stability, where when a system is asymptotically stable all solutions that start near xe ðtÞ converge to xe ðtÞ. The definition of asymptotic stability does not quantify the rate of convergence and this is important for applications.

1.3 Nonlinear Dynamic Systems Behaviour

15

Exponential stability: This is a stronger form of stability, which demands an exponential rate of convergence. That is, the notion of exponential stability guarantees a minimal rate of decay. That is, an estimate of how quickly the solutions converge. Global stability: Local definitions of stability, like those above, describe system behaviour near an equilibrium point. Global stability is more desirable, but is more difficult to achieve. It is often called stability in the large. The system is taken to be stable for all initial conditions x0 2 Rn . That is, the system is globally asymptotically stable if it is locally asymptotically stable for all initial conditions x(t0) and xðtÞ ! 0 as t ! 1. The initial conditions can affect stability, unlike for linear systems, and so can the presence of external input signals. Input–output stability: Useful since it enables the stability of a system to be assessed by considering the relative size of signals at input and output, without knowing the internal state of the system. Bounded-input, bounded-output stability: A feedback system, with reference input r(t), input u(t) and output y(t), is Bounded-Input, Bounded-Output (BIBO) stable if there exist constants c1 ; c2 ; c3 [ 0, so that any bounded reference yields a bounded-output and control signal. That is, jrðtÞj  c1 \1 ! jyðtÞj  c2 \1 and juðtÞj  c3 \1. If a system is BIBO stable, then the output will be bounded for every input to the system that is itself bounded [8]. Finite-gain stability: A system is a Finite-Gain Stable system if there exists a positive gain c such that kys k  ckus k for all t, where us and ys are the input and output signals of the system, respectively, and k : k is a norm such as the 2-norm or ∞-norm. The us represents a truncated signal equal to u(t) for t 2 ½0; s and zero for t > s [9]. The connections between finite-gain and asymptotic stability were explored by Hill and Moylan [10, 11], and they also considered situations where input–output stability is taken to mean finite-gain input–output stability.

1.4

Nonlinear Control Systems Analysis Methods

Much of the early work on nonlinear control systems involved analysis, rather than design methods. The phase-plane method for second-order systems is a plot with the axes being the values of the two state variables x1 ðtÞ and x2 ðtÞ. It provided a graphical method to analyse systems and to determine the existence of limit cycles. It had some limited value in applications and helps when explaining the behaviour of nonlinear systems. The more important describing function methods (described below) enable classical frequency response intuition to be invoked.

16

1.4.1

1 Introduction to Nonlinear Systems Modelling and Control

Stability and Robustness Analysis Methods

Some of the more recent methods for analysing the stability and robustness of feedback systems include the following: Passivity analysis: Passivity captures the physical notion of a system that does not generate internal energy. For example, electrical resistance involves absorption and dissipation, but a resistor does not produce energy. An asymptotically stable SISO linear system G(s) is passive if and only if ReðGðjxÞÞ  0 for all x [ 0. A system is strictly passive if and only if there exists e [ 0 such that ReðGðjxÞ  eÞ  0

x[0

for all

The type of results obtained can be seen from the use of the feedback system shown in Fig. 1.8. If S1 is strictly passive and S2 is passive then the system is BIBO stable. Passivity implies stability. Thor Fossen and co-workers have utilized such concepts in the analysis and design of marine control systems [12]. Lyapunov methods: These methods were developed at the end of the nineteenth century, based on the definition of an energy-related function, for stability analysis. Lyapunov studied at St. Petersburg and was a Professor at the University of Kharkiv. His work on the General Problem of Motion Stability was first published in 1892 [13] and included two methods for stability analysis: the so-called linearization and the direct methods. Lur’e problem: Some stability analysis approaches are based on Lur’e problem. The feedback system model that is assumed has the form shown in Fig. 1.10. The forward path is assumed linear and time-invariant, and is represented by a transfer-function or a state-space model. The feedback path is assumed to contain a memoryless (possibly time-varying), sector-bounded nonlinearity (Fig. 1.9). Popov stability: The Popov stability criterion may be applied in the Lur’e problem. It provides a frequency-domain stability test for feedback systems, consisting of a linear component described by a transfer-function, preceding a nonlinear component, involving a time-varying memoryless nonlinearity subject to Fig. 1.8 Feedback system for analysis by small-gain theorem

r1

+ +

e1

S1

S2

Fig. 1.9 Sector-bounded nonlinearity for sector ½a1 ; a2 

Output y(t)

+ r2 +

e2

y = α2 u y = α1 u u(t)

1.4 Nonlinear Control Systems Analysis Methods Fig. 1.10 Closed-loop feedback system with sector-bounded nonlinearity

17 r(t) + -

y(t)

e(t) G(s)

f (.)

Im (G(jω))

Fig. 1.11 Circle criterion stability test

Re (G(jω)) G(jω)

-1/α1

-1/α2

sector conditions. This enables a so-called absolute stability criterion to be defined. This type of stability criterion is generally stated for a linear system, and it applies to every element of a specified class of nonlinearities. Absolute stability theory provides sufficient conditions for robust stability for a given class of uncertain elements. The control loop in Fig. 1.10 is absolutely stable for a given sector [0, a], if the closed-loop has only one global asymptotic equilibrium for any nonlinearity f(.) lying in this sector [14]. Circle criterion: The circle criterion is for systems with a sector-bounded nonlinearity f ð:Þ, such that 0\a1  f ðxÞ=x  a2 . The input–output characteristic of the assumed nonlinearity lies between the two lines y ¼ a1 u and y ¼ a2 u, as shown in Fig. 1.9. Then the feedback loop, shown in Fig. 1.10, with stable linear transfer G(s), is BIBO stable if the Nyquist plot of G(s) does not intersect or encircle the circle defined by the points 1=a1 and 1=a2 , as shown in Fig. 1.11. Small-gain theorem: The small-gain theorem was derived by George Zames in 1966 [15], at McGill University and can be used to establish input–output stability conditions for a feedback system. It provides a sufficient condition for finite gain Lp stability of the closed-loop system. If two input–output stable systems S1 and S2 are connected, as shown in the feedback-loop in Fig. 1.8, then the closed-loop is input– output stable if the loop gain kS1 k:kS2 k\1, where the norm used is any induced norm. For example, if a system has a transfer-function G(s) the L2 gain of G(s) is the spectral norm or largest singular value of G(s), or supx2R kGðjxÞk. The result is often employed where S1 is linear and S2 is a nonlinear system model, and to deal with unstable signals the space Lp;e is used where the upper limit of the norm integral is finite [15].

18

1.5

1 Introduction to Nonlinear Systems Modelling and Control

Nonlinear Control Design Methods

Control design techniques for nonlinear systems are well-established [16]. They include several empirical techniques, and some of the nonlinear control design methods are linear system techniques in disguise. For example, a nonlinear system might be considered at a number of operating points and linear models assumed to hold for a small range of operation. Since linearization is an approximation in the neighbourhood of an operating point, it can only predict the local behaviour of the nonlinear system. It cannot predict the nonlocal or global behaviour. In fact, a linearization procedure is valid only when the deviations from nominal trajectories and inputs are small. Other nonlinear control design methods attempt direct compensation of the nonlinearity so that the system can be treated as a linear system for control design purposes. Nonlinear compensated PID control: For some applications, industrial PID controllers include nonlinear compensation terms. For example, it is common for automotive engine controls to have PID gains that are scheduled using engine map information. These controllers are discussed further in the control design Chap. 5 (Sect. 5.1.1). Internal model control: The internal model control structure and design philosophy has much in common with Smith Predictors (see Chap. 5 (Sect. 5.5.1)). Similar results apply to nonlinear systems and one form of the family of Nonlinear Generalized Minimum Variance (NGMV) controllers (described in Chap. 4) can be related to this structure. This topic is discussed further later in Chap. 5 (Sect. 5.1.3). The nonlinear control design methods that are not explored in more detail in later chapters are discussed briefly below.

1.5.1

Describing Function Methods

Describing functions provide an approximate analysis method to determine whether the feedback-loop will include persistent oscillations. This provides an opportunity to avoid such oscillations by good control loop design but the approach is heuristic and offers no guarantees. These methods provide a procedure for analysing nonlinear control problems based on quasi-linearization [17–20]. It involves the replacement of the nonlinear system model by a system that is linear except for a dependence on the amplitude of the input signal waveform. The describing function method involves computing an approximation to the response of nonlinear subsystems. The system is assumed to be driven by sinusoidal inputs of given amplitude and frequency, and the output is then approximated by the first harmonic. This approximation is only valid if the plant model includes low-pass filtering action. Using the describing function approach the possibility of sustained oscillations in the feedback loop can be explored. These will arise when the loop has a gain of unity and a phase shift of p radians occurs. It can provide an estimate for the

1.5 Nonlinear Control Design Methods

19

frequency and amplitude of limit cycle oscillations. The method can be useful for systems analysis and suggests ways of changing the gain to avoid such oscillations. The system response at different operating points may be represented by what might be loosely termed an amplitude-dependent transfer-function that can be denoted NðA; jxÞ. If a system is quasi-linear with an input that is a single sine wave, the output will be a sine wave of the same frequency but with phase and gain determined by the appropriate sinusoidal-input describing function NðA; jxÞ. This SIDF is, of course, a generalization of the transfer-function GðjxÞ that is used to characterize a linear system. Describing functions may also be defined for other types of the input signal. For example, the so-called random-input describing functions (RIDF) may be invoked for systems with stochastic Gaussian noise inputs. Unfortunately, results involve an approximation and stability is not guaranteed.

1.5.2

Gain Scheduling

The simple idea of gain scheduling involves switching between linear controllers, each of which provides satisfactory control at a particular operating point of the system. Gain scheduling is popular, where the controller gains can be based on the current value of a so-called “scheduling variable”. However, a linearized model may not be very representative or even controllable at certain operating points. Moreover, the certification of systems at a large number of operating points is an expensive process (a problem in flight control design). Scheduling in given operating ranges is very common for commercial industrial controllers. More generally, different nonlinear controllers can be scheduled that depend on scheduling variables, or some measure of the system state. This topic is discussed again later in this chapter.

1.5.3

Variable Structure and Sliding Mode Control

Variable Structure Systems (VSS) often represent a state-dependent switching feedback control that intentionally changes the structure of the system. It follows from the implementation of a discontinuous feedback control law. A variable structure control system can be composed of independent structures and switching logic. Consider a dynamic system of the form x_ ¼ f ðx; tÞ, where f ðx; tÞ has discontinuities with respect to some arguments. Such systems occur in problems of physics, control engineering and mathematics, and in some physical devices such as electric motors and power converters. The resulting state-feedback control laws are naturally discontinuous and involve a time-varying control solution. A property of these systems is that they often exhibit a behaviour referred to as sliding motion.

20

1 Introduction to Nonlinear Systems Modelling and Control

This is characterized by the fact that the commutation between the different system structures takes place at infinite frequency. Sliding Mode Control (SMC) is a special class of variable structure systems in which the exclusive feature is the sliding mode. The theory was developed in the late 1950s, in the former USSR, led by Professors Utkin and Emelyanov (see [21– 23]). They received the Lenin Prize which was one of the most prestigious awards for Soviet Scientists. The main idea is to force the system trajectory to reach a sliding manifold and to keep it there by means of high-frequency switching control. A valuable characteristic of the sliding modes is their insensitivity to disturbances. The design of state-feedback controllers for nonlinear systems using sliding mode control forces the system trajectories to reach a desirable surface in the statespace in a finite time. The system response is forced to “slide” along a predefined trajectory in a phase-plane by a switching algorithm, despite parameter variations or disturbances. The state-feedback control law is not a continuous function of time. It switches from one continuous structure to another depending on the position in the state-space. A sliding mode controller is, therefore, a variable structure control method, where the dynamics of the nonlinear system are changed by the use of high-frequency switching [24, 25]. Such switching systems are special cases of hybrid control systems. A manifold in the state-space is found for which the system dynamics have a simple stable form. This is the sliding surface or switching surface, where the system trajectory has the desired behaviour. The state is driven towards a subset of the state-space referred to as the sliding set and the state trajectory is then moved asymptotically towards a value consistent with the desired equilibrium. The switching can involve switching between constant gain values or between state and output dependent values. The gain in the feedback path may, for example, be switched between two values according to some switching rule. The switching occurs with respect to a switching surface in the state-space and aims to ensure asymptotic stability. The design of a switching controller involves the definition of a switching condition and then the choice of the feedback control law inside the areas separated by the switching condition. The controllers with switching elements are normally easy to realize. The feedback gains are determined so that the trajectories reach the manifold in a finite time and then stay on that manifold. A generalized Lyapunov function can be used where for each switched control structure, the “gains” are chosen so that the derivative of the Lyapunov function is negative definite, guaranteeing motion of the state trajectory on the surface. To summarize, sliding mode control includes two phases: • Reaching phase: The system state is driven from any initial state to reach the switching manifolds (the anticipated sliding modes) in finite time. In the reaching mode, the trajectory reaches the sliding surface. • Sliding-mode phase: The system has sliding motion on the switching manifolds and the switching manifolds become an attractor. The trajectory on reaching

1.5 Nonlinear Control Design Methods

21

the sliding surface remains there for all time and evolves according to the dynamics specified by the sliding surface. Practical Aspects and Sliding Mode Control Design The main practical problem that arises in design is that due to delays there will be no ideal sliding on the manifold. The result is that chattering occurs due to the discontinuous behaviour around the switching surface. The high-frequency switching is called chattering because of the sound made by mechanical relays or switches. This can result in power losses, excite unmodelled dynamics and cause instability. In practice, it will, therefore, require some smoothing of the control signal. A possible way to reduce chattering, though maintaining a very high switching frequency, is based on the use of observers for the modelled part of the system Shi et al. [26]. These two design stages correspond to the following steps: • Switching manifold selection: A set of switching manifolds are selected with prescribed desirable dynamical characteristics. Common candidates are linear hyperplanes. • Discontinuous control design: A discontinuous control strategy is constructed to ensure the finite-time reachability of the switching manifolds. The controller may be either local or global, depending upon specific control requirements. Example 1.1: Sliding Mode Control Consider the double integrator plant model shown in Fig. 1.12 and in the feedback loop in Fig. 1.13. Assume that a linear, static, output feedback controller is used to stabilize the system as shown in Fig. 1.13. Also assume that two switched gains are used in this feedback system, where neither structure is asymptotically stable. Only signs of both states are needed and it can be seen that the total system is asymptotically stable. Let k = 4. Then

dx1 dx2

¼  4xx21 and 4x1 dx1 þ x2 dx2 ¼ 0 or 0:3x21 2

4x21 2

þ

x22 2

¼ const (see

x22 2

Fig. 1.14a). Let k = 0.3. Then þ ¼ const (see Fig. 1.14b). Combine these two using the two controller gains and switching:  k¼

4 if 0:3 if

x1 x2 [ 0 x1 x2 \0

The result in Fig. 1.14c shows the trajectory spiralling into the origin, providing a stable switching control.

Fig. 1.12 Double integrator

u(t)

Plant

1 s2

y(t)

y = u and y = x1 x1 = x2 and x2 = u

22

1 Introduction to Nonlinear Systems Modelling and Control r(t) = 0

+

1 s2

–

y(t)

x1 = x2 , x2 = − kx1 = − ky

and k = constant.

k

Fig. 1.13 Double integrator feedback

(a)

(b)

(c)

Fig. 1.14 System trajectories for Example 1.1

1.5.4

Lyapunov Control Design

In the Lyapunov direct method, an energy-like function is defined. Then if the total energy is dissipated, the system must be stable in some sense. A design starts with the specification of a candidate Lyapunov function and then involves the choice of a feedback control law to ensure desirable properties are obtained that guarantee stability. A stable system is synthesized by first choosing a candidate Lyapunov function V(t). A state-feedback control law is then be selected that renders the derivative of V(t) negative. This has the advantage that the nonlinear differential equations representing the system do not have to be solved to determine if the system is stable. The main difficulty is how to find a Lyapunov function for a very diverse range of control applications.

1.5.5

Feedback Linearization

The nonlinearities in a system can sometimes be cancelled in such a way that the closed-loop dynamics that result are in a linear form. One of the drawbacks with feedback linearization methods, discussed below, is that exact cancellation of nonlinear terms may not be possible due to, e.g. parameter uncertainties. However, this is one of the most popular nonlinear control design methods in applications such as robotics [16]. The idea is to transform the nonlinear system model into a linear model, or into a partially linear model so that linear control techniques can be

1.5 Nonlinear Control Design Methods

23

applied. Feedback linearization is usually applied to nonlinear state and output equation models of the form: x_ ðtÞ ¼ f ðxðtÞÞ þ gðxðtÞÞuðtÞ

ð1:4Þ

yðtÞ ¼ hðxðtÞÞ

ð1:5Þ

The full state equation can be linearized, or input–output linearization may be applied, where the goal is to develop a control signal input u(t) that renders the input–output map linear. Feedback linearization in its original form does not allow the control amplitudes to be controlled as easily as in optimal control methods, but it may be more sensitive to modelling errors, since it relies on the cancellation of terms. However, it has been applied successfully to a number of practical nonlinear control problems and it can be used for both stabilization and tracking control problems. Unfortunately, the basic method has the following limitations: • Cannot be used for all types of nonlinear system models. • The state is assumed accessible and measured. • Robustness is not guaranteed in the presence of parameter uncertainty or unmodelled dynamics.

1.5.6

High-Gain Control

The beneficial effects of high gain in feedback systems are well understood regarding insensitivity to parameter variations and the rejection of disturbances [2]. The difficulty is, of course, the destabilizing effects that high-gain feedback can introduce, and intuition suggests that the problem will be exacerbated for nonlinear systems. However, high gain in feedback loops, if stability is preserved, has a linearizing effect. Internal feedback around very nonlinear actuators can make the resulting outer-loop system appear linear for control purposes, which is clearly desirable. Consider for example the sector-bounded nonlinearity in Fig. 1.9, and the feedback loop shown in Fig. 1.15. Then if a1 eðtÞ  f ðeðtÞÞ  a2 eðtÞ, we obtain: a1 rðtÞ=ð1 þ a1 kÞ  yðtÞ  a2 rðtÞ=ð1 þ a2 kÞ and if a1 k and a2 k  1 then yðtÞ rðtÞ=k. Thus, irrespective of the nonlinearity the input–output behaviour can become linearized and insensitive to the form of the nonlinearity. A high-gain observer can be used with, for example, a variable structure controller for tracking in the presence of unknown time-varying disturbances and modelling errors. The theory of high-gain observers for use in nonlinear feedback control has been developed over more than two decades [2].

24

1 Introduction to Nonlinear Systems Modelling and Control r(t) + –

e(t)

f(.)

y(t)

k

Fig. 1.15 Feedback in a high-gain system

1.5.7

Backstepping

Backstepping is a technique developed in the 1990s and is a design approach whose roots lie in the theory of feedback linearization of the 1980s [27]. It can be applied to systems where feedback linearization is not suitable. The system states are assumed available for feedback and the backstepping method involves creating additional nonlinearities so that the undesirable nonlinearities can be eliminated. The design process can start at a subsystem that can be stabilized, and then new controllers can be introduced that progressively stabilize each of the outer subsystems. The process terminates when the final external control is reached in a process known as backstepping. An integrator backstepping approach involves starting with a system that is stabilizable with a known feedback control law and a Lyapunov function. An integrator can then be added to the input and a new stabilizing feedback control law designed for the augmented system. This must be shown to be stabilizing for a new Lyapunov function and the process is repeated. Unfortunately, as with some of the other design methods, it is unlikely the full state vector will be available for feedback, and hence nonlinear observers or filters are needed which introduce other design issues. If only the outputs are available for feedback, the backstepping results are more complex. Adaptive state-feedback control of nonlinear plants has also been proposed using backstepping procedures.

1.5.8

Dynamic Inversion

In the case of dynamic inversion, the plant dynamics are assumed known perfectly and a suitable inverse is assumed available. However, non-minimum phase elements cannot be cancelled without creating unstable hidden modes and certain types of nonlinearity have no inverse. Moreover, system models include uncertainties and measurements include bias and noise so even when perfect dynamic inversion is possible it may not be practical. Nevertheless, nonlinear dynamic inversion has been used in flight control systems using models of the nonlinear dynamics of the aircraft. The benefit is that the nonlinear controller can be valid for the whole flight envelope without the need to apply gain scheduling techniques [28]. There are, of course, many applications where the models of the plant are much more uncertain than in the aerospace industry, which may cause difficulties.

1.5 Nonlinear Control Design Methods

1.5.9

25

Nonlinear Filtering

Nonlinear controllers can be constructed for systems represented in nonlinear state-equation form that are based on the use of nonlinear state estimators and a state-estimate feedback control solution. Several types of nonlinear optimal control may be used that employ an Extended Kalman filter (EKF). The resulting separation principle structure is similar to that in Linear Quadratic Gaussian (LQG) design. However, such solutions have not become popular, even though the EKF for state estimation normally works well. An extended Kalman filter has a similar structure to that of the linear Kalman filter but the model employed is represented by a nonlinear state equation. The computation of the EKF gain is similar to that for a Kalman filter but it involves an online calculation using the current linearized model. The linearization is performed around the previously computed state estimate. The EKF approach may easily be justified intuitively, but there is not a rigorous theoretical derivation of the EKF. Nevertheless, it has been found to be valuable in some applications such as ship positioning systems. The EKF can also be used for parameter estimation if the unknown parameters are represented by slowly varying or constant state variables [29]. The resulting augmented state-space system, with these additional states is nonlinear even if the plant model is linear. The same EKF gain calculation algorithm may be applied when some parameters are to be estimated. Unfortunately, the resulting EKF is more likely to diverge and is not as reliable as one that only has to cope with the plant nonlinearities. The EKF uses a Taylor-series expansion approximation to linearize the nonlinear function for the EKF gain calculation. It is, therefore, a suboptimal solution and can result in divergence when both parameter and state estimation is attempted. The Unscented Kalman Filter (UKF) may also be used in such a state-estimate feedback control solution. It was proposed by Julier and Uhlmann [30] and is a derivative-free alternative to the EKF. It utilizes a novel method, called an unscented transformation, for calculating the statistics (the mean and covariance) of the states that undergo a nonlinear transformation. The UKF often outperforms the EKF in terms of estimation accuracy and it is discussed further in the state estimation Chap. 13 (Sect. 13.1).

1.5.10 Adaptive Control Adaptive control involves modifying the control law parameters, and possibly structure, to compensate for slowly varying uncertainties. Linear and nonlinear system identification algorithms are often part of the solution [31–33]. The resulting adaptive control strategies will be nonlinear regardless of whether the plant is linear or includes nonlinearities. Adaptive control has the advantage that it does not

26

1 Introduction to Nonlinear Systems Modelling and Control

usually require a priori information on uncertainty bounds or precise knowledge on the form of the uncertainty. It offers the possibility of improved control in both linear and nonlinear systems but a major limitation of adaptive systems is the lack of robustness and risk introduced. This risk can be reduced by considering simple adaptive controls for very specific configurations and applications. The term limited authority adaptive control is used to refer to adaptive systems that are tailored to an application with a limited ability to change controller parameters or structure [34]. The effects of uncertainties can also be mitigated using robust control law design, which is explored in Chap. 6. This is simpler than in adaptive control that involves online changes to the control law. However, adaptation still offers considerable promise for the future (see [35–38]). One possible path to increase the reliability of adaptive controls is to limit the authority to change conditions as suggested in Grimble et al. [34].

1.6

Linearization, Piecewise-Affine Systems and Scheduling

It is not surprising that early attempts to design controllers for nonlinear systems involved linearizing the plant model so that well-known and trusted linear control design methods could be applied. The use of piecewise-linear functions to approximate nonlinearities followed, and the popular control law scheduling techniques that are widely used in current industrial systems. The Linear Parameter-Varying modelling approach in the next section can be thought of as a further evolution of these ideas.

1.6.1

Linearization Methods

Linearization methods are easy to understand and provide intuitive insights into the behaviour of a nonlinear system about some operating conditions. However, they have limitations: • A control design based on linearized system dynamics can have stability problems when operating away from the equilibrium point or trajectory. • The equilibrium points and the trajectories must be known in advance and this knowledge is often not available. Piecewise-linear functions are sometimes used in deriving approximate models for nonlinear control. A piecewise-linear function is a real-valued function defined over the real numbers, with a graph that has straight-line sections. It is a function where each of the sections is an affine function. The use of piecewise-linear functions to approximate a nonlinearity, like saturation, is illustrated in Fig. 1.16.

1.6 Linearization, Piecewise-Affine Systems and Scheduling

27

Fig. 1.16 Piecewise-linear function

f ( x) Slope S 3

x Slope S 1

1.6.2

Slope S 2

Piecewise-Affine Dynamical Systems

Some processes and electromechanical systems can be modelled, or approximated, by continuous dynamics that within each discrete mode are affine and where the mode switching occurs at specific subsets of the state-space that are known a priori [39]. A Piecewise-Affine (PWA) dynamical system is a finite-dimensional, nonlinear system, where the functions describing the differential and output equations for the system are piecewise affine. A piecewise-affine system involves a collection of finite-dimensional affine systems together with a partition of the product of the state-space and input space into regions. Each of the regions is associated with a particular affine model, and the dynamics are determined by the region in which the state and input vector appear at a certain time. The dynamical behaviour switches as the state-input vector changes from one region to another enabling nonlinear characteristics to be modelled [40]. Piecewise-affine systems have been studied extensively since they represent the simplest extension of linear systems models that can represent hybrid phenomena (see Sect. 1.8). The PWA system models provide a way of approximating nonlinear systems, since many nonlinearities can be approximated as piecewise-affine functions, such as actuator saturation. They can, therefore, model a rich class of hybrid and nonlinear processes, involving systems where the dynamical behaviour is described by a finite number of discrete-time affine models, together with a set of logical switching rules [39]. A PWA system can be defined by partitioning the state and/or input space of the system into a finite number of polyhedral regions, associating a different affine dynamic with each region (a polyhedron is the intersection of a finite number of half-spaces). Piecewise-Affine (PWA) systems can represent switched systems that are defined by using a polyhedral partition of the state-space [39]. Such a system can represent a restricted class of hybrid systems. The family of Nonlinear Generalized Minimum Variance (NGMV) control laws, introduced in Chap. 4, can be used for the control of Piecewise-Affine (PWA) systems, and by implication for a limited set of hybrid control problems [41].

28

1.6.3

1 Introduction to Nonlinear Systems Modelling and Control

Gain Scheduling

If a nonlinear system is linearized about an equilibrium point, a time-invariant dynamic linear system model is obtained that represents the system locally. The small change model will include a deviation of the control input signal and returns the deviation of the output signal as the output. The small change linear model is illustrated in Fig. 1.17. Linearization about a trajectory results in a linear time-varying model. The gain scheduling approach is one of the simplest nonlinear control techniques. A nonlinear system can be linearized at a number of operating points and local controllers can then be computed for each, using classical or advanced linear control methods. The resulting set of controllers will depend upon the system parameters. The closest controller, in some sense, to the actual operating point for the system can be applied. However, there are problems in how to switch between different controllers, since switching can cause instability. One cause of the instability is the excessive transients that can arise and hence bumpless transfer methods should be employed. The variables to be used for Gain Scheduling will depend upon the application, but Mach number and dynamic pressure are used in flight controls, engine speed and load in automotive controls and average wind speed in wind turbines (see [42–46]). Gain-scheduled control approaches provide a set of controllers corresponding to a set of parameter values, such that for all frozen values of the parameters, the closed-loop has the desired feedback properties. However, since the controller parameters are actually time-varying, these desirable properties will not necessarily be reflected in the resulting closed-loop system. Shamma [45] summarized the conditions which guarantee that the closed-loop will retain the feedback properties of the frozen-time design. His results revealed that a time-varying system would retain its frozen-time exponential stability providing that the time variations of the closed-loop dynamics are sufficiently slow. The computational burden of linearization-based scheduling approaches is normally less than for other nonlinear control design approaches. There are well-known limitations of conventional gain-scheduling methods. These are only effective near equilibrium points of operation, since they are designed based on an equilibrium linearization. The so-called velocity-based gain-scheduling approach overcomes many of the deficiencies of conventional gain scheduling. In this case, a linear system obtained from the velocity-based linearization is associated with every operating point of the nonlinear system and not

u0

Fig. 1.17 Locally linearized system model u

– +

y0

δu

Locally Linearized System Model

δy

+ +

y

1.6 Linearization, Piecewise-Affine Systems and Scheduling

29

just at the equilibrium points. This family of linearized models describes the entire dynamics of the nonlinear system, and is an alternative representation, which is valid globally. It does not involve any restriction to the vicinity of the equilibrium points and large transients and sustained non-equilibrium operation can both be modelled accurately. Velocity-based model scheduling has the advantage over conventional scheduling that the approach does not require an assumption that plant variations with time are slow. The possible drawbacks are the sensitivity to initial conditions, problems with numerical differentiation and understanding how best to use the models in control design (see [47–53]). Gain scheduling usually involves a number of ad hoc steps, beginning with the problem formulation. This can be straightforward in some cases, but is increasingly difficult for systems that are more complex. Unfortunately, stability can only be assured locally if it is assumed the variations with operating points are sufficient. There are also no performance guarantees.

1.7

LPV System Modelling and Control

There have been an increasing interest in Linear Parameter Varying (LPV) models over the past two decades, exploiting the fact that the dynamics of many physical systems can be modelled as linear equations that are functions of varying parameters. Linear Time-Varying (LTV) systems are related but in this case, the variation of system parameters is assumed known and not dependent on some measured or estimated signal. For LPV control law synthesis a model is required that describes the varying dynamic behaviour of the plant in terms of the scheduling parameters. It is normally assumed that the parameters are measurable, but they are not known in advance. Polynomial LPV models for continuous-time systems may be defined to have the input–output form Gðs; pÞ ¼ Nðs; pÞ=Mðs; pÞ, where p(t) is a vector of known time-varying parameters. A related modelling problem arises when a system can be represented by a set of linear models, which is sometimes referred to as multiple model control design. A simple control strategy can be derived by representing the plant at a family of operating points, where the system can be described by a linear model. A linear controller can then be designed for each of the regions to try to ensure good performance and robustness. The total control law can then be implemented by interpolating between these linear controllers in different regions. Gain-scheduled control for LPV systems has received attention because it provides a systematic way of deriving controllers for nonlinear and parameter-dependent systems. The LPV model idea was applied to study the behaviour of gain-scheduled control systems by Shamma in his Ph.D. thesis [45]. A linear parameter-varying system is a linear system, whose coefficients depend on some exogenous time-varying parameter, or parameter vector. For gain-scheduled control, an LPV model can be defined with respect to a scheduling time-varying parameter p(t). This can be used to describe a nonlinear plant by a collection of

30

1 Introduction to Nonlinear Systems Modelling and Control

indexed Linear Time-Invariant (LTI) models. The “exogenous parameter” p(t) is not known a priori but is assumed calculable or measurable in real time using sensors. For example, the wind speed and direction determine some of the parameters in a wind turbine model (see example in Chap. 14). However, they can be treated as known parameters since they are measured during the operation of the wind turbine. The LPV systems are often represented by state-space models with known functions of the parameter vector p(t). The time variation of each of the parameters is not known in advance, but it is assumed to be measurable in real time. The controller for such a system can be assumed to be linear and possibly time varying. Unfortunately, some of the properties of linear time-varying systems add to the difficulty of control loop design. For example, the state-space model of a continuous-time linear time-invariant system is asymptotically stable if all the eigenvalues of the system matrix A are strictly negative. However, the asymptotic stability (or uniform stability) for time-varying systems cannot be characterized by the location of the eigenvalues of the state matrix A(t). The system can be unstable even if all the eigenvalues of A(t) have negative real parts for all time t. Control design methods based on LPV models have become increasingly popular over the past few decades and are a powerful and flexible tool when dealing with nonlinear systems (see [54–56]). The rationale behind the approach is to represent the system dynamics in a quasi-linear form, to allow the rich collection of linear control techniques to be utilized. The linear control design methods are easier to use and understand, and engineers have experience and intuition to draw upon. For systems where there are significant changes in dynamics with operating point, gain-scheduling methods may be applied. There are, for example, gain scheduling methods designed for LPV models based on Linear Matrix Inequalities (LMIs). This is a topic that is discussed further in Chap. 5 (Sect. 5.6.4). The use of LPV modelling methods in predictive controls is discussed in Chap. 11 (Sect. 11.6.4). It is clearly desirable to obtain guarantees of stability and performance when analysing these control systems and a number of papers have appeared over the past decade concerning LPV control law design (see [57–64]).

1.7.1

Modelling LPV Systems

There are many examples of physical parameters that influence control law design including inertia, stiffness or viscosity coefficients in mechanical systems, aerodynamic coefficients in flight control, resistance and capacitor values in electrical circuits, and so on. A Linear Parameter-Varying (LPV) model will treat the parameters that have significant variations as parts of a time-varying vector of parameters p(t). In some cases, an LPV model will arise naturally from the physical equations of the problem. In other problems, the equations may be manipulated or transformed into an LPV type of structure. The parameter p(t) is termed an exogenous parameter when it is not related to the dynamic evolution of the system

1.7 LPV System Modelling and Control

31

states, and an endogenous parameter when it can be represented by some state-variable relationship. The former case relates to the LPV model case discussed above, and the latter case is considered in more detail below. An LPV model may also be obtained when approximating a nonlinear system. Assume, for example, that a continuous-time nonlinear state-space model of a plant is available that may be written as x_ ðtÞ ¼ f ðxðtÞ; uðtÞÞ

ð1:6Þ

yðtÞ ¼ gðxðtÞ; uðtÞÞ

ð1:7Þ

where f ðxðtÞ; uðtÞÞ and gðxðtÞ; uðtÞÞ are differentiable nonlinear functions with Lipschitz continuous first derivatives, and where xðtÞ 2 Rn , uðtÞ 2 Rm , yðtÞ 2 Rp denote the state, input to the plant and the output, respectively. Since the LPV control design approach is easier to use than nonlinear control methods, a transformation of the equations or an approximation of the nonlinear system (1.6), (1.7) is required. There are different ways to obtain the LPV model of a system involving Jacobian linearization or quasi-LPV methods. An LPV model for a continuous-time system has the state-equation form: x_ ðtÞ ¼ AðqÞxðtÞ þ BðqÞuðtÞ

ð1:8Þ

yðtÞ ¼ CðqÞxðtÞ þ DðqÞuðtÞ

ð1:9Þ

where xðtÞ 2 Rn , uðtÞ 2 Rm , yðtÞ 2 Rp and qðtÞ is an external quantity, sometimes called the scheduling variable, which is assumed to be measurable in real time. For a flight control, the time-varying parameter vector qðtÞ might involve airspeed and altitude, and for an automotive vehicle, it might include engine speed. A wind turbine model is dependent on a parameter representing the wind speed, as mentioned above and in Chap. 14. State dependent: State-dependent models are a special form of this type of model, where qðtÞ only involves system states and no other parameters. Pearson [65] first noted that a linear state-dependent system may be used to approximate a nonlinear and non-stationary system. He proposed minimizing a quadratic performance index, where the system is taken to be linear and stationary instantaneously. This is the approach taken in the so-called State-Dependent Riccati Equation (SDRE) approach that was extended by Wernli and Cook [66] and is now popular and is described further in Chap. 10 (Sect. 10.2.2). Scheduling: In the simplest case, a scheduled control law may be applied to this type of system. The “scheduling variable” (from which the name originates) can be used to select the appropriate controller based on the linearized model at a particular operating point. However, the full benefits of LPV control will not be obtained when the plant model is restricted to being only a set of scheduled linearized models. It is often more desirable to base the controller on what might be termed a “global model” so that the solution applies across the operating envelope of the

32

1 Introduction to Nonlinear Systems Modelling and Control

design. The scheduling variable qðtÞ may simply involve a time-varying known parameter p(t), but in general qðtÞ will represent a vector function involving parameters, states and inputs. It may, for example, include a rate of change of parameter term dpðtÞ=dt (see [47, 48]). Quasi-LPV model: The scheduling variable q(t) and system depicted in Fig. 1.18 represent a general function of a parameter vector p(t) and possibly both states and inputs. For a fixed value of time, the model matrices describe a Linear Time-Invariant (LTI) system. An LPV system reduces to a Linear Time-Varying (LTV) system if the trajectory for the parameter variations is known, and to an LTI system on trajectories that are constant (see [67, 68]). If the scheduling variable qðtÞ includes the states of the system x(t), or inputs u(t), so that the model is nonlinear, then this class of systems is often called quasi-LPV, which is denoted qLPV for simplicity. In the following the term, qLPV is used to describe the systems considered in Chaps. 10 and 11. These may be nonlinear since the models may be state dependent. The term endogenous variable is sometimes used as a classification of a variable, whose value is determined by the states or other variables in the system. If the model contains endogenous variables, it is termed a qLPV model in this text. Such a model may be an exact model of a nonlinear system or it may be obtained from an approximation to a nonlinear process.

1.7.2

Deriving an LPV Model by Jacobian Linearization

Jacobian linearization involves the calculation of the first-order Taylor-series approximation of a nonlinear system. The Jacobian linearization approach is one of the simplest methods of obtaining a type of LPV model. It can be used for any nonlinear system, which admits linearization at its equilibrium points of interest, and can be based on a family of linearized models computed at different equilibrium points. The set of equilibrium points will then represent the working space. The resulting model is a local approximation of the dynamics around this set of points. The operating points can be either a discrete set or a continuous trajectory of points. The LPV-based controllers can be designed or calculated for each of these equilibrium points or trajectory. The controller then requires a way to determine the nearest operating point corresponding to the state of the system. The Jacobian linearization approach to computing an LPV model involves linearizing the nonlinear system at different equilibrium operating points, and using interpolation to obtain the LPV model in the operating region. To illustrate the derivation of such a model assume an equilibrium point ðxe ; ue Þ has been

Fig. 1.18 Representation of LPV system and signals

p(t), x(t), u(t) u(t)

y(t) S( ρ (t))

1.7 LPV System Modelling and Control

33

determined for the nonlinear system of Eqs. (1.8), (1.9) by setting x_ ¼ 0. Let q(t) denote a scheduling variable involving a time-varying parameter, and define the equilibrium points for the plant on a set S, where f ðxe ðt; qÞ; ue ðt; qÞÞ ¼ 0 for q 2 S. The small change LPV plant model may be written in the following form: 

  x_ d ðtÞ AðqÞ ¼ yd ðtÞ C ðqÞ

BðqÞ DðqÞ



 xd ðtÞ ; ud ðtÞ

q 2 S:

ð1:10Þ

where @f ðxe ðt; qÞ; ue ðt; qÞÞ; @x @g C ðqÞ ¼ ðxe ðt; qÞ; ue ðt; qÞÞ; @x AðqÞ ¼

@f ðxe ðt; qÞ; ue ðt; qÞÞ @u @g DðqÞ ¼ ðxe ðt; qÞ; ue ðt; qÞÞ @u

BðqÞ ¼

The state, output and the control signals are clearly related to the small change variables and in this LPV model xðtÞ ¼ xd ðtÞ þ xe ðt; qÞ, yðtÞ ¼ yd ðtÞ þ ye ðt; qÞ and uðtÞ ¼ ud ðtÞ þ ue ðt; qÞ. The behaviour of the system between points can be approximated using interpolation. Example 1.2 Consider a nonlinear system described by the following state and output equations: x_ ðtÞ ¼ xðtÞ þ uðtÞ yðtÞ ¼ tanhðxðtÞÞ At the equilibrium point x_ ¼ 0 then xe þ ue ¼ 0 giving xe ¼ ue and hence ye ¼ tanhðxe Þ and xe ¼ tanh1 ðye Þ. If the scheduling variable is taken equal to the output q ¼ ye , then xe ¼ ue ¼ tanh1 ðqÞ. Evaluating the components of the LPV matrix: df df ðxe ; ue Þ ¼ 1; BðqÞ ¼ ðxe ; ue Þ ¼ 1 dx du dg dg C ðqÞ ¼ ðxe ; ue Þ ¼ 1  tanh2 ðxe Þ ¼ 1  q2 ; DðqÞ ¼ ðxe ; ue Þ ¼ 0 dx du AðqÞ ¼

Writing the system in terms of the scheduling parameter q: 

x_ d yd



 ¼

1 1  q2

1 0



xd ud



for q 2 ð1; 1Þ. The small-signal LPV model is then obtained as x_ d ¼ xd þ ud and ■ yd ¼ ð1  q2 Þxd . Unfortunately, there is usually no direct mapping between a nonlinear system and a suitable LPV or qLPV model. This linearization-based approach may, therefore,

34

1 Introduction to Nonlinear Systems Modelling and Control

result in similar problems that arise when gain scheduling is used, and may result in poor performance or even unstable operation. However, there are other methods of deriving LPV models as mentioned in the sections below. Modified Jacobian Linearization The Jacobian linearization method relies upon linearization at equilibrium points but the same idea may be employed using other operating points, determined by say ðx0 ; u0 Þ. This will be referred to as the Modified Jacobian LPV model approach that also involves the calculation of a first-order Taylor-series approximation of the nonlinear system (introduced by mathematician Brook Taylor). In this modelling approach, the operating points ðx0 ; u0 Þ do not need to be at equilibrium points. Recall the first terms in the Taylor-series expansion: f ðx; uÞ ¼ f ðx0 ; u0 Þ þ

@f @f ð x 0 ; u0 Þ ð x  x 0 Þ þ ðx0 ; u0 Þðu  u0 Þ @x @u

Unlike the previous case the first term on the right-hand side of this equation is unlikely to be null. That is, the computation is unlikely to be at an equilibrium point, because ðx0 ; u0 Þ may be any point on the trajectory, and hence f ðx0 ; u0 Þ 6¼ 0. The modified Jacobian LPV model may now be written in two apparently equivalent ways: x_ ðtÞ ¼ f ðx0 ; u0 Þ þ

@f @f ðx0 ; u0 Þðx  x0 Þ þ ð x 0 ; u0 Þ ð u  u0 Þ @x @u

ð1:11Þ

or x_ ðtÞ ¼ ðf ðx0 ; u0 Þ 

@f @f @f @f ðx0 ; u0 Þx0  ðx0 ; u0 Þu0 Þ þ ðx0 ; u0 Þx þ ðx0 ; u0 Þu @x @u @x @u

This latter equation is clearly in a type of LPV form that may be written as x_ ðtÞ ¼ d0 ðqðtÞÞ þ A0 ðqðtÞÞxðtÞ þ B0 ðqðtÞÞuðtÞ

ð1:12Þ

This is similar to Eq. (1.8) except for the signal d0 ðqðtÞÞ that might be treated as an additional disturbance term. An advantage of this modified Jacobian LPV approach is that a single global model is obtained approximating the nonlinear system dynamics at arbitrary operating points. The original nonlinear equations of the system may be utilized to find a Taylor-series expansion, where the first derivative terms provide an approximation to the LPV model. The procedure is straightforward but there is an approximation in neglecting higher order terms. The algorithm has the advantage that it does not require a lot of engineering experience to obtain the model. The LPV controllers may be designed in continuous time but are very likely to be implemented digitally. The computation of discrete-time LPV controllers based upon the continuous models has been considered by Tóth et al. [69].

1.7 LPV System Modelling and Control

1.7.3

35

Quasi-LPV and State-Dependent Models

The quasi-LPV (qLPV) system models can represent a reasonably wide class of nonlinear systems. Linear control algorithms (possibly time-varying) can be used with these models for controller synthesis. This requires accepting the approximations involved, rather than the more demanding nonlinear control design methods. The qLPV models can often provide a close approximation to the behaviour of a nonlinear system and it simplifies the design problem when well-known and understood linear control methods can be applied. The qLPV system equations are of the same form as those for a linear time-varying system. In predictive control, for example, the prediction of future states and outputs may be introduced by exploiting the linear structure. This is particularly important since predictive control is one of the most useful of the model-based modern control laws. The qLPV models are used in different ways by different parts of the control community. That is the state-dependent Riccati equation exponents are concerned with nonlinear systems and control, whilst the LPV design community produce linear possibly time-varying control solutions. The scheduling variable in the qLPV model will sometimes change relatively slowly. In fact, the original meaning of a quasi-LPV model is to cover systems, which are not really LPV but can be treated as LPV if the time scale of the state-dependent scheduling variables is much longer than the time constants in the process to be controlled. From the perspective of the generation of the control laws and the design procedures, there is no need to differentiate between the LPV, qLPV or state-dependent models. However, there is a significant difference when analysing stability properties, since LPV models are a special form of time-varying linear system, and state-dependent models are actually nonlinear system representations. The qLPV models introduced in Chaps. 10 and 11 are intended to cover LPV models, where the optimal control solutions are a particular form of linear multivariable control. The qLPV models also describe state-dependent and input-dependent models that cover nonlinear state-space systems, where the optimal control laws obtained are nonlinear. Such models are normally obtained from physical plant modelling followed by manipulation to put the system into a qLPV form. Approximations may or may not be required in obtaining a qLPV model based on the physical system equations. In fact, it may be pragmatic to introduce approximations to obtain an LPV model to avoid some of the theoretical problems that arise due to nonlinearities (stability conditions being easier to establish for LPV systems rather than for the more general nonlinear qLPV systems). Assume, for example, that a discrete-time state-space model is obtained, as an approximation to a complex nonlinear system. If the A matrix is found to be a function of the state x(t), then the term A(x(t))x(t) might contain the products of states, so the system is state dependent and nonlinear. If now the model for control design (referred to as the design model) is represented as Að^xðtÞÞ, then since the state estimate ^xðtÞ is a signal that may be assumed known at time t, this model becomes similar to a time-varying linear system. The design model may then be

36

1 Introduction to Nonlinear Systems Modelling and Control

treated as a form of LPV system, which simplifies some of the theoretical questions regarding the behaviour of the optimal control. It does not, of course, avoid the problem of showing the system is stable on the true underlying nonlinear system (possibly involving a state-dependent model). This text does, of course, focus on control design issues rather than questions of stability, and this subtle difference in the significance of such models will often be neglected but should not be forgotten.

1.7.4

Derivation of Quasi-LPV Models

As noted, the quasi-LPV model approach requires the physical nonlinear equations for the industrial process to be manipulated and possibly approximated to obtain a suitable qLPV related form. It may, therefore, require a good understanding of the physical system and model. In fact, it is not very clear what constitutes a “good” qLPV model (controllability, low sensitivity, links to the physical equations and so on). However, Paijmans et al. [70] noted that a good LPV model should satisfy a set of requirements that are often conflicting, such as • The model should represent the dynamic behaviour of the system throughout the parameter range. • The complexity of the model should be limited. • The numerical conditioning of the resulting LPV model should be good. State-space models for linear systems are not, of course, unique and there is not a unique qLPV model, which adds to the modelling and model selection problem. However, algorithms based on such models have provided some of the most promising robust control solutions, for applications ranging from automotive systems to wind turbines. To illustrate the ideas consider the continuous-time system Eqs. (1.8) and (1.9) again. At each time t and scheduling variable q(t), the local behaviour of the nonlinear plant may be determined. If one or more of the “parameters” of the system matrices depend upon states, than the system is classed as qLPV. If, for example, the state vector is partitioned as xT ¼ xT1 ; xT2 and the state matrices only depend on the group of states included in the vector x1 , the quasi-LPV model may be written as 

  x_ ðtÞ Aðx1 Þ ¼ yðtÞ C ð x1 Þ

Bðx1 Þ Dðx1 Þ



xðtÞ uðtÞ

 ð1:13Þ

where qðtÞ ¼ x1 ðtÞ. Since the states x1 are included in the scheduling parameters, they must, of course, be assumed to be measured or estimated. Example 1.3 Consider a nonlinear two state system described by the following equations:

1.7 LPV System Modelling and Control

37

x_ 1 ¼ sinðx1 Þ þ x2 x_ 2 ¼ x1 x2 þ u There are many ways to write the system in a qLPV equation form, and two can be illustrated. Write the qLPV model as  x_ ¼ AðxÞx þ Bu ¼

sinðx1 Þ=x1 x2

   1 0 xþ u 0 1

with the scheduling parameter qðtÞ ¼ ½x1 ðtÞ; x2 ðtÞT . This model may be suitable, however, if say only x1 is available, a more useful form of qLPV model is given as  x_ ¼

sinðx1 Þ=x1 0

   1 0 xþ u x1 1

where the scheduling parameter qðtÞ ¼ x1 ðtÞ.

■

More generally, consider a system that has a continuous-time qLPV state-space form, whose state-equation matrices depend upon the first subset of states. Also, let p(t) denote a vector of external scheduling variables so that the scheduling  T parameter is a vector qðtÞ ¼ xT1 ðtÞ pT ðtÞ . The system can then be described in continuous-time state-equation form as 2

3 2 32 3 x1 A11 ðqÞ A12 ðqÞ B1 ðqÞ x_ 1 4 x_ 2 5 ¼ 4 A21 ðqÞ A22 ðqÞ B2 ðqÞ 5 4 x2 5 C1 ðqÞ C2 ðqÞ DðqÞ y u

ð1:14Þ

In this case the signal x2 ðtÞ is a subset of the system state vector  T xðtÞ ¼ xT1 ðtÞ xT2 ðtÞ , which does not enter or influence the system matrices. Since qLPV systems arise when a scheduling variable q(t) contains some states of the system, this state dependency can result in nonlinear feedback through the system matrices. By treating the scheduling parameters as independent variables, the techniques used to design LPV controllers can be applied to qLPV models. The algebraic manipulations in the control law derivations that follow in other chapters do not normally require any distinction between the two cases. However, a system that requires an qLPV model will be harder to tune, and more difficult to control, than one which uses a strict LPV model.

1.7.5

Velocity-Based LPV Models

The LPV models were found from Jacobian linearization in Sect. 1.7.2 but involve approximations. Other methods of deriving LPV models, using analytic

38

1 Introduction to Nonlinear Systems Modelling and Control

transformations, are not suitable when the nonlinear system is of high order or contains lookup tables. The velocity-based method of scheduling provides a general approach to transform systems into LPV or quasi-LPV form. The Velocity-Based LPV (VBLPV) methods have the advantage that the nonlinear equations representing the system can be differentiated to obtain this type of LPV model without using approximations. There is no need to manipulate the equations to put them into an LPV form. The basic computational procedure involves taking derivatives of the nonlinear function and this provides the LPV form directly. Engineering judgment and experience are not essential in deriving these models. The velocity-based linearization approach provides a direct theoretical relationship between the nonlinear plant and LPV system model. It was developed originally to provide more reliable scheduling methods (see [49–53, 71–75]). The basic structure of a VBLPV model is illustrated in Fig. 1.19. Generation of a velocity dependent model: Consider again the nonlinear system defined in the state Eqs. (1.6) and (1.7) where gðxðtÞ; uðtÞÞ and f ðxðtÞ; uðtÞÞ are differentiable nonlinear functions. A straightforward analytical method of obtaining the continuous-time LPV model is required. This may be written in terms of the vector of scheduling parameters as follows: x_ ðtÞ ¼ AðqðtÞÞxðtÞ þ BðqðtÞÞuðtÞ

ð1:15Þ

yðtÞ ¼ CðqðtÞÞxðtÞ þ DðqðtÞÞuðtÞ

ð1:16Þ

Unfortunately, there is not a direct method of obtaining these equations from the nonlinear system in (1.6) and (1.7). However, to obtain the velocity-based linearization, first differentiate the nonlinear system model to obtain: €xðtÞ ¼ AðqðtÞÞ_xðtÞ þ BðqðtÞÞuðtÞ _

ð1:17Þ

_ y_ ðtÞ ¼ CðqðtÞÞ_xðtÞ þ DðqðtÞÞuðtÞ

ð1:18Þ

where xðtÞ 2 Rn , uðtÞ 2 Rm , yðtÞ 2 Rp . The qðtÞ is a scheduling variable that can involve estimated or measured variables. The matrices in these equations may be obtained from the following Jacobian computations: AðqðtÞÞ ¼

u d dt

df ðx; uÞ; dx

du dt

BðqðtÞÞ ¼

p

Fig. 1.19 Velocity-dependent LPV model

dy dt

LTI model

df ðx; uÞ; du

ð1:19Þ

y

∫

1.7 LPV System Modelling and Control

CðqðtÞÞ ¼

39

dg ðx; uÞ; dx

DðqðtÞÞ ¼

dg ðx; uÞ du

ð1:20Þ

which are numerically the same as in the Jacobian model (1.10). The LPV model that results from (1.17) and (1.18) may be portrayed as shown as in Fig. 1.20, where the differentiation on u(t) involves a high-pass filter to be realizable and practical. Two integrators are needed to give the desired output and states rather than the derivative terms. These integrators must be initialized and hence the presence of the initial condition input terms. Augmented Velocity-Based LPV form: The additional states introduced from the output integrators can be included in a quasi-LPV structure that is in an augmented form. Including a vector of time-varying external parameters p(t) the nonlinear model (1.17), (1.18) can be differentiated to obtain: @f ðx; u; pÞ @f ðx; u; pÞ x_ þ u_ @x @u @gðx; u; pÞ @gðx; u; pÞ y_ ¼ x_ þ u_ @x @u

€x ¼

These equations may be written in the familiar form: z_ ¼ Aðx; u; pÞz þ Bðx; u; pÞu_ y ¼ ½0

1 z

where the extended state and LPV system matrices follow as   x_ z¼ ; y

" Aðx; u; pÞ ¼

@f ðx;u;pÞ @x @gðx;u;pÞ @x

# 0 ; 0

" Bðx; u; pÞ ¼

@f ðx;u;pÞ @u @gðx;u;pÞ @u

#

 T Because of the extended state z ¼ x_ T yT the order of the model is equal to the order of the nonlinear system plus the number of outputs.

x(0) x x

1 s

y

1 s

x x (t ) = A ( ρ (t )) x (t ) + B ( ρ (t )) u (t ) u

s Ts + 1

Approximate differentiation

y (t ) = C ( ρ (t )) x (t ) + D ( ρ (t )) u (t )

u Velocity dependent model

Fig. 1.20 LPV model derived from velocity-dependent model

y (0)

x

y

40

1 Introduction to Nonlinear Systems Modelling and Control

The velocity-based linearization methodology does not involve linearization and the equilibrium points are not required (see [47–53, 71–75]). The model represents the system throughout the range of operation. In gain-scheduling methods, a controller may be obtained to respond to changing operating conditions, but these conditions must be assumed to vary slowly when using the LPV or qLPV approaches. In the velocity-based LPV or qLPV method, this is not an issue. Example 1.4: Different Linear Parameter-Varying Models To illustrate the differences between the mathematical representations of the qLPV, Modified Jacobian LPV (MJLPV) and Velocity-Dependent LPV (VDLPV) models, consider the following nonlinear system: x_ ¼ f ðx; uÞ ¼ x2 þ cosðxÞu with a general operating point ðx0 ; u0 Þ. The LPV models for this system can be defined as below: qLPV: x_ ¼ f ðx; uÞ ¼ ½x  x þ ½cosðxÞ  u

ð1:21Þ

MJLPV: x_ ¼ f ðx0 ; u0 Þ þ ½2x0  sinðx0 Þ  u0   ðx  x0 Þ þ ½cosðx0 Þ  ðu  u0 Þ  ¼ ½2x0  sinðx0 Þ  u0   x þ ½cosðx0 Þ  u þ x20 þ sinðx0 Þ  x0 u0

ð1:22Þ

VDLPV: €x ¼ ½2x0  sinðx0 Þu0   x_ þ ½cosðx0 Þ  u_

ð1:23Þ

Note that the qLPV model formulation is not unique, and if f ðx0 ; u0 Þ ¼ 0 then the MJLPV model describes local dynamics around an equilibrium point. All three models describe the variations of the actual system states, rather than their deviations around the operating point. ■

1.7.6

Discrete-Time LPV State-Space Model

The discrete-time Linear Parameter-Varying (LPV) structure is useful for many applications, including automotive engine control. In fact, this is a good example of a system whose dynamics can be approximated well by an LPV model. There are now often more inputs and outputs to be controlled, which requires the use of more advanced control methods. The vector of parameters p(t) in this case may include engine speed, intake manifold pressure, CAM angles and other variables that describe the operating conditions of the engine. Note that engine control design often uses an event-based model, rather than a traditional discrete-time model,

1.7 LPV System Modelling and Control

41

where the sample rate is synchronized with the engine cycle, and is, therefore, a function of the engine speed. Wind turbine models are also very dependent on a changing parameter, namely the wind speed, and aircraft control models are dependent on altitude and airspeed. A discrete state-space LPV model, where the vector of measured (or estimated) system parameters is denoted p(t), may be written in the form: xðt þ 1Þ ¼ AðpðtÞÞxðtÞ þ BðpðtÞÞuðtÞ

ð1:24Þ

yðtÞ ¼ CðpðtÞÞxðtÞ þ DðpðtÞÞuðtÞ

ð1:25Þ

It is often convenient to use a related approximate numerical LPV model, which replaces an LPV model possibly obtained from the physical system equations by a family of models: ~ ~ Dxðt þ 1Þ ¼ AðpðtÞÞDxðtÞ þ BðpðtÞÞDuðtÞ

ð1:26Þ

~ ~ DyðtÞ ¼ CðpðtÞÞDxðtÞ þ DðpðtÞÞDuðtÞ

ð1:27Þ

This can be defined for deviations from the steady-state operating conditions determined by the scheduling parameters. The matrices in the model (1.24) and (1.25) will be different to those in (1.26) and (1.27). They can be computed, as for continuous systems, using the Jacobian of the underlying nonlinear model, which may not have a natural LPV structure. An advantage is that the LPV model is easily calculated by this procedure (assuming the Jacobian exists). This avoids the difficult problem of deriving the LPV model by trying to rewrite the nonlinear equations in an LPV form, but it is an approximation and introduces some uncertainty. The Velocity-Based LPV (VBLPV) modelling approach is an example of a state transformation approach (acting on state derivatives rather than actual states). The calculations are quite similar to the modified Jacobian LPV modelling approach. The “velocity” aspect comes in through the differentiation with respect to time, of all states, inputs and outputs of the nonlinear system model. As for the MJLPV model, this is easier in continuous time (ignoring time delays), followed by discretization and re-introduction of the delays. A theoretical advantage of the VBLPV models, over the MJLPV approach, is that the VBLPV model is an exact representation of the original system. Moreover, no first-order approximation is involved, if the original nonlinear system is continuously differentiable. However, this advantage is somewhat counterbalanced by the need to specify an additional set of states (of the same order as the original model), together with the need to specify initial conditions for these additional states.

42

1.8

1 Introduction to Nonlinear Systems Modelling and Control

Introduction to Hybrid System Models and Control

A hybrid control system involves the control of plants or processes that involve both continuous-time dynamics and discrete-event system models that may involve switching or logical decision-making. Complex industrial systems normally involve a hierarchical structure, with continuous variable dynamics at the lowest level and management decision-making at the highest level. In this latter case, the system may involve logic for start-up, shut-down and emergency response switching. This is a typical hybrid system that has continuous dynamics at lower levels and discrete-event dynamics at the supervisory level. Hybrid system models can be used to describe, in a rigorous mathematical framework, systems characterized by discrete and continuous dynamics, including the interactions that arise. This enables Hybrid Control philosophies to be applied that often involve system optimization. In addition to the technical contribution to our understanding of such complex systems, the subject could have a significant effect on design teams and working practice, since a hybrid control philosophy merges some of the work of systems engineers, software engineers, with that of control-engineering designers. The hybrid control approach was introduced for applications where some decision variables in the system could only assume discrete (often integer) values, for example, where there are multiple possible modes of operation. Previous control approaches to systems with continuous control and discrete decision-making employed ad hoc switching and logic rules. The hybrid modelling and control approach offers a unified integrated framework, which should lead to superior overall control performance. Hybrid systems normally involve continuous system dynamics governed by physical laws and discrete-event dynamics. They are modelled by continuous-time differential, or discrete-time difference equations, together with discrete-event models that depend upon logic, rules and switching [76]. The system responses often evolve in continuous time at the lower levels of the control hierarchy but are subject to decision-making and mode switching at higher levels [77]. Hybrid systems include processes represented by discontinuous differential equations and control laws that switch between different modes. For example, controls might switch between time optimal control during large setpoint changes and linear control when operating close to setpoints. Continuous and discrete states: The control of an industrial process may involve the following: • Continuous states: Angle, position, velocity, temperature, pressure. • Discrete states: On/off variables, logic sequencing, controller modes, loss of actuators or sensors, relays. This combination of discrete and continuous behaviour leads to the class of systems that are termed hybrid systems. Most of the theory of control systems involves linear or nonlinear models, where the transitions between operating points are smooth. This applies whether they are represented by differential or difference

1.8 Introduction to Hybrid System Models and Control

43

equations. However, in some applications, there are problems where switching is involved that can result in abrupt changes in plant behaviour. This may, for example, arise due to the action of safety systems that cause a degree of reconfiguration. Switched systems can be considered a particular class of hybrid system, where a switching law specifies the subsystems that are active at a particular time (see Sect. 1.6.2). Recall that a system represented by a nonlinear state equation is normally assumed to be described by functions satisfying the Lipschitz condition (the function f(x) satisfies a Lipschitz condition on [a, b] if there exists a constant c, such that jf ðx1 Þ  f ðx2 Þj\cjx1  x2 j). Such a system can move continuously from one real-value state to another. Hybrid systems containing jumps can represent impulsive behaviour in the continuous model state. However, if the states are subject to continuous behaviour followed by a jump, the system does not satisfy the Lipschitz condition. The behaviour of the system at the time of the jump can be represented by discrete state values. Discrete-event dynamic system: A discrete-event dynamic system is part of the hybrid system model. In a discrete-state event-driven system, the state evolution depends upon the occurrence of asynchronous discrete events over time. It involves signals that have values taken from a finite or infinite discrete value set. Hybrid LPV dynamical systems: Hybrid dynamical systems include both continuous- and discrete-state variables, which can sometimes be modelled using an LPV structure (described in the previous section). If the parameters are discrete-valued, so that hðtÞ 2 fh1 ; . . .; hn g then the LPV model represents a hybrid dynamical system, where the continuous dynamics are linear and the discrete switching dynamics are assumed exogenous to the system. Automotive Example: There is a need, discussed in the following chapters, for model-based control algorithms in automotive engine control that can be implemented via embedded controllers. The aim is normally to reduce emissions and fuel consumption whilst maintaining performance. Antsaklis and Koutsoukos [78], demonstrated that a model of a four-stroke gasoline engine has a natural hybrid representation. The engine and air dynamics are continuous-time processes, whilst the pistons have four modes of operation (air/fuel intake, compression, ignition power stroke and exhaust). The behaviour of the piston is a discrete-event process that can be represented by a finite-state machine. The previous practice was to convert the discrete part of the engine behaviour into a continuous model, where the average values of the physical quantities are modelled. However, the time and event-based behaviours can be represented more accurately using hybrid models, and optimization in a hybrid system setting should, therefore, be more effective.

1.8.1

Hybrid Phenomena and Definitions

A hybrid system is a dynamic system that involves both continuous- and discrete-valued variables, where the evolution of the state trajectory depends on

44

1 Introduction to Nonlinear Systems Modelling and Control

both, and involves both continuous and jump behaviour. Branicky et al. [79] described four phenomena that are called hybrid. The hybrid phenomena can be characterized by continuous dynamics that are modelled by a differential equation x_ ðtÞ ¼ nðtÞ, t  0, where x(t) represents the continuous component of the state and nðtÞ is a controlled vector field that generally depends upon x(t), the continuous component of the control command u(t) and the discrete phenomena involved. The four categories included the following: 1. Autonomous switching: In this case, the vector field nðtÞ changes discontinuously or switches if the state x(t) reaches given boundaries or limits. 2. Autonomous impulses: The continuous state x(t) of the system may jump after it has reached a specified region of the state-space or a threshold/limit. 3. Controlled switching: In this type of problem, the vector field nðtÞ changes in response to a control command u(t) and this may be with an associated cost. 4. Controlled impulses: In this case, the state of the system may change discontinuously if the continuous input u(t) reaches a given bound. Boolean variables: A Boolean variable Xn can be defined as a Boolean function namely of Boolean variables f : ftrue; falsegn1 ! ftrue; falseg, Xn ¼ f ðX1 ; X2 ; . . .; Xn1 Þ. The following correspondence between a Boolean variable X and its associated binary variable d can be used: X ¼ true , d ¼ 1 and X ¼ false , d ¼ 0. A Boolean variable dn 2 f0; 1g can then be defined as a Boolean function of Boolean variables: f : f0; 1gnl ! f0; 1g and dn ¼ f ðd1 ; d2 ; . . .; dn1 Þ. The function f is a combination of the different logical operators. Boolean variables and switching: The hybrid systems approach for systems that exhibit both discrete state and continuous state behaviour has become the standard modelling paradigm for a variety of applications that require dynamic models that include embedded discontinuities. Propositional logic problems may be transformed into equivalent sets of linear inequalities for the solution of hybrid control problems but this transformation process is not unique.

Switching

Controller

u1 u

Controller

signal

Switching logic

Environment

2

Plant u Controller

m

Fig. 1.21 Switching signal for scheduling controllers

1.8 Introduction to Hybrid System Models and Control

45

Switching: As an example of controlled switching assume a collection of nonlinear state-feedback controllers exists, as illustrated in Fig. 1.21, where u1 ðtÞ ¼ f1 ðxðtÞÞ;

u2 ðtÞ ¼ f2 ðxðtÞÞ;

. . .;

uq ðtÞ ¼ fq ðxðtÞÞ

and f1 ; f2 ; . . .; fq are chosen continuous functions. The approach involves the use of switching in controllers or state-feedback gains is a form of scheduling. A reminder of some useful concepts is required before the next section. Linear inequalities and polyhedra: A convex set X Rn given as an intersection of a finite number of m closed half-spaces X, fx 2 Rn jAx  Bg, where A 2 Rmn , B 2 Rm , is a polyhedron. Depending on the actual inequalities, the region can be unbounded, or bounded, and in the latter case, it is called a polytope.

1.8.2

Modelling Hybrid Systems

There are three main approaches for modelling hybrid systems and the first involves discrete formalisms that are extended by continuous variables, which evolve according to differential equations associated with discrete states [80]. Finite-state machines and Petri nets are examples of such discrete models. A hybrid automata model integrates discrete finite automata with time-dependent continuous variables. The second approach is to employ a discrete model and a continuous model, and couple them by interfaces that transform continuous-valued measurements into discrete-event signals and vice versa. For example, the MATLAB® Stateflow is a block diagram based simulation tool for modelling and simulating event-driven systems. It provides a method for designing embedded systems and can represent supervisory logic, finite-state machines, discrete-event systems and hybrid automata. The third approach, which is of interest here, involves the application of optimization techniques to the solution of hybrid control problems. One of the main approaches is the use of predictive control but other optimization/optimal control solutions can also be used. Piecewise-affine systems: It was noted in Sect. 1.6.2 that Piecewise-Affine (PWA) systems can be considered as hybrid systems, as they can involve switching between a number of predefined linear models, as operating conditions change. Piecewise-affine models can be defined by partitioning the input-state space into polyhedral regions, associating a different linear state equation with each region [81]. They can provide a useful approximation for a large class of nonlinear systems. For example, the following PWA system can be used as an approximation of a nonlinear system model: xðt þ 1Þ ¼ Ai xðtÞ þ Bi uðtÞ þ fi yðtÞ ¼ Ci xðtÞ þ Di uðtÞ þ gi

46

1 Introduction to Nonlinear Systems Modelling and Control



  

xðtÞ x for 2 Xi and Xi ¼ : Hi x þ Ji u  Ki for i ¼ 1; . . .; s, where t is the uðtÞ u time instant. The X is a polytope and the set Xi defines the polyhedral partition fXi gsi¼1 in the state-input space. The PWA model provides the “simplest” extension of linear system models that can represent nonlinear and non-smooth processes with arbitrary accuracy. It is capable of handling various hybrid phenomena. In fact, the PWA model is reasonably general, covering discrete-time linear systems with static piecewise-linear characteristics, discrete-time linear systems with logic states and inputs, and switching systems. Piecewise-affine systems provide a useful framework for modelling hybrid systems, and for the approximation of many nonlinear systems [82]. Discrete-time PWA systems are equivalent to interconnections of linear systems and finite automata, and can be shown to be equivalent to hybrid systems described in the Mixed Logic Dynamical (MLD) form. However, PWA models are not in a suitable form for transforming control synthesis problems into compact optimization problems. The MLD models described below were developed for use in the solution of hybrid control problems, such as mixed-integer linear quadratic optimization problems [81]. Mixed logical dynamical systems: One of most common hybrid models, MLD systems, are discrete-time systems described by physical laws (with linear dynamics), logic rules (if-then-else rules) and operating constraints. This model is a useful compromise between applicability and complexity. It is very suitable for engineering applications, and includes finite-state machines, discontinuous and piecewise discrete-time linear systems. They can represent continuous and discrete-valued states, constraints, nonlinearities and logic statements. The MLD approach involves embedding the logical expressions in the state equations by transforming Boolean variables into the integers 0 or 1. These relations are then expressed as mixed-integer linear inequalities. The general MLD form of a hybrid system is xðt þ 1Þ ¼ AxðtÞ þ B1 uðtÞ þ B2 dðtÞ þ B3 zðtÞ

ð1:28Þ

yðtÞ ¼ CxðtÞ þ D1 uðtÞ þ D2 dðtÞ þ D3 zðtÞ

ð1:29Þ

E1 xðtÞ þ E2 uðtÞ þ E3 dðtÞ þ E4 zðtÞ  g5

ð1:30Þ

where x(t) denotes the states, u(t) the inputs and y(t) the outputs with real and binary  T components. The state of the system may be written as x ¼ xTc xTl , where the continuous part of the state xc 2 Rnc and the logical or discrete part of the state xl 2 f0; 1gnl . The binary variables are denoted d(t), and the auxiliary continuous variables z(t), respectively. These variables are introduced when translating propositional logic, or PWA functions, into linear inequalities. All constraints on the states, inputs, outputs and auxiliary variables can be represented in the mixed-integer linear inequality form. Mixed logical dynamical systems provide a useful computational framework for hybrid systems. It consists

1.8 Introduction to Hybrid System Models and Control

47

of a collection of linear difference equations involving both real and Boolean (i.e. 1 or 0) variables, subject to a set of linear inequalities. A range of problems may be transformed into the MLD form, which can represent a wide class of hybrid dynamical systems. These include sequential logical systems, discrete-event systems and Linear Complementarity (LC) systems described below. Linear complementarity systems: The LC systems are also related to PWA systems. The time evolution of a LC system will involve a number of continuous phases but with events that are governed by inequalities. These events can cause a change in the dynamics and possibly jumps in the state vector. The model covers switching control systems, variable structure systems and problems in hydraulic systems such as those containing non-return or one-way valves. The general LC form of a hybrid system is xðt þ 1Þ ¼ AxðtÞ þ B1 uðtÞ þ B2 wðtÞ

ð1:31Þ

yðtÞ ¼ CxðtÞ þ D1 uðtÞ þ D2 wðtÞ

ð1:32Þ

vðtÞ ¼ E1 xðtÞ þ E2 uðtÞ þ E3 wðtÞ þ E4

ð1:33Þ

The vðtÞ and wðtÞ signals are the so-called complementarity variables and if ? denotes the orthogonality of these variables then 0  vðtÞ ? wðtÞ  0.

1.8.3

Optimization Methods and Hybrid Control Problems

Hybrid systems can be used to optimize the continuous control action together with the discrete-event/logical decision-making actions. It provides a unified theory to encompass switching, discrete-event systems and continuous control. Within the hybrid systems theory, two popular categories of systems and control problems can be defined using the Piecewise-Affine and Mixed Logical Dynamical system models. The interest in hybrid system modelling has been complemented by advances in optimization theory. Lee and Barton [83] described developments in the global optimization of linear time-varying hybrid systems, where the transition times and the sequence of modes are fixed. By some reformulation of the problem, developments in convexity theory for linear time-varying continuous systems can be exploited and a mixed-integer dynamic optimization problem solved. The global hybrid solution can then be obtained in a finite number of iterations. Mixed-integer dynamic optimization problems: A class of algorithms that employ mixed-integer programming techniques, known as Mixed-Integer Linear Programming (MILP) and Mixed-Integer Quadratic Programming (MIQP), can be employed for the solution of predictive control problems. In fact, the MLD modelling approach may be applied to the solution of a range of control and estimation problems involving MILP’s or MIQP’s, for which efficient algorithms are available.

48

1 Introduction to Nonlinear Systems Modelling and Control

The main classes of multi-parametric programming are represented by multi-parametric MILP for performance indexes based on the 1-norm or ∞-norm, and multi-parametric MIQP for performance indexes based on the 2-norm. The MIQP problems are classified as NP-hard, signifying that in the worst case the solution time grows exponentially with the problem size (i.e. the number of integer/ binary variables). Model Predictive Control (MPC): This is the most popular control approach for hybrid systems represented by MLD models. The solutions to this class of predictive control problem can be found using mixed-integer programming. Multi-parametric programming is used for efficient online implementation by obtaining the equivalent piecewise-linear control form of MPC [84–86]. Much of the computational effort is moved off-line, which is the main advantage of the technique. A multi-parametric programming approach can be used to calculate the optimal state-feedback control law u(x(t)), as an explicit function of the state variables x(t). The online optimization involves function evaluations, at regular time intervals, depending on the given state of the plant.

1.8.4

Hybrid Optimal Control and Optimization

There are numerous control applications in which some decision variables (controller outputs), are inherently discrete. These normally include switching during start-up and shutdown procedures but may also be present during normal system operation. For example, a number of distinct possible modes of operation will involve logical decision-making, or perhaps an actuator may assume a finite set of discrete values. Let the functions f(.), g(.) and h(.) represent different classes of hybrid systems, such as the Piecewise-Affine (PWA) systems, Mixed Logical Dynamical (MLD) systems and Linear Complementarity (LC) systems. A discrete-time hybrid system can then be modeled in the general discrete form: xðt þ 1Þ ¼ f ðxðtÞ; uðtÞ; wðtÞÞ

ð1:34Þ

yðtÞ ¼ gðxðtÞ; uðtÞ; wðtÞÞ

ð1:35Þ

0  hðxðtÞ; uðtÞ; wðtÞÞ

ð1:36Þ

where uðtÞ 2 Rm , xðtÞ 2 Rn , yðtÞ 2 Rl denote the input, the state and the output, and wðtÞ 2 Rr is a vector of auxiliary variables. Optimal Control Problem: Consider a PWA system that may be subject to hard input and state constraints of the form ExðtÞ þ LuðtÞ  M; for t  0, with a state equation:

1.8 Introduction to Hybrid System Models and Control

xðt þ 1Þ ¼ Ai xðtÞ þ Bi uðtÞ þ fi if



49

xðtÞT

uðtÞT

T

~i 2X

ð1:37Þ

 s where X~i i¼1 is a new polyhedral partition of the state plus input space. It is given by the intersection of the polyhedrons Xi with the polyhedron described by the constraints ExðtÞ þ LuðtÞ  M: Defining the cost-function with the time horizon N: J ðUN ; xð0ÞÞ ¼ kPxðNÞkp þ

N 1 X

kQxðtÞkp þ kRuðtÞkp

ð1:38Þ

k¼0

The constrained finite-time minimization problem is defined as J ðxð0ÞÞ ¼ min J ðUN ; xð0ÞÞ fUN g

ð1:39Þ

subject to the state model (1.37) and for T UN ¼ ½uT ð0Þ; . . .; uT ðN  1Þ 2 Rnc N  f0; 1gnl N . The state term in the cost-function involves kQxkp denotes the p-norm of the weighted state vector with the weighting matrix Q. For p = 2, the term kQxkp ¼ xQT Qx, where the weightings Q ¼ QT  0, R ¼ RT [ 0, P  0 [81, 87]. The state-feedback solution to the finite-time optimal control problem, based on quadratic or linear cost-function norms, is a time-varying piecewise-affine feedback control law. Transformed Optimal Control Problem: One approach to computing the control gains is to first translate the PWA system into a set of inequalities with integer variables that represent the switching between the different dynamics. The Mixed Logical Dynamical (MLD) systems modelling method introduced above provides such a framework [88]. The problem then becomes min J ðUN ; xð0ÞÞ ¼ kPxðNÞkp þ fU0 g

kQxðtÞkp þ kRuðtÞkp

ð1:40Þ

t¼0

 subject to

N 1 X

xðt þ 1Þ ¼ UxðtÞ þ G1 tðtÞ þ G2 dðtÞ þ G3 zðtÞ E2 dðtÞ þ E3 zðtÞ  E1 tðtÞ þ E4 xðtÞ þ E5

ð1:41Þ

The optimal control problem can be formulated as a mixed-integer quadratic program when the squared Euclidean norm p = 2 is used, or as a mixed-integer linear program, when p = ∞ or p = 1. Multi-parametric Programming: Multi-parametric mixed-integer nonlinear programming problems have the general form: J ðxÞ ¼ inf ðf ðz; xÞÞ; such that gðz; xÞ  0 z

ð1:42Þ

where z 2 Rs denotes the optimization variable, x 2 Rn is the parameter, f : Rs  Rn ! R is the cost-function and g : Rs  Rn ! Rng are the constraints.

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1 Introduction to Nonlinear Systems Modelling and Control

Multi-parametric programming can be used to compute the explicit form of the optimal state-feedback control law. The main advantage is that a complete map of all the optimal solutions can be generated off-line. Then, as the operating conditions change, there is no need to optimize online, since the optimal control is a known function of the parameters (or the new set of conditions). Hybrid predictive control problems can be recast as MIQP optimization problems, and the main difference between hybrid predictive control and standard MPC, in terms of the challenges involved in the solution, amounts to the computational overhead of MIQP solvers, as compared with the standard quadratic programming. The hybrid modelling paradigm is expected to play an important role in the design, analysis and control of future industrial, cyber-physical and complex systems. Because of the combination of discrete and continuous states, hybrid models represent a valuable tool to describe complex systems, which will facilitate future performance analysis, data-driven modelling, controller design and synthesis and fault-detection methods.

1.9

Nonlinear System Identification and Auto-tuning

If physical system models are not available, and are too difficult or expensive to obtain, nonlinear system identification methods may be needed. In some cases, the physical model structure is known but parameters must be estimated. If a system is Linear Time-Invariant (LTI) in the region of operation there are reliable system identification methods available but they do need very careful validation and verification. Validation is the task of demonstrating that the model is a good representation of the real process and responds to typical inputs with representative behaviour; having sufficient fidelity to satisfy subsequent control design needs. Verification is the action of assessing the validity or accuracy of the model by using test procedures, or by exploiting engineering experience, to check that the model behaves in a suitable manner. The identification of models for Linear Parameter-Varying (LPV) systems is more difficult because of the time variation of the parameters and larger range of possible operating conditions and responses. Plant models must be assumed identifiable since this is a prerequisite for model identification. Identifiability is a property that concerns the uniqueness of the model parameters that can be determined from input–output data under the ideal conditions of noise-free measurements and error-free model structure. Statistically, a system is identifiable if it is theoretically possible to compute the true values of the model parameters given an infinite number of observations from the system. From a practical point of view, a necessary condition for identifiability is that the system representation is minimal [89]. The system models may be defined in terms of various shades of grey, when they contain parameters that have unknown or uncertain numerical values [90]:

1.9 Nonlinear System Identification and Auto-tuning

51

• White-box models: Model obtained by careful and extensive physical system modelling from first principles. • Black box models: Some structure is assumed like Autoregressive Moving Average (AMA), and models are generated from input–output data. • Grey-box models: Formed from white-box models, but where some parameters are unknown, or where certain subsystems are modelled as black-box.

1.9.1

Nonlinear System Parameter Estimation Methods

Gradient methods are very successful in parameter estimation for linear and nonlinear systems. In this approach, the parameters are computed recursively to minimize a cost-function, using first-order gradient information. Popular parameter estimation methods include Gradient Descent, Gauss–Newton and Conjugate Gradient methods. Gradient Descent Methods: These are also known as steepest descent methods. To find the minimum of the cost-function, the gradient descent method takes steps proportional to the negative gradient of the cost-function at the current solution estimate. The method is based on first-order derivatives and is simple and easy to implement. It is computationally efficient, but it can take a long time to converge when the estimation error cost-function is badly conditioned. Gauss–Newton Methods: Rather than using the first-order information, as in the gradient descent method, a more efficient strategy is to utilize both first and second-order derivatives in the optimization step as in Newton’s method. The Gauss–Newton method is a modification of Newton’s method, which approximates the second-order information using the first-order derivatives. This is useful because the computation of the second-order derivatives can be expensive numerically or even infeasible for practical problems. The use of the gradient and the (approximated) second-order Hessian information provides a more efficient numerical algorithm than the gradient descent method for the optimum search. However, it involves a comparable level of computational cost. Conjugate Gradient Methods: These are more efficient than the gradient descent methods, since the conjugate gradient algorithms can locate the minimum efficiently by searching along a set of conjugate directions. These are constructed by conjugation of the residuals, instead of the local gradient. The conjugate gradient method can follow narrow valleys, where the steepest descent method slows down and follows a criss-cross pattern. The minimum can be reached in far fewer steps compared with gradient descent methods. Other recursive algorithms, such as the Extended Kalman Filter (EKF) or the Unscented Kalman Filter (UKF), can also be applied to parameter estimation for nonlinear models, if the structure of the dynamic model is known (these estimation methods are described briefly in Chap. 13). Although for many practical problems the gradient-based approaches may converge slowly or stop at local minima, they

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1 Introduction to Nonlinear Systems Modelling and Control

are usually very efficient computationally. The EKF-based methods may have superior estimation properties (e.g. more stable, faster), but they require knowledge of the system structure.

1.9.2

Role of Nonlinear System Identification Methods

Reliable methods of identifying nonlinear systems are available for some nonlinear systems using bespoke software or commercial toolboxes [30–33]. There are, of course, limitations on the structure of the nonlinear systems that can be identified, but such methods can be valuable for the following: • To identify industrial processes that have been in operation for some years where it is difficult to obtain plant information and to produce good physical models. • To use identification methods to obtain approximate models for nonlinear systems like automotive engines, where models may be available but are too complicated for control loop design. Grey-box models may also be used to obtain simplified models for control design, where the identification is applied only to the subsystems that are difficult to model physically (such as combustion torque generation) • To use for online adaptation or condition and fault monitoring systems. Input–output dynamic models may be developed based on Wiener or Volterra models. However, if the structure of a system is known, state-equation models may be preferable. In this case, one option is to use an extended Kalman filter to cope with the nonlinearities and to provide parameter estimation capabilities. An alternative approach is to use subspace identification methods, but in this case, the underlying model, whose physical state-space structure may be known, will not be related directly to the subspace state matrices obtained.

1.9.3

LPV or qLPV System Identification

Even when a good physical model say an automotive engine is available, it may be desirable to represent the system in a quasi-LPV or state-dependent form which is very convenient for nonlinear control design. In some cases, physical systems can be represented in such a form directly but more commonly, such models are used to approximate the underlying physical system model. Nonlinear system identification methods may be used to generate such models, without the time-consuming detailed analysis that might otherwise be required. These model calculations are part of the off-line control design stages but nonlinear identification methods can also be exploited online to introduce an adaptive capability. One of the most useful

1.9 Nonlinear System Identification and Auto-tuning

53

roles for nonlinear system identification is in the derivation of state-dependent or linear parameter-varying models, that are employed in later chapters for controller design (Chaps. 10 and 11). It was noted earlier that a class of discrete-time nonlinear systems can be represented as LPV systems, where the scheduling parameters are measured, and the functional dependence of the system coefficients on the parameters is known. A simple approach to identify LPV systems can be obtained by reducing the problem to linear regression, and compact formulae may be derived using Least Mean Squares or Recursive Least Squares algorithms. Assume the scheduling parameters qðtÞ are equal to a vector of known time-varying parameters qðtÞ ¼ pðtÞ. There are two main approaches to the LPV system identification problem [67]: Local methods: • Local linear models are obtained for fixed scheduling parameters pðtÞ ¼ ci . • Global data is used to interpolate results in an LPV model. Global methods: • Involves determining a global LPV model structure. • Using global data to estimate an LPV model. Some problems are avoided when using global identification methods relative to local approaches. For example, interpolating local linear state-space models is difficult when the McMillan degree varies over the set of local models. Input–output models: Given a parameter vector p(t) a discrete-time input–output model, that can be used for LPV system identification, may be represented in the scalar case as yðtÞ ¼ 

na X i¼1

ai ðpðtÞÞyðt  iÞ þ

nb X

bi ðpðtÞÞuðt  iÞ þ nðtÞ

ð1:43Þ

i¼1

where u(t) is the input, y(t) is the output and nðtÞ denotes process noise. The model may easily be generalized to the multivariable control case. There are various choices for parameter dependencies, with polynomial functions being the most common. It is desirable to define a parameterization so that linearity in the parameters is retained. The parameters may then be estimated using a simple recursive least squares algorithm. State-space models: State-space methods provide a more general framework that is often better suited for higher dimensional systems. For a LPV continuous-time system, the state model may be represented in the form: x_ ðtÞ ¼ AðpðtÞÞxðtÞ þ BðpðtÞÞuðtÞ þ EðpðtÞÞnðtÞ

ð1:44Þ

yðtÞ ¼ CðpðtÞÞxðtÞ þ DðpðtÞÞuðtÞ

ð1:45Þ

54

1 Introduction to Nonlinear Systems Modelling and Control

The model states do not need to be the same as the physical model process states. There are many ways to parameterize an LPV model of an unknown system to capture the underlying parameter variations or nonlinearities. This involves a bespoke approach for each problem, utilizing any prior knowledge about the system. However, a simple generic choice like a linear, quadratic or polynomial parameter dependence on each model coefficient can be used as an initial guess. An affine parameter dependency is often assumed to simplify the parameter estimation problem and to use for control design. In this case: AðpÞ ¼ A0 þ

np X

A i pi

ð1:46Þ

i¼1

and similarly for the other state-equation matrices. Subspace identification: Subspace system identification methods are often used for state-space model estimation [91]. In the time-invariant case, the modelling structures (1.43) and ((1.44), (1.45)) can be made equivalent, but for LPV models this is not generally the case [67, 68]. The plant models for applications in the following chapters are often given in continuous-time form, since they represent the physical nonlinear processes. The models are needed in discrete-time form for control design purposes. Subspace identification methods provide one option to generate discrete-time state-space design models. Series expansion models: Series expansion discrete-time models can be represented in a general discrete-time system form as follows: yðtÞ ¼

nf X

wi ðpðtÞÞFi ðz1 ÞuðtÞ

ð1:47Þ

i¼1

The filters Fi ðz1 Þ are usually chosen to be orthogonal, or orthonormal functions, such as the Laguerre (first-order) or Kautz (second-order) filters that constitute a basis for the signal space. If the filter is chosen as Fi ðz1 Þ ¼ zi the Nonlinear Finite Impulse Response (NFIR) model is obtained. The weighting functions wi ðpðtÞÞ can be parameterized, as for the case of the input–output LPV models. The number of terms nf needed to represent the system adequately depends on the choice of filters, or basis functions. This is influenced by prior knowledge about the system dynamics. The well-known Orthonormal Basis Function (OBF) method can use this modelling approach [67].

1.9.4

Orthonormal Basis Functions

Recall that a basis function is an element of a particular basis for a function space, and every continuous function in the function space can be represented as a linear

1.9 Nonlinear System Identification and Auto-tuning

55

combination of basis functions. Assume the basis functions /i are complex-valued. They are said to be orthogonal over some interval ðT1  t  T2 Þ if for all j: Z

T2 T1

/i ðtÞ/ j ðtÞdt

 ¼

0; i 6¼ j kj ; i ¼ j

If moreover kj ¼ 1 for all j then the basis functions are said to be orthonormal. The orthonormal basis function system identification approach was developed by Toth and co-workers [67, 68]. The OBF model has the form: ^yðtÞ ¼

m X

wi ðpÞ/i ðpÞuðtÞ

ð1:48Þ

i¼1

where the weighting functions wi are parameterized with the scheduling variables p, and the basis functions /i ðpÞ are generated by a cascaded network of stable all-pass filters. Note that the LPV system identification method is a global approach. That is, a single model is estimated based on the data from a single identification experiment or dataset. However, the OBF method includes a preliminary step that builds in prior knowledge and involves a number of local experiments. This determines the pole locations of the all-pass filters that define the basis functions. The OBF method consists of the selection of the orthonormal basis functions, followed by parameterization with the scheduling variables. The steps in the application of the identification algorithm can be listed as follows: 1. Determination of the system poles (from an existing model or local identification experiments). 2. Poles clustering and computation of orthonormal basis functions. 3. LPV model parameterization. 4. Model estimation and validation. This model structure is good for representing a range of system types and involves a limited number of parameters. It is useful for control loop design and the identification procedure is a relatively low complexity problem. One of the most valuable features of the use of orthogonal basis functions for the identification of LPV systems is the fact that only a relatively small number of coefficients need to be estimated for accurate models to be obtained.

1.9.5

Model Assessment

The quality of plant models used for control design is probably the most important influence on the success of the implementation of model-based controls. It is, therefore, common for the dataset used for system identification to be split into two

56

1 Introduction to Nonlinear Systems Modelling and Control

halves. These may then be used, respectively, for model estimation and validation. A measure of the quality of identified models is the Best Fit Rate (BFR), defined as

 kyðtÞ  ^yðtÞk2 BFR ¼ 100%  1  kyðtÞ  yðtÞk2 where y is the mean of the output y of the system and ^y is the simulated output of the model calculated using the validation data. The BFR is normalized so that 100% indicates a perfect fit ð^y ¼ yÞ. If the size of the misfit kyðtÞ  ^yðtÞk2 equals the signal variance then clearly the BFR = 0%. Worse fits can result in negative BFR. The BFR is useful for LPV model assessment but it does not always provide a meaningful figure. For example, if the model output follows the measured output exactly but with an offset comparable with the signal variance, then the BFR value will be close to zero, which is clearly not representative of the quality of the model. The calculation of the BFR should, therefore, be combined with a visual assessment of the model validation results.

1.9.6

Relay Auto-tuning

There follows a situation where nonlinearity is introduced that is actually beneficial to the control engineer. The idea of identifying a plant model and using that for control design was the basis of the self-tuning control principle. Unfortunately, the simplistic application of this procedure did not lead to reliable adaptive controllers, even for linear systems. Relay auto-tuning is an example of introducing a nonlinearity in a control loop to achieve a benefit but such cases are unusual. Unlike the self-tuning experience, this approach has been very successful. It is, of course, a simpler problem since auto-tuning is only used for the initial tuning and the controller gains are then normally fixed. Åström and Hägglund [92, 93] developed the first important auto-tuning method. They obtained a simple method of tuning PID controllers that exploited a result from describing function analysis of feedback controls that contain a relay. The feedback controller was temporarily replaced by a relay. By introducing an ideal relay in the feedback loop, the resulting system had a natural sustained oscillation. This enabled them to apply a sustained oscillation PID tuning rule based on the period of oscillation observed and the so-called ultimate gain. The describing function results for a relay feedback system provided the link between the ultimate gain, the frequency of oscillation and the test results. The auto-tuning test procedure requires a step to be introduced into the feedback loop to initiate the limit cycle. The plant is then subject to a square wave input of magnitude h, where h denotes the relay output magnitude. The amplitude of oscillations A on the plant output signal and the period of oscillations Tu can then be recorded. The stability and the frequency of oscillations are determined by the describing function NðA; jxÞ, based on the harmonic balance equation

1.9 Nonlinear System Identification and Auto-tuning

57

ð1 þ NðA; jxÞGðjxÞÞ ¼ 0. The point of self-oscillations in the complex plane is the crossing point of the Nyquist frequency response plot of GðjxÞ and the inverse of the describing function NðA; jxÞ1 . Recall for a relay characteristic the describing function is given by NðA; jxÞ ¼ 4h=ðpAÞ and in Ziegler–Nichols PID control terminology the ultimate or critical gain is given by the describing function at the intersection, namely kc ¼ 4h=ðpAÞ. It is an approximation since the signals contain harmonics, but the ultimate gain can be taken as the relative amplitude of the two oscillations multiplied by 4/p. The time required to complete a single oscillation (the ultimate period Tu) and the ultimate gain kc follow. Given the period of oscillation, the frequency of oscillation xu ¼ 2p=Tu can also be determined, and the Ziegler–Nichols tuning rules (or related), can then be applied to tune the PID controller. A deadband can be used to avoid the frequent switching that may be caused by measurement noise in this test procedure. The steps are similar to the Ziegler–Nichols sustained oscillation PID tuning method, but instead of increasing the gain to obtain continuous cycling, it uses the relay with deadzone to establish the oscillations. The relay introduces a periodic signal with good frequency content, and introduces a transient that provides helpful information. The tuning rules are built into commercial controller algorithms so that the relay tests can be performed and the controller can automatically be retuned (see Fig. 1.22). Advantages: Relay auto-tuning methods have several advantages compared with the usual continuous cycling PID tuning methods that involve increasing the proportional gain. Only a single experiment test is required instead of a trial-and-error procedure. The amplitude of the process output A can also be limited by adjusting the relay amplitude h. More importantly, the process is not forced to a stability limit.

Tuning rules

PID controller

r(t)

+ +h -h

Fig. 1.22 Relay auto-tuning

G(jω) u(t)

Linear subsystem

y(t)

58

1 Introduction to Nonlinear Systems Modelling and Control

Auto-tuning is used extensively in the process industries, mostly involving standard commercial solutions. However, it has not yet been applied in large numbers in other industries. For example, calibration engineers in the automotive industry or servo-system suppliers might simplify the time-consuming tuning process by using bespoke auto-tuning solutions. There are also obvious applications in marine systems that use PID controls, such as setting up autopilots or fin roll stabilization systems.

1.10

Concluding Remarks

In practice, no system or component is actually linear, but when the deviations from a steady-state operating point are small, many systems can be analysed and designed very successfully using a linear model and applying a scheduling algorithm. There may not be a significant loss of accuracy in the computed system input–output responses when a linear model is used, because of the linearizing effect of feedback action. However, linear analysis and design methods are not so suitable for machines or processes that have discontinuities in their characteristics. The problems that arise become more serious for severe nonlinearities. Some nonlinear components can, of course, be avoided, because they are due to bad design. For example, a poorly sized pump or valve that is always working at one end of its range will cause problems and may need to be replaced. Unfortunately, the control system designer cannot always influence such decisions. The general nonlinear control techniques described in this chapter provide valuable analysis and design tools, but in the main, they do not provide synthesis techniques. Formal synthesis methods for designing nonlinear control systems are very desirable so that a well-defined design strategy can be established for particular applications. Optimal control solutions provide a formal synthesis framework and this opens up a very wide range of different design approaches. However, for the purposes of this work synthesis and design approaches that are easy to understand and implement are of most interest. The family of controllers, which are sometimes termed Nonlinear Generalized Minimum Variance (NGMV) controllers are relatively simple and are introduced. The simplest nonlinear predictive control strategy termed Nonlinear Generalized Predictive Control (NGPC) is also considered and more general predictive algorithms. These controllers seem very suitable for industrial applications, particularly when used in conjunction with quasi-LPV modelling methods. Most of the applications of these and the other control design methods considered involve experience from design studies on real applications. Most of the following text will focus upon the use of discrete-time system models and the corresponding control design methods, since this is how the majority of advanced controllers will be implemented [94]. The nonlinear system modelling techniques will include nonlinear operator, state-dependent, LPV and qLPV methods [95–107]. The focus in the text is mainly on the controller synthesis problems, the control design procedures and finally on the application studies.

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References 1. Nijmeijer H, van der Schaft AJ (2006) Nonlinear dynamical control systems, 3rd edn. Springer, NY 2. Khalil H (2002) Nonlinear systems, 3rd edn (international edition). Pearson Ed 3. Isidori A (1995) Nonlinear control systems, 3rd edn. Springer, London 4. Atherton DP (1982) Nonlinear control engineering. Van Nostrand Reinhold, New York 5. Vukic Z, Kuljaca L, Donlagic D, Tesnjak S (2003) Nonlinear control systems. Marcel Dekker 6. Fradkov AL, Evans R, Andrievsky BR (2006) Control of chaos: methods and applications in mechanics. Philos Trans R Soc A 364:2279–2307 7. Vidyasagar M (1993) Nonlinear system analysis, 2nd edn. Prentice-Hall 8. Desoer CA, Vidyasagar M (1975) Feedback systems: input-output properties. Academic Press 9. Hoseinnezhad R, Harding P (2005) A novel hybrid angle tracking observer for resolver to digital conversion. In: IEEE conference on decision and control, and European control conference, Seville, pp 12–15 10. Hill D, Moylan P (1980) Dissipative dynamical systems: basic input output and state properties. J Franklin Inst 309:327–357 11. Hill D, Moylan P (1980) Connections between finite-gain and asymptotic stability. IEEE Trans Autom Control 25(5):931–936 12. Fossen TI, Strand JP (1999) Passive nonlinear observer design for ships using Lyapunov methods: full-scale experiments with a supply vessel. Automatica 35(1):3–16 13. Lyapunov AM (1892) The general problem of the stability of motion (in Russian). Doctoral dissertation, Univ. Kharkov, English translation: (1) Stability of motion. Academic Press, New-York and London (1966), (2) The general problem of the stability of motion. Taylor & Francis, London (1992) 14. Haddad WM, Kapil V (1995) Absolute stability criteria for multiple slope-restricted monotonic nonlinearities. IEEE Trans Autom Control 40(2) 15. Zames G (1966) On the input-output stability of time-varying nonlinear feedback systems, Part one: Conditions derived using concepts of loop gain, conicity and positivity. IEEE Trans Automa Control 11(2):228–238 16. Slotine JJE, Li W (1991) Applied nonlinear control. Prentice-Hall 17. Taylor JH, Strobel KL (1984) Applications of a nonlinear controller design approach based on quasilinear system models. In: American control conference, San Diego, CA 18. Taylor JH, Strobel KL (1985) Nonlinear compensator synthesis via sinusoidal-input describing functions. In: American control conference, pp 1242–1247, Boston, MA 19. Taylor JH, O’ Donnell JR (1990) Synthesis of nonlinear controllers with rate feedback via SIDF methods. In: American control conference, San Diego, CA, pp 2217–2222 20. Taylor JH, Lu J (1995) Robust nonlinear control system synthesis method for electro-mechanical pointing systems with flexible modes. J Syst Eng (Special issue) 21. Utkin VI (1992) Sliding modes in control optimization. Springer, Berlin 22. Utkin VI, Lee H (2006) Chattering problem in sliding mode control systems. In: 2nd IFAC conference on analysis and design of hybrid systems, Alghero, Italy 23. Young KD, Utkin VI, Özgüner Ü (1999) A control engineer’s guide to sliding mode control. IEEE Trans Control Syst Technol 7:328–342 24. DeCarlo RA, Zak SH, Mathews GP (1988) Variable structure control of nonlinear multivariable systems: a tutorial. Proc IEEE 76(3) 25. Edwards C, Spurgeon SK (1998) Sliding mode control. Taylor and Francis, London 26. Shi S-L, Li J-X, Fang Y-M (2018) Extended-state-observer-based chattering free sliding mode control for nonlinear systems with mismatched disturbance. IEEE Access 6:22952– 22957

60

1 Introduction to Nonlinear Systems Modelling and Control

27. Kristic M, Kokotovic PV (1995) Estimation-based adaptive backstepping designs for linear systems. In: 34th IEEE conference on decision and control, vol 4, pp 3935–3940 28. Papageorgiou C, Glover K (2004) Robustness analysis of nonlinear dynamic inversion control laws with application to flight control. In: 43rd IEEE conference on decision and control, Bahamas, pp 3485–3490 29. Watson PK (1983) Kalman filtering as an alternative to ordinary least squares—some theoretical considerations and empirical results. Empir Econ 8:71–85 30. Julier SJ, Uhlmann JK (1997) New extension of the Kalman filter to nonlinear systems. In: Signal processing, sensor fusion, and target recognition VI, SPIE proceeding series, pp 182– 193 31. Billings SA, Leontaritis IJ (1982) Parameter estimation techniques for nonlinear systems. In: 6th IFAC symposium on identification and system parameter estimation, Washington DC, pp 427–432 32. Haber R, Keviczky L (1974) The identification of the discrete time Hammerstein model. Period Polytech Elect Eng 1:71–84 33. Haber R, Keviczky L (1974) Nonlinear structures for system identification. Period Polytech 18(4):393–404 34. Grimble MJ, Johnson MA, Brasca CL, Arcidiacono V (1999) Self-tuning control and industrial applications: the limited authority control concept. Int J Adapt Control Signal Process 13(6):521–536 35. Anbumani K, Sarma IG, Patnaik LM (1981) Self-tuning control of nonlinear systems characterised by Hammerstein models. IEEE Trans Autom Control 26:959–961 36. Agarawal M, Seborg DE (1987) Self-tuning controllers for nonlinear systems. Automatica 23(2):209–214 37. Zhang J, Lang S (1988) Indirect adaptive suboptimal control for linear dynamic systems having polynomial nonlinearities. IEEE Trans Autom Control 33:389–392 38. Korpela K, Koivo HN (1982) On self-tuning control of a class of nonlinear scalar processes. In: 6th IFAC symposium on identification and system parameter estimation, Washington, USA 39. Sontag E (1981) Nonlinear regulation: the piecewise linear approach. IEEE Trans Automat Control 26(2):346–358 40. Le Quang T (2013) Piecewise affine dynamical systems: well-posedness, controllability and stabilizability. Dutch Institute of Systems and Control report, University of Groningen 41. Pang Y, Grimble MJ (2010) NGMV control of delayed piecewise affine systems. IEEE Trans Autom Control 55(12):2817–2821 42. Rugh WJ, Shamma JS (2000) Research on gain scheduling. Automatica 36(10):1401–1425 43. Shamma JS, Athans M (1990) Analysis of gain scheduled control for nonlinear plants. IEEE Trans Autom Control 35(8):898–907 44. Shamma JS, Athans M (1991) Guaranteed properties of gain scheduled control for linear parameter-varying plants. Automatica 27(3):559–564 45. Shamma JS (1988) Analysis and design of gain scheduled control systems. PhD thesis, Lab. for Information and Decision Sciences, MIT, Cambridge, MA 46. Shamma JS, Athans M (1992) Gain scheduling: potential hazards and possible remedies. IEEE Control Syst Mag 101–107 47. Cai GB, Duan GR, Hu CH (2010) A velocity-based LPV modelling and control framework for nonlinear tracking. In: 29th Chinese control conference, Beijing, pp 286–291 48. Cai GB, Duan GR, Hu CH (2011) A velocity-based LPV modelling and control framework for an air breathing hypersonic vehicle. Int J Innov Comput Inf Control 7(5):2269–2281 49. Leith DJ, Leithead WE (1996) Can an appropriate gain-scheduling realisation enhance performance? In: UKACC international conference on control 96, vol 1, pp 563–568 50. Leith DJ, Leithead WE (1997) Analysis of gain-scheduled and nonlinear systems by velocity-based linearization families. In: 36th IEEE conference on decision and control, vol 1, pp 424–429

References

61

51. Leith DJ, Leithead WE (1998) Gain scheduled and nonlinear systems: dynamic analysis by velocity-based linearization families. Int J Control 70(2):289–317 52. Leith DJ, Leithead WE (1998) Gain-scheduled controller design: an analytic framework directly incorporating non-equilibrium plant dynamics. Int J Control 70(2):249–269 53. Leith DJ, Leithead WE (1998) Comments on gain scheduling dynamics linear controllers for a nonlinear plant. Automatica 34(8):1041–1043 54. Wu F (1995) Control of linear parameter varying systems. PhD thesis, Mechanical Engineering, University of California, Berkeley, CA, 94720 55. Wu F, Packard A, Balas V (1995) LPV control design for pitch-axis missile autopilots. In: 34th IEEE conference on decision and control, pp 188–193, New Orleans 56. Balas GJ (2002) Linear parameter-varying control and its application to aerospace systems. In: 23rd congress of international council of the aeronautical sciences, ICAS, Toronto, Canada, pp 541.1–541.9 57. Xie W, Eisaka T (2004) Design of LPV control systems based on Youla parameterization. IEE Proc Control Theory Appl 151(4):465–472 58. Wu F (2001) A generalized LPV system analysis and control synthesis framework. Int J Control 74(7):745–759 59. Xie W, Eisaka T (2008) Two-degree-of-freedom controller design for linear parameter-varying systems. Asian J Control 10(1):115–120 60. Khargonekar PP, Petersen IR, Zhou K (1990) Robust stabilization of uncertain linear systems: quadratic stabilizability and H∞ control theory. IEEE Trans Autom Control 35 (3):356–361 61. Feron E, Apkarian P, Gahinet P (1996) Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions. IEEE Trans Autom Control 41(7):1041–1046 62. Xie W (2007) Quadratic stabilization of LPV system by an LTI controller based on ILMI algorithm. In: Mathematical problems in engineering. Hindawi Publishing Corporation 63. Becker G, Packard A, Philbrick D, Blas G (1993) Control of parametrically dependent linear systems: a single quadratic Lyapunov approach. In: American control conference, pp 2795– 2799 64. Apkarian P, Gahinet P, Becker G (1995) Self-scheduled H∞ control of linear parameter-varying systems: a design example. Automatica 31(9):1251–1261 65. Pearson JD (1962) Approximation methods in optimal control I. Sub-optimal Control. J Electron Control 13:453–469 66. Wernli A, Cook G (1975) Suboptimal control for the nonlinear quadratic regulator problem. Automatica 11:75–84 67. Tóth R (2008) Modelling and identification of linear parameter-varying systems: an orthonormal basis function approach. Doctoral dissertation, Delft University of Technology 68. Tóth R, Heuberger PSC, Van den Hof PMJ (2007) LPV system identification with globally fixed orthonormal basis functions. In: 46th IEEE conference on decision and control, New Orleans 69. Toth R, Heuberger PSC, Van den Hof PMJ (2010) Discretisation of linear parameter-varying state-space representations. Control Theory Appl IET 4(10):2082–2096 70. Paijmans B, Symens W, Van Brussel H, Swevers J (2008) Identification of interpolating affine LPV models for mechatronic systems with one varying parameter. Eur J Control 1:16– 29 71. Leith DJ, Leithead WE (1999) Input-output linearization by velocity-based gain-scheduling. Int J Control 72(3):229–246 72. Leith DJ, Leithead WE (1999) Analytic framework for blended multiple model systems using linear local models. Int J Control 72(7):605–619 73. Leithead WE, Leith DJ, Murray-Smith R (2000) A Gaussian process prior/velocity-based framework for nonlinear modelling and control. In: Irish signals and systems conference, Dublin

62

1 Introduction to Nonlinear Systems Modelling and Control

74. Leith DJ, Leithead WE (2001) Gain-scheduled control of a skid-to-turn missile: relaxing slow variation requirements by velocity-based design. In: American control conference, vol 1, pp 500–505 75. Leith DJ, Leithead WE (2001) Velocity-based gain-scheduled lateral auto-pilot for an agile missile. Control Eng Pract Arlington 9(10):1079–1093 76. Heemels W, Schutter BD, Bemporad A (2001) Equivalence of hybrid dynamical models. Automatica 37(7):1085–1091 77. Balbis L, Ordys AW, Grimble MJ, Pang Y (2007) Tutorial introduction to the modelling and control of hybrid systems. Int J Model Identif Control 2(4):259–272 78. Antsaklis PJ, Koutsoukos XD (2001) Hybrid systems control. Research report, Department of Electrical Engineering, University of Notre Dame, IN 46556 79. Branicky MS (1998) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control 43(4):475–482 80. Engell S, Frehse G, Schnieder E (eds) (2002) Modelling, analysis and design of hybrid systems. Springer 81. Bemporad A, Borrelli F, Morari M (2002) On the optimal control law for linear discrete time hybrid systems. In: International workshop on hybrid systems: computation and control. Springer, pp 105–119 82. Rowe C, Maciejowski J (2003) Robust constrained receding horizon control of PWA systems with norm-bounded input uncertainty. In: American control conference, Denver, pp 3949–3954 83. Lee CK, Barton PI (2003) Global dynamic optimization of linear hybrid systems. In: Floudas CA, Pardalos PM (eds) Frontiers in global optimization. Kluwer, Santorini, pp 289– 312 84. Bemporad A, Morari M (1999) Control of systems integrating logic, dynamics, and constraints. Automatica 35(3):407–427 85. Bemporad A, Morari M (1999) Verification of hybrid systems via mathematical programming. In: Vaandrager FW, van Schuppen JH (eds) Hybrid systems: computation and control. HSCC 1999. Lecture notes in computer science, vol 1569. Springer, Berlin, Heidelberg 86. Bemporad A, Morari M (2001) Optimization-based hybrid control tools. In: American control conference, Arlington, vol 2, pp 1689–1703 87. Zhu F, Antsaklis PJ (2013) Optimal control of switched hybrid systems: a brief survey. Technical report: ISIS-2013–007, University of Notre Dame 88. Bemporad A, Ferrari-Trecate G, Mignone D, Morari M, Torrisi FD (1999) Model predictive control: ideas for the next generation. In: European control conference, Karlsruhe, Germany 89. Gaspar P, Szabo Z, Bokor J (2005) Gray-box continuous-time parameter identification for LPV models with vehicle dynamics applications. In: 13th mediterranean conference on control and automation, Limassol, Cyprus 90. Ljung L (2010) Approaches to identification of nonlinear systems. In: 29th Chinese control conference, Beijing, China 91. Dos Santos PL, Ramos JA, de Carvalho JLM (2008) Subspace identification of linear parameter varying systems with innovation-type noise models driven by general inputs and a measurable white noise time-varying parameter vector. Int J Syst Sci 39(9):897–911 92. Åström KJ, Hägglund T (1984) Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 20:645–651 93. Åström KJ, Hägglund T (1995) PID controllers: theory, design and tuning. Instrument Society of America, Research Triangle Park, NC, USA 94. Youssef A, Grimble M, Ordys A, Dutka A, Anderson D (2003) Robust nonlinear predictive flight control. In: European control conference, 1–4 Sept 2003, Cambridge, England 95. Herrnberger M, Lohmann B (2010) Nonlinear control design for an active suspension using velocity-based linearisations. In: 6th IFAC symposium on advances in automotive control, Munich, Germany, vol 43, No 7, pp 330–335

References

63

96. Gao R, O’Dwyer A, McLoone S, Coyle E (2004) State feedback integral control by velocity-based multiple model networks. In: American control conference, pp 2039–2044 97. Toivonen HT, Sandström KV, Nyström RH (2003) Internal model control of nonlinear systems described by velocity-based linearizations. J Process Control 13(3):215–224 98. Toivonen HT (2003) State-dependent parameter models of non-linear sampled-data systems: a velocity-based linearization approach. Int J Control 76(18):1823–1832 99. McLoone SC, Irwin GW (2003) On velocity-based local model networks for nonlinear identification. Asian J Control 5(2):309–315 100. Halas M, Huba M, Žakova K (2003) The exact velocity linearization method. In: IFAC conference on control systems design, vol 36, No 18, pp 259–264 101. McLoone SC, Irwin GW, McLoone SF (2002) On Gaussian weighting functions for velocity-based local model networks. In: Irish signals and systems conference, Cork, pp 135–140 102. McLoone SC, Irwin GW (2001) On nonlinear modelling using velocity-based multiple model networks. In: American control conference, vol 4 103. McLoone SC, Irwin GW, McLoone SF (2001) Blending local models within a velocity-based multiple model network. In: IFAC conference on emerging computing technologies, Hong Kong, pp 117–122 104. McLoone SC, Irwin GW, McLoone SF (2001) Constructing networks of continuous-time velocity-based local models. IEE Proc Control Theory Appl 148(5):397–405 105. Tsourdos A, Leith DJ, Leithead WE, White BA (2001) A velocity-based framework for the robust stability analysis of dynamic inversion flight controllers. In: American control conference, vol 5, pp 3341–3345 106. Herrnberger M, Lohmann B (2010) Nonlinear control design for an active suspension using velocity-based linearisations. In: 6th IFAC symposium advances in automotive control, Munich, Germany 107. McLoone SC, Irwin GW (2000) Nonlinear identification using velocity-based local model networks. In: Proceedings of the UKACC international control conference on control, Cambridge

Chapter 2

Review of Linear Optimal Control Laws

Abstract A review of linear optimal control laws is provided that are related to the nonlinear optimal control methods to be described in later chapters. The Minimum Variance, Generalized Minimum Variance, Linear Quadratic Gaussian and H∞ control design approaches are summarized and useful properties of the algorithms are highlighted. A useful introduction is provided to the solution procedures that are needed in a modified form for nonlinear systems. The solutions themselves are also required since they are the same as the limiting cases of the related nonlinear control problems when the plant is actually linear. The design of low-order or so-called restricted structure controllers is also considered where one controller can stabilize a set of linear models. This provides an empirical method for stabilizing a nonlinear system. The chapter ends with a number of ways of controlling uncertain and nonlinear processes.

2.1

Introduction

The linear optimal controllers introduced in this chapter are based on a polynomial systems modelling approach. In fact, the first few chapters are concerned with systems represented in polynomial form or in the multivariable case by polynomial system matrices. The linear optimal control problems considered are those that motivate the nonlinear systems problems in later chapters. Various aspects of linear stochastic control systems theory are reviewed, beginning with Åström’s well-known Minimum Variance (MV) controller [1]. The Generalized Minimum Variance (GMV) controller that was developed by Hastings-James [2], and by Clarke and Hastings-James [3] is then introduced. It was used by Clarke and Gawthrop [4] in a self-tuning control algorithm that became popular in the 1970s. The design of optimal controllers to minimize a quadratic criterion, for a linear stochastic system, is also considered. The Linear Quadratic Gaussian (LQG) controller is introduced in its less familiar polynomial system form. The so-called Generalized Linear Quadratic Gaussian (GLQG) controller

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_2

65

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2 Review of Linear Optimal Control Laws

that contains cross-product cost-function terms is also described and links between these different control problems are explored. The minimum variance and generalized minimum variance results will be touched upon throughout the book. Understanding the linear case will make it easier to understand the analysis and design results for the nonlinear feedback controller introduced in Chap. 4. This is referred to as a Nonlinear Generalized Minimum Variance (NGMV) control law and there are significant parallels with the linear case. The plant in Chap. 4 is assumed to include both linear and nonlinear subsystems. The linear subsystem is represented by a polynomial system model, which is similar to that in the section to follow. The use of low-order controllers for systems in polynomial form is also considered, under the heading of Restricted Structure (RS) control design and ways to treat nonlinearities and uncertainties are considered briefly.

2.2

Stochastic System Polynomial Models and Control

The feedback control system, represented by discrete-time polynomial models, and shown in Fig. 2.1, will first be considered [5, 6]. The stochastic disturbance dðtÞ and reference rðtÞ signals shown in this figure are modelled by linear dynamic systems driven by white noise. This is appropriate for stochastic systems; however, deterministic inputs can also be included in a similar mathematical framework. In fact, deterministic signals with random magnitudes can also be included in the minimum variance problem description if required. Discrete-time models of the process or plant will normally be used, where the system transfers are in terms of the indeterminate z1 . The symbol z1 will normally denote the unit-delay time-domain operator but in this chapter, it can also Pce Fcu

+ + Error weighting

Pc

Reference

Wr

+ r -

e

Fc

Control weighting

Controller

Plant

C0

W0

u

Wd

Disturbance model

d m

+ +

Fig. 2.1 Generic linear single degree-of-freedom stochastic feedback control system

y

2.2 Stochastic System Polynomial Models and Control

67

represent the inverse of the Z-transform complex number. It should be clear from the context of its use which interpretation is appropriate. By default, all the time-domain polynomial operators are assumed to be in terms of the backward shift operator z1 . This single-step unit-delay operator when applied to the control signal u(t) adds a delay of one sample instant (denoted as uðt  1Þ ¼ z1 uðtÞ). Generalized or inferred output: The generalized output to be controlled, is shown by the dotted lines in Fig. 2.1, and has the form: /ðtÞ ¼ Pc ðz1 ÞeðtÞ þ F c ðz1 ÞuðtÞ This signal includes dynamically weighted error and control signals, where the dynamic weightings Pc ðz1 Þ and F c ðz1 Þ are transfer-function terms operating on the error and control signals. Since it is not a physical plant output, but is constructed using the weighting functions, it is often referred to as an inferred output. In the minimum variance control problem, /ðtÞ is equal to the error signal eðtÞ (when the transfers Pc ¼ 1 and Fc ¼ 0Þ. The plant or process model W 0 and the control weighting F c can both be nonlinear models and in that case the convention used is to denote these nonlinear system models by script fonts W 0 or F c . This convention is adopted throughout the text for nonlinear operators. Matrix fractions: In this section, polynomial matrix fractions (equivalent to transfer-functions in the SISO discrete-time case) are used to represent the linear plant subsystem models. Thus, let the following models represent the linear discrete-time polynomial description of the system:

W0 ðz1 Þ Wd ðz1 Þ Wr ðz1 Þ ¼ Aðz1 Þ1 Bðz1 Þ Cd ðz1 Þ

Er ðz1 Þ



ð2:1Þ

A common denominator matrix A can be assumed without loss of generality. For notational simplicity, the arguments of the polynomial matrix terms are often omitted, so that the above equation will be written as ½W0

2.2.1

Wd

Wr  ¼ A1 ½B

Cd

Er 

ð2:2Þ

Minimum Variance Control Fundamentals

The optimal controllers considered in this chapter will mostly be those minimizing quadratic cost-functions such as Minimum Variance (MV), Generalized Minimum Variance (GMV), Linear Quadratic Gaussian (LQG) and Generalized Linear Quadratic Gaussian (GLQG) (see Refs. [7–9]). The Minimum variance (MV) controller was obtained from the solution of a linear stochastic optimal control problem using a polynomial systems approach by Professor Åström [1, 10]. The aim was to develop a simple and practical industrial control strategy that would

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2 Review of Linear Optimal Control Laws

minimize the variance of a process variable such as an output or tracking error signal. In paper manufacture, a suspension of fibres and water is deposited onto a moving web and the water is then removed using gravity, a vacuum and heat. The paper is then coiled and transported. The heat that is used to dry the paper is applied by steam-heated rolls in the drying section of a paper machine. In this problem, the basis-weight (mass of fibres per unit area) and the moisture content must be controlled. The moisture content is important for improving both the energy efficiency and the quality of the paper produced. The minimum variance control strategy due to Åström [10] was applied successfully in a paper machine to minimize the variance of the moisture content of the paper. Following this early work, the MV control technique was generalized for use in different applications. The MV control solution is directly related to GMV control and to a class of Model Predictive Control (MPC) techniques, known as Generalized Predictive Control (GPC) algorithms described in Chap. 7. In the simple case considered in the next section, the block diagram of Fig. 2.1 can represent a Single-Input Single-Output (SISO) linear unity-feedback control system. However, the polynomial equations for the system are similar for the Multi-Input Multi-Output (MIMO) problem.

2.2.2

Minimum-Variance Control Law Derivation

The theory of linear minimum variance control is now summarized since the optimization approach constitutes a recurring theme in the following, albeit in a more generalized form. It is also important to describe the approach for historical reasons, and to put later developments into perspective [11]. Consider a single-input single-output linear time-invariant system and the feedback control shown in the block diagram in Fig. 2.2. The output represents a stationary stochastic process described by a Box–Jenkins model, as follows:

Disturbance model

Reference

r=0

+

-

e

Controller

Plant

C0

W0

u

Wd d

+ m +

Fig. 2.2 Simplified one degree-of-freedom SISO feedback control system

y

2.2 Stochastic System Polynomial Models and Control

yðtÞ ¼ W0 ðz1 ÞuðtÞ þ Wd ðz1 ÞnðtÞ ¼

69

Bðz1 Þ Cðz1 Þ uðtÞ þ nðtÞ Aðz1 Þ Dðz1 Þ

ð2:3Þ

The plant is controlled by a linear feedback controller denoted C0 ðz1 Þ: Using negative feedback convention and the unit-delay operator: uðtÞ ¼ C0 ðz1 ÞyðtÞ

ð2:4Þ

The signal yðtÞ represents the output signal for a linear time-invariant system, or the variations of the output signal around a given steady-state operating point for a linearized nonlinear system. The control signal is denoted uðtÞ; and nðtÞ denotes a zero-mean white noise signal of unity-variance. The setpoint or reference signal rðtÞ is initially assumed to be zero, and the more general case is considered later in Sect. 2.3.1. Polynomial system model: The terms Aðz1 Þ; Bðz1 Þ; Cðz1 Þ and Dðz1 Þ are polynomials in the backward shift operator z1 and for simplicity, the roots of Cðz1 Þ and Bðz1 Þ are assumed to lie strictly within the unit-circle in the z–domain. The polynomial Bðz1 Þ will include any explicit k-steps transport delay, so that Bðz1 Þ ¼ zk Bk ðz1 Þ: The linear system equations to be solved in this polynomial-based system are referred to as Diophantine equations, and are described in Kucera [5] and Grimble and Kucera [12]. The solution of the minimum variance control problem will be found by first computing the optimal least squares predictor for the output. The output at time t þ k can be written as yðt þ kÞ ¼

Bk ðz1 Þ Cðz1 Þ uðtÞ þ nðt þ kÞ 1 Aðz Þ Dðz1 Þ

ð2:5Þ

The following polynomial identity (Diophantine equation) can be introduced that splits the stochastic disturbance signal Dðz1 Þ1 Cðz1 ÞnðtÞ into two disjoint timeframes: Cðz1 Þ ¼ Dðz1 ÞFðz1 Þ þ zk Gðz1 Þ

ð2:6Þ

The polynomials Fðz1 Þ and Gðz1 Þ in this equation are of the form: Fðz1 Þ ¼ 1 þ

k1 X i¼1

fi zi

and Gðz1 Þ ¼

nG X

gi zi

ð2:7Þ

i¼0

with nG ¼ maxðnC  k; nD  1Þ: Substituting into the process Eq. (2.5) now yields:

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2 Review of Linear Optimal Control Laws

Bk C Bk G uðtÞ þ nðt þ kÞ ¼ uðtÞ þ nðtÞ þ Fnðt þ kÞ D A A D Bk G B k ¼ uðtÞ þ yðtÞ  zk uðtÞ þ Fnðt þ kÞ C A A   Bk FD G uðtÞ þ yðtÞ þ Fnðt þ kÞ ¼ AC C

yðt þ kÞ ¼

ð2:8Þ

The prediction error ~yðt þ kjtÞ ¼ yðt þ kÞ  ^yðt þ kjtÞ now follows as ~yðt þ kjtÞ ¼

  Bk FD G uðtÞ þ yðtÞ  ^yðt þ kjtÞ þ Fnðt þ kÞ AC C

ð2:9Þ

Output predictor cost-index: The variance of the prediction error for this scalar case: n o n o Jy ðtÞ ¼ E ðyðt þ kÞ  ^yðt þ kjtÞÞ2 ¼ E ~yðt þ kjtÞ2

ð2:10Þ

where Ef:g denotes the expectation operator. To derive the optimal predictor note that the control signal at time t must be a function of the information available at time t, which includes the past values of the control signal, and the past and current output. The disturbance noise driven term Fðz1 Þnðt þ kÞ in (2.9) involves only future values of white noise. It follows that the term in the round brackets in (2.9) is uncorrelated with the final term, which is independent of the choice of the predictor. To minimize the variance of the estimation error (2.10) the round bracketed term must, therefore, be set to zero. This then results in the k-steps-ahead prediction equation: ^yðt þ kjtÞ ¼

Bk FD G uðtÞ þ yðtÞ AC C

ð2:11Þ

This linear prediction equation is actually useful for predicting the output of a system in addition to its value in solving the MV control problem. Minimum-variance control problem: The minimum variance controller can be obtained by minimizing the variance of the output at time t þ k: That is, n o JðtÞ ¼ E yðt þ kÞ2

ð2:12Þ

Note that the expectation of the product of the first two terms on the right-hand side of Eq. (2.8) with the third term is null, since the system is driven by zero-mean white noise and they refer to signals in mutually exclusive time frames. The variance of the output r2y ¼ E y2 ðt þ kÞ may, therefore, be obtained, noting yðt þ kÞ ¼ ^yðt þ kjtÞ þ ~yðt þ kjtÞ; as

2.2 Stochastic System Polynomial Models and Control

71

      JðtÞ ¼ E y2 ðt þ kÞ ¼ E ^y2 ðt þ kjtÞ þ E ~y2 ðt þ kjtÞ ( 2 ) n o Bk FD G uðtÞ þ yðtÞ ¼E þ E ðFnðt þ kÞÞ2 AC C

ð2:13Þ

The expression for the variance can clearly be split into the following two terms: J ¼ J0 þ Jmin

ð2:14Þ

n o   Jmin ¼ E ~y2 ðt þ kjtÞ ¼ E ðFnðt þ kÞÞ2

ð2:15Þ

where the minimum cost:

The component of the cost-function that depends upon control action:   J0 ¼ E ^y2 ðt þ kjtÞ ¼ E

(

Bk FD G uðtÞ þ yðtÞ AC C

2 ) ð2:16Þ

Optimal output: The expression (2.14) is interesting since the minimum achievable cost Jmin is independent of the choice of control action, but the term J0 is dependent on the choice of control law. To achieve the minimum variance of the output variable, the cost term J0 must be set to zero. The optimal control follows by setting (2.16) to zero: uðtÞ ¼ 

GA yðtÞ Bk FD

ð2:17Þ

This is, of course, the control action to set the predicted value of the output ksteps-ahead to zero. Substituting for the resulting minimum variance control (2.17) into the expression (2.8) yields the optimal output: yðtÞ ¼ Fðz1 ÞnðtÞ Optimal controller: The optimal control may, therefore, be expressed, using (2.17), in terms of the total transfer-function from the disturbance input, as uðtÞ ¼ 

GA nðtÞ Bk D

ð2:18Þ

The minimum variance control law may be viewed as the stochastic equivalent of deadbeat control in the sense that it places all of the closed-loop poles at the origin. This observation may be confirmed by recalling the optimal closed-loop output ymin ðtÞ ¼ Fðz1 ÞnðtÞ and in terms of a transfer-function in z it has all the poles at the origin. This good control is achieved by cancelling the process

72

2 Review of Linear Optimal Control Laws

dynamics, which explains why the minimum variance control is aggressive and sensitive to process-model mismatch. Stability: Note from (2.18) that any attempt to control a non-minimum-phase plant with this version of the minimum variance controller leads to unbounded control signal variations and to instability. Note that if the plant is open loop unstable the closed-loop can only be stabilized if the disturbance model includes the same unstable mode. For example, an unstable plant driven by white noise at the control input would ensure the unstable mode could be cancelled before implementing (2.17). Before proceeding, several issues relating to the models used in the MV control solution need clarification. Integral disturbance: Under the assumption of a stationary disturbance, both the output and control signals are stationary stochastic signals (recall the Dðz1 Þ polynomial is assumed to be strictly Schur). More commonly, the process is subject to a non-stationary, drifting, disturbance. In that case, the Dðz1 Þ polynomial includes a factor D ¼ ð1  z1 Þ (introducing an integrator), and from Eq. (2.17) the control signal becomes non-stationary. If the disturbance has large low-frequency content, the controller will include an integrator to ensure the output is a moving average process yðtÞ ¼ Fðz1 ÞnðtÞ: Unstable plant model: If the plant is open-loop unstable, the plant denominator polynomial A in the control law (2.18) may attempt to cancel the unstable plant modes and thereby create unstable hidden modes leading to instability. However, if the plant model has a common denominator ðAðz1 Þ ¼ Dðz1 ÞÞ with the disturbance model any unstable modes in Aðz1 Þ and Dðz1 Þ may be cancelled when computing the minimal realization of the control law (2.18) and thereby avoid the stability problem. This suggests that any unstable modes in the plant must be included in the disturbance model. To ensure this condition is always satisfied it is sufficient to assume the plant input also includes a white noise disturbance component. In state-space system modelling terms, recall that for a steady-state Kalman filter to be stable it must be observable in terms of the measured output matrix and controllable in terms of the process noise input matrix. This latter condition would not be satisfied if unstable plant modes were not affected by the noise inputs and an LQG controller could not be shown to be stabilizing. A general rule of thumb is that it should be assumed that all plant and disturbance model states are driven by noise. The covariance matrix elements can be chosen to be small when there is little actual noise present. Such a choice often benefits robustness.

2.2.3

MV Control Law for Non-minimum-phase Systems

A second form of minimum variance control law was introduced by Åström, shortly after the original MV solution was developed. This controller could stabilize non-minimum-phase systems. The solution involved factorization of the Bk

2.2 Stochastic System Polynomial Models and Control

73

polynomial into strictly minimum phase Bkþ and non-minimum-phase B k terms [7].  e This enables the cost-index to be redefined using a polynomial B k defined to have zero locations that are the mirror image to those of B k , so that the polynomial is  e minimum phase. The Z-transfer function B =B is then an all-pass function and the k k modified cost-function can be written as 8 !2 9 < B = e k JðtÞ ¼ E yðt þ kÞ : B ; k

ð2:19Þ

The expression for the cost-function includes the inverse of the non-minimum-phase zeros term, which in the time domain can be considered a non-causal weighting. Since this only affects the cost-function weighting definition, which is not part of the physical process modelled, it does not cause a problem in realization. From the plant model (2.5), we obtain: e  B þ ðz1 Þ e e  Cðz1 Þ B B B k uðtÞ þ nðt þ kÞ yðt þ kÞ ¼ B Dðz1 Þ Aðz1 Þ B

ð2:20Þ

Introduce a Diophantine equation: e  Cðz1 Þ Dðz1 ÞFðz1 Þ þ zk B Gðz1 Þ ¼ B

ð2:21Þ

e  C=ðB DÞ ¼ F=B þ zk G=D: Thence, in (2.20) we obtain: so that B e B þ e B F G B k uðtÞ þ  nðt þ kÞ þ nðtÞ yðt þ kÞ ¼  B D B A   e B þ B F G Bk k ¼  nðt þ kÞ þ yðtÞ  zk uðtÞ uðtÞ þ B C A A F 1  ¼  nðt þ kÞ þ GAyðtÞ þ DFBkþ uðtÞ B CA By similar arguments to those above, the optimal control is that which sets the final bracketed term to zero. The resulting optimal minimum variance control law no longer contains the offending cancelling zeros: uðtÞ ¼ 

GA yðtÞ Bkþ FD

ð2:22Þ

This controller will, therefore, stabilize a non-minimum-phase system. Tracking: The above derivation of the MV control law was presented for the case of an output regulator with a zero setpoint signal. A non-zero setpoint with one degree-of-freedom control law involves a similar expression to (2.22), but applied to the error signal rðtÞ  yðtÞ; rather than to the output. A trajectory following

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2 Review of Linear Optimal Control Laws

(servo control) requires a modification of the cost-function to include the error variance: n o n o J ¼ E ðeðt þ kÞÞ2 ¼ E ðrðt þ kÞ  yðt þ kÞÞ2 A two degrees-of-freedom controller structure involves a controller with separate output and reference signal inputs (rather than the error signal). It can provide additional design and tuning freedom in tracking problems. In the remainder of the chapter, the focus is mainly on one degree-of-freedom controllers but two degree-of-freedom solutions are used in later chapters for predictive controls.

2.2.4

Stochastic Versus Deterministic Signal Models

The traditional steady-state or infinite-horizon Linear Quadratic Gaussian (LQG) stochastic and deterministic optimal control problems have a closer relationship than might be expected. Consider the solution of a stochastic output feedback discrete-time optimal control problem for a reference model involving a white noise signal feeding a time-lag term. The solution of the deterministic Linear Quadratic (LQ) optimal control problem, where the reference is modelled by a pulse into the same time constant term, and for the same plant and cost-function weightings, will result in the same optimal controller. This is easy to show using polynomial or transfer-functions, since the models for the Z-transfer function of the pulse and the white noise signal covariance are the same. The stochastic minimum–variance controller and the deterministic mean square error controller have the same form if the underlying input signal models are equivalent. The distribution of the white noise generating sources does not affect the controller design in this type of least squares optimization problem. The idea is illustrated in Fig. 2.3 for the case of white noise filtered by an integrator. The distribution of the white noise signal e1 ðtÞ is Gaussian and the signal y1 ðtÞ is a first-order non-stationary process. On the other hand, if the underlying probability density function for the noise is as depicted in the upper right-hand plot of Fig. 2.3, the signal e2 ðtÞ will mostly be zero except for occasional random pulses, and consequently the signal y2 ðtÞ will represent step changes occurring at random times. It turns out that the MV controller for the two cases has the same transfer-function. Other signals such as sinusoids, or ramps with random slopes and switching times, can be related in a similar way to their stochastic equivalents. It follows that when using LQ methods for industrial control either stochastic or deterministic signal models may be used. The frequency content of the signal shaping filters is what is important for control design [13].

2.3 Generalized Minimum Variance Control

75

Fig. 2.3 Stochastic and deterministic signal equivalence

2.3

Generalized Minimum Variance Control

Some difficulties inherent in the original minimum variance control law formulation have already been described. In particular, the original MV control technique cannot be used for processes that are non-minimum phase, since it introduces unstable uncontrollable/unobservable modes. This problem was addressed by Åström [1]. As explained above, the solution was to factorize the polynomial Bðz1 Þ into stable and unstable parts and to use a modified cost-index. A related controller was derived by Grimble [14]. In the following, an alternative and a rather different solution is considered, called the Generalized Minimum Variance (GMV) control law. It is the direct linear counterpart of the nonlinear controller to be introduced in Chap. 4, and it is very relevant to what follows in the remainder of the text. The Åström minimum variance controller generates the control u(t) to minimize the MV criterion (2.12), representing the variance of the output k-steps into the future. However, it was noted in Ordys et al. [15], that the resulting control uo ðtÞ ¼ uðtÞ is also optimal with respect to the more general performance criterion that averages the output variances over the past and the future. That is:

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2 Review of Linear Optimal Control Laws

(

n o 2 u ðtÞ ¼ arg min E yðt þ kÞ ¼ arg min E lim o

T X 1 yðt þ kÞ2 T!1 T  k þ 1 j¼k

)!

The arg min notation used here denotes the points, where the function reaches its minimum value. This result suggests that Linear Quadratic Gaussian controllers lead to the same solution when considering the limiting case of this MV problem. The generalized minimum variance algorithm was developed by Hastings-James [2] and by Clarke and Hastings-James [3]. They added a scalar control signal weighting function into the minimum variance performance index to enable both the variance of the error and the control signal to be costed. That is, n o Jtg ¼ E yðt þ kÞ2 þ kuðtÞ2

ð2:23Þ

The new MV controller with control costing was termed a Generalized Minimum Variance (GMV) controller. Superficially, the GMV controller appeared to solve the same problem as for the Linear Quadratic Gaussian (LQG) problem. However, MacGregor and Tidwell [16] noted that the problems solved were rather different because the GMV solution applied to the case where the criterion involved a conditional expectation. The cost-function they minimized to obtain the GMV controller was of the conditional cost form: n o Jt ¼ E yðt þ kÞ2 þ kuðtÞ2 t

ð2:24Þ

where k  0 is a scalar and the expectation is conditioned on all past measurements and controls. It was valuable since the authors also noted that although the controller did not minimize the equivalent LQG criterion (2.23), involving the unconditional expectation, it still appeared to lead to reasonable controllers that optimized the related criterion (2.24). Unfortunately, the GMV controller does not have the valuable feature of the Åström MV controller. That is,

n o uðtÞg ¼ arg min E yðt þ kÞ2 þ kuðtÞ2 t ( )! T

 X 1 q 2 2 6¼ uðtÞ ¼ arg min E lim yðt þ jÞ þ kuðt þ j  kÞ T!1 T  k þ 1 j¼k ð2:25Þ In the limiting case of zero control costing (when a minimum variance controller results), the two cost-function problems (2.24) and (2.25) lead to the same solution if the system is minimum phase. That is, when the control cost-weighting tends to zero in the GMV control problem the same controller is obtained as found from an

2.3 Generalized Minimum Variance Control

77

LQG problem, and this is guaranteed to be stabilizing (using either polynomial or state-space solutions). In steady-state operation, the criterion (2.23) will have a larger value when using the GMV control ug ðtÞ rather than the time-averaged optimal control uq ðtÞ [16]. The optimal control to minimize a cost-function in steady-state conditions with the conditional expectation will not, therefore, in general minimize the same cost-function using an unconditional expectation. The GMV control law was termed “short-sighted” or instantaneous optimal control [16], because it does not take into account the effect that the present control action will have on future outputs at times greater than k. The GMV cost-minimization problem (2.24), therefore, yields a static or single-stage optimization repeated at each time instant t, whilst the infinite horizon in the performance index leads to a time-averaged dynamic optimization Ordys et al. [15]. The solution obtained via static optimization, at any time instant is optimal with respect to the state of the system at this time, but this does not necessarily imply global optimality. The above discussion is important because the model predictive controllers and particularly the Generalized Predictive Controller (GPC) also involves a static optimization. The minimization procedure separates the dynamic problem into individual time steps so that a solution can be computed at each sample instant. The limiting form of the GPC controller, as the prediction horizon shrinks to one step, is a GMV control solution. This is useful to know since it indicates the type of behaviour expected on non-minimum phase systems with low control costing and it is a reminder there are implications for the minimum cost achievable. This is not to imply that GMV and GPC controllers are unsuitable, since both have the virtue of simplicity and a good record in applications.

2.3.1

Scalar GMV Control Law Derivation

The GMV controller for linear systems is derived below using a polynomial approach. It is useful to understand since it minimizes a cost-function that relates to many of the nonlinear optimal control problems in the following chapters. Consider the feedback system shown in Fig. 2.1 with the plant model W0 ¼ A1 B: A reference model Wr ðz1 Þ driven by white noise may be introduced for generating the reference signal, and without loss of generality, a common denominator may be used for plant, disturbance and reference signals (Eq. (2.2)). This enables a so-called innovations signal model spectral-factor Yf ðz1 Þ ¼ Df ðz1 Þ=Aðz1 Þ to be introduced (see [17, 18]). A signal may now be defined as f ðtÞ ¼ rðtÞ  dðtÞ ¼ Yf ðz1 ÞeðtÞ

ð2:26Þ

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2 Review of Linear Optimal Control Laws

The error can now be expressed in the innovations signal form: B eðtÞ ¼ rðtÞ  yðtÞ ¼ Yf eðtÞ  uðtÞ A

ð2:27Þ

This represents the combined effect of all the exogenous inputs acting on the system, where eðtÞ denotes a unity-variance white noise source. The spectral-factor can be obtained from the reference and disturbance signal models by noting that the spectrum of the sum of two uncorrelated stochastic signals equals the sum of the respective spectra: Uff ¼ Urr þ Udd Using the subsystem transfer-functions: Yf Yf ¼ Wr Wr þ Wd Wd

ð2:28Þ

The polynomial Df can be obtained as the solution of the following spectral-factorization problem: Df Df ¼ Er Er þ Cd Cd

ð2:29Þ

The GMV controller may now be derived to minimize the variance of a generalized output signal of the form: /ðtÞ ¼ Pc ðz1 ÞeðtÞ þ Fc ðz1 ÞuðtÞ

ð2:30Þ

where the dynamic cost-function weightings are given as 1 1 Pc ðz1 Þ ¼ P1 cd ðz ÞPcn ðz Þ and

1 1 Fc ðz1 Þ ¼ Fcd ðz ÞFcn ðz1 Þ

ð2:31Þ

These weightings can be represented by stable transfer-functions. GMV cost-index: In the light of the comments at the start of Sect. 2.3 the GMV cost-function will be defined in terms of the conditional expectation as   JGMV ¼ E /2 ðtÞjt

ð2:32Þ

The form of this criterion is discussed further in the next section. Generalized output: The control action uðtÞ affects the output after a k-steps delay, and hence the dynamic control signal weighting Fc is defined to include the delay term as Fc ¼ zk Fck ¼ zk Fcnk =Fcd . The generalized or inferred output, ksteps into the future, may now be represented as /ðt þ kÞ ¼ Pc ðz1 Þeðt þ kÞ þ Fck ðz1 ÞuðtÞ

ð2:33Þ

2.3 Generalized Minimum Variance Control

79

This is not, of course, an actual output but represents the signal to be minimized “inferred” from other signals. From (2.27), the predicted error eðt þ kÞ ¼ A1 Df eðt þ kÞ  A1 Bk uðtÞ and the last equation may be written as   Df Bk eðt þ kÞ  uðtÞ þ Fck uðtÞ A A   Pc Df Pc Bk eðt þ kÞ þ Fck  ¼ uðtÞ A A

/ðt þ kÞ ¼ Pc

ð2:34Þ

By analogy with the MV control law derivation, the first noise driven term Pc Yf eðt þ kÞ can be split into unpredictable and predictable components using a Diophantine equation of the form: Pcn Df ¼ APcd F þ zk G

ð2:35Þ

The polynomial Fðz1 Þ has a minimum degree of k  1: Equation (2.34) can now be rewritten as

   G Fcnk Pcn Bk /ðt þ kÞ ¼ Feðt þ kÞ þ eðtÞ þ  uðtÞ APcd Fcd Pcd A

ð2:36Þ

The first term in this expression is independent of control action and because of the order of the F polynomial; the first and remaining terms in (2.34) are uncorrelated. The expression for the optimal controller, therefore, follows by setting the term in the square brackets to zero. The GMV optimal control follows. GMV optimal control: ugmv ðtÞ ¼

GFcd eðtÞ Pcn Fcd Bk  Fcnk Pcd A

ð2:37Þ

The generalized output under optimal control follows from the first term in (2.36): /gmv ðtÞ ¼ Fðz1 ÞeðtÞ

ð2:38Þ

After noting eðtÞ ¼ Yf1 ðeðtÞ þ Wðz1 ÞuðtÞÞ and some straightforward algebraic manipulations (see remarks below), the expression for the controller can be obtained from (2.37) as ugmv ðtÞ ¼

GFcd eðtÞ HPcd

where the denominator polynomial Hðz1 Þ is defined as

ð2:39Þ

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2 Review of Linear Optimal Control Laws

H ¼ Bk Fcd F  Fcnk Df

ð2:40Þ

Solution: The steps in deriving the expression (2.39) for the generalized minimum variance controller, given (2.37) are straightforward, and similar to the above MV problem solution. The steps in the solution need only be summarized as follows: (a) f ðtÞ ¼ rðtÞ  dðtÞ ¼ Yf ðz1 ÞeðtÞ (b) eðtÞ ¼ rðtÞ  yðtÞ ¼ rðtÞ  dðtÞ  Wðz1 ÞuðtÞ ¼ Yf ðz1 ÞeðtÞ  Wðz1 ÞuðtÞ (c) eðtÞ ¼ Yf1 ðeðtÞ þ Wðz1 ÞuðtÞÞ (d) (e) (f) (g) (h) (i)

uðtÞ ¼ ðPcn Fcd Bk  Fcnk Pcd AÞ1 GFcd Yf1 ðeðtÞ þ Wðz1 ÞuðtÞÞ ðPcn Fcd Bk  Fcnk Pcd AÞuðtÞ ¼ ðGFcd Yf1 eðtÞ þ GFcd Yf1 Wðz1 ÞuðtÞÞ ðPcn Fcd Bk Df  Fcnk Pcd ADf  GFcd BÞuðtÞ ¼ GFcd AeðtÞ From (2.35) Pcn Df ¼ APcd F þ zk G ðFcd Bk Pcd F  Fcnk Pcd Df ÞuðtÞ ¼ GFcd eðtÞ uðtÞ ¼ ðFcd Bk F  Fcnk Df Þ1 P1 cd GFcd eðtÞ

Stability: As already mentioned, the MV control law (2.17) cannot be used to control a non-minimum phase plant. This would result in an unbounded control action and instability. A similar effect occurs in GMV control when the control signal costing is small. It also applies under a large control signal costing if the plant is open-loop unstable (for a confirmation see the denominator of Eq. (2.37)). However, it is useful to note that (2.36) may be written as /ðtÞ ¼ FeðtÞ þ ½ðG=ðAPcd ÞÞeðt  kÞ þ ðFc  Pc W0 ÞuðtÞ A necessary condition for stability is that the weighted plant model ðFc  Pc W0 Þ be minimum phase. This is sometimes referred to as the generalized plant in these problems. This stability result may be confirmed by noting the denominator of (2.37) depends upon this term. This simple solution highlights a result that applies to a number of the control laws discussed in the following. That is because of the square of sum nature of the cost-function if the generalized plant is minimum phase and internal pole-zero cancellations are avoided in synthesizing the controller then the system will be stable. This class of problems is similar to the first minimum variance control problem of Åström [1] for minimum-phase systems. The minimum variance achieved is, therefore, the same for a cost-function involving either the conditional or the unconditional expectation operators. Cost-function weighting relationships: The dynamic cost-function weightings for the GMV optimal control cost-function can be chosen by broadly similar rules as for LQG problems but there are some special relationships, noted below: • By setting Pc ¼ 1 and Fck ¼ 0 in (2.39) and (2.40), the original MV controller (2.17) is obtained. However, the dynamic weighting Pc may be used to penalize the error in a specified frequency range. Choosing Pc ðz1 Þ ¼ ð1  z1 Þ1 Pc0

2.3 Generalized Minimum Variance Control

81

introduces integral action into the controller to provide high gain at low frequencies. • The weighting Fck ðz1 Þ may be used to adjust the speed of response of the controller. For example, increasing this weighting can reduce the gain to address the problem of actuator saturation. • If optimality is not of importance, another method of selecting the weightings involves pole placement. That is, the roots of the characteristic equation of the system can be specified to determine the closed-loop poles. From (2.37), the characteristic equation can be confirmed as Pcn Fcd Bk  Fcnk Pcd A ¼ 0

ð2:41Þ

The weightings Pc and Fck can be selected so that (2.41) equals a chosen polynomial, so that the closed-loop poles are located at desired positions in the complex Z-plane.

2.3.2

Remarks on the GMV Criterion

The GMV cost-function defined by Eqs. (2.30) and (2.32) is not the same as that used by Clarke and Gawthrop in their influential work [4]. The cost-function they employed, for the case of a scalar control weighting k [ 0; and a conditional expectation, had the sum of squares form:   b J GMV ¼ E y2 ðt þ kÞ þ ku2 ðtÞjt

ð2:42Þ

This is obviously different to the expected value of the square of a sum cost-terms: n o JGMV ¼ E ðyðtÞ þ kuðtÞÞ2 jt

ð2:43Þ

Both (2.42) and (2.43) behave like single-stage type criteria although the former is sometimes confused with an LQG type of infinite-horizon cost-function. The differences between these different formulations, based on the probabilistic definition of the expectation operator, were explained in the paper by MacGregor and Tidwell [16]. A way to improve the stability properties of both Minimum variance and Generalized Minimum Variance control laws is to use the so-called factorized versions of these control laws although this can introduce additional complexity. These can stabilize non-minimum phase systems even with small or zero control cost-weightings [19].

82

2.4

2 Review of Linear Optimal Control Laws

LQG, GLQG and GMV Control Laws

Linear Quadratic Gaussian (LQG) control has a natural robustness if all the states can be measured, but this is rather impractical. A continuous-time state feedback LQ or LQG control solution has guaranteed gain and phase margins and a certain tolerance to nonlinearities. It can even be improved by the use of dynamic cost-function weightings. Unfortunately, in practice, a state estimator is required, which may affect these guaranteed stability margins. In the polynomial approach, the state estimator is not evident in the solution but it is embedded in the equations used to compute the polynomial version of a feedback controller. An LQG approach is particularly appropriate for systems where disturbances are the dominant problem. It provides a very flexible design capability for a range of industrial applications, when dynamic weighting functions that might be time variable are included. Quadratic cost-function: The LQG cost-function for an optimal discrete-time output-regulator control problem, with output y(t) and control u(t), can be written in the time domain as ( J ¼ lim

T!1

( )) T X 1 T T E y ðtÞQc yðtÞ þ u ðtÞRc uðtÞ 2T t¼T

where Qc and Rc are constant symmetric weighting matrices (positive semi-definite and positive definite, respectively). A useful extension is to use a dynamically weighted cost-index that has the form: ( J ¼ lim

T!1

( )) T X 1 T T E ðHq yÞ ðtÞðHq yÞðtÞ þ ðHr uÞ ðtÞðHr uÞðtÞ 2T t¼T

where Hq(z−1) and Hr(z−1) are transfer-functions or more precisely dynamic weighting operators in the unit-delay operator z−1. The output weighting Hq is a rational operator in z−1, which is often chosen as an integrator to obtain integral action in the controller. The control weighting Hr is often chosen as a lead–lag term. The usual linear optimal control philosophies, including LQG control, which use an unconditional cost-index [20–22] can be found in terms of the solution of an optimal control problem, which is very similar to dynamically weighted Generalized Minimum Variance (GMV) control. That is, the so-called Generalized Linear Quadratic Gaussian (GLQG) problem formulation, discussed below, can be related to the conditional cost, generalized minimum variance problem. This also applies to one-block H∞ problem solutions [23, 24].

2.4 LQG, GLQG and GMV Control Laws

2.4.1

83

Multivariable Linear System Description

Consider the r-output and m-input discrete-time multivariable system shown in Fig. 2.4. This system is assumed to be finite dimensional, linear and time-invariant and to be represented in the following polynomial matrix form (matrix fraction description involves two polynomial matrices describing a matrix transfer-function), as follows: Plant: W ¼ A1 B Input disturbance: Wd ¼ A1 Cd Output disturbance/measurement noise: Wn ¼ A1 Cn Reference signal: Wr ¼ A1 E Note that the input and output disturbance models enter at the same point in Fig. 2.4 but the output of the system to be controlled y(t) includes the disturbance from Wd and not the output disturbance and measurement noise from Wn . This explains the input and output disturbance terminology. Observe that the output disturbance affects the optimal solution in the same manner as for the measurement noise represented by the model Wn . Output disturbance and measurement noise

Input disturbance

Wd

1

A Cd

ω

d Reference

Wr

A 1E

r + –

Plant

Controller

e0

C0

C0d1C0n

u

W

A 1B

m + +

Wn y + +

A 1Cn n

Fig. 2.4 Canonical linear feedback system including disturbances and measurement noise

z

84

2 Review of Linear Optimal Control Laws

The system equations, which define the main time-responses, become: Plant output equation:  yðtÞ ¼ Aðz1 Þ1 Bðz1 ÞuðtÞ þ Cd ðz1 ÞnðtÞ

ð2:44Þ

Controller input signal: e0 ðtÞ ¼ rðtÞ  yðtÞ  nðtÞ Reference and measurement noise: rðtÞ ¼ Aðz1 Þ1 Eðz1 ÞfðtÞ and

nðtÞ ¼ Aðz1 Þ1 Cn ðz1 ÞxðtÞ

There is no loss in generality in using a common denominator Aðz1 Þ polynomial. The various subsystems are assumed free of unstable hidden modes and are represented by left-coprime matrix fraction decomposition: ½W

Wd

Wr

Wn  ¼ A1 ½ B

Cd

E

Cn 

ð2:45Þ

An alternative right-coprime representation of the plant becomes Wðz1 Þ ¼ Aðz1 Þ1 Bðz1 Þ ¼ B10 ðz1 ÞA10 ðz1 Þ1 Here a right divisor U of B10 and A10 can be defined so that B10 = B1U and A10 = A1U. Stochastic Signals: The white driving noise sources n(t), x(t) and f(t) are assumed zero-mean and statistically independent (they are uncorrelated). Without loss of generality, the covariance’s of these signals may be assumed equal to the identity. The linear reference, disturbance and noise system models can again be represented in an innovations signal form, where e(t) denotes white noise with identity covariance matrix and zero-mean. The combined stochastic driving noise can be represented by the signal f ðtÞ ¼ rðtÞ  dðtÞ  nðtÞ ¼ Yf ðz1 ÞeðtÞ. The generalized spectral-factor Yf ðz1 Þ is again obtained by factoring the total exogenous signal spectrum. This can be related to the return-difference matrix for a Kalman filter loop and it is sometimes referred to as the filter spectral-factor (Chap. 13, Sect. 13.3.1).

2.4.2

Sensitivity Matrices and Signal Spectra

Assume the controller has the following discrete-time multivariable polynomial system form:

2.4 LQG, GLQG and GMV Control Laws

85

Controller expression: 1 1 C0 ¼ Cod Con ¼ C1n C1d

The sensitivity and complementary sensitivity matrices enter the equations naturally and relate to the sensitivity and robustness properties of the system. There are two sensitivity functions for a multivariable system defined, respectively, as Sr ¼ ðIr þ WC0 Þ1 and Sm ¼ ðIm þ C0 WÞ1 and corresponding complementary sensitivity matrices Tr ¼ Ir Sr ¼ WC0 Sr ¼ WSm C0 and Tm ¼ Im Sm ¼ C0 WSm ¼ C0 Sr W. The control sensitivity matrix (sometimes called input sensitivity matrix) is defined as M ¼ C0 Sr ¼ Sm C0 and these matrices determine the spectra of the signals in the system, defined as follows: Error signal: e ¼ r  y ¼ Sr ðr  dÞ þ WMn Control signal: u ¼ Co Sr ðr  d  nÞ ¼ Mðr  d  nÞ The control spectrum Uuu ¼ MUff M  and the total exogenous signal spectrum: Uff ðz1 Þ ¼ Urr ðz1 Þ þ Udd ðz1 Þ þ Unn ðz1 Þ This may be spectrally factored into the form Yf Yf ¼ Uff , where Yf is the filter spectral-factor. This relates to the Kalman filter section of the LQG control law as described in Chap. 13 (Sect. 13.3.1).

2.4.3

Solution of the Scalar LQG Cross-Product Problem

For simplicity, consider a scalar system, without measurement noise, and the minimization of an LQG cost-index with cross-product terms. The cost-function can be expressed with dynamic cost-weightings in the complex integral form: J¼

1 2pj

I jzj¼1



Qc0 Uee þ Rc0 Uuu þ Gc0 Uue þ Gc0 Ueu

 dz z

ð2:46Þ

The dynamic cost-function weightings in the criterion have the polynomial forms (using a common denominator):

86

2 Review of Linear Optimal Control Laws

Qc0 ¼

Qn ; Aw Aw

Rc0 ¼

Rn ; Aw Aw

Gc0 ¼

Gn Aw Aw

ð2:47Þ

This general cross-product cost-function is needed for some of the special cases considered below. The solution of the problem including measurement noise was described in Grimble [25]. Based on this solution the LQG controller for the system, shown in Fig. 2.4 (without measurement noise), may be computed from the following equations. Spectral-factors: Compute the Schur spectral-factors Dc and Df using Dc Dc ¼ B Qn B þ A Rn A  B Gn A  A Gn B

ð2:48Þ

Df Df ¼ Cd Cd þ Cr Cr

ð2:49Þ

Feedback equation: Compute the polynomials (G, H, F), with F of smallest degree: Dc Gzg þ FAAw ¼ ðB Qn  A Gn ÞDf zg

ð2:50Þ

Dc Hzg  FBAw ¼ ðA Rn  B Gn ÞDf zg

ð2:51Þ

The scalar g is chosen to be the smallest positive integer to ensure the above equations are polynomial in the indeterminate z1 . That is, g  deg(Dc) and the One DOF LQG controller is given as Feedback controller: 1 1 C0 ðz1 Þ ¼ C0d ðz ÞC0n ðz1 Þ ¼ Hðz1 Þ1 Gðz1 Þ

ð2:52Þ

The characteristic polynomial and the related implied equation that determine closed-loop stability, are given, respectively, as qc ¼ AC0d þ BC0n and AH þ BG ¼ Dc Df . Sensitivity, Control Sensitivity, Complementary Sensitivity: S¼

AH ; Dc Df

M¼

AG ; Dc Df

T¼

BG Dc Df

The proof of these results was given in Grimble [9]. They will be used to demonstrate a link below that is between optimal control problems that are apparently very different, but have many features in common.

2.4 LQG, GLQG and GMV Control Laws

2.4.4

87

Scalar Linear Optimal Control Examples

In this section, six simple SISO control design methods are applied to four different plant models covering minimum-phase, non-minimum phase and stable and open loop unstable plant models. The four open-loop transfer-function plant models and their step responses are shown in Table 2.1. The following discrete-time controllers have been designed for each of these linear plant models: 1. 2. 3. 4. 5. 6.

Minimum Variance Control Factorized Minimum Variance Generalized Minimum Variance Factorized Generalized Minimum Variance Generalized LQG Control LQG Control.

Table 2.1 Four plant models and their open loop step responses Minimum phase

Non-minimum phase

Stable

þ 2Þ G1 ðsÞ ¼ ðs þðs3Þðs þ 4Þ

G2 ðsÞ ¼ ðs þðs2Þ 3Þðs þ 4Þ

Unstable

ðs þ 2Þ G3 ðsÞ ¼ ðs1Þðs þ 4Þ

ðs2Þ G4 ðsÞ ¼ ðs1Þðs þ 4Þ

88

2 Review of Linear Optimal Control Laws

The basic principles of the control laws 2 and 4 are described in [19]. These control laws can stabilize non-minimum-phase linear systems. The control designs first involved discretizing the continuous-time plant models, using a zero-order-hold and a sampling period of Ts = 0.1 s. The model-based controller synthesis problems were then solved. Wherever possible, the same dynamic weighting functions were used (except in minimum variance cases). Simulations were conducted to test disturbance rejection and the tracking performance of the different controllers. Comments: Unstable poles and zeros make it more difficult to achieve good control-loop performance. Non-minimum phase zeros cannot be cancelled and have a significant effect on the closed-loop responses. Compare, for example, the time-responses, using LQG control, for the four plants (with the same weightings), shown in Fig. 2.5. The initial negative transient of the two non-minimum phase cases is evident. The LQG solution guarantees closed-loop stability (if not arbitrary performance), as illustrated in Fig. 2.5.

Fig. 2.5 Disturbance rejection and reference tracking for four different plants, under LQG control with the same cost-function weightings

2.4 LQG, GLQG and GMV Control Laws

89

As explained above the basic MV controller cannot be used to stabilize a non-minimum-phase system. However, a factorized version is available that is stabilizing [14]. Alternatively, a control weighting can be introduced into the MV cost-index as in the GMV or LQG cost-functions. The closed-loop responses for the non-minimum phase plant G2 are shown in Fig. 2.6, comparing the performance of the Factorized MV, GMV and LQG controllers. The MV controller cannot be used to stabilize this system, and is not, therefore, shown. It is difficult to choose the constant GMV controller cost-function weightings, to stabilize an unstable non-minimum phase plant. However, the factorized version can be used to stabilize the unstable non-minimum phase plant G4 as shown in Fig. 2.7. The factorized solution involves more off-line computations but does allow a stabilizing solution to be found. The stability issues are more difficult in the multivariable control case and controllers like LQG can be more complex and numerically sensitive. A simpler controller such as GMV, with carefully selected weighting functions, may sometimes provide similar performance in a more practical option.

Fig. 2.6 Control of non-minimum phase plant (G2) with three different controllers

90

2 Review of Linear Optimal Control Laws

Fig. 2.7 Factorized GMV control of an unstable non-minimum phase plant (G4)

2.5

Multivariable LQG, GLQG, MV and H∞ Cost Problems

For simplicity, the solution of a scalar LQG problem was presented above but the system was actually introduced in the multivariable form in Sect. 2.4.1. Attention will now turn to the multivariable control problem using the same system description. The relationship between the usual LQG optimal control problem [20] involving a sum of squares and the minimum variance of a signal type of problem that involves the square of a sum is established next. This latter class of problem involves a particular form of cost-index that has cross-product terms. It is referred to here as a Generalized Linear Quadratic Gaussian (GLQG) problem. In this class of problems, the unconditional expectation is used to compute the variance of a combination of signals. For simplicity, the measurement noise is again assumed null, since it is simple to ensure the controller attenuates measurement noise through the suitable definition of the control signal dynamic cost-function weighting. LQG Cost-Function: The usual LQG cost-function, without cross-product terms, but with dynamic weightings, can be expressed in the complex integral form: I

 dz 1 J¼ ð2:53Þ tracefHq Uee Hq g þ tracefHr Uuu Hr g 2pj z jzj¼1

2.5 Multivariable LQG, GLQG, MV and H∞ Cost Problems

91

where the dynamic error Hq(z−1) and control Hr(z−1) signal weightings have the right matrix fraction form: rr 1 mm 1 ðz Þ and Hr ¼ Br A1 ðz Þ Hq ¼ Bq A1 q 2R r 2 R

These weightings satisfy Hq Hq  0 and Hr Hr [ 0 on |z| = 1. The “denominator” weighting polynomials Aq and Ar are assumed strictly Schur, and the right-coprime 1 representation of the system A1 q A BAr is defined as 1 1 rm 1 ðz Þ A1 q A BAr ¼ B1 A1 2 R

ð2:54Þ

Sum of squares LQG cost-function: The so-called sum of squares LQG costfunction may be expressed in the form: I  dz 1 ð2:55Þ J ¼ Ef/T0 ðtÞ/0 ðtÞg ¼ tracefU/0 /0 ðz1 Þg 2pj z jzj¼1


 where /T0 ðtÞ ¼ ðHq eðtÞÞT ðHr uðtÞÞT and U/0 /0 denotes the power spectrum of the signal /0 ðtÞ: The polynomial solution of sum of squares LQG problems of this type are well known, via Bongiorno [26], Youla et al. [27], Kucera [17], Shaked [18, 28] and Grimble [29].

2.5.1

Multivariable GLQG Cost-Function Problem

The Generalized Linear Quadratic Gaussian (GLQG) cost-function [30], may be defined as follows: 1 J ¼ Ef/ ðtÞ/ðtÞg ¼ 2pj

I

T



tracefU// ðz1 Þg

jzj¼1

dz z

ð2:56Þ

where the signal being penalized: /ðtÞ ¼ Pc ðz1 ÞeðtÞ þ Fc ðz1 ÞuðtÞ

ð2:57Þ

is defined in terms of the dynamic cost-function weightings Pc ðz1 Þ 2 Rmr ðz1 Þ and Fc ðz1 Þ 2 Rmm ðz1 Þ: In the scalar case, this represents a square of sums cost-function. The error Pc ðz1 Þ and control Fc ðz1 Þ cost-function weightings may be assumed to have the following polynomial matrix forms:

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2 Review of Linear Optimal Control Laws

Pc ¼ P1 Pcn

and Fc ¼ P1 Fcn

where Pc is assumed to be strictly Schur and P(0) is full-rank to ensure causality. These weightings can be used to define the polynomial matrix: L ¼ Pcn B  Fcn A

ð2:58Þ

and the following right-coprime polynomial matrix factorizations be introduced: 1 L0 P1 0 ¼ P L;

1 Fn P1 3 ¼ P Fcn ;

1 1 D2 A1 2 ¼ P2 A Df ;

1 B3 D1 3 ¼ Df BP3

This GLQG type of criterion appears similar to that often used in the rest of the text. Observe that this “square of sums” form is not the same as the cost-function (2.55), which has a “sum of squares” form. These terms really only relate to the scalar cases, but are used as a convenient shorthand in the multivariable problem. One implication for this choice of criterion (2.56) is that it has the same number of weighted error and weighted control terms. That is, the row dimensions of Pc and Fc must be the same. This restriction is not needed for general LQG cost-functions but in the GLQG cases considered later, it leads to a simplification of the algorithms.

2.5.2

Solution of the Multivariable GLQG Design Problem

A time-domain solution for the GLQG feedback controller will now be obtained. When a system is represented in polynomial form, the solution is normally found in the s or z domains [26]. However, the time-domain optimization approach is used here for the nonlinear system problems considered in later chapters. The innovations signal model was introduced in Sect. 2.4.1 where the spectral-factor Yf ðz1 Þ ¼ A1 ðz1 ÞDf ðz1 Þ (the polynomial matrices are in terms of the unit-delay operator in this section). The controller input signal (denoted e0(t)) may be written in terms of the innovations signal model as e0 ðtÞ ¼ rðtÞ  zðtÞ ¼ A1 Df eðtÞ  A1 BuðtÞ

ð2:59Þ

Multiplying this signal by the error weighting: P1 Pcn e0 ðtÞ ¼ P1 Pcn A1 Df eðtÞ  P1 Pcn A1 BuðtÞ: 1 1 Write P1 Pcn ¼ Pn P1 and P1 then P1 Pcn e0 ðtÞ ¼ 2 2 A Df ¼ D2 A2 , 1 1 1 Pn D2 A2 eðtÞ  Pn P2 A BuðtÞ; and let the right-coprime polynomial matrix 1 B10 A1 10 ¼ A B so that a polynomial matrix L may be defined as

2.5 Multivariable LQG, GLQG, MV and H∞ Cost Problems

L ¼ Pcn B10  Fcn A10

and

P1 L ¼ L0 P1 0

93

ð2:60Þ

First Diophantine equation: Introduce the first Diophantine equation, in terms of the unknowns G and F: D0 zg G þ FA2 ¼ L0 Pn D2 zg

ð2:61Þ

where the scalar g is chosen to be the smallest positive integer that ensures the equations are polynomials in z−1. Also, assume the polynomial matrix D0 is Schur and satisfies D0 D0 ¼ L0 L0

ð2:62Þ

This Diophantine equation can be written as  1 g 1 1 L1 0 D0 GA2 þ z L0 F ¼ Pn D2 A2

ð2:63Þ

and the weighted error signal may be expressed, using (2.63), as  1 g 1 1 1 P1 Pcn e0 ðtÞ ¼ L1 0 D0 GA2 eðtÞ þ z L0 FeðtÞ  Pn P2 A BuðtÞ 1  1  1 P Pcn e0 ðtÞ þ Pn P1 ¼ L1 0 D0 GðPn D2 Þ 2 A BuðtÞ 1 g 1  Pn P1 2 A BuðtÞ þ z L0 FeðtÞ  1 1 1  1 1 ¼ L1 0 D0 GD2 P2 e0 ðtÞ þ L0 ðD0 GA2 Df B 1 g 1  L0 Pn D2 A1 2 Df BÞuðtÞ þ z L0 FeðtÞ

Substituting from Eq. (2.61):  1 1 1 g 1 1 g 1 P1 Pcn e0 ðtÞ ¼ L1 0 D0 GD2 P2 e0 ðtÞ þ L0 ðFA2 z ÞA2 Df BuðtÞ þ z L0 FeðtÞ

 1 1 g g ¼ L1 D0 GD1 0 2 P2 e0 ðtÞ  FDf Bz uðtÞ þ z FeðtÞ

ð2:64Þ Signal minimized: The signal, /(t), to be minimized in the GLQG problem, was defined in (2.57). If the measurement noise n(t) is null then substituting from (2.64): /ðtÞ ¼ Pc ðz1 ÞeðtÞ þ Fc ðz1 ÞuðtÞ   1 1 D0 GD2 P2 e0 ðtÞ þ zg FeðtÞ ¼ L1 0

 1 1 g  L FD Bz þ Fn P1 uðtÞ 3 0 f

ð2:65Þ

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2 Review of Linear Optimal Control Laws

Second Diophantine equation: The second Diophantine equation, in terms of the unknown polynomials H and F, may also be introduced as D0 zg H  FB3 ¼ L0 Fn D3 zg

ð2:66Þ

This equation may be written in the form: g 1  Fn D3  L1 0 FB3 z ¼ L0 D0 H

ð2:67Þ

1 1 g 1  1 1 and Fn P1 3  L0 FDf Bz ¼ L0 D0 HD3 P3 . The signal /(t) can now be written, from (2.65) and (2.66), as  1 1 1 1 1 g /ðtÞ ¼ L1 0 D0 ½GD2 P2 e0 ðtÞ  HD3 P3 uðtÞ þ L0 z FeðtÞ

ð2:68Þ

 1 Now D0 D0 ¼ L0 L0 ) L1 0 D0 ¼ L0 D0 and hence,


1 1  1 1 1 g /ðtÞ ¼ L0 D1 0 GD2 P2 e0 ðtÞ  HD3 P3 uðtÞ þ ðL0 z FeðtÞÞ

ð2:69Þ

This signal can be written as the sum of terms /ðtÞ ¼ /a ðtÞ þ /b ðtÞ; where 1 g /a ðtÞ ¼ L0 D1 0 ½: and /b ðtÞ ¼ ðL0 z FeðtÞÞ: The first two terms (in the square brackets) involve only stable time-domain operators. The final term involves a strictly unstable system. The operators can therefore be expressed as convergent sequences in terms of the delay operator z1 and the advance operator z, respectively. Thus, the final term in (2.69) involves future values of e(t) and the first two terms depend only on past values of e(t). Thence, since these (white noise) terms are uncorrelated, the variance of /(t) can be written as Ef/T ðtÞ/ðtÞg ¼ Ef/Ta ðtÞ/a ðtÞg þ Ef/Tb ðtÞ/b ðtÞg

ð2:70Þ

The second term in (2.70), representing the variance of the signal /b ðtÞ; does not depend upon the control signal. Thence, the optimal control to minimize the ~ ðtÞ ¼ variance of the signal /(t) must minimize the variance of /a ðtÞ: Writing / a 1 1  1 ðL0 D1 0 Þ /a ðtÞ and noting D0 L0 L0 D0 ¼ I; the cost term to be minimized ~ T ðtÞ/ ~ ðtÞg. The optimal control must, therefore, set / ~ ðtÞ ¼ Ef/Ta ðtÞ/a ðtÞg ¼ Ef/ a a a 0; where ~ ðtÞ ¼ GD1 P1 e0 ðtÞ  HD1 P1 uðtÞ / a 2 2 3 3

ð2:71Þ

Optimal control: The optimal control follows as 1 1 1 1 uðtÞ ¼ ðHD1 3 P3 Þ GD2 P2 e0 ðtÞ

ð2:72Þ

2.5 Multivariable LQG, GLQG, MV and H∞ Cost Problems

95

The cost at the minimum follows as 9 8 > > I <  T   T   1  1 dz= 1 Ju ¼ E / ðtÞ/ðtÞ ¼ E /b ðtÞ/b ðtÞ ¼ trace L0 FF L0 >2pj z> ; : jzj¼1

ð2:73Þ Implied equation: The implied equation, which determines closed-loop stability, may be derived from Eqs. (2.66) and (2.67) since D0 zg G þ FA2 ¼ L0 Pn D2 zg and D0 zg H  FB3 ¼ L0 Fn D3 zg . Right multiplying the first of these equations by 1 1 1 D1 2 P2 A BP3 , and the second equation by D3 and adding gives 1 1 1  1 1 D0 ðHD1 3 þ GD2 P2 A BP3 Þ ¼ L0 ðPn P2 A BP3  Fn Þ

Multiplying this equation on the right by P1 3 A10 gives: 1 1 1 1  1 1 Do ðHD1 3 P3 A10 þ GD2 P2 A BA10 Þ ¼ Lo ðPn P2 B10  Fn P3 A10 Þ

¼ L0 P1 ðPcn B10  Fcn A10 Þ ¼ L0 P1 L ¼ D0 D0 P1 0

ð2:74Þ

After division, obtain the so-called implied equation, which determines the system stability: 1 1 1 1 HD1 3 P3 A10 þ GD2 P2 B10 ¼ D0 P0

ð2:75Þ

The expression (2.72) for the control signal (2.75) is clearly the characteristic polynomial matrix equation multiplied by the denominator of the weighting matrix. The system is guaranteed to be asymptotically stable because of the strictly Schur definitions of D0, D2 and D3.

2.5.3

LQG with Cross-Product Criterion and Links

The solution of the LQG problem with cross-product terms in the criterion is the more general problem and can provide a solution to the GLQG problem. Observe that the GLQG criterion: 1 J ¼ Ef/ ðtÞ/ðtÞg ¼ 2pj

I

T

jzj¼1



tracefU// ðz1 Þg

dz z

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2 Review of Linear Optimal Control Laws

where the signal /(t) = Pc(z−1)e(t) + Fc(z−1)u(t), may be expanded so it has the form of an LQG cost-index with cross-product terms: I  dz 1 ð2:76Þ tracefPc Uee Pc þ Fc Uuu Fc þ Pc Ueu Fu þ Fc Uue Pc Þg J¼ 2pj z jzj¼1

This has the same form as the GLQG criterion if the weightings in (2.76) satisfy Qc0 ¼ Pc Pc , Rc0 ¼ Fc Fc , Gc0 ¼ Pc Fc . To demonstrate the links it is sufficient to return to the scalar case that is more transparent. The results of the LQG problem with cross-product terms were obtained for the scalar case in Sect. 2.4.3. The results for the GLQG problem may now be obtained. If the GLQG weightings are written as Pc ¼ P1 Pcn and Fc ¼  Fcn and Gn ¼ Pcn Fcn . The P1 Fcn identify Aw ¼ P; Qn ¼ Pcn Pcn , Rn ¼ Fcn Diophantine equation s for the LQG problem may, therefore, be written, from (2.66) and (2.67), in the form:  Dc Gzg þ FAAw ¼ ðB Pcn Pcn  A Fcn Pcn ÞDf zg  Dc Hzg  FBAw ¼ ðA Fcn Fcn  B Pcn Fcn ÞDf zg 1 1 1 1 1 1 1 Note P1 Pcn ¼ Pn P1 2 , P2 A Df ¼ D2 A2 , B3 D3 ¼ Df BP3 , Fn P3 ¼ P Fcn and hence in the scalar problem Pn ¼ Pcn , Fn ¼ Fcn , B10 ¼ B, A10 ¼ A, P2 ¼ P3 ¼ P ¼ Aw , D2 ¼ D3 ¼ Df so that L ¼ Pcn B10  Fcn A10 ¼ Pn B  Fn A: The Diophantine equations to be solved above become

Dc Gzg þ FAP ¼ L Pn Df zg Dc Hzg  FBP ¼ L Fn Df zg The solution for H and G defines the LQG optimal controller for this special weighting and scalar problem: 1 1 C0 ðz1 Þ ¼ C0d ðz ÞC0n ðz1 Þ ¼ Hðz1 Þ1 Gðz1 Þ

ð2:77Þ

These equations provide the same solution as the GLQG solution as Eqs. (2.61), (2.66) and (2.72). To confirm this result, note after simplification the Diophantine equations: D0 zg G þ FAP ¼ L Pn Df zg

ð2:78Þ

D0 zg H  FBP ¼ L Fn Df zg

ð2:79Þ

2.5 Multivariable LQG, GLQG, MV and H∞ Cost Problems

97

After cancellation of terms, the GLQG optimal control law becomes 1 1 1 1 1 uðtÞ ¼ ðHD1 3 P3 Þ GD2 P2 e0 ðtÞ ¼ H Ge0 ðtÞ

Minimum Phase Special Case There is a special case where the GLQG problem provides the same solution as a GMV control law, even though the cost indices and expectation operators are different. This is of interest because of the importance of the nonlinear version of the GMV controller that arises frequently in later chapters. If L ¼ Pcn B  Fcn A happens to be minimum phase, then from (2.74) D0 D0 ¼ L0 L0 ) Do ¼ Lk zk and the Diophantine equations become D0 zg G þ FAP ¼ Lk Pn Df zg þ k D0 zg H  FBP ¼ Lk Fn Df zg þ k The polynomial F may be written as F ¼ D0 zg þ k F0 and the first Diophantine equation then becomes D0 zg G þ D0 zg þ k F0 AP ¼ Lk Pn Df zg þ k . After cancellation of terms, the first equation becomes zk G þ F0 AP ¼ Pn Df . The second Diophantine equation D0 zg H  D0 zg þ k F0 BP ¼ Lk Fn Df zg þ k can also be simplified as H  zk F0 BP ¼ Fn Df zk or H ¼ F0 Bk P  Fnk Df . GLQG Controller: The Diophantine equations that define the GLQG controller may be collected as follows: zk G þ F0 AP ¼ Pn Df

ð2:80Þ

H ¼ F0 Bk P  Fnk Df

ð2:81Þ

Solving for G and H the GLQG controller is given as C0 ¼ H 1 G

ð2:82Þ

The weightings in the GLQG problem above use a common denominator Pcd ¼ Fcd and the term L ¼ Pcn B  Fcn A, apart from the sign of the term, is what was referred to as the numerator of the generalized plant ðFc  Pc W0 Þ in Sect. 2.3.1. Comparing the GLQG controller expression in this case with the GMV control law (2.39) the observation made that these results are the same. In fact, the note on stability that follows (2.39) explains that this type of problem is related to the Åström minimum variance controller for minimum-phase systems. That is why, in this special case, the conditional expectation cost-function and the unconditional cost used in the GLQG problems lead to the same stabilizing control and minimum cost variance. This link is interesting because it provides some confidence in the class of GMV control laws that are considered later in the more difficult nonlinear control problems.

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2 Review of Linear Optimal Control Laws

2.5.4

Multivariable Minimum Variance Control Problem

A further relationship between the GLQG and LQG solutions may be determined, which is also exploited for nonlinear systems control in Chap. 6 (the NLQG problem). Most of the techniques for solving the nonlinear problems rely upon the solution of GMV or the related GLQG control problems, but it is also valuable to link the solutions in special cases to the LQG control problem solution, since the behaviour is so well known. The results that follow demonstrate a rather contrived and fictitious GLQG problem that can be posed to derive a related LQG controller. First introduce the right polynomial matrix spectral-factor Dc that satisfies Dc Dc ¼ B1 Bq Bq B1 þ A1 Br Br A1

ð2:83Þ

The signal /(t) which is minimized in this special GLQG control problem may now be defined as

   1 /ðtÞ ¼ D1 B1 Bq Bq A1 c q eðtÞ  A1 Br Br Ar uðtÞ

ð2:84Þ

where the tracking error e = r – (d + Wu) = c – Wu and the signal representing the difference between the reference and disturbance c(t) = r(t) – d(t). Substituting in (2.84):

   1   1 1   1 /ðtÞ ¼ D1 B B B A cðtÞ  ðB B B A A B þ A B B A ÞuðtÞ q q r c 1 q q 1 q q 1 r r 1 1 Recalling A1 obtain q A BAr ¼ B1 A1  1 1 Dc Dc A1 Ar . The signal /(t) can therefore be written as

  1 ðB1 Bq Bq B1 A1 1 þ A1 Br Br ÞAr ¼

  1 1 1 /ðtÞ ¼ D1 c B1 Bq Bq Aq cðtÞ  Dc A1 Ar uðtÞ

ð2:85Þ

This equation can be considered as the output expression for a strictly minimum-phase system with input u(t) and disturbance input c(t). Since it is assumed that a finite minimum to the cost-function exists the term D1 in (2.85) c may be interpreted as a non-causal stable weighting function. The power spectrum of the signal /(t) can now be expressed as

  1 1  1 1 1 1 1  U// ¼ D1 Dc c B1 Bq Bq Aq Ucc Aq Bq Bq B1 Dc þ Dc A1 Ar Uuu Ar A   1 1 1  1 1 1  1 D1 c B1 Bq Bq Aq Ucu Ar A1 Dc  Dc A1 Ar Uuc Aq Bq Bq B1 Dc



Thence, the integrand of the GLQG cost-index, which determines the variance of the signal /(t), becomes

2.5 Multivariable LQG, GLQG, MV and H∞ Cost Problems

99

1 1 1    1 1 1 Iu ¼ tracefUuu g ¼ tracefDc A1 1 Ar Uuu Ar A1 Dc  B1 Bq Bq Aq Ucu Ar A1 1 1  1   1 1  1  A1 1 Ar Uuc Aq Bq Bq B1 g þ tracefDc B1 Bq Bq Aq Ucc Aq Bq Bq B1 Dc g

ð2:86Þ Conventional LQG Cost-Function: It will now be interesting to look at the equivalent expression for the LQG problem case. Expand the spectral terms in the integrand of the conventional LQG cost-function (2.53) to obtain Uee ¼ Ucc þ WUuu W   WUuc  Ucu W  and 1  1 1 1 1   1 1  Bq A1 q Uee Aq Bq ¼ Bq B1 A1 Ar Uuu Ar A1 B1 Bq þ Bq Aq Ucc Aq Bq 1 1  1 1 1    Bq B1 A1 1 Ar Uuc Aq Bq  Bq Aq Ucu Ar A1 B1 Bq

Thence, the integrand of the LQG cost-index may be expanded as 1  1 1  I‘ ¼ tracefBq A1 q Uee Aq Bq g þ tracefBr Ar Uuu Ar Br g 1 1 1     ¼ tracefA1 1 Ar Uuu Ar A1 ðB1 Bq Bq B1 þ A1 Br Br A1 Þ 1 1    1 1 1  A1 1 Ar Uuc Aq Bq Bq B1  B1 Bq Bq Aq Ucu Ar A1 g 1  þ tracefBq A1 q Ucc Aq Bq g 1 1 1  ¼ tracefDc A1 1 Ar Uuu Ar A1 Dc 1 1 1 1   B1 Bq Bq A1  A1 q Ucu Ar A1 1 Ar Uuc Aq Bq Bq B1 g 1  þ tracefBq A1 q Ucc Aq Bq g

ð2:87Þ

Observe that the special minimum variance cost-function integrand (2.86) has the same terms as the LQG cost-function integrand (2.87), with the exception of the final terms in each of the expressions, namely:   1 1  1 I/0 ¼ tracefD1 c B1 Bq Bq Aq Ucc Aq Bq Bq B1 Dc g

ð2:88Þ

1  I‘0 ¼ tracefBq A1 q Ucc Aq Bq g

ð2:89Þ

and

These two cost terms are both independent of the choice of control signal and hence they do not change the optimal control solution. The two problems, therefore, have the same optimal controller, although the minimum values of the cost-functions differ due to the difference between these control independent terms.

100

2.5.5

2 Review of Linear Optimal Control Laws

Equivalence of the Multivariable GLQG and LQG Solutions

The GLQG cost-function is not changed if the signal /(t) is multiplied by an  g 1  g all-pass function D1 c1 Dc z , where g = deg(Dc). Let the signal /1 ¼ Dc1 Dc z /,   where Dc1 is strictly Schur and satisfies Dc1 Dc1 ¼ Dc Dc . In this case 1  tracefU/1 /1 g ¼ tracefDc D1 c1 Dc1 Dc U// g ¼ tracefU// g. It follows that minimizing the variance of the signal /1(t) in a rather special GLQG problem results in an optimal controller, which is the same as the solution of the usual LQG design problem. Also, note that the signal u1(t) can be written as   1 g  1 1 g u1 ðtÞ ¼ ðD1 c1 B1 Bq Bq Aq z ÞcðtÞ  ðDc1 A1 Ar z ÞuðtÞ

ð2:90Þ

The first term, recalling c(t) = r(t) – d(t), represents the filtered reference and disturbance signal components and the second term represents a modified plant model having the same denominator polynomial matrix as the actual plant model W. These results may be summarized in the following lemma. Lemma 2.1: Equivalence Between the LQG and Minimum Variance Problems The optimal controller that minimizes the LQG criterion (2.53) is the same as the optimal control to minimize the following Generalized Linear Quadratic Gaussian (GLQG) cost function: J ¼ Ef/T1 ðtÞ/1 ðtÞg ¼

1 2pj

I jzj¼1



tracefU/1 /1 ðz1 Þg

dz z

ð2:91Þ

where the signal /1 ðtÞ is given by (2.90). The difference in cost values, between the minimum values of the two cost-functions, can be computed as 9 8 > = < 1 I

 dz>   1 trace tracefD1 B B U B B D  U g ‘0 c1 1 q ‘0 q 1 c > z> ; :2pj jzj¼1

1  where the spectrum U‘0 ¼ Bq A1 q Ucc Aq Bq .

■

Proof By collecting the above results.

■

2.5.6

H∞ Control Design and Relationship to GLQG

The early promise of H∞ design to provide a truly robust control has not perhaps been totally realized, but it does provide a mechanism to deal with certain classes of

2.5 Multivariable LQG, GLQG, MV and H∞ Cost Problems

101

uncertainty, and it usually provides remarkably good transient responses and decoupled behaviour. The use of H∞ design does not automatically lead to more robust control solutions, but it does provide a direct method of minimizing sensitivities that is useful. These affect robustness to parameter variations and disturbance attenuation properties. There are close links between H∞ design and dynamically weighted LQG solutions regarding the system equations involving the sensitivity functions. There are, of course, multi-criterion minimization methods, which now provide an easy way to a trade-off between both types of solution. Kwakernaak [31] first considered the H∞ mixed sensitivity control design problem using a polynomial systems approach. Interestingly, this can be solved via an LQG or H2 minimization problem with special cost-function weighting terms. That is, the H∞ optimal controller can be derived from the solution of special dynamically weighted LQG control problem [23, 24]. A LQG controller can itself be derived by solving a special Minimum-Variance control problem as discussed above. These results suggest that a special minimum variance or GLQG design problem may be solved [30], to provide the solution of the one block or so-called Generalized H∞(GH∞) control problem. That is, the GH∞ design problem can be solved by embedding the problem in a related GLQG design problem. This leads to a technique to try to influence robustness properties in the nonlinear control designs in Chap. 6.

2.6

Restricted Structure Control Systems

Although high-performance applications with tight design specifications usually require advanced model-based control schemes, a simple PID controller is often adequate to provide stable, robust operation, particularly in process control systems. It is estimated that more than 90% of all flow, pressure, temperature and level controllers are of a PID structure in the process industries (usually just a PI form). The practical importance of the PID control methods has resulted in innumerable papers, tuning methods and patents. Most model-based optimal control systems have a higher order, determined by the order of the plant model and the two cost-function weightings. An obvious question is whether some of the performance benefits of optimal advanced control can be obtained but still using a classical “low-order” controller structure. The subject of Restricted Structure (RS) controller design is introduced in the following sections and later in Chap. 11. Such a controller has a prespecified structure with gains or parameters that are to be determined. They are normally of a lower order than the full-order optimal controllers, and often have a simple classical control structure. The RS controller structure might be PID, cascaded lead–lag or simply a transfer-function or state-space based controller of low order. The RS optimal control problem is to find tuning parameters that minimize a suitable cost-function (like the LQG, GMV or H1 cost-functions discussed above). The RS

102

2 Review of Linear Optimal Control Laws

controllers will be simpler to implement and low-order designs often have good robustness properties to parameter variations (see [32–46]). The RS solution approach applies to SISO or MIMO systems. In the case of multivariable systems, the approach may also be used to assess the best control-loop structure of the multivariable feedback control system. It can be used to choose optimal input–output pairings, where the word “optimal” is normally used in terms of an LQG or similar cost-index. The model of the process is assumed linear for the present discussion. However, the use of this approach to RS controller design for nonlinear systems is considered in Sect. 2.8 using a multiple-model approach. This provides a link with the nonlinear control design topics covered in the remainder of the book. We will return to this topic in Chap. 11 where a very different approach is followed for RS control based on nonlinear predictive control.

2.6.1

Controller Order Reduction

Before proceeding with the polynomial approach to RS optimization, the general approaches to deriving low-order optimal controllers will be discussed briefly. The order of most optimal controllers is at least as high as the combined orders of the plant, disturbance, reference models and the dynamic cost-function weightings. This controller order is often large, relative to classical designs like PID control. This can cause problems with the cost of implementation, system complexity, execution speed and robustness. There are three possible ways, given the system model and cost criterion, to obtain low-order controllers: A. Design a full-order controller and then perform controller model reduction, B. Perform model reduction on the system model and then design a full-order controller for the low-order approximate plant model, or C. Compute a low-order controller directly, using parametric optimization, based on the original system model, criterion, and the specified controller structure. These three methods are illustrated in Fig. 2.8. Fig. 2.8 Strategies for obtaining reduced-order optimal controllers

Full-order model A

Full-order controller

B

C

A

Reduced-order model B

Reduced-order controller

2.6 Restricted Structure Control Systems

103

The methods involving the paths A and B involve a direct model reduction step. Different approaches to this problem are possible but all of them eventually amount to solving an optimization problem, where the cost is defined as the “error” in some sense, between the full-order model and parameterized lower order approximation (for example, minimization of the error between the frequency response characteristics). The direct route C should clearly provide the best solution, given the low-order structure chosen for the controller. This is the Restricted Structure design route followed below.

2.6.2

LQG Restricted Structure Control for SISO Systems

A polynomial systems approach to the so-called Restricted Structure (RS) optimal control and tuning problem is discussed in the next few sections [32]. It can be used to optimize the parameters for different low-order classical controller structures; however, the PID structure is so popular it will be used for illustrative purposes. Before discussing applications of the restricted-structure control technique in multivariable systems, an overview of the basic ideas and algorithms will be presented for the Single-Input Single-Output (SISO) problem. The goal will be to find the optimal PID controller parameters that minimize an LQG cost-function. This result will then be used to benchmark an existing PID controller. To obtain the restricted structure solution, an optimal control problem is defined and a simple cost-minimization problem is established, where the controller parameters are optimized directly to achieve the low-order optimal control solution. The optimization approach involves a transformation into the frequency domain and a numerical optimization of the integral cost-function. We again refer to the feedback system shown in Fig. 2.4, which is assumed linear, continuous-time and single-input, single-output. The continuous-time case is considered here since the results are slightly simpler and the optimization is based on approximating the cost-function in frequency-domain form. The solution for discrete-time MIMO systems will be presented later in the chapter. The system equations, assuming for simplicity that the measurement noise is null, may be listed as System output: yðsÞ ¼ W0 ðsÞuðsÞ þ dðsÞ Control error: eðsÞ ¼ rðsÞ  yðsÞ

104

2 Review of Linear Optimal Control Laws

Reference generator: rðsÞ ¼ Wr ðsÞfðsÞ Disturbance model: dðsÞ ¼ Wd ðsÞnðsÞ Control signal: uðsÞ ¼ C0 ðsÞeðsÞ The spectrum for the signal f ðtÞ ¼ rðtÞ  dðtÞ is denoted by Uff ðsÞ and the generalized spectral factor Yf ðsÞ satisfies Yf Yf ¼ Uff ¼ Urr þ Udd . In polynomial system matrix form Yf ¼ A1 Df . The disturbance model is assumed to be such that Df is strictly Hurwitz (strictly minimum phase), and satisfies Df Df ¼ Er Er þ Cd Cd . The signal f ¼rd may again be written in the form f ðsÞ ¼ Yf ðsÞeðsÞ ¼ A1 ðsÞDf ðsÞeðsÞ, where eðsÞ represents the transform of a zero-mean white noise signal of unity-variance. Full-Order LQG Controller: The first step in the design procedure is to obtain the optimal full-order controller. This result also provides a simplified criterion for the subsequent RS numerical optimization. A linear plant model must be available (possibly linearized around an important operating point). An objective function must also be defined, which will be taken here to be of the LQG type, as in (2.53). The results presented in this section are for the scalar case and based on Grimble [30, 32]. The initial steps of the solution for the full-order optimal controller follow Kucera’s very influential polynomial systems modelling and control approach [5]. The full-order optimal controller is first obtained to minimize the following LQG cost-function: JLQG ¼

1 2pj

I ðQc ðsÞUee ðsÞ þ Rc ðsÞUuu ðsÞÞds

ð2:92Þ

D

where the cost-function weightings Qc ðsÞ and Rc ðsÞ are dynamic, and act on the spectra of the error and control signals, respectively. They may be represented as Rc ¼ Rn =ðAr Ar Þ and Qc ¼ Qn =ðAq Aq Þ: The minimum cost is independent of the choice of controller. The optimal controller of restricted structure, to optimize the cost-function (2.92), can therefore be found by minimizing only that part of the criterion that depends upon the choice of controller, namely:

2.6 Restricted Structure Control Systems

1 J0 ¼ 2pj

I



T0 ðsÞT0 ðsÞ

1 ds ¼ 2p

D

105

Z1 ðT0 ðjxÞT0 ðjxÞÞdx

ð2:93Þ

1

where the transfer-function term [32]: T0 ¼

H0 Aq C0n  G0 Ar C0d Aq Ar ðAC0d þ BC0n Þ

ð2:94Þ

and G0 and H0 are the solutions of a system of two coupled Diophantine equations: Dc G0 þ F0 AAq ¼ B Ar Qn Df and Dc H0  F0 BAr ¼ A Aq Rn Df

ð2:95Þ

with F0 of minimum degree. The solution of these two equations ensures the implied equation AAq H0 þ BAr G0 ¼ Dc Df is satisfied. The polynomial Dc in the above equations is the Hurwitz solution of the following spectral factorization problem: Dc Dc ¼ B Ar Qn Ar B þ A Aq Rn Aq A: The full-order controller follows by setting (2.94) to zero, yielding C0 ¼ ðH0 Aq Þ1 G0 Ar

ð2:96Þ

This optimal full-order controller sets the term T0 in the integral (2.93) to zero, and the minimum value of the LQG cost (2.92) may then be found as Jmin

1 ¼ 2pj

I



F0 ðsÞF0 ðsÞ=ðDc ðsÞDc ðsÞÞ ds

ð2:97Þ

D

For an arbitrary nonoptimal controller, the integral term (2.93) will be nonzero, and this part of the expression for the cost therefore serves as a simplified criterion for optimizing the RS controller.

2.6.3

Parametric Optimization Problem

A parametric optimization problem provides the parameters for this scalar RS controller. To show how this is constructed, assume that the plant has an existing or desired PID feedback controller with a filtered derivative term:

106

2 Review of Linear Optimal Control Laws

C0 ðsÞ ¼ k0 þ k1 =s þ k2 s=ð1 þ hsÞ

ð2:98Þ

Assume the PID gains are to be found to minimize the LQG cost-function and the resulting minimum achievable cost is required. The RS control design problem is to find the values of the PID gains k0 , k1 and k2 that minimize (2.93). Since the parameters appear both in the numerator and denominator of the cost term (2.94), this is a nonlinear optimization problem. It can be solved iteratively using a successive approximation, or back-substitution, procedure. There is, of course, an implicit assumption that a stabilizing control law actually exists for the assumed controller structure. Optimization problem: To set up the optimization problem, rewrite (2.94): T0 ¼ ðC0n L1  C0d L2 Þ=ðC0n L3 þ C0d L4 Þ ¼ C0n Ln1 þ C0d Ln2 where the terms L1 ¼ H0 Aq , L2 ¼ G0 Ar , L3 ¼ Aq Ar B, L4 ¼ Aq Ar A and Ln1 ðsÞ ¼ L1 =ðC0n L3 þ C0d L4 Þ, Ln2 ðsÞ ¼ L2 =ðC0n L3 þ C0d L4 Þ: The controller numerator and denominator in Ln1 and Ln2 can initially be assumed to be known (computed from the results of the previous iteration). Evaluating the complex-function T0 ðjxÞ, the real and imaginary parts of T0 can be written as T0 ¼ T0r þ jT0i . By separating the optimization parameters, the following expression is obtained: T0 ¼ k0 ½ðar0 Lrn1  ai0 Lin1 Þ þ jðai0 Lrn1  ar0 Lin1 Þ þ k1 ½ðar1 Lrn1  ai1 Lin1 Þ þ jðai1 Lrn1  ar1 Lin1 Þ þ k2 ½ðar2 Lrn1  ai2 Lin1 Þ þ jðai2 Lrn1  ar2 Lin1 Þ

ð2:99Þ

r r i r i i  ½ðC0d Ln2  C0d Lin2 Þ  jðC0d Ln2  C0d Lrn2 Þ

Thence, the function T0, defined by this equation, can be written in the form:



 T0 ¼ k0 f0r þ j  f0i þ k1 f1r þ j  f1i þ k2 f2r þ j  f2i  gr þ j  gi

ð2:100Þ

where the definitions of fi ðjxÞ and gðjxÞ terms are obvious from (2.99). For numerical optimization, the cost-function (2.93) needs to be approximated using a finite summation: J0 ffi J0 0 ¼

N 1 X T0 ðjxl ÞT0 ðjxl ÞDx 2p l¼1

ð2:101Þ

The integer N is the number of frequency points used to approximate the integral. Evaluating the terms in the square brackets in (2.96) at each frequency point, separately for the real and imaginary parts, the above equation can be represented in the following matrix format:

2.6 Restricted Structure Control Systems

2

r r ðx1 Þ f12 ðx1 Þ f11 .. .. 6 6 .

r 6 r . r 6 f11 ðxN Þ f12 T0 ðx NÞ ¼6 6 f i ðx1 Þ f i ðx1 Þ T0i 11 12 6 6 .. .. 4 . . i i f11 ðxN Þ f12 ðxN Þ

107

3 3 2 r r f13 ðx1 Þ g ðx1 Þ .. 7 6 . 7 72 3 6 .. 7 . 7 7 6 r k 0 r 6 g ðxN Þ 7 f13 ðxN ÞÞ 7 7 7 6 4 5  k 1 i 6 gi ðx1 Þ 7 ¼ Fx  g f13 ðx1 Þ 7 7 k2 7 6 7 6 . 7 .. 5 4 .. 5 . i f13 ðxN Þ

gi ðxN Þ

ð2:102Þ where x ¼ ½k0 ; k1 ; k2 T . The optimization can now be performed by minimizing the sum of squares of J00 formed from the terms at each frequency point, as in (2.101). The cost term to be minimized may, therefore, be written as J00 ¼ ðFx  gÞT ðFx  gÞ

ð2:103Þ

Finally, the optimal gain parameters for the current iteration can be obtained as ½k0 ; k1 ; k2 T ¼ ðF T FÞ1 F T g

ð2:104Þ

The procedure requires the vector of initial optimization parameters to be provided, and is repeated until a stopping criterion is satisfied, as illustrated in Fig. 2.9.

Start: initialize C0

Calculate denominator of T0

Perform optimization assuming constant denominator

YES

Solution converged or maximum number of iterations reached?

Return C0 and stop

Fig. 2.9 Successive approximation algorithm

NO

108

2 Review of Linear Optimal Control Laws

The successive approximation procedure is only one of a number of possible optimization methods that might be used. It is very simple to implement and the number of iterations required is normally relatively small (10 is typical), but monotonic convergence to the global minimum is not guaranteed. Alternatively, gradient or conjugate gradient search methods may be used to minimize the cost-function (2.101) directly.

2.6.4

RS Controllers for Benchmarking Performance

The RS optimal control method is useful for benchmarking control systems. Restricted structure, low-order controllers, based on say the Generalized Minimum Variance or LQG benchmarks, will have good low-frequency gain, roll-off at high frequencies and will limit the variances of the outputs, which are often related to economic performance. The minimum cost for the RS optimal control can provide a benchmark figure of merit, which an engineer on a process plant with a classical controller, may try to achieve by skilful tuning. Techniques for benchmarking the performance of controllers have been used routinely; particularly in the process industries. However, commercial benchmarking tools often use minimum variance criteria for benchmarking systems but they do not take into account the fact that the industrial controller to be assessed is normally a low-order design and cannot approach the ideal minimum cost of a high-order minimum variance or other optimal controllers. The benefit of the RS benchmarking approach is that it provides figures of merit, which reveal the performance improvement that is possible for the actual low-order controller structure being used (see Refs. [47–51]). A classical controller on a plant can, therefore, be assessed against an optimal controller of the same structure. It provides a more realistic benchmark than a comparison with an ideal controller that is of minimum variance type and is of high order, with an unrealistic high-gain high-frequency response. The value of the cost-function for both the optimal RS controller J0RS and the actual controller J0Act can be evaluated by substituting the optimal controller transfer-functions into the expanded cost-function expressions. The difference between the costs of the full and RS solutions will be due to the cost term in (2.93). To compute this difference a numerical integration can be performed or a successive approximation algorithm can be used Grimble [32]. Controller Performance Index (CPI): A controller performance index can be defined as j ¼ J0RS =J0Act

ð2:105Þ

2.6 Restricted Structure Control Systems

109

The performance index j lies in the range ½0; 1, where 0 corresponds to very poor control, and unity to the optimal RS control. An alternative formulation of the controller performance index is to define j2 ¼ ðJmin þ J0RS Þ=ðJmin þ J0Act Þ

ð2:106Þ

where Jmin is the minimum achievable cost, corresponding to the optimal controller. This latter expression requires the additional calculation of Jmin . A CPI value of unity can be achieved by a well-tuned existing controller. Moreover, the optimal tuning parameters are readily available as a by-product of the benchmarking algorithm. A drawback, relative to minimum variance benchmarking, is that plant model information is essential.

2.6.5

LQG Restricted Structure Control for MIMO Systems

Restricted Structure controller design and benchmarking of multivariable systems is based on similar ideas to those presented for the scalar case, except that it is also possible to assess the input–output structure of the controller as well as its tuning. In the case of an n  n multivariable plant, all n2 elements of the controller transfer-function matrix can have a specified low-order structure. It is possible to restrict the controller structure further, by forcing some of the gain parameters to be zero (no feedback then exists between the output–input pair). By computing the costs corresponding to various configurations, it is possible to determine the potential benefits gained by introducing additional multivariable controller off-diagonal elements in the system. The procedure may also be used for the optimal tuning of controllers. Multivariable Discrete-Time LQG Controller: The equations defining the polynomial LQG controller for the discrete-time MIMO case are very similar to those presented above for SISO systems; the only differences result from the polynomial matrix representation of the system model and criterion. The details of the derivation are omitted here but are similar to those given for the GLQG case above, or in Grimble [32]. The component of the cost-function that is influenced by the control action is given as 1 J0 ¼ 2pj

I jzj¼1

tracefT0 ðz1 ÞT0 ðz1 Þg

dz z

ð2:107Þ

where 1 1 1 1 T0 ¼ ðGD1 2 Aq C0d  HD3 Ar C0n ÞðAC0d þ BC0n Þ Df

ð2:108Þ

110

2 Review of Linear Optimal Control Laws

The optimal full-order controller sets the integral (2.107) to zero, however, when the controller structure is restricted, the minimum value of (2.107) will normally be non-zero, and the minimum will depend on the controller structure chosen and tuning. The polynomial matrices G; H; D2 ; D3 satisfy the following relationships: 1 1 1 1 1 1 A1 q A BAr ¼ B1 A1 , Df AAq ¼ A2 D2 , Df BAr ¼ B2 D3 . The Diophantine equations to be solved may be listed as Dc G0 zg þ F0 A2 ¼ B Ar Qn D2 zg Dc H0 zg  F0 B2 ¼ A Aq Rn D3 zg The Diophantine equations are identical to those in the continuous-time SISO case (2.95), when D2 ¼ D3 ¼ Df , with the exception of the delay operator zg where the integer g > 0 is chosen to ensure the equations are polynomials in unit-delay z1 terms. Desired structure: Assume that the desired Restricted Structure (RS) controller structure is of a filtered PID form, and for notational simplicity, consider only square n  n systems. The multivariable PID controller structure can be represented in a multivariable form as C0 ¼ K0 þ K1

1 ð1  z1 Þ þ K2 1 1z 1  sd z1

ð2:109Þ

where 2

i k11

6 6 ki 6 11 6 Ki ¼ 6 .. 6 . 6 .. 4 . i kn1

i k12 i k22 .. . .. .

...

3 i . . . k1n . 7 . . . .. 7 7 .. 7 .. . . 7 7 . 7 .. . .. 5 i . . . knn

ð2:110Þ

The controller can also be represented in the equivalent right matrix fraction 1 form as C0 ¼ C0n C0d , where C0d ðz1 Þ ¼ ð1  z1 Þð1  sd z1 Þ  In

ð2:111Þ

C0n ðz1 Þ ¼ a0 ðz1 ÞK0 þ a1 ðz1 ÞK1 þ a2 ðz1 ÞK2

ð2:112Þ

and

with a0 ðz1 Þ ¼ ð1  z1 Þð1  sd z1 Þ, a1 ðz1 Þ ¼ 1  sd z1 and a2 ðz1 Þ ¼ ð1  z1 Þ2 .

2.6 Restricted Structure Control Systems

111

A parametric optimization algorithm can be used to minimize the cost-function (2.107), with respect to the restricted structure controller parameters. The controller performance index (2.105) can be computed by substituting into the cost-function to obtain the theoretical minimum value of the cost, which can then be compared with the value of the cost-function, obtained using an existing classical controller.

2.6.6

Solution of the Parametric Optimization Problem

The algorithm is a direct generalization from the SISO case and involves representing the integral (2.107) in the frequency-domain form: J0 ¼ min C0

8 2p 0 steps ahead, providing some future reference knowledge. The controller involves the zp term in (3.22) and if there is no future information then let p = 0. This provides some limited predictive control action.

3.3 NGMV Open-Loop and Feedforward Control

141

Fig. 3.5 Tracking and feedforward controller modules

(d) The class of problems considered are those for which a solution to the Diophantine equation [16] can be found, where F0 ðz1 Þ and F1 ðz1 Þ have minimum row degrees of less than the delay path magnitudes fk1 ; k2 ; . . .; kr g and fk1 þ q; k2 þ q; . . .; kr þ qg, respectively. (e) An unusual feature of the NGMV solution applies to the full family of both feedforward and feedback controllers. The nonlinear input plant subsystem W 1k ð:; :Þ can be a black-box model with unknown content. All that is required is the ability to compute an output mk ðtÞ ¼ ðW 1k uÞðtÞ for a given control input uðtÞ. As noted such a black-box model might, for example, be described in C code, or could be represented by a neural network or a fuzzy neural network [10].

3.3.2

Solution of the Open-Loop/Feedforward Problem

Consider the minimization of the signal /0 ðtÞ, which represents the sum of weighted error and control signals. This signal is assumed to have the same number of outputs as the input signal dimension, so that F c may be defined as either a linear or a nonlinear invertible operator. It can be considered either a convenient cost-function signal, to be minimized in a variance sense, or an inferred output. The signal is defined as /0 ðtÞ ¼ Pc ðz1 ÞeðtÞ þ ðF c uÞðtÞ where Pc ðz1 Þ is a transfer-function matrix and F c may be a nonlinear operator that is high pass. The error signal follows from Eqs. (3.15) and (3.16) as e ¼ ri  y ¼ ri  d  W u, and hence: /0 ðtÞ ¼ Pc ðri  d  WuÞ þ F c u ¼ Pc ðri  d 0  d 1 Þ  Pc ðWuÞ þ ðF c uÞ ¼ Pc d0 þ Pc ðri  d 1 Þ  ðPc W  F c Þu This last term involves nonlinear operators that may be expanded as

ð3:26Þ

142

3 Open-Loop and Feedforward Nonlinear Control

ðPc W  F c Þu ¼ Pc ðWuÞðtÞ  ðF c uÞðtÞ

ð3:27Þ

Before continuing with the solution, recall the reference and measurable disturbance signals have stochastic and deterministic components. Thus, introduce the models and signals: Wd1 ¼ zi Wdi ;

Wri ¼ Wi Wr

and rWD ðtÞ ¼ Wi rd ðtÞ;

dWD ðt þ iÞ ¼ Wdi dd ðtÞ

Hence, we obtain: ri ðt þ pÞ ¼ Wi ðWr xðtÞ þ rd ðt þ pÞÞ ¼ Wri xðtÞ þ rWD ðt þ pÞ

ð3:28Þ

d1 ðt þ iÞ ¼ Wd1 ðWd2 fðt þ iÞ þ dd ðt þ iÞÞ ¼ Wdi Wd2 fðtÞ þ Wdi dd ðtÞ ¼ Wdi Wd2 fðtÞ þ dWD ðt þ iÞ

ð3:29Þ

That is, these equations may be written as ri ðtÞ ¼ Wri xðt  pÞ þ rWD ðtÞ

ð3:30Þ

d1 ðtÞ ¼ Wdi Wd2 fðt  iÞ þ dWD ðtÞ

ð3:31Þ

and

By introducing the reference and disturbance generalized spectral-factors [17, 18], the ideal response model output ri and measurable disturbance d1 can be modelled in an innovations signal form as ri ðtÞ ¼ zp Yri e0 ðtÞ þ rWD ðtÞ and d1 ðtÞ ¼ zi þ j Yd1 e1 ðtÞ þ dWD ðtÞ, respectively. The driving white noise signals e0 ðtÞ and e1 ðtÞ denote zero-mean white noise innovations sequences, with identity covariance matrices. The inferred output, which represents the signal to be minimized, may now be written from (3.26), as /0 ðtÞ ¼ Pc d0 þ Pc ðri  d 1 Þ  ðPc W  F c Þu   ¼ Pc d0 þ Pc zp Yri e0 ðtÞ þ rWD ðtÞ  zi þ j Yd1 e1 ðtÞ  dWD ðtÞ  ðPc W  F c Þu ¼ Pc d0 þ gPD ðtÞ þ Pc zp Yri e0  Pc zi þ j Yd1 e1  ðPc W  F c Þu

ð3:32Þ

gPD ðtÞ ¼ Pc ðrWD ðtÞ  dWD ðtÞÞ ¼ Pc ðWi rd ðtÞ  Wdi dd ðt  iÞÞ

ð3:33Þ

where

3.3 NGMV Open-Loop and Feedforward Control

3.3.3

143

Polynomial Models for System

The assumption that the disturbances could be represented by linear models may cause a degree of sub-optimality in the disturbance rejection performance. Models for disturbances may be represented by nonlinear power spectra, like sea wave or wind spectra. However, these are only representative in typical circumstances, and in most industrial applications, stochastic disturbances can be approximated adequately by linear disturbance models driven by white noise [18]. The reference signal spectral-factor Yri may be assumed to have the polynomial matrix form in (3.7), where based on the system description Dr and Dr1 are strictly Schur polynomial matrices. The tracking and feedforward disturbance model spectral-factors (3.7) and (3.8) follow as 1 Yri ¼ Dr1 A1 r1 and Yd1 ¼ Dd1 Ad1

ð3:34Þ

Introduce the right-coprime polynomial matrix models: 1 Dr0 P1 rd ¼ Pcd Dr1

ð3:35Þ

1 Dd0 P1 dd ¼ Pcd Dd1

ð3:36Þ

Thence, the tracking and disturbance terms become 1 1 1 Pc Yri e0 ¼ Pcn P1 cd Dr1 Ar1 e0 ¼ Pcn Dr0 Prd Ar1 e0

ð3:37Þ

1 1 1 Pc Yd1 e1 ¼ Pcn P1 cd Dd1 Ad1 e1 ¼ Pcn Dd0 Pdd Ad1 e1

ð3:38Þ

Collecting the above results, the weighted error and control signal to be optimized has the form: /0 ðtÞ ¼ Pc d0 þ gPD þ Pc zp Yri e0  Pc zi þ j Yd1 e1  ðPc W  F c Þu 1 i þ j 1 ¼ Pc d0 þ zp Pcn Dr0 P1 Pcn Dd0 P1 rd Ar1 e0  z dd Ad1 e1 þ gPD

 ðPc W  F c Þu

3.3.4

ð3:39Þ

Diophantine Equations

The solution of the optimization problem depends upon the solution of two Diophantine equations. These can be used to expand the linear white noise driven terms into two groups, depending upon the length of the output channel delay. These linear polynomial based equations can be solved by equating coefficients of the polynomial matrix terms. This results in a vector matrix equation whose

144

3 Open-Loop and Feedforward Nonlinear Control

solution provides the constant coefficients in the polynomial equations. See Kucera [16] for polynomial systems algebra, Diophantine equations and their solution. Commercial software is available to compute the solution of such equations, such as The Polynomial ToolboxTM for Matlab developed by PolyX. The tracking or open-loop Diophantine equation may be introduced as F 0 Ar1 Prd þ zK0 G0 ¼ zp Pcn Dr0

ð3:40Þ

and this may be written as F 0 þ zK0 G0 ðAr1 Prd Þ1 ¼ zp Pcn Dr0 ðAr1 Prd Þ1

ð3:41Þ

Similarly, for the feedforward Diophantine equation: F1 Ad1 Pdd þ zK0 qI G1 ¼ zi þ jq Pcn Dd0

ð3:42Þ

that may be written as zq F1 þ zK0 G1 ðAd1 Pdd Þ1 ¼ zi þ j Pcn Dd0 ðAd1 Pdd Þ1

ð3:43Þ

The integer q is the smallest possible positive integer such that (3.42) involves polynomials in powers of z1 . Substituting Eqs. (3.41) and (3.43) into (3.39), obtain the inferred output as 1 i þ j 1 /0 ðtÞ ¼ Pc d0 þ zp Pcn Dr0 P1 Pcn Dd0 P1 rd Ar1 e0  z dd Ad1 e1 þ gPD  ðPc W  F c Þu   1 1 1 ¼ Pc d0 þ F0 e0  zq F1 e1 þ zK0 G0 P1 rd Ar1 e0  G1 Pdd Ad1 e1

þ gPD  ðPc W  F c Þu

ð3:44Þ

Note for later use that row n of the inferred output vector (3.44) may be separated into unit-delay terms of up to zkn þ 1 and of greater order.

3.3.5

Optimization and Signals

The optimization to find the optimal control is straightforward but some manipulation of the reference and disturbance signal terms is first required. These signals have the respective forms: ri ðtÞ ¼ zp Yri e0 ðtÞ þ rWD ðtÞ ¼ zp Dr1 A1 r1 e0 ðtÞ þ rWD ðtÞ and

ð3:45Þ

3.3 NGMV Open-Loop and Feedforward Control

d1 ðtÞ ¼ zi þ j Yd1 e1 ðtÞ þ dWD ðtÞ ¼ zi þ j Dd1 A1 d1 e1 ðtÞ þ dWD ðtÞ

145

ð3:46Þ

This may be rewritten as the two terms: 1 1 D1 r1 ri ðt þ pÞ ¼ Ar1 e0 ðtÞ þ Dr1 rWD ðt þ pÞ

ð3:47Þ

1 1 D1 d1 d1 ðt þ i  jÞ ¼ Ad1 e1 ðtÞ þ Dd1 dWD ðt þ i  jÞ

ð3:48Þ

and

  Now from (3.18) d1 ðt þ i  jÞ ¼ Wdi Wdj1 yf ðtÞ  bf ðtÞ and dWD ðt þ iÞ ¼ Wdi dd ðtÞ. Thence, define:   d~1 ðtÞ ¼ d1 ðt þ i  jÞ  dWD ðt þ i  jÞ ¼ Wdi Wdj1 yf ðtÞ  bf ðtÞ  Wdi dd ðt  jÞ   ð3:49Þ ¼ Wdi Wdj1 yf ðtÞ  Wd3 dd ðtÞ  Wdi Wdj1 bf ðtÞ Also, we define: ~ri ðtÞ ¼ ri ðtÞ  rWD ðtÞ

ð3:50Þ

From (3.47) and (3.48), and the two expressions (3.49) and (3.50), we obtain: 1 1 1 A1 ri ðt þ pÞ r1 e0 ðtÞ ¼ Dr1 ri ðt þ pÞ  Dr1 rWD ðt þ pÞ ¼ Dr1 ~ 1 1 1 ~ A1 d1 e1 ðtÞ ¼ Dd1 d1 ðt þ i  jÞ  Dd1 dWD ðt þ i  jÞ ¼ Dd1 d1 ðtÞ

ð3:51Þ ð3:52Þ

Substituting these results in (3.44), we obtain the following equation for the inferred output being minimized: 1 K0 1 /0 ðtÞ ¼ Pc d0 þ F0 e0  zq F1 e1 þ zK0 G0 P1 G1 P1 rd Ar1 e0  z dd Ad1 e1 þ gPD q  ðPc W  F c Þu ¼ Pc d0 þ F0 e0  z F1 e1   K0 1 1 ~ þz G0 P1 ð3:53Þ ri ðt þ pÞ  G1 P1 rd Dr1 ~ dd Dd1 d1 ðtÞ þ gPD  ðPc W  F c Þu

For the optimization, observe that the control signal at time t affects the jth system output at time t þ kj . Hence, the control signal costing F c is defined to include a delay of kj steps for the jth output channel, and is defined to be of the form F c ¼ zK0 F ck : Also assume that Pc and zK0 commute, then (3.53) becomes

146

3 Open-Loop and Feedforward Nonlinear Control

þz

 K0

/0 ðtÞ ¼ Pc d 0 ðtÞ þ F0 e0 ðtÞ  zq F1 e1 ðtÞ  G0 P1 D1~ri ðt þ pÞ  G1 P1 D1 d~1 ðtÞ þ gPD  zK0 ðPc W k  F ck Þu rd

r1

dd

d1

ð3:54Þ Note from (3.50) and (3.49) that the signals ~ri ðt þ pÞ and d~1 ðtÞ represent the stochastic components of the reference and measurable disturbance signals. Inferred output: The weighted error and control signals may be written, using (3.54), in the form: /0 ðtÞ ¼ ½F0 e0 ðtÞ  zq F1 e1 ðtÞ  Pc d 0 ðtÞ   1 1 ~ þ zK0 G0 P1 ri ðt þ pÞ  G1 P1 rd Dr1 ~ dd Dd1 d1 ðtÞ þ gPD ðt þ K0 Þ  ðPc W k  F ck ÞuðtÞ

ð3:55Þ The signal Pc ðz1 Þd0 ðtÞ is an unknown stochastic signal that is statistically independent of the other signal sources. Also note that for row j the deg(F0) < kj , and the first term is therefore dependent upon the values of the white noise signal components eðtÞ, … ,eðt  kj þ 1Þ, for the jth output channel. The remaining terms in the expression are delayed by at least kj steps, and depend upon earlier values eðt  kj Þ; eðt  kj  1Þ;…. Similar reasoning applies to the e1 ðtÞ white noise signal terms. Condition for optimality: The first three terms in the square brackets [.] and the remaining terms in the round brackets (.) in (3.55) are statistically independent. The variance of the signal /0 ðtÞ , therefore, involves the sum of the variances of each term since the variance of the cross-product terms is null. Also, note that the first three terms in (3.55) are independent of the control action and only the terms within the round brackets are control dependent. It follows that the smallest variance is achieved when the final group of terms is set to zero. That is, the optimal control to minimize the variance of the signal may be computed by setting the round bracketed term in (3.55) to zero. Thence the condition for optimality becomes 1 1 ~ G0 P1 ri ðt þ pÞ  G1 P1 rd Dr1 ~ dd Dd1 d1 ðtÞ þ gPD ðt þ K0 Þ  Pc W k uðtÞ þ F ck uðtÞ ¼ 0

ð3:56Þ Optimal open-loop and feedforward: If it is assumed that the inverse of the control weighting operator F ck exists then the following expression may be obtained for the optimal control:   1 1 1 ~ uðtÞ ¼ F 1 ri ðt þ pÞ  G1 P1 ck G0 Prd Dr1 ~ dd Dd1 d1 ðtÞ  Pc W k uðtÞ þ gPD ðt þ K0 Þ ð3:57Þ

3.3 NGMV Open-Loop and Feedforward Control

147

An alternative expression may be obtained from the condition for optimality (3.56):   1 1 ~ uðtÞ ¼ ðPc W k  F ck Þ1 G0 P1 ri ðt þ pÞ  G1 P1 rd Dr1 ~ dd Dd1 d1 ðtÞ þ gPD ðt þ K0 Þ ð3:58Þ The numerators of the spectral-factors for the reference and disturbance signals are represented by the polynomial matrices Dr1 and Dd1 and these are by definition normal full rank and minimum phase. However, for simplicity assume that the definition of the system models is such that these matrices are strictly minimum phase.

3.4

Design and Implementation Issues

As with most of the optimal control design methods, the cost-function weightings can normally be parameterized to include simple tuning variables on the error and control signal costing terms. The optimal control signal to be minimized is shown dotted in Fig. 3.4 since it is not normally a physical signal. This signal is constructed from the weighted error and control signals as follows: /0 ðtÞ ¼ Pc ðz1 ÞeðtÞ þ ðF c uÞðtÞ ¼ Pc ðz1 ÞðWi rðtÞ  yðtÞÞ þ ðF c uÞðtÞ A necessary condition for stability may be derived by inspecting the expression (3.58) for the control signal in terms of exogenous inputs. That is, the operator ðPc W k  F ck Þ1 must be assumed to be finite-gain stable for the control signal to be generated from stable operators. This must be ensured from the choice of cost-function weightings.

3.4.1

Benchmarking Performance

The minimum value of the cost-function is clearly due to the first set of linear time-invariant (LTI) terms in (3.55) (since the remaining terms are null). It follows that the cost Jmin depends only on the reference and disturbance signal models that were assumed linear time-invariant systems:   Jmin ¼ E ðF0 e0 ðtÞÞT ðF0 e0 ðtÞÞ þ ðF1 e1 ðtÞÞT ðF1 e1 ðtÞÞ þ ðPc d 0 ðtÞÞT ðPc d 0 ðtÞÞjt ð3:59Þ This minimum-cost expression can provide a benchmark cost for nonlinear controller design [19, 20]. Benchmarking is important in the process industries, where there are often large numbers of controllers that are poorly tuned (see Chap. 2, Sect. 2.6 and Svrcek et al. [21], Smith et al. [22]).

148

3 Open-Loop and Feedforward Nonlinear Control

3.4.2

Implementation Issues

One of the problems with the implementation of the control law is an “algebraic loop” that is formed when computing the control signal. Observe from Fig. 3.5, the control signal u(t) at time t depends upon u(t) itself. To avoid this problem first note the signal wðtÞ in Fig. 3.5 may be defined as wðtÞ ¼ ðPc W k  F ck Þu ¼ Pc ðW k uÞðtÞ  ðF ck uÞðtÞ

ð3:60Þ

Assuming the existence of the inverse of the nonlinear control signal weighting, the optimal control signal uðtÞ ¼ F 1 ck ðPc ðW k uÞðtÞ  wðtÞÞ. Unfortunately, the right-hand side of this equation also includes the control signal uðtÞ, which is the cause of the algebraic-loop problem. It is possible to compute uðtÞ by using successive approximation iterations but this is not very efficient. To avoid the need to perform iterations, the operator Pc W k (that is a function of z1 ), may be split into two parts. These involve a term without a delay N 0 and a term that depends upon past values of the control action z1 N 1 . Writing   pðtÞ ¼ ðPc W k uÞðtÞ ¼ ðN 0 uÞðtÞ þ z1 N 1 u ðtÞ Then we obtain:   wðtÞ ¼ ðPc W k  F ck Þu ¼ ðN 0 uÞðtÞ þ z1 N 1 u ðtÞ  ðF ck uÞðtÞ

ð3:61Þ

The algebraic loop is, therefore, removed in the computation of the optimal control signal:   uðtÞ ¼ ðF ck  N 0 Þ1 ðz1 N 1 uÞðtÞ  wðtÞ

ð3:62Þ

Since the weightings may be chosen freely the pulse response operator ðF ck  N 0 Þ can be assumed full rank to ensure the existence of the inverse. However, recall the weightings must also be chosen to ensure the existence of a stable causal inverse for the nonlinear operator ðPc W k  F ck Þ. Example 3.1 To be able to use (3.62) for the computation of the control signal, an explicit form for the two operators ðN 0 uÞðtÞ and ðz1 N 1 uÞðtÞ is required. This may be obtained from a nonlinear state-space model. By way of example assume that the nonlinear subsystem pðtÞ ¼ ðPc W k uÞðtÞ can be represented by an affine (with respect to u(t)) nonlinear state-space model of the form: xðt þ 1Þ ¼ f ðxðtÞÞ þ gðxðtÞÞuðtÞ

ð3:63Þ

yp ðtÞ ¼ cðxðt  1ÞÞxðtÞ

ð3:64Þ

3.4 Design and Implementation Issues

149

Fig. 3.6 Nonlinear operator non-delay and unit-delayed terms

pðtÞ ¼ yp ðt þ 1Þ

ð3:65Þ

pðt  1Þ ¼ cðxðt  1ÞÞxðtÞ ¼ cðxðt  1ÞÞðf ðxðt  1ÞÞ þ gðxðt  1ÞÞuðt  1ÞÞ ¼ z1 ðcðxðtÞÞðf ðxðtÞÞ þ gðxðtÞÞuðtÞÞÞ or pðtÞ ¼ ðPc W k uÞðtÞ ¼ ðN 0 uÞðtÞ þ ðz1 N 1 uÞðtÞ ¼ cðxðtÞÞðf ðxðtÞÞ þ gðxðtÞÞuðtÞÞ Note a unit pulse dðtÞ does not affect the output from the term f ðxðtÞÞ at time t, and the two operators may, therefore, be split into two terms: ðN 0 uÞðtÞ ¼ gðxðtÞÞuðtÞ

ð3:66Þ

ðz1 N 1 uÞðtÞ ¼ cðxðtÞÞðf ðxðtÞÞ

ð3:67Þ

These results suggest the method of implementing the operators ðN 0 uÞðtÞ and ðz1 N 1 uÞðtÞ illustrated in Fig. 3.6.

3.4.3

Existence of the Nonlinear Operator Inverse

The optimal control (3.58) indicated that a necessary condition for stability is that the operator ðPc W k  F ck Þ must have a stable inverse. For the case of linear systems, the operator should be of normal full rank, and strictly minimum phase [20]. Clearly one of the restrictions on the choice of the cost-weightings is that this stability condition must be fulfilled. An important question is whether sensible choices of the weightings will lead to this condition. To show this is true of a very wide class of systems, consider the case where F ck is linear and negative F ck ¼ Fk . Then, we obtain: ðPc W k þ Fk Þu ¼ Fk ðFk1 Pc W k þ IÞu

150

3 Open-Loop and Feedforward Nonlinear Control

The term ðI þ Fk1 Pc W k Þ is the same as the return-difference operator for a fictitious system with plant W k and with a feedback control Kc ¼ Fk1 Pc . The stability of the open-loop and feedforward control system is clearly related to the stability of this loop, which can be ensured by an appropriate choice of weightings. For example, assume a PID controller exists to stabilize the closed-loop with plant model W k that may be found by simulation or some other method. Then a starting point for weighting selection, that will ensure ðPc W k þ Fk Þ is stably invertible, is to define the weightings to satisfy Fk1 P ¼ Kc : However, note that W k is not the same as the plant model because of the absence of the explicit delay term.

3.4.4

Weightings and Scheduling

The selection of the optimal control cost-function weightings will determine the types of time and frequency responses obtained, but this provides a rather indirect way of changing performance. Another possible way of shaping responses is to choose an ideal response model in (3.15). When the control signal weighting becomes small, the responses of the system will then tend to those of the ideal response model Wi ðz1 Þ. It is common practice in engine controls to base feedforward controls on engine maps obtained by calibration engineers. The gains are, therefore, often scheduled and this suggests some form of scheduling on the cost-function weightings may also be needed. This topic is discussed further in Chap. 4 but in the context of feedback systems.

3.4.5

Benefits of NGMV Open-Loop/Feedforward Control

The benefits of using the NGMV approach for open-loop and feedforward control can be listed as follows: 1. A clear measure of performance is involved and the cost-function can be used to benchmark performance. 2. Tracking or disturbance rejection properties can be optimized by choice of weightings. 3. The cost-function tuning parameters have a direct effect on traditional properties like speed of response. 4. The compensation for nonlinearities is introduced in a natural manner and a pure inverse is not attempted even when possible. 5. The NGMV compensator has dynamics that are dependent upon the system and cost-function descriptions and is not therefore as naive as the traditional compensators that use simple filters and ad hoc solutions.

3.5 Example of Feedforward and Tracking Control

3.5

151

Example of Feedforward and Tracking Control

The aim of the following example is to illustrate the design and tuning mechanisms for the open-loop tracking control, and for the feedforward (disturbance rejection) control. The example illustrates the type of results that are obtained and the benefits of optimal feedforward and tracking control. A discrete-time plant model with nominal sample time Ts = 1 s and a delay of k ¼ 6 samples is included in the measurable disturbance to output path. The output signal measured is not assumed to be delayed, but the use of feedforward action should still significantly improve the performance [23]. State-Space Nonlinear Plant Model: The nonlinear dynamic system is given in the state-space form: x2 ðtÞ x1 ðt þ 1Þ ¼ þ u1 ðtÞ 1 þ x21 ðtÞ x2 ðt þ 1Þ ¼ 0:9x2 ðtÞex1 ðtÞ þ u2 ðtÞ 2

yðtÞ ¼ xðt  kÞ

ð3:68Þ

Let the initial state xð0Þ ¼ ½0; 0T (the stable equilibrium point of the autonomous system). Both outputs are assumed to include a transport-delay of k ¼ 6 samples, and the time-delay matrix has the form: zK0 ¼ diagfzk ; zk g Other linear subsystems may be defined as follows: Linear plant subsystem: 2 3 1 0:1 6 0:2z1 1  0:04z1 7 W0k ¼ 4 1 0:12 5 1 1  0:3z1 1  0:4z1 Reference model:



1 1 ; Wr ¼ diag 1  z1 1  z1



Ideal response model: Wi ¼ diagfð1  a1 Þ=ð1  a1 z1 Þ;

ð1  a2 Þ=ð1  a2 z1 Þg

Unmeasurable disturbance:  Wd0 ¼ diag

0:05 0:1 ; 1  0:3z1 1  0:5z1



152

3 Open-Loop and Feedforward Nonlinear Control

Measurable disturbance: Wd1



ð1  0:05z1 Þ ; ¼ z Wdi ¼ z diag ð1  0:2z1 Þ i

6

ð1  0:1z1 Þ ð1  0:3z1 Þ



Measurable disturbance: 

Wd2

ð1  0:1z1 Þ ð1  0:1z1 Þ ; ¼ 0:3 diag ð1  z1 Þ ð1  z1 Þ



Measurable disturbance: Wd3 ¼ zj Wdj ¼ z5 diag



ð1  0:1z1 Þ ; ð1  0:3z1 Þ

ð1  0:4z1 Þ ð1  0:2z1 Þ



The “ideal response model” includes the tuning variables a1 and a2 that are chosen as 0.92 for reasonable nominal time responses, and the transfer Wri ¼ Wr Wi . Dynamic Weightings: The dynamic cost-function weightings were selected as follows:

Fck ¼

Pcn ¼

2ð1  0:1z1 Þ 0 0 2ð1  0:1z1 Þ

2ð1  0:9z1 Þ 0 0 2ð1  0:9z1 Þ



 Fcd ¼

and

 and Pcd ¼

1  z1 0

1 0

0 1



0 1  z1



The weightings can be chosen in the usual way by choosing a low-pass error weighting to compensate for low-frequency tracking errors, and a lead control weighting term to penalize high-frequency control action. The gains and the corner frequencies were chosen rather arbitrarily, tuned and checked by simulation (see Chap. 5 for further discussion on weighting selection procedures for feedback controls). Results: For simulation purposes, the reference and measurable disturbance step changes are introduced in both channels. Initially, no stochastic noise is introduced to the system and only the transient performance is assessed. This is compatible with selecting the reference and the measurable disturbance models as integrators. A bias on the output sensor measurements is assumed in the simulation trials of 0.06 and 0.08, respectively. This illustrates the type of behaviour when a modelling error occurs.

3.5 Example of Feedforward and Tracking Control

153

Transient performance: The time responses of the NGMV controllers (open-loop only, feedforward only and open-loop tracking plus feedforward) are compared, initially without stochastic system inputs. The time responses for the nonlinear system are shown in Fig. 3.7 for the two outputs and two inputs. This figure is for the ideal case where there is no mismatch and hence the disturbance is rejected totally in the steady state for either of the cases where feedforward is present. The open-loop control with no feedforward does, of course, include the disturbances without change at the outputs. Although the controller is optimal for the inferred output signal /0 ðtÞ, rather than just the actual output or control signals, there is a visible improvement in the transient error and control responses with the introduction of feedforward action. If there is a mismatch in the system models the “perfect responses” obtained above are compromised as shown in Fig. 3.8. The mismatch affects the steady-state characteristics that a feedback loop could correct. Stochastic performance: Stochastic inputs can be included and the performance of the controllers, in terms of the steady-state variances, can be considered. Stochastic disturbances have been added, feeding the disturbance models Wd and Wd0. The simulations were run for 1000 s, and the steady-state variances for the three controller configurations were calculated from the time responses for both the case of no mismatch and with mismatch. In this latter case, the disturbance and plant model parameters were all changed by about 10% to determine robustness. Observe from the variances collected in Tables 3.1 and 3.2 and the responses

1.4

1.2

1.2

1 Open-loop + Feedforward

1 Open-loop

0.8

0.6

y2

y1

0.8

0.4

Feedforward

0.6 0.4

0.2

0.2

0

0

-0.2

-0.2

0

100

200

300

400

500

600

1

0

100

200

0

100

200

300

400

500

600

300

400

500

600

0.3 Open-loop Feedforward Open-loop + Feedforward

0.5

0.25 0.2

u2

u1

0.15 0

0.1 0.05 0

-0.5

-0.05 -1 0

100

200

300

400

time [samples]

500

600

-0.1

time [samples]

Fig. 3.7 Deterministic tracking and feedforward control without model mismatch

154

3 Open-Loop and Feedforward Nonlinear Control 1.2

1.4

1

1.2 1

Open-loop + Feedforward

0.8

0.6

y2

y1

0.8

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

0

100

200

300

400

500

600

0

100

200

Open-loop Feedforward Open-loop + Feedforward

0.6 0.4

-0.1 0

100

200

400

500

600

300

400

500

600

0.25 0.2 0.15

u2

0.2 0

0.1 0.05

-0.2 -0.4

0

-0.6

-0.05

-0.8 0

300

0.3

0.8

u1

Open-loop

0.6

Feedforward

100

200

300

400

500

600

time [samples]

time [samples]

Fig. 3.8 Deterministic tracking and feedforward control with model mismatch

Table 3.1 Stochastic ideal performance results (no mismatch) Controller

trace(Var[e])

trace(Var[u])

trace(Var[/0 ])

NGMV open-loop NGMV open-loop and feedforward

0.174 0.053

0.285 0.277

3378.6 1.6

Table 3.2 Stochastic performance results (with mismatch) Controller

trace(Var[e])

trace(Var[u])

trace(Var[/0 ])

NGMV open-loop NGMV open-loop and feedforward

0.143 0.112

0.281 0.302

2810.6 646.9

shown in Figs. 3.9 and 3.10, that the feedforward control provides a considerable improvement, particularly in the case of model mismatch. Note that the error variances can change quite significantly between simulations having different noise seeds, because of the nonlinearities. Nevertheless, the addition of feedforward always provides a significant improvement.

3.5 Example of Feedforward and Tracking Control 1.5

155

1.5 Open-loop + Feedforward

1

1

y2

0.5

y1

0

0.5

-0.5 0 -1

Open-loop

-1.5 0

-0.5 0

100 200 300 400 500 600 700 800 900 1000

100 200 300 400 500 600 700 800 900 1000

0.8

1.5 Open-loop Open-loop + Feedforward

1

0.6 0.4

u1

u2

0.5 0

0.2

-0.5

0

-1 0

-0.2 0

100 200 300 400 500 600 700 800 900 1000

100 200 300 400 500 600 700 800 900 1000

time [samples]

time [samples]

Fig. 3.9 Stochastic tracking and feedforward control without model mismatch

2

1.5 Open-loop + Feedforward

1

1.5 1

y2

y1

0.5 0

0.5 0

-0.5 -1 -1.5

-0.5

Open-loop

0

-1 0

100 200 300 400 500 600 700 800 900 1000

1.2

1.5 1

100 200 300 400 500 600 700 800 900 1000

Open-loop Open-loop + Feedforward

1 0.8

u2

u1

0.5

0.6 0.4

0

0.2 -0.5 -1 0

0 100 200 300 400 500 600 700 800 900 1000

time [samples]

-0.2 0

100 200 300 400 500 600 700 800 900 1000

time [samples]

Fig. 3.10 Stochastic tracking and feedforward control with model mismatch

156

3.6

3 Open-Loop and Feedforward Nonlinear Control

Concluding Remarks

Feedforward control and inverse compensation methods are used regularly in industrial systems, and these sometimes involve nonlinear compensation. These may involve large approximations and empirical design methods (see Refs. [24– 29]). In this chapter, a simple optimal open-loop and feedforward control approach for nonlinear multivariable systems was introduced. Although feedforward control and open-loop control are well established in a range of applications, there has been limited theoretical analysis of the problem when systems contain nonlinear elements. A NGMV optimal control approach to the problem was described. Classical compensators often involve a pure inverse but even when one is possible, it is not usually desirable, since it suggests very violent control action. The proposed design approach enables more reasonable control action to be achieved that is easily tuned via the control signal costing weighting term. Another advantage is that the effects of disturbances and reference signals are allowed for in the solution, and disturbances are rejected or signals are tracked in an optimal manner. This is an improvement over the classical control design where ad hoc methods are used. If necessary, the results can be extended to include feedback action. However, results are simplified if the feedback and feedforward controls are derived independently. This leads to some lack of optimality, but since the approach is mainly to provide a formal design philosophy, the absolute minimum value of the cost-function is not so important. The assumptions made in the definition of the system reference and disturbance models, and the specification of the cost-index, are all aimed at obtaining a simple control philosophy. However, the plant description is reasonably general. In fact, the nonlinear plant model may even include a Neural Network, providing a natural adaptive control capability. The system models were chosen so that the main polynomial matrix equations to be solved, relating to disturbance and reference models, are all linear. The resulting open-loop or feedforward controller is simple to compute and to implement. A solution for the combined feedforward and tracking problems, where a feedback controller is computed as part of an integrated solution will be discussed in the next chapter [20, 30]. The difficulty with this approach is that the solution obtained is more complicated and it obscures some of the useful features of the feedforward control problem. The approach taken in the current chapter was, therefore, to solve the open-loop and feedforward control problems by assuming no measurement for feedback was directly available.

References

157

References 1. Grimble MJ (1986) Feedback and feedforward LQG controller design. In: American control conference. Seattle, pp 63–69 2. Sternad M, Söderström T (1988) LQG-optimal feedforward regulators. Automatica 24 (4):557–561 3. Hunt KJ (1988) General polynomial solution to the optimal feedback/feedforward stochastic tracking problem. Int J Control 48(3):1057–1073 4. Grimble MJ (1988) Two-degrees of freedom feedback and feedforward optimal control of multivariable stochastic systems. Automatica 24(6):809–817 5. Hai-Ping L, Bongiorno JJ Jr (1993) Wiener-Hoff design of optimal decoupled multivariable feedback control systems. IEEE Trans Autom Control 38(12):1838–1843 6. Marchetti G, Massimiliano B, Jovanovic L, Zisser H, Seborg D (2008) A feedforward-feedback glucose control strategy for type 1 diabetes mellitus. J Process Control 18(2):149–162 7. Thomas B, Soleimani-Mohseni M, Fahlen P (2005) Feedforward in temperature control of buildings. Energy Build 37:755–761 8. Luyben WL, Gester JA (1964) Feedforward control of distillation columns. Ind Eng Chem Process Dev 3(4):374–381 9. Bang J, Agustriyanto R (2001) Inferential feedforward control of a distillation column. In: American control conference. Arlington, pp 25–27 10. Zhu Q, Ma Z, Warwick K (1999) Neural network enhanced generalised minimum variance self-tuning controller for nonlinear discrete-time systems. IEE Proc Control Theory Appl 146 (4):319–326 11. Hussain A, Zayed A, Smith L (2001) A new neural network and pole placement based adaptive composite self-tuning controller. In: 5th IEEE international multi-topic conference. Lahore, pp 267–271 12. Zayed A, Hussain A, Grimble MJ (2006) A non-linear PID-based multiple controller incorporating a multi-layered neural network learning sub-model. Int J Control Intell Syst 34 (3):201–1499 13. Åström KJ (1979) Introduction to stochastic control theory. Academic Press, London 14. Hastings-James R (1970) A linear stochastic controller for regulation of systems with pure time delay. University of Cambridge, Department of Engineering, Research Report No CN/ 70/3 15. Clarke DW, Hastings-James R (1971) Design of digital controllers for randomly disturbed systems. IEE Proc 118(10):1502–1506 16. Kucera V (1979) Discrete linear control. Wiley, Chichester 17. Shaked U (1979) A transfer function approach to the linear discrete stationary filtering and the steady-state discrete optimal control problems. Int J Control 29(2):279–291 18. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 19. Grimble MJ (1994) Robust industrial control. Prentice Hall, Hemel Hempstead 20. Grimble MJ (2005) Non-linear generalised minimum variance feedback, feedforward and tracking control. Automatica 41:957–969 21. Svrcek WY, Mahoney DP, Young BR (2006) A real-time approach to process control. Wiley 22. Smith CA, Corripio AB (1997) Principles and practice of automatic process control. Wiley 23. Marconi L, Naldi R (2007) Robust full degree-of-freedom tracking control of a helicopter. Automatica 43(11):1909–1920 24. Devasia S, Chen D, Paden B (1996) Nonlinear inversion-based output tracking. IEEE Trans Autom Control 41(7):930–942 25. Galinaitis WS, Rogers RC (1998) Control of a hysteretic actuator using inverse hysteresis compensation. In: SPIE conference on mathematics and control in smart structures, vol 33, No 23, pp 267–277

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26. Devasia S (2002) Should model-based inverse input be used as feedforward under plant uncertainty. IEEE Trans Automat Control 47(11):1865–1871 27. Leang KK, Devasia S (2007) Feedback-linearized inverse feedforward for creep, hysteresis, and vibration compensation in AFM piezo actuators. IEEE Trans Control Syst Technol 15(5):927–935 28. Zou Q (2007) Optimal preview-based stable-inversion for output tracking of non-minimum phase linear systems. In: IEEE conference on decision and control. New Orleans, pp 5258–5263 29. Wu Y, Zou Q (2007) Robust-inversion-based 2DOF-control design for output tracking: piezoelectric actuator example. In: IEEE conference on decision and control. New Orleans, pp 2451–2457 30. Gongyou T, Yandong Z, Baolin Z (2006) Feedforward and feedback optimal control for linear time-varying systems with persistent disturbances. J Syst Eng Electron 17(2):350–354

Chapter 4

Nonlinear GMV Feedback Optimal Control

Abstract The nonlinear feedback optimal control problem considered in this chapter is fundamental to many of the results that follow. It uses a polynomial systems description for output and disturbance subsystems and includes a nonlinear black-box operator term to represent nonlinear elements. A generalized minimum variance cost-function is minimized and the resulting feedback controller is shown to have an intuitive well-defined block structure. Later in the chapter, a more general system description is used and feedforward and tracking controllers are included. The important lessons from this chapter are the simple solution strategy and the resulting synthesis approach that provides a practical controller for nonlinear systems.

4.1

Introduction

There has been a need for a practical industrial controller that is simple to understand and to implement, and which may be used on very nonlinear systems. A relatively very effective control design paradigm for the control of nonlinear systems is introduced in this chapter, referred to as Nonlinear Generalized Minimum Variance (NGMV) feedback control. This was built upon a rich heritage including the work of Åström, who introduced the Minimum Variance (MV) controller for industrial applications. As described in Chap. 2 Åström initially assumed the plant was linear and minimum phase but when the plant was non-minimum phase, the solution was clearly not stabilizing, since the controller attempted to cancel unstable poles and zeros. A little later Åström introduced an MV controller for processes that could be non-minimum phase. This would stabilize non-minimum-phase processes but it did require factorization of the numerator of the plant numerator polynomial (in the scalar case). This was summarized in Åström’s 1979 seminal textbook [1]. In a later development Hastings-James and later Clarke and Hastings-James [2], modified the first of these control laws by adding a control signal costing term in the cost-index. This was termed a Generalized Minimum Variance (GMV) control law © Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_4

159

160

4 Nonlinear GMV Feedback Optimal Control

that also enabled non-minimum-phase processes to be stabilized by careful choice of cost-function weightings. However, when the control weighting tended to zero the control law reverted to the first MV algorithm of Åström (unstable on non-minimum-phase systems). The NGMV control law introduced in this chapter is related to this GMV controller. The GMV control law was interesting since it had similar characteristics to Linear Quadratic Gaussian (LQG) design for some types of system and was simpler to implement [3, 4]. This simplicity was exploited in the development of the GMV self-tuning controller [5]. The introduction of dynamic cost-function weightings in the GMV cost-index provided additional design flexibility and this was exploited to derive a Generalized H∞ controller [6, 7], described in Chap. 6. Simple control laws like the NGMV solution that uses a polynomial system description are very suitable for use in adaptive systems. Anbumani et al. [8] developed a polynomial systems approach for nonlinear systems to solve a minimum variance control problem that was used in a self-tuning control algorithm. This was applied to the control of a stochastic nonlinear system described by a Hammerstein model with unknown model parameters. Later Grimble [9] and Grimble and Carr [10], developed a so-called Observations Weighted minimum variance controller, for a restricted class of nonlinear systems, and Hernandez and Arkun [11] developed a controller for nonlinear systems based on Autoregressive Moving Average (ARMA) models.

4.1.1

Strategy and Problem Formulation

The aim in the following is to introduce the GMV controller for nonlinear multivariable processes. It uses dynamic cost-function weightings and is related to the results for linear systems presented in Chap. 2. The structure of the system and the cost-function are chosen so that a simple feedback controller and solution are obtained. As with most of the algorithms introduced in the text, the nonlinear control problems are defined so that in the limiting case of a linear model the results revert to the familiar and well-known linear control law solutions. In the NGMV control law case, when the system is linear the results revert to those for the GMV controller described in Chap. 2 [6]. The nonlinear optimal control problem considered here is defined so that the resulting NGMV control solution is easy to understand and implement. For industrial applications, this was considered a priority even before performance requirements were considered. Black-box model: The plant model can again be in a very general nonlinear operator form, which is unusual for a model-based control approach. This is referred to as the “black-box” model that was described in the previous chapter. This model can include non-smooth and severe static nonlinearities, complex nonlinear dynamic equations, nonlinear state-space equations and transfer-operators [12]. It might represent valves, or a servo-system that has a model in software but not in a known equation form. The optimal solution reveals that the model

4.1 Introduction

161

equations do not need to be known within this subsystem. All that is needed is the ability to compute an output for a given input to this subsystem. With so little information assumed, it is not surprising that this subsystem must be assumed open-loop stable if closed-loop stability is to be ensured. The ability to introduce a very general plant subsystem is an advantage of the method. In fact, on a more philosophical point, it is an unusual example of a model-based control design approach that can incorporate models without a precise known mathematical description. Simplicity and properties: A major benefit is that the solution is very simple, which is a feature that only applies to a few nonlinear control methods such as feedback linearization (Chap. 1 and [13]). However, feedback linearization methods do not provide a general solution for disturbance rejection and tracking. An advantage of the NGMV solution is that it applies to a wider class of problems. For example, the plant model does not need to be affine in the control. Degrees of freedom: The results that are presented first concentrate on a single-degree -of-freedom feedback control problem and the design/implementation issues. This is, of course, a classical feedback controller type of structure where the controller is driven by a tracking error signal. In this case, using a polynomial approach, only one unilateral Diophantine equation must be solved. The result is a nonlinear controller that is straightforward to use in applications. Later in the chapter attention turns to the more general Feedforward and Tracking NGMV control problem, introduced in Grimble [14]. This requires the solution of three bilateral Diophantine equations. In practice, a simple solution would probably be employed for the feedforward control by adding a non-optimal “empirical” feedforward control action.

4.2

System Description

The actual nonlinear plant model will normally be a continuous-time system but one that is to be controlled digitally. The problems of discretization, sample time selection and the selection of anti-aliasing filters will, therefore, arise. However, for current purposes a discrete-time system model will be assumed available, where the sampling interval Ts is assumed to be small enough that the disturbance inputs can be approximated by piecewise constant functions during the sampling interval. A one Degree-of-Freedom (DOF) feedback control system is considered in this first part of the chapter. The input to the controller is formed from the difference of the reference (or setpoint), and the plant output or observations signal. This is illustrated in Fig. 4.1, which shows the multivariable system with nonlinear plant model and linear reference and disturbance models. Attention will turn to the two or three degrees of freedom solution in the second part of the chapter in Sect. 4.5. The system description is of restricted generality so that simple results are obtained but as mentioned the plant can be nonlinear and may have quite a general operator form. The reference and disturbance signals are assumed to have linear

162

4 Nonlinear GMV Feedback Optimal Control

Fig. 4.1 Single-degree-of-freedom feedback control structure

time-invariant model representations. This is often valid since in many applications the stochastic disturbance and reference signals are represented by Linear TimeInvariant (LTI) approximations. In this chapter, these linear subsystems are represented by polynomial matrix models. There is no loss of generality in assuming that the zero-mean white noise sources xðtÞ and nðtÞ have identity covariance matrices, and there is no requirement to specify the distribution of the noise sources, since the structure of the system leads to a prediction equation, which is only dependent upon the linear disturbance and reference models. For stability analysis, the time functions can be considered to be contained in extensions of the discrete Marcinkiewicz space m2 , as discussed in Jukes and Grimble [15] and Grimble et al. [16]. This is the space of time sequences with time-averaged square summable signals, which have finite power. The aim of the nonlinear control design is then to ensure certain input–output maps are finite-gain m2 stable and the cost-index is minimized [17]. Linear plant models: The linear disturbance, reference and plant output models have the left-coprime polynomial matrix representation:



 Wd ðz1 Þ; Wr ðz1 Þ; W0k ðz1 Þ ¼ Aðz1 Þ1 Cd ðz1 Þ; Er ðz1 Þ; B0k ðz1 Þ ð4:1Þ

Nonlinear plant model: The r  m multivariable plant model can be assumed to be of the form: ðWuÞðtÞ ¼ zK0 ðW k uÞðtÞ

ð4:2Þ

where zK0 ¼ diag fzk1 ; zk2 ; . . .; zkr g denotes the block of transport-delay elements in the respective output signal paths and it is assumed k1 [ 0; . . .; kr [ 0.

4.2 System Description

163

A strength of the method is that the model equations need not be known for the nonlinear subsystem ðW k uÞðtÞ.

4.2.1

Plant Equations

In some problems, it is physically more informative to write the nonlinear plant model W in the form of two cascaded subsystems. That is, the plant model can be separated into an input subsystem W 1k and a linear output subsystem W0k . The nonlinearities are then all associated with the input subsystem, whilst the output subsystem is linear. In the case of a multivariable system, this output subsystem often includes the main interactions. If, for example, the plant is open loop unstable, a decomposition into the two subsystems is essential. In this case, the input nonlinear subsystem W 1k is assumed stable. Any unstable modes of the linear plant subsystem are included in a linear time-invariant block of polynomial matrix form W0k ¼ A1 B0k . This approach enables open-loop unstable processes to be stabilized. However, for stable plant models, a breakdown into the two subsystems is not necessary, although it may be more representative of the physical system. The operator forms of the plant model may now be introduced. Delay-free plant model: ðW k uÞðtÞ ¼ W0k ðW 1k uÞðtÞ ¼ A1 B0k ðW 1k uÞðtÞ

ð4:3Þ

Total plant model: ðWuÞðtÞ ¼ zK0 W0k ðW 1k uÞðtÞ

ð4:4Þ

It is assumed that the input subsystem nonlinear model W 1k is finite-gain stable. There is some loss of accuracy in assuming that the reference and disturbance models are represented by linear subsystems but this is normally acceptable. The signals shown in the system model of Fig. 4.1 may be listed as Error signal: eð t Þ ¼ r ð t Þ  yð t Þ

ð4:5Þ

yðtÞ ¼ d ðtÞ þ ðWuÞðtÞ

ð4:6Þ

Plant output:

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4 Nonlinear GMV Feedback Optimal Control

Reference: r ðtÞ ¼ Wr xðtÞ

ð4:7Þ

d ð t Þ ¼ W d nð t Þ

ð4:8Þ

f ðt Þ ¼ r ðt Þ  d ðt Þ

ð4:9Þ

Disturbance signal:

Combined signal:

The power spectrum for the difference of the reference and disturbance signals f (t) = r(t) – d(t): Uff ¼ Urr þ Udd ¼ Wr Wr þ Wd Wd

ð4:10Þ

Define the so-called generalized spectral-factor Yf using Yf Yf ¼ Uff

ð4:11Þ

For simplicity, the reference and disturbance system models are assumed asymptotically stable so that the generalized spectral-factor Yf is strictly minimum phase. The spectral-factor Yf may be written in a polynomial matrix form using the “common denominator” A matrix in (4.1). Thus, we write: Yf ¼ A1 Df

ð4:12Þ

Measurement noise: A measurement noise model has not been included here (to simplify the equations). This is a practical assumption since the control signal cost-function weighting, defined in the next section, can be used to ensure the controller rolls-off at high frequencies.

4.3

Nonlinear Generalized Minimum Variance Control

The NGMV feedback control problem considered in the first part of the chapter will be the simplest, where the total plant operator ðWuÞðtÞ ¼ zK0 ðW k uÞðtÞ is assumed stable and there is no need to consider separate input and output subsystems. The solution only requires knowledge of the total black-box plant model ðWuÞðtÞ, whereas in the second part of the chapter the solution assumes the plant model to be written in the form of the two subsystems in (4.4). This is a more general case, since the output subsystem can be open loop unstable. It is considered later in Sect. 4.5.

4.3 Nonlinear Generalized Minimum Variance Control

165

Weighted error and control: The optimal control problem involves the minimization of the variance of the signal /0 ðtÞ 2 Rm shown in Fig. 4.1. The signal whose variance is to be minimized is defined as /0 ðtÞ ¼ Pc ðz1 ÞeðtÞ þ ðF c uÞðtÞ

ð4:13Þ

As noted, this is a generalization of the so-called GMV controller cost-index and includes an ðm  rÞ dynamic cost-function weighting matrix Pc ðz1 Þ acting on the error signal and an m-square, nonlinear dynamic control signal costing term ðF c uÞðtÞ. This signal can be treated as an inferred output that does not exist in the physical system, but can be constructed. In some applications, the signal /0 ðtÞ may represent the output of a real system that cannot be measured but is to be controlled. For example, in the steel industry the centre temperature of an ingot cannot be measured but the temperature may be inferred from other measurements and can, therefore, be controlled. Error weighting: The error weighting function Pc ðz1 Þ in (4.13) is typically a low-pass transfer-function matrix and it is represented below by linear polynomial 1 1 matrices Pc ðz1 Þ ¼ P1 cd ðz ÞPcn ðz Þ. Control weighting: The control weighting F c in (4.13) is a nonlinear operator, which is normally chosen to be linear and has a high-pass characteristic. If the smallest of the delays in each output channel of the plant has integer values fk1 ; k2 ; . . .; kr g then the control signal at time t affects the jth output at least kj steps later. To allow for delay terms the m-square control signal costing can be defined to be of the form: ðF c uÞðtÞ ¼ zK0 ðF ck uÞðtÞ

ð4:14Þ

The control-weighting operator F ck is assumed to be normal full rank and to be invertible [14]. This weighting will often be chosen to be a linear operator but it may also be nonlinear. It can be used to compensate for the plant input nonlinearities in appropriate cases. Cost-function: In terms of the inferred output signal in (4.13), the cost-function can be defined as       J ¼ E /T0 ðtÞ/0 ðtÞjt ¼ E trace /0 ðtÞ/T0 ðtÞ jt

ð4:15Þ

where Efjtg denotes the conditional expectation operator. The solution to this problem is presented in the following theorem and the proof is given in the sections that follow. Theorem 4.1: NGMV Optimal Controller Assume that the plant model W k is finite gain stable and the nonlinear operator ðPc W k  F ck Þ has a finite gain stable causal inverse, due to the choice of the weighting operators Pc and F c , and assume that the reference and disturbance models are chosen so that the spectral-factor polynomial matrix Df is strictly Schur. The NGMV optimal controller to minimize

166

4 Nonlinear GMV Feedback Optimal Control

the variance of the weighted error and control signals may be computed from the solution of polynomial equations. The smallest degree solution ðG0 ; F0 Þ, with respect to F0 , must be computed from the polynomial matrix unilateral Diophantine equation: Ap Pcd F0 þ zK0 G0 ¼ Pcf Df

ð4:16Þ

The left-coprime polynomial matrices satisfy 1 A1 p Pcf ¼ Pcn A

ð4:17Þ

The NGMV optimal controller should be implemented in a minimal form and the optimal control:   1 1 1 uðtÞ ¼ F 1 ck F0 Yf ðW k uÞðtÞ  ðAp Pcd Þ G0 Yf eðtÞ

ð4:18Þ

An alternative form for the NGMV optimal controller is given as  1 ðAp Pcd Þ1 G0 Yf1 eðtÞ uðtÞ ¼ F0 Yf1 W k  F ck

ð4:19Þ ■

Proof The proof involves collecting the results in the section that follows.

■

Remarks on the solution 1. The solution is simplified if zK0 and the weighting Pc and Yf commute. This assumption is valid if the delay elements are the same in each signal channel zK0 ¼ zk I or if Pc and Yf are diagonal. 2. The class of problems considered are those for which a solution to the Diophantine equation can be found where the row degrees of F0 ðz1 Þ are less than the delay path magnitudes fk1 ; k2 ; . . .; kr g. This is ensured under the conditions listed in the previous remark. 3. The external disturbance and reference signal models are assumed stable subsystems and the controller to be implemented in its minimal form. 4. The assumption was that the explicit delay terms in (4.2) were not zero in any channel. This assumption is necessary for the polynomial solution provided here so that a unique solution to (4.16) can be defined.

4.3 Nonlinear Generalized Minimum Variance Control

4.3.1

167

NGMV Optimal Feedback Control Solution

A simple optimization argument is used in the following that is similar to that in the previous chapter. The signal to be minimized is shown to consist of both linear and nonlinear terms. However, the stochastic part of the problem involves linear models so that a prediction equation may be derived easily. This enables the signal to be written in terms of future and past white noise driven terms. The optimal causal solution is shown to be that which sets the terms depending on the past events to zero. Consider the minimization of the signal /0 ðtÞ that represents the weighted sum of error and control signals. This inferred output /0 ðtÞ ¼ Pc ðz1 ÞeðtÞ þ ðF c uÞðtÞ is the same dimension as the input signal. It is defined in terms of dynamic weightings, where Pc is assumed linear, and F c can be a linear or a nonlinear operator. From the equations in Sect. 4.2 the error e ¼ r  y ¼ r  d  Wu and substituting /0 ðtÞ ¼ Pc ðrðtÞ  dðtÞÞ  ðPc W  F c ÞuðtÞ

ð4:20Þ

Assumption: Recall the assumption that the two models involved in the definition of the signal f ¼ r  d are linear. This enables the spectral-factor of this signal to be computed. Spectral-factor: An innovations signal model may be defined that has the form f ðtÞ ¼ Yf ðz1 ÞeðtÞ, where Yf ðz1 Þ is a linear transfer-function matrix, and eðtÞ denotes a zero-mean white noise signal of identity covariance matrix. This follows from the above spectral-factor computation in terms of the disturbance Wd and the reference Wr signal models. The polynomial matrix form of the spectral-factor Yf ¼ A1 Df , where for stability purposes, it is assumed the system description ensures Df is strictly Schur (rather than simply Schur). Signal to be optimized: From the first term in (4.20): 1 Pc ðrðtÞ  dðtÞÞ ¼ Pc f ðtÞ ¼ P1 cd Pcn A Df eðtÞ

Now introduce the left-coprime matrices Ap and Pcf , where Pcn A1 ¼ A1 p Pcf , and we obtain Pc f ðtÞ ¼ ðAp Pcd Þ1 Pcf Df eðtÞ Thus, from (4.20) the weighted error and control signals: /0 ðtÞ ¼ ðAp Pcd Þ1 Pcf Df eðtÞ  ðPc W  F c ÞuðtÞ

ð4:21Þ

The main linear system equation to be solved is a Diophantine equation [18, 19]. This equation is introduced to expand the combined weighted disturbance and reference model into two groups of terms:

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4 Nonlinear GMV Feedback Optimal Control

Ap Pcd F0 þ zK0 G0 ¼ Pcf Df

ð4:22Þ

where the solution for ðF0 ; G0 Þ satisfies the row j degree of F0 \kj : Hence, Pc Yf ¼ ðAp Pcd Þ1 Pcf Df ¼ F0 þ ðAp Pcd Þ1 zK0 G0

ð4:23Þ

Observe that the first polynomial matrix includes delay elements in the jth channel, up to and including zðkj 1Þ and the last term involves delay elements greater than or equal to kj in each channel. Substituting into (4.21) /0 ðtÞ ¼ F0 eðtÞ þ ðAp Pcd Þ1 zK0 G0 eðtÞ  ðPc W  F c ÞuðtÞ

ð4:24Þ

Writing eðtÞ ¼ Yf1 f ðtÞ ¼ Yf1 ðrðtÞ  dðtÞÞ and substituting in (4.24) /0 ðtÞ ¼ F0 eðtÞ þ ðAp Pcd Þ1 zK0 G0 Yf1 eðtÞ   þ ðAp Pcd Þ1 ðzK0 G0  Ap Pcd Pc Yf ÞYf1 W þ F c uðtÞ

ð4:25Þ

But Pcf ¼ Ap Pcn A1 and Ap Pcd Pc Yf ¼ Ap Pcn A1 Df ¼ Pcf Df , and hence (4.22) gives zK0 G0  Pcf Df ¼ Ap Pcd F0 Weighted error and control signal: Equations (4.23) and (4.25) give the desired inferred output or weighted error and control signals, as    1 /0 ðtÞ ¼ F0 eðtÞ þ Ap Pcd zK0 G0 Yf1 eðtÞ þ F c  F0 Yf1 W uðtÞ

ð4:26Þ

The control signal at time t affects the jth system output at time t + kj and hence the control signal costing term F c should include a delay of kj steps, so that F c ¼ zK0 F ck . Moreover, since in general the control signal costing is required on each signal channel, the F ck weighting is defined to be full rank and invertible. Equation (4.26) may be simplified further if Ap Pcd and zK0 , and also F0 Yf1 and zK0 commute, which is certainly the case under the assumptions on Pc and Yf discussed at the end of the last section. The inferred output may now be written, from (4.26), as /0 ðtÞ ¼ F0 eðtÞ þ zK0 ððF ck uÞðtÞ  F0 Yf1 ðW k uÞðtÞ þ ðAp Pcd Þ1 G0 Yf1 eðtÞÞ

ð4:27Þ

To compute the optimal control signal inspect the form of the weighted error and control signals in (4.27). Since the row degrees of F0 are required to be less than kj

4.3 Nonlinear Generalized Minimum Variance Control

169

(the magnitude of the delay in the jth channel), the jth row of the first term is dependent upon the values of the white noise components eðtÞ,…, eðt  kj þ 1Þ. The remaining terms in the expression for the jth row are all delayed by at least kj steps, and therefore depend upon the earlier values of eðt  kj Þ, eðt  kj  1Þ; … and it follows the first and remaining terms in (4.27) are statistically independent. Condition for optimality: The first term on the right of (4.27) is independent of the control action and hence the smallest variance is achieved when the remaining terms are zero. That is, the condition for optimality becomes   F ck  F0 Yf1 W k uðtÞ þ ðAp Pcd Þ1 G0 Yf1 eðtÞ ¼ 0

ð4:28Þ

Optimal control signal: It follows that the NGMV optimal control signal must satisfy    uðtÞ ¼ F ck  F0 Yf1 W k ðAp Pcd Þ1 G0 Yf1 eðtÞ

ð4:29Þ

or more usefully:   1 1 1 uðtÞ ¼ F 1 ck F0 Yf ðW k uÞðtÞ  ðAp Pcd Þ G0 Yf eðtÞ

ð4:30Þ

Controller structure: This control signal may be generated by the controller shown in the feedback loop of Fig. 4.2. The signal /0 ¼ Pc e þ F c u ¼ Pc ðr  yÞ þ F c u involves a weighting F c that normally has a negative zero-frequency gain. The forward path gain of the controller block is therefore usually positive at low frequencies. There are some subtleties involved with implementing such a feedback controller but these will be discussed in the next chapter.

Fig. 4.2 NGMV control signal generation and controller

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4 Nonlinear GMV Feedback Optimal Control

4.3.2

Non-square Systems

The approach applies to some non-square system problems but care is needed with the cost-weighting design choices, and to ensure the problem is physically realistic. The need for caution arises because the cost-function is not a “sum of squares” form. To illustrate the problem consider the case of a two input and one output linear system, and recall from (4.13) /0 ðtÞ ¼ Pc ðz1 ÞeðtÞ þ ðF c uÞðtÞ. In this case, the variance of the following signal is to be minimized:





/01 ðtÞ Pc11 ðz1 Þ F ðz1 Þ ¼ eðtÞ þ c11 1 /02 ðtÞ Pc22 ðz Þ 0

0 Fc22 ðz1 Þ



u1 ðtÞ u2 ðtÞ

 ð4:31Þ

In a sum of squares type cost-function, only one weighted error term would be involved. In the square-of-sum NGMV cost-function, used here, this suggests treating Pc22 ðz1 Þ in (4.31) as a dummy cost, which can be chosen to be small. This is not as natural as for standard quadratic cost-function problems (GPC or LQG), however, the square-of-sum NGMV cost-function, leads to a simple controller structure for nonlinear systems, which is valuable. If there are more outputs than inputs (r > m) then the simplest option is to choose the dimension of Pc ðz1 Þ to be m  r so that a smaller number of outputs or errors are costed directly. Alternatively, a slightly modified solution can be used with a minor change in some systems with more outputs than inputs. In this case, the control signal cost-weighting F c dimension can be as r  m. From the condition for optimality (4.28): ðF ck uÞðtÞ  F0 Yf1 ðW k uÞðtÞ þ ðAp Pcd Þ1 G0 Yf1 eðtÞ ¼ 0 y To compute the optimal control the existence of a pseudo-inverse F ck for the full-rank control-weighting operator ðF ck uÞðtÞ must be assumed. The computation of this pseudo-inverse will be simple in some problems, where F ck is a constant full-rank matrix or a minimum-phase Z-transfer-function (linear weighting model). The solution then follows as   y uðtÞ ¼ F ck ðAp Pcd Þ1 G0 Yf1 eðtÞ þ F0 Yf1 ðW k uÞðtÞ As in the example in Eq. (4.31) above, the case with less outputs than inputs (m > r) can be solved by an appropriate definition of the m  r error weighting matrix Pc ðz1 Þ. This matrix does not have to be full rank or square. Alternatively, another output to be controlled may be added which may be based on a model rather than a real measurement. For example, a useful generalization is to add a dynamic cost-function weighting term Wu ðz1 Þu0 ðtÞ on the input term to the linear subsystem u0 ðtÞ. This is not the same as the control signal input u(t) to the plant, but it may be useful to limit actuator control activity in a certain frequency range.

4.3 Nonlinear Generalized Minimum Variance Control

4.3.3

171

Stability and Weighting Choice

It was noted in Chap. 2 that for linear systems the stability of a GMV-controlled system is ensured when the combination of a control-weighting function and an error weighted plant model is strictly minimum phase. It will be shown below that for nonlinear systems a related operator equation must have a stable inverse. An alternative expression for the control signal, in terms of the exogenous signals, is first required for the stability analysis. Assume for simplicity that the explicit delays are the same in each signal path zK0 ¼ zk I, for k > 0. Then from the condition for optimality (4.28):   F ck  F0 Yf1 W k uðtÞ þ ðAp Pcd Þ1 G0 Yf1 ðr ðtÞ  d ðtÞ  ðWuÞðtÞÞ ¼ 0  1 Substituting from (4.23) F0 Yf1 þ Ap Pcd zK0 G0 Yf1 ¼ P1 cd Pcn and Pc ¼ P1 P the condition becomes cd cn 

 F ck  F0 Yf1 W k  ðAp Pcd Þ1 G0 Yf1 zk W k uðtÞ þ ðAp Pcd Þ1 G0 Yf1 ðr ðtÞ  d ðtÞÞ ¼ 0

ðF ck  Pc W k ÞuðtÞ þ ðAp Pcd Þ1 G0 Yf1 ðr ðtÞ  d ðtÞÞ ¼ 0

ð4:32Þ

The optimal control, in terms of the exogenous signals, that is needed for the stability analysis follows from Eq. (4.32). There are two alternative forms and both are of interest:   1 1 ð Þ ð Þ ð ð Þ ð Þ Þ uðtÞ ¼ F 1 P W u t  ðA P Þ G Y r t  d t c k p cd 0 ck f 1 1  1 uðtÞ ¼ ðPc W k  F ck Þ Ap Pcd G0 Yf ðr ðtÞ  d ðtÞÞ

ð4:33Þ

The disturbance-free output may be obtained from the last result as:  1 ðW k uÞðtÞ ¼ W k ðPc W k  F ck Þ1 Ap Pcd G0 Yf1 ðr ðtÞ  d ðtÞÞ

ð4:34Þ

 Generalized plant: If the transport delays are the same in each path zk I the operator ðPc W k  F ck Þ in these equations represents the signal path in Fig. 4.1 from the control signal to minus the inferred output signal /0 ðt þ kÞ. The variance of this signal is to be minimized. Recall that in the initial minimum variance control algorithm developed (Chap. 2), the plant must be minimum phase for the controller to be stabilizing. The NGMV problem can be thought of as a minimum variance control problem where the plant model is modified by the weightings to become a new effective plant model zk ðPc W k  F ck Þ. This can be referred to as the generalized plant model for such problems.

172

4 Nonlinear GMV Feedback Optimal Control

Stability and design: The previous two expressions (4.33) and (4.34) can be written in a useful form when the control weighting is linear and written as F ck ¼ Fk ðz1 Þ. It then follows that the optimal control and disturbance-free output may be written as  1  1 uðtÞ ¼ I þ Fk1 Pc W k Fk1 Ap Pcd G0 Yf1 ðr ðtÞ  d ðtÞÞ

ð4:35Þ

 1  1 ðW k uÞðtÞ ¼ W k I þ Fk1 Pc W k Fk1 Ap Pcd G0 Yf1 ðr ðtÞ  d ðtÞÞ

ð4:36Þ

The operator ðI þ Fk1 Pc W k Þ in the above equations is similar to a return-difference matrix equation defined for linear systems. This matrix determines stability, since the zeros of the matrix determine the poles of the closed-loop system. These have a significant effect on the form of the transient response of the system. Design: The next chapter (Sect. 5.2.2) is concerned with the design of optimal nonlinear feedback control systems. It is suggested that one method of selecting the cost-function weightings in the NGMV control problem is by equating the ratio of the weightings to a classical simple controller Fk1 Pc ¼ Kc ðz1 Þ that stabilizes the loop. This is often an existing PID controller and the method is motivated by the similarity of the operator ðI þ Fk1 Pc W k Þ to a return-difference matrix equation. If the ratio of the weightings is chosen in this way, the NGMV controller responses are usually similar to those for a nonlinear plant W k with a feedback controller Fk1 Pc . This choice of weightings is not intended to improve upon a linear controller that stabilizes such a nonlinear system; however, it provides a useful starting point for weighting selection. This is illustrated in the process tank level-control example discussed later in Sect. 4.4.

4.3.4

Stability of the System

Returning to the stability issue consider again the expressions for the control signal and the disturbance-free output, defined by Eqs. (4.33) and (4.34). These represent the closed-loop expressions given in terms of the exogenous inputs. A necessary condition for stability is clearly the existence of a finite-gain stable inverse for the nonlinear operator ðPc W k  F ck Þ. As mentioned above for a linear system, the requirement is that the equivalent linear operator should be strictly minimum phase. In fact by reference to the results in Chap. 2, it may be shown that the stability of the closed-loop system with the GMV controller depends upon the zeros of the polynomial ðPcn Fcd Bk  Fcnk Pcd AÞ, and these depend on the cost-function weightings [6]. The choice of weightings to ensure the nonlinear operator ðPc W k  F ck Þ has a stable inverse is not so clear, although one method was mentioned briefly in the

4.3 Nonlinear Generalized Minimum Variance Control

173

previous section. However, the necessary condition for stability can be stated in terms of this generalized plant operator ðPc W k  F ck Þ being minimum phase. An assumption made in this first part of the chapter was that the plant model W k absorbs both linear and nonlinear subsystems and is stable. This assumption and the existence of a finite-gain stable causal inverse of the nonlinear operator ðPc W k  F ck Þ ensures the control action and output is that for a stable system. One of the restrictions on the choice of cost-function weightings is that this “generalized plant” stability condition be fulfilled. The zeros of a system are often considered a linear system concept but zero dynamics can be defined to be the internal dynamics of the system when the system output is kept at zero by the input. Similar to the linear case, a nonlinear system whose zero dynamics are asymptotically stable can be called a minimum-phase system [20]. Adopting this terminology the necessary condition for stability is that the generalized plant ðPc W k  F ck Þ be minimum phase. For stability, the controller must not involve the cancellation of any unstable modes that in the more general case considered later can be present in the linear subsystem W0k ðz1 Þ. In the present discussion, the full plant operator W k was assumed stable so that any linear block W0k ðz1 Þ in the plant can be assumed stable. The cascaded controller subsystem ðAp Pcd Þ1 G0 Yf1 in Eq. (4.18) or (4.19) does not, therefore, contain unstable modes. However, when the linear subsystem is unstable then this controller block must be implemented in its minimal form, otherwise, the control in (4.29) will attempt an unstable pole-zero cancellation. To provide further insights into stability the nonlinear plant can be assumed to have a nonlinear operator polynomial form. This unrealistic but provides a useful example. The system output, without the disturbance signal, is denoted as mðtÞ ¼ ðWuÞðtÞ that includes a possible black-box subsystem. To show the output is stable a further assumption on the system structure will be made. Assume that the open-loop plant is “stable” and has a polynomial operator representation, which may be written as W k ¼ Bk1 A1 k1 , where the matrix-fraction terms involve nonlinear functions Grimble [14]. Consider the input–output response: 1 hðtÞ ¼ ðPc W k  F ck Þ1 pðtÞ ¼ ðPc Bk1 A1 k1  F ck Þ pðtÞ

so that that may be written as pðtÞ ¼ ðPc Bk1  F ck Ak1 ÞA1 k1 hðtÞ, hðtÞ ¼ Ak1 ðPc Bk1  F ck Ak1 Þ1 pðtÞ, where under the assumption the stable operator ðPc W k  F ck Þ1 ¼ Ak1 ðPc Bk1  F ck Ak1 Þ1 . Since the open loop plant is stable, it follows that ðPc Bk1  F ck Ak1 Þ1 is a stable operator. The output (4.34) simplifies as  1  1 W k uðtÞ ¼ Bk1 Pc Bk1  F ck Ak1 Ap Pcd G0 Yf1 ðr ðtÞ  d ðtÞÞ

174

4 Nonlinear GMV Feedback Optimal Control

It follows that under the assumptions on the closed-loop operator and the minimal realization of the controller; both the plant output and control signal expressions contain only stable operators. The operator ðPc W k  F ck Þ representing the generalized plant must, of course, be designed to be stably invertible, and this requires a careful choice of cost weightings, which is a design issue discussed further in Chap. 5. These stability results provide some comfort that the issue has been considered for this class of algorithms but as with other results in later chapters from a practical point of view, they are not so helpful. The more pressing problem is to obtain a design procedure to find the cost-function weightings that will ensure stability for the full range of operation. This problem is addressed in the following chapter but is not completely solved. Guarantees on stability for a real application are elusive and best practice suggests the use of extensive simulation testing to validate a design. Robustness: The expressions for the system responses (4.33) and (4.34) or (4.35) and (4.36), are for the ideal case when the prediction is perfect from the models. There will inevitably be a mismatch and the return-difference operator will include the controller and the “actual” plant W model including explicit delay terms. Robustness issues are considered again in Chap. 5 (Sect. 5.6).

4.3.5

Asymptotic Conditions and Inverse Plant

The asymptotic behaviour is interesting since the controller behaves like an idealized classical solution. As the control signal costing tends to zero the internal loop within the controller in Fig. 4.2 tends to a high-gain feedback loop. Providing the stability of this block is maintained the result is the insertion of an inverse of the plant model nonlinearity (consider the limiting case F ck ! 0 of the expression (4.34)). The controller, therefore, does what would be expected. If the control weightings are small, signifying large control actions are permissible; an inverse plant characteristic is introduced. This does, of course, depend upon whether the inverse actually exists for small control weightings. Similar ideas arise in classical open loop or feedforward control design, where approximate inverse plant models are assumed. When the control action is limited (by increasing the control signal costing), the inverse which occurs is naturally a detuned inverse solution.

4.3.6

Benchmarking Nonlinear Controllers

All the major Supervisory Control And Data Acquisition (SCADA) control equipment manufacturers now include techniques for controller benchmarking in their systems [21]. The methods are normally based on linear system models as described in Chap. 2 (Sect. 2.6). An expression for the minimum cost is obtained below that can be used to provide a method for benchmarking nonlinear controllers.

4.3 Nonlinear Generalized Minimum Variance Control

175

Assume for simplicity that  the transport delays are the same in each channel of magnitude k zK0 ¼ zk . From Eq. (4.16), the polynomial F0 will have the order k – 1 and be of the form: F0 ¼ f0 þ f1 z1 þ    þ fk1 zðk1Þ Thence, from (4.27): /0 ðtÞ ¼ F0 eðtÞ þ zk fðF ck  F0 Yf1 W k ÞuðtÞ þ ðAp Pcd Þ1 G0 Yf1 eðtÞg

ð4:37Þ

At the optimum, the term within the braces in (4.37) is null and /0 min ðtÞ ¼ F0 ðz1 ÞeðtÞ. The minimum cost follows as k1   X Jmin ¼ E ðF0 eðtÞÞT ðF0 eðtÞÞjt ¼ fjT fj

ð4:38Þ

j¼0

The expression (4.38) for the minimum cost provides a benchmark for nonlinear control. It depends only on the reference and disturbance signal models that are assumed Linear Time-Invariant (LTI). This arises because the control action effectively removes the nonlinear plant model from the prediction of the inferred output signal /0 ðtÞ, whose variance is being minimized. It is important to recognize that this does not suggest the controller linearizes the feedback loop but it demonstrates that the path from the disturbance and reference inputs to the fictitious output /0 ðtÞ becomes linear when the optimal control action is applied. Despite these rather curious properties, if an NGMV controller can be tuned to have good properties for the application, then it can provide a benchmark minimum cost by which other controllers might be judged.

4.4

Process Tank Level Nonlinear Control

Buffer tanks with liquid level control are widely used in the process industries to prevent upstream flows from upsetting the downstream processes. Liquid-level control is one of the most common control problems in a pharmaceutical, chemical, petrochemical, biological/biochemical and power plant. For example, liquid level is important in heat exchanger control, where heat and mass transfer rates can be controlled using the amount of liquid covering the tubes. It is also important in distillation columns, mixing tanks, surge tanks, condensers and steam drums. In the example that follows, the NGMV control design procedure is applied to the control of the fluid level in a process tank. Consider the control of liquid level h(t) in the spherical tank shown in Fig. 4.3. Spherical tanks arise in applications in

176 Fig. 4.3 Nonlinear fluid level control in a spherical tank problem

4 Nonlinear GMV Feedback Optimal Control

q (t) in

h(t)

q (t) out

gas plants and many other processes. Control of the level is more challenging because of the nonlinearity associated with the change in cross section with level. The control input to the tank is the inflow rate qin ðtÞ. The outflow from the tank depends on the head level and the pipe dimensions. The differential equation describing the dynamic behaviour of this physical system is given by AðhÞ

pffiffiffiffiffiffiffiffi dh ¼ qin  a 2gh dt

ð4:39Þ

where the cross-sectional area of the tank is defined as AðhÞ ¼ ph2 , and a is the cross-sectional area of the outlet pipe. The level required is set to hð0Þ ¼ 0:1 and the aim is to regulate the level to a given setpoint. As explained above the control of liquid level in a spherical tank involves a nonlinear process. However, the system is self-regulating and stable. The behaviour of the plant can be understood by looking at the open loop response to a series of steps in the control input, as shown in Fig. 4.4. It is clear from the open loop responses that the system is both nonlinear and stable. Stability is a necessary condition for the NGMV controller if the total plant in this problem is to be included in the black-box subsystem ðW k uÞðtÞ. Discrete-time plant model: A discrete-time model of the system is required, since the controller is to be implemented digitally. Such a model may be obtained from data by nonlinear system identification techniques, or estimated as a neural network. However, if a physical model of the system is available the model can be discretized directly: ht þ 1 ¼ ht þ

pffiffiffiffiffiffiffiffiffi Ts ðut  a 2ght Þ 2 pht

ð4:40Þ

where Ts is the sample time in seconds. Observe that the discretization process introduces a single step sample delay into the system, and since no other explicit transport delay is assumed, let the delay k = 1 in the NGMV control design. The choice of a near-integrator for the reference model is common, since such a model represents random step changes in the reference signal. On the other hand, the disturbance is usually modelled by a low-pass filter, based on the expected/identified

4.4 Process Tank Level Nonlinear Control

177

Tank open-loop response

1 0.9 0.8

tank level

0.7 0.6 0.5 0.4 0.3 0.2 0.1

inflow rate

0 0

2

4

6

8

10

12

time [sec]

Fig. 4.4 Open-loop step responses

noise spectrum. Since the “true” model is often hard to ascertain the parameters in the disturbance model Wd may be considered tuning parameters. The linear system models for the reference and disturbance signals can be assumed to have the form: Wr ðz1 Þ ¼

4.4.1

1 ; 1  0:999z1

Wd ðz1 Þ ¼

0:01 1  0:5z1

ð4:41Þ

NGMV Feedback Control Design Results

Assume that the plant is controlled by the nominal discrete-time PI controller Kc ðz1 Þ ¼ Kp þ Ki =ð1  z1 Þ, with the tuning parameters Kp ¼ 5 and Ki ¼ 4, and the sample time Ts ¼ 0:01 seconds. As discussed in Sect. 4.3 and in the following Chapter (Sect. 5.2.2), the initial choice of dynamic cost-function weightings for the control design may be defined in terms of this controller by defining Pc ðz1 Þ ¼ Kc ðz1 Þ and a linear control weighting F ck ðz1 Þ ¼ Fck ðz1 Þ ¼ 1. The frequency responses of these weightings are shown in Fig. 4.5. The presence of the large low-frequency gain in the error weighting will provide near integral action in the controller to ensure almost zero steady-state error in the step responses. Computation of the NGMV controller: The cost-function weightings implied by the existing PI controller, in polynomial form, become Pcn ðz1 Þ ¼ 5  4:96z1 ,

178

4 Nonlinear GMV Feedback Optimal Control Bode Diagram 80 70

Pc

Magnitude (dB)

60 50 40

Fnom ck

30 20

Flead ck

10 0 -10 10 -4

10 -2

10 0

10 2

Frequency (rad/s)

Fig. 4.5 Frequency responses of the cost function weightings

Pcd ðz1 Þ ¼ 1  z1 , Fck ¼ 1. The Diophantine equation (4.22) has the solution (in the scalar case Ap ¼ A and Pcf ¼ Pcd ): F0 ¼ 5:0

and

G0 ¼ 5:04  7:51z1 þ 2:50z2

Also compute the solution of the spectral-factorization problem (4.12), Df ¼ 1  0:5z1 . The expression for the optimal control law may now be found from (4.30) as    1 1 1 uðtÞ ¼ F 1 F Y ð W u Þ ð t Þ  A P G Y e ð t Þ 0 k p cd 0 ck f f The linear dynamic blocks in Fig. 4.2, that act like cascade and feedback compensators, can be computed as Ccascade ðz1 Þ ¼ ðAp Pcd Þ1 G0 Yf1 ¼ ð5:035  7:512z1 þ 2:497z2 Þ=ð1  1:5z1 þ 0:5z2 Þ Cloop ðz1 Þ ¼ F0 Yf1 ¼ ð5  7:495z1 þ 2:498z2 Þ=ð1  0:5z1 Þ Recall in the above control expression W k denotes the system model (4.40) without the explicit time-delay term. This needs to be taken into account in the implementation stage, since an algebraic loop is introduced. The way that this can be addressed is explained in Chap. 5 (Sect. 5.4) that follows.

4.4 Process Tank Level Nonlinear Control

179

Liquid level

1

0.5

0

set-point PID NGMV

0

2

4

6

8

10

12

8

10

12

inflow rate

2

1

0

-1

0

2

4

6

time [sec]

Fig. 4.6 Time responses for nominal PID and NGMV controllers

Results: The reference tracking of a sequence of steps for the two nominal controllers is shown in Fig. 4.6. Observe that the nominal PI tuning parameters have not been optimized in any sense. However, this controller is useful in that it can provide “initial” design parameters for the NGMV controller that will stabilize the closed-loop plant, i.e. make the nonlinear operator stable and invertible. As can be seen from Fig. 4.6, the performance of the initial nonlinear controller design is close to that of the original PI controller and this will usually be the case. As explained in Sect. 4.3.3 choosing the weightings in this way will lead to similar results to a PI controlled nonlinear plant. However, the benefit of using the NGMV controller is that the weighting functions can then be “retuned” to maximize performance. The resulting NGMV controller includes the nonlinear plant model knowledge and the cost weightings are dynamic. In fact, the control weighting can also be nonlinear (possibly scheduled). There is, therefore, a greater likelihood that performance can be improved, without destabilizing the system, than is possible using a basic PID controller. This procedure provides a straightforward way to obtain an initial choice of the cost-function weightings. In this case, the weighting can be parameterized as a linear lead term of the form F ck ¼ Fck ¼ qð1  cz1 Þ (also shown in Fig. 4.5). The scalar q is positive and c is a scalar c 2 ½0; 1. Such a parameterization is useful since it reduces the high-frequency gain of the controller. The parameterization of the weightings involves only two tuning parameters, greatly simplifying the design. For the nominal design q ¼ 1 and c ¼ 0. Decreasing the value of q (reducing the control

180

4 Nonlinear GMV Feedback Optimal Control Liquid level

1

= 0.1 = 0.5 =1 = 2.5

0.5

0

2

3

4

5

6

7

8

9

10

11

12

8

9

10

11

12

Inflow rate

10

5

0

-5

2

3

4

5

6

7

time [sec]

Fig. 4.7 Time responses for q ¼ 0:1; 0:5; 1; 2:5

weighting) leads to a faster response and a more violent control action, which can often be improved using a lead term. Figure 4.7 shows the simulation results for different values of q. Increasing q results in a slower response providing a simple tuning mechanism. Comparison with PID: It seems relatively easy to obtain an NGMV control design very close (possibly better) than an existing PI/PID control design using the above procedure. It is shown in Chap. 5 (Sect. 5.2) that if there exists say a PID controller that will stabilize the nonlinear system, without transport delay elements, then a set of cost weightings can be defined to guarantee the existence of this inverse and thereby ensure the stability of the closed-loop. The NGMV controller can then be “retuned” to improve performance and achieve the type of step responses needed. The approach is not in competition with PID control of course. There is every reason in all applications to use the simplest possible controller. The NGMV controller has the advantage that it may be used when the plant is high-order, multivariable, nonlinear and contains difficult dynamics. In such cases, a stabilizing PID control law may not exist. As the controller depends upon the nonlinear model of the plant, it should be more robust against changes of the operating point, whereas any linear controller may have problems regulating across the complete operating range [22].

4.5 Feedback, Tracking and Feedforward Control

4.5

181

Feedback, Tracking and Feedforward Control

Attention now turns to the derivation of a control law for nonlinear systems with a more general controller structure (several degrees of freedom). The nonlinear system is again defined so that a simple feedback controller structure and solution are obtained. However, as described in Chap. 3, if the plant has an additional measurement, to enable some components of the disturbance to be estimated, then feedforward control may be included. In this second part of the chapter, separate tracking and feedforward controllers are added to the system. The solution for the feedback optimal control law is again obtained in the time-domain using a nonlinear operator representation of the process. The plant model is assumed separated into linear and nonlinear subsystem terms. There is some loss of generality in assuming the reference and disturbance models are represented by linear subsystems but this is not important in many practical applications. Nonlinear System Description: As for the NGMV control problem above, the plant model can be in a nonlinear operator form. The plant model W is assumed multivariable ðr  mÞ and is subject to both measurable and unmeasurable disturbances [23]. This more general system model is shown in Fig. 4.8. Systems of this type have separable nonlinearities that may be identified using well-established nonlinear estimation techniques. The input subsystem is black-box and is represented by the nonlinear operator W 1k . However, it will be assumed that any unstable modes of the plant are included in a stable/unstable linear time-invariant output subsystem W0k . The reference, setpoint and disturbance signals are assumed

Fig. 4.8 System models and measurable and unmeasurable disturbances

182

4 Nonlinear GMV Feedback Optimal Control

to have linear time-invariant model representations, which is not very restrictive, since in many applications the models for the disturbance and reference signals are linear time-invariant (LTI) model approximations. Disturbance measurement: The measurement of the disturbance, used for feedforward control, is shown in Fig. 4.9. The measurement yf ðtÞ is not assumed to be the same as the disturbance on the plant output d1(t). The disturbance models are assumed to be such that only an estimate of this signal d1 ðtÞ can be obtained from the measured disturbance component. The measured disturbance model in Fig. 4.9 is therefore separated into the two components Wd1 ¼ W d11 Wd12 , to illustrate the difference between the signal that is measured and the disturbance signal that affects the plant outputs. Signals: The reference or setpoint signal is again denoted rðtÞ. There is no loss of generality in assuming that the zero-mean white noise sources shown in Fig. 4.10 (denoted xðtÞ, fðtÞ and nðtÞ) have identity covariance matrices. A measurement noise model has not been included so that the equations are simplified. This is reasonable so long as the control signal cost-function weighting is chosen to ensure the feedback controller rolls-off at high frequencies. The exogenous signals may include non-zero mean bias signals [14]. Control strategy: Reference tracking is included, using the so-called two and half degrees of freedom tracking controller structure shown in Fig. 4.10, [6]. In this structure, the reference enters both a feedback controller element C0 ðz1 Þ and the tracking controller C1 ðz1 Þ. Recall that,

Fig. 4.9 Nonlinear plant and disturbance model showing feedforward subsystems

4.5 Feedback, Tracking and Feedforward Control

183

Fig. 4.10 Closed-loop feedback control system for the nonlinear plant

1. A one degree of freedom controller gives a zero steady-state error to a reference step change when an integrator is employed in the feedback controller. 2. A two degrees of freedom control structure provides a separate choice of tracking and feedback controller, and hence a more flexible design facility. The structure in Fig. 4.10 provides the same features and benefits as both the one-degree of freedom controller and the two-degrees of freedom controller structure. The feedforward action is introduced in the optimisation problem by adding the measurement of the disturbance signal. The compensator shown in Fig. 4.10 also includes a feedforward of the measurable disturbance through the controller term C2 ðz1 Þ.

4.5.1

Linear and Nonlinear Subsystem Models

The linear subsystems will again be represented by polynomial matrix models. The disturbance, setpoint and linear plant subsystem models have the following left-coprime polynomial matrix representation: ½Wd0 ðz1 Þ; Wd1 ðz1 Þ; Wr ðz1 Þ; W0k ðz1 Þ ¼ A1 ðz1 Þ½Cd0 ðz1 Þ; Cd1 ðz1 Þ; Er ðz1 Þ; B0k ðz1 Þ

ð4:42Þ

Unmeasurable disturbance: Wd0 ðz1 Þ ¼ A1 ðz1 ÞCd0 ðz1 Þ

ð4:43Þ

184

4 Nonlinear GMV Feedback Optimal Control

Measurable disturbance: Wd1 ðz1 Þ ¼ A1 ðz1 ÞCd1 ðz1 Þ

ð4:44Þ

Wr ðz1 Þ ¼ A1 ðz1 Þ Er ðz1 Þ

ð4:45Þ

Reference model:

Without loss of generality, these models again have the common denominator polynomial matrix Aðz1 Þ. Nonlinear plant model: The total plant model, including linear and nonlinear subsystems: ðWuÞðtÞ ¼ zk W0k ðW 1k uÞðtÞ

ð4:46Þ

where the plant model without the explicit delay term: ðW k uÞðtÞ ¼ W0k ðW 1k uÞðtÞ

ð4:47Þ

The nonlinear, possibly time-varying, elements are assumed included in the input subsystem W 1k . Saturation in electrical machines and nonlinearities in process control valves can be included in the input channels. This may often be a diagonal matrix of static nonlinearities. The output subsystem W0k can represent the multivariable plant dynamics and interactions in the process. For simplicity the nonlinear subsystem W 1k is finite-gain stable but the linear subsystem W0k is allowed to contain unstable modes. Delay model structure: To simplify the analysis assume the delay model is a simple diagonal matrix of channel delay elements: zk I ¼ diagfzk ; zk ; . . .; zk g

ð4:48Þ

This matrix represents the main delay elements in the output signal paths. Spectral-factor: The power spectrum for the combined reference and disturbance signal f ¼ r  d0  d1 can be calculated, noting these are linear subsystems, using:   þ Wd1 Wd1 Uff ¼ Urr þ Ud0d0 þ Ud1d1 ¼ Wr Wr þ Wd0 Wd0

ð4:49Þ

The generalized spectral-factor Y f may be computed to satisfy Yf Yf ¼ Uff and the system models are assumed to be such that the spectral-factor Y f is strictly minimum phase. The spectral-factor Yf is written in polynomial matrix form as: Yf ¼ A1 Df

ð4:50Þ

4.5 Feedback, Tracking and Feedforward Control

185

The signals that are shown in the above system diagrams, Figs. 4.8, 4.9 and 4.10, may be listed as follows: Tracking error signal: eð t Þ ¼ r ð t Þ  yð t Þ

ð4:51Þ

yðtÞ ¼ d ðtÞ þ ðWuÞðtÞ

ð4:52Þ

r ðtÞ ¼ Wr xðtÞ

ð4:53Þ

d0 ðtÞ ¼ Wd0 nðtÞ

ð4:54Þ

d1 ðtÞ ¼ Wd11 d2 ðtÞ and d2 ðtÞ ¼ Wd12 fðtÞ

ð4:55Þ

Plant output:

Reference:

Unmeasurable disturbance:

Measurable disturbances:

Total disturbance: dðtÞ ¼ d0 ðtÞ þ d1 ðtÞ

ð4:56Þ

yf ðtÞ ¼ Wd2 d2 ðtÞ

ð4:57Þ

f ðtÞ ¼ r ðtÞ  d0 ðtÞ  d1 ðtÞ

ð4:58Þ

Feedforward signal:

Combined signal:

Control signal components: uðtÞ ¼ ðC0 eÞðtÞ þ C1 ðz1 ÞrðtÞ þ C2 ðz1 Þyf ðtÞ

ð4:59Þ

The feedback controller input e ðtÞ ¼ rðtÞ  yðtÞ ¼ f ðtÞ  ðWuÞðtÞ and the feedback component uc0 ðtÞ ¼ ðC0 eÞðtÞ. The tracking control signal uc1 ðtÞ ¼ C1 ðz1 ÞrðtÞ and the feedforward control uc2 ðtÞ ¼ C2 ðz1 Þyf ðtÞ. Disturbance modelling: The assumption that the disturbances can be represented by linear models does not affect the stability properties, but may cause a degree of sub-optimality in disturbance rejection. Disturbance models are often determined by nonlinear power spectrum models but can easily be approximated by a linear system driven by white noise. A typical example of a linear disturbance model

186

4 Nonlinear GMV Feedback Optimal Control

arises in ship positioning systems for oil rig drilling vessels, covered in Chap. 15. These include feedback from measured position and wind feedforward action. The wave disturbance motion is modelled by a nonlinear sea spectrum for simulation purposes (usually the Pierson–Moskowitz spectrum escribed in [24]). However, for the control system design the wave motion is normally approximated by a lightly damped second or fourth-order linear system, driven by white noise.

4.5.2

Nonlinear GMV Control Problem with Tracking and Feedforward

It will be assumed here that the tracking C1 ðz1 Þ and feedforward C2 ðz1 Þ controllers shown in Fig. 4.10 are based upon classical design or some other design method. This is unlike previous work where all three controllers including the feedback, feedforward and tracking controllers were chosen optimally Grimble [25]. The reason for this assumption is that the full optimal solution results in high-order tracking and feedforward controllers, which is often unnecessary [6]. The feedback controller must of course compensate for steady-state tracking errors, ensure adequate stability margins and provide a degree of performance and disturbance rejection robustness. A high-order feedback controller may therefore be needed. However, simple low order feedforward and tracking controllers can normally provide reasonable performance in conjunction with the feedback control. Note that if the system is reduced to one DOF and without measurable disturbances, the results obtained will be similar to those for the NGMV controller introduced in the first part of the chapter. The optimal NGMV control problem involves the minimisation of the variance of the following signal: /0 ðtÞ ¼ Pc eðtÞ þ ðF c uc0 ÞðtÞ

ð4:60Þ

This includes a dynamic cost-function weighting on the error signal, represented by a linear polynomial matrix fraction Pc ¼ P1 cd Pcn , and a nonlinear dynamic control costing operator term ðF c uc0 ÞðtÞ, where uc0 ðtÞ represents the component of control action due to the feedback controller. The signal is to be minimized in a variance sense, so that the cost-index to be minimized:

      J ¼ E /T0 ðtÞ/0 ðtÞ t ¼ E trace /0 ðtÞ/T0 ðtÞ t

ð4:61Þ

where Ef:jtg again denotes the conditional expectation operator and the control signal costing must include the delay and is defined to have the form:

4.5 Feedback, Tracking and Feedforward Control

ðF c uc0 ÞðtÞ ¼ zk ðF ck uc0 ÞðtÞ

187

ð4:62Þ

As in the first part of the chapter the assumption is made that the nonlinear, possibly time-varying operator ðPc W k  F ck Þ has a finite-gain stable causal inverse, due to the choice of weighting operators Pc and F c . This is the so-called generalized plant for the problem and the assumption is a necessary condition for stability. Theorem 4.2: NGMV Feedforward and Feedback Optimal Controller The NGMV optimal controller to minimize the variance of the weighted error and control signals may be computed from the following Diophantine and spectral-factor equations. The smallest degree polynomial solution (G0, F0), with respect to F0, must be computed from the polynomial matrix equation: Ap Pcd F0 þ zk G0 ¼ Pcf Df

ð4:63Þ

1 where the left-coprime polynomial matrices Ap and Pcf satisfy A1 p Pcf ¼ Pcn A and the spectral-factor Yf is written in the polynomial matrix form Yf ¼ A1 Df . The matrix zk I denotes a diagonal matrix of delay elements in output signal channels and the degrees of the polynomial matrices F0 are less than k. Nonlinear Optimal Control: The nonlinear optimal control signal may be computed as:

  1 1 1 uc0 ðtÞ ¼ F 1 ck F0 Df B0k ðW 1k uÞðtÞ  ðAp Pcd Þ G0 Yf eðtÞ where the total control uðtÞ ¼ uc0 ðtÞ þ C1 ðz1 ÞrðtÞ þ C2 ðz1 Þyf ðtÞ.

ð4:64Þ ■

Remark The NGMV feedback control defined above is chosen to minimize the cost-index but note the tracking and feedforward controls are assumed given; possibly designed by a classical method to simplify the solution.

4.5.3

Solution for Feedback, Tracking and Feedforward Controls

Consider the minimisation of the variance of the inferred output signal: /0 ðtÞ ¼ Pc eðtÞ þ ðF c uc0 ÞðtÞ

ð4:65Þ

where Pc is assumed linear, and F c can be a linear or nonlinear operator. Noting the error signal, from Eqs. (4.51) and (4.52):

188

4 Nonlinear GMV Feedback Optimal Control

/0 ðtÞ ¼ Pc ðr  d  WuÞ þ F c uc0 ¼ Pc ðr  dÞ  Pc ðWuÞ þ ðF c uc0 Þ

ð4:66Þ

The signal f ¼ r  d may be modelled as f ðtÞ ¼ Yf eðtÞ, where eðtÞ denotes a zero-mean white noise innovations sequence, with identity covariance matrix. The inferred output may therefore be written, substituting for the reference and disturbance models, as: /0 ðtÞ ¼ Pc Yf eðtÞ  Pc WuðtÞ þ F c uc0 ðtÞ

ð4:67Þ

The spectral-factor Yf will be assumed to have a polynomial matrix form (defined in (4.50)), where the system description ensures Df is strictly Schur. The first 1 term in (4.67), therefore, involves the term Pc Yf eðtÞ¼ P1 cd Pcn A Df eðtÞ. Introducing the left-coprime polynomial matrices Ap and Pcf satisfying Pcn A1 ¼ 1 A1 p Pcf gives Pc f ðtÞ ¼ ðAp Pcd Þ Pcf Df eðtÞ. Then, from (4.67):

/0 ðtÞ ¼ ðAp Pcd Þ1 Pcf Df eðtÞ  Pc WuðtÞ þ F c uc0 ðtÞ

ð4:68Þ

Before proceeding with the solution this equation must be simplified using Diophantine equations and these are introduce in the next section.

4.5.4

Introduction of the Diophantine Equations

It is an important feature of the problem description that the reference and disturbance subsystems are approximated by linear subsystems. Many of the resulting equations to be solved are therefore linear. The first Diophantine equation is used to expand the linear terms driven by white noise into two groups, depending upon the output channel delay. That is, the combined disturbance and setpoint model can be expanded using the Diophantine equation (4.63): Ap Pcd F0 þ zk G0 ¼ Pcf Df

ð4:69Þ

where the solution for ðF0 ; G0 Þ satisfies the condition degðF0 Þ\k and hence: 

Ap Pcd

1

 1 Pcf Df ¼ F0 þ Ap Pcd zk G0

ð4:70Þ

The polynomial operator matrix F0 includes delay elements up to and including zðk1Þ , and the last term involves delay elements greater than or equal to k. Substituting for Eq. (4.70), obtain the inferred output (4.68), separated into terms of order up to zðk1Þ and of order zk and greater, as:

4.5 Feedback, Tracking and Feedforward Control

/0 ðtÞ ¼ F0 eðtÞ þ zk ðAp Pcd Þ1 G0 eðtÞ  Pc WuðtÞ þ F c uc0 ðtÞ

189

ð4:71Þ

Recall, from the models in Sect. 4.5, the innovations signal model Yf eðtÞ ¼ rðtÞ  dðtÞ or eðtÞ ¼ Yf1 ðrðtÞ  dðtÞÞ. Substituting into Eq. (4.71): /0 ðtÞ ¼ F0 eðtÞ  Pc WuðtÞ þ F c uc0 ðtÞ þ zk ðAp Pcd Þ1 G0 Yf1 ðrðtÞ  dðtÞÞ ð4:72Þ The control signal at time t affects the system output at time (t + k) and hence the control signal costing term F c should include a delay of k steps, so that F c ¼ zk F ck : The control signal costing is required on each signal channel and the F ck weighting should be defined to be full rank and invertible. Thence, from (4.72): /0 ðtÞ

¼ F0 eðtÞ  Pc WuðtÞ þ F c uc0 ðtÞ þ zk ðAp Pcd Þ1 G0 Yf1 ðrðtÞ  yðtÞ þ WuðtÞÞ ¼ F0 eðtÞ þ zk ðAp Pcd Þ1 G0 Yf1 ðrðtÞ  yðtÞÞ þ zk ððAp Pcd Þ1 G0 Yf1 zk  Pc ÞW k uðtÞ þ F ck uc0 ðtÞÞ

ð4:73Þ Two of the terms in (4.73) may be combined. Recalling (4.47),  1  k ðAp Pcd Þ1 G0 Yf1 zk  Pc ¼ Ap Pcd z G0  Ap Pcd Pc Yf Yf1 Noting Pcn A1 ¼ A1 p Pcf and (4.69) the following result is obtained: 

   1  k ðAp Pcd Þ1 G0 Yf1 zk  Pc ¼ Ap Pcd z G0  Pcf Df Yf1 ¼ F0 Yf1

Substituting in (4.73): /0 ðtÞ ¼ F0 eðtÞ þ zk ððAp Pcd Þ1 G0 Yf1 ðrðtÞ  yðtÞÞ  F0 Yf1 W k uðtÞ þ F ck uc0 ðtÞÞ

4.5.5

ð4:74Þ

Optimization and Benchmarking

To compute the optimal control signal observe that the weighted error and control signals may be written, using (4.74) and noting Yf1 W k ¼ D1 f B1k W 1k in the k step-ahead form:

190

4 Nonlinear GMV Feedback Optimal Control

/0 ðt þ kÞ ¼ F0 eðt þ kÞ  þ ðAp Pcd Þ G0 Yf1 eðtÞ  F0 D1 f B0k W 1k uðtÞ þ F ck uc0 ðtÞ 

1

ð4:75Þ

Since deg ðF0 Þ < k, the first term is dependent upon the future values of the white noise signal components eðt þ kÞ; . . .; eðt þ 1Þ. The remaining terms are dependent upon the past values eðtÞ; eðt  1Þ; . . . . It follows that the first term and the remaining terms in (4.75) are statistically independent. Also observe that the first term in (4.75) is independent of the control action. The smallest variance and optimal control signal is therefore achieved when the remaining terms are set to zero. Optimal controller: If it is assumed that the inverse of the control-weighting operator F ck exists then the following expression may be obtained for the optimal control:   1 1 1 uc0 ðtÞ ¼ F 1 F D B ðW uÞðtÞ  ðA P Þ G Y eðtÞ 0 0k 1k p cd 0 ck f f

ð4:76Þ

The spectral-factor numerator polynomial matrices are square, of normal full rank, and strictly minimum phase. The optimal control signal may be realized using the system model shown in Fig. 4.11. Note that in the scalar problem the signal /0 ¼ Pc e þ F c uc0 ¼ Pc ðr  yÞ þ F c uc0 involves a weighting F c that normally has a negative sign to ensure /0 is minimized by a control signal introducing negative feedback.

Fig. 4.11 Feedback, tracking and feedforward control signal generation

4.5 Feedback, Tracking and Feedforward Control

191

Minimum value of criterion: The minimum cost is clearly due to the first, linear time-invariant (LTI) term in (4.75). That is: k1   X f0jT ðtÞf0j ðtÞ Jmin ¼ E ðF0 eðtÞÞT ðF0 eðtÞÞjt ¼

ð4:77Þ

j¼0

This expression for the minimum value of the cost-function can provide a benchmark for nonlinear controller design as discussed in Sect. 4.3.6. It depends only on the reference and disturbance signal models that are LTI. This arises because the control action effectively removes the nonlinear plant model from the prediction of the signal /ðtÞ; whose variance is being minimized. This provides a possible method for benchmarking and assessing the performance of nonlinear processes.

4.5.6

Stability Conditions

The tracking and feedforward controller elements C1 ðz1 Þ and C2 ðz1 Þ are outside the feedback loop, and will be assumed to be chosen using classical ideas to affect the transient characteristics of the system. Thus, even though they will affect operating points and thereby stability, assume for the moment that these signals are null. As in the first part of the chapter, an alternative expression for the control signal in terms of the exogenous signals is useful for stability analysis. From (4.71) the k steps-ahead form:   /0 ðt þ kÞ ¼ F0 eðt þ kÞ þ ðAp Pcd Þ1 G0 eðtÞ  Pc W k uðtÞ þ F ck uc0 ðtÞ

ð4:78Þ

By a similar argument to that in the first half of the chapter (see Sect. 4.3.1), the optimal feedback control signal, may be obtained by setting the round bracketed term in (4.78) to zero:   1 1 uc0 ðtÞ ¼ F 1 ck Pc ðW k uÞðtÞ  ðAp Pcd Þ G0 Yf ðrðtÞ  dðtÞÞ

ð4:79Þ

In terms of the nonlinear plant operator inverse the optimal feedback control signal becomes:   uc0 ðtÞ ¼ ðPc W k  F ck Þ1 ðAp Pcd Þ1 G0 Yf1 ðrðtÞ  dðtÞÞ

ð4:80Þ

Stability: As in the previous discussion, assume the existence of a finite-gain stable causal inverse of the nonlinear generalized plant operator ðPc W k  F ck Þ. One of the restrictions on the choice of cost weightings is that this necessary condition for stability be fulfilled (similar remarks apply to those earlier in Sect. 4.3.3). In addition, when the plant is open loop unstable (A1 and A1 p

192

4 Nonlinear GMV Feedback Optimal Control

unstable terms), the controller must be implemented in its minimal form (4.76) to avoid unstable hidden modes. Black-box term stability: The way the controller (4.76) must be implemented sheds light on the necessary assumption that the black-box term W 1k must be stable. Note the term D1 f B0k ðW 1k uÞðtÞ in the controller shown in Fig. 4.11. This 1 replaces the term Yf W k in Fig. 4.2 for the first problem discussed in Sect. 4.3, where the plant was assumed stable. Recalling Yf1 ¼ D1 f A the difference in the two cases lies in the cancellation involved in Yf1 W k ¼ D1 f A W k . Clearly if the first controller solution was used on an unstable open-loop problem, there would be an unstable pole and zero cancellation resulting in an unstable system. The solution in (4.76) avoids such a cancellation.

4.6

Feedforward and Tracking Control Design Example

The following example is a modification of the one used in the previous chapter (Sect. 3.5), but in this case the feedback action is also included. It illustrates the type of results that are obtained from the algorithm and the benefits of using feedforward and tracking control. A large delay is present in the measurable disturbance to output path; however, the signal measured is not delayed. The use of feedforward action should, therefore, significantly improve the controller performance. For the tracking model the reference model Wr is chosen as an integrator, which is the stochastic equivalent of step reference changes, and knowledge of p future values of the reference is assumed (by default p = 4). Nonlinear State-Space Plant Model: The nonlinear dynamic system is given in the state-space form: x1 ðt þ 1Þ ¼

x2 ðtÞ þ u1 ðtÞ 1 þ x21 ðtÞ

x2 ðt þ 1Þ ¼ 0:9x2 ðtÞex1 ðtÞ þ u2 ðtÞ 2

yðtÞ ¼ xðtÞ with the initial state xð0Þ ¼ 0 (the stable equilibrium point of the autonomous system). Both outputs are assumed to include a transport delay of k ¼ 6 samples so the time-delay matrix has the form zk ¼ z6 I. Linear disturbance and reference models: The linear reference and disturbance models are defined as Reference model:

Wr ¼

1 1z1

0

0 1 1z1



4.6 Feedforward and Tracking Control Design Example

193

Unmeasurable disturbance:

Wd0 ¼

0:1 10:5z1

0



0 0:1 10:5z1

Measurable disturbance: 6



Wd ¼ z

1 1z1

0



0 1 1z1

Dynamic cost-weightings: The dynamic cost-function weightings were selected as follows:

5 Fck ¼ 0

1  0:1z1 Pcn ¼ 0



 0 1 0 and Fcd ¼ 5 0 1

0 1  0:1z1





and Pcd

1  z1 ¼ 0

0 1  z1



The nominal settings of a low-pass error weighting and a constant control weighting were chosen, and then the gains and the corner frequencies were found so that an acceptable design was achieved [6].

4.6.1

Results for Feedforward and Feedback Control Design

For simulation purposes, the reference and measurable disturbance step changes are introduced in both channels (at times 10 and 50, and 100 and 150, respectively). Initially, noise is not introduced into the system and only the transient performance is assessed. Transient performance: The time responses for the nonlinear system are shown in Fig. 4.12. The responses for the different NGMV control structures (feedback only, feedback and feedforward; and feedback and feedforward and tracking) are compared in Fig. 4.12. Although the controller is optimal for the inferred output signal /0 ðtÞ, rather than the actual output or control signals, there is a visible improvement in the transient error and control responses with the introduction of feedforward action. The improved tracking performance when adding the tracking controller is also evident. Stochastic performance: The performance of the controllers, in terms of the steady-state variances, will be considered after adding stochastic inputs. For that purpose, the deterministic setpoint and disturbance have been set to zero and instead stochastic disturbances have been added, feeding the models Wd and Wd0.

194

4 Nonlinear GMV Feedback Optimal Control 2

2 Fbk Fbk + Ffwd Fbk + Ffwd + Tr

1.5

y2

y1

1.5

1

0.5

0.5

0

1

0

50

100

150

0

200

0

50

0

50

100

150

200

100

150

200

1.5

1.5

1

u1

u2

1

0.5

0.5

0 0

0

50

100

150

200

-0.5

time [samples]

time [samples]

Fig. 4.12 Deterministic scenario comparing feedback, feedback and feedforward, and feedback, feedforward and tracking controllers

The simulations were run for 1000 s and the steady-state variances for the three controller configurations, were calculated from the time responses, and are presented in Table 4.1. Note that while the variances of the signal /0 ðtÞ for the given case remain relatively unchanged, the error variances can change quite significantly between simulations (different seeds), because of nonlinearities. However, the signal /0 ðtÞ is only linearly dependent on the noise sources. Predictive action: Incorporating the knowledge of the future reference signal, as in Grimble [14], may be thought of as leading to a type of predictive control with a single-stage criterion. Unlike the multi-stage model predictive control (MPC) cost-functions, the future setpoint knowledge only has an effect on the control for p  k, however, it still improves controller performance. This is illustrated in

Table 4.1 Stochastic performance results

Controller

Var(e)

Var(u)

Var(/0)

NGMV FB NGMV FB + FF NGMV FB + FF + TR

0.075 0.068 0.034

0.143 0.136 0.107

0.730 0.609 0.037

4.6 Feedforward and Tracking Control Design Example

1.5

1.5

y2

2

y1

2

1

1 0.5

0.5 0

195

0

50

100

150

0

200

0

50

100

150

1.5

1.5

Fbk only Fbk+Tr, p=2 Fbk+Tr, p=4 Fbk+Tr, p=6 Fbk+Tr, p=8

1

u1

u2

1 0.5

200

0.5 0

0 0

50

100

150

time [samples]

200

-0.5 0

50

100

150

200

time [samples]

Fig. 4.13 Feedback only control and predictive action with feedback plus tracking controller (with horizons p = 2, 4, 6, 8)

Fig. 4.13, for increasing values of the prediction interval p. Unfortunately, no advantage is gained with this present form of control law for any future reference information with p greater than k.

4.7

Concluding Remarks

The NGMV feedback control approach that was the focus of this chapter is the development of the seminal work of Åström, who derived stochastic control laws for applications such as paper machine drives [26]. The design method provides a relatively simple controller for nonlinear multivariable systems. The assumptions made in the definition of the system and the specification of the cost-function were all aimed at obtaining a simple control solution. However, the nonlinear plant model itself can be very general since it includes a nonlinear operator or black-box term. An advantage of the solution in some applications is that the equations for the nonlinear subsystem of the plant model are not required, only the ability to compute an output for a given control input. The controller is also very simple to understand, compute and implement, which was a feature of Åström’s [26] original minimum variance controller. The closed-loop stability of the system was shown to depend upon the existence of a stable inverse of a particular loop operator. This operator depended upon the cost-weighting definitions. Soft constraints may be applied through an appropriate definition of the error and control dynamic weighting functions. Penalties on control action can, for example, be increased by increasing the magnitude of the

196

4 Nonlinear GMV Feedback Optimal Control

control-weighting term. Hard constraints on input actuators, like mechanical bending limits, can be considered using a so-called barrier function, which may be absorbed into the plant model as a further nonlinearity. However, constraint handling is better performed using a predictive type of control algorithm and this type algorithm and this will be discussed in Chaps. 7, 9 and 11 to follow. A type of integral windup protection, discussed in the following chapter, can also be handled in a very simple manner by changing the control-weighting function [27]. Other nonlinear control methods do of course provide alternative approaches [28, 29]. However, there are not many control design methods that are simple to understand, are general, and stem from a formal synthesis theory.

References 1. Åström KJ (1979) Introduction to stochastic control theory. Academic Press, London 2. Clark DW, Hastings-James R (1971) Design of digital controllers for randomly disturbed systems. IEE Proc 118(10):1502–1506 3. Grimble MJ (1981) A control weighted minimum-variance controller for non-minimum phase systems. Int J Control 33(4):751–762 4. Grimble MJ (1988) Generalized minimum-variance control law revisited. Optim Control Appl Methods 9:63–77 5. Clark DW, Gawthrop PJ (1975) Self-tuning controllers. IEE Proc 122(9):929–934 6. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 7. Grimble MJ (1993) H∞ Multivariable control law synthesis. IEE Proc Control Theory Appl 140(5):353–363 8. Anbumani K, Patnaik LM, Sarma IG (1981) Self-tuning minimum-variance control of nonlinear systems of the Hammerstein model. IEEE Trans Autom Control 26(4) 9. Grimble MJ (1986) Observations-weighted minimum-variance control of nonlinear systems. IEE Proc Pt D 133(4):172–176 10. Grimble MJ, Carr S (1991) Observations weighted optimal control of a class of nonlinear systems. IEE Proc Pt D 138(2):160–164 11. Hernandez E, Arkun Y (1993) Control of nonlinear systems using polynomial ARMA models. Am Inst Chem Eng AIChE 39:446–460 12. Isidori A (1995) Non-linear control systems, 3rd edn. Springer, Berlin 13. Goodwin G, Rojas O, Takata H (2001) Nonlinear control via generalized feedback linearization using neural networks. Asian J Control 3(2):79–88 14. Grimble MJ (2005) Non-linear generalised minimum variance feedback, feedforward and tracking control. Automatica 41(6):957–969 15. Jukes KA, Grimble MJ (1981) A note on a compatriot of the real Marcinkiewicz space. Int J Control 33(1):187–189 16. Grimble MJ, Jukes KA, Goodall DP (1984) Nonlinear filters and operators and the constant gain extended Kalman filter. IMA J Math, Control Inf 1:359–386 17. Safonov MG, Athans M (1978) Robustness and computational aspects of nonlinear stochastic estimators and regulators. IEEE Trans Autom Control 23:717–725 18. Kucera V (1979) Discrete linear control. Wiley, New York 19. Grimble MJ, Kucera V (eds) (1994) Polynomial methods for control systems design. Springer, London 20. Byrnes CI, Isidori A (1988) Local stabilization of minimum-phase nonlinear systems. Syst Control Lett 11:9–17

References

197

21. Grimble MJ (2002) Controller performance benchmarking and tuning using generalised minimum variance control. Automatica 38(12):2111–2119 22. Slotine JJ, Weiping L (1991) Applied nonlinear control. Prentice Hall International 23. Gibson JE (1963) Nonlinear automatic control. McGraw-Hill Book Company, Tokyo 24. Grimble MJ, Johnson MA (1988) Optimal multivariable control and estimation theory: theory and applications. Volumes I and II. Wiley, London 25. Grimble MJ (2006) Design of generalized minimum variance controllers for nonlinear systems. Int J Control Autom Syst 4(3):1–12 26. Åström KJ (1967) Computer control of a paper machine—an application of linear stochastic control theory. IBM J Res Dev 11:389–405 27. Grimble MJ, Majecki P (2005) Nonlinear generalised minimum variance control under actuator saturation. IFAC World Congress, Prague, vol 38, no 1, pp 993–998 28. Khalil HK (1996) Nonlinear systems, 2nd edn. Prentice Hall, New Jersey 29. Atherton DP (1982) Nonlinear control engineering. Van Nostrand Reinhold, New York

Chapter 5

Nonlinear Control Law Design and Implementation

Abstract The critically important topic of control law design is considered in this chapter. The approach described applies to the nonlinear generalized minimum variance control law but the general strategy is applicable to the other optimal control problems considered later. The definition of the criterion to be minimized and the different methods for choosing the dynamic cost-function weightings are considered. The problems in implementing the optimal controller and the relationship to the well-known Smith-predictor transport-delay compensator are described. Finally, the presence of system uncertainties and various aspects of the sensitivity minimization and the robust control problems are explored.

5.1

Introduction

There are a number of factors that determine what is achievable in feedback control systems design including non-minimum-phase zeros, open-loop unstable poles, transport delays, actuator and measurement system nonlinearities, uncertainties and nonlinear plant dynamics. These design limitations will impose constraints on, for example, the placement of the closed-loop poles of a linear system. The design problem will, therefore, involve trade-offs between robustness and closed-loop performance, subject to these and other limitations. There are many objectives of a closed-loop control design and one of the most important is the speed of response. The factors limiting the speed of response of a system, or the closed-loop bandwidth, can be listed as • Measurement noise characteristics. • Disturbances that should not be rejected in a mid- or high-frequency range. • Structural limitations (non-minimum-phase zeros, unstable poles and transportdelay terms) • Modelling errors and uncertainties. • Actuator limits including saturation and nonlinear friction characteristics.

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_5

199

200

5 Nonlinear Control Law Design and Implementation

To reject disturbances adequately, there must be adequate gain in the frequency range where the disturbances are dominant. This suggests that the gain should normally be high at low frequencies, where most of the significant disturbances normally arise. The bandwidth should be greater than a lower limit that is determined by the frequency response of the significant disturbances. Some disturbances like the first-order wave forces in ship positioning should not be rejected (Chap. 15) and in this less familiar case, gains must be small in the region of the dominant wave frequency, limiting the bandwidth. Modelling errors usually increase as the frequency increases and this imposes an upper bound on the closed-loop bandwidth. The measurement noise is also usually a high-frequency phenomenon that will limit the closed-loop bandwidth that can be achieved. Nonlinear systems exhibit complex behaviour, including limit-cycle responses and chaotic behaviour that cannot arise in linear systems (see Chap. 1). The classical approach to nonlinear control problems is to design a linear controller based on a model of the process that is linearized about a steady-state operating point. However, in many critical applications this linearization approach will not be adequate (e.g. for the control of highly nonlinear servo-systems or batch processes), and ad hoc nonlinear compensation will be needed. There is, therefore, a need for general nonlinear control strategies like the NGMV method described in the previous chapter. This approach has rather practical features and capabilities that will be explored in this chapter.

5.1.1

Desirable Characteristics of an Industrial Controller

Before describing possible nonlinear control laws, the characteristics of a desirable industrial controller that can provide improved performance, will be noted. These are based on the comments of Dr. Chen-Fang Chang of General Motors Global R&D. The requirements of a good industrial control law can be listed as 1. Should be easy to understand the basic strategy and to motivate the way it operates. 2. Solution must be robust to disturbances and modelling errors. 3. Should be intuitive to calibrate and not be too sensitive to tuning variables. 4. It should have limited computational requirements and load. 5. Be easy to carry over to a similar system (extendable).

5.1.2

Nonlinear Controllers and Design Methods

Some well-known nonlinear analysis and control design methods were described in Chap. 1 (Sect. 1.5). There now follows a brief summary of simple and practical nonlinear control design approaches that are used in applications. Attention then

5.1 Introduction

201

turns to the design questions and procedures for the NGMV family of controllers. These design guidelines will also apply in a similar form to most of the other control methods described in the text. This includes the Nonlinear Generalized Predictive Controllers, and the related optimal predictive nonlinear controllers for polynomial and state-space based systems. Nonlinear PID Control: The use of a scheduled PID controller is probably the simplest feedback control solution for nonlinear systems. However, a simple nonlinear controller can also be synthesized by modifying a PID controller structure [1]. The linear and nonlinear versions of the PID controller for a continuous-time system may be defined to have the form: Linear PID:

  Z t uðtÞ ¼ Kc eðtÞ þ Ki eðtÞdt þ Kd e_ ðtÞ 0

 Z t   Nonlinear PID: uðtÞ ¼ Kc f ðeðtÞÞ þ Ki g eðsÞds þ Kd hðe_ ðtÞÞ 0

where the functions f (.), g (.) and h (.) are chosen according to the needs of the physical problem. The gains or tuning parameters are sometimes modified as a function of the error. In some nonlinear PID designs, a small linear region is included in the nonlinear functions so that when the error e(t) is small in magnitude the closed-loop system behaves like a linear system [32]. Nonlinear Transformations: Nonlinear transformations may be used in some cases to “linearize” the input–output mappings and linear PID control can then be used. This is common practice in automotive applications and one reason extensive engine mapping is required. In process control, the logarithm of composition can be used when developing the inferential models needed in distillation columns (on-line accurate measurement of composition is often difficult). However, in general, the cancellation of nonlinearities is usually impossible or just undesirable for several reasons, including the presence of uncertainties. They should not normally be cancelled by use of an inverse mapping, even when this is physically possible and the nonlinearity is static. Backstepping: An advantage of backstepping design procedures is that it avoids the direct cancellation of nonlinearities. This nonlinear control law was developed by Kokotovic and others in the 1990s [2], for a special class of nonlinear dynamical systems. The state model for such a system may be available naturally or after manipulation in a lower triangular form. The class of systems are those that build out from an irreducible subsystem that can be stabilized by some method. The backstepping procedure involves the design of successive new controllers that stabilize each outer subsystem. A Lyapunov function is constructed for each subsystem so that the final control signal is obtained recursively through a series of what might be termed virtual control signals. This process continues until the external control is reached and this gives rise to the term backstepping (Chap. 1 in Sect. 1.5.7).

202

5.1.3

5 Nonlinear Control Law Design and Implementation

Optimal Nonlinear Control

The family of nonlinear generalized minimum variance (NGMV) controllers provides a global solution, in the sense that the controller is optimal for the full range of operation. It does not rely on any form of local linearization. The results in the previous chapter focussed on the theoretical derivation of the NGMV controller, rather than on the design and implementation issues. In this chapter, aspects of the design and implementation problem are explored, mostly in terms of the NGMV controller [3]. However, the design guidelines, although not directly applicable, will have some relevance to the other optimal control design approaches described in subsequent chapters. The structure of the controller, shown in Fig. 4.2, reveals that the computational complexity increases with the model order and with the number of inputs and outputs. This is because the controller structure includes the plant model (without explicit delay elements). One of the objectives of this chapter is to clarify some issues on the implementation of the algorithms on computing platforms that will have a limited capacity. A crucial aspect of the nonlinear optimal controller design is the selection of cost-function weightings and the flexibility different choices may provide. For example, integral windup protection is often needed and can be introduced by exploiting the nonlinear control signal cost-function weighting term in the NGMV cost-function (see Sect. 5.3.2). The NGMV controller structure is related to a nonlinear version of a Smith-Predictor structure. The Smith predictor is, of course, well known in industrial and other applications. This provides some confidence in the NGMV design approach, and emphasizes its role in providing transport-delay compensation. This relationship is discussed in Sect. 5.5.1, and is followed by a discussion of robust control issues. The Smith predictor removes the effect of transport delays from the closed-loop design when the plant model matches the real plant. It is equivalent to Internal Model Control (IMC) in the sense that the delayed plant model is used to cancel the plant output to obtain an estimate of the disturbance that can then be fed back. The IMC philosophy relies on the internal model principle that is introduced below.

5.1.4

Internal Model Control Principle

The internal model principle was first presented by Francis and Wonham [4]. The internal model principle suggests that the control system should encapsulate, either implicitly or explicitly, some representation of the process to be controlled. For disturbance rejection recall that in order to be able to reject a disturbance, a model of the disturbance should be incorporated in the controller. To be more explicit for a scalar system represented in transfer function form a sufficient condition for

5.1 Introduction

203

steady-state disturbance rejection is that the polynomial generating the disturbance is included in the controller denominator. For the systems of interest, this translates into the need to include stochastic disturbance models in the system description. It is interesting that including the poles of the dominant disturbance model in the dynamic cost-weighting on the error signal will also introduce these poles in the controller. A common example arises when the weighting Pc ðz1 Þ includes an integrator, since this will result in integral action being introduced in the controller.

5.2

Optimal Cost-Function Weighting Selection

The selection of the cost-function weightings for the NGMV controller is discussed in this section but the guidelines will be relevant to any of the model-based control methods that include dynamic cost-function weightings. In the linear case, it is possible to relate the design of dynamic cost-weighting transfer functions to traditional frequency response loop-shaping ideas. This is not possible so easily for nonlinear systems; however, some useful guidelines can still be formulated. The use of dynamic cost-function weightings in optimal linear control problems is very well established. In fact, dynamic weightings are particularly important for H∞ controllers, where the use of constant weightings can easily result in unacceptable control designs. The properties of an NGMV controller are often critically dependent on the selection of the dynamic cost-function weighting functions Pc ðz1 Þ and F ck ðz1 Þ. These weightings are, of course, the means of communicating the desired performance requirements to the optimization algorithm. The selection of weightings for NGMV designs is discussed below and it will be found that similar rules will apply to the dynamic error and control weightings used in the predictive controllers in later chapters. The desired frequency response characteristics of the weightings are often similar for linear and nonlinear systems. In the simple case of NGMV control, by setting Pc ðz1 Þ ¼ I and F ck ð:Þ ¼ 0, a nonlinear version of the minimum variance controller is obtained (Chap. 2). Whilst this result may be useful to explain limiting properties, it does not provide a practical solution. This is due to the aggressive control action that results when there is no weighting on the control signal. The controller may also have poor robustness properties due to the high gain employed. Moreover, since the basic MV controller attempts to cancel the plant dynamics, the plant model must be stably invertible. Recall from Chap. 2 that the standard MV feedback control is unstable for non-minimum-phase linear processes, because it attempts to cancel plant zeros.

204

5.2.1

5 Nonlinear Control Law Design and Implementation

NGMV Free Weighting Choice

The basic principles of NGMV cost-weighting selection are now illustrated for single-input single-output (SISO) systems. However, when the number of outputs to be costed is the same as the number of control inputs u0 ðtÞ the cost-weightings may easily be generalized to the multivariable case by assuming a diagonal weighting structure. In this case, each diagonal element acts on a specific error and input channels. Frequency-dependent weightings can be used to penalize different frequency ranges for the error and control signals. There is seldom any need to use off-diagonal weightings in such cases. As explained in Sect. 4.3.2, the error weighting Pc ðz1 Þ can be non-square in some problems, and in this case the choice of weightings is not as obvious as for the diagonal weightings case. Error weighting: The controller is normally required to include integral action, and this can be achieved by including an integrator on the error-weighting signal of the form: Pc ðz1 Þ ¼ Pcn ðz1 Þ=ð1  z1 Þ

ð5:1Þ

The transfer-function numerator term Pcn ðz1 Þ may be constant, or it may have the form ð1  az1 Þ, where 0\a\1 is a tuning parameter. The frequency at which integral action behaviour is cut-off is determined by the choice of a. The general effect of using integral error weighting is to introduce high gain into the feedback loop at low frequencies. In fact, as in the linear case (Chap. 2), the controller poles will normally include the poles of the dynamic error weighting. Since the system sensitivity function gets smaller with increased loop gain, the low-frequency disturbances (such as offsets) will be rejected asymptotically. The problems associated with using integral-action are discussed in more detail later in Sect. 5.3. Control weighting: The control-weighting function can often be assumed linear, and chosen as a lead term. The lead term will ensure the controller rolls-off at high frequencies and does not amplify measurement noise. An alternative way to attenuate measurement noise is to include a measurement noise block explicitly in the system model, as normally occurs in GMV, LQG or H2 designs. However, there is a more direct control of the controller roll-off when a dynamic control signal cost-function weighting is used. Control-weighting parameterization: To parameterize the dynamic weighting function on control action a scaling factor may be used to change the balance of the steady-state variances of the error and control signals. A linear scalar control weighting F ck ¼ Fck ðz1 Þ can be parameterized as Fck ðz1 Þ ¼ q or alternatively, as

ð5:2Þ

5.2 Optimal Cost-Function Weighting Selection

Fck ðz1 Þ ¼ qð1  cz1 Þ

205

ð5:3Þ

where q and c 2 ð0; 1Þ may be considered as control law tuning parameters. The sign of the parameter q, and the DC gain (low-frequency gain) of the control weighting, will normally be negative for stability reasons. This can be explained for scalar systems rather simplistically in the following way. Consider the minimization of the variance of a signal /0 ðtÞ ¼ rðtÞ  yðtÞ þ quðtÞ, with zero reference r = 0 and with positive weighting q. The output and control signals can then be very large, and the value of the cost can be zero. This is clearly an ill-posed problem that results in positive feedback. If the sign of q is negative and the cost is minimized the optimal control involves negative feedback. Recall from Chap. 4 (Sect. 4.3) in the more general nonlinear weighting case that the inferred output: /0 ðtÞ ¼ Pc ðz1 Þ ðrðtÞ  yðtÞÞ þ ðF c uÞðtÞ 1 1 and the DC gain of the ratio F 1 ck ðz ÞPc ðz Þ should be negative using a similar argument. Multivariable weightings: The multivariable case is more complicated than the Single-Input Single-Output (SISO) problem, but the dynamic weightings are often diagonal matrices and the weighting-function frequency responses can be chosen in a similar manner to the above. Typical frequency responses of the NGMV weightings for a 2  2 problem are shown in Fig. 5.1. Note that the individual frequency responses of the weightings could be displayed, or the singular values, as in this case. The former is useful when choosing the weightings and the latter are of interest when analysing weighted sensitivities and robustness. Scaling: The weightings that are needed are very dependent on the plant model scaling (choice of units for signals), and this has a great influence on the numerical reliability and accuracy of computations. The scaling should be such that the signal inputs and outputs from the plant are normalized to similar orders of magnitude. This is not a precise science since a plant may contain integrators, for example, and the DC gains cannot be scaled in an obvious manner. However, a simulation can be run with a baseline (classical) controller, and given representative reference and disturbance conditions. The plant and controller can then be scaled to provide similar levels of inputs and outputs. High accuracy is not needed, since the objective is to ensure the range of numbers in the computations is limited to a reasonable range. This particularly helps the choice of weightings in the multivariable case, where the dynamic error weightings will be of similar order and changing the relative magnitudes in the design will be more meaningful. Square of sums cost-function: The special form of the NGMV cost-function (square of a linear combination of signals), requires care to be taken over the sign of the control weighting as explained for the weighting parameterization discussed above. It is often found that the low-frequency gain of the control weighting should be negative. Simple first-order cost-function weighting parameterizations normally lead to reasonable results. Weightings that are more complex can be used, although

206

5 Nonlinear Control Law Design and Implementation Free choice of weightings for NGMV control 90 Pc error costing Fck control costing

85

Singular Values (dB)

80 75 70 65 60 55 50 45 40 -1 10

10

0

1

10

10

2

10

3

Frequency (rad/s)

Fig. 5.1 Weighting frequency responses for multivariable problem

this will obviously increase the order and complexity of the controller. In fact, this also applies when nonlinearities are introduced in the control signal weighting term (an example of the use of a nonlinear control signal weighting is presented in Sect. 5.3).

5.2.2

NGMV PID Motivated Weightings

The cost-function weightings for an NGMV controller must be chosen so that the operator that determines stability of the closed-loop system has a finite gain stable inverse. This was discussed in Chap. 4 (Sect. 4.3.4). The Eq. (4.32) indicated that a necessary condition for stability is that the operator ðPc W k  F ck Þ must have a stable inverse. The case considered in the previous chapter was the most common situation where the control weighting F ck is actually linear and the sign is negative. That is, F ck ¼ Fk so that the operator determining stability may be written as ðPc W k  F ck ÞuðtÞ ¼ Fk ðFk1 Pc W k þ IÞuðtÞ

5.2 Optimal Cost-Function Weighting Selection

207

It was noted that the term ðI þ Fk1 Pc W k Þ could represent the return-difference operator for a system with a plant model W k and feedback controller C0 ðz1 Þ ¼ Fk1 Pc . Thus, consider the plant model W k and assume a PID controller Kc ðz1 Þ can be found to stabilize the closed-loop delay-free system. A choice of weightings, that will ensure ðPc W k þ Fk Þ is stably invertible, is clearly Fk1 Pc ¼ Kc ðz1 Þ. The return-difference operator term has an important role in determining system responses, and further insights can be gained if the NGMV controller is represented in an equivalent Smith-Predictor form (only relevant for open-loop stable systems). That is, it is shown later that the NGMV controller when expressed in this form has an internal feedback loop in the controller (see Sect. 5.5 and Fig. 5.17) with the same return-difference operator ðI þ Fk1 Pc W k Þ. This result does not apply for open-loop unstable systems since the controller cannot be restructured in this way. The link to the Smith predictor emphasizes the controller is acting as a k-steps predictor, where k is the number of time steps in the common plant delay term. Example 5.1: Parameterizing a PID control Law Following the previous discussion if the plant already has a PID controller Kc ðz1 Þ, that stabilizes a delay-free plant model (often follows from a PID controller that stabilizes the loop with the time-delay), then the cost-weightings can be chosen as Pc ¼ P1 cd Pcn ¼ Kc

and

F ck ¼ I

ð5:4Þ

For example, a discrete PID controller has the form: k1 þ k2 ð1  z1 Þ Kc ðz1 Þ ¼ k0 þ ð1  z1 Þ   ¼ ðk0 þ k1 þ k2 Þ  ðk0 þ 2k2 Þz1 þ k2 z2 =ð1  z1 Þ

ð5:5Þ

If the PID gains are positive numbers, with a derivative gain that tends to zero, then the Pcn term: Pcn ¼ ðk0 þ k1 Þ  k0 z1 is minimum phase if k1 [ 0. The error weighting Pc ðz1 Þ ¼ Kc ðz1 Þ will, therefore, be minimum phase and include integral action. From the viewpoint of the weighted error signal, a minimum-phase error weighting is desirable. NGMV performance with PID motivated weightings: The optimal controller that results from such a choice of PID control inspired cost-weightings will lead to similar NGMV controller characteristics, which is not perhaps surprising. These will include a high gain at low frequencies and a high gain at high frequencies. This suggests that some modification will be necessary to limit the gain at high

208

5 Nonlinear Control Law Design and Implementation

frequencies. Using the equivalence to a filtered PID controller leads to weightings of similar characteristics but with a limited gain at high frequencies. There are normally reasonable low-order cost-weightings that may be chosen to ensure a stable inverse for the operator Pc W k  F ck . Starting point for design: The choice of the weightings to be equal to a stabilizing PID control law is only a starting point for design, where closed-loop stability is easy to achieve. However, the control weighting will normally require some further adjustment, such as an additional lead term (or alternatively a high-frequency lag term may be added to the error Pc ðz1 Þ weighting). The high-frequency characteristics of the optimal controller will then have a more realistic roll-off. This may not be necessary if the PID solution already has a low-pass filter for noise attenuation. After the initial weighting choice, as in linear control design, the magnitude of the control-weighting function may be varied to change the speed of response of the feedback system. As usual, a large control signal costing will normally slow down the closed-loop system step-response, and reducing the control signal costing will widen the bandwidth and speed up step responses (assuming stability is maintained). Plant scaling problem: A possible NGMV control design procedure is, therefore, to use the transfer of an existing classically designed controller to define the cost-function weightings. Since the initial design will probably give reasonable responses this procedure reduces the risk of lengthy commissioning time on the plant. An indirect benefit is that the problem of scaling the cost-weightings as described in the above section Sect. 5.2.1 is avoided. That is, although the frequency shape of the error and the control weightings is often easy to choose, the relative magnitudes may not be so obvious. Clearly, a system, which has different physical parameters, will require different cost-function weightings. By utilizing the existing controller structure, to define the initial choices of cost-weightings, this scaling problem is mostly avoided. However, the type of transient response characteristics obtained for the initial optimal control solution will probably be similar to those for the classical PID control design. This approach therefore only provides a starting point for choosing the cost-function weightings.

5.2.3

NGMV a-PID Motivated Weightings

The PID motivated weighting approach can be put in a simple form that will be referred to as the a-PID weighting selection approach. For simplicity neglect the derivative term since it is used to influence high-frequency behaviour. The main requirements are usually that the low-frequency tracking is accurate with good disturbance rejection properties, and the mid-frequency characteristic is such that the transient response, in terms of step responses, is appropriate. It is, therefore, reasonable to choose a PI structure to determine the form of the frequency

5.2 Optimal Cost-Function Weighting Selection

209

weightings that will shape the low-frequency characteristics, even if a derivative term is added later. In choosing the PI motivated weightings it is usually a requirement to have integral control, either pure or approximate, at low frequency and to have a mid-frequency breakpoint term so that excessive phase shift is not introduced in the important mid-frequency region that determines stability. The frequency where the breakpoint should occur can be specified by the control designer, who will have an idea of a realistic closed-loop bandwidth. Similarly, if a pure integrator is not to be used but an approximate integrator (may help the numerical conditioning), a suitable factor to define this “near integrator” approximation can be chosen. This allows the PI weighting function to be defined apart from a gain. That is, in a scalar problem only one tuning gain parameter is required. A simple formula is provided below that enables the integral gain to be defined in terms of the proportional gain so that only the proportional gain needs to be varied in tuning. This approach is particularly convenient for discrete-time weighting function definitions. This is because the lag–lead term of the resulting weighting has zeros and poles clustered around the z ¼ 1 stability point, and it is not, therefore, obvious how to choose the proportional and integral gains to achieve the desired mid-frequency breakpoint. For say a 2  2 multivariable system two gains must, of course, be selected to define the weighting functions but this is straightforward, by slightly generalizing the results below. Example 5.2: Continuous-Time Case The a-PID weighting approach is to fix the lead term in the weighting, which means that in the scalar case only one gain needs to be determined. Let Pc be written as Pc ¼ kp þ ki =ðs þ bÞ, where b is usually zero or small. Then,   Pc ¼ kp þ ki =ðs þ bÞ ¼ ðkp b þ ki Þ þ kp s =ðs þ bÞ ¼ ðkp b þ ki Þð1 þ asÞ =ðs þ bÞ where a ¼ kp =ðkp b þ ki Þ. The lead term can be fixed by the designer to roll-off the integral term in mid-frequency. Let 1=a be chosen as the desired break-frequency x 0 (usually a simple choice with starting point one decade below the desired bandwidth). Then the breakpoint ax0 ¼ 1 or a ¼ 1 =x0 , and a ¼ kp =ðkp b þ ki Þ, giving ki ¼ kp ð1  abÞ=a. It follows that only one gain kp needs to be calculated. The a and b are easy to select using the approach suggested. Example 5.3: Discrete-Time Case For the discrete-time Pc ¼ kp þ ki =ð1  bz1 Þ, where b is usually close to unity. Then,

case

let

        Pc ¼ kp ð1  bz1 Þ þ ki = 1  bz1 ¼ kp þ ki  kp bz1 = 1  bz1       kp b z1 Þ= 1  bz1 ¼ ðkp þ ki Þ 1  az1 = 1  bz1 ¼ ðkp þ ki Þð1  ðkp þ ki Þ

210

5 Nonlinear Control Law Design and Implementation

where a ¼ kp b=ðkp þ ki Þ, for the breakpoint aex0 T ¼ 1, so a is known and hence ki ¼ kp ðb  a Þ=a: Letting c ¼ ðb  a Þ=a, we obtain:         Pc ¼ ðkp þ ki Þ 1  az1 = 1  bz1 ¼ kp ð1 þ cÞ 1  az1 = 1  bz1 where ki ¼ kp ðb  a Þ=a or ki ¼ kp c. As for the continuous-time case, only one gain kp needs to be selected.

5.2.4

General Weighting Selection Issues

The cost-function weighting selection methods for the NGMV control laws also apply to GMV and may be useful for LQG cost-weighting selection methods. This follows since with reasonable choices of dynamic cost-function weightings, a GMV controller gives similar responses to those of an LQG controller. Thus, a weighting selection method that works for GMV (or NGMV) designs are also likely to apply to LQG or NLQG design methods (discussed in Chap. 6). It has been seen in the previous section that if a scalar delay- free system is already controlled by say a PID controller, or some other well-defined classical control structure, then a starting choice of NGMV cost-function weighting is to choose the ratio of the error weighting divided by the control weighting equal to this controller. There are several assumptions to make this result valid but it is a useful starting point for the design. Moreover, it provides realistic frequency response characteristics. For example, a PID controller clearly has high gain at low frequency and if it includes a filter then it will have low gain at high frequencies. This is exactly the type of response that is needed for the ratio between the error and control weightings, and even applies for H∞ control designs. As previously discussed if a stabilizing existing controller structure is known then the required weightings follow almost immediately, and it provides a very fast way of generating the starting values of cost-weighting functions. It is true that some adjustment will then be necessary after this initial selection. The magnitude of the control-weighting function often needs to be reduced, to speed up the system. In this way, the initial design will often be close to the existing classical controller but the design can usually be improved by reducing the value of the control signal weighting term. Experience reveals that halving the control-weighting gain is a useful starting point. Since the initial design will probably give reasonable responses this procedure reduces the chance of possible commissioning problems. A number of simulation trials may, of course, be needed with different weightings to tune such a system. Free weighting choice: An alternative method of choosing the cost-function weightings is the so-called free weighting choice approach. In this case, the form of the error and the control weightings can be defined to have a specified structure as described in Sect. 5.2.1 above. That is the weightings will be defined to have a

5.2 Optimal Cost-Function Weighting Selection

211

particular dynamic system form or structure but with some unknown parameters. The actual magnitude of the cost-function weightings will depend upon the speed of response and performance required from the system. Magnitude of weightings: If the system is to be made faster, the magnitude of the control weighting should generally be reduced. One method of finding a good cost-weighting gain is to try small control weightings and then a much larger control weighting and interpolate between the two to obtain the type of response required. For example, if the small control weighting gives a one-second response and the large control weighting gives a 50-s response, then something in between should give an intermediate value for the dominant time constant. Such a procedure does, of course, require iteration, and in real systems, it may not be possible to attempt low control weightings that might lead to severe actuator movements. The alternative methods of choosing the cost-function weightings described above are illustrated in the examples in the rest of the text. The PID motivated method, is the easiest to use for those unfamiliar with dynamically weighted optimal control designs. However, the free weighting choice approach, where the dynamic cost-function weights are parameterized and the designs are then chosen based more on desired frequency response characteristics are often the most effective. These two main approaches do of course have different merits, which should also be evident from the examples.

5.2.5

Nonlinear GMV Control Design Example

The design of the NGMV controller based on an existing, or simulated, PID controller is now illustrated for the design of a scalar nonlinear dynamic system. Consider an open-loop stable system described in terms of the following nonlinear discrete-time state-space model:   x1 ðt þ 1Þ ¼ x1 ðtÞ  x2 ðtÞ=ð1 þ 0:8x21 ðtÞÞ þ uðtÞ

ð5:6Þ

x2 ðt þ 1Þ ¼ ð0:9ex1 ðtÞ j x2 ðtÞ j Þ þ 0:64uðtÞ 2

ð5:7Þ

yðtÞ ¼ x1 ðt  3Þ

ð5:8Þ

Let the initial state xð0Þ ¼ ½0; 0T (the stable equilibrium point of the autonomous system). Observe that the output yðtÞ includes a transport delay of k ¼ 4 samples. The open-loop system response to a series of steps is shown in Fig. 5.2. The nonlinearity present in the system is evident from the different output step sizes at different operating points, and the wide range of dynamic responses. For the nonlinear GMV controller design, the linear reference model was defined as Wr ¼ 0:05=ð1  0:99z1 Þ, which is the stochastic equivalent of a reference

212

5 Nonlinear Control Law Design and Implementation 2.5 input output

2

plant input u,output y

1.5 1 0.5 0 -0.5 -1 -1.5 -2

0

20

40

60

80

100

120

140

160

180

time (samples)

Fig. 5.2 Open-loop plant step responses for different operating regions

signal with near step changes (see Chap. 2 for discussion of the similarity between stochastic and deterministic signal models). The model of the additive linear disturbance acting on the system output is also a low-frequency disturbance, defined as Wd ¼ 0:05=ð1  0:8z1 Þ. Existing control: Assume that the plant can be controlled by an existing stabilizing PID controller, denoted CPID ðz1 Þ, with a filtered derivative term: 

1 Td ð1  z1 Þ þ CPID ðz Þ ¼ K 1 þ Ti ð1  z1 Þ ð1  sd z1 Þ 1

 ð5:9Þ

Here the gain and tuning parameters are K ¼ 0:1, Ti ¼ 4s, Td ¼ 1s, and sd ¼ 0:5s. Initial design: As explained above, using the PID motivated weighting selection procedure, the nominal dynamic weightings for the NGMV control design may be defined in terms of (5.9) as Pc ðz1 Þ ¼ CPID ðz1 Þ and F c ¼ z4 . The reference tracking of a sequence of steps, and the step disturbance rejection, for the two nominal controllers, is illustrated in Fig. 5.3. Note that the nominal PID tuning parameters have only been found to stabilize the delay-free plant and are not “optimized”. However, this controller is useful in that it can provide initial design parameters for the NGMV controller that will stabilize the plant. As can be seen from Fig. 5.3, the performance of this initial nonlinear controller design is close to that of the original PID solution, which often occurs.

5.2 Optimal Cost-Function Weighting Selection

213

output

2

0

-2

-4 0

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PID NGMV

control

0 -2 -4

-6

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400

500

600

700

800

time [samples] Fig. 5.3 Time responses of PID and nominal NGMV controllers

Improving the initial design: The nominal NGMV control design can often be improved by changing the control cost-function weighting. Parameterize the control weighting as F ck ¼ qð1  cz1 Þ, where q is a positive scalar and c 2 ½0; 1 introduces a lead term in the weighting. The lead term on the control signal weighting is useful to reduce the high-frequency gain of the controller. For the nominal constant-weighting design q ¼ 1 and c ¼ 0. The magnitude responses for some combinations of these two parameters are shown in Fig. 5.4 for a range of frequencies. Control-weighting parameterization: The control-weighting parameterization involves only two tuning parameters and this is meant to simplify the design task. For example, decreasing the value of q (reducing the control weighting), leads to a faster response and a more aggressive control action. On the other hand, adding a lead term to the control weighting, by means of the parameter c influences the controller high-frequency gain. For example, Fig. 5.5 shows the simulation results for different values of the tuning scalars q, and for c ¼ 0:3. As expected, increasing q results in a slower step-response, and this provides a simple tuning mechanism. For comparison purposes, the nominal PID controller was retuned and its performance compared is with that of the NGMV controller with design parameters

214

5 Nonlinear Control Law Design and Implementation Bode Diagram

15

Increasing

Magnitude (dB)

10

5

0

-5

Increasing

-10

P

c

-15

-20 -2 10

10

-1

10 0

10

1

Frequency (rad/s)

Fig. 5.4 Frequency responses of the dynamic cost-function weightings (Pc (solid), F ck with c ¼ 0 (dashed), F ck with q ¼ 1 (dotted))

q ¼ 0:5 and c ¼ 0. A set of PID parameters were obtained that were close to the NGMV design in terms of the speed of response, but the plant nonlinearity caused oscillatory behaviour in the PID control design responses at some operating points. This is a common phenomenon and to obtain a better performance would require a scheduled PID control design. Increasing transport delay: In the final test, the plant time-delay was increased from 4 to 10 samples. The same weightings were used for the design but the NGMV controller obtained was, of course, different, reflecting the change in the time-delay. The MATLAB® Nonlinear Control Design Blockset was then used to find the optimal PID parameters, given the desired response. The boundary constraints were relaxed until a feasible set of parameters were found. However, it was not possible to tune the PID controller to provide satisfactory responses across the operating range. Figure 5.6 shows the response of the NGMV controller and the two PID controllers obtained. The dynamic response of the NGMV controller is very close to the original one, despite the significant increase in the time-delay. It was not possible to obtain PID controllers (without some form of additional time-delay compensation), that provided fast transient responses and acceptable robustness margins

5.2 Optimal Cost-Function Weighting Selection

215

output

2

rho=0.5 rho=0.7 rho=1 rho=2

0

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-4

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800

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2

control

0 -2 -4 -6

time [samples] Fig. 5.5 Time responses for weighting parameters (q ¼ 0:5; 0:7; 1; 2 and c ¼ 0:3)

across the operating range. The example demonstrates the potential of the NGMV controller for the control of nonlinear plants with significant transport delays. Comments: The NGMV approach has the advantage that it is naturally multivariable and the cost-function weightings provide a simple mechanism to improve robustness. It seems easy to obtain an NGMV design very close, and normally better, than the performance of an existing PID controller. This nominal design can then be modified to achieve further improvement. The NGMV approach is not in competition with a PID control philosophy of course. There is every reason to use the simplest possible controller that will meet the performance requirements. The NGMV was only compared with the PID design for this low-order problem, but it has the advantage of high-order systems, where a stabilizing PID control law may not even exist. As the NGMV controller includes the nonlinear model of the plant, it should be more robust to changes of the operating point, whereas a linear controller will generally have difficulty regulating consistently across the full operating range. The above results are, of course, for a scalar system and do not include plant model mismatch.

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5 Nonlinear Control Law Design and Implementation

output

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NGMV PID PID 2

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-2

-4

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control

0 -2 -4 -6

time [samples] Fig. 5.6 Time-responses for time-delay increased from 4 to 10 samples

5.2.6

Scheduled Weightings

Further design freedom is available if the cost-function weightings, static or dynamic, can be changed on-line. This device may be used with all the optimal methods described but it is less ad hoc if the system model is in a quasi-linear parameter varying form. The system description can then absorb the dynamic weightings, which can be switched according to say wind speed in a wind turbine, or engine speed in a vehicle. It is unlikely that one set of weightings will be suitable for the full range of operation of a nonlinear system. In the case of a wind turbine, there are, for example, two distinct regions of operation depending on wind speed where the control has different objectives (maximize power in below-rated power region and limit power captured to the maximum allowed for higher wind speeds). For an automotive engine control, above idle speed, there is no major change in requirements. However, to maintain the best performance above this speed, some use of scheduled weightings may be desirable.

5.3 NGMV Control of Plants with Input Saturation

5.3

217

NGMV Control of Plants with Input Saturation

Actuator saturation is a common cause of poor control loop performance and may even lead to instability in a feedback system [6]. Saturation characteristics are similar to limits that may be on voltage or current in converter systems that provide power to the actuators. These limits can ensure that there are not problems with overloading or overcurrent leading to motor burnouts. Two forms of mechanical limits are also often involved: • Magnitude limits that constrain the amplitude of the demand to the actuators. • Slew-rate limits that limit the rate of change of the actuators’ output. Although such limits are introduced for good reasons, they result in similar performance and stability problems as saturation characteristics at the plant inputs. The control design method should, therefore, allow for saturation effects and electromechanical limits. For example, to avoid entering regions where there are slew-rate limiting problems, an upper limit on the closed-loop bandwidth may be imposed in the control design. Note that when an actuator hits its physical limits, or a controller output limit is reached, then the closed-loop system behaves as though it were open loop with a constant input. These effects are particularly severe when the controller includes integral action. Under saturation conditions, the integrator state increases without bound due to a non-diminishing error signal, and it requires time to return to the working range when the sign of the error reverses. This so-called “windup” problem usually results in very undesirable and unpredictable transient behaviour, involving a sluggish response to reference changes and even instability. In this section, the class of nonlinear systems that are characterized by limits introduced on the magnitude of the input signals is considered. This is illustrated in Fig. 5.7, where the output plant subsystem W 0 can either be a linear system (Hammerstein model), or can be a nonlinear system model. All real-world signals are of course limited in magnitude due to physical actuator constraints (e.g. valves) or finite energy resources, and this is, therefore, a useful practical case to explore [5].

5.3.1

Classical Anti-windup Protection

Whilst it is not possible to compensate fully for the decrease in gain caused by saturation, a number of standard heuristic methods have been developed to avoid

Fig. 5.7 Plant with an input limit or saturation

u(t)

y(t)

u0 (t ) usat (t ) 0

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5 Nonlinear Control Law Design and Implementation

integrator windup. When implementing anti-windup schemes in digital systems, it is common to use logic to determine when to clamp the integrator output. This approach can be used with the NGMV controller or some of the other controllers discussed in the following. However, it is useful to introduce the classical anti-windup scheme shown in Fig. 5.8, since it provides an intuitive and simple solution. Consider a continuous-time linear plant W0 ðsÞ and the simple case of a PI controller and a hard limit or actuator saturation. Compute the difference between the output and input of the limit or model of the saturation and feed it back to the input of the integrator. In the case when the saturation effect is due to the presence of an actuator nonlinearity, both the output and input of the actuator may be measurable and the model of the saturation in Fig. 5.8 may not be needed. When the system is not in saturation the difference of the signals around the saturation block may be scaled so that the difference is null and no action is taken. When the slope (linear gain) of the saturation characteristic is unity and the system is in saturation a signal is fed back and the sign of the signal is such that the integrator is discharged. The result is that the integrator output is maintained near the saturation limit. When the error changes sign, the integrator does not then have to come down from a value far in excess of the saturation limit. The consequence is that the integrator output changes quickly, so that the system enters the linear region of operation and this considerably reduces the delay due to windup. The speed of response of the controller windup loop can be adjusted using the parameter Tt .

5.3.2

NGMV Solution of the Anti-windup Problem

It will be shown in the following that a similar anti-windup mechanism to that shown in Fig. 5.8 can be obtained via the NGMV controller. This arises quite naturally thanks to a special choice of the control signal weighting function. The heuristic structure shown in Fig. 5.8 can then be justified as the solution to a formal optimization problem. There is another benefit of considering this approach and that is it demonstrates the value of including a nonlinear control signal cost term in the NGMV performance index [7].

e(t)

K Ti

v(t)

+

K

+

+

1 s

+

–

+

1 Tt

Fig. 5.8 Classical PI control and anti-windup scheme

Compensator

u(t)

y(t)

W0 ( s )

5.3 NGMV Control of Plants with Input Saturation

219

It has been noted previously that if the NGMV cost-function error weighting contains an integrator (denominator Pcd ðz1 Þ ¼ 1  z1 ) then the controller will also include an integrator. In fact, this rule usually applies to other optimal control laws that include dynamic cost-function weightings. If the plant includes a saturation, integral windup can then occur due to the presence of the integrator in the controller. Anti-windup protection must, therefore, be introduced, based on empirical (non-optimal) methods or by introducing such a solution naturally [8]. It is this latter “optimal” method that will now be described [9]. Let the function satðÞ denote a saturation characteristic, which may represent hard constraints on the input signals to the plant subsystem. For simplicity, the gain in the linear region of this function will be taken as unity. Denote a “nominal” ^c ¼ F ^ ck zk with invertible operator term F ^ ck . choice of the control weighting as F Consider the following modification to the control cost-weighting operator:   ^ ck u ðtÞ þ ðF ck uÞðtÞ ¼ F

q ðuðtÞ  satðuðtÞÞÞ ð1  z1 Þ

ð5:10Þ

where q is a positive scalar gain. The second term in (5.10) does not affect the overall weighted control signal in the linear region but becomes active under saturation conditions. The rationale behind such a choice of the weighting characteristic is to penalize excessive control action when it arises and thereby prevent the integral windup. Modified control: The NGMV optimal control action (4.30), may be written in the form:  ðF ck uÞðtÞ ¼ F0 Yf1 ðW k uÞðtÞ  ðAp Pcd Þ1 G0 Yf1 eðtÞ The effect on this solution when the control weighting is defined as in (5.10) may now be determined. Equating the expression with (5.10): 

 ^ ck u ðtÞ þ F

 q 1 1 1 ð uðtÞ  satðuðtÞÞ Þ ¼ F Y ð W u Þ ð t Þ  ðA P Þ G Y eðtÞ 0 k p cd 0 f f ð1  z1 Þ

After manipulation: ^ 1 ðF0 Y 1 ðW k uÞðtÞ  ðAp Pcd Þ1 G0 Y 1 eðtÞ uðtÞ ¼ F ck f f q ð uðtÞ  satðuðtÞÞ ÞÞ  ð1  z1 Þ

ð5:11Þ

Assume the error weighting Pc includes an integrator to penalize steady-state errors, and define a transfer-function without the integrator R0 ¼ ð1  z1 ÞðAp Pcd Þ1 G0 , or R0 =ð1  z1 Þ ¼ ðAp Pcd Þ1 G0 . Then the optimal control action (4.30), may be written as

220

5 Nonlinear Control Law Design and Implementation

 1 ^ uðtÞ ¼ F ck F0 Yf1 ðW k uÞðtÞ 

R0 q Y 1 eðtÞ  ðuðtÞ  satðuðtÞÞÞ ð1  z1 Þ ð1  z1 Þ f



ð5:12Þ  ^ 1 ð1  z1 Þ1 ð1  z1 ÞF0 Y 1 ðW k uÞðtÞ  R0 Y 1 eðtÞ  qu1 ðtÞ ¼F f f ck

ð5:13Þ

where u1 ðtÞ ¼ uðtÞ  satðuðtÞÞ. When implementing this solution note that there is a problem due to the so-called “algebraic loop” that arises but this is discussed in Sect. 5.4 to follow. Figure 5.9 shows the resulting modified control solution to account for windup, where W k is the delay-free model of the plant. This assumes the output of the saturation can be measured which is certainly the case when it is due to a limit that is applied. Note that both integrators in (5.12) have been moved inside the “anti-windup” loop in (5.13) and the feedback is only active under saturation. It prevents the integrator state from increasing or winding up [10]. The speed of response is determined by the parameter q in this loop. In the linear region, the control law (5.13) collapses to the nominal case. The optimal control action, obtained by introducing the modified control weighting, has a similar structure to a classical anti-windup scheme (compare Fig. 5.8 and Fig. 5.9).

5.3.3

Example: Control of Plant with Saturation

To illustrate the effect of the parameter q in the anti-windup solution, the NGMV control of a linear plant with an input saturation will be considered. The linear part of the plant will be modelled by a simple first-order lag in series with an integrator WðsÞ ¼ K=ðsðss þ 1ÞÞ, with s ¼ 0:5s and K ¼ 1. This may represent the dynamics of a DC motor, where the input is the voltage applied to the motor, and the output is the shaft position. There will normally be a maximum permissible voltage or current that can be applied, and this constraint can be modelled using an ideal saturation term. The saturation function for this example is symmetric and u1  t

U

e(t) R0Y

1 f

– +

–

ˆ 1 ck

–

+

1 1  z 1

u(t)

(1  z 1 ) F0Y f1 Fig. 5.9 NGMV controller modules including anti-windup

y(t)

k

5.3 NGMV Control of Plants with Input Saturation

221

constrains the actual input to the plant to the range ½a; a, with the nominal setting a ¼ 1. Since the controller includes an integrator, arbitrary position setpoints are achievable; however, the transient performance may deteriorate due to integral windup. The modification of the control weighting described above can be used to tackle this problem. A linear GMV controller design has been performed for this system, resulting in ^ck ¼ 0:2. The the following weighting choices Pc ¼ ð1  0:8z1 Þ=ð1  z1 Þ and F final design includes the nonlinear modification as in Eq. (5.13). The NGMV controller contains the discretized model of the plant with a sampling period Ts ¼ 0:1s. Figure 5.10 shows the simulation results for a step change in the reference and the values of the parameter q ranging from 0.005 to 50. The horizontal dash-dotted lines in the control signal plot correspond to the saturation limits. The value q ¼ 0:005 is close to the “no anti-windup” protection case, and the larger that q becomes, the more effective is the correcting action that prevents the state of the integrator from increasing. The final set of results were obtained for a first-order plant with input constraints limiting the control action to the interval ð3; 3Þ. The parameter q in this case was varied from 104 (no anti-windup) to 0.5, and the resulting responses are shown in Fig. 5.11. There is an optimal value of q, corresponding to the best control, which can be determined from the plot as q ¼ 0:025.

Output with different levels of windup

r(t) and y(t)

8 6 4 2 0

0

2

4

6

8

10

12

14

16

18

20

time [s] 60

rho = 0.005 rho = 0.01 rho = 0.1 rho = 50

u(t)

40 20 0 -20 0

2

4

6

8

10

12

14

16

18

20

time [s] Fig. 5.10 Simulation results for an integrating plant with different values of the parameter q

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5 Nonlinear Control Law Design and Implementation

Output

4

setpoint rho = 0.0001 rho = 0.01 rho = 0.025 rho = 0.5

r,y

2 0 -2 -4

0

1

2

3

4

5

6

4

5

6

Control 20

u

10 0 -10 -20 0

1

2

3

time [s] Fig. 5.11 Simulation for a self-regulating plant with PI control, and varying q

5.3.4

Other Input Nonlinearities

A similar approach can be used to handle other input nonlinearities. For example, actuators may have rate constraints limiting their bandwidth. The rate can be computed using a difference operator D ¼ 1  z1 , and the suitable modification of the control weighting follows as   ^ck u ðtÞ þ ðF ck uÞðtÞ ¼ F

q ½DuðtÞ  SatðDut Þ ð1  z1 Þ

ð5:14Þ

This will introduce rate limits in the controller, providing an implicit way of dealing with such constraints. The same approach might be applied to other optimal and predictive control methods by modifying the cost-function to include a similar time-varying weighting, however, this would be an empirical solution. The NGMV control design method is optimal for the nonlinear control costing involved, which provides a little more confidence in its use.

5.4 Algebraic Loop and Implementation

5.4

223

Algebraic Loop and Implementation

One of the problems with the implementation of NGMV controllers is the algebraic loop formed when computing the control signal in Fig. 4.2. This problem arises when both sides of an equation involve the same unknown variable to be computed. To illustrate this problem recall from Eq. (4.19) that the optimal control signal is given as uðtÞ ¼ ðF0 Yf1 W k  F ck Þ1 wp ðtÞ

ð5:15Þ

wp ðtÞ ¼ ðAp Pcd Þ1 G0 Yf1 eðtÞ

ð5:16Þ

where

This algebraic loop is related to the computation of a stable causal inverse for the nonlinear operator: wp ðtÞ ¼ ðF0 Yf1 W k  F ck Þu ¼ F0 Yf1 ðW k uÞðtÞ  ðF ck uÞðtÞ

ð5:17Þ

The optimal control can also be expressed in terms of this signal as 1 uðtÞ ¼ F 1 ck ½F0 Yf ðW k uÞðtÞ  wp ðtÞ

ð5:18Þ

Since F0 , Yf1 and W k do not include delay, the control signal uðtÞ to be calculated appears on both sides of (5.18), creating an algebraic loop when implementing the controller. There are several possible approaches to the solution of this problem: 1. Solve the loop iteratively on-line, as in commercial simulation packages. 2. Break the algebraic loop by introducing an additional delay/memory block by either: • An ad hoc modification to introduce a single-step delay z1 in the controller loop or • Redefinition of the control weighting F c ¼ zK0 þ I F ck , so that (4.27) becomes /0 ðtÞ ¼ F0 eðtÞ þ zK0 þ I ððF ck uÞðtÞ  zI F0 Yf1 ðW k uÞðtÞ þ zI ðAp Pcd Þ1 G0 Yf1 eðtÞÞ and the resulting solution includes the additional delay term in the controller internal loop. 3. Transform this problem into an equivalent problem without an algebraic loop, which does not introduce an approximation. This can be achieved by rearranging the equations so that the computed control signal u(t) only depends upon past values of the signal.

224

5 Nonlinear Control Law Design and Implementation

The first solution involves the use of a numerical solver that at each simulation step iteratively solves a system of nonlinear equations, and this may prevent use for real-time implementation. The second method avoids the algebraic loop because of the zI term but it is a sub-optimal solution and may lead to some degradation in performance (see Sect. 5.4.5). The last approach is in principle optimal and is considered in the following section.

5.4.1

Algebraic Loops in Implementing the Controller

Consider the third of the above approaches, which must be considered on a case-by-case basis. To avoid the algebraic-loop problem, the operator N ¼ ðF0 Yf1 W k  F ck Þ in (5.17) can often be split into two parts involving a term without a delay N 0 and a term that depends upon past values of the control action, denoted N 1 . That is, wp ðtÞ ¼ ðF0 Yf1 W k  F ck ÞuðtÞ ¼ ðN 0 uÞðtÞ þ ðN 1 uÞðt  1Þ

ð5:19Þ

The modified expression for the control signal can then be written as uðtÞ ¼ N 1 0 ðwp ðtÞ  ðN 1 uÞðt  1ÞÞ

ð5:20Þ

The controller structure is shown in Fig. 5.12, where it can be seen that the static term N 0 must be an invertible operator in the assumed region of operation (implying N 0 ðuÞ must be a monotonic function in that region). For implementation, the dynamic nonlinear operator N ¼ ðF0 Yf1 W k  F ck Þ may be represented as N ¼ L G  F where L ¼ F0 Yf1 is a linear transfer, G ¼ W k and F ¼ F ck are nonlinear operators. Assume these may be split into static and delay dependent parts as follows: L ¼ L0 þ L1 z1 ;

G ¼ G0 þ G1 z1 ;

F ¼ F 0 þ F 1 z1

Inverse operator

Fig. 5.12 Implementation of the nonlinear inverse operator

\p ( t )

+ _

u (t )

1 0

1

z 1

5.4 Algebraic Loop and Implementation Fig. 5.13 Optimal controller and implementation of nonlinear inverse operator (to avoid the algebraic loop)

225

p(t) + –

+

 L0

0



0



u(t)

1

z 1

+ 1

–

L0

1

L1

0

The expression for the optimal control may then be written as uðtÞ ¼ ðL0 G0  F 0 Þ1 ½wp ðtÞ  ðL0 G1 þ L1 G0  F 1 Þuðt  1Þ

ð5:21Þ

The optimal control solution is shown in Fig. 5.13 where the inner loop contains a unit-delay to avoid the algebraic-loop problem. In the scalar case, it is straightforward to invert the static characteristic N 0 ¼ L0 G0  F 0 , provided it is monotonic. In fact, the formulation of the control-weighting operator F ck is intended to guarantee this condition. The inverse of N 0 may be implemented as a function inverse or using a look-up table with interpolation. The multivariable case is more complicated as it involves the inverse of a static mapping from n inputs to n outputs. Recall the operator N ¼ ðF0 Yf1 W k  F ck Þ is square even for non-square plants. The inverse may be obtained by means of a look-up table; however, for a high-dimensional system the size of the look-up table may grow rapidly.

5.4.2

Example for Nonlinear State-Space Model

To be able to use (5.20) for the computation of the control signal, an explicit form for the two operators ðN 0 uÞðtÞ and ðN 1 z1 uÞðtÞ is required. This may be obtained from a nonlinear state-space model. Assume the nonlinear subsystem: wp ðtÞ ¼ ðF0 Yf1 W k  F ck ÞuðtÞ ¼ ðN 0 uÞðtÞ þ ðN 1 uÞðt  1Þ This can be represented by an affine control nonlinear state-space model of the form:

226

5 Nonlinear Control Law Design and Implementation

xðt þ 1Þ ¼ f ðxðtÞÞ þ gðxðtÞÞuðtÞ

ð5:22Þ

yp ðtÞ ¼ cðxðt  1ÞÞxðtÞ

ð5:23Þ

wp ðtÞ ¼ yp ðt þ 1Þ

ð5:24Þ

Thence we obtain: wp ðt  1Þ ¼ cðxðt  1ÞÞxðtÞ ¼ cðxðt  1ÞÞðf ðxðt  1ÞÞ þ gðxðt  1ÞÞuðt  1ÞÞ ¼ z1 ðcðxðtÞÞðf ðxðtÞÞ þ gðxðtÞÞuðtÞÞÞ Hence, wp ðtÞ ¼ ðN 0 uÞðtÞ þ ðN 1 z1 uÞðtÞ ¼ cðxðtÞÞf ðxðtÞÞ þ cðxðtÞÞgðxðtÞÞuðtÞ A unit-pulse dðtÞ does not generate an output from the term f ðxðtÞÞ and the two operators may, therefore, be split into a through term and a term that involves a unit-delay:

5.4.3

ðN 0 uÞðtÞ ¼ cðxðtÞÞgðxðtÞÞuðtÞ

ð5:25Þ

ðN 1 z1 uÞðtÞ ¼ cðxðtÞÞðf ðxðtÞÞ

ð5:26Þ

Use of Scaling in the Algebraic-Loop Solution

The term L0 introduced above is not diagonal and this complicates the computation 1 of N 0 which involves the inverse of the operator ðL0 G0  F 0 Þ. This computation can be simplified if an appropriate scaling is used. Introducing a non-singular constant scaling matrix Y0 and following the argument above, the control signal may be written, using (4.19) as  1 1 1 1 uðtÞ ¼ ðY01 F0 D1 Ap Pcd G0 Yf1 eÞðtÞ f B0k W 1k  Y0 F ck Þ ðY0 or  1 ðY01 F0 D1 G0 Yf1 eðtÞ f B0k W 1k  F cy ÞuðtÞ ¼ Ap Pcd Y0

ð5:27Þ

where the scaling on the control weighting F ck ¼ Y0 F cy . Now redefine the scaled signal wp ðtÞ using (5.27) as

5.4 Algebraic Loop and Implementation

227

   1 wp ðtÞ ¼ Ap Pcd Y0 G0 Yf1 eðtÞ ¼ Y01 F0 D1 f B0k ðW 1k uÞðt Þ  F cy u ðtÞ ð5:28Þ The operator in (5.27) may be defined as ðN uÞðtÞ ¼ ðY01 F0 D1 f B0k W 1k  F cy ÞuðtÞ

ð5:29Þ

This operator N may again be split into two parts involving a through term, without a delay N 0 and a term that depends upon past values of the control action, denoted N 1 z1 . That is, 1 wp ðtÞ ¼ ðY01 F0 D1 f B0k W 1k  F cy ÞuðtÞ ¼ ðN 0 uÞðtÞ þ ðN 1 z uÞðtÞ

Thus, from Eq. (5.27): ðN 0 uÞðtÞ þ ðN 1 z1 uÞðtÞ ¼ wp ðtÞ The optimal control then follows, without the presence of the algebraic loop, as 1

uð t Þ ¼ N 0



wp ðtÞ  ðN 1 z1 uÞðtÞ



ð5:30Þ

The operator and its inverse can, therefore, be illustrated, as shown in Fig. 5.14. The crucial condition is the ability to define the through term and the remaining terms involving the explicit delay.

5.4.4

Avoiding the Algebraic Loop and Scaling

Further insights may be gained into the ways to avoid the algebraic loop in the implementation of the NGMV controller discussed in the previous section. Assume the nonlinear plant subsystem model can be written as W 1k ¼ G0 þ z1 G1 and let

Inverse operator

Operator to be inverted

u(t)

1k

Y0 1F0 D f1B0k

+ _

 p (t )

u(t)

+ _

1 0

cy 1

Fig. 5.14 Nonlinear operator and its inverse

z 1

228

5 Nonlinear Control Law Design and Implementation

1 the linear terms Y01 F0 D1 f B0k ¼ L0 þ z L1 , where these nonlinear and linear terms G0 and L0 include no delay terms. Let F cy be split into a non-dynamic, through term F c0 and a delayed term z1 F c1 (that is F cy ¼ F c0 þ z1 F c1 ). A more explicit description of N 0 and N 1 may now be obtained. From (5.29):

  1 wp ðtÞ ¼ N u ¼ ðY01 F0 D1 f B0k W 1k  F cy Þu ¼ L0 G0  F cy þ z ðG1 þ L1 W 1k Þ u ¼ ðN 0 uÞðtÞ þ ðz1 N 1 uÞðtÞ ð5:31Þ The following operators may be identified from this equation: ðN 0 uÞðtÞ ¼ ðL0 G0  F c0 ÞuðtÞ

ð5:32Þ

ðN 1 uÞðtÞ ¼ ðG1 þ L1 W 1k  F c1 ÞuðtÞ

ð5:33Þ

and

The algebraic loop is not present in the implementation of the inverse operator shown in Fig. 5.14 if the inverse of ðL0 G0  F c0 Þ can be computed. This problem can be simplified using a scaling matrix. Scaling: The effect of scaling matrices may now be considered. Let Y0 denote a constant full-rank matrix Y0 ¼ F0 ð0ÞDf ð0Þ1 B0k ð0Þ. This may be used to normalize and diagonalize the first (constant) term in the denominator of the expression for the optimal control (5.27). It is interesting that the scaling matrix Y0 may be related to the weighted plant model. That is, the Diophantine equation in (4.26) may be written in the form: 1 1 K 1 F0 D1 G0 D1 f þ Pcd Ap z f ¼ Pc A

Thence, we obtain: Y0 ¼ F0 ð0ÞDf ð0Þ1 B0k ð0Þ ¼ Pc ð0ÞW0k ð0Þ

ð5:34Þ

Recall that in the previous section, the term Y01 F0 D1 f B0k was expanded into a constant matrix and terms that are delayed by at least one time-step: 1 Y01 F0 D1 f B0k ¼ L0 þ z L1

ð5:35Þ

When the scaling in (5.34) is applied, the constant matrix L0 ¼ I. Clearly, F c0 can also be taken as a diagonal matrix function so that in the majority of problems the operator ðN 0 uÞðtÞ ¼ ðG0  F c0 ÞuðtÞ in (5.30) is easy to invert, even noting these terms may involve a nonlinear and multivariable process.

5.4 Algebraic Loop and Implementation

5.4.5

229

Modified Weighting to Avoid Algebraic Loop

A rather artificial but simple way of avoiding the algebraic-loop problem is now introduced. This involves changing the optimization problem and performance will, therefore, be degraded somewhat. Assume for simplicity that the delays are the same in each signal path and are of the same magnitude k, so that zK ¼ zk I. The change involves redefining the control signal cost-function weighting to be of the form ðF c uÞðtÞ ¼ zm ðF cm uÞðtÞ for 1  m  k. The solution to the modified problem may be found by repeating the main steps in the solution given in Sect. 4.2.1. Thus, introduce the modified Diophantine equation: Ap Pcd F0m þ zm G0m ¼ Pcf Df The solution for ðFm ; Gm Þ satisfies the row j degree of F0m \m  1: Hence, Pc Yf ¼ ðAp Pcd Þ1 Pcf Df ¼ F0m þ ðAp Pcd Þ1 zm G0m Substituting from this result, we obtain: /0 ¼ ðAp Pcd Þ1 Pcf Df e  ðPc W  F c Þu ¼ F0m e þ ðAp Pcd Þ1 zm G0m e  ðPc W  F c Þu Recall e(t) ¼ Yf1 f (t) ¼ Yf1 ðr(t)  d(t)Þ and substituting: /0 ¼ F0m e þ ðAp Pcd Þ1 zm G0m ðYf1 ðr  dÞÞ  ðPc W  F c Þu 

¼ F0m e þ ðAp Pcd Þ1 zm G0m Yf1 e

þ ðAp Pcd Þ1 ðzm G0m  Ap Pcd Pc Yf ÞYf1 W þ F c u Let Pcf ¼ Ap Pcn A1 and Ap Pcd Pc Yf ¼ Ap Pcn A1 Df ¼ Pcf Df , and hence zm G0m  Pcf Df ¼ Ap Pcd F0m . These two equations then give the desired weighted error and control signal as 

/0 ¼ F0m e þ ðAp Pcd Þ1 zm G0m Yf1 e

þ ðAp Pcd Þ1 ðzm G0m  Ap Pcd Pc Yf ÞYf1 W þ F c u   1 ¼ F0m e þ Ap Pcd zm G0m Yf1 e þ F c  F0m Yf1 W u

230

5 Nonlinear Control Law Design and Implementation

The expression for the inferred output becomes /0 ðtÞ ¼ F0m eðtÞ þ zm ðF cm uðtÞ  F0m Yf1 ðW k zmk uÞðtÞ þ ðAp Pcd Þ1 G0m Yf1 eðtÞÞ

ð5:36Þ

Since the row degrees of F0m are required to be  m  1, the first term is dependent upon the values of the white noise signal components eðtÞ,…, eðt  m þ 1Þ. The remaining terms in the expression are all delayed by at least msteps. These terms depend upon the earlier values eðtÞ, eðt  1Þ; . . . and it follows the first and remaining terms are statistically independent. The same optimization argument may, therefore, be used as previously to generate the condition for optimality, which involves setting the round bracketed term in (5.36) to zero. The optimal control in (5.37) below follows. Optimal control: The optimal control signal must, therefore, satisfy  1 1 mk 1 uðtÞ ¼ F 1 F Y ðW z uÞðtÞ  ðA P Þ G Y eðtÞ 0m k p cd 0m cm f f

ð5:37Þ

Clearly this redefinition of F ck avoids the algebraic loop, when the integer m\k (since the control signal on the right side of the expression (5.37) involves only values in the past). Cost implications: The physical significance of the redefinition ðF c uÞðtÞ ¼ zm ðF cm uÞðtÞ is that the weighted error at time t is being minimized and the weighted control action at the time t – m. Now by costing the weighted control uðt  mÞ, this clearly has an influence on the current control. That is, if the control signal is very heavily weighted so the norm of uðt  mÞ is to be small then uðtÞ will also be small. Using the optimal control (5.37) gives /min ðtÞ ¼ F0m eðtÞ and the minimum cost: m

 X Jmin ðtÞ ¼ E ðF0m eðtÞÞT ðF0m eðtÞÞjt ¼ f0jT f0j j¼1

Choice of m: It seems best to choose m = k – 1 to avoid the algebraic loop. If there are a large number of delay steps, the difference in the minimal cost values for the original NGMV criterion with m = k or the above-modified cost-weighting with m = k – 1 may be small. Generalized plant: Recall that the transfer-operator between the control input and minus the inferred output signal /0(t) was referred to as the generalized plant model in Chap. 4 (Sect. 4.3.3), for this class of problems. In the usual case the generalized plant model ðPc W0k zk W 1k  zk F ck Þ but with the modified weighting this becomes ðPc W0k zk W 1k  zm F cm Þ. In the low- to mid-frequency region, which is important to performance, the Pc weighting will normally dominate and in this region, the two models will appear similar. There will be a need to retune the weightings when the control weighting is modified in this way. There is also some

5.4 Algebraic Loop and Implementation

231

impact on stability since, as described in Chap. 4 (Sect. 4.3.3), the necessary condition for optimality is that these operators are stably invertible (minimum phase in the linear case).

5.5

Transport-Delay Compensation Structures

Transport delays, or deadtime, destabilizes and confuses control loops since the effect of control action is not effective until the deadtime is exceeded. Transportdelays are present in many systems and are introduced by • The transportation of material, such as liquid in pipes, or steel strip in a rolling mill. • Measurement and sensor systems. • Digital data communication channels, scan time and computing delays. • Indirectly due to valve stiction, integral windup and deadzone effects. The effect of dead time in a linear system generally decreases the gain and phase stability margins. A lower controller gain must then be applied, which decreases control-loop performance. The result is that classical controllers must normally be detuned particularly when integral action is involved. Transport delays limit the performance that is achievable, and are common in process control systems. The delays cause a deterioration in the disturbance rejection properties, since a delay occurs in the control action, so that the disturbances cannot be countered so effectively. The delays also limit the bandwidth that can be achieved and they reduce robustness to modelling uncertainties. Nonlinearities in a feedback system are not only destabilizing but they can also add to the apparent transport-delay effects.

5.5.1

The Smith Predictor

The Smith Predictor is a well-known transport-delay compensation method used mainly in the chemical and petrochemical process industries, but also used in the steel industry, automotive applications and machinery controls. It has been over half a century since Otto J. M. Smith, a Professor of Mechanical Engineering at the University of California at Berkeley, invented the “predictor” feedback controller. The aim was to provide a controller structure to compensate for significant known transport delays possibly due to the actuators or the process [22]. The Smith Predictor is a particular case of the so-called Internal Model Control principle, which uses an internal model of the plant to estimate the disturbances. The Smith Predictor was developed for stable linear systems. The structure is illustrated in Fig. 5.15. It consists of a feedback controller (a classical controller like a PID design) and an internal feedback loop. This contains a process or plant model, but without the plant transport-delay term. The controller in this internal

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5 Nonlinear Control Law Design and Implementation

d Setpoint +

Controller C0(z -1)

-

Delay free loop

h

+

Delay free model

+

Wk (z -1)

Plant W (z -1)

Plant model W (z -1)

+

Plant Output

+

y

+ -

Fig. 5.15 Smith predictor structure for stable linear systems

loop can be designed as though the time-delay in the process is absent. This considerably simplifies the controller-tuning problem. The design of this internal feedback loop must ensure that this “ideal” closed-loop system, without the time-delay, will have acceptable responses. Since the same control signal is input to the real plant, the plant output will theoretically be the same as the ideal loop but delayed. This is, of course, assuming no modelling errors or disturbances. The Smith predictor provides a measure of disturbance rejection based on the model information. To illustrate the approach for rejecting disturbances let the plant with delay be denoted Wðz1 Þ ¼ zk Wk ðz1 Þ, where Wk ðz1 Þ represents the plant dynamics without deadtime, and zk denotes a k-steps transport delay. The internal loop in the Smith predictor controller structure in Fig. 5.15 generates the main component of the control action. This loop contains a classically designed compensator C0 ðz1 Þ and is tuned assuming the plant involves the delay-free block Wk ðz1 Þ. The premise is that the plant without time-delays will be easy to tune for good reference tracking. The control signal computed in response to a reference change is, therefore, for the loop without the plant delay, and may, therefore, be considered a k-steps ahead control action for the actual plant. This control signal action does not take account of the disturbance. This is the role of the outer dotted feedback path in Fig. 5.15, which includes the estimate of the ^ (formed from the difference of the real plant output and the disturbance signal dðtÞ modelled plant output). If the assumption is made that the disturbance remains the same for k future steps then the total control action produced u(t) can be considered the required future or predicted control, giving rise to the term Smith predictor. Thus in principle the design process is simple and involves designing a classical controller for a delay-free linear stable plant model. The solution assumes there is no model error and the Smith predictor will, therefore, be sensitive to plant model mismatch. Recall one of the assumptions made for the Smith predictor is that the plant is open-loop stable. This is a necessary assumption because of the parallel structure of the true plant and the modelled plant in Fig. 5.15. This parallel path, involving the difference of the plant and modelled outputs, is needed to provide the

5.5 Transport-Delay Compensation Structures

233

^ disturbance estimate dðtÞ. When there is no model mismatch, the control action ^ does not affect the outer loop feedback signal (shown as dðtÞ). There is not, therefore, feedback control in the usual sense. If the open-loop plant were unstable, the parallel path would create unstable hidden modes. It follows that even in the case of ideal models, with no mismatch; unstable plants cannot be stabilized using the Smith predictor controller structure.

5.5.2

Nonlinear Smith Predictor

The controllers in the NGMV family can be expressed in a similar form to that of a Smith predictor, and what might be termed a Nonlinear Smith Predictor will now be derived. However, note that the introduction of this structure limits the application of the solution to open-loop stable systems. That is, although the structure to be derived illustrates a useful link between the NGMV controller and the Smith time-delay compensator, it also has the same disadvantage, that it may only be used on open-loop stable systems. This restriction does not apply to the basic NGMV controller structure in Chap. 4. To demonstrate the link between the Smith predictor and the NGMV controller first assume that the delays are the same in different signal paths, so that the delay block zK ¼ zk Im . Then observe that the control system shown in Fig. 4.2 may be redrawn as in Fig. 5.16, where wp ðtÞ ¼ ðAp Pcd Þ1 G0 Yf1 eðtÞ The changes were made to the linear subsystems in Fig. 5.16 by adding and subtracting equivalent terms. d(t) r(t) + e(t) –

( Ap Pcd ) 1 G0Y f1

 p (t ) +

–

–

+

1 ck

u(t)

+ k

F0Y f1

mk(t)

( Ap Pcd ) 1 G0Y f1 z  k z k I m – +

Fig. 5.16 Modifications to the NGMV controller structure

m(t) +

y(t)

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5 Nonlinear Control Law Design and Implementation

Combine the two inner-loop linear blocks in Fig. 5.16 by first defining the signal: mk ðtÞ ¼ ðW k uÞðtÞ

ð5:38Þ

and writing: 

 1   F0 Yf1 þ ðAp Pcd Þ1 G0 Yf1 Dk mk ¼ Ap Pcd Ap Pcd F0 þ Dk G0 Yf1 mk ð5:39Þ Thence, from (5.38) and (5.39), noting zk Im and G0 commute: ðF0 Yf1 þ ðAp Pcd Þ1 G0 Yf1 Dk Þmk ¼ Pc mk

ð5:40Þ

The system may now be redrawn using this result, as shown in Fig. 5.17, where the control action satisfies Eq. (4.30). This represents a form of nonlinear Smith predictor. If Pc includes an integrator the “denominator” matrix P1 cd must be placed in the inner error channel, rather than in the individual input and feedback blocks (as in Fig. 5.18 with w0 ¼ Pcd wp ). The compensator may, therefore, be rearranged, as shown in Fig. 5.18 to obtain a more practical form of nonlinear Smith predictor. Thus, it has been established that the NGMV controller shown in Fig. 5.18 has a Smith predictor related structure. However, it is also related to the internal model control (IMC) design method discussed in Sect. 5.1.4. The plant model is introduced to estimate the disturbance term and for the structure shown the signal p(t) should be equal to the disturbance term d(t) if there are no modelling errors. Note that although the link to the Smith-predictor time-delay compensator was made, the NGMV approach has the significant advantage that it provides an optimal stochastic control design procedure that should be good for disturbance rejection or tracking varying reference signals. The basic Smith predictor has the limitation that d(t)

r(t)

+ –

e(t)

( Ap Pcd ) 1 G0Y f1

 p (t ) –

+

Plant u(t)

1 ck

+ k

Pc Controller subsystems p(t)

Fig. 5.17 Nonlinear Smith predictor compensator

m(t) +

mk(t)

z k Im +

–

y(t)

5.5 Transport-Delay Compensation Structures

235

it only provides a structure for time-delay compensation. It provides no guidance on how to design the delay-free loop controller C0 ðz1 Þ in Fig. 5.15 for either “good” disturbance rejection or tracking. The NGMV control design approach, therefore, provides what might be thought of as a nonlinear version of the Smith Predictor. It has the advantage that it not only defines a structure for a transport-delay compensator but it also provides a design method. The optimal control minimizes a cost-function that is related to the weighted variance of the error and control terms, so that the stochastic disturbance rejection and tracking properties can be tuned.

5.5.3

Smith Predictor and PID-Motivated Weightings

By writing the NGMV controller in the Smith predictor structure shown in Fig. 5.17, the behaviour of the system can be predicted, particularly when the weightings are selected using the PID motivated procedure suggested in Sect. 5.2.2. That is, define the ratio of the weightings Fk1 Pc ¼ Kc ðz1 Þ, where Kc ðz1 Þ is a stabilizing PID controller [34]. In this case, the inner loop in the Smith predictor form of compensator in Fig. 5.18 has the loop operator Kc W k . The return-difference operator is, therefore, the same as for a delay-free plant with the PID controller Kc ðz1 Þ. In an ideal case with no modelling error, the control signal u(t) does not affect the signal p(t) in Fig. 5.18, which emphasizes that the outer loop does not provide feedback control action. It is only used to provide the prediction of the disturbance. It is not, therefore, surprising that the reference to output response will have similar dynamics to a unity feedback PID controlled system with plant W. In terms of a linear system, the poles of the NGMV controlled system will include those of the PID controlled inner loop in Fig. 5.18. The result is that when the weightings are

d(t)

r(t)

+ –

e(t)

Ap1G0Y f1

(t) – +

1 1 ck Pcd

u(t)

m(t) + +

k

mk(t)

Pcn

z k Im p(t)

+

–

Fig. 5.18 Nonlinear Smith-predictor compensator and internal model ðw0 ¼ Pcd wp Þ

y(t)

236

5 Nonlinear Control Law Design and Implementation

chosen in this way, the NGMV-controlled system has a very similar behaviour to a PID design. These links add to an understanding of the NGMV controller, but it should not be implemented in this form, which is not robust. The controller weightings will also normally require further adjustment. The final NGMV system responses will therefore normally be faster and an improvement on the PID inspired cost-weightings.

5.6

Robustness Analysis and Uncertainty

For linear systems, a simple model that describes the process dynamics, in the region around the unity-gain crossover frequency, is often sufficient for design. However, uncertainties in nonlinear systems can cause unpredictable behaviour in operation and result in performance limitations. These limitations arise due to the lack of fidelity in the nominal plant model that is used for the control systems design. Modelling accuracy is normally good at low frequencies but can deteriorate as the frequency increases, and this limits the bandwidth that can be used. A control system is considered robust if it is insensitive to differences between the real plant and the model used to design the controller, and if it is insensitive to disturbances, and other exogenous signals. Good robust control design practice for linear systems can often be translated into procedures for nonlinear controls, even if additional problems arise [20, 21]. The parameters in a linear design model, approximating a nonlinear system, will, of course, vary due to the nonlinearities and changes in operating conditions. There are two main classes of plant model uncertainty: • Parametric uncertainty: Some of the parameters in the model are uncertain but the model order and structure are assumed known. • Unmodelled dynamics and unstructured uncertainty: The design model can have missing dynamics (possibly at high frequencies), either through the deliberate use of a low-order model, or because of errors in modelling the physical process. Methods of improving robustness that are effective in practice often result from careful enhancements of system models and cost-functions tailored to the application. These application-specific improvements are not very easy to build into a standard control design theory. Mismatch in models and disturbance knowledge often results in low-frequency DC or bias signals associated with plant inputs or outputs. In this case, integrators can be added at the disturbance input or output points of the plant model driven by white noise. The observer or Kalman filter employed in the state-space solution will then include additional integrator states to provide estimates of these terms. Similar remarks apply to polynomial solutions but in this case, the observer/filter is absorbed in the controller polynomial equations. This bias or disturbance compensation technique is also discussed in Sect. 13.1.1, where the Kalman filter algorithm for time-varying systems is summarized.

5.6 Robustness Analysis and Uncertainty

5.6.1

237

Control Sensitivity Functions and Uncertainty

Physical systems have limitations on available bandwidth and performance due to measurement noise, actuator characteristics, right half-plane zeros, time-delays and right half-plane poles. Sensitivity improvements in one frequency range must be paid for with sensitivity deterioration in another frequency range, and the price is higher if the plant is open-loop unstable. The limitations imposed on the sensitivity characteristics by the system description applies to every controller, no matter how it was designed [27]. The sensitivity functions feature large in robustness studies and the NGMV design is reasonably transparent in the relationship between sensitivities and weighting terms. Recall the expression for the NGMV controller in Chap. 4 and defined in Eq. (4.32). That is,  uðtÞ ¼ ðPc W k  F ck Þ1 ðAp Pcd Þ1 G0 Yf1 ðrðtÞ  dðtÞÞ

ð5:41Þ

Stability depends on this inverse term ðPc W k  F ck Þ1 , which in the “linear” scalar problem depends upon the poles of the operator or zeros of the generalized plant ðPc W k  F ck Þ. Stability is, therefore, directly related to the choice of cost-function weightings. The performance and the form of the time-responses depends upon the numerator term ðAp Pcd Þ1 G0 Yf1 , which in turn depends upon the error weighting, reference and disturbance models. The actuator responses are determined by (5.41), that enables a control sensitivity function to be defined as follows: M ¼ ðPc W k  F ck Þ1 ðAp Pcd Þ1 G0 Yf1

ð5:42Þ

This sensitivity is related to the weighting choices and indirectly to the plant model, noise and disturbance signals. The sensitivity properties can, therefore, be tuned using the cost-function weightings and the choice of disturbance models (there is often freedom in the choice of such models that are by their very nature uncertain). To be able to improve robustness via these tuning mechanisms, a method for analysing the stability and performance properties in the presence of uncertainty is required. Robustness properties may be analysed based on the previous Smith predictor version structure, which is, of course, only applicable to stable open-loop plants. Assume the plant has an additive modelling error, so that W k ! W k þ DW k . Then the last figure may be redrawn, as shown in Fig. 5.19. This reveals the compensator designed using the nominal (uncertainty free) model is simply in cascade with the additive error model. The robust control design problem requires that the two cascaded compensator blocks (one involving the internal feedback loop) together with the “plant” zk DW k to satisfy the small gain theorem, for stability (Chap. 1). However, to ensure good stability margins, the weightings must be chosen to shape

238

5 Nonlinear Control Law Design and Implementation

e(t)

r(t) – d(t) +

–

Ap1G0Y f1

(t) – +

1 1 ck Pcd

u(t)

z k 

p(t) k

k

Compensator

Pcn

mk(t)

Fig. 5.19 Nonlinear Smith predictor compensator for uncertain plant model

the frequency responses, particularly in the mid-frequency region. The most desirable loop shapes will depend upon the particular application. A useful property of the NGMV design approach is that the control weighting does not affect the first of the compensator blocks, and only influences the internal feedback loop terms. Thus, frequency shaping is more straightforward using this weighting term, even allowing for the fact the system is nonlinear. Moreover, this control-weighting function may itself be nonlinear, which could be important since the uncertainty model may also be nonlinear.

5.6.2

Uncertainty and Fuzzy/Neural Networks

One method of dealing with uncertainty is to introduce a form of adaptation. This text does not cover formal methods of adaptive control but in applications, adaptation is often introduced using bespoke solutions with limited adaptive range. One way of introducing adaptation or learning is to introduce machine learning via a fuzzy model and/or a neural network. The approach taken throughout the text is to use model-based control design methods, but the NGMV approach is one of the few that can provide a simple bridge to heuristic modelling techniques. The plant model for the NGMV controller includes the nonlinear operator or black-box model. This allows a fuzzy model or neural network to be included in the black-box nonlinear input model. It provides a valuable way of merging a model-based control design method with qualitative modelling approaches. Artificial intelligence (AI), Neural Networks (NN) and nonlinear empirical models can provide a useful modelling approach in some applications [26]. Unfortunately, such models do not extrapolate and NN models do not usually involve physically meaningful parameters. Moreover, the accuracy of models obtained depends upon the quality of the data used for model fitting. However, with reasonable training data, a neural network model is likely to perform well, whilst it stays within the range covered by the training data.

5.6 Robustness Analysis and Uncertainty

239

Fuzzy set theory can be used with neural networks to provide a more intelligent modelling capability. It can be used in a rigorous manner to capture qualitative knowledge. The theoretical foundations of the fuzzy logic approach were developed by Professor Lofti A. Zadeh, using the concept of fuzzy sets (generalization of conventional set theory). Fuzzy sets are related to ideas in multi-valued logic, probability theory, artificial intelligence and artificial neural networks. The most popular kinds of fuzzy system models are the Mandani, Takagi-Sugeno-Kang, Tsukamoto and Singleton models. Fuzzy control invokes the mapping between a continuous and a fuzzy space. It can emulate human reactions and experience in a decision-making process involving a natural language. Fuzzy controllers are implemented using fuzzy rules that can reduce the number of computations compared with conventional controllers. Implementation can be easier than with most conventional controllers, which is useful for consumer goods having limited computing capacity. Fuzzy systems have been used in the control and identification of industrial processes and in many other fields. The fuzzy logic approach, artificial neural networks, genetic algorithms and expert systems are often referred to as intelligent control methods. Fuzzy system theory remains to be fully exploited in both signal processing and control, but it does have real limitations if high performance is needed. The use of AI to determine actions to be taken has obvious applications in automotive systems, such as autonomous vehicles. Measurements of sound or images to recognize the environment enables a map of the activities around a vehicle to be determined. The decisions and actions to be taken can be assisted by AI methods learning from human behaviour. The control of the vehicle then becomes a more traditional control problem that can be solved by optimization and MPC approaches. The black-box input subsystem, in the NGMV plant model, or the related predictive controllers described in the following chapters, can be used to include the neural network or fuzzy models. Note the control law depends upon this black-box term in the block diagram implementation but it does not affect the other computations and is therefore easy to introduce on-line. This can, therefore, provide a useful form of learning/adaptation in the presence of modelling uncertainties (see Refs. [23–25]). One of the potential application areas includes automotive engine modelling, as described in the fuzzy-neural modelling approach taken in the thesis by Sergio Enrique Pinto Castillo [26].

5.6.3

Quantitative Feedback Theory

The use of gain and phase margins in linear systems control is one of the few reliable methods of analysing and improving the robustness of feedback control loops. It was, therefore, a natural next step to develop a frequency response approach to robust control systems design. Horowitz introduced a technique in the 1960s that provided insights into “real robustness”, called quantitative feedback theory (QFT). This is a frequency-domain design method for systems with

240

5 Nonlinear Control Law Design and Implementation

structured uncertainties (see [11–13]). The QFT philosophy is a natural extension of classical frequency-domain design methods. It was initially developed for scalar systems using a Nichols diagram. All requirements, including robust stability and robust performance, can be translated into bounds that are portrayed on the Nichols chart. The uncertainties (structured and unstructured) can be translated into areas in the Nichols chart called templates. There is a clear understanding of the controller gains that will be applied in QFT control design. From a design perspective, Horowitz emphasized the cost of control, which is a topic that had received little attention previously. The cost measure was in terms of the gain used. The amount of feedback-gain must be chosen depending upon the following: • Amount of plant gain. • Magnitude of disturbances. • Level of the performance specifications. QFT is one of a small number of formal robust control design methods. It is very different in nature to H∞ design, which also aims to provide robust control. Unfortunately, QFT design requires classical control engineering expertise in its use and the multivariable controllers were difficult to design. However, multivariable systems can be decomposed into multi-input single-output problems, and the coupling can be treated like disturbances. New software tools are also now available considerably simplifying the use of the technique. A QFT controller can be designed to satisfy closed-loop stability and performance specifications over a range of plant uncertainty. It can provide guaranteed stability properties for a range of uncertainties including those due to parametric variations. The order of the controller may also be fixed by the designer, and controllers are usually of low order and have restricted complexity. The QFT approach is fundamentally different to LQG and H∞ control design. It is based on the classical control idea of frequency shaping the open-loop transfer-function. It also differs in the way the uncertainty is characterized using the gain-phase variations or templates on a Nichols chart. The inclusion of phase uncertainty gives less conservative solutions than with H∞ design or µ- synthesis methods. A benefit of the QFT approach is that it is easy to see the amount of uncertainty, which can be present in the frequency responses of the plant before the system becomes unstable or performance degrades. QFT was initially developed as a design method for linear systems. Nonlinear QFT is often based on the linearization of the uncertain nonlinear plant with respect to a set of closed-loop inputs (due to given references and/or disturbances). The nonlinearities have to be translated into requirements on the QFT templates or bounds. One approach is referred to as a Nonlinear Equivalent Disturbance Attenuation (NLEDA) technique. A second is the so-called Linear Time-Invariant Equivalent (LTIE) method for the QFT design of a nonlinear plant. A robust quantitative feedback theory (robust QFT) design approach that includes a graphical stability criterion for switching linear systems was applied to the design of an

5.6 Robustness Analysis and Uncertainty

241

advanced pitch control system for a Doubly Fed Induction Generator wind turbine by Garcia-Sanz and Elso [15]. The QFT design method is more transparent regarding robustness than most other approaches, and it offers a direct insight into the tradeoffs between the following: • Stability and performance specifications. • Plant and disturbance uncertainty. • Controller complexity and controller bandwidth. Most of the control techniques covered in this text are optimal or predictive controls that are not so transparent in this respect. However, QFT can be used with such methods in an analysis role. It can provide useful insights into the effects of uncertainties at given operating points. This approach may provide a better understanding of how to design effective robust controls. Modern frequency design software tools make the approach much easier to apply and more accessible. Constantine Houpis and colleagues [14] provided clear insights into the QFT design methods and contributed to the development of the software tools. Mario Garcia-Sanz has produced a very user-friendly package for MATLAB called the QFT Control Toolbox (QFTCT).

5.6.4

Linear Matrix Inequalities

Linear Matrix Inequalities (LMI’s) have been used for the solution of a wide range of control problems. Various forms of uncertainty can be included in the problem description and the LMI approach can be used to find the best robust control solution. The early work considered the stability of linear autonomous systems, and employed LQG and H∞ control problem formulations, that can be expressed in terms of LMI’s. The Linear Parameter Varying (LPV) system control problems and some classes of mild nonlinear control problems may also be solved using LMI’s. The use of LMI’s for solving problems in systems theory reaches back to the work of Lyapunov in the 1890s. More recently, Stephen Boyd (Stanford) and co-workers were very influential in promoting the use of LMI methods for the solution of control and signal processing problems (Boyd et al. [28]). A Linear Matrix Inequality (LMI) in the variable x 2 Rn can be written [28] as F ð xÞ ¼ F0 þ x1 F1 þ x2 F2 þ    þ xn Fn ¼ F0 þ

n X

xi Fi [ 0

ð5:43Þ

i¼1

where x ¼ ½ x1 x2    xn T and F0 2 Rmm ; F1 2 Rmm ; . . .; Fn 2 Rmm are given m-square constant symmetric matrices. This matrix inequality is, therefore, linear in the variables xi . The matrix FðxÞ : Rn ! Rmm is an affine function in the

242

5 Nonlinear Control Law Design and Implementation

variable x. When the matrices Fi are diagonal, the LMI F(x) > 0 is just a set of linear inequalities. Multiple LMI’s can always be written as a single LMI of larger dimension. There is, therefore, no need to distinguish between a set of LMI’s and a single LMI. As an example, a set of linear inequalities aTi x  ci for i = 1, 2,…, p, can be written as the following LMI: 2 6 6 6 6 4

c1  aT1 x 0 .. . 0

0 .. .

 0

0 .. .

0 

cp1  aTp1 x 0

0 cp  aTp x

3 7 7 70 7 5

A set of inequalities can, therefore, be written, using a diagonal matrix, as an LMI of the form: diag c1  aT1 x

c2  aT2 x

   cp  aTp x

0

Matrix inequalities can often be transformed into LMI’s by a simple change of variables or congruence transformations. Schur Complement Inequality: The Schur Complement Inequality can be used to transform quadratic matrix inequalities into linear matrix inequalities. For example, the Riccati inequality involved in H∞ control is not an LMI because of the quadratic term. However, the quadratic inequality is convex and it may be transformed into an LMI using Schur’s result. This shows that the following statements are equivalent: " ðiÞ ðiiÞ U22 \0

U¼

U11

U12

UT12

U22

# \0

T and U11  U12 U1 22 U12 \0:

Nonlinear convex inequalities can be converted to an LMI form using this result. Interior point methods may also be applied to obtain the solution of convex optimization problems (Nesterov and Nemirovski [30]). The consequence of this work was that problems, whose solutions could be found numerically, with reasonable computational effort, could be considered solved in some sense, even though there was no analytical solution available. Two types of LMI problems can be defined: • The LMI feasibility problem is to test whether there exist x1 ; . . .; xn so that the LMI’s are satisfied. • The LMI optimization problem is to minimize F0 þ x1 F1 þ x2 F2 þ    þ xn Fn over all x1 ; . . .; xn that satisfy the LMI’s.

5.6 Robustness Analysis and Uncertainty

243

The solution of LMI problems requires the minimization of the linear function under nonlinear and non-smooth but convex constraints with respect to a vector of decision variables. The related convex optimization problems can be considered as semi-definite programs, and there are powerful semi-definite programming tools for solving LMI problems. The LMI toolbox produced by Gahinet and co-workers [29] provided convenient numerical solutions that enabled application studies to progress. With advances in software tools, LMI’s can now be solved very efficiently (solved in the sense that x can be obtained to satisfy the LMI or it can be determined that no solution exists). The book by Boyd et al. [28] illustrated that a wide variety of problems arising in system and control theory could be reduced to a few standard convex or quasi-convex optimization problems involving LMI’s. Traditional control design problems with additional requirements can be put in an LMI-based form, such as LQR, LQG, H2, H∞ and multi-criterion control problems. A Lyapunov inequality: AT P þ PA þ Q  0

ð5:44Þ

can be written as an LMI in the variable P, so that the function: FðPÞ ¼ AT P  PA  Q is affine in P. The use of LMI’s for the design of robust controls is particularly appealing, since it tackles a problem that remains problematic for control system designers [33].

5.6.5

Restricted Structure Nonlinear Controllers

Low-order controllers have a natural robustness and the ubiquitous PID controller is probably the best example. The NGMV controller structure is simple to understand and implement, but in some industries, the experience gained at using a PID parameterization of a controller is so important it suggests replacing the cascade linear block in the NGMV design by a low-order parameterized model. This could involve similar ideas to the so-called Restrictive Structure (RS) control design method for linear systems, introduced in Chap. 2 and Grimble [16, 17] and Majecki and Grimble [18]). This approach is suitable for the design of RS-controllers for systems represented by multiple linear models. Such models can, of course, be used to approximate a mildly nonlinear system. An alternative approach to restricted structure control linking the PID control structures to predictive control was discussed by Uduehi et al. [31]). The development of RS predictive control methods that can be related to this approach is discussed further in the last part of Chap. 11 (Sect. 11.10). A reduced-order NGMV controller has also been developed which retains many of the properties of the

244

5 Nonlinear Control Law Design and Implementation

full-order NGMV controller but enables the controller structure and order to be pre-specified (Grimble [35]). This is suitable for the quasi-Linear Parameter-Varying models discussed in Chap. 1.

5.7

Concluding Remarks

This chapter has covered design issues that are mostly relevant to the NGMV controller, but are also applicable to the other optimal and predictive control design methods described in later chapters. The problem of optimal cost-function weighting definition was considered and various design approaches were described to try to formalize the design procedure. One approach was to parameterise the cost-weightings, so that only a few parameters need be chosen. A second approach was to base the cost-function error weighting on a PID controller structure. If a PID controller exists to stabilize a delay-free plant model, this ensures the existence of a set of weightings to ensure closed-loop stability. If an existing industrial process already uses a PID controller, this can motivate the specification of the NGMV cost-function weightings. This method often provides a reliable starting point for cost-function weighting selection. A nonlinear control problem that often arises in industrial processes is that due to saturation characteristics in actuators. This causes integral windup problems, so that when the system enters saturation, the integrator builds up to produce a large control action and when the error reverses it then takes a long time to recover. In this period no effective feedback control action occurs. Both the classical and optimal ways of solving this problem were discussed. It was shown that the NGMV controller with suitable control cost-function weighting provided an anti-windup capability, which can be related to the traditional classical anti-windup solution. A simpler method that can be applied in the state-space modelling case was also discussed, which involves clamping the state associated with the integrator. This integral action state is contained in the Kalman filter for the state-space version of NGMV controller (Chap. 8). When implementing an NGMV controller the algebraic loop that is present in the standard solution must be avoided, otherwise numerical problems may arise. Several ways to avoid this problem when implementing the controller were discussed. These involve either a minor approximation, or solutions that are dependent on the type of nonlinear system model that is involved. An interesting link to classical transport-delay compensation methods was also discussed in Sect. 5.5, where the NGMV control solution was related to that of the well-known Smith predictor. The NGMV controller can be represented in a similar structure for pedagogical reasons but does not suffer the same disadvantages since in its original form it can stabilize open-loop unstable systems. Various ways to tackle robustness issues were also considered, summarizing the different approaches to improving robustness by different design procedures. This problem is discussed further in later chapters including Chap. 6 that covering H∞

5.7 Concluding Remarks

245

control design. A different approach to attempt to improve robustness is to use low-order restricted structure controllers, as discussed here and in Chaps. 2 and 11. The main message from this chapter was that the design of the family of NGMV controllers is usually straightforward and the problems that can arise when applying the approach have been explored.

References 1. Jutan A (1989) A non-linear PI (D) controller. Canadian J Chem Eng 67(3):485–493 2. Kokotovic P, Arcak M (2001) Constructive nonlinear control: a historical perspective. Automatica 37(5):637–662 3. Grimble MJ (2004) GMV control of nonlinear multivariable systems. In: UKACC control 2004 conference, University of Bath 4. Francis BA, Wonham WM (1976) The internal model principle of control theory. Automatica 12(5):457–465 5. Goodwin GC (1972) Amplitude-constrained minimum variance controller. Electron Lett 8 (7):181–182 6. Goodwin GC, Graebe SF, Levine WS (1993) Internal model control of linear systems with saturating actuators. In: European control conference, pp 1072–1077 7. Majecki P, Grimble MJ (2004) Controller design and performance assessment using nonlinear generalized minimum variance benchmark: scalar case. In: UKACC control conference, Bath, vol 38, No. 1, pp 993–998 8. Hanus R, Kinnaert M, Henrotte JL (1987) Conditioning technique, a general anti-windup and bumpless transfer methods. Automatica 23(6):729–739 9. Grimble MJ, Majecki P (2005) Nonlinear generalised minimum variance control under actuator saturation. In: IFAC world congress, Prague, pp 993–998 10. Seron MM, Graebe SF, Goodwin GC (1994) All stabilizing controllers, feedback linearization and anti-windup: a unified review. In: American control conference, Baltimore, vol 2, pp 1685–1689 11. Horowitz I (1963) Synthesis of feedback systems. Academic Press, New York 12. Horowitz I, Sidi M (1972) Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances. Int J Control 16(2):287–309 13. Horowitz I (1991) Survey of quantitative feedback theory (QFT). Int J Control 53(2):255–291 14. Houpis CH, Rasmussen SJ, Garcia-Sanz M (2006) Quantitative feedback theory: fundamentals and applications. CRC Press, Taylor & Francis, Boca Raton 15. Garcia-Sanz M, Elso J (2009) Beyond the linear limitations by combining Switching and QFT: application to Wind Turbines Pitch Control Systems. Int J Robust Nonlinear Control 19 (1):40–58 16. Grimble MJ (2000) Restricted structure LQG optimal control for continuous-time systems. IEE Proc Pt D 147(2):185–195 17. Grimble MJ (2002) Restricted structure control loop performance assessment for state-space systems. In: American control conference, vol 2. Anchorage, pp 1633–1638 18. Majecki P, Grimble MJ (2004) GMV and restricted-structure GMV controller performance assessment - multivariable case. In: American control conference, Boston, pp 697–702 19. Grimble MJ (2001) Restricted structure control of multiple model systems with series 2 DOF tracking and feedforward action. Opt Control Appl Methods 22(4):157–196 20. Maciejowski JM (1991) Multivariable feedback design. Addison-Wesley Publishing Company, Boston 21. Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice Hall, Upper Saddle River

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22. Smith OJM (1957) Closer control of loops with dead time. Chem Eng Progr 53(5):217–219 23. Zhu Q, Ma Z, Warwick K (1999) Neural network enhanced generalised minimum variance self-tuning controller for nonlinear discrete-time systems. IEE Proc Control Theory Appl 146 (4):319–326 24. Hussain A, Zayed AS, Smith LS (2001) A new neural network and pole placement based adaptive composite self-tuning. In: 5th IEEE international multi-topic conference, Lahore, pp 267–271 25. Hussain A, Grimble MJ, Zayed AS (2006) A nonlinear PID-based multiple controller incorporating a multi-layered neural network learning submodel. Control Intell Syst 34 (3):201–1499 26. Castillo SEP (2011) New fuzzy control architectures applied to industrial applications, Ph.D., Thesis, University of Strathclyde, Glasgow 27. Stein G (1989) Respect the unstable, Bode lecture. IEEE CDC, Tampa, Florida (IEEE Control Syst Mag 23(4):12–25, 2003) 28. Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia 29. Gahinet P, Nemirovskii A, Laub AJ, Chilali M (1994) The LMI control toolbox. In: 33rd IEEE conference on decision and control, Lake Buena Vista, Florida, vol 3, pp 2038–2041 30. Nesterov Y, Nemirovsky A (1988) A general approach to polynomial-time algorithms design for convex programming, Technical report, Economic and Mathematical Institute, USSR Academy of Science, Moscow, USSR 31. Uduehi D, Ordys A, Grimble MJ (2002) Multivariable PID controller design using online generalised predictive control optimisation. In: IEEE international conference on control applications, Glasgow, vol 1, pp 272–277 32. Han J (2009) From PID to active disturbance rejection control. IEEE Trans Ind Electron 56 (3):900–906 33. Kawai F, Vinther K, Andersen P, Bendtsen JD (2017) An LMI approach for robust design of disturbance feedback control. In: SICE annual conference, Kanazawa University, Japan, pp 1447–1452 34. Åström KJ, Wittenmark B (1996) Computer controlled systems, 3rd edn. Prentice Hall, Englewood Cliffs 35. Grimble MJ (2018) Reduced order nonlinear generalized minimum variance control for qLPV systems. IET Proc Pt D 12(18):2495–2506

Chapter 6

Nonlinear Quadratic Gaussian and H∞ Robust Control

Abstract The results in this chapter relate to well-known linear system control laws. It is important to establish this type of link to demonstrate the credibility of the nonlinear controllers presented. An optimization problem is first constructed for a nonlinear system that is shown to provide an LQG controller when the plant is actually linear. The second problem is related, but it involves H∞ cost-minimization. The examples illustrate the difference in the computations required for the two algorithms and the type of behaviour on nonlinear systems. The important lesson is to choose the most suitable control law for a problem that may be dominated by stochastic disturbances or influenced by plant model uncertainties. Even for cases where by careful tuning very similar results can be achieved, it can be very beneficial to have a design method that seems natural for the type of physical system involved.

6.1

Introduction

Two of the more unusual optimal control problems are considered in this chapter. These problems were formulated to obtain controllers having useful properties in special limiting cases. The main premise is that there is more confidence in controllers for nonlinear systems if they reduce to well-known and popular controllers when the system dynamics are linear. The implication is such controllers will be easier to understand, design, tune and use if they have a familiar behaviour when the plant is close to linear or is operating in linear regions. The nonlinear multivariable discrete-time system of interest will again be assumed to contain linear and nonlinear subsystems, where the linear section is represented in a polynomial matrix form. The aim in the first part of the chapter is to introduce a controller for nonlinear stochastic systems that is related to the Linear Quadratic Gaussian (LQG) controller. The Nonlinear Quadratic Gaussian Minimum Variance (NQGMV) control law has the useful property that when the plant model is linear the controller reverts to a LQG design (discussed in Chap. 2). The cost-index for the NQGMV control law © Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_6

247

6 Nonlinear Quadratic Gaussian …

248

involves both dynamically weighted error and control signal costing terms. The solution is useful for applications that are dominated by stochastic disturbances or have relatively high levels of measurement noise. The aim in the second part of the chapter is to develop a so-called Nonlinear Generalized H∞ (NGH∞) controller. The NGH∞ controller minimizes a H∞ norm and enables sensitivity functions to be frequency shaped. The cost-function involves a penalty on both error and control signals, which are related to the sensitivity and control signal sensitivity costing terms. The NGH∞ controller is a special form of the Nonlinear Generalized Minimum Variance (NGMV) controller, introduced in Chap. 4. The link between the NGMV and NGH∞ control problems is established in a lemma that relates the cost-function weightings for the two problems. The H∞ control solution may be useful for systems that are dominated by uncertainties in plant models and disturbances. The need for robustness was discussed briefly in the previous Chap. 5 (Sect. 5.6) on control design. Both of the controllers (NQGMV and NGH∞) have a similar structure; they include an internal model of the process, and are relatively simple to implement. They can both be related to the basic NGMV optimal control problem solution.

6.2

Introduction to NQGMV Control Problem

The main advantage of the NQGMV control design method is that it reduces to an LQG control design in the limiting case when the system is linear. This is one of the most popular optimal control design philosophies [1]. It provides a useful solution for many real applications [2]. This limiting case can enable the weighting choice in the first stage of the design process to be based on the linear plant subsystem. The nonlinearities may then be considered in the second stage of the design process, where the control signal costing (possibly nonlinear) is selected, and the stability issues are considered. An extension over the basic NGMV control law involves a modification to the cost-index that enables the input to the linear subsystem to be penalized (at the output of the input nonlinearity).

6.2.1

Nonlinear System Description

The discrete-time plant model can include severe dynamic nonlinearities of a very general nature. However, the rest of the system description is again defined so that simple results are obtained. The reference and disturbance signals are assumed to have linear time-invariant model representations. The system, shown in Fig. 6.1, includes the nonlinear plant model and the reference, measurement noise and disturbance models. A white measurement noise signal vðtÞ is included in this case and is assumed to have a covariance matrix R ¼ RT  0. There is no loss of

6.2 Introduction to NQGMV Control Problem

249

Fig. 6.1 Closed-loop feedback control system for nonlinear plant

generality in assuming that the zero-mean white noise sources fðtÞ and nðtÞ have identity covariance matrices. The effect of bias on the measurement noise or disturbance signals can also easily be accommodated [4]. Linear plant model: It will be assumed that any unstable modes of the plant are included in the stable/unstable linear time-invariant output subsystem, denoted W0 ðz1 Þ. The disturbance, reference and linear subsystem models may be represented by the left-coprime polynomial matrix expression: ½Wd0 ðz1 Þ; Wr ðz1 Þ; W0 ðz1 Þ ¼ A1 ðz1 Þ½Cd ðz1 Þ; Er ðz1 Þ; Bðz1 Þ

ð6:1Þ

Without loss of generality, these models have the “common denominator” polynomial matrix Aðz1 Þ. The arguments of the polynomial matrices will often be omitted for simplicity. The plant output subsystem with channel delays may be written as ðW0 u0 ÞðtÞ ¼ zK0 ðW0k u0 ÞðtÞ

ð6:2Þ

where zK0 ¼ diagfzk1 ; zk2 ; . . .; zkr g represent the transport delay elements in the output signal paths. Total nonlinear plant model: ðWuÞðtÞ ¼ W0 ðW 1 uÞðtÞ

ð6:3Þ

The input nonlinear subsystem may also include explicit synchronous transport delays, denoted as

6 Nonlinear Quadratic Gaussian …

250

ðW 1 uÞðtÞ ¼ zk ðW 1k uÞðtÞ

ð6:4Þ

The output of the nonlinear subsystem W 1k is denoted as u0 ðtÞ ¼ ðW 1k uÞðtÞ, which may or may not represent a physical signal. The partitioning of the system into the input and output blocks may be conceptual and not represent actual physical subsystems. If the total plant only contains the nonlinear block then the linear output subsystem may be defined as W0k ðz1 Þ ¼ I. Nonlinear System Stability: The nonlinear subsystem W 1k is assumed to be finite gain stable, and the linear subsystem W0k is assumed to contain any unstable modes [5]. For stability analysis, the time sequences are contained in extensions of the discrete Marcinkiewicz space.

6.2.2

Signals in the Feedback System

The signals shown in the closed-loop system model of Fig. 6.1 may be listed as follows: Error signal: eð t Þ ¼ r ð t Þ  yð t Þ

ð6:5Þ

yðtÞ ¼ d ðtÞ þ ðWuÞðtÞ

ð6:6Þ

r ðtÞ ¼ Wr xðtÞ

ð6:7Þ

d ð t Þ ¼ W d nð t Þ

ð6:8Þ

f ð t Þ ¼ r ð t Þ  d ð t Þ  vð t Þ

ð6:9Þ

e0 ðtÞ ¼ r ðtÞ  zðtÞ ¼ eðtÞ  vðtÞ

ð6:10Þ

Plant output:

Reference:

Disturbance signal:

Combined signal:

Controller input:

Note for later use there is a difference between the tracking error eðtÞ, and the controller input e0 ðtÞ, due to the white measurement noise v(t).

6.2 Introduction to NQGMV Control Problem

6.2.3

251

Filter Spectral Factor

The power spectrum for the combined reference, disturbance and noise models can be computed, noting these are all linear subsystems. That is, Uff ¼ Urr þ Udd þ Uvv ¼ Wr Wr þ Wd Wd þ R where the notation for the adjoint of Wr implies Wr ðz1 Þ ¼ WrT ðzÞ. The generalized spectral factor Yf may be defined to satisfy Yf Yf ¼ Uff , where Yf ¼ A1 Df . The matrix Df is a strictly Schur polynomial matrix that satisfies Df Df ¼ Cd Cd þ Er Er þ R

ð6:11Þ

As in the previous chapters the signal f ðtÞ in (6.9) may be modelled as f ðtÞ ¼ Yf eðtÞ, where eðtÞ is a unity covariance white noise signal.

6.2.4

Control Spectral Factor

The dynamic cost-function error and input signal cost terms will now be introduced that are related to the underlying LQG problem that will be a special case. These and Hr ¼ Br A1 have the right-coprime polynomial matrix forms Hq ¼ Bq A1 q r , respectively. They define the weighted linear plant subsystem model: 1 1 1 B1 A1 1 ¼ Aq A BAr ¼ ðAAq Þ BAr

ð6:12Þ

where B1 and A1 are right-coprime polynomial matrices. A generalized control spectral factor Yc may be defined that satisfies 1     1 1 Yc Yc ¼ W0 Hq Hq W0 þ Hr Hr ¼ A1 r A1 ðB1 Bq Bq B1 þ A1 Br Br A1 ÞA1 Ar 1 1 where Yc ¼ Dc A1 1 Ar ¼ Dc ðAr A1 Þ . The control spectral factor Dc is a polynomial matrix that satisfies

Dc Dc ¼ B1 Bq Bq B1 þ A1 Br Br A1

ð6:13Þ

The polynomial spectral factor Dc depends on the weighting and system descriptions, and it is assumed strictly Schur.

6 Nonlinear Quadratic Gaussian …

252

6.3

Nonlinear Quadratic Generalized Minimum Variance Control

The NQGMV optimal control problem involves the minimization of the variance of a signal /0 ðtÞ shown in Fig. 6.1. This signal depends on a dynamic error weighting matrix term ðHq eÞðtÞ, where the weighting: 1 Hq ðz1 Þ ¼ Bq ðz1 ÞA1 q ðz Þ

ð6:14Þ

and a dynamic input signal weighting matrix ðHr u0 ÞðtÞ, where the weighting 1 Hr ðz1 Þ ¼ Br ðz1 ÞA1 r ðz Þ

ð6:15Þ

It also includes an m-square, nonlinear dynamic control signal cost term ðF c uÞðtÞ, similar to that in the NGMV control problem. The choice of these dynamic cost-function weightings is important. Typically, Hq is a low-pass transfer-function and both F c and Hr have high-pass characteristics [5]. The cost-index to be minimized involves the variance of the inferred output:

 J ¼ E /T0 ðtÞ/0 ðtÞ ¼ E tracef/0 ðtÞ/T0 ðtÞg

ð6:16Þ

where Ef:g denotes the unconditional expectation operator, and the signal to be minimized /0 ðtÞ ¼ Pc eðtÞ þ Fc0 u0 ðtÞ þ ðF c uÞðtÞ

ð6:17Þ

The cost weightings in this expression will now be defined in a rather unusual form, which is only justified when the properties of the resulting controller are discussed below. That is, define the linear cost weightings Pc and Fc0 in terms of the chosen Hq and Hr , respectively, as Pc ¼ Dc 1 B1 Bq Hq

ð6:18Þ

Fc0 ¼ Dc 1 A1 Br Hr zk :

ð6:19Þ

and

Observe that if the system has m control inputs the terms in the inferred output (6.17) will also have m outputs. The dynamic weighting F c may be linear or nonlinear, must be defined so that a stable inverse exists. It will be written in the msquare form ðF c uÞðtÞ ¼ zk ðF ck uÞðtÞ

ð6:20Þ

6.3 Nonlinear Quadratic Generalized Minimum Variance Control

253

where the cost weighting term F ck is dynamic but excludes the explicit transport delay elements. Cost-function definition: Note that the weightings Pc and Fc0 in (6.17) do not represent physical subsystems. They define the criterion to be minimized but do not appear as part of the controller. They may be thought of as representing stable non-causal models, and the motivation for the form of these terms is only apparent in the properties of the controller derived below. That is, for linear systems, where W 1 ¼ I, in the limiting case, where F c ! 0, a LQG controller is obtained. The cost-function in the limiting case involves the variance of the signal /0 ðtÞ ¼ Pc eðtÞ þ Fc0 u0 ðtÞ. The resulting controller is identical to an LQG controller that minimizes the more familiar sum of squares cost terms
 J0 ¼ E ðHq eÞT ðtÞðHq eÞðtÞ þ ðHr u0 ÞT ðtÞðHr u0 ÞðtÞ

ð6:21Þ

(see Grimble [3]). Thus, in the special case of a linear system the optimal control to minimize Eq. (6.16), will have the asymptotic property that as the control costing F c ! 0 the control signal becomes LQG optimal. This is a valuable property since it is often the best controller for linear stochastic systems. For nonlinear systems, the controller obtained is a form of NGMV controller. Input cost weighting: The costing on the input signal u0 ðtÞ that appears in (6.17), shown in Fig. 6.1, can be thought of from two viewpoints. It represents a cost-function weighting on the input to the linear subsystem in the plant. Note that in the special case when the nonlinear subsystem W 1 has a stable realizable inverse and the control weighting F c on the actual control signal u tends to zero, then the input signal u0 will equal the optimal control for the equivalent linear system. This is because in this physically unrealistic case the controller will attempt to cancel the input nonlinearity. Another interpretation of the cost weighting Fc0 that acts on the signal u0 is that it represents the cost on the output of the nonlinear input subsystem W 1 . This may, for example, represent a penalty on the output of the actuators, which are often very nonlinear but may have their own feedback loops. The signal u0 will then represent a measured signal, which might provide the feedback around process control valves (to provide more precise positioning). The main results are summarized in the following theorem and the proof is provided in the sections that follow. Theorem 6.1: NQGMV Optimal Feedback Controller The cost-function (6.16) to be minimized, contains the weighted error, input and control signals, where the 1 error weighting Hq ¼ Bq A1 q , the input weighting Hr ¼ Br Ar , and the control signal weighting ðF c uÞðtÞ¼ zk ðF ck uÞðtÞ. It will be assumed that the operator ðDc ðAr A1 Þ1 W 1k  F ck Þ has a finite gain stable inverse, due to the choice of the weightings Hq , Hr and F c . The optimal control then follows from the solution of

254

6 Nonlinear Quadratic Gaussian …

spectral factorization and Diophantine equations. Let the right-coprime decom1 1 position of the weighted plant B1 A1 1 ¼ Aq A BAr and the spectral factors Df and Dc satisfy Df Df ¼ Cd Cd þ Er Er þ R

ð6:22Þ

Dc Dc ¼ B1 Bq Bq B1 þ A1 Br Br A1

ð6:23Þ

The smallest degree solution ðG0 ; H0 ; F0 Þ, with respect to F0 , must be computed to satisfy the following Diophantine equations zgk Dc G0 þ F0 A2 ¼ zg B1 Bq Bq D2

ð6:24Þ

zg Dc H0  F0 B3 ¼ zg A1 Br Br D3

ð6:25Þ

where the solution satisfies the degree F0 \g þ k, and the right-coprime polynomial 1 1 1 matrices D2 , A2 satisfy A2 D1 2 ¼ Df AAq and B3 , D3 satisfy B3 D3 ¼ Df BAr , respectively. Nonlinear optimal control: The optimal NQGMV control can be computed as   1 1 uðtÞ ¼ F 1 ck H0 ðAr D3 Þ ðW 1k uÞðtÞ  G0 ðAq D2 Þ e0 ðtÞ

ð6:26Þ

The following alternative operator expression is useful for establishing the system properties  1 G0 ðAq D2 Þ1 e0 ðtÞ uðtÞ ¼ H0 ðAr D3 Þ1 W 1k  F ck Proof The proof of this result is presented in the following section.

ð6:27Þ ■

Remarks on the solution: There are many insights to be gained from the results in the above Theorem (a) The control signal costing F ck will often be a full-rank linear operator, but it may be a nonlinear function [6]. (b) The NQGMV control law may be implemented based on Eq. (6.26) and the solution is then as shown in Fig. 6.2, which is easy to implement (to avoid the algebraic loop problem the techniques described in Chap. 5, Sect. 5.4 may be used). (c) The signal /0 ðtÞ ¼ Pc eðtÞ þ Fc0 u0 ðtÞ þ ðF c uÞðtÞ involves a weighting function F c that normally has a negative sign to ensure /0 ðtÞ is minimized by a signal u(t) with negative feedback. The forward path gain of the controller block in Fig. 6.2 is, therefore, usually positive.

6.3 Nonlinear Quadratic Generalized Minimum Variance Control

255

Fig. 6.2 NQGMV control signal generation and controller modules

(d) As arranged in the problem construction, when the system is linear (W 1k ¼ I) and the limit F ck ! 0, the problem reduces to a LQG control problem, where the cost-function weightings Hq and Hr represent the dynamic error and control signal LQG cost-function terms. The optimal control from (6.27), is then the same as the polynomial solution of the dynamically weighted LQG problem (Chap. 2 and Grimble [3]). (e) The solution (6.27) may also be represented, as shown in Fig. 6.3, but this is not practical for implementation in nonlinear problems, because of the required inverse of the nonlinear operator. The structure of the controller in Fig. 6.2 avoids the need to compute this inverse. All that is needed is to compute the output of the actuator model W 1k (as with other members of the NGMV control family). The properties of the control law may be summarized in the following lemma. Lemma 6.1: Properties of the NQGMV Optimal Control Law The minimum cost, when measurement noise is null, is due to the time-invariant linear term that is dependent on the reference and disturbance models. Using the solution of the Diophantine equations (6.24) and (6.25) the minimum cost may be computed as

Fig. 6.3 Equivalent NQGMV controller structure

6 Nonlinear Quadratic Gaussian …

256

I Jmin ¼ jzj¼1

n o dz trace ðDc Dc Þ1 F0 ðz1 ÞF0 ðz1 Þ z

ð6:28Þ

For the special case, when the nonlinear plant subsystem W 1k has a stable inverse and the control weighting F ck tends to the null matrix, the system reverts to a LQG design with dynamically weighted error and control signal weighting terms Qc ¼ Hq Hq and Rc ¼ Hr Hr . The implied equation that determines stability in this limiting case 1 zk G0 D1 2 B1 þ H0 D3 A1 ¼ Dc

ð6:29Þ

and the degree of stability is determined by the return-difference matrix I þ Ar D3 H01 zk G0 ðAq D2 Þ1 W0 ¼ Ar D3 H01 Dc ðAr A1 Þ1

ð6:30Þ ■

Proof By collecting results in the proof that follows:

■

Remarks • The expression (6.28) for the minimum cost can provide a benchmark as discussed in Chap. 4. • A measurement noise signal v(t) need not be included in the model, so long as the control signal cost weightings ensures the controller rolls-off at high frequencies. • The nonlinear subsystem ðW 1k uÞðtÞ in the plant affects the control action through the inner feedback loop in Fig. 6.2. Any unstable modes in the plant model must be included in the output linear subsystem model W0 , but the input nonlinear subsystem model W 1k must be assumed stable. As in the previous chapters, this black-box part of the plant model W 1k need not be available in traditional model-based equation form.

6.3.1

Solution NQGMV Optimal Control Problem

To obtain the solution of the NQGMV control problem the inferred output /0 ðtÞ, representing the weighted sum of the error, input and control signals, is to be minimized. The inferred output signal in (6.17) was defined in terms of the chosen weighting functions (6.18) and (6.19). From the equations in Sect. 6.2, the tracking error e ¼ r  y ¼ r  d  Wu and hence,

6.3 Nonlinear Quadratic Generalized Minimum Variance Control

/0 ðtÞ ¼ Pc ðrðtÞ  dðtÞ  ðWuÞðtÞÞ þ ðFc0 u0 ÞðtÞ þ ðF c uÞðtÞ

257

ð6:31Þ

The model for the signal r  d is linear. This is an assumption that may cause a degree of suboptimality in the disturbance rejection properties but in practice, disturbance models are often approximated by linear systems driven by white noise. The signal f ðtÞ may be modelled in innovations form as f ðtÞ ¼ Yf ðz1 ÞeðtÞ, where Yf ¼ A1 Df is defined by the spectral-factorization relationship (6.11). The spectral factor Df is assumed strictly Schur, and the driving noise sequence eðtÞ denotes a white noise signal of zero-mean and identity covariance matrix [1]. The weighted reference and disturbance terms may be expressed, as in the previous chapter, as Pc ðrðtÞ  dðtÞÞ ¼ Pc Yf eðtÞ þ Pc vðtÞ. The expression for the inferred output may now be derived, recalling the output of the nonlinear subsystem, was denoted by u0 ðtÞ ¼ ðW 1k uÞðtÞ. Equation (6.31) may now be written in the form   /0 ðtÞ ¼ Pc r  d  W0 zk u0 þ ðFc0 u0 ÞðtÞ þ ðF c uÞðtÞ ¼ Pc ðYf e þ vÞ þ ðFc0  Pc W0 zk Þu0 þ F c u

ð6:32Þ

The matrix Fc0  Pc W0 zk in this expression may be simplified by substituting 1 1 1 1 from (6.17), and noting Hq ¼ Bq A1 q , Hr ¼ Br Ar and W0 ¼ A B ¼ Aq B1 A1 Ar . Hence, we obtain 

   Fc0  Pc W0 zk ¼ ðzg Dc Þ1 zg A1 Br Hr zk þ zg B1 Bq Hq W0 zk 1 ¼ ðzg Dc Þ1 zgk ðA1 Br Br A1 þ B1 Bq Bq B1 ÞA1 1 Ar 1 k ¼ ðDc A1 1 Ar z Þ

The desired simplified linear polynomial matrix equation: 1 k Fc0  Pc W0 zk ¼ ðDc A1 1 Ar z Þ

ð6:33Þ

Substituting (6.33) into (6.32): 1 k /0 ðtÞ ¼ Pc ðYf e þ vÞ  ðDc A1 1 Ar z Þu0 þ F c u

where u0 ðtÞ ¼ ðW 1k uÞðtÞ. The first term in (6.34) includes Pc Yf ¼ ðzg Dc Þ1 zg B1 Bq Hq Yf ¼ ðzg Dc Þ1 ðzg B1 Bq Bq D2 A1 2 Þ

ð6:34Þ

6 Nonlinear Quadratic Gaussian …

258

Substituting in (6.34) 1 1 k /0 ðtÞ ¼ ðzg Dc Þ1 zg B1 Bq Bq D2 A1 2 e þ Pc v  ðDc A1 Ar z Þu0 þ F c u

ð6:35Þ

Before continuing with the solution two Diophantine equations must be introduced that will enable the first term in this expression to be written as two terms in terms of future and past values of eðtÞ.

6.3.2

Diophantine Equations

To expand the weighted combined disturbance and reference model (representing the first term in (6.35)), introduce the first Diophantine equation zgk Dc G0 þ F0 A2 ¼ zg B1 Bq Bq D2

ð6:36Þ

where the solution for ðF0 ; G0 Þ, satisfies the degree F0 \g þ k. Equation (6.36) can be written as g  1 g  1 g   1 zk G0 A1 2 þ ðz Dc Þ F0 ¼ ðz Dc Þ z B1 Bq Bq D2 A2

ð6:37Þ

This enables the first term in (6.35) to be simplified. A second Diophantine equation is also needed for the stability analysis in the asymptotic situation where the control weighting F c tends to zero. To motivate this second equation, note that (6.37) may be written as 1 g   zgk Dc G0 D1 2 B1 þ F0 Df AAq B1 ¼ z B1 Bq Bq B1

ð6:38Þ

Also, observe that the term on the right of this equation is the same as the first group of terms in (6.13). A second Diophantine equation will also be introduced that will provide an equation related to the second group of terms A1 Br Br A1 in the spectral factor expression. The form of the second equation will be postulated and the required implied equation result will then be obtained. Let ðF0 ; H0 Þ satisfy the second Diophantine equation: zg Dc H0  F0 B3 ¼ zg A1 Br Br D3

ð6:39Þ

1 where B3 D1 3 ¼ Df BAr . Equation (6.39) may now be written as 1 g   zg Dc H0 D1 3  F0 B3 D3 ¼ z A1 Br Br

or

1 g   zg Dc H0 D1 3 A1  F0 Df BAr A1 ¼ z A1 Br Br A1

ð6:40Þ

6.3 Nonlinear Quadratic Generalized Minimum Variance Control

259

Adding the Eqs. (6.38) and (6.40), and noting that AAq B1 ¼ BAr A1 and (6.13):   1 g  zg Dc zk G0 D1 2 B1 þ H0 D3 A1 ¼ z Dc Dc Implied equation: Cancelling terms in this equation obtain the desired “implied” equation 1 zk G0 D1 2 B1 þ H0 D3 A1 ¼ Dc

ð6:41Þ

It is shown in Sect. 6.4 that when the nonlinear dynamics are invertible, and F c ! 0, this equation determines the stability of the resulting linear system.

6.3.3

Simplification of the Inferred Signal

Continuing with the solution of the NQGMV problem, the Diophantine equation (6.37) may be written as k 1 g  1 ðzg Dc Þ1 zg B1 Bq Bq D2 A1 2 ¼ z G0 A2 þ ðz Dc Þ F0

Substituting into (6.35), the expression for the inferred output signal /0 ðtÞ: g  1 1 1 k /0 ðtÞ ¼ ðzk G0 A1 2 þ ðz Dc Þ F0 Þe þ Pc v  ðDc A1 Ar z Þu0 þ F c u g  1 1 1 ¼ ðzk G0 A1 2 þ ðz Dc Þ F0 Þe þ Pc v þ ðF c  ðDc A1 Ar Þ W 1 Þu

ð6:42Þ

but e ¼ Yf1 f ¼ Yf1 ðr  d  vÞ ¼ Yf1 ðe0 þ WuÞ. Substituting into Eq. (6.42): 1 g  1 1 1 /0 ðtÞ ¼ zk G0 A1 2 Yf ðe0 þ WuÞ þ ðz Dc Þ F0 e þ Pc v þ ðF c  ðDc A1 Ar Þ W 1 Þu 1 ¼ ðzg Dc Þ1 F0 e þ Pc v þ zk G0 A1 2 Y f e0 1 1 1 þ ðF c þ zk G0 A1 2 Yf W  ðDc A1 Ar ÞW 1 Þu

ð6:43Þ

To simplify this equation note the relationship: 1 1 1 k 1 1 1 1 ðzk G0 A1 2 Yf W  ðDc A1 Ar ÞW 1 Þ ¼ ðz G0 A2 Yf W0  ðDc A1 Ar ÞÞW 1 1 1 1 1 1 1 1 1 1 Now G0 A1 2 Yf W0 ¼ G0 A2 Df B ¼ G0 D2 Aq A B ¼ G0 D2 B1 A1 Ar from (6.29) we obtain

and

6 Nonlinear Quadratic Gaussian …

260



  k  1 1 1 1 1 1 1 1 zk G0 A1 2 Yf W0  ðDc A1 Ar Þ W 1 ¼ z G0 D2 B1 A1 Ar  ðDc A1 Ar Þ W 1 1 ¼ H0 D1 3 Ar W 1

Substituting this result in (6.43) and advancing the inferred output k-steps gives h

gþk /ðt þ kÞ ¼ D1 F0 eðtÞ þ Pc vðt þ kÞ c z

i 1 þ G0 ðAq D2 Þ1 e0 ðtÞ þ ðF ck  H0 D1 3 Ar W 1k ÞuðtÞ

6.3.4

ð6:44Þ

Optimization

To obtain the expression for the NQGMV optimal control inspect the form of the weighted error and control signals in (6.44). The signal F0 eðtÞ is multiplied by the matrix ðDc Þ1 and recall that the matrix Dc is strictly non-Schur. Also from the Diophantine equation solution the degree of F0 is less than g + k. The polynomial matrix term ðDc Þ1 zg þ k F0 in Eq. (6.44) may, therefore, be expanded as a convergent sequence of terms in the unit advance operator fz; z2 ; z3 ; z4 ; . . .g. It follows that the first term in (6.44) is dependent upon the future values of the white noise signal components { eðt þ 1Þ, eðt þ 2Þ, eðt þ 3Þ,…}. The measurement noise term vðtÞ in (6.44) slightly complicates the analysis and hence two cases will be considered (a) zero measurement noise and (b) where the error weighting Hq is assumed to be a polynomial matrix of degree less than the number of explicit output channel delays. Either of these cases ensures the term Pc vðt þ kÞ is statistically independent of the remaining terms in (6.44) and simplifies the problem. The final group of terms (in the square brackets in (6.44)) are all dependent upon the past values of the white noise signals, whereas the first two terms only depend upon the future values. It follows that these two groups of terms are statistically independent and the expected value of the cross-terms is null. Moreover, the first two terms on the right-hand side of (6.44) are independent of the control action. The smallest variance is achieved when the remaining terms are set to zero. The optimal control signal is, therefore, obtained by setting the second group of terms to zero to obtain the condition for optimality: 1 G0 ðAq D2 Þ1 e0 ðtÞ þ ðF ck  H0 D1 3 Ar W 1k ÞuðtÞ ¼ 0

ð6:45Þ

This gives rise to two expressions for the NQGMV optimal control with the first being Eq. (6.26) in Theorem 6.1. The optimal controller may be represented in the block diagram form shown in Fig. 6.2. Alternatively, setting the same group of terms to zero gives expression (6.27) in the theorem, and the controller structure is as shown in Fig. 6.3. The former is the more practical solution for implementation, since it may not be possible to generate the nonlinear operator inverse in Fig. 6.3.

6.4 NQGMV Design and Stability

6.4

261

NQGMV Design and Stability

The solution presented in Theorem 6.1 provides control in terms of the noisy error input e0 ðtÞ. An alternative expression for the NQGMV optimal control signal may be found that is dependent on the exogenous signals. This provides the equation describing the closed-loop behaviour and is useful for analysing performance and stability. Substituting for e0 ðtÞ in the condition for optimality (6.45): 1 G0 ðAq D2 Þ1 ðr ðtÞ  d ðtÞ  vðtÞ  ðWuÞðtÞÞ þ ðF ck  H0 D1 3 Ar W 1k ÞuðtÞ ¼ 0

ð6:46Þ 1 Noting B1 A1 1 ¼ Aq W0 Ar the implied equation (6.41) may be written as

zk G0 ðAq D2 Þ1 W0 þ H0 ðAr D3 Þ1 ¼ Dc ðAr A1 Þ1

ð6:47Þ

Simplifying (6.46) using (6.47) now obtain the condition for optimality in the form G0 ðAq D2 Þ1 ðr ðtÞ  d ðtÞ  vðtÞÞ   1  zk G0 ðAq D2 Þ1 W0 þ H0 D1 ðW 1k uÞðtÞ þ F ck uðtÞ ¼ 0 3 Ar or

G0 ðAq D2 Þ1 ðr ðtÞ  d ðtÞ  vðtÞÞ  Dc ðAr A1 Þ1 ðW 1k uÞðtÞ þ F ck uðtÞ ¼ 0 ð6:48Þ

The optimal control signal now follows from (6.48), in terms of the exogenous inputs   1  uðtÞ ¼ Dc ðAr A1 Þ1 W 1k  F ck G0 ðAq D2 Þ1 ðr ðtÞ  d ðtÞ  vðtÞÞ

ð6:49Þ

The disturbance free output follows as   1  G0 ðAq D2 Þ1 ðr ðtÞ  d ðtÞ  vðtÞÞ ðWuÞðtÞ ¼ W Dc ðAr A1 Þ1 W 1k  F ck ð6:50Þ These expressions may be simplified by defining the signal pðtÞ as pðtÞ ¼ G0 ðAq D2 Þ1 ðr ðtÞ  d ðtÞ  vðtÞÞ

ð6:51Þ

The optimal control and disturbance free output may now be written, respectively, as

6 Nonlinear Quadratic Gaussian …

262

 1 uðtÞ ¼ Dc ðAr A1 Þ1 W 1k  F ck pðtÞ

ð6:52Þ

 1 pðtÞ ðWuÞðtÞ ¼ W Dc ðAr A1 Þ1 W 1k  F ck

ð6:53Þ

and

It is interesting to consider the special case when the control signal weighting function is linear and is written in the form F ck ¼ Fk ðz1 Þ (as in Chap. 4, Sect. 4.3.3). Then (6.51) may be written as 1 pðtÞ ¼ ðDc ðAr A1 Þ1 W 1k  F ck ÞuðtÞ ¼ F k ðI þ F 1 k Dc ðAr A1 Þ W 1k ÞuðtÞ

Thus, obtain the control and disturbance free output, in terms of the exogenous inputs, in the form 1 1 1 uðtÞ ¼ ðI þ F 1 k Dc ðAr A1 Þ W 1k Þ F k pðtÞ 1 1 1 W0 zk W 1k uðtÞ ¼ zk W0 W 1k ðI þ F 1 k Dc ðAr A1 Þ W 1k Þ F k pðtÞ

ð6:54Þ ð6:55Þ

1 The operator ðI þ F 1 k Dc ðAr A1 Þ W 1k Þ, in the above equations, acts like a returndifference matrix equation, which is important for stability and controller design.

6.4.1

Closed-Loop Stability

From the expression for the optimal control (6.52) a necessary condition for stability is that the nonlinear operator ðDc ðAr A1 Þ1 W 1k  F ck Þ1 is stable. Thus, assume that the cost-function weightings are chosen to ensure this operator is finite gain stable, and consider the case where W0 is stable. As in Chap. 4 to obtain a sufficient condition for stability, consider the case where there is more structural information. Assume the weighted plant has a polynomial operator representation 1 Bk1 A1 k1 ¼ ðAr A1 Þ W 1k . From (6.52) 1 uðtÞ ¼ ðDc ðAr A1 Þ1 W 1k  F ck Þ1 pðtÞ ¼ ðDc Bk1 A1 k1  F ck Þ pðtÞ 1 where pðtÞ ¼ ðDc Bk1 A1 k1  F ck ÞuðtÞ ¼ ðDc Bk1  F ck Ak1 ÞAk1 uðtÞ 1 This may be written as uðtÞ ¼ Ak1 ðDc Bk1  F ck Ak1 Þ pðtÞ but B1 A1 1 ¼ 1 1 1 Aq W0 Ar thus W0 ¼ Aq B1 A1 Ar . The disturbance free plant output in (6.53) follows as:

6.4 NQGMV Design and Stability

263

1 k ðWuÞðtÞ ¼ W0 zk W 1k uðtÞ ¼ Aq B1 A1 1 Ar z W 1k uðtÞ

¼ zk Aq B1 Bk1 ðDc Bk1  F ck Ak1 Þ1 pðtÞ

ð6:56Þ

Note that the operator that was assumed stable can be written as ðDc ðAr A1 Þ1 W 1k  F ck Þ1 ¼ Ak1 ðDc Bk1  F ck Ak1 Þ1 and hence both the disturbance free output (6.56) and control are stable.

6.4.2

Treatment of Explicit Transport Delay Terms

In some cases, the model cannot be obtained in terms of an explicit transport delay of k-steps and the remaining plant model. The actual nonlinear plant model may involve a set of equations containing delay terms but where the explicit output or input delays just cannot be removed. In this case, the delay k can be set to zero in Theorem 6.1. The optimal control will still account for any delays contained in W 1k . However, it is better for efficient implementation to remove any delay elements from the plant model when this is possible. The cost-function (6.16) also accounts for the difference between the responses of the control and error terms in a more physically justified manner. That is, it allows for the fact that a change in control action will not affect the plant output for at least k time steps.

6.4.3

Example of NQGMV Control Design

The computational steps and properties of a Nonlinear Quadratic GMV controller are illustrated in the following example. The nonlinear system of interest is assumed discrete-time with a sampling period of Ts ¼ 1 second and the plant includes k = 3 steps delay. The input nonlinearity is static, as shown in Fig. 6.4, and the model has a Hammerstein model form (Chap. 1). The disturbance and reference models are defined as being near-integrators. That is, the controller will “expect” low frequency stochastic variations (random walk) or step changes in the reference and disturbance signals. To illustrate the advantages of the quadratic NGMV control design approach, consider the following two cases of the output subsystem linear plant dynamics • Stable and minimum phase (Case 1), • Stable but including a non-minimum phase factor (Case 2). Recall that both the linear GMV and the NGMV controls become unstable in the second case when the control weighting tends to zero (Chap. 2). However, the

6 Nonlinear Quadratic Gaussian …

264 Fig. 6.4 Static input nonlinearity

NQGMV approach includes spectral factorization and provides a stabilizing control law, even with a small penalty on the control weighting. The linear plant models in this example are defined as Disturbance and reference models: Wd ¼ 0:1=ð1  0:99z1 Þ and Wr ¼ 1=ð1  0:99z1 Þ Linear plant subsystem Case 1 (minimum phase): W0k1 ¼

4  103 ð1  0:75z1 Þ ð1  0:9z1 Þ2 ð1  0:8z1 Þð1  0:1z1 Þ

Linear plant subsystem Case 2 (non-minimum phase): W0k2 ¼

4  103 ð1  1:25z1 Þ ð1  0:9z1 Þ2 ð1  0:8z1 Þð1  0:1z1 Þ

Note that the plant models were selected to have the same DC gain, facilitating the controller design comparisons. The open-loop step responses for both plants are shown in Fig. 6.5, and both the NMP behaviour and the nonlinear gain are evident from the plots. The effect of the NMP zero is similar to introducing a delay into the system. This effect is in addition to the existing explicit delay of k = 3 samples, and limits the allowable controller gain, and hence the achievable speed of response. Explicit delay terms: To illustrate the problem discussed in Sect. 6.4.2 assume that explicit delays cannot be removed from the model for the nonlinear black-box subsystem. The Theorem 6.1 then applies by setting k = 0. In this case the model W 1k will include both the static input nonlinearity and the 3 steps delay. Nominal PID and basic NGMV control: A PID controller that contains a low-pass filter on the derivative term is introduced below for comparison purposes

6.4 NQGMV Design and Stability

265

Fig. 6.5 Open-loop responses

C0 ðz1 Þ ¼ Kp þ Kp =Ti =ð1  z1 Þ þ Kp Td ð1  z1 Þ=ð1  sd z1 Þ The discrete-time pole of the derivative filter was set to sd ¼ 0:5, and the PID controller had the following parameters Kp ¼ 8; Ti ¼ 10s; Td ¼ 2:5s. The weightings for the basic NGMV controller have been selected as Pc ¼ ð1  0:95z1 Þ=ð1  0:99z1 Þ and Fck ¼ 0:1. The error weighting includes a (near) integrator that should provide good low frequency reference tracking and rejection of low frequency disturbance signals. A comparison of PID and NGMV responses for a sequence of reference step changes is shown in Fig. 6.6 for Case 1. In Case 2, the addition of the NMP zero causes some significant deterioration in the responses. The PID loop becomes unstable, whilst the NGMV solution approaches the verge of instability (not shown). These results were obtained for the same design parameters, without retuning the controllers. Detuning the PID controller by reducing the gains, would, of course, improve stability at the expense of slower response. Quadratic NGMV design for NMP dynamics: In the case of NQGMV design, there is no need to detune the controller for Case 2, since the algorithm automatically handles any unstable zeros in the system dynamics. For a meaningful comparison, the same models and weightings were used and the linear control weighting was set to a small value. The responses for Case 1 are indeed similar, as shown in Fig. 6.7. However, as shown in Fig. 6.8 the NQGMV result for Case 2 is greatly improved as compared with the NGMV design, which is unstable. The NMP behaviour when the reference has step changes is still present, as is the explicit delay. However, the closed-loop system is very stable and the response speed is almost the same as the first (minimum phase) case. Comments: Both the NGMV and NQGMV controllers in the example take account of the system nonlinearities and do not have the same operating point

266

6 Nonlinear Quadratic Gaussian …

Fig. 6.6 PID and NGMV responses (Case 1)

Fig. 6.7 NGMV and NQGMV responses (Case 1)

dependency as linear controllers. However, the NQGMV solution can stabilize some systems that cannot be controlled by NGMV without detuning the controller gains. This was the case for the non-minimum phase (NMP) problem, with low control cost weighting. The NQGMV design method should, therefore, provide improved performance in applications such as the rudder-roll stabilization of ships and others exhibiting NMP dynamics.

6.5 Robust Control Design Philosophy

267

Fig. 6.8 NGMV and NQGMV responses (Case 2)

6.5

Robust Control Design Philosophy

When a controller is designed, numerous assumptions are made about the system description and these are incorporated in a mathematical model. Unfortunately, any model will involve modelling errors and uncertainties. There will be two sources of uncertainty due to external signals and due to the model itself. Disturbance signals may be added to the plant inputs, states and outputs and a noise signal is often included on the sensor output. The system model may include unmodelled dynamics and system equations with uncertain time-varying parameters. A control system is considered robust when the changes in performance due to model changes or inaccuracies are within desired bounds. For industrial systems, the performance and robustness requirements of the main process variables or outputs may be quite demanding, but the lower-level loops may have relatively poor performance without affecting the overall quality of control too adversely. A robust control system should maintain the desired performance despite the presence of significant plant uncertainty. In classical terms, the system may be considered robust if changes of loop gains leave the system with an adequate stability margin assessed in terms of gain and phase margins. The issues of the robustness of linear feedback systems became a particular focus of attention from the late 70s to the late 80s [7]. The problems with LQG control design were for example considered by Safonov and Athans [8, 9]. The lack of guaranteed stability margins in LQG designs, when a Kalman filter was needed for state estimation, provoked much debate, and was one of the factors that drove interest in robust and H∞ control design methods. However, “guaranteed” robustness properties are not available in most design approaches except in special cases, such as the Linear Quadratic Regulator state feedback control problem.

6 Nonlinear Quadratic Gaussian …

268

6.5.1

H∞ Control Methods

The aim in the remainder of the chapter is to develop a relatively simple H∞ controller for nonlinear multivariable processes. The interest in minimizing a H∞ criterion stems from the work of Zames and Francis [39]. This early work was for linear systems (Chap. 2 and Sect. 2.5.6) and involved the minimization of sensitivity functions in terms of the H∞ norm to improve robustness. With the perspective of time it seems clear real robustness is an elusive prey and the benefits of H∞ design over dynamically weighted least squares methods (LQ, LQR, LQG), is not so clear. However, the alternative design philosophy and focus can be valuable and provide a different perspective. It is, therefore, useful to derive a nonlinear H∞ controller [38], but with the same requirement as previously. That is, the controller must be as simple as possible to understand and implement. The H∞ robust control design methods that were developed for linear and nonlinear multivariable systems, involved a number of different solution approaches. The state-space H∞ solution methods for linear systems are by far the most popular [10–13]. The polynomial systems approach to H∞ control design for linear systems [14–17] is particularly appropriate for systems where models are derived from system identification methods, and for adaptive or learning control systems [18, 19]. The solution of the multivariable polynomial based H∞ control design problem was first obtained by Kwakernaak [20]. The approach by Grimble [17, 21] involved the construction of a fictitious LQG problem, which provided the solution of the desired H∞ control problem. This technique for solving one and two-block H∞ problems was referred to as LQG embedding [1]. A modified version is employed below for the nonlinear control design problem. Other approaches to minimax tracking, disturbance rejection and sensitivity minimization problems were described by Mosca et al. [24–26]. The so-called Generalized H∞ (denoted GH∞) control problem explored below is a simple class of H∞ problem that is referred to as a one-block problem. This approach was originally developed for linear systems described in polynomial system form. It required the solution of an eigenvalue-eigenvector problem [20, 22, 23]. A similar approach is followed below but for nonlinear processes. The strategy for solving the Nonlinear Generalized H∞ (denoted NGH∞) robust control design problem is to embed it within a NGMV optimal control problem. A linking lemma is established so that the NGH∞ controller can be found by minimizing a modified NGMV cost-index. This requires the introduction of a fictitious weighting function, and follows a similar strategy to that for linear systems [1].

6.5 Robust Control Design Philosophy

6.5.2

269

Problem Description

The ultimate aim is to improve the robustness of control designs in the presence of model uncertainties. However, the nominal system model is assumed known. The discrete-time system description is the same as described at the start of the Chapter (in Sect. 6.2). In this case, the transport delays are assumed to be the same in all channels and to be in the diagonal form zk I. The measurement noise is also assumed null (the measurement noise attenuation is arranged via the cost-function control signal weighting choice). The nonlinear subsystem W 1k is assumed finite gain stable but the linear subsystem W0k is again allowed to contain unstable modes. The total plant model is represented in operator form as W ¼ zk W0k W 1k and the signals and closed-loop system of interest are as shown in Fig. 6.1, but with a different inferred output to be minimized. The generalized spectral factor for the combined reference and disturbance model can be computed, noting these are linear subsystems, using Yf Yf ¼ Uff ¼ Wr Wr þ Wd Wd . The system models are assumed to ensure Yf is strictly minimum phase.

6.5.3

Uncertainty Descriptions

If there is uncertainty in the system, it may be possible to represent this uncertainty by lumping it into a perturbation block, using one of the unstructured uncertainty models. In the case of a linear, time-invariant discrete system, an unstructured uncertainty block Dðz1 Þ may be represented by an unknown transfer-function matrix. Additive, multiplicative and coprime factor structures are popular, and can be expressed as Additive Perturbation:       G z1 ¼ G0 z1 þ DGA z1 Multiplicative Perturbation:       G z1 ¼ G0 z1 1 þ DGM z1 Coprime Factor Uncertainty:     1 G z1 ¼ Nðz1 Þ þ DN ðz1 Þ Mðz1 Þ þ DM ðz1 Þ The nominal system model may be denoted G0 ðz1 Þ or Nðz1 ÞMðz1 Þ1 and the uncertainty block or perturbation is usually assumed to be norm-bounded. These uncertainty models all require different sensitivity-functions to be minimized to

6 Nonlinear Quadratic Gaussian …

270

account for the uncertainties. The use of H∞ design in sensitivity minimization problems is discussed later in this chapter. Some of the possible uncertainty structures are as shown in Fig. 6.9, but there are many options particularly for multivariable systems. The use of linear fractional transformations and standard system descriptions (shown in Fig. 6.9e), to represent the various uncertainty structures, are valuable. The upper block in the standard system model represents the uncertainty, which is assumed to be norm-bounded. For multivariable systems rmax ðDðejx ÞÞ  dðejx Þ, where dðejx Þ denotes a scalar frequency response function and x 2 ½0; p is normally specified by the designer based on the physical system description or experience.

6.5.4

Nonlinear Generalized Minimum Variance

To solve the NGH∞ control problem in the next section the results of the NGMV control problem considered in Chap. 4, will be revisited. However, in this chapter, the measurement noise will be assumed zero. This will provide the results in a suitable form to derive the NGH∞ controller. The NGMV optimal controller

Fig. 6.9 Typical norm-bounded uncertainty structures

6.5 Robust Control Design Philosophy

271

involves the minimization of the variance of the signal /0 ðtÞ ¼ Pc eðtÞ þ ðF c uÞðtÞ (as shown in Fig. 4.1 but without the Fc0 u0 ðtÞ cost term). The dynamic error cost-function weighting is represented by a polynomial matrix Pc ¼ Pcn P1 cd , and the inferred output cost term includes the nonlinear dynamic control weighting function ðF c uÞðtÞ ¼ zk ðF ck uÞðtÞ, where the F ck weighting is full-rank and invertible. The NGMV cost-index will be defined in terms of the unconditional expectation
 as J ¼ Ef/T0 ðtÞ/0 ðtÞg ¼ E tracef/0 ðtÞ/T0 ðtÞg to provide a link to the LQG type of results needed in the Lemma discussed below. The solution is similar to that in Chap. 4, but the equations are in a more convenient form for the subsequent H∞ 1 analysis. Introduce the right-coprime matrices where D2 A1 2 ¼ ðAPcd Þ Df . The model for the signal f ¼ r  d is linear and the spectral factor, corresponding to this signal model, is assumed to be strictly Schur. The assumption is also made that the nonlinear (possibly time-varying) generalized plant operator ðPc W k  F ck Þ has a finite gain stable causal inverse, due to the choice of weighting operators Pc and F ck . The results required are summarized in the theorem below, which is similar to Theorem 4.1 in Sect. 4.3, but uses the system model and notation employed in the current problem. Theorem 6.2: Nonlinear Generalized Minimum Variance Controller The NGMV optimal controller to minimize the variance of the weighted error and control signals may be found by computing the smallest degree solution (G0, F0), with respect to F0, of the Diophantine equation F0 A2 þ zk G0 ¼ Pcn D2

ð6:57Þ

The right-coprime polynomial matrices A2 and D2 satisfy 1 D2 A1 2 ¼ ðAPcd Þ Df

ð6:58Þ

The spectral factor Yf is written in the polynomial matrix form Yf ¼ A1 Df : The optimal control action can be computed as Optimal control:   1 1 ð Þ ð Þ ð Þ uðtÞ ¼ F 1 F D B W u t  G ðP D Þ e t 0 0k 1k 0 cd 2 ck f

ð6:59Þ

Minimum cost: Jmin


 1 ¼ E ðF0 eðtÞÞT ðF0 eðtÞÞ ¼ 2p

Zp


 trace F0 ðejx ÞF0T ðejx Þ dx

ð6:60Þ

p

■

6 Nonlinear Quadratic Gaussian …

272

6.6

NGH∞ Optimal Control Problem

The Nonlinear Generalized H∞ (NGH∞) cost-minimization problem will now be considered. The results will demonstrate how sensitivity terms (linked to robustness), may be minimized. The approach is to exploit the previous NGMV control results above. In the linear case, the problem is known as a one-block H∞ control design problem. The results are not as valuable as for the two-block H∞ control design problem, but the controller is easier to compute and implement. The strategy is to use a so-called NGMV embedding procedure [27]. This enables most of the attention to be focused on the auxiliary NGMV problem, whose solution is available. The NGH∞ cost-function to be minimized includes the same type of weighted error and weighted control terms as in the NGMV criterion but the norm minimized is the H∞ norm. Recall the NGMV cost-index was defined in terms of the variance of the signal /0 ðtÞ ¼ Pc eðtÞ þ ðF c uÞðtÞ. Since this signal is dependent upon the exogenous signals, the /0 ðtÞ may be written (in terms of a closed-loop operator), in the form /0 ðtÞ ¼ ðHc eÞðtÞ, where the mapping from eðtÞ to /0 ðtÞ, is nonlinear. The form of this operator Hc ðz1 Þ is established below. Denote the nonlinear controller as C0 ðz1 Þ and note /0 ðtÞ may be written as /0 ðtÞ ¼ Pc eðtÞ þ ðF c uÞðtÞ ¼ ðPc þ ðF c C0 ÞÞeðtÞ In terms of the nonlinear sensitivity S and the control sensitivity M functions, this signal may be represented as /0 ðtÞ ¼ ðHc eÞðtÞ ¼ ðPc S þ F c MÞ Yf eðtÞ

ð6:61Þ

The NGH1 cost-function is then defined as   J1 ¼  Hc ðz1 Þ  21

6.6.1

ð6:62Þ

Linking Lemma

Consider the system described in Sect. 6.2 that includes the linear and nonlinear subsystems. The variance of the signal /0r ðtÞ ¼ ðWr Hc eÞðtÞ, where Wr may be considered an additional cost-function weighting, is to be minimized. It is a property of the class of NGMV controllers, that the operator Hc is linear when the cost is minimized. Thus at the optimum denote the operator Hc ðz1 Þ ¼ Hc ðz1 Þ to emphasize the linear form. It is shown in the following lemma that if the weighting Wr ðz1 Þ is such that the function Xðz1 Þ ¼ Hc ðz1 ÞHc ðz1 Þ equals a constant diagonal constant matrix k2 I, then the controller also minimizes the H∞ norm of Hc ðz1 Þ. This solution, where the maximum singular value is a constant scalar, is

6.6 NGH∞ Optimal Control Problem

273

referred to as an equalizing solution. The solution where Xðz1 Þ is a diagonal constant matrix is called a super-optimal solution when all singular values of the cost-index are minimized. Lemma 6.2: Auxiliary Problem and Linking Lemma Consider the auxiliary problem of minimizing the variance J of the signal /0r ðtÞ ¼ ðWr Hc eÞðtÞ. Suppose that for some real rational weighting function matrix Wr ðz1 Þ, where Wr ðz1 ÞWr ðz1 Þ [ 0, that the criterion J is minimized by a NGMV controller 2 1 1 1  ðz1 C01 Þ, for which Xðz Þ ¼ Hc ðz ÞHc ðz Þ ¼ k Ir (a real constant matrix) on z ¼ 1. Then this controller also minimizes the H∞ norm of the function Hc ðz1 Þ. ■ Proof The proof of this result is by contradiction and is similar to that in Kwakernaak for linear systems [14, 15]. It uses the fact that any controller that minimizes the variance of the cost for the system of interest is a member of the class of NGMV controllers. These have the important property that the path between the input eðtÞ and the inferred output /0r ðtÞ ¼ ðWr Hc eÞðtÞ becomes linear at the optimum, so that the operator Hc ! Hc may be denoted by a transfer-function Hc ðz1 Þ. This linear behaviour at the optimum enables Parseval’s theorem to be invoked when evaluating the minimum cost. Thus, assume there exists a NGMV controller C0 ðz1 Þ that minimizes the variance of the signal /0r ðtÞ ¼ ðWr Hc eÞðtÞ, for some cost weighting Wr . Then the cost at the optimum Jmin

1 ¼ 2p

Zp


 trace Wr ðejx ÞHc ðejx ÞHcT ðejx ÞWrT ðejx Þ dx

ð6:63Þ

p

where the function at the minimum Xðz1 Þ ¼ Hc ðz1 ÞHc ðz1 Þ ¼ K KT ¼ k2 Ir . The K matrix is a diagonal matrix that at the optimum reduces to kIr in the solution for the H∞ controller, where k is a scalar. According to the Lemma, this controller C0 ðz1 Þ also minimizes the H∞ norm of the operator. Now assume that this is not the optimum controller. This implies there exists another NGMV controller that also minimizes the variance of the cost J, but results in a lower H∞ norm. Thus at the satisfies optimum, the resulting function, denoted X0 ðz1 Þ, 2 jx T jx Hc0 ðe ÞHc0 ðe Þ\ k I. Thence, the minimum variance for this second controller Jmin0

1 ¼ 2p

Zp


 T trace Wr ðejx ÞHc0 ðejx ÞHc0 ðejx ÞWrT ðejx Þ dx \ Jmin

p

This contradicts the assumption that the controller C0 ðz1 Þ minimizes the variance of the cost J. ■

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6.6.2

Robustness Weighting Function

To obtain the NGH∞ optimal control law the error and control signal weightings are now modified by the presence of the fictitious minimum phase, asymptotically stable, weighting filter Wr , where Wr ¼ A1 r Br and Ar , Br are polynomial matrices to be determined [17, 1]. The solution of the NGMV control problem is then obtained for the modified system model, illustrated in Fig. 6.10, which includes the Wr term. If the weighting filter Wr is chosen, so that the conditions at the optimum provide an equalizing solution, then according to the above lemma the controller minimizes the NGH∞ cost-function for the system in Fig. 6.10. To be able to invoke the above lemma, the integrand of the cost-function, at the optimum, must equal I1 ¼ k2 Wr Wr . The first step is, therefore, to introduce a filter Wr ðz1 Þ on the cost-function weightings that will lead to the NGH∞ optimum cost in (6.63). This requires a modification to the weightings so that the error weighting and control weighting become Wr Pc and Wr F c , respectively. The conditions of the lemma will then be satisfied if the weighting Wr is chosen so that the two integrands in (6.60) and (6.63) are equal at the optimum: Wr K KT Wr ¼ k2 Wr Wr ¼ F0 F0

ð6:64Þ

The significance of the diagonal matrix K is discussed below and at the optimum K ¼ kIr . Clearly Eq. (6.64) is satisfied if Wr K ¼ F0s , where Wr can be written in and Ar and Br can be defined as the right-coprime matrix form Wr ¼ Br A1 r 1  Ar = I and Br ¼ F0s K , where F0s satisfies F0s F0s ¼ F0 F0 and F0s is a Schur full-rank polynomial matrix. The solution of the NGMV problem in Theorem 6.2 may, therefore, be used to solve the NGH∞ problem by substituting for the modified cost weightings.

Fig. 6.10 System with additional robustness cost weighting filter

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275

The error weighting becomes Pc ! Wr Pc ¼ F0s K1 Pc and the Eq. (6.57) to be solved has the form F0 A2 þ zk G0 ¼ F0s K1 Pcn D2

ð6:65Þ

The calculations are straightforward because (6.65) can be solved by equating the coefficients of polynomial matrix terms leading to a system of equations that represent an eigenvalue-eigenvector problem [22]. Eigenvalue equation solution: A minimal degree polynomial solution of (6.65) may be found where F0 may be assumed strictly non-minimum phase (any minimum phase component in F0 and F0s can be cancelled). The smallest degree polynomial matrix F0s is, therefore, strictly minimum phase and satisfies F0s ðz1 Þ ¼ F0 ðzÞznf , where nf ¼ degðF0 Þ. The coefficient matrix F0 ð0Þ may be scaled to the identity matrix in (6.65) by multiplying F0 , F0s and G0 by F01 ð0Þ. If the non-Schur polynomial matrix F0 ðz1 Þ is written as F0 ðz1 Þ ¼ Ir þ F1 z1 þ    þ Fnf 1 znf þ 1 þ Fnf znf Then the Schur matrix F0s ðz1 Þ can be written as F0s ðz1 Þ ¼ Fnf þ Fnf 1 z1 þ    þ F1 znf þ 1 þ Ir znf The solution required is that where the magnitude of the eigenvalue jkj is a minimum and the smallest degree solution for F0 is strictly non-Schur. By definition, the related F0s ðz1 Þ is strictly Schur. Optimum function: It may also be noted that the optimum inferred output for the NGMV optimal control problem, has the form /0r ðtÞ ¼ ð Wr Hc eÞðtÞ ¼ F0s K1 Hc eðtÞ ¼ F0 eðtÞ, and at the optimum, the transfer Hc becomes the linear 1 1 F0 ¼ kF0s F0 . all-pass function Hc ¼ KF0s

6.6.3

Solution of the NGH∞ Control Problem

The NGH∞ optimal control problem assumes a stochastic system description. The H∞ aspect of the problem, involves minimizing sensitivities that affect robustness properties. The solution of the NGMV optimal control problem, given in Theorem 6.2; may be invoked using the definitions of Ar and Br derived above (invoking Lemma 6.2). The following theorem may now be established by collecting results. Theorem 6.3: NGH∞ Optimal Controller The NGH∞ optimal controller, defined as a member of the class of NGMV controllers, is required to minimize the criterion (6.62), for the system shown in Fig. 6.10. It can be computed from the solution ðG0 ; F0 ; F0s ; kÞ of the generalized eigenvalue problem defined by the equation

6 Nonlinear Quadratic Gaussian …

276

F0 A2 þ Dk G0 ¼ F0s K1 Pcn D2

ð6:66Þ

where F0 is of minimal degree ðk  1Þ and jkj is a minimum. The strictly Schur spectral factors Df and F0s satisfy Df Df ¼ Er Er þ Cd Cd

 and F0s F0s ¼ F0 F0

ð6:67Þ

The optimal control signal follows as 1 uðtÞ ¼ ðF0s K1 F ck Þ1 ðF0 D1 f ðB0k W 1k uÞðtÞ  G0 ðPcd D2 Þ eðtÞÞ

ð6:68Þ

Minimal cost:  2  2 J1 ¼  Hc ðz1 Þ  1 ¼  Hc ðz1 Þ  1 ¼ k2 Optimum function: Xmin ¼ k2 Ir

ð6:69Þ

Optimum inferred output signal: 1 F0 eðtÞ /0 ðtÞ ¼ kF0s

ð6:70Þ

where eðtÞ may represent a unity variance zero-mean white noise source, and the 1 ■ right-coprime polynomial matrices D2 and A2 satisfy D2 A1 2 ¼ ðAPcd Þ Df . Proof By collecting the above results and noting Pc ! F0s K1 Pc or Pcn ! F0s K1 Pcn in the Diophantine equation (6.57). Also let F ck ! F0s K1 F ck , where K ¼ kI is to be substituted into the expression for the optimal control (6.59). ■

Fig. 6.11 Control signal generation and NGH∞ controller modules

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277

Fig. 6.12 Equivalent single degree of freedom NGH∞ controller structure

Remarks 1. The optimal control solution (6.68) is illustrated in Fig. 6.11 and this reveals that the solution involves an internal model of the process, including the nonlinear block. 2. An alternative solution, following from the condition for optimality, is illustrated in Fig. 6.12, and is determined by the expression 1 1 1 uðtÞ ¼ ðF0 D1 f B0k W 1k  F0s K F ck Þ ðG0 ðPcd D2 Þ eðtÞÞ

3. The necessary condition for stability may be established by relating the results back to the NGMV solution described in Chap. 4 and presented in Theorem 6.2 for the current problem. 4. In the case when the nonlinear system is actually linear and the cost-functions are the same, the solution obtained corresponds to that for the GH∞ controller for linear systems [19]. 5. A super-optimal solution may be defined, with little additional complication, learning from the results in [22].

6.6.4

Super-Optimality and Eigenvalue Problem

The H∞ controller is not unique and the design can be improved by taking advantage of the additional freedom that is available. The special structure of this problem enables a so-called super-optimal H∞ design to be achieved. That is, the 1 equation F0 A2 þ Dk G0 ¼ F0s K1 Pcn D2 where D2 A1 2 ¼ ðAPcd Þ Df can normally be assumed to be a diagonal matrix since the disturbance and reference models are usually diagonal. In this case, the problem reduces to r simple eigenvalue problems with eigenvalues fk1 ; k2 ; . . .; kr g collected in the matrix K ¼ diagfk1 ; k2 ; . . .; kr g. Thus, all the eigenvalues can be minimized and not simply the maximum value.

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278

6.6.5

Cost-Function Weighting Selection and Robustness

The choice of dynamic cost-function weightings in H∞ control problems is much more important than in for example LQG problems, where constant weightings will often suffice. The dynamic cost weightings in H∞ design influence the sensitives more directly than with H2 or LQG problems, because of the norms used. It is. therefore. easier to shape the frequency response characteristics to achieve required bandwidths, and influence the low, middle and high frequency characteristics. To ensure that the one-block H∞ cost-function is well posed and to give good controller characteristics Pc is normally chosen to be low-pass and F c is normally selected as a high-pass transfer-function [2], as discussed in Chap. 5. If Pc is selected to include an integrator, the controller will include integral action. If F c is chosen to be a high-pass filter the controller roll-off can be chosen to improve stability properties, robustness or measurement noise rejection. For a fast, wide bandwidth, closed-loop system the coefficients of F c must normally be small in magnitude relative to those for the error weighting Pc. The actual levels depend upon the plant model scaling. To analyse robustness the sensitivity functions for the nonlinear system may be defined as Sensitivity:  1 Sðz1 Þ ¼ I þ Wðz1 Þ C0 ðz1 Þ Control sensitivity:  1 Mðz1 Þ ¼ C0 ðz1 Þ I þ Wðz1 Þ C0 ðz1 Þ

ð6:71Þ

In terms of these sensitivities, the signals e ¼ r  d  Wu ¼ r  d  W C0 e ¼ ðI þ W C0 Þ1 ðr  dÞ ¼ S ðr  dÞ ð6:72Þ u ¼ C0 ðI þ W C0 Þ1 ðr  dÞ ¼ M ðr  dÞ

ð6:73Þ

The inferred output signal in terms of these nonlinear operators becomes /0 ðtÞ ¼ Pc eðtÞ þ F c uðtÞ ¼ ðPc þ F c C0 Þ eðtÞ ¼ ðPc þ F c C0 Þ ðI þ W C0 Þ1 ðrðtÞ  dðtÞÞ ¼ ðPc S þ F c MÞ Yf eðtÞ

ð6:74Þ

This signal contains the weighted sensitivities, where the weightings include Pc , F c and Yf . The latter can be determined by the stochastic disturbances in the physical problem or treated as a cost weighting function to be chosen.

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279

H∞ cost-function: The above results reveal that the NGH∞ problem can be interpreted in terms of weighted power spectrum minimization, or weighted mixed-sensitivity optimization, where the cost term to be minimized  2 J1 ¼  ðPc S þ F c MÞYf  1

ð6:75Þ

The weighted sensitivity S and control sensitivity M requires the appropriate selection of the cost weightings Pc and F c . The disturbance model that determines the spectral factor Yf also has an impact on the effective sensitivity costing. For an example of a ship fin roll-stabilization example, where the minimization of sensitivity is paramount, see Hickey et al. [35].

6.6.6

GH∞ Controller Example

The computational procedure for the linear and nonlinear problems involves some similar stages. Thus, to illustrate the main parts of the computational procedure first assume the models are in linear parameterized model form. Consider the following scalar linear system and cost-function weighting definitions Plant: W ¼ A1 B ¼

z2 ðb0 þ b1 z1 Þ ð1 þ a1 z1 þ a2 z1 Þ

Weighting functions: Pc ¼ Pcn P1 cd ¼

ðpn0 þ pn1 z1 Þ ð1 þ pd1 z1 Þ

and

1 Fc ¼ Fcn Fcd ¼

ðfn0 þ fn1 z1 Þ 2 z ð1 þ fd1 z1 Þ

Disturbance model:

Wd ¼ A1 Cd ¼ ðc0 þ c1 z1 þ c2 z1 Þ Aðz1 Þ Reference model: Wr ¼ A1 E ¼

ðe0 þ e1 z1 þ e2 z1 Þ Aðz1 Þ

where the filter spectral factor has the form Df ¼ dfo þ df 1 z1 þ df 2 z1 . Solution: Recall (6.66) F0 A2 þ zk G0 ¼ F0s Pcn D2 =k, where D2 A1 2 ¼ ðAPcd Þ1 Df and in this problem D2 ¼ Df ¼ d0 þ d1 z1 þ d2 z2 and A2 ¼ APcd ¼ a20 þ a21 z1 þ a22 z2 þ a23 z3 . Since the polynomial F0 must be of smallest degree deg(F0 ) < k = 2, and F0 must have the form F0 ¼ f0 þ f1 z1 . Thence, F0s ¼ f1 þ f0 z1

6 Nonlinear Quadratic Gaussian …

280

and hence to balance the degrees of the polynomials G0 must be of second-order G0 ¼ g0 þ g1 z1 þ g2 z2 . Let Pcn D2 ¼ pd0 þ pd1 z1 þ pd2 z2 þ pd3 z3 and then the polynomial Eq. (6.65) to be solved becomes ðf0 þ f1 z1 Þða20 þ a21 z1 þ a22 z2 þ a23 z3 Þ þ z2 ðg0 þ g1 z1 þ g2 z2 Þ ¼ ðf1 þ f0 z1 Þðpd0 þ pd1 z1 þ pd2 z2 þ pd3 z3 Þ=k Equating the coefficients of terms in z0 ; z1 ; z2 ; z3 ; z4 note that the resulting equations may be written in terms of the matrix pencil T1  kT2 02

0 B6 0 B6 B6 1 B6 @4 0 0

0 0 0 1 0

0 0 0 0 1

a20 a21 a22 a23 0

3 2 0 0 60 a20 7 7 6 1 6 a21 7 7  k 60 40 a22 5 a23 0

0 0 0 0 0

0 0 0 0 0

0 pd0 pd1 pd2 pd3

3 12 3 pd0 g0 C 6 g1 7 pd1 7 7 C6 7 C6 7 pd2 7 7 C 6 g2 7 ¼ 0 5 pd3 A4 f0 5 0 f1

If this equation is written as ðT1  kT2 Þx ¼ 0 then T1 has a Toeplitz form, which ensures the inverse exists and the equation may be written as ðk0 I  T0 Þx ¼ 0, where T0 ¼ T11 T2 . This is clearly an eigenvalue and eigenvector equation, which may be solved for the eigenvalue of smallest magnitude (smallest jkj). The corresponding eigenvector determines the desired solution xT ¼ ½ g0 g1 g2 f0 f1 , where F0 ¼ f0 þ f1 z1 is non-Schur.

6.6.6.1

Computation of GH∞ Controller for Stable and Unstable NMP Plants

A numerical example will now follow, where the system is initially assumed linear. First Case: Unstable linear NMP plant Consider the system model and weights defined below Plant model: W¼

B ð1  4z1 Þzk ¼ ; A ð1  2z1 Þð1  0:1z1 Þ

Wd ¼

Cd Df ð1  0:5z1 Þ ; ¼ ¼ A A A

Wr ¼ 0

Weighting choice: The weightings were chosen to place two of the poles of the system that depend upon the cost weightings at z = 0.5 and z = 0.2. The control weighting is linear so F c ¼ Fc and the weightings: Pc ðz1 Þ ¼ ð0:5641  0:4282z1 Þ=ð1  0:99z1 Þ and

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281

Fc ðz1 Þ ¼ 0:4359zk =ð1  0:99z1 Þ Solution: To compare the results in Grimble [18] consider the case where the delay k = 1. The plant polynomials may be defined as A ¼ð1  2z1 Þð1  0:1z1 Þ ¼ ð1  2:1z1 þ 0:2z2 Þ; B ¼ ð1  4z1 Þz1 ; Df ¼ ð1  0:5z1 Þ ¼ Cd The solution of the eigenvalue equations gives the minimum cost k = 0.7051. The computed controller C0 ðz1 Þ ¼

1:7659ð1  0:7932z1 Þð1  0:09967z1 Þ ð1 þ 3:47z1 Þð1  0:8046z1 Þ

The resulting return-difference function becomes ð1  0:5006z1 Þð1  0:5z1 Þð1  0:1994z1 Þ ðð1 þ 3:47z1 Þð1  2z1 Þð1  0:8046z1 Þð1  0:1z1 ÞÞ with zeros at almost 0.5 and 0.2 signifying, the system is stable. The usual rule is that the weightings should have frequency responses that are large at low frequency for Pc and are large at high frequency for Fc, which is not satisfied in this case, so the steady state errors are large and the performance is poor. Second Case: Stable linear NMP plant Plant model: W¼

B 0:1ð1  1:25z1 Þzk ¼ ; A ð1  0:8z1 Þð1  0:1z1 Þ

Disturbance model: Wd ¼

Cd Df ð1  0:13z1 Þ2 ¼ ¼ ; A A A

Wr ¼ 0

Consider the case where explicit delay k = 2. Weighting choice: Pc ¼

3ð1  0:5z1 Þ ð1  0:999999z1 Þ

and Fc ¼ 30ð1  0:9z1 Þzk

The solution of the eigenvalue equations gives the minimum-cost k = 5.1631. The computed controller

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282

C0 ðz1 Þ ¼

ðð1 

z1 Þð1

0:20154ð1  0:6669z1 Þð1  0:09978z1 Þ  0:8913z1 Þð1 þ 0:5716z1 Þð1  0:1612z1 Þð1  0:1087z1 ÞÞ

The differences between NGMV and NGH∞ is small for such systems since the use of an integrator in the error weighting forces similar behaviour except at high frequencies. Third Case:Nonlinear Scalar System with Stable NMP Plant Consider the same scalar linear system as in the second case, but with a much greater delay k = 14, and assume the system is nonlinear. New weightings are needed: Pc ¼ 7ð1  0:1z1 Þ=ð1  0:999999z1 Þ and

Fc ¼ 10ð1  0:5z1 Þzk

These weightings have the usual desired frequency response form shown in Fig. 6.13. The magnitude spectrum of /ðtÞ is uniform since the controller is H∞ optimal. The computed k = 176.56 = 44.938 dB’s and it may easily be confirmed that the sum of weighted sensitivities / ¼ ðPc S þ Fc MÞYf is equalizing and of magnitude k. This result confirms that the form of the sensitivities S and M are related to the inverse of the weighting functions (Pc Yf ) and (Fc Yf ), respectively. Frequency responses: The sensitivity and control sensitivity functions and controller frequency response are shown in Fig. 6.14 for the corresponding GH∞ controller. Nonlinear Plant Results Example: The static nonlinearity shown in Fig. 6.15 was introduced and the NGH∞ controlled time responses computed. The open-loop responses for the system are as shown in Fig. 6.16. The input staircase function had equal steps, but the responses are of course operating point dependent. The closed-loop time responses are shown in Fig. 6.17 for different control signal costings. That is, the control costing term Fc was multiplied by increasing Fig. 6.13 Frequency response of cost-function weightings

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283

Fig. 6.14 Frequency response of controller and sensitivity functions in the linear case

Fig. 6.15 Nonlinear input subsystem

values of q (1, 1.2, 1.3 and 1.4) so that the control action was reduced and the outputs became slower. The transport delay are large and of the same order as the dominant time constant for the system, so it is difficult to obtain a fast response. The benefit of feedback and near integral action is, of course, the zero steady state error achieved. However, integral windup protection may be needed because of the saturation characteristic. One of the mechanisms described in Chap. 5 may be used, to improve the recovery time when the reference reverses and the system departs the saturation region. If robustness is a priority, the tuning should be related to the sensitivity function peaks since for linear systems these are related directly to the stability margins and similar behaviour is observed for nonlinear systems. A simple approach is to

6 Nonlinear Quadratic Gaussian …

284 Fig. 6.16 Open-loop step responses for staircase function into the nonlinear system

Fig. 6.17 NGH∞ Closed-loop step responses for different values of parameter q

consider the main operating points and to use linearized models to assess the “local” sensitivities.

6.6.7

Multivariable NGH∞ Control Example

Consider the case of a square non-minimum phase 2  2 nonlinear system, with the following linear subsystems:

6.6 NGH∞ Optimal Control Problem

285

Fig. 6.18 Nonlinearities in input channels

" W0 ¼ z10

1:5ð1 þ 1:2 z1 Þ 1  0:85 z1 2:3 1  0:7 z1

0:15 1  0:91 z1 1 þ 4 z1 1  0:86 z1

# ;

n Wd ¼ diag 1  0:05 1 ; n 0:95 z 0:1 Wr ¼ diag 1  0:90 z1 ;

0:07 1  0:95 z1 0:2 1  0:90 z1

o

o

The static input nonlinearities are as shown in Fig. 6.18 and the system have the form of a Hammerstein model (Chap. 1). The cost weightings are assumed the same for the NGMV and new NGH∞ controllers, as shown in Fig. 6.19. They are chosen using the typical design guidelines, with a lead term added to the control weighting.

Fig. 6.19 Singular values of the cost weightings

286

6 Nonlinear Quadratic Gaussian …

Fig. 6.20 Sensitivity function frequency responses

Fig. 6.21 Comparison of output responses for nonlinear system

The sensitivity plots for the two cases are as shown in Fig. 6.20. They indicate the slightly better robustness of the H∞ design (as measured by the singular value

6.6 NGH∞ Optimal Control Problem

287

peaks), although the difference is not significant. However, it translates into smaller overshoots on the step and disturbance responses, as shown in Fig. 6.21. Further improvements are limited by the non-minimum phase dynamics of the system. A possible remedy to this problem is discussed in the next section.

6.7

Improving the Stability and Performance of H∞ Designs

The above H∞ design solution is simply relative to the existing H∞ nonlinear control algorithms. However, there are a number of disadvantages. In the limiting case, as the control weighting goes to zero, the controller assumes the characteristics of a minimum variance controller, which is unstable on non-minimum phase systems (in its basic form). Not only does this provide a problem of stability but it also limits performance, since there is a restricted range of cost-function weightings that may be employed. In fact, some cost weightings may even lead to an unstable closed-loop design. A way around this problem is similar to that followed for the original minimum variance controller design, described in Chap. 2, of Åström. In the second type of MV controller introduced by Åström (also described in Chap. 2) the numerator B polynomial is factorized into minimum phase and non-minimum phase terms, and then the corresponding optimal control problem are solved. The resulting controller is more complicated than the original but it has the property that it can stabilize non-minimum phase systems. That is, it avoids the unstable pole zero cancellation, present in the original solution. The first step to improving the H∞ design approach above is, therefore, to generate a so-called factorized NGMV controller, and to use the embedding approach described above. The resulting H∞ solution and design procedure provides much greater freedom in the choice of cost-function weightings and, therefore, enables higher performance to be obtained [29]. Moreover, stability properties are better which suggests that the robustness of designs will be improved. Consider the situation where the system is close to being non-minimum phase so that the solution presented above (Theorem 6.3) is on the verge of instability. The factorized solution is stabilizing whether the system is minimum or non-minimum phase and this suggests improved robustness in such sensitive situations. For further details of the factorized NGMV controller, and the factorized H∞ design solution, see Grimble [30]. Probabilistic approach: Another promising approach to improving robustness is to represent unknown parameters probabilistically. That is, the parameters in the plant or disturbance models can be assigned known means and variances and the resulting optimal control problem can then be solved. This approach has been used for solving linear control and estimation problems using H∞ or H2 control techniques [31, 32].

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Restricted structure control: Low order controllers often have a natural robustness as discussed in Chap. 2 (Sect. 2.6) and Chap. 11 (Sect. 11.10) but this behaviour is not captured by the H∞ theory. However, an interesting link between PID and H∞ control was established by Saeki and Aimoto [33]. The related problem of controller tuning to improve robustness, using H2, H∞ and l objective functions, was considered by Rivera and Morari [34].

6.8

Concluding Remarks

Two optimal control approaches for the design of nonlinear multivariable discrete-time systems were introduced. They were related to the NGMV control methods described in the previous chapters but the problems were constructed to have particularly useful properties in stochastic or uncertain systems. The nonlinear plant model is reasonably general for both the NQGMV and the NGH∞ control solutions. The solutions did not require that the structure of the total nonlinear plant model W k be known. The controllers can, therefore, be calculated without all the usual model information required in model-based control law design. The NQGMV controller is particularly suitable for stochastic systems where disturbances dominate, and the NGH∞ controller is useful for sensitivity minimization. The chapter began with the solution of the Nonlinear Quadratic Generalized Minimum Variance (NQGMV) control problem. The cost weightings may be chosen freely, but they are transformed so that when the weighting on the control action tends to zero, the controller becomes equal to an LQG controller. This is unrealistic since the controller attempts to cancel any nonlinear plant dynamics. This is, of course, impractical even when it is possible since it leads to excessive control action. However, the result is useful when the plant is almost linear and it suggests a starting point for cost-function weighting selection using a two-stage design process. The first stage considers only the linear subsystem with the nonlinear subsystem set to the identity, and F c set to zero. The weightings Hq and Hr may then be selected, using a LQG control design procedures [36]. The second stage of the design procedure introduces the weighting F c to retune the closed-loop system to accommodate the nonlinearity. The NGH∞ controller was derived in the second part of the chapter specifically for uncertain systems. The cost-function was defined so that a so-called one-block H∞ control problem resulted, which has a relatively simple solution. This solution was obtained by invoking a link between NGH∞ and NGMV control problems. The solution accounted for the nonlinearity but the system model included linear reference and disturbance models. The polynomial matrix equations to be solved were all linear and the calculation of the NGH∞ controller mainly involved the solution of a standard eigenvalue-eigenvector problem. The controller is simple to understand, compute and implement. Unfortunately, an approach to design that will achieve real robustness in an industrial application remains problematic [28]. The

6.8 Concluding Remarks

289

dynamic cost-function weighting choices were important for a good design and in the future automated tuning may be possible [37].

References 1. Grimble MJ (2006) Robust industrial control: optimal design approach for polynomial systems. Wiley, Chichester 2. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 3. Grimble MJ (1984) LQG multivariable controllers: minimum variance interpretation for use in self-tuning systems. Int J Control 40(4):831–842 4. Grimble MJ (2005) Non-linear generalized minimum variance feedback, feedforward and tracking control. Automatica 41(6):957–969 5. Grimble MJ (2004) GMV control of nonlinear multivariable systems. In: UKACC conference control, University of Bath, 6–9 September 6. Grimble MJ, Majecki P (2005) Nonlinear generalized minimum variance control under actuator saturation. In: 16th IFAC world congress, Prague, pp 993–998 7. Safonov MG (1980) Stability and robustness of multivariable feedback systems. In: MIT Press, Cambridge, MA. (Based on author’s PhD Thesis, Robustness and stability aspects of stochastic multivariable feedback system design, MIT, 1977) 8. Safonov MG, Athans M (1977) Gain and phase margin for multi-loop LQG regulators. IEEE Trans Autom Control 22:173–178 9. Safonov MG, Athans M (1978) Robustness and computational aspects of nonlinear stochastic estimators and regulators. IEEE Trans Autom Control 23:717–725 10. Doyle JC, Glover K, Khargonekar PP, Francis BA (1989) State-space solutions to standard H2 and H∞ control problems. IEEE Trans Autom Control 34(8):831–846 11. Francis BA (1987) Lecture notes in control and information sciences: a course in H∞ control theory. Springer, Berlin 12. Glover K, McFarlane D (1989) Robust stabilization of normalized coprime factor plant descriptions with H∞ bounded uncertainty. IEEE Trans Autom Control 34(8):821–830 13. Safonov MG, Limebeer DJN (1988) Simplifying the H∞ theory via loop shifting. In: IEEE conference on decision and control, Austin, Texas 14. Kwakernaak H (1983) Robustness optimization of linear feedback systems. In: 22nd IEEE conference on decision and control, San Antonio, Texas 15. Kwakernaak H (1985) Minimax frequency domain performance and robustness optimization of linear feedback systems. IEEE Trans Autom Control 30:994–1004 16. Kwakernaak H (1986) A polynomial approach to minimax frequency domain optimization of multivariable feedback systems. Int J Control 44(1):117–156 17. Grimble MJ (1986) Optimal H∞ robustness and the relationship to LQG design problems. Int J Control 43(2):351–372 18. Grimble MJ (1987) H∞ robust controller for self-tuning control applications, Part 1: controller design. Int J Control 46(4):1429–1444 19. Grimble MJ (1987) H∞ robust controller for self-tuning control applications, Part 2: self-tuning and robustness. Int J Control 46(5):1819–1840 20. Kwakernaak H (1984) Minimax frequency domain optimization of multivariable linear feedback systems. IFAC World Congress, Budapest, Hungary 21. Grimble MJ (1993) H∞ Multivariable control law synthesis. IEE Proc, Pt D, 140(5):353–363 22. Grimble MJ (1989) Extensions to H∞ multivariable robust controllers and the relationship to LQG design problems. Int J Control 50(1):309–338 23. Grimble MJ (1989) Minimization of a combined H∞ and LQG cost function for two-degrees of freedom control design. Automatica 25(3):635–638

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24. Casavola A, Mosca E (1989) On the polynomial solution of the H∞ generalized sensitivity minimization problem. In: 29th IEEE conference on decision and control, Tampa, Florida, pp 1500–1505 25. Mosca E, Giarre L (1989) Minimax LQ stochastic tracking and disturbance rejection problems. In: 28th IEEE conference on decision and control, Tampa, Florida, pp 1473–1476 26. Mosca E, Casavola A, Giarre L (1990) Minimax LQ stochastic tracking and servo problems. IEEE Trans Autom Control 35(1):95–97 27. Grimble MJ (1988) Minimax design of optimal stochastic multivariable systems. IEE Proc Pt. D. 135(6):436–440 28. Hvostov HS (1990) On the structure of H∞ controllers. In: American control conference, San Diego, California, pp 2484–2489 29. Casavola A, Grimble MJ, Mosca E (1996) Extensions to generalized LQG and H∞ multivariable controllers. Int J Control 63(3):507–517 30. Grimble MJ (2012) Factorized H∞ control of nonlinear systems. Int J Control 85(7):964–982 31. Grimble MJ (1982) Optimal control of linear uncertain multivariable stochastic systems. IEE Proc, Pt D, Control Theory Appl 129(6):263–270 32. Grimble MJ (1984) Wiener and Kalman filters for systems with random parameters. IEEE Trans 29(6):552–554 33. Saeki M, Aimoto K (2000) PID controller optimization for H∞ control by linear programming. Int J Robust Nonlinear Control 10:83–99 34. Rivera DE, Morari M (1990) Low order SISO controller tuning methods for the H2, H∞ and l objective functions. Automatica 26(2):361–369 35. Hickey NA, Grimble MJ, Johnson MA, Katebi MR, Melville R (1997) Robust fin roll stabilisation of surface ships. In: 36th IEEE conference on decision and control, vol 5, San Diego, pp 4225–4230 36. Grimble MJ (1998) Multi-step H∞ predictive control. J Dyn Control, Kluwer 8:303–339 37. Grimble MJ, Majecki P (2006) Automated tuning of LQG cost function weightings: scalar case. In: IEEE international conference on control applications munich, Germany, pp 1886–1891 38. Isidori A, Astolfi A (1992) Disturbance attenuation and H∞ control via measurement feedback in nonlinear systems. IEEE Trans Autom Control 37(9):1283–1293 39. Zames G, Francis BA (1983) Feedback, minimax sensitivity, and optimal robustness. IEEE Trans Autom Control 28(5):585–601

Chapter 7

Linear and Nonlinear Predictive Optimal Control

Abstract This chapter introduces the most popular modern control paradigm, namely model predictive control. The introduction includes a historical perspective on predictive control and discusses the implicit and explicit form of the algorithms. The unconstrained solution to the linear system problem is first presented, which is similar to that for most predictive control approaches. The more interesting solution is presented in the second part of the chapter where a nonlinear operator term is included in the plant model. This is the last of the control chapters where polynomial system models are used to represent the linear subsystem. Stability issues are explored after the algorithms have been defined and the use of both hard and soft constraints are also considered. The importance of constraints in applications is illustrated in the ship roll stabilization multivariable control example presented.

7.1

Introduction

In the chemical and petrochemical industries, Advanced Process Control (APC) methods are used to deliver optimal plant performance by optimizing setpoints, reducing interaction effects, and improving control system responses. The APC methods are normally based upon Model Predictive Control (MPC) algorithms. In this type of algorithm a dynamic model of the plant is used to predict, and to optimize, the future outputs of the process. The term Model Based Predictive Control (MBPC) is sometimes used to describe this type of control law. The MPC algorithms are particularly suitable for large multivariable systems that have significant interactions and disturbances. The system models can include nonlinearities and hard constraints on both inputs and outputs. The MPC approach is an optimization based control law that provides the only well-established method of treating hard constraints on input or output signals. The MPC control methods were initially developed assuming linear system models and were adopted relatively quickly for industrial control applications in the 70s and 80s. The approach became very popular in the petrochemical and chemical industries. They are often used for critical processes in oil refineries or chemical © Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_7

291

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plants where the constraint handling features are important and there are hundreds of inputs and outputs. The model predictive control philosophy was proposed by two groups completely independently but around the same time period. The two initial approaches were as follows: • Model Algorithmic Control (MAC) introduced by Richalet in France (1978, [1]). • Dynamic Matrix Control (DMC) proposed by Cutler and Ramaker, whilst working for Shell Oil in the United States (1979, [2]). There have been many variants of the basic MPC approach, because of the different plant model descriptions (impulse-response, step-response, state-space, polynomial models), and the different disturbance and reference models used (deterministic and stochastic). Adaptation to time-varying models and scheduling to cope with nonlinearities has often been used. The MPC solutions are mostly based on the minimization of a quadratic cost-function. The mathematical analysis and numerical methods needed are simplified by using a quadratic performance criterion. Efficient optimization algorithms can then be developed for many types of plant model and system description. The potential benefits of MPC can be summarized as follows: • Great success in process plant control and more recently in faster electromechanical systems. • Constraint handling allows traditional safety margins to be reduced so that the plant can be operated near operational boundaries that is often more profitable. • Can often provide improved quality and/or greater efficiency. • Very suitable for large-scale complex systems and for supervisory control systems. A traditional predictive controller uses future reference signal or setpoint information and minimises a multi-step quadratic cost-function. The predictive algorithms compute an open-loop sequence of the controls (manipulated variables) at each sample instant, to optimize the future outputs (process variables). The first computed value of the control in this optimal sequence is used as the actual control action at the current time. The optimization procedure is then repeated at the subsequent sample points. This process is called the receding horizon principle. The resulting control action is not of course the same as in the previously computed control sequence based on finite-horizon optimization, since it is updated at each sample instant. The control actually implemented depends upon the outputs or state measurements at each sample time, and therefore provides a form of feedback control. Predictive controls have been used to improve the performance in difficult systems containing long delay times, time-varying system parameters or strong multivariable interactions. These algorithms were initially applied on relatively slow systems, such as thermal processes, for the chemical, petrochemical, food and cement industries. They are now being applied or assessed for use on much faster systems, such as servo and hydraulic systems, gas turbine applications, automotive

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293

powertrains and robotics control problems. The different predictive control approaches and major historical contributions are described in the next section.

7.1.1

Historical Perspective

The Model Predictive Control (MPC) approach to controller design involves on-line optimization calculations that take account of system dynamics, control objectives, soft and hard constraints. The most popular predictive control algorithms, for linear time invariant systems, are probably Dynamic Matrix Control (DMC), due to Cutler and Raemaker [2]; Generalized Predictive Control (GPC), due to Clarke et al. [3, 4, 5], and the algorithms of Richalet [1, 6]. The GPC optimal control law is very representative of many of the MPC solutions for linear systems. An introduction to GPC and the advantages and limitations was provided in Grimble [7]. The relationship between the popular optimal Linear Quadratic (LQ) and predictive control methods was explored in the seminal text by Bitmead et al. [8]. The GPC controller was originally obtained in a polynomial system form by Clarke and co-workers (see [3–5]), and this was convenient for early applications [9]. A state-space version of GPC that is more suitable for larger systems, was obtained later by Ordys and Clarke [10]. The basis of DMC was developed by Charles Cutler, whilst working for Shell Oil in the United States. He applied the technique very successfully before leaving the company and forming the DMC Corporation in 1984. Cutler was the CEO and president for 12 years until it became part of Aspen Technology in 1996. Most large industrial systems now include advanced condition monitoring and a form of model based predictive control. Supervisory Control and Data Acquisition (SCADA) systems are often used for process control applications covering a wide geographical location and they focus upon the collection of data. The so-called Distributed Control Systems (DCS) are more concerned with a single plant and the design approach is process oriented. They normally include both upper level static optimization and intermediate level dynamic optimization features with predictive control options. Supervisory level control systems set the reference signals, or setpoints, for the lower level regulating loops that are often PI or PID controllers. The early industrial applications of predictive control involved chemical or petrochemical processes that have slow sampling rates of say a minutes’ duration but recent machine control applications can have millisecond sample rates. The model predictive control approach has been applied very successfully in the process industries, where it has improved the profitability and competitiveness of production plant. However, most of the designs have been based on linear control design approaches, often implemented using a scheduling algorithm to cope with nonlinearities.

294

7.1.2

7 Linear and Nonlinear Predictive Optimal Control

Predictive Functional Control

The idea of projecting the manipulated variable, or control signal, onto a functional basis can be found in Richalet [11]. This was given the name Predictive Functional Control (PFC) and had the advantage of simplicity [12]. The Richalet approach to predictive control involves a discrete-time model of the open-loop system and a specified future response that will move the system from the current output to close to the setpoint at some future time point. The criterion is usually a quadratic-function of the error between the reference trajectory and the predicted output, taken over a so-called coincidence horizon. The desired and the predicted future outputs are required to coincide at only a subset of points in the prediction horizon, and not over the whole prediction horizon. An error signal is defined between these signals, and an objective function is minimised. The future PFC control signal trajectory is parameterized in terms of a set of basis functions (selected set of time-functions), multiplied by weighting factors. These weighting factors are selected to minimise the objective function. The future P b control signal uðt þ iÞ ¼ Nj¼1 lj ðtÞbj ðiÞ is represented by a linear combination of the basis functions, denoted bj ðiÞ, and the control signal is implemented as: uðtÞ ¼

Nb X

lj ðtÞbj ð0Þ

j¼1

This PFC approach can be considered a special case of the more general subject of what might be termed Generalized Predictive Functional Control (GPFC). The first PFC applications took place in the early 70s, and since that time there have been many successful applications [13].

7.1.3

Nonlinear Model Predictive Control

Linear MPC theory can be considered as a mature subject; however, many applications require the plant to operate over a wide region, rather than in the neighbourhood of an operating point. For processes that are highly nonlinear, the performance of a MPC control, based on a linear model can be poor. Most commercially available MPC algorithms are based on a linear model of the process. The DCS and SCADA manufacturers therefore provide a scheduling facility to account for gentle nonlinearities (without sharp discontinuities or spikes), and this is normally suitable for chemical and petrochemical processes. The problems in applications that arise due to the presence of nonlinearities has motivated the development of Nonlinear Model Predictive Control (NMPC), where a nonlinear model of the plant is used for both prediction and optimization. Most of the current NMPC schemes are based on a physical model of the process, but in

7.1 Introduction

295

some cases, physical models are difficult and expensive to obtain. A model can in this case be computed from input-output measurements using nonlinear system identification (Chap. 1). Approximations to nonlinear model predictive control have proven successful in some applications, based upon simple scheduling and anti-wind-up methods. This applies to the large and complex chemical processes where sampling times are of the order of a few seconds, or minutes, and the setpoints for systems move across operating regions slowly. Other applications like servo-systems and combustion engines reveal highly nonlinear behaviour and require sampling times of a few milliseconds. These very challenging control problems require tailored bespoke predictive control methods. The Nonlinear MPC (NMPC) control methods are so important they will be the focus of this chapter, and of Chaps. 9 and 11. The NMPC laws offer the potential to improve process operation, but the theoretical and practical problems are substantially more challenging than those associated with a linear MPC. There are difficulties associated with the nonlinear optimization program for a general nonlinear and non-convex problem. This nonlinear optimization problem must be solved on-line at each sample instant to generate the optimal control sequence. This is in contrast to linear MPC algorithms, where the optimization algorithm normally uses a Quadratic Program (QP), for which fast and reliable algorithms exist. The optimization problem for nonlinear MPC involves an iterative solution on a finite prediction horizon. A NMPC requires the use of nonlinear system models in the prediction calculations, and the repeated on-line solution of a nonlinear optimal control problem. This is expensive computationally, which is a particular problem for fast processes, and is one of the main limiting factors for the practical application of NMPC. The optimization problem for linear MPC is convex which simplifies the optimization problem and algorithm. This is not the case for the true NMPC problems. Unfortunately they usually require more sophisticated algorithms with greater computational loading and they have less predictable behaviour. The stability theory for NMPC is also more difficult and this sometimes leads to rather restrictive assumptions and simplifications. The basic formulation of a typical “continuous-time” general NMPC problem has the form: tZþ Tp

min JðxðtÞ; uðÞ; Tc ; Tp Þ for JðxðtÞ; uðÞ; Tc ; Tp Þ ¼

F ðxðsÞ; uðsÞÞds

uðÞ

t

subject to: x_ ðtÞ ¼ f ðxðtÞ; uðtÞÞ;

xðtÞ ¼ x0

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and yðtÞ ¼ gðxðtÞ; uðtÞÞ with uðsÞ 2 U, s 2 ½t; t þ Tc , uðsÞ ¼ uðs þ Tc Þ; s 2 ½t þ Tc ; t þ Tp  and xðsÞ 2 X; s 2 ½t; t þ Tp . The cost is usually discretized and a receding horizon NMPC control strategy is then applied. Applications: Some systems have strongly nonlinear behaviour with constraints and involve large operating regions with transitions between different local linear models. The performance criteria can also involve nonlinear terms to represent physical phenomena such as fatigue loading in wind turbines, or less tangibly “drivability” in automotive systems. This type of general system and cost-function provides some motivation for considering the use of NMPC. In fact nonlinear MPC has been applied to a wide range of applications, including applications in automotive systems. The NMPC control of engines has been an active area of research for some years. The control of turbocharged diesel engines has been explored with very promising results and this is a particularly difficult nonlinear control design problem [14]. The predictive controllers are very suitable for multivariable systems and in applications like automotive many more actuators and sensors are available. The use of predictive control for actuator allocation and control in automotive engines was considered by Vermillion et al. [15, 16]. In some situations, there are severe nonlinearities, which are crucial to stability (such as titration-curves in pH control), where nonlinear effects justify the use of NMPC. There are also tracking problems with frequent transitions, usually away from the steady state, and systems with nonlinearities that are never in the steady state (such as batch processes). The numerical implementation of a nonlinear and constrained predictive control algorithm can be particularly challenging for fast industrial processes that may have to converge reliably to the optimum in less than a few milliseconds. The main benefit of the types of nonlinear predictive control technique, described in this and later chapters, is the theoretical simplicity, and the limited computational complexity of the algorithms. The constrained optimization problem is relatively simple if the plant is linear, state-dependent or linear parameter varying, and the cost-function is quadratic, and the constraints are linear. The convex optimization problem is not so difficult to solve numerically.

7.1.4

Implicit and Explicit MPC and Algorithms

There are two broad categories of model predictive controllers, namely Implicit and Explicit algorithms.

7.1 Introduction

7.1.4.1

297

Implicit MPC Algorithms

This class of algorithms do not generate a feedback control law, but rather an open-loop control signal trajectory for the future. This is the approach used in the following, where the optimization problem is solved at each time step. For constrained systems, the optimal control can be found using a quadratic program (QP) solver (see Sect. 7.6.1). The plant output is an implicit nonlinear function of the controller state, and other variables, such as the current output reference values [17]. Due to the complexity of the computations, implicit MPC can be more difficult to use for fast systems with short sample times. The main advantage of the implicit algorithms is the conceptual simplicity and the intuitive way in which constraints can be incorporated. However, they involve a significant amount of online computations, since an optimization problem is solved at each sampling time. Most applications have therefore involved processes with slow dynamics, although the methods to be presented in the following chapters limit the computational complexity.

7.1.4.2

Explicit MPC Algorithms

There has been rapid progress in multi-parametric programming methods so that the solution to some optimization problems can be found in an “explicit form” where the control law is given as a function of parameters [18]. Before preceding a short description of the multi-parametric quadratic programming problem will be given. Multi-Parametric Quadratic Programming Problem: The constrained MPC problems can often be manipulated into a form where they can be solved using a multi-parametric quadratic program. A multi-parametric Quadratic Programming (mpQP) problem has the form: min U

1 T ðx Zx þ U T HU þ 2xT GUÞ 2

ð7:1Þ

Such that VU  h þ Sx

ð7:2Þ

The constant matrices and vectors in this problem H, Z, G, V, S and h are of compatible dimensions. The solution involves a search over the set of feasible parameters X* of all x 2 X for which the problem has a solution that satisfies (7.2). The objective in such a case is to solve a Quadratic Program off-line for all x 2 X to obtain the optimized variable U*. The problem is strictly convex if the weighting matrices satisfy:

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7 Linear and Nonlinear Predictive Optimal Control



H GT

 G 0 Z

and H ¼ H T [ 0

It may be shown that the set X* of parameters x for which the problem is feasible is a polyhedral set and a linear MPC controller is then a continuous function of the state [19]. The explicit MPC algorithms enable the majority of the computations to be performed offline. However, the complexity of the solution is determined by the number of regions that form the explicit solution. Unfortunately, multi-parametric programming methods suffer from the curse of dimensionality. For systems with many states and constraints, the complexity of the explicit control law can increase significantly and the approach may not be practical unless the number of regions can be reduced. When the prediction horizon N increases the number of computations grows exponentially and at some point the solution will become becomes impractical. The solution of MPC problems by explicit methods has the benefit that it leads to online control laws that do not need to solve an optimization problem at each time step. In this type of problem there is a performance index to optimise, a vector of constraints and a vector of parameters. The solution is then obtained for the optimization variables, as a function of the parameters and the regions in the space of parameters where these functions are valid. The multi-parametric quadratic programming approach is an alternative means of implementing conventional predictive control algorithms where much of the computational load is in offline calculations. A set of linear MPC controller gains may be pre-calculated for a range of different operating points, and stored for on-line use. The on-line algorithm then chooses the nearest gains corresponding to the current operating point and switches them into use accordingly. This approach considerably reduces the on-line computations [19]. In summary, the explicit MPC algorithms have some important advantages: (a) The online computational time can be reduced to microseconds, so that explicit MPC is attractive for fast-embedded systems. (b) The real-time code for the online algorithm is simple to implement, short and efficient. Explicit MPC algorithms also have some disadvantages: (a) Less attractive when the number of regions grows, since too much memory is required. (b) Only practical for low-order systems where the number of states is limited to say ten or less. The optimization problem is solved parametrically in explicit MPC controllers, such as Hovland [20]. In referring to explicit MPC the authors noted: “good performance is achieved by tuning based on exhaustive simulations for ranges of operating conditions. In many cases, this approach leads to better performance

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299

than using robust MPC techniques. Choosing the right robust MPC technique is an art, and much experience is necessary to make it work.”

7.1.5

Simple Nonlinear Model Predictive Control

There are many difficulties with general Nonlinear MPC problems. The derivation of a plant model from first principles is not always possible. Moreover, the generation of nonlinear models from data is problematic and there is a lack of generic and reliable nonlinear system identification techniques. The optimization problem may be non-convex, giving rise to local minima and stability problems, and an increase in computational time. The study of the stability and robustness of NMPC designs are also of course much more complex problems than for linear MPC. A relatively simple linear predictive controller is introduced in the first part of this chapter that is a modification of the very popular Generalized Predictive Control (GPC) algorithm for linear time-invariant systems. The system model is initially assumed to be linear and time-invariant (LTI). The GPC controller is derived assuming a dynamically weighted cost-function. This problem is interesting in its own right, however, it also provides the first stage in the solution for a nonlinear predictive controller. In this case the nonlinear plant is assumed to include an input subsystem represented by a general nonlinear operator. The predictive control paradigm that is the main contribution in the later parts of this chapter is referred to as Nonlinear Predictive Generalised Minimum Variance (NPGMV) control. This control strategy can be related to previous results on Nonlinear Generalised Minimum Variance (NGMV) control introduced in the earlier chapters and in Grimble [21, 22], and Grimble and Majecki [23]. The major difference with the NGMV control problem involves an extension of the NGMV cost-index to include the future tracking error and control signal-costing terms. These terms can be the same as in the GPC cost-minimization problem. When the system is linear and the cost functions have the same quadratic form, the NPGMV controller becomes the same as the GPC controller. This is a valuable feature providing some confidence in the nonlinear control solution for this limiting linear plant case. The NPGMV solution is derived in polynomial system form in this chapter, and the relationship to the GPC controller is explored [24]. In later chapters, the state-space versions of these controllers are introduced.

7.2

Nonlinear System Description

The plant model relating input and output can be grossly nonlinear, dynamic and may have a very general form but the disturbance model is chosen to be linear, so that a simple solution is obtained. The nonlinear plant model and the linear disturbance model are shown in Fig. 7.1. The signals vðtÞ and nðtÞ are vector

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7 Linear and Nonlinear Predictive Optimal Control

Fig. 7.1 Two-degrees of freedom feedback control for plant with linear and nonlinear subsystems

zero-mean, independent, white noise signals. The measurement noise signal vðtÞ is assumed to have a constant covariance matrix Rf ¼ RTf  0. The disturbance white noise source nðtÞ has an identity covariance matrix. The reference r(t) and disturbance dD ðtÞ signals, are assumed to be deterministic, and future values are assumed known throughout the prediction horizon. As previously there is no need to specify the distribution of the noise sources, since the prediction equation is only dependent upon the linear models. The various signals in the system may be listed as follows: u0 ðtÞ vector of m0 input signals to the linear subsystem. uðtÞ vector of m control signals applied to the nonlinear subsystem, representing the plant input. yðtÞ vector of r plant output signals that are to be controlled. zðtÞ vector of r observations or measured plant output signals including measurement noise. rðtÞ vector of r plant set point or reference signal values. yp ðtÞ vector of m inferred output signals to be controlled including the cost weighting. rp ðtÞ vector of r plant setpoint or reference signals including cost weighting Nonlinear plant subsystem: The plant model will be assumed to have an input subsystem of a general nonlinear operator form:

7.2 Nonlinear System Description

ðW 1 uÞðtÞ ¼ zk ðW 1k uÞðtÞ

301

ð7:3Þ

where zk I denotes a diagonal matrix of the common delay elements in the output signal paths. The output of the non-linear subsystem W 1k is denoted u0 ðtÞ ¼ ðW 1k uÞðtÞ. The nonlinear subsystem W 1 is assumed to be finite gain stable. The linear subsystem, denoted W0 ðz1 Þ ¼ zk W0k ðz1 Þ, is introduced below and can contain any unstable modes. If the decomposition into a nonlinear and a linear subsystem is not relevant then the linear subsystem can be neglected by defining W0k ¼ I. The whole plant model in this case may be included in the W 1k subsystem. The generalisation to different delays in different signal paths is straightforward [22].

7.2.1

Linear Subsystem Polynomial Matrix Models

The system models for the linear multivariable output subsystem may now be introduced in polynomial matrix form. These subsystems are associated with any linear subsystem W0 in the plant model and the linear disturbance model. The linear subsystem of the plant may be represented by a Controlled Auto-Regressive Moving Average (CARMA) model. The non-square r  m plant model may be written as: Aðz1 Þm0 ðtÞ ¼ B0k ðz1 Þu0 ðt  kÞ þ Cd ðz1 ÞnðtÞ

ð7:4Þ

where nðtÞ denotes a white noise disturbance and the input signal channels in the plant model include a k-steps transport-delay (k  0) and B0 ðz1 Þ ¼ B0k ðz1 Þzk . The delay-free transfer-function of the linear plant subsystem and the stochastic disturbance model may be defined in the following left-coprime operator form: ½W0k ðz1 Þ Wd ðz1 Þ ¼ Aðz1 Þ1 ½B0k ðz1 Þ Cd ðz1 Þ

ð7:5Þ

The plant output, including a deterministic disturbance component, is written as yðtÞ ¼ m0 ðtÞ þ dD ðtÞ. Also introduce a stable cost-function error-weighting model, 1 1 Pcn ðz1 Þ. The in left coprime polynomial matrix form, as Pc ðz1 Þ ¼ P1 cd ðz Þ weighted output yp ðtÞ ¼ Pc ðz1 Þ yðtÞ may now be written as: yp ðtÞ ¼ Pc ðz1 Þ ðdD ðtÞ þ Aðz1 Þ1 ðB0k ðz1 Þu0 ðt  kÞ þ Cd ðz1 ÞnðtÞÞÞ

ð7:6Þ

Note that the arguments of these polynomial matrices are often omitted in what follows for simplicity. The model for the disturbance signal is linear, which is an assumption that does not affect stability properties, but may cause a degree of sub-optimality in the disturbance rejection properties.

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7 Linear and Nonlinear Predictive Optimal Control

The power-spectrum for the combined stochastic disturbance and noise f ¼ d þ v ¼ Wd n þ v can be computed as Uff ¼ Udd þ Uvv ¼ Wd Wd þ Rf . Recall the notation used for the polynomial computations implies Wd ðz1 Þ ¼ WdT ðzÞ where in this case z represents the z-domain complex number. The generalized spectralfactor Yf satisfies Yf Yf ¼ Uff , where Yf ¼ A1 Df . The system models are such that Df is a strictly Schur polynomial matrix that satisfies [25, 26]: Df Df ¼ Cd Cd þ ARf A

ð7:7Þ

Innovations signal model: The signal f ¼ d þ v may be modelled in a so-called innovations signal form as f ðtÞ ¼ Yf eðtÞ, where Yf ¼ A1 Df is defined using (7.7). The signal eðtÞ denotes a white noise signal of zero-mean and identity covariance matrix. The observations signal may now be written, using (7.4), as: zðtÞ ¼ yðtÞ þ vðtÞ ¼ dD ðtÞ þ Aðz1 Þ1 B0k ðz1 Þu0 ðt  kÞ þ Aðz1 Þ1 Cd ðz1 ÞnðtÞ þ vðtÞ ¼ dD ðtÞ þ A1 ðz1 ÞB0k ðz1 Þu0 ðt  kÞ þ Yf ðz1 ÞeðtÞ ð7:8Þ Define the right-coprime model for the weighted spectral factor: 1 Pc ðz1 ÞYf ðz1 Þ ¼ Dfp ðz1 ÞA1 f ðz Þ

ð7:9Þ

The weighted observations signal zp ðtÞ ¼ Pc ðz1 ÞzðtÞ may be written as: 1 zp ðtÞ ¼ Pc ðz1 ÞdD ðtÞ þ Pc ðz1 ÞW0k ðz1 Þu0 ðt  kÞ þ Dfp ðz1 ÞA1 f ðz ÞeðtÞ ð7:10Þ

7.2.2

Optimal Linear Prediction Problem

The solution of the optimal control problem requires the introduction of a least squares predictor. This enables the estimated inferred output y(t) at future times t + k + 1, t + k + 2, … to be calculated. The prediction of signals is an interesting topic for different applications in signal processing. The optimal least-squares predictor is defined to minimise the cost-function with weighted estimation error: J ¼ Ef~yTp ðt þ jjtÞ~yp ðt þ jjtÞjtg

ð7:11Þ

where the estimation error, for j  0: ~yp ðt þ jjtÞ ¼ yp ðt þ jÞ  ^yp ðt þ jjtÞ

ð7:12Þ

7.2 Nonlinear System Description

303

and ^yp ðt þ j jtÞ denotes the predicted value of yp ðtÞ at a time j steps ahead. To generate the prediction, a Diophantine equation must be solved for ðEj ; Hj Þ, with Ej of smallest degree degðEj ðz1 ÞÞ\j þ k. First Diophantine equation: Ej ðz1 ÞAf ðz1 Þ þ zjk Hj ðz1 Þ ¼ Dfp ðz1 Þ

ð7:13Þ

This equation may be written in the form: 1 1 1 1 Ej ðz1 Þ þ zjk Hj ðz1 ÞA1 f ðz Þ ¼ Dfp ðz ÞAf ðz Þ

ð7:14Þ

and since j can be zero the integer k is assumed k [ 0. Prediction equation: Substituting from (7.13) the expression for the weighted observations signal (7.10): 1 zp ðtÞ ¼ Pc ðz1 ÞdD ðtÞ þ Pc ðz1 ÞW0k ðz1 Þu0 ðt  kÞ þ Dfp ðz1 ÞA1 f ðz ÞeðtÞ 1 ¼ dPD ðtÞ þ Pc ðz1 ÞW0k ðz1 Þu0 ðt  kÞ þ ðEj ðz1 Þ þ zjk Hj ðz1 ÞA1 f ðz ÞÞeðtÞ

where the weighted disturbance signal dPD ðtÞ ¼ Pc ðz1 ÞdD ðtÞ. Substituting from the innovations signal (7.8) eðtÞ ¼ Yf1 zðtÞ  D1 f B0k u0 ðt  kÞ obtain: zp ðtÞ ¼ dPD ðtÞ þ Pc ðz1 ÞW0k ðz1 Þu0 ðt  kÞ þ Ej ðz1 ÞeðtÞ   1 1 1 1 1 1 þ zjk Hj ðz1 ÞA1 f ðz Þ Yf ðz ÞzðtÞ  Df ðz ÞB0k ðz Þu0 ðt  kÞ 1 1 1 1 1 1 Recall A1 ¼ D1 f Yf fp Pc and noting Pc ðz ÞYf ðz Þ ¼ Dfp ðz ÞAf ðz Þ, the weighted observations signal may be obtained as: 1 1 zp ðtÞ ¼ dPD ðtÞ þ Ej ðz1 ÞeðtÞ þ zjk Hj ðz1 ÞD1 fp ðz ÞPc ðz ÞzðtÞ



 1 1 1 1 þ Pc ðz1 ÞA1 ðz1 ÞB0k ðz1 Þ  zjk Hj ðz1 ÞA1 f ðz ÞDf ðz ÞB0k ðz Þ u0 ðt  kÞ

Weighted output signal: To obtain the expression for the weighted output yp ðtÞ, note zp ðtÞ ¼ Pc zðtÞ ¼ yp ðtÞ þ vp ðtÞ, where yp ðtÞ ¼ Pc yðtÞ and vp ðtÞ ¼ Pc vðtÞ. Thence, 1 zp ðtÞ ¼ dPD ðtÞ þ Ej ðz1 ÞeðtÞ þ zjk Hj ðz1 ÞD1 fp ðz Þzp ðtÞ  1 1 1 1 þ Pc ðz1 ÞYf ðz1 ÞAf ðz1 Þ  zjk Hj ðz1 Þ A1 f ðz ÞDf ðz ÞB0k ðz Þu0 ðt  kÞ



ð7:15Þ

304

7 Linear and Nonlinear Predictive Optimal Control

From (7.9) Pc Yf Af ¼ Dfp and from (7.14) and (7.15): 1 yp ðtÞ ¼ dPD ðtÞ þ Ej ðz1 ÞeðtÞ  vp ðtÞ þ zjk Hj ðz1 ÞD1 fp ðz Þzp ðtÞ   1 1 1 1 þ Dfp ðz1 Þ  zjk Hj ðz1 Þ A1 f ðz ÞDf ðz ÞB0k ðz Þu0 ðt  kÞ

Future values weighted output: Using (7.13), the j þ k steps ahead weighted output signal: yp ðt þ j þ kÞ ¼ dPD ðt þ j þ kÞ þ Ej ðz1 Þeðt þ j þ kÞ  vp ðt þ j þ kÞ 1 1 1 1 1 þ Hj ðz1 ÞD1 fp ðz Þzp ðtÞ þ Ej ðz ÞDf ðz ÞB0k ðz Þu0 ðt þ jÞ

ð7:16Þ Recalling D1 is assumed to be stable, define the right coprime model: f 1 1 1 1 B1k ðz1 ÞD1 f 1 ðz Þ ¼ Df ðz ÞB0k ðz Þ

ð7:17Þ

1 Also let the signal uf ðtÞ ¼ D1 f 1 ðz Þu0 ðtÞ, then Eq. (7.16) may be written as:

  yp ðt þ j þ kÞ ¼ Ej ðz1 Þeðt þ j þ kÞ  vp ðt þ j þ kÞ h i 1 1 1 þ dPD ðt þ j þ kÞ þ Hj ðz1 ÞD1 ðz Þz ðtÞ þ E ðz ÞB ðz Þu ðt þ jÞ p j 1k f fp ð7:18Þ The maximum degree of the polynomial matrix Ej is j þ k  1, and hence the noise components in Ej eðt þ j þ kÞ includes eðt þ j þ kÞ ; . . .; eðt þ 1Þ at future times. That is, Ej eðt þ j þ kÞ denotes the weighted sum of future white noise signal components.

7.2.3

Derivation of the Predictor

The polynomial form of the linear predictor, needed for this problem, is defined below using the above prediction equation. This polynomial form of predictor is useful for non-control applications. Consider first the case where the measurement noise term vðtÞ is zero. The optimal predictor for the output at time t + j + k, given observations up to time t, can now be derived. The observations, up to time t can be assumed known. The future values of the control inputs u0 ðtÞ; . . .; u0 ðt þ jÞ used in the predictor, will be assumed known at time t. The future control input is independent of the future disturbance and noise sequence. It follows that the expected value of the round (.) and square [.] bracketed terms in Eq. (7.18) must be zero. The optimal predictor must minimise the cost-function (7.11) at time t + j + k. After a little manipulation, noting the expectation of the cross-terms is null, the cost-function simplifies as:

7.2 Nonlinear System Description

¼E

n

305

J ¼ Ef~yTp ðt þ k þ jjtÞ~yp ðt þ k þ jjtÞjtg T   o  ½:  ^yp ðt þ k þ jjtÞ ½:  ^yp ðt þ k þ jjtÞ jt þ E ð:ÞT ð:Þjt

The last term is independent of the predicted value ^yp ðt þ k þ j jtÞ and hence the optimal predictor sets the first cost term to zero. That is,^yp ðt þ k þ j jtÞ ¼ ½ : , where the square bracketed term in (7.18) is defined as: h i 1 1 1 ½: ¼ dPD ðt þ j þ kÞ þ Hj ðz1 ÞD1 fp ðz Þzp ðtÞ þ Ej ðz ÞB1k ðz Þuf ðt þ jÞ Optimal predictor: 1 ^yp ðt þ j þ kjtÞ ¼ dPD ðt þ j þ kÞ þ Hj ðz1 ÞD1 fp ðz Þzp ðtÞ þ Ej ðz1 ÞB1k ðz1 Þuf ðt þ jÞ

ð7:19Þ

Measurement noise case: Consider the prediction problem when the measurement noise is non-zero so that the weighted noise vp ðt þ j þ kÞ¼ Pc ðz1 Þvðt þ j þ kÞ. If the weighting Pc ðz1 Þ is a constant, which is usual in GPC control, or if it is assumed a polynomial matrix of degree j þ k  1, then the future noise vp ðt þ j þ kÞ is only dependent on future white measurement noise terms. In this situation the expected value of such a term and the square-bracketed terms in Eq. (7.18) must be zero. It follows that the optimal linear predictor is given by (7.19), and from (7.18), the prediction error:   ~yp ðt þ j þ kjtÞ ¼ Ej ðz1 Þeðt þ j þ kÞ  vp ðt þ j þ kÞ

ð7:20Þ

Second Diophantine equation: A second Diophantine equation may now be introduced to separate the term Ej ðz1 ÞB1k ðz1 Þ into a part with a j + 1 step delay 1 and a part depending on Df 1 ðz1 Þ (recall uf ðtÞ ¼ D1 f 1 ðz Þu0 ðtÞ). Thus, for j  0, introduce the following equation, which has a solution ðGj ; Sj Þ, of smallest degree for Gj . Gj ðz1 ÞDf 1 ðz1 Þ þ zj1 Sj ðz1 Þ ¼ Ej ðz1 ÞB1k ðz1 Þ

ð7:21Þ

where degðGj ðz1 ÞÞ ¼ j. The prediction equation, from Eq. (7.19) may now be obtained (for j  0) as: 1 ^yp ðt þ j þ kjtÞ ¼ dPD ðt þ j þ kÞ þ Hj ðz1 ÞD1 fp ðz Þzp ðtÞ

þ Gj ðz1 Þu0 ðt þ jÞ þ Sj ðz1 Þuf ðt  1Þ

ð7:22Þ

Time partitioned prediction: Note that the degree of Gj ðz1 Þ is j and the third term in (7.22) therefore involves the future inputs u0 ðt þ jÞ; . . .; u0 ðtÞ. The filtered

306

7 Linear and Nonlinear Predictive Optimal Control

observations and control may now be denoted as fj ðtÞ, which may be written in terms of past outputs and inputs: 1 1 fj ðtÞ ¼ Hj ðz1 ÞD1 fp ðz Þzp ðtÞ þ Sj ðz Þuf ðt  1Þ

ð7:23Þ

Thus, the predicted weighted output (7.22) may be written, for j  0, as: ^yp ðt þ j þ kjtÞ ¼ dPD ðt þ j þ kÞ þ Gj ðz1 Þu0 ðt þ jÞ þ fj ðtÞ

ð7:24Þ

The signal fj ðtÞ determines the free response prediction of the weighted output yp ðt þ j þ kÞ, assuming that the inputs u0 ðt þ iÞ, for i  0, are zero. The coefficients of the polynomial matrix Gj ðz1 Þ have a physical importance. Note from (7.21) and (7.13): Gj ðz1 ÞDf 1 ðz1 Þ þ zj1 Sj ðz1 Þ ¼ Ej ðz1 ÞB1k ðz1 Þ Ej ðz1 ÞB1k ðz1 Þ þ zjk Hj ðz1 ÞAf ðz1 Þ1 B1k ðz1 Þ ¼ Dfp ðz1 ÞAf ðz1 Þ1 B1k ðz1 Þ and from (7.17) From these two equations, noting (7.9) Pc Yf ¼ Dfp A1 f Df B1k ¼ B0k Df 1 , obtain: j1 Sj Gj Df 1 ¼ Pc Yf B1k  zjk Hj A1 f B1k  z

or j1 Gj ¼ Pc A1 B0k  ðzjk Hj A1 Sj ÞD1 f B1k þ z f1

Physical significance: The polynomial matrix Gj ðz1 Þ, therefore includes the first j + 1 Markov parameters gj of the weighted plant transfer term Gj ¼ Pc W0k . Thus, let Gj ðz1 Þ ¼ g0 þ g1 z1 þ    þ gj zj where degðGj ðz1 ÞÞ ¼ j and the matrix G0 ðz1 Þ ¼ g0 .

7.2.4

Matrix Representation of the Prediction Equations

The prediction equations need to be in a concise vector-matrix form for use in the predictive control law computations. The matrices are introduced below. The weighted outputs are to be predicted for inputs computed in the interval s 2 ½t; t þ N, where N  0. Equation (7.24) may be used to obtain the following weighted output vector equation at N þ 1 future times:

7.2 Nonlinear System Description

3 2 3 2 3 2g 0 ^yp ðt þ k jtÞ dPD ðt þ kÞ f0 ðtÞ g1 6 ^yp ðt þ 1 þ k jtÞ 7 6 dPD ðt þ 1 þ kÞ 7 6 f1 ðtÞ 7 6 6 7 6 7 6 7 6 .. 7 6 7 6 .. 7 6 6 .. .. . 7¼6 7þ6 . 7þ6 6 . . 6 7 6 7 6 7 6 6 5 4 5 4 5 4 4 ^yp ðt þ N þ k jtÞ dPD ðt þ N þ kÞ fN ðtÞ gN 2

307

0 g0

 0

g1 .. .

g0 .. .

gN1

...

0  .. . .. .

32 3 u0 ðtÞ 76 u ðt þ 1Þ 7 76 0 7 76 7 .. 76 7 . 76 7 74 5 5 ðt þ NÞ u 0 g0 0 0 .. .

ð7:25Þ Introducing an obvious definition for the matrices in Eq. (7.25), the vector form of the prediction for weighted outputs may be written as: 0 Y^t þ k;N ¼ Dt þ k;N þ Ft;N þ GN Ut;N

ð7:26Þ

The vector of free response predictions Ft;N in (7.26) may be expanded, using (7.23), as follows: 2

Ft;N

3 2 3 2 3 f0 ðtÞ H0 ðz1 Þ S0 ðz1 Þ 6 f1 ðtÞ 7 6 H1 ðz1 Þ 7 6 S1 ðz1 Þ 7 6 7 6 7 1 1 6 7 Dfp ðz Þzp ðtÞ þ 6 ¼6 . 7¼6 7 7uf ðt  1Þ . .. .. 4 .. 5 4 5 4 5 . fN ðtÞ

HN ðz1 Þ

SN ðz1 Þ

This equation may be written in a more convenient vector-matrix form as in (7.27). The transfer-operators HNZ ðz1 Þ and SNZ ðz1 Þ can be defined from the matrices in this equation and Ft;N can be written as: Ft;N ¼ HNZ ðz1 Þzp ðtÞ þ SNZ ðz1 Þuf ðt  1Þ

ð7:27Þ

The vector form of the prediction error Ej ðz1 Þeðt þ j þ kÞ  vp ðt þ j þ kÞ may also be written, recalling degðEj ðz1 ÞÞ\j þ k, as: 2 6 6 Y~t þ k;N ¼ 6 4

e0 eðt þ kÞ þ . . . þ ek1 eðt þ 1Þ  vp ðt þ kÞ e0 eðt þ 1 þ kÞ þ . . . þ ek eðt þ 1Þ  vp ðt þ 1 þ kÞ .. .

3 7 7 7 5

ð7:28Þ

e0 eðt þ N þ kÞ þ . . . þ eN þ k1 eðt þ 1Þ  vp ðt þ N þ kÞ Future set point knowledge: It is reasonable to assume for many applications that the future variations of the setpoint or reference signal rðtÞ are predetermined, at least over a fixed future horizon of N steps. For example, batch operations in the petrochemical industry often involve processes with known future desired setpoint trajectories. Recall the weighted reference is assumed to include a stable error weighting function rp ðtÞ ¼ Pc ðz1 ÞrðtÞ. The vectors of known future weighted reference, output and input signals may now be defined as follows:

308

7 Linear and Nonlinear Predictive Optimal Control

2

Rt;N

3 rp ðtÞ 6 rp ðt þ 1Þ 7 6 7 ¼6 7; .. 4 5 . rp ðt þ NÞ

2

Yt;N

3 yp ðtÞ 6 yp ðt þ 1Þ 7 6 7 ¼6 7; .. 4 5 . yp ðt þ NÞ

2

0 Ut;N

3 u0 ðtÞ 6 u0 ðt þ 1Þ 7 6 7 ¼6 7 .. 4 5 .

ð7:29Þ

u0 ðt þ NÞ

Tracking error: The k steps-ahead vector of future weighted outputs can be written as Yt þ k;N ¼ Y^t þ k;N þ Y~t þ k;N and the future tracking error, that includes any dynamic error weighting, may be written in vector form as: Et þ k;N ¼ Rt þ k;N  Yt þ k;N ¼ Rt þ k;N  ðY^t þ k;N þ Y~t þ k;N Þ

ð7:30Þ

The vector of predicted signals Y^t þ k;N in (7.30) and the prediction error Y~t;N are orthogonal.

7.3

Generalized Predictive Control for Linear Systems

Model Predictive Control (MPC) based on linear systems theory has been very successful in applications where the processes are mildly nonlinear or the control can be scheduled to regulate about an operating point. Generalized Predictive Control (GPC), mentioned in the introduction, is one of the best-known predictive control algorithms. It is thought to be attractive to engineers in industry, since the basic process of prediction and optimization, based on current conditions (receding horizon concept), and subsequent correction by feedback, is intuitive and can be portrayed as mimicking a natural process. The GPC algorithm is representative of many MPC methods, and for present purposes, the terms can be used almost interchangeably. The GPC/MPC methods have many attractive features including: • • • • • • • • •

A cost-index is minimised, providing a benchmark figure of merit. Future reference or setpoint information is included. A cost can be placed on future predicted tracking errors. Ability to limit future predicted control actuator variations. Suitable for multivariable systems with high degrees of interaction. Will accommodate transport delays on inputs and outputs efficiently. Can model and thereby reduce the effects of unmeasured disturbances. Can include measured disturbances, providing a feedforward capability. Can allow for hard constraints on system inputs, states, and outputs.

The last capability listed is arguably the most important in many applications. The constraint-handling feature can be provided by using quadratic dynamic programming.

7.3 Generalized Predictive Control for Linear Systems

309

Because of the importance of GPC/MPC methods in linear systems, a review of the derivation of the GPC controller is provided first below. The control input u(t) to the system will therefore be taken to be that to the linear subsystem, shown in Fig. 7.1, denoted u0 ðtÞ. This is the same as defining the black-box nonlinear input block W 1k as the identity. In the second part of the chapter, when the nonlinear case is considered, this nonlinear subsystem is reintroduced and uðtÞ 6¼ u0 ðtÞ. The GPC control solution for a linear system is of interest since it is so popular. However, the results are also needed to motivate the definition of the nonlinear predictive control problem of interest. In this case, the control input to the plant uðtÞ is the input to the nonlinear input block W 1k . GPC Performance Index: The performance index to be minimised is multi-step and may be defined as: J ¼ Ef

N X

ep ðt þ j þ kÞT ep ðt þ j þ kÞ þ u0 ðt þ jÞT k2j u0 ðt þ jÞÞjtg

ð7:31Þ

j¼0

where Ef:jtg denotes the conditional expectation, conditioned on measurements up to time t. The term kj denotes a scalar control signal weighting factor, or a diagonal matrix, and the vector of future weighted error signal values can be written as: ep ðt þ j þ kÞ ¼ Pc ðz1 Þ ðrðt þ j þ kÞ  yðt þ j þ kÞÞ The future optimal control signal is to be calculated for the interval s 2 ½t; t þ N. The weighting Pc ðz1 Þ can include an integrator so that the cost-function includes the integral of error term ep ðtÞ. This normally introduces integral action in the controller. The physical intuitive reason this occurs is similar to that for other optimal controllers using integral error weightings. That is, the error signal cannot tend to a non-zero constant, as time advances; otherwise, the weighted error signal ep ðtÞ would grow without bound. The consequence is that the “optimal” controller is forced to include some form of integral action.

7.3.1

GPC Optimal Control Solution

The multi-step cost-function may be written, in a more concise vector form, by introducing the vectors of future signals (7.29). Thence, substituting in (7.31): 0T 2 0 KN Ut;N jtg J ¼ EfJt g ¼ EfðRt þ k;N  Yt þ k;N ÞT ðRt þ k;N  Yt þ k;N Þ þ Ut;N

ð7:32Þ

where the cost weightings on the future inputs u0 ðtÞ are written as K2N ¼ diagfk20 ; k21 ; . . .; k2N g. Introduce the optimal predictor, using (7.30) and (7.32), and note that the GPC cost-index can then be written as:

310

7 Linear and Nonlinear Predictive Optimal Control

J ¼ EfðRt þ k;N  ðY^t þ k;N þ Y~t þ k;N ÞÞT ðRt þ k;N  ðY^t þ k;N þ Y~t þ k;N ÞÞ 0T 2 0 þ Ut;N KN Ut;N jtg

ð7:33Þ

The terms in the performance index can be simplified by noting the prediction errors in Y~t þ k;N depend on future values of the white noise innovations signal e(t), which is independent of the future controls. The estimate Y^t þ k;N is therefore orthogonal to the estimation error Y~t þ k;N . Also, note that the future reference or setpoint trajectory Rt þ k;N is assumed known over N + 1 future steps. The vector/matrix form of the cost-expression may therefore be simplified, exploiting orthogonality, to obtain: 0T 2 0 J ¼ ðRt þ k;N  Y^t þ k;N ÞT ðRt þ k;N  Y^t þ k;N Þ þ Ut;N KN Ut;N þ J0

ð7:34Þ

where the final cost term J0 ¼ EfY~tTþ k;N Y~t þ k;N jtg is independent of the control signal action. Substituting (7.26) into (7.35), expand the cost-index as: 0 J ¼ ðRt þ k;N  Dt þ k;N  ðGN Ut;N þ Ft;N þ Dt þ k;N ÞÞT 0 0T 2 0 ðRt þ k;N  Dt þ k;N  ðGN Ut;N þ Ft;N þ Dt þ k;N ÞÞ þ Ut;N KN Ut;N þ J0

To simplify notation, define the modified vector of future reference signals as: ~ t þ k;N ¼ Rt þ k;N  Dt þ k;N  Ft;N R

ð7:35Þ

Substituting into the multi-step cost expression now obtain: 0 T ~ 0 0T 2 0 ~ t þ k;N  GN Ut;N Þ ðRt þ k;N  GN Ut;N Þ þ Ut;N KN Ut;N þ J0 J ¼ ðR   0 T 0T T T 0 0T T ~ t þ k;N  Ut;N GN R ~ t þ k;N  R ~ t þ k;N R ~ t þ k;N GN Ut;N þ Ut;N GN GN þ K2N Ut;N ¼R þ J0

ð7:36Þ where the final J0 term is independent of the control action. The procedure for minimising this conditional cost-function term is similar to that when the signals are all deterministic. That is, the gradient of the cost-function can be set to zero, to obtain the vector of future optimal control signals. For an appropriate weighting, choice the Hessian ðGTN GN þ K2N Þ can be shown to be positive-definite, which ensures a unique minimum exists. Let this matrix be denoted as: XN ¼ GTN GN þ K2N

ð7:37Þ

7.3 Generalized Predictive Control for Linear Systems

7.3.1.1

311

Gradient Minimization

The minimization of the quadratic form (7.36) for this convex optimization problem may be performed using standard matrix algebra. Consider a criterion of the general vector/matrix form: J ¼ kZ  BU k2 þ kU k2XN ¼ ðZ  BUÞT ðZ  BUÞ þ U T XN U ¼ Z T Z  U T BT Z  Z T BU þ U T XN U

ð7:38Þ

where Z and U are vectors. Then note U T BT Z ¼ Z T BU and hence: J ¼ Z T Z  2U T BT Z þ U T XN U Recall the gradient rU U T L ¼ L and rU U T XN U ¼ 2XN U if X is symmetric. The Z term is independent of the control action and the gradient follows as @J=@U ¼ 2BT Z þ 2XN U. The necessary condition for optimality is therefore obtained by setting the gradient to zero: BT Z þ XN U ¼ 0

ð7:39Þ

2 The second-derivative or Hessian matrix @ 2 J @U ¼ 2XN and the sufficient condition is therefore XN [ 0. The control costing is always assumed a positive-definite matrix in the following chapters on predictive control and this sufficient condition can be assumed satisfied. GPC Vector of optimal controls: The result (7.39) can be applied to the ~ t þ k;N , minimization of the quadratic cost-function (7.36), and in this case Z ¼ R 0 U ¼ Ut;N and B ¼ GN . The condition for optimality becomes:   0 ~ t þ k;N þ GTN GN þ K2N Ut;N GTN R ¼0

ð7:40Þ

The vector of future controls to minimise the quadratic cost-function therefore follows as:  1 0 ~ t þ k;N ¼ GTN GN þ K2N GTN R Ut;N The actual GPC optimal control signal at time t is computed based on the receding horizon principle, and the optimal control is taken as the first element in 0 . The receding horizon philosophy is introduced in the vector of future controls Ut;N greater detail in the next section. The results can be summarised in the following theorem. Theorem 7.1: Generalized Predictive Control Consider the linear system represented in polynomial matrix form described in Sect. 7.2, with the nonlinear input subsystem set to the identity ðW 1k ¼ I). The cost-function (7.32) is to be minimized

312

7 Linear and Nonlinear Predictive Optimal Control

for given error Pc ðz1 Þ and control KN cost-function weightings. Given the solution to the Diophantine equations (7.13) and (7.21), and the spectral factorization (7.7) the matrices Dt þ k;N ; Ft;N ; GN may be defined from (7.26) and the GPC optimal control can be computed from the first element of the vector of future controls:  1   0 ¼ GTN GN þ K2N GTN Rt þ k;N  Dt þ k;N  Ft;N Ut;N

ð7:41Þ

where the cost-function weighting definitions ensure the matrix GTN GN þ K2N is non-singular. ■ Proof By collecting the above results.

7.3.2

■

Receding Horizon Principle

The receding horizon principle that is used in traditional model predictive control was described in a number of early papers by Kwon and Pearson [27, 28, 29]. In the receding horizon optimal control philosophy the control and output cost intervals move, or recede, with time. The advantage is that the solutions are simpler to compute than Linear Quadratic (LQ) optimal controls. It was shown that a stabilizing control law could be derived for a continuous-time system represented by a state-space, possibly time varying model, in the paper by Kwon and Pearson [28]. They considered a finite-time quadratic optimal control problem with initial and final times ðt0 ; tf Þ, and with an endpoint constraint. The LQ type of control law they introduced involved replacing the initial time t0 by the current time t and the final time tf by the time t + T. This control law was a state-feedback solution and resulted in a uniform asymptotically stable closed-loop system. For a linear time-invariant system, the control law became a fixed state-feedback gain and that depended on the solution of a Riccati equation at the time T. The feedback gains were found by integrating a Riccati equation backwards in time over a finite interval. It was shown that the control law obtained was stabilizing. This was without needing to consider an infinite time cost-horizon. The later paper by Kwon and Pearson [29] was for a discrete system and paralleled the continuous-time results. The solution of the discrete optimal control problem was considered with the following criterion: J¼

tþ N1 X

yT ðsÞQðsÞyðsÞ þ uT ðsÞRðsÞuðsÞ

s¼t

subject to the moving horizon end constraint x(t + N) = 0. They showed that the solution of this receding horizon optimal control problem tends to the solution of the steady-state LQ optimal control problem when the horizon N is sufficiently

7.3 Generalized Predictive Control for Linear Systems

313

large. Thus, this receding horizon optimal control problem can provide an approximation to the LQ solution whilst still guaranteeing asymptotic stability. These results are often used to motivate the use of the “receding horizon control principle” in predictive control problems.

7.3.3

Choice of Prediction and Control Horizons

The predictive control cost-function can involve a different output or prediction horizon N, and a control horizon Nu. For a Linear Time-Invariant (LTI) processes, the prediction horizon may be defined to be larger than the dominant time-constant of the plant dynamics, or close to the open-loop settling time of the system. The settling time is taken here as the time for the output to a step-input to enter and stay within a 5% error band of the steady-state value. The prediction horizon N can be used as a tuning parameter but is usually not a critical choice so long as it is sufficiently large. The control horizon Nu is often chosen to be small (5 to 10 time steps), since this limits the computations. This is a substantial advantage numerically but from a physical viewpoint it is not so obvious why the number of control actions should be different to the number of outputs or errors costed. For the predictive-control problems considered here, the same control and prediction horizons are often used (Nu = N). This is helpful when the predictive control is to be compared with LQ design. It is often found that increasing the prediction horizon N reduces step-response overshoots and by implication the sensitivity functions (so robustness is improved). There is a caveat on the last point concerning scaling. If the horizon N increases and the error and control terms in the cost-function are not equally important then the speed of response of the system will change in addition to the effect on robustness. If say the error weighting is much more significant (meaning the transfer gain from disturbances/reference to error is greater than for the path to control), then as the horizon increases the overall penalty on error effectively increases. The result is the form of the closed-loop responses will change but the time responses will also become faster as well. If the prediction horizon N is large, the performance gets closer to the LQ infinite-time or steady-state solution (see the book by Bitmead et al. [8]). As the horizon N increases the closed-loop sensitivity functions usually improve but computations increase. If the horizon N is small, the solution of an optimal control problem does not always lead to a stabilizing control law (Chap. 2). An optimal predictive control that minimizes a short horizon cost-function may not ensure stability. In the original GPC or MPC algorithms, the onus was on the technicians or engineers that chose the cost-horizons and tuned the cost-weightings to try to ensure stability and achieve reasonable performance.

314

7 Linear and Nonlinear Predictive Optimal Control

If the prediction horizon N is too small it may be less than the dominant time-constant of the plant, which suggests poor results will be obtained. In fact, the predictive control will then approach a single-stage (minimum-variance) control behaviour. From the results discussed in Chap. 2, the system will be unstable if the control weighting tends to zero, and the plant is non-minimum phase. Thus, if N becomes small poor performance can be obtained using GPC algorithms. It is useful to remember that an optimal predictive control is not necessarily stable even when plant models are linear and known exactly.

7.3.4

Equivalent GPC Cost Minimization Problem

The GPC controller may also be derived from the solution of a “square of sums” cost-minimisation control problem. Recall the NGMV controllers in the previous chapters minimized such a cost-index. This equivalent cost-minimization problem that leads to the same solution as the GPC problem is demonstrated below. It is an interesting property but is also needed to motivate the criterion minimized in the Nonlinear Predictive Generalized Minimum Variance (NPGMV) problem considered in later sections of this chapter (starting in Sect. 7.4). Equivalent cost index: Let the constant positive-definite, real symmetric matrix GTN GN þ K2N that enters the GPC solution be factorised into the form: Y T Y ¼ XN ¼ GTN GN þ K2N

ð7:42Þ

Then, observe that by completing the squares in Eq. (7.36) the cost-function may be written as: 0T T ~ 0 0T T 0 ~ ~ Ttþ k;N R ~ Ttþ k;N GN Ut;N J¼R  Ut;N GN Rt þ k;N  R þ Ut;N Y YUt;N þ J0  t þ k;N   T 1 0T T T T 0 ~ t þ k;N  YUt;N ~ t þ k;N GN Y  Ut þ k;N Y ¼ R Y GN R

~ t þ k;N þ J0 ~ Ttþ k;N ðI  GN Y 1 Y T GTN ÞR þR Clearly, this cost-function may be written in an equivalent form as: ^ ^T J¼U t þ k;N Ut þ k;N þ J10 ðtÞ

ð7:43Þ

In this case the signal to be optimized: 0 ^ t þ k;N ¼ Y T GT R ~ U N t þ k;N  YUt;N 0 ¼ Y T GTN ðRt þ k;N  Dt þ k;N  Ft;N Þ  YUt;N

ð7:44Þ

The terms that are independent of the control action may be written as J10 ðtÞ ¼ J0 þ J1 ðtÞ where,

7.3 Generalized Predictive Control for Linear Systems

~ Ttþ k;N ðI  GN XN1 GTN ÞR ~ t þ k;N J1 ðtÞ ¼ R

315

ð7:45Þ

Writing the cost-function in this alternative form does not of course change the optimal control solution. This may be confirmed by noting last term J10 ðtÞ in Eq. (7.43) does not depend upon control action. The optimal control is found by setting the first term in the criterion to zero, which implies setting (7.44) to zero. The expression for the solution is clearly the same as the GPC optimal control defined in Eq. (7.41).

7.3.5

Modified Cost-Index Giving GPC Control

The above discussion motivates the definition of a new multi-step minimum variance cost minimization problem. This also has a solution for the optimal controller that is the same as the GPC design. Moreover, it is in the form needed for the nonlinear control problem considered in the next section. There are some mathematical preliminaries and the required result is then presented. Consider a new signal to be minimised involving a weighted sum of error and input signals of the form: /ðtÞ ¼ Pc ðz1 Þ ðrðtÞ  yðtÞÞ þ Fc0 u0 ðtÞ

ð7:46Þ

The vector of future values of this signal, for a multi-step cost index, may be written as: 0 0 Ut;N ¼ PCN Et;N þ FCN Ut;N

ð7:47Þ

where the cost-function weightings, based on the original GPC weightings, are defined as follows: PCN ¼ GTN

0 and FCN ¼ K2N

ð7:48Þ

The results in the previous section lead to these definitions. However, the reason for this definition of the cost terms becomes more apparent when the Theorem 7.2 is consulted that is presented below. Minimum variance multi-step criterion: Motivated by the preceding analysis define a minimum variance multi-step cost-function, using the vector of signals: ~JðtÞ ¼ Ef~Jt g ¼ EfUTtþ k;N Ut þ k;N jtg

ð7:49Þ

316

7 Linear and Nonlinear Predictive Optimal Control

Predicting forward k-steps: 0 0 Ut þ k;N ¼ PCN ðRt þ k;N  Yt þ k;N Þ þ FCN Ut;N

ð7:50Þ

Now consider the signal Ut þ k;N and substitute for the vector of outputs Yt þ k;N ¼ Y^t þ k;N þ Y~t þ k;N . Then from (7.50) obtain the inferred output as: 0 0 Ut;N Ut þ k;N ¼ PCN ðRt þ k;N  ðY^t þ k;N þ Y~t þ k;N ÞÞ þ FCN 0 0 ¼ PCN ðRt þ k;N  Y^t þ k;N Þ þ FCN Ut;N  PCN Y~t þ k;N

ð7:51Þ

This expression for the inferred output may be written in terms of an estimate and estimation error vector as: ^ ~ Ut þ k;N ¼ U t þ k;N þ Ut þ k;N

ð7:52Þ

The estimated component: 0 0 ^ ^ U t þ k;N ¼ PCN ðRt þ k;N  Yt þ k;N Þ þ FCN Ut;N

and the prediction error: ~ ~ U t þ k;N ¼ PCN Yt þ k;N

ð7:53Þ

The performance index (7.49) may now be expanded as follows: T ^ ^ ~ ~ ~J ¼ Ef~Jt g ¼ EfUTtþ k;N Ut þ k;N jtg ¼ EfðU t þ k;N þ Ut þ k;N Þ ðUt þ k;N þ Ut þ k;N Þjtg

The terms in the performance index (7.49) can again be simplified, recalling the optimal estimate Y^t þ k;N and the estimation error Y~t þ k;N are orthogonal, and the future reference or setpoint trajectory Rt þ k;N is a known signal. Expanding the cost expression, and invoking the orthogonality of signals, obtain: ^T ^ ~ ^T ~J ¼ EfU t þ k;N Ut þ k;N jtg þ EfUt þ k;N Ut þ k;N jtg ~T ^ ~ ~T þ EfU U U jtg þ EfU jtg t þ k;N

t þ k;N

t þ k;N

t þ k;N

^T ^ ~ ~T ¼U t þ k;N Ut þ k;N þ EfUt þ k;N Ut þ k;N jtg

ð7:54Þ

Cost-minimization: The cost-function may therefore be written in the form: ^T ^ ~JðtÞ ¼ U ~ t þ k;N Ut þ k;N þ J1 ðtÞ

ð7:55Þ

The final cost term is independent of the control action and may be written as:

7.3 Generalized Predictive Control for Linear Systems T ~ ~T ~J1 ðtÞ ¼ EfU ~ ~T t þ k;N Ut þ k;N jtg ¼ EfYt þ k;N PCN PCN Yt þ k;N jtg

317

ð7:56Þ

^ The vector of predicted signals U t þ k;N that defines the optimal control may be ^ simplified by substituting for Yt þ k;N from (7.26), and using (7.42) and (7.48) to obtain: 0 0 ^ ^ U t þ k;N ¼ PCN ðRt þ k;N  Yt þ k;N Þ þ FCN Ut;N   0 0 0 ¼ PCN Rt þ k;N  Dt þ k;N  PCN ðGN Ut;N þ Ft;N Þ þ FCN Ut;N 0 ¼ PCN ðRt þ k;N  Dt þ k;N  Ft;N Þ  ðK2N þ GTN GN ÞUt;N

Substituting from (7.42) obtain the predicted future inferred outputs: 0 ^ U t þ k;N ¼ PCN ðRt þ k;N  Dt þ k;N  Ft;N Þ  XN Ut;N

ð7:57Þ

From a similar argument to that in the previous section, the optimal multi-step minimum-variance control sets the first squared term in (7.55) to zero or ^ t þ k;N ¼ 0. The vector of future optimal controls follows as: U 0 Ut;N ¼ XN1 PCN ðRt þ k;N  Dt þ k;N  Ft;N Þ

This vector is the same as the vector of future controls (7.41) for the GPC solution. These results may be summarised in the theorem that follows. Theorem 7.2: Equivalent Minimum Variance Control Problem Consider the minimisation of the GPC cost-index (7.31) for the system and assumptions introduced in Sect. 7.2, where the nonlinear subsystem W 1k ¼ I and the vector of optimal GPC controls is given by (7.41). Let the cost-index be redefined to have the following multi-step minimum variance form: ~JðtÞ ¼ EfUTtþ k;N Ut þ k;N jtg 0 0 Also let Ut þ k;N ¼ PCN ðRt þ k;N  Yt þ k;N Þ þ FCN Ut;N and define the cost-function 0 ¼ K2N . Then the vector of future optimal weightings as PCN ¼ GTN and FCN controls is identical to the GPC law, defined in (7.41). ■

Solution The proof follows by collecting the above results.

7.4

■

NPGMV Control Problem

There is a rich history of recent research on nonlinear predictive control [30, 42], but the present development is somewhat different, since it is motivated from a model based control design and performance perspective, rather than a method built

318

7 Linear and Nonlinear Predictive Optimal Control

around guaranteed stability requirements. Stability is of course important but the philosophy here is that an optimal control solution is needed, where the cost weightings are parameterized so the controller can easily be tuned to provide good performance. It is accepted that for practical applications robustness and stability will then require very careful assessment over the full operating range. The so-called Nonlinear Predictive Generalised Minimum Variance (NPGMV) control problem is derived in this section [24]. This will be linked to the GPC problem, described above, but it involves a more general cost-function and a nonlinear system description similar to that in the NGMV control problems. The nonlinear input subsystem W 1k in (7.3) is now assumed to be included in the analysis for the rest of the chapter. In this case the actual input to the system is the control signal input denoted uðtÞ and shown in Fig. 7.1, rather than the input to the linear subsystem that is denoted u0 ðtÞ. The cost-function for the nonlinear control problem will therefore need an additional control signal costing term acting on u(t). The cost-function weighting on the intermediate signal u0 ðtÞ will be retained in the criterion to examine certain limiting weighting cases, and to provide an actuator output cost weighting term. This option is useful in machinery and in process control problems. For example, in the case of a valve, the control signal input uðtÞ would represent the position reference to the valve and the signal u0 ðtÞ would represent the position movement of the valve, determining the flow of the process. It is sometimes useful to be able to penalise both of these signals in the cost-function. Control signal weighting: If the smallest delay in each output channel of the plant is of magnitude k the control signal u(t) affects the output at least k-steps later. The control signal costing should therefore be defined to have the dynamic nonlinear operator form: ðF c uÞðtÞ ¼ zk ðF ck uÞðtÞ

ð7:58Þ

Typically this weighting on the nonlinear subsystem input will be a linear minimum-phase transfer-function, but it may also be chosen to be nonlinear to attempt to cancel the plant input nonlinearities in appropriate cases, and it may also be used to introduce an anti-windup capability (Chap. 5 and [23]). The control-weighting operator F ck will be assumed to be square and full rank and to have an inverse. This inverse will normally be chosen to be linear and stable. The signal whose variance is to be minimised, involving a weighted sum of error, linear subsystem input and the control signal, may now be defined to have the form: /0 ðtÞ ¼ Pc eðtÞ þ Fc0 u0 ðtÞ þ ðF c uÞðtÞ

ð7:59Þ

If not needed the weighting Fc0 on the actuator signal u0 ðtÞ can be assumed null. In analogy with the previous GPC problem a multi-step cost-index may be defined that is an extension of the cost-function in (7.49).

7.4 NPGMV Control Problem

319

Extended multi-step cost-function: 0 ~Jp ¼ EfU0T t þ k;N Ut þ k;N jtg

ð7:60Þ

Noting (7.47) the signal U0t þ k;N may be defined that is extended to include the additional future control signal costing term F ck;N Ut; N . Thus, the vector form of future inferred outputs may be expressed as follows: 0 0 Ut;N þ ðF ck;N Ut;N Þ U0t þ k;N ¼ PCN Et þ k;N þ FCN 0 0 ¼ PCN ðRt þ k;N  Yt þ k;N Þ þ FCN Ut;N þ ðF ck;N Ut;N Þ

ð7:61Þ

The nonlinear function F ck;N Ut; N can be defined to have the simple diagonal matrix operator form: ðF ck;N Ut;N Þ ¼ diagfðF ck uÞðtÞ; ðF ck uÞðt þ 1Þ; . . .; ðF ck uÞðt þ N Þg

ð7:62Þ

0 Similarly, Ut;N ¼ ðW 1k;N Ut;N Þ, where W 1k;N it also a block diagonal matrix of the form:

ðW 1k;N Ut;N Þ ¼ diagfW 1k ; W 1k ; . . .; W 1k gUt;N ¼ ½ðW 1k uÞðtÞT ; . . .; ðW 1k uÞðt þ NÞT T

ð7:63Þ

Remarks • The problem simplifies when N = 0 to the single-stage (non-predictive) control problem, which is essentially the same as the NGMV control problem discussed in Chaps. 4 and 5 [22] but for a two degrees of freedom solution. • A particularly useful special case of this algorithm is considered in Chap. 11, where the input subsystem W 1k is set to the identity, and the output subsystem is generalised to have a Linear Parameter Varying (LPV) or state-dependent. This provides a very simple NPGMV controller for a reasonably general system model.

7.4.1

Solution of the NPGMV Optimal Control Problem

The solution for the vector of future controls follows from very similar steps to those in Sect. 7.3.5 and will therefore be summarised only briefly below. Based on ^0 ~0 the estimate U t þ k;N and the estimation error Ut þ k;N the vector of future cost terms ^0 ~0 ¼U þU . From (7.50) this signal may be can be written as U0 t þ k;N

t þ k;N

t þ k;N

written as U0t;N ¼ Ut;N þ zk ðF ck;N Ut;N Þ so that the prediction:

320

7 Linear and Nonlinear Predictive Optimal Control

^0 ^ U t þ k;N ¼ Ut þ k;N þ ðF ck;N Ut;N Þ 0 0 Ut;N þ ðF ck;N Ut;N Þ ¼ PCN ðRt þ k;N  Y^t þ k;N Þ þ FCN

ð7:64Þ

The estimation error for the vector of predicted inferred outputs is given as: T~ ~0 ~ U t þ k;N ¼ Ut þ k;N ¼ GN Yt þ k;N

ð7:65Þ

^0 The future predicted values in the signal U t þ k;N involve the estimated vector of ^ weighted outputs Yt þ k;N and these are orthogonal to Y~t þ k;N . The estimation error is zero-mean and hence the expected value of the product with any known signal is null. The cost-function (7.60) may now be written as: ^0 ^ 0T U ~Jp ðtÞ ¼ U ~ t þ k;N t þ k;N þ J1 ðtÞ

ð7:66Þ

^0 The condition for optimality may therefore be written as U t þ k;N ¼ 0 or from (7.64): 0 PCN ðRt þ k;N  Y^t þ k;N Þ þ ðF ck;N þ FCN W 1k;N ÞUt;N ¼ 0

ð7:67Þ

The alternative solutions for the vector of future optimal controls become:   Ut;N ¼ ðF ck;N  K2N W 1k;N Þ1 PCN ðRt þ k;N  Y^t þ k;N Þ

ð7:68Þ

  2 ^ Ut;N ¼ F 1 ck;N PCN ðRt þ k;N  Yt þ k;N Þ þ KN W 1k;N Ut;N

ð7:69Þ

and

The optimal predictive control law is nonlinear, since it involves the nonlinear control signal costing term F ck;N and the nonlinear model for the plant W 1k;N . Further simplification is possible by substituting from (7.26) for the estimate Y^t þ k;N . The condition for optimality in (7.67) may then be written as:   0 0 Þ þ ðF ck;N þ FCN W 1k;N ÞUt;N ¼ 0 PCN Rt þ k;N  Dt þ k;N  ðFt;N þ GN Ut;N or     0 PCN Rt þ k;N  Dt þ k;N  Ft;N þ F ck;N  ðPCN GN  FCN ÞW 1k;N Ut;N ¼ 0

7.4 NPGMV Control Problem

321

~ t þ k;N from (7.35) and noting from (7.48): Substituting for R 0 ¼ GTN GN þ K2N XN ¼ PCN GN  FCN

The condition for optimality becomes: ~ t þ k;N þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0 PCN R

ð7:70Þ

The vector of future optimal controls, to minimise the cost-index (7.66), follows from the condition for optimality in (7.70), and satisfies:   ~ t þ k;N Ut;N ¼ ðF ck;N  XN W 1k;N Þ1 PCN R

ð7:71Þ

An alternative solution of (7.70), in a more useful form for implementation, follows as:   ~ Ut;N ¼ F 1 R P þ X W U CN t þ k;N N 1k;N t;N ck;N

ð7:72Þ

Remarks The following observations may be made on these last two results: • The NPGMV control law in Eq. (7.71) or (7.72) is clearly model-based and include an internal model of the nonlinear process. The control law is to be implemented using a receding horizon philosophy, and from the preceding discussion it becomes identical to the GPC controller (7.41) in the linear case when the control costing tends to zero (F ck;N ! 0, W 1k;N ¼ I). • The problem construction enables an important asymptotic property to be predicted, and confirmed from (7.71). That is, if the control weighting F ck;N ! 0 then Ut;N should introduce the inverse of the plant model W 1k;N (if one exists) 0 and the resulting vector of future controls Ut;N will then be the same as the GPC control law for the resulting linear system. This is not suggesting that this is a useful or practical solution but it does provide some confidence that the control in this limiting ideal case is desirable. The following theorem may therefore be stated for the solution of the NPGMV control problem. Theorem 7.3: Nonlinear Predictive GMV Control Consider the system described in Sect. 7.2 with the nonlinear finite gain stable nonlinear plant dynamics W 1k . Define the multi-step predictive control cost-function as a sum of future predicted cost terms: 0 Jp ¼ EfU0T t þ k;N Ut þ k;N jtg

ð7:73Þ

322

7 Linear and Nonlinear Predictive Optimal Control

where the signal U0t þ k;N includes the following vector of future error, input and control signal costing terms: 0 0 U0t þ k;N ¼ PCN Et þ k;N þ FCN Ut;N þ ðF ck;N Ut;N Þ

ð7:74Þ

The control signal cost-function weighting F ck;N is a diagonal control weighting (7.62) and is full rank and invertible. The dynamic error PCN and the input 0 are specified in terms of the GPC cost-index (7.31) where the weighting FCN 0 ¼ K2N are defined in (7.48). This ensures the weightings satisfy PCN ¼ GTN , FCN control in the limiting case of a linear system with zero control input costing becomes equal to the GPC solution. The NPGMV optimal control law to minimise the variance (7.73) follows as:   Ut;N ¼ ðF ck;N  XN W 1k;N Þ1 PCN ðRt þ k;N  Dt þ k;N  Ft;N Þ

ð7:75Þ

where the constant matrix XN ¼ GTN GN þ K2N . The current control is found using the receding horizon philosophy, from the first element of the vector. The preferred expression for implementation of the vector of future optimal controls becomes:   Ut;N ¼ F 1 ck;N PCN ðRt þ k;N  Dt þ k;N  Ft;N Þ þ XN W 1k;N Ut;N

ð7:76Þ

where the signals Ft;N ¼ HNZ ðz1 Þzp ðtÞ þ SNZ ðz1 Þuf ðt  1Þ, uf ðtÞ ¼ 1 1 1 1 1 1 Df 1 ðz Þu0 ðtÞ and where Df 1 satisfies B1k ðz1 ÞD1 ðz Þ ¼ D ðz ÞB ðz Þ. ■ 0k f1 f Proof The proof of the NPGMV optimal control solution follows by collecting the results in the above section. The closed-loop stability of the system is considered in the analysis that follows in Sect. 7.5. ■ Remarks The following observations may be made: • The known component of the disturbance enters the equations in a similar way to the known future setpoint information, and it provides a degree of feedforward action [43]. If future disturbance information is not available, then only the stochastic disturbance model need be used. If the disturbance only has a known component at time t, then the deterministic component can be approximated as being constant into the future. • The two expressions for the NPGMV control signal (7.75) and (7.76) lead to two alternative structures for the optimal controller. The first solution is of conceptual interest, and is shown in Fig. 7.2. The second is more practical for implementation, and is discussed in the next section.

7.4 NPGMV Control Problem

323

Fig. 7.2 First form of the NPGMV polynomial controller structure

7.4.2

Implementation of the NPGMV Optimal Controller

The way in which the nonlinear controller may now be implemented in principle will now be considered. There are other issues to be considered such as avoiding the algebraic loop but these will be discussed later in Chap. 11 (Sect. 11.7). The control at time t is computed for N > 0 from the vector of current and future controls. It is convenient to introduce the matrix: CI0 ¼ ½I; 0; . . .:; 0

ð7:77Þ

The control at time t can then be found from the vector of controls as: uðtÞ ¼ ½I; 0; . . .:; 0Ut;N

ð7:78Þ

Similarly, to compute the vector of future controls (for t > 0), introduce the matrix:

f Ut;N ¼ C0I Ut;N

C0I ¼ ½ 0 IN  2 uðtÞ 6 .. ¼ ½ 0 IN  4 .

uðt þ NÞ

ð7:79Þ 3 uðt þ 1Þ 7 7 6 .. 5 5¼4 . 3

2

ð7:80Þ

uðt þ NÞ

The vector of future controls can now be computed, from (7.76), as:   f ~ Ut;N ¼ CI0 F 1 ck;N PCN Rt þ k;N þ XN W 1k;N Ut;N

ð7:81Þ

~ t þ k;N ¼ Rt þ k;N  Dt þ k;N  Ft;N . The vector W 1k;N Ut N may be written, where R noting (7.63), by partitioning the current and future terms, in the form:

324

7 Linear and Nonlinear Predictive Optimal Control

ðW 1k;N Ut;N Þ ¼ ½ðW 1k uÞðtÞT ; . . .; ðW 1k uÞðt þ NÞT T f T T ¼ ½ðW 1k uÞðtÞT ; ðW 1k;N1 Ut;N Þ 

ð7:82Þ

Write the constant symmetric matrix XN , using a related partition, in the form: XN ¼ GTN GN þ K2N ¼ ½Y1

Y2 

ð7:83Þ

The first column Y1 has the same number of columns as the number of control inputs m0 . Thence obtain, f XN W 1k;N Ut;N ¼ ½Y1 Y2 W 1k;N Ut;N ¼ Y1 ðW 1k uÞðtÞ þ Y2 ðW 1k;N1 Ut;N Þ

ð7:84Þ

Using these results the current and future NPGMV optimal controls may be expressed, as: Current control:   f ~ uðtÞ ¼ CI0 F 1 ck;N PCN Rt þ k;N þ Y1 ðW 1k uÞðtÞ þ Y2 ðW 1k;N1 Ut;N Þ

ð7:85Þ

Future controls:   f f ~ R ¼ C0I F 1 P þ Y ðW uÞðtÞ þ Y ðW U Þ Ut;N CN t þ k;N 1 1k 2 1k;N1 t;N ck;N

ð7:86Þ

Controller structure: The second equation (7.76) in Theorem 7.3 for implementing the optimal control may therefore be split into the current and future controls (7.85) and (7.86), and implemented in the more practical form shown in Fig. 7.3. These expressions can be simplified further noting the control costing F ck; N has a block-diagonal matrix structure. In this case, from (7.77) and from (7.79), respectively:

Fig. 7.3 Second form of NPGMV controller (including future predicted control signal generation)

7.4 NPGMV Control Problem

325

1 1 CI0 F 1 ck;N ¼ ½F ck ; 0; . . .; 0 ¼ F ck CI0

 C0I F 1 ck;N ¼ 0ðN þ 1Þm;m

7.4.3

h IN F 1 0ðN þ 1Þm;m ¼ ck;N

ð7:87Þ F 1 ck;N1

i

ð7:88Þ

Properties of the Predictive Optimal Controller

The above results may be summarised in the following lemma that also introduces a necessary condition for stability. Lemma 7.1: NPGMV Optimal Control Law Properties and Implementation Consider the nonlinear system described in Sect. 7.2 where the nonlinear plant operator W 1k is finite gain stable. Assume in this case that the control weighting term F ck; N is linear and the NPGMV control cost-index to be minimised is given by (7.60). Also, assume for stability, that the following operator has a finite gain stable causal inverse: 

ðXN þ PCN WNk CI0 ÞW 1k;N  F ck;N



ð7:89Þ

If the control signal weighting term is block diagonal then by partitioning the optimal control to be applied at time t (invoking the receding horizon principle), the vector of future predicted controls, may be computed as:   f uðtÞ ¼ CI0 F 1 ck;N PCN ðRt þ k;N  Dt þ k;N  Ft;N Þ þ Y1 ðW 1k uÞðtÞ þ Y2 ðW 1k;N1 Ut;N Þ

ð7:90Þ and   f f Ut;N ¼ C0I F 1 ck;N PCN ðRt þ k;N  Dt þ k;N  Ft;N Þ þ Y1 ðW 1k uÞðtÞ þ Y2 ðW 1k;N1 Ut;N Þ

ð7:91Þ where the constant matrices CI0 ¼ ½I; 0; . . .; 0, C0I ¼ ½ 0 IN  and XN ¼ ½Y1 Y2 . If the error and input cost-function weightings are defined in the GPC motivated form 0 PCN ¼ GTN and FCN ¼ K2N then for a linear system (W 1k ¼ I) the limiting form of ■ the optimal control when F c ! 0 is equal to a GPC optimal control law. Proof The different forms of the control law follow by collecting the results in the section above. The relationship to linear GPC control was established from the definitions of the weightings in Sect. 7.3.3 and the results in Theorem 7.3. The assumption required to ensure closed-loop stability is considered in the analysis of stability that follows in the next section. ■

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7 Linear and Nonlinear Predictive Optimal Control

Fig. 7.4 Second form of NPGMV controller

Remarks • The computation of the current control in (7.90) is illustrated in the form shown in Fig. 7.4. • If the output weighting Pc includes a near integrator, it appears both in the feedback and reference channels and for implementation; it is desirable to move this integrator term into the common path that has the difference of signals. This is simple to achieve if the controller is implemented as in Fig. 7.4, where the integrator can clearly be removed from the signals Rt þ k;N  Dt þ k;N and zp ðtÞ ¼ Pc ðz1 ÞzðtÞ into the common channel feeding the error weighting PCN (note (7.27)). Robustness: Simulation of predictive controls is often valuable to reduce commissioning time and to optimise performance and robustness. Confirmation of the importance of simulation and the link to tuning is evident from researchers at Cummins working on diesel engine controls. They used a switched MPC approach for the diesel engine air-handling system [44, 45] and employed extensive MonteCarlo based simulation trials to tune the system and improve robustness.

7.5

Stability of the Closed-Loop System

As noted for the GPC controller due to the finite horizon cost-function, stability is not guaranteed with many predictive controls and stability must be ensured by careful tuning of the weightings and the cost-horizons. Some predictive control methods were developed with guaranteed stability properties in the 1990s. One method involved the use of terminal constraints. In one of the more significant early results [28] showed that finite-time LQ optimal control problems could be stabilized using a receding horizon based controller with an end-state constraint xðTf Þ ¼ 0. The use of an end-state weighting in the cost-index to improve stability properties is explored further in Chap. 9. The stability of infinite horizon and moving horizon discrete-time optimal control laws was considered by Keerthi and Gilbert in a

7.5 Stability of the Closed-Loop System

327

seminal paper [46]. They established that under weak conditions on the cost-function and constraints uniform asymptotic stability could be guaranteed. Despite progress, there remain stability and robustness issues still to explore for both linear and nonlinear predictive controls [47]. The approach to the analysis of stability conditions taken here is rather different to most of the work on predictive control. The industrial applications of interest are so complex, uncertain and nonlinear, that it is not very useful to have theoretical guarantees of stability even though this is desirable and comforting. In practice, cost-function weightings will have to be selected and disturbance models modified to try to ensure stability and adequate robustness. It is therefore useful to explore how these weightings will affect stability. Recall for linear GMV control design stability is ensured when the combination of a control weighting and an error weighted plant model is strictly minimum-phase. For the proposed nonlinear predictive controller it is shown below that a necessary condition for stability is that the nonlinear operator  ðXN þ PCN WNk CI0 ÞW 1k;N  F ck;N has a stable inverse. The type of nonlinear stability condition that is established depends upon the assumption of stability for the nonlinear plant subsystem W 1k . This black-box nonlinear term is often assumed finite gain stable. Assumption for stability analysis: The terms in the control action need to be simplified further for the stability analysis. Also assume for this discussion that the stochastic exogenous inputs and known disturbance signals are null, and the only input is due to the known future reference (setpoint) signal. Condition for optimality: The condition for optimality in (7.70) may now be written, setting the aforementioned signals to zero, as follows: PCN ðRt þ k;N  Ft;N Þ þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0 1 Noting uf ðtÞ ¼ D1 f 1 ðz Þu0 ðtÞ then from (7.27):

Ft;N ¼ HNZ zp ðtÞ þ SNZ D1 f 1 u0 ðt  1Þ Also from (7.10) for this problem zp ðtÞ ¼ Pc ðz1 ÞW0k ðz1 Þu0 ðt  kÞ and substituting:   1 PCN Ft;N ¼ PCN HNZ Pc W0k zk þ PCN SNZ D1 u0 ðtÞ f1 z Thence, after substitution, the condition for optimality becomes:   1 PCN Rt þ k;N  PCN HNZ Pc W0k zk þ SNZ D1 CI0 W 1k;N Ut;N f1 z þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0

ð7:92Þ

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7 Linear and Nonlinear Predictive Optimal Control

The second group of terms in this expression may be written as: k WNk ¼ z1 SNZ D1 f 1 þ z HNZ Pc W0k

ð7:93Þ

The condition for optimality (7.92) now follows from (7.93) as:   PCN Rt þ k;N þ F ck;N  ðXN þ PCN WNk CI0 ÞW 1k;N Ut;N ¼ 0

ð7:94Þ

From this condition, the vector of future controls and outputs, given the vector of future reference values, may be expressed as follows. Future controls:  1 Ut;N ¼ ðXN þ PCN WNk CI0 ÞW 1k;N  F ck;N PCN Rt þ k;N

ð7:95Þ

NL plant future outputs:  1 W 1k;N Ut;N ¼ W 1k;N ðXN þ PCN WNk CI0 ÞW 1k;N  F ck;N PCN Rt þ k;N

ð7:96Þ

Total NL future plant outputs:  1 W k;N Ut;N ¼ W k;N ðXN þ PCN WNk CI0 ÞW 1k;N  F ck;N PCN Rt þ k;N

ð7:97Þ

where the total future predicted plant block structure has the form: ðW k;N Ut;N Þ ¼ ½ðW0 W 1k uÞðtÞT ; . . .; ðW0 W 1k uÞðt þ NÞT T

ð7:98Þ

The output signal involves the unstructured nonlinear input block, which is assumed stable, but the linear output subsystem can be unstable. If it is assumed that the cost-weightings can be chosen so that the inverse of the operator ðXN þ PCN WNk CI0 ÞW 1k;N  F ck;N is finite-gain m2 stable, then the control signal (7.95) depends only on stable operators. The stability of the inverse of this operator is a necessary condition for stability. Some observations may be made on these results: • The necessary condition for stability is that the operator (7.89) has a stable inverse. This plays a similar role to the “generalized plant” in the NGMV control solutions (Chap. 4). • The future predicted controls feed into the optimal solution as in Fig. 7.4. • In the limiting case when the nonlinear control weighting F ck tends to zero, and the plant is linear, the control signal becomes equal to that of a linear system GPC control design. 0 ¼ K2N goes to • In the limiting case when the multi-step control weighting FCN zero and the number of steps is unity the controller reverts to a two degrees of freedom NGMV control solution.

7.5 Stability of the Closed-Loop System

7.5.1

329

Relationship to the Smith Predictor and Stability

The Predictive controller can be related to the Smith Predictor, in much the same way as was undertaken in previous chapters for the NGMV controller. The introduction of this type of structure limits the applications to open-loop stable systems. That is, although the structure illustrates a useful link between the new solution and the Smith time-delay compensator, it also has the same disadvantage, that it may only be used on open-loop stable systems [43]. This Nonlinear Smith Predictor will now be considered briefly. First, observe that the controller structure shown in Fig. 7.4 may be redrawn as shown in Fig. 7.5. The linear subsystems may also be changed by adding and subtracting equivalent terms. Combining the terms in the three linear inner-loop blocks with the common input u0 ðtÞ ¼ ðW 1k uÞðtÞ, and using the result in (7.93): k ðz1 PCN SNZ D1 f 1 þ z PCN HNZ Pc W0k þ Y1 Þ ¼ ðPCN WNk þ Y1 Þ

This reveals that the controller may be implemented in the Smith Predictor form shown in Fig. 7.6. Stability in Smith form: In the Smith Predictor version, the transfer in Fig. 7.6 from the control signal u to the feedback signal p is null when the model zk W k ¼ zk W0k W 1k matches the actual plant model. It follows that the control action, due to the reference signal changes, is not due to feedback but involves the open-loop stable compensator with the block PCN HNZ Pc and the inner nonlinear feedback subsystem. The assumption is made that this subsystem is stable because of the weighting choice. The NPGMV solution shown in Fig. 7.6 is therefore stable, because the plant is assumed stable (in this case), the inner-loop is stable and there are only stable terms in the input block.

Fig. 7.5 NPGMV feedback control signal generation and controller modules

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7 Linear and Nonlinear Predictive Optimal Control

Fig. 7.6 Nonlinear Smith-predictor version of NPGMV controller and internal model

7.6

Design, Limits and Constraints

The linear predictive control approach is very successful in industrial applications mainly due to its flexibility and limited complexity in the unconstrained case. The main drawback of MPC is that a model for the process, which describes the input to output behaviour of the process, is needed. This is of course a feature of all model based control laws, but for MPC the model is needed for prediction, leading to concerns about model accuracy. For example, it is usually true that as the prediction horizon increases the control tends to be more robust (in terms of sensitivity functions). Unfortunately, over a certain prediction horizon this trend can reverse due to the effect of uncertainty in the prediction models (caused by a build-up of modelling errors). The choice of the cost-function weightings and cost-horizon are important to the robustness properties. Recall one of the methods for generating cost-function weightings for NGMV controllers, was that based on the PID parameterization, described in Chap. 5. The approach employed a PID structure that provided a starting point for design and guaranteed a stabilising solution. A similar method for choosing the initial values of the dynamic cost-weightings may be used in NPGMV control. Note that the NPGMV control law reduces to an NGMV controller in the special case of a single-stage criterion. This link can be exploited and the initial values of the dynamic cost-function weightings for NPGMV design may be based on a PID structure, although strictly the results for the predictive control problem are only valid in this special limiting case. The constraint handling features of predictive controls are welcomed by designers, since they provide a direct connection between the engineering design process and the plant performance. Such constraints may not correspond with physical barriers. That is, constraints or limits may be introduced using software to protect electro-mechanical systems from overloading due to excessive current, speed, temperature, torque or loading.

7.6 Design, Limits and Constraints

331

The natural multivariable capabilities of predictive controls is also a valuable feature. Large multivariable systems often arise in process control applications. The number of manipulated variables (inputs) and controlled variables (outputs) is often different in process applications and only a small number of outputs may have fixed setpoints.

7.6.1

Limits, Constraints and Quadratic Programming

Nonlinearities of some form are always present in real systems and these can relate to constraints like current limits in converters. One of the main features of predictive controllers is that constraints in the system may be included in the optimization problem. The MPC control structure has been used in the heavy industries for several decades and it has become an industry standard in the petrochemical industry. This is due to its ability to handle constraints and to cope with the multivariable interacting nature of systems. Constraints often represent critical limits that can be related to the economic performance of a process. In fact, all physical systems can have hard limits on inputs, outputs and states, such as the rate limits on actuators. These limits can be addressed using quadratic or linear programming, possibly at an upper level, or in the intermediate level predictive controller optimization algorithm. As an example of hard limits, consider a continuous stirred tank reactor (CSTR) with an exothermic reaction and a cooling coil. The composition of the reactant in the effluent is to be controlled, but the temperature must be limited to below a maximum limit to prevent damaging the glass lining of the reactor. Hard constraints may also be due to mechanical limits like the rudder limits in ships [48–50]. Even classical PID-based control systems involve compensation for limits, since they include ‘jacketing software’ like anti-windup schemes. Some common constraints may be listed as: • Safety limits and constraints, which mostly restrict the state variations and trajectories. • Actuator and input saturation constraints, which usually restrict the control signal amplitude. • Hardware constraints, which influence the real-time implementation of the algorithm • Performance constraints that usually restrict the time and frequency responses. • Product quality constraints. Nonlinear cost-function weighting terms in MPC provide a mechanism to handle so-called soft constraints indirectly. However, one of the main advantages of MPC design is that hard constraints on process variables can be treated explicitly [51–53]. Input amplitude constraints, and input rate of change constraints are common and can be accommodated very efficiently using MPC. This usually involves adding an

332

7 Linear and Nonlinear Predictive Optimal Control

inequality constraint of the form, Au(t + k)  b, which is introduced into the optimization problem. In fact, lower and upper bounds can be treated as linear inequality constraints in this standard form. In most MPC algorithms the predictions are made online at each control instant, and it is therefore possible to choose future control inputs so that constraints can be satisfied. There is not of course a guarantee in practice that the control implemented using a receding horizon philosophy will ensure the output constraints are always respected. Quadratic programming: If a MPC involves a Linear Quadratic (LQ) cost-function and the system is linear, with linear constraints, then the control solution leads to a Quadratic Programming (QP) optimization problem (Sect. 7.1.4). It has the following standard description:

Min U

1 T ðU XUÞ þ F T U 2

 subject to AU  b

ð7:99Þ

A QP problem can be solved within a finite number of numerical operations. It is a convex optimization problem with a unique minimum. If the process model is non-linear then the prediction model will also be non-linear, which requires a non-linear optimization method that may be solved using Sequential Quadratic Programming (SQP). It is well known a non-linear optimization problem can have problems with local minima and convergence. The solution of MPC problems with constraint handling can also be very computationally demanding compared with the unconstrained case. Consider the general quadratic cost-index (7.38) in Sect. 7.3.1, and note that to minimize the criterion the following cost was to be minimized: J ¼ Z T Z  U T BT Z  Z T BU þ U T XN U which can be written in the QP form (7.99). Minimizing this quadratic function with linear constraints on U is therefore a standard QP problem. This can be solved by algorithms such as active set or interior point methods, and since the matrix XN is positive-semidefinite, any local solution of the QP problem is a global solution (since it is a convex QP problem). The need for an on-line solution to a constrained optimization problem restricts the application of MPC for systems with very fast dynamics. This has led to the use of application-specific controllers for control systems that need fast sampling and short sampling periods [54]. The wide-scale adoption by industry of model predictive control is due to its ability to handle hard constraints in multivariable systems on controls, control increments, states and outputs (both measured and unmeasured). These constraints are particularly important in the petro-chemical industry, where optimization of set points results in steady-state operation close to the boundary of the set of permissible states. The traditional method of handling hard constraints is to use a predictive control law, applying a numerical solution using quadratic dynamic programming. In this type of application, the plant models are sufficiently ‘slow’ to permit the use of such algorithms.

7.6 Design, Limits and Constraints

333

Soft constraints: Soft constraints may be introduced in optimal control problems by increasing the penalty on control action. This soft form of constraint will enable the variance of the control signal to be limited. Another way of addressing constraints was considered in Chap. 4. That is, a way of introducing anti-wind up compensation for controllers that have integral action was introduced. Embedding procedures for constraint satisfaction: The flexibility demonstrated of the so-called NGMV family of control laws to utilise nonlinear input subsystems and cost weightings on control action may also be exploited in this NPGMV case. For example, if a converter fed drive has current limits then these can be treated as a hard saturation nonlinearity. This is not quite the same as the situation in a mechanical system where there are actual hard limits on the movement of an actuator. Nevertheless, as far as the nonlinear control law design is concerned it believes there is a hard limit present in the system and the solution for the optimal control should accommodate this limitation. The problem of dealing with input constraints in this way for a flight control has been considered by Youssef et al. [55]. The constraints were approximated by means of a smooth limiting function that was included in the plant dynamics. This method of incorporating constraints by introducing them within the nonlinear system model is a so-called constraint embedding approach. This has an advantage over techniques such as quadratic programming because a clear controller structure exists that can be justified intuitively. The well-defined controller structure enables its behaviour to be analysed and predicted. Unfortunately, it is not so precise at satisfying constraints as QP methods. Numerical optimisation algorithms like QP solutions can be very effective, but they do not allow intuitive insights to be obtained, and they are more difficult to validate in a range of possible operating conditions. There has therefore been pressure to find alternative means of ensuring constraints are satisfied. The reference governor approach to handling constraints described in the next section has the advantage that it is both intuitive and numerically efficient.

7.6.2

Reference Governors

Constrained optimization adds to computational complexity and makes stability and robustness margins more difficult to quantify. Another way of dealing with constraints is referred to as a reference governor (see [46, 56]). In this approach, if the system senses the control or output will exceed given limits, then the reference trajectory is adjusted accordingly. For example, the magnitude of the reference input signal might be reduced, if it is clear future controls will exceed saturation limits. The benefit of this idea is that the adjustment occurs outside the feedback control loop. Thus, if the system is linear but has constraints then such modifications do not change its linear behaviour. There is therefore a smaller chance, even with a nonlinear system, that such changes will be destabilising.

334

7 Linear and Nonlinear Predictive Optimal Control

Compared with a numerical optimisation algorithm the reference governor approach is simple to understand. The reference governor can specify constraints pointwise in time. The limits and the nonlinearities, like actuator saturation can be taken into account by modifying the reference or setpoint signal. The constraints can be imposed indirectly. The main benefit is that when the constraints are satisfied, the closed-loop control system will respond linearly. Poor responses due to constraints and nonlinearities should then be avoided. Reference governor schemes are convenient because they augment, rather than replace an existing controller. Constraint handling summary: The different ways that constraints or limits can be handled may now be summarized: 1. Cost-function weighting changes to introduce so called soft constraints. 2. The use of nonlinear cost-function weightings to introduce hard limits. 3. Embedding constraints within the nonlinear system models so that the constraint is introduced indirectly by the resulting nonlinear control law. 4. Use of reference signal or setpoint adaption to introduce constraints due to reference signal variations. 5. Direct numerical constrained optimization, such as QP methods.

7.7

Ship Roll Stabilization Using Fins and Rudder

As an illustration for the proposed nonlinear predictive control algorithm, the control of the yaw and the rolling motions of a ship will be considered. The rudder angle and the fin roll stabilizer positions will be the control inputs. The fins are mounted beneath the waterline and are connected to the hull. They are mainly used to reduce a ship’s rolling motion in a rough sea state due to high waves and strong wind. The fin angles are controllable in an active fin roll stabilization control system, so that the rolling forces can be countered. These are necessary because the period of the waves at sea can sometimes be close to the natural roll period of the vessel. The rolling motion can therefore become magnified, which is unacceptable to passengers in cruise vessels and can degrade the performance of military vessels, or cause containers to fall overboard. A ship’s heading or yaw angle is controlled by the rudder but this also affects the roll of the vessel. The rolling motion that is caused by the first-order wave force disturbances can be counteracted by use of both the rudder and the fin roll stabilizers. A number of commercial rudder-roll stabilization systems have been developed. They require high-performance rudder machinery, but they have the advantage that they can provide improved performance, or enable smaller fins to be used for roll stabilization (see [57–59]). Large cruise vessels may have four roll stabilizers, two on each side. A vessel with conventional angular motion notation, is shown in Fig. 7.7. The roll stabilizers and rudder represent the actuators and the outputs to be controlled are the yaw and roll angles.

7.7 Ship Roll Stabilization Using Fins and Rudder

335

Fig. 7.7 Standard ship motion description

Fig. 7.8 Block diagram of the ship model and control structure

The basic dynamics of the ship roll and yaw motion [60], with respect to the fin and rudder, for particular ship speed and encounter angle, are shown in Fig. 7.8. Ship models are very nonlinear and include problems like non-minimum phase behaviour but are often well defined if ship tank testing has been performed [57– 59]. This problem is also discussed in Chap. 15 but using a simpler single-loop control design strategy and a non-predictive (NGMV) controller. Model Definitions The transfer-functions, for the components of the model, are defined as: 2

ð0:8Þ Roll model: G/ ðsÞ ¼ s2 þ 20:20:8s þ ð0:8Þ2 0:2 Yaw model: Gw ðsÞ ¼ sð10s þ 1Þ

336

7 Linear and Nonlinear Predictive Optimal Control

Bode Diagram Magnitude (dB)

50 0 -50 -100

Roll/Fin Yaw/Rudder Roll/Rudder Wave and Roll

Phase (deg)

-150 360 180 0 -180 -360 10 -3

10 -2

10 -1

10 0

10 1

10 2

Frequency (rad/s) Fig. 7.9 Frequency responses of the system models and wave spectra

Rudder to roll interaction model: Gd/ ðsÞ ¼ 0:1ð14sÞ ð6s þ 1Þ The model includes non-minimum phase interaction from the rudder to roll motion. There is an integrator in the yaw model. The roll characteristics of the ship are modelled using a resonant second-order system, with a natural frequency of 0.8 radians/s and a low damping factor. The frequency responses of these models are as shown in Fig. 7.9. Fin and rudder constraints: The fin and rudder actuators Ga and Gd have hard constraints on the angle and rate. The system can be represented by the block diagram in Fig. 7.10. The actuator amplitude and rate limits are 25° and 10°/s for the fins, and 30° and 7°/s for the rudder servo, respectively. Classical controllers: The classical controllers have roll rate and yaw tracking error inputs, respectively: Fin-roll controller: Cfin-roll ðsÞ ¼ ð17s2 þ 4216s þ 133:3Þ=ð100s2 þ 1000s þ 1Þ Course control autopilot: Cautopilot ðs) = ð2:5 s + 0:25Þ=ðs + 1Þ

7.7 Ship Roll Stabilization Using Fins and Rudder

337

Fig. 7.10 Fin and rudder servo models with constraints

Yaw angle response

Roll angle response 30

50 Uncontrolled Classical GPC

40

10

30

(deg)

(deg)

20

0

20

-10

10

-20

0

-30 0

100

200

300

400

500

-10 0

600

100

500

600

(deg)

50

10

0

c

(deg)

400

100

20

c

300

Rudder command

Fin command 30

0

-50

-10 -20 0

200

100

200

300

400

500

600

-100 0

100

200

300

400

500

600

time (s)

time (s)

Fig. 7.11 Comparison of nominal GPC (N = 10) and classical control results for the unconstrained case

Disturbances: The effect of the wave disturbance on the roll and yaw motion is represented in Fig. 7.11 by the signals d/ and dw, where d/ ¼

5s s2 þ 2  0:1  0:7s þ ð0:7Þ2

and

dw ¼

0:5 f s

338

7 Linear and Nonlinear Predictive Optimal Control

and n and f are white noise sequences. The model for the roll wave disturbance provides a second order linear approximation to the Pierson-Moskowitz spectrum. The yaw disturbance is assumed to be of a dominant low-frequency nature and is modelled by an integrator driven by white noise. Discretized plant and disturbance models: For the purpose of controller design, the continuous-time models of the system have been discretized using the sample time of 0.5 s, and the resulting discrete-time models were obtained as: " W0 ¼ z

1

0:0749 þ 0:07099z1 11:706z1 þ 0:8521z2

0:01549 þ 0:003845z1 þ 0:01554z2 12:626z1 þ 2:422z2 0:7840z3 0:002459 þ 0:002418z1 11:951z1 þ 0:9512z2

0 "

Wd ¼

0:06208 þ 0:1692z1 0:1759z2 0:05534z3 13:522z1 þ 4:882z2 3:138z3 þ 0:7945z4

0

#

# 0 0:2499 1z1

Control objectives: The main control objectives are the reduction of rolling motion and the tracking of the heading setpoint. The former can be characterized by the so-called Roll Reduction Ratio (RRR), which can be defined as: RRR ¼

  closed-loop roll variance 1  100 % open-loop roll variance

This ratio describes the improvement in roll reduction achieved by using feedback control, with 100% corresponding to the ideal zero roll motion. The yaw tracking performance can be measured using conventional measures such as IAE or ISE. In a classical control scheme, rolling motion is regulated using fin stabilizers, and the heading is controlled with the rudder, forming a 2  2 multi-loop system. On the other hand, a multivariable control scheme will consider the system interaction, allowing the rudder to actively attenuate the roll and yaw motions, which is possible because of the separation in the roll and yaw frequency responses [59]. Simulation results: In the simulations, the ship yaw angle is required to follow a known trajectory consisting of two-step changes, while minimizing the roll motion, according to the specified criterion. In the limiting case when W 1k ¼ I and F ck ! 0, the NPGMV controller collapses to a version of the standard GPC controller but with weighted output and reference signals in the cost criterion. The results for the nominal settings of N ¼ 10; K0 ¼ diagf0:005; 0:01g are shown in Fig. 7.11. The Pc weighting was chosen based on a multi-loop classical controller (see Ref. [9]), the performance of which is also shown. The figures of merit for this and subsequent cases are collected in Table 7.1. The GPC results for the roll attenuation in this unconstrained case are comparable with the classic design results but are somewhat unrealistic. Detuning the controller (increasing k weighting) is normally needed in the real system. Note also the slightly higher bandwidth in the yaw-tracking channel. The predictive action can also be utilized when the future yaw trajectory is known, and this is illustrated in Fig. 7.12 (the stochastic noise has been removed and the time scale magnified to

7.7 Ship Roll Stabilization Using Fins and Rudder

339

Table 7.1 Figures of merit for different cases Controller

RRR (%)

Yaw IAE

Nominal GPC (N = 10) and classical control (unconstrained case) GPC 95.5 3380 Classical 97.5 3376 GPC control: effect of varying the prediction horizon N (constrained case) N=1 1.8 2706 N=5 8.4 2245 N = 10 32.7 1898 N = 25 55.8 1504 N = 50 59.0 1520 Final retuned designs (constrained case, N = 5) GPC 63.7 2225 NPGMV 90.5 2477

Yaw angle response (unconstrained)

200

240

260

300

200

200

0

220

240

260

N=1 N=5 N = 10 N = 25 280N = 50

300

Rudder command

20 0

c

c

80 60 40 20 0 -20 180

40

(deg)

(deg)

220

N=1 N=5 N = 10 N = 25 280N = 50

Rudder command

400

-200 180

Yaw angle response (constrained)

(deg)

(deg)

80 60 40 20 0 -20 180

-20 200

220

240

time (s)

260

280

300

-40 180

200

220

240

260

280

300

time (s)

Fig. 7.12 GPC results: effect of varying the prediction horizon (N = 1, 5, 10, 25, 50) for the unconstrained case (left) and the constrained case (right)

show the predictive action more clearly). Increasing values of N lead to increasingly more damped responses, and actuator action that takes account of future reference changes. The longer prediction horizon often seems to increase the speed of response and improve the robustness of the solution, which is clear from the reduced overshoots in the yaw angle responses. The actuator constraints impose a limit to the response speed that can be achieved. A comparison of the unconstrained and constrained system responses is shown in Fig. 7.12. When the constraints are present, the GPC controller needs to be detuned to maintain stability. The nonlinearities can be accounted for more effectively by introducing the nonlinear control weighting F ck into the NPGMV control structure. After tuning, the results are shown in Fig. 7.13 (for the case of N = 5), where the

340

7 Linear and Nonlinear Predictive Optimal Control Roll angle response

Yaw angle response

30

50 Uncontrolled GPC NPGMV

0

20 10

-20

0 100

200

300

400

500

-10 0

600

Fin command

100

300

400

500

600

500

600

0

c

0

200

Rudder command

50

(deg)

50

(deg)

30

-10

-30 0

c

40

10

(deg)

(deg)

20

-50 0

100

200

300

400

500

time (s)

600

-50 0

100

200

300

400

time (s)

Fig. 7.13 NPGMV and linear GPC responses for the nonlinear system (N = 5)

NPGMV satisfies the rudder angle limits that are exceeded (with command signals truncated) by the GPC design. The Table 7.1 reveals that the GPC controller performance deteriorates when constraints are applied, as might be expected. The improved control performance with increasing cost-horizons is also typical behaviour. Finally, the NPGMV controller seems to offer some benefits because of its greater flexibility in cost-function definitions.

7.8

Concluding Remarks

There are many nonlinear predictive control strategies based on ideas such as linearization around a trajectory and others. However, the aim was to try to produce a control law that is simple to implement and use. The overall solution approach above was not of course based on a rigorous stability theory for nonlinear systems, since this can lead to algorithms that are difficult to understand and apply. The focus here was more on the development of an optimal control design methodology that was linked to well established, model based control design approaches.

7.8 Concluding Remarks

341

The stability issue [22, 23] was not neglected but it does require further study, particularly when qLPV or state-dependent models are used (see Chap. 11). The predictive control methods described in this chapter were based on polynomial system methods to represent the linear part of the plant model [61]. The generalized predictive controller was the first of the MPC solutions presented. This relates to linear predictive controls like dynamic matrix control. These have been hugely successful in applications. The GPC controller minimised a multi-step predictive control cost-function and assumed future setpoint information was available. The most interesting version of this GPC type solution is given later in Chap. 11, where a quasi-LPV or state-dependent/LPV plant structure is used. The Nonlinear-GPC (NGPC) structure described in Chap. 11 includes various other generalizations. The Nonlinear Predictive Generalized Minimum Variance control design problem was considered in the later part of the chapter (Sect. 7.4 onwards). This was is a development of the nonlinear generalised minimum variance (NGMV) design approach, described in Chap. 4. It has the useful property that if the system is linear then the control reverts to the GPC design method, which is very well accepted in industry. That is, the NPGPC control design method reduces to that of GPC control design when the weighting F ck tends to zero and the system is linear (W 1k ¼ I). If the predictive control cost-horizon reduces to a single-step, then the NPGMV control law approaches a so-called two degree of freedom NGMV control solution. The relationship of the NPGMV control design to a Smith Predictor was established to provide some confidence in the solution. The NPGMV controller can however, stabilize an open-loop unstable plant, unlike the Smith controller. The NPGMV control law includes an internal model of the process but many of the computations, as in traditional polynomial equation based predictive control, simply involve the solution of Diophantine equations and matrix multiplications. The polynomial solution is more suitable for use in applications such as multivariable adaptive controllers [62, 63]. The use of state-space models in predictive control is introduced in Chap. 9. A more general solution to nonlinear predictive control problems, using quasi-LPV or state-dependent models, is provided in Chap. 11.

References 1. Richalet J, Rault A, Testud JL, Papon J (1978) Model predictive heuristic control: applications to industrial processes. Automatica 14:413–428 2. Cutler CR, Ramaker BL (1979) Dynamic matrix control—a computer control algorithm. In: 86th national meeting, A.I.C.H.E 3. Clarke DW, Mohtadi C, Tuffs PS (1987) Generalized predictive control: part 1. The basic algorithm. Automatica 23(2):137–148 4. Clarke DW, Mohtadi C, Tuffs PS (1987) Generalized predictive control: part II. Extensions and interpretations. Automatica 23(2):149–160

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5. Clarke DW, Montadi C (1989) Properties of generalised predictive control. Automatica 25 (6):859–875 6. Richalet J (1993) Industrial applications of model based predictive control. Automatica 29 (8):1251–1274 7. Grimble MJ (1992) Generalized predictive optimal control: an introduction to the advantages and limitations. Int J Syst Sci 23(1):85–98 8. Bitmead R, Gevers M, Wertz V (1989) Adaptive optimal control: the thinking man’s GPC. Prentice Hall, Englewood Cliffs 9. Grimble MJ (2006) Robust industrial control. Wiley, Chichester 10. Ordys AW, Clarke DW (1993) A state-space description for GPC controllers. Int J Syst Sci 24 (9):1727–1744 11. Richalet J (1998) La commande prédictive, Techniques de l’Ingénieur Traite Mesure et Control, R7 423, pp 1–17 12. Rossiter JA, Richalet J (2002) Handling constraints with predictive functional control of unstable processes. In: American control conference, Anchorage, Alaska, pp 4746–4751 13. Khadir MT, Ringwood JV (2008) Extension of first order predictive functional controllers to handle higher order internal models. Int J Appl Math Comput Sci 18(2):229–239 14. Herceg M, Raff T, Findeisen R, Allgöwer F (2006) Nonlinear model predictive control of a turbocharged engine. In: IEEE international conference on control algorithms, Munich, Germany, pp 2766–2771 15. Vermillion C, Sun J, Butts K (2007) Model predictive control allocation for over actuated systems—stability and performance. In: 46th IEEE conference on decision and control, New Orleans, pp 1251–1257 16. Vermillion C, Sun J, Butts K (2009) Model predictive control allocation—design and experimental results on a thermal management system. In: American control conference, St. Louis, pp 1365–1370 17. Bemporad A, Morari M, Ricker NL (2014) Model predictive toolbox, user’s guide. The MathWorks 18. Herceg M (2009) Real-time explicit model predictive control of processes, PhD dissertation, Slovak University of Technology, Bratislava 19. Bemporad A, Morari M, Dua V, Pistikopoulos E (2000) The explicit solution of Model predictive control via multiparametric quadratic programming. In: American control conference, Chicago, Illinois, pp 872–876 20. Hovland S, Gravdahl JT, Willcox KE (2008) Explicit model predictive control for large-scale systems via model reduction. AIAA J Guid Control Dyn 31:918–926 21. Grimble MJ (2004) GMV control of nonlinear multivariable systems. In: UKACC conference control, University of Bath 22. Grimble MJ (2005) Non-linear generalised minimum variance feedback, feedforward and tracking control. Automatica 41:957–969 23. Grimble MJ, Majecki P (2005) Nonlinear generalised minimum variance control under actuator saturation. IFAC World Congress, Prague 24. Grimble MJ, Majecki P, Giovanini L (2007) Polynomial approach to nonlinear predictive GMV control. In: European control conference, Koss, Greece 25. Shaked U (1976) A general transfer-function approach to the steady state linear quadratic Gaussian stochastic control problem. Int J Control 24(6):771–800 26. Kucera V (1979) Discrete linear control. Wiley, Chichester 27. Kwon WH, Pearson AE (1975) On the stabilization of a discrete constant linear system. IEEE Trans Autom Control 20(6):800–801 28. Kwon WH, Pearson AE (1977) A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Trans Autom Control 22(5):838–842 29. Kwon WH, Pearson AE (1978) On feedback stabilization of time-varying discrete linear systems. IEEE Trans Autom Control 23(3):479–481 30. Cannon M, Kouvaritakis B (2001) Open-loop and closed-loop optimality in interpolation MPC. Nonlinear predictive control: theory and practice. IEE, London, pp 131–149

References

343

31. Camacho EF (1993) Constrained generalized predictive control. IEEE Trans Autom Control 38:327–332 32. Cannon M, Kouvaritakis B (2002) Efficient constrained model predictive control with asymptotic optimality. SIAM J. Control Optim 41(1):60–82 33. Cannon M, Kouvaritakis B, Lee YI, Brooms AC (2001) Efficient nonlinear predictive control. Int J Control 74(4):361–372 34. Cannon M, Deshmukh V, Kouvaritakis B (2003) Nonlinear model predictive control with polytopic invariant sets. Automatica 39(8):1487–1494 35. Kothare MV, Balakrishnan V, Morari M (1996) Robust constrained model predictive control using linear matrix inequalities. Automatica 32(10):1361–1379 36. Michalska H, Mayne DQ (1993) Robust receding horizon control of constrained non-linear systems. IEEE Trans Autom Control 38:1623–1633 37. Shamma JS, Athans M (1990) Analysis of gain scheduled control for non-linear plants. IEEE Trans Autom Control 35:898–907 38. Kouvaritakis B, Cannon M, Rossiter JA (1999) Nonlinear model based predictive control. Int J Control 72(10):919–928 39. Mayne DQ, Rawlings JB, Rao CV, Scokaert POM (2000) Constrained model predictive control: stability and optimality. Automatica 36(6):789–814 40. Scokaert POM, Mayne DQ, Rawlings JB (1999) Suboptimal model predictive control (feasibility implies stability). IEEE Trans Autom Control 44(3):648–654 41. Brooms AC, Kouvaritakis B (2000) Successive constrained optimisation and interpolation in non-linear model based predictive control. Int J Control 73(4):312–316 42. Allgower F, Findeisen R (1998) Non-linear predictive control of a distillation column. International symposium on non-linear model predictive control, Ascona, Switzerland 43. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 44. Borhan H, Hodzen E (2014) A robust design optimization framework for systematic model based calibration of engine control system. In: ASME 2014 internal combustion engine division fall technical conference, Columbus, Indiana 45. Borhan H, Kothandaraman G (2015) Air handling control of a diesel engine with a complex dual-loop EGR and VGT air system using MPC. In: American control conference, Chicago, Illinois, pp 4509–4516 46. Keerthi SS, Gilbert EG (1988) Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations. J Optim Theory Appl 57(2):265–293 47. Kouvaritaki B, Rossiter JA, Chang AOT (1992) Stable generalised predictive control: an algorithm with guaranteed stability. IEE Proc Pt D 139(4):349–362 48. Giovanini L, Grimble MJ (2004) Robust predictive feedback control for constrained systems. Int J Control Autom Syst 2(4):407–422 49. Maciejowski JM (2001) Predictive control with constraints. Prentice Hall, UK 50. Kock P, Ordys AW, Grimble MJ (1998) Constrained predictive control design for multivariable systems. In: SIAM conference on control and its applications, Jacksonville, Florida 51. Lee YI, Kouvaritakis B, Cannon M (2003) Constrained receding horizon predictive control for nonlinear systems. Automatica 38(12):2093–2102 52. Lee YI, Kouvaritakis B, Cannon M (1999) Constrained receding horizon predictive control for nonlinear systems. In: 38th IEEE conference on decision and control, Phoenix, AZ, vol 4, pp 3370–3375 53. Oliveira SL (1996) Model predictive control for constrained nonlinear systems. PhD thesis, Caltech 54. Basterretxea K, Benkrid K (2011) Embedded high-speed model predictive controller on a FPGA. In: NASA/ESA conference on adaptive hardware and systems, San Diego, CA 55. Youssef A, Grimble MJ, Ordys A, Dutka A, Anderson D (2003) Robust nonlinear predictive flight control. In: European control conference, Cambridge, pp 09-01–09-04

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56. Keerthi SS, Gilbert EG (1988) Optimal infinite-horizon feedback laws for a class of constrained discrete time systems. J Opt Theory Appl 57:265–293 57. Hickey NA, Johnson MA, Katebi MR, Grimble MJ (1999) PID controller optimisation for fin roll stabilisation. In: IEEE international conference on control applications, Hawaii, pp 1780– 1785 58. Hickey NA, Grimble MJ, Johnson M, Katebi R, Wood D (1995) H∞ fin roll-stabilization control system design. In: 3rd IFAC workshop on control applications in marine systems, Trondheim, Norway, pp 304–311 59. Katebi R (2004) A two-layer controller for integrated fin and rudder roll stabilization. In: Control applications of marine systems (CAMS) conference, Ancona, Italy, pp 101–106 60. Perez T (2005) Ship motion control. Springer, London 61. Grimble MJ, Majecki P (2010) Polynomial approach to nonlinear predictive generalized minimum variance control. IET Control Theory Appl 4(3):411–424 62. Duan J, Grimble MJ, Johnson MA (1997) Multivariable weighted predictive control. J. Proc Control 7(3):219–235 63. Morningred JD, Paden BE, Seborg DE, Mellichamp DA (1992) An adaptive nonlinear predictive controller. Chem Eng Sci 47(4):755–762

Part III

State-Space Systems Nonlinear Control

Chapter 8

State-Space Approach to Nonlinear Optimal Control

Abstract The generalized minimum variance control problem is considered for systems that include a nonlinear black-box subsystem. This is the first of the chapters to use state-space models for representing the linear subsystems. The Kalman filter is introduced for state estimation and its properties analysed. The stability of the system is discussed and the relationship of the controller structure to a Smith predictor is established. A multivariable control design example is provided that includes a plant in a Hammerstein model form that is often a reasonable approximation to use in applications.

8.1

Introduction

The aim of this chapter is to consider optimal control laws for a nonlinear plant where the input subsystem is nonlinear and the output subsystem is linear and is represented by a state equation based model. The models for most physical systems are described more naturally in state-space rather than in polynomial system form. State equations are also more versatile than polynomial models that are often limited to linear time-invariant systems. The state-space models are particularly recommended for large systems since the numerical algorithms are more reliable. The total plant structure for nonlinear multivariable processes will be related to that considered in previous chapters but the linear subsystem model will be assumed to have a state-space model form: xðt þ 1Þ ¼ A0 xðtÞ þ B0 /ðx; uÞ yðtÞ ¼ C0 xðtÞ The nonlinear function /ðx; uÞ is normally defined to be a continuous function when deriving a nonlinear optimal control. However, the Generalized Minimum Variance (GMV) controller to be introduced involves a general nonlinear input subsystem. The structure of the system and the criterion are again chosen so that a simple controller structure and solution are obtained. When the system is linear the results revert to those for a state-space version of the GMV controller [1]. © Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_8

347

348

8 State-Space Approach to Nonlinear Optimal Control

The solution for the state-space version of the Nonlinear Generalized Minimum Variance (NGMV) optimal control law is based upon a straightforward strategy that parallels the results in the polynomial system based chapters. The signal to be minimized /0 ðtÞ involves both the weighted error and the weighted control action and it includes any explicit transport delays associated with the outputs. The input subsystem, which may represent most of the process, or plant model, is assumed nonlinear, and is represented in a general operator form. The output of this subsystem is denoted u0 ðtÞ and represents the input to any linear subsystem dynamics. This linear subsystem may be associated with some of the process dynamics and the measurement devices, and the total plant output is then formed from this signal together with a disturbance signal. The assumption is again made that the disturbance and reference models are represented by linear subsystems. The states of the linear plant output subsystem and stochastic disturbance and reference models can be estimated using a Kalman filter with inputs from the observations or error signals and from the signal input to the linear plant subsystem u0 ðtÞ: When a Kalman prediction stage is introduced, the predicted values of the signal /0 ðtÞ may be obtained. The optimal control law is then chosen to set the predicted values of /0 ðtÞ to zero to minimize the variance of this signal. Although the Kalman predictor involves linear state equations, it is driven from the nonlinear subsystem and the resulting control action is nonlinear. The control law obtained is related to the polynomial based Nonlinear Generalized Minimum Variance (NGMV) optimal control strategy described in earlier chapters [2]. The control law is simple to implement and does not involve feedback linearization, nor does it attempt to cancel the nonlinearities.

8.2

System Description

The system description is carefully chosen so that simple results are obtained. The nonlinear component of the plant model can be quite general since it is represented by a nonlinear operator, which can denote the full plant model. The nonlinear plant model has an explicit transport delay term ðWuÞðtÞ ¼ zk ðW k uÞðtÞ. This model may contain a linear output subsystem W0k (without delay) that can be combined with the linear disturbance and reference models in an augmented state-space system model to be defined below. The system is shown in Fig. 8.1 including the nonlinear plant model and any linear plant subsystem W0k dynamics. The stochastic reference and disturbance signals are assumed to have linear state equation model representations. If the system includes a linear dynamic error weighting term Pc ðz1 Þ in the cost-function (introduced later) this should also be added in a state equation form, as shown in Fig. 8.1. In this chapter the output measured is taken to be the same as that to be controlled, and the error weighting in the cost-function (defined later), acts on the difference of the reference signal and this output signal.

8.2 System Description

349 Disturbance model

ξd

+

Dd

z −1

+

Error weighting Pc ( z −1 )

xd

Cd

+

Bp

d

z −1

+

ξ0

u (t )

u0

1k (.,.)

Control Signal c

(.,.)

z − k B0

+ +

z −1

+

x0

C0

_

+

y0

+

_

A0

uc

Control weighting

Output y

Linear subsystem dynamics

D0

Cp

Ap

Ad NL system dynamics

xp

+

Ep

+

+

e

yp

e0

Error +

_

uc

v Noise

+ +

z − k E0

ω

Br

+ +

z −1 Ar

xr

Cr

r

φ0 (t ) = Pc e(t ) +

c u (t )

Reference

Reference generation model

Fig. 8.1 Nonlinear and linear plant subsystems, linear output disturbance and reference model

The observations or measurements are assumed to be corrupted by zero-mean white measurement noise, denoted vðtÞ that has a covariance matrix Rf . The zero-mean, white noise signals xðtÞ, nd ðtÞ and n0 ðtÞ, that feed the reference, disturbance and linear plant models can be assumed, without loss of generality, to have identity covariance matrices. If there are also deterministic reference and disturbance signal components they may also be included, as described in Chap. 11. The signals shown in the system model of Fig. 8.1 may be listed as follows: Error signal: eð t Þ ¼ r ð t Þ  yð t Þ

ð8:1Þ

yð t Þ ¼ d ð t Þ þ y0 ð t Þ

ð8:2Þ

r ðtÞ ¼ Wr xðtÞ

ð8:3Þ

d ð t Þ ¼ W d nd ð t Þ

ð8:4Þ

z ð t Þ ¼ yð t Þ þ vð t Þ

ð8:5Þ

Plant output:

Reference signal:

Output disturbance:

Observations signal:

350

8 State-Space Approach to Nonlinear Optimal Control

Noisy error signal: e0 ð t Þ ¼ r ð t Þ  z ð t Þ

ð8:6Þ

yp ðtÞ ¼ Pc eðtÞ

ð8:7Þ

Weighted error:

8.2.1

System Models

The state-space system models, for the (r  m) multivariable system, shown in Fig. 8.1, may now be introduced. The full nonlinear plant model may be put in the black-box input subsystem W 1k but it adds flexibility to include a linear plant subsystem model with states x0 ðtÞ. This has the virtue that it can represent open-loop unstable systems, whereas the nonlinear operator W 1k subsystem must be assumed stable. Linear plant subsystem: If the plant has a linear output subsystem, possibly unstable, it may be included as x0 ðt þ 1Þ ¼ A0 x0 ðtÞ þ B0 u0 ðt  kÞ þ D0 n0 ðtÞ; x0 ðtÞ 2 Rn0

ð8:8Þ

y0 ðtÞ ¼ C0 x0 ðtÞ þ E0 u0 ðt  kÞ

ð8:9Þ

W0k ðz1 Þ ¼ C0 ðzI  A0 Þ1 B0 þ E0 , and where W0 ðz1 Þ ¼ zk W0k ðz1 Þ, 1 1 W0d ðz Þ ¼ C0 ðzI  A0 Þ D0 . If the E0 matrix is assumed full-rank then the W0 ðz1 Þ subsystem has a total of k-steps transport delay. The case where this matrix E0 is null, and an additional one-step delay is introduced, is discussed in Chap. 11. The linear state-space disturbance, reference and the cost-function error weighting models are shown in Fig. 8.1, and may be listed as follows: Disturbance: xd ðt þ 1Þ ¼ Ad xd ðtÞ þ Dd nd ðtÞ; xd ðtÞ 2 Rnd

ð8:10Þ

dðtÞ ¼ Cd xd ðtÞ and Wd ðz1 Þ ¼ Cd ðzI  Ad Þ1 Dd

ð8:11Þ

xr ðt þ 1Þ ¼ Ar xr ðtÞ þ Dr xðtÞ; xr ðtÞ 2 Rnr

ð8:12Þ

Reference:

rðtÞ ¼ Cr xr ðtÞ

and

Wr ðz1 Þ ¼ Cr ðzI  Ar Þ1 Dr

ð8:13Þ

8.2 System Description

351

Cost-function error weighting: xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp ðrðtÞ  dðtÞ  y0 ðtÞÞ;

xp ðtÞ 2 Rnp

yp ðtÞ ¼ Cp xp ðtÞ þ Ep ðrðtÞ  dðtÞ  y0 ðtÞÞ

ð8:14Þ ð8:15Þ

The last of these equations, for the cost-function error weighting term, may be written as xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp ðCr xr ðtÞ  Cd xd ðtÞ  C0 x0 ðtÞ  E0 u0 ðt  kÞÞ

ð8:16Þ

yp ðtÞ ¼ Cp xp ðtÞ þ Ep ðCr xr ðtÞ  Cd xd ðtÞ  C0 x0 ðtÞ  E0 u0 ðt  kÞÞ

ð8:17Þ

Nonlinear plant model: Let the integer k denotes the magnitude of the common delay elements in the output signal paths. The plant model is formed from the cascade of nonlinear and linear subsystems. The nonlinear input subsystem W 1k is assumed to be finite gain stable, but the linear state-space output subsystem W0k can represent an open-loop unstable plant model. The total nonlinear plant model: ðWuÞðtÞ ¼ zk W0k ðW 1k uÞðtÞ

ð8:18Þ

The output signal from the nonlinear plant subsystem is denoted u0 ðtÞ ¼ ðW 1k uÞðtÞ and the delay free plant model ðW k uÞðtÞ ¼ W 0k ðW 1k uÞðtÞ. If the full plant model is to be included in the “black-box” operator term W 1k the linear block is not needed and hence define W0k ¼ I. Augmented state equation system model: Combining the equations in Sect. 8.2.1, the augmented state equations for the total system can be represented as xðt þ 1Þ ¼ AxðtÞ þ Bu0 ðt  kÞ þ DnðtÞ

ð8:19Þ

yðtÞ ¼ CxðtÞ þ Eu0 ðt  kÞ;

ð8:20Þ

xðtÞ 2 Rn

zðtÞ ¼ yðtÞ þ vðtÞ ðmeasurementsÞ

ð8:21Þ

yp ðtÞ ¼ C/ xðtÞ þ E/ u0 ðt  kÞ ðoutputs to be minimizedÞ

ð8:22Þ

In most problems it will not be practical to measure all the state variables and a Kalman filter will be needed for state estimation. This must have a structure based on the augmented state equation model for the system shown in the figure and in Eqs. (8.19)–(8.22). The augmented state-space model is assumed to be controllable and observable or the less onerous requirement of stabilizable and detectable. The observability or detectability is a necessary condition for proving the stability of the Kalman filter. If the resolvent operator is denoted Uðz1 Þ ¼ ðzI  AÞ1 then the state-vector xðtÞ ¼ Uðz1 ÞðBu0 ðt  kÞ þ DnðtÞÞ. In terms of these models the input subsystem may be represented as W 0k ¼ E þ CUB and the transfer from the control input to

352

8 State-Space Approach to Nonlinear Optimal Control

the weighted output yp ðtÞ is of the form Pc W0k ¼ C/ UB  E/ , where the negative sign arises because the input to the Pc weighting in (8.15) is the error signal.

8.2.2

Kalman Filter and Observers for State Estimation

The states of a system cannot usually be measured but can be estimated using a state observer. This is a dynamic system that is used to estimate the states of a system (or some of the states), given noisy measurements of the system output. An observer can be in open-loop, or closed-loop form. A closed-loop full-state observer, used to estimate all the states in a linear system has the same structure as a Kalman filter. The Luenberger observer is the best known of the non-optimal state observers for linear systems [3]. The Kalman filter is an optimal estimator that minimizes the variance of the state estimation error. The system model, noise and disturbance covariance’s are assumed to be known, and the Kalman filter gains are chosen to minimize a cost-function involving the variance of the state estimation error. The discrete version of the filter may be implemented in a recursive form and it is particularly suitable for discrete-time implementation. This is discussed further in Chap. 13. Practical experience across a range of industrial sectors, reveals that the Kalman filter is very robust and reliable in operation. Knowledge of the process models and noise variances does not have to be as good as might be expected. There can be reasonably large errors in noise and disturbance variance estimates, without adversely affecting the performance of the filter. However, the performance can be more unpredictable if the system dynamics are seriously mismatched with the linear, possibly time-varying, plant dynamics used in computing the filter. A Kalman Filter is often used as part of a state estimate feedback control system, and provides the state estimates in the GMV and NGMV solutions to be described in the following sections. It usually works well, as part of a state estimate feedback control system, even with relatively poor noise and disturbance information. It requires plant model knowledge but it gives good estimates unless there is a significant difference between the plant model and the actual plant. It is versatile and can be used for applications ranging from control applications to condition monitoring. It is used in various forms of fault monitoring system. The Kalman Filter is particularly valuable for inferential estimation to give estimates of parameters or signals that cannot be measured easily. For the problem to be considered in this chapter a constant gain Kalman filter, based on the Linear Time-Invariant (LTI) augmented system model, is used. In later chapters, a time-varying Kalman filter is introduced for use with the quasi Linear Parameter Varying (qLPV) models that are needed. An extended Kalman filter (EKF) is similar to a Kalman filter but it uses a nonlinear model for the system, and a gain calculation based on a small signal linearized model (evaluated at the current state estimate). It is valuable in some

8.2 System Description

353

nonlinear control problems, and if no simultaneous parameter estimation is included, it is usually reliable. However, an EKF is not an optimal estimator since it involves an approximation when using the linearized plant model to compute the gain matrix. It is useful for some nonlinear state estimate feedback control problems and it is discussed further in Chap. 13. Robustness: Changes to a Kalman filter can sometimes be made to improve the robustness of an overall control system. This may cause stochastic estimation properties to deteriorate but improve system robustness (like the use of loop-transfer recovery methods in LQG control). However, the filter seems to be inherently robust and there is normally not a need to modify the filter to improve the robustness of the state estimates themselves. If a nonlinear filter is needed a constant gain extended Kalman filter may be an option to provide a suboptimal, robust and practical solution [4].

8.2.3

Kalman Filter Estimation Equations

The Kalman filter for continuous-time systems was introduced in a seminal contribution by Kalman and Bucy [5]. The duality with Linear Quadratic Regulators and the use of the separation principle to define the Linear Quadratic Gaussian controller were valuable associated results and the paper had a significant impact. The discrete-time filter was introduced a year earlier by Kalman [6] in a form that was actually more useful for digital implementation. The discrete-time recursive Kalman estimator equations are defined below. The state-space model is assumed to be controllable and observable, and the results presented below have been extended in an obvious way to accommodate the delays on input channels and any through terms. Note that such changes, which affect the known input signal, do not affect the basic stochastic relationships and hence the gain of the optimal filters. The gain of a Kalman filter for finite-time estimation problems varies with time but computations are straightforward via a discrete-time Riccati equation [7, 8]. See Chap. 13 for further details and the algorithm for computing the discrete-time Kalman filter gain. Before introducing the estimation equations note that the reference here is assumed to be modelled by a random signal that is known at the current time t. The input to the filter is the signal e0 ðtÞ ¼ rðtÞ  zðtÞ so that the controller structure is one degree of freedom design, which is different to the two degree of freedom predictive control case discussed in the next chapter. Writing the constant matrices of the error channel map Ce ¼ C the estimator equations can now be derived. Controller input: Note that the filter is to be applied to the error channel signal: e0 ðtÞ ¼ rðtÞ  zðtÞ ¼ rðtÞ  ðCxðtÞ þ E0 u0 ðt  kÞ þ vðtÞÞ ¼ Ce xðtÞ  E0 u0 ðt  kÞ  vðtÞ

ð8:23Þ

354

8 State-Space Approach to Nonlinear Optimal Control

Estimator in predictor corrector form: ^xðt þ 1jtÞ ¼ A^xðtjtÞ þ zk Bu0 ðtÞ

ðPredictorÞ

^xðt þ 1jt þ 1Þ ¼ ^xðt þ 1jtÞ þ Kf ðe0 ðt þ 1Þ  ^e0 ðt þ 1jtÞÞ ðCorrectorÞ

ð8:24Þ ð8:25Þ

where ^e0 ðt þ 1jtÞ ¼ C e^xðt þ 1jtÞ  zk E0 u0 ðt þ 1Þ

ð8:26Þ

Kalman filter prediction equation: Manipulating these equations we obtain

 ^xðt þ 1jt þ 1Þ ¼ A^xðtjtÞ þ zk Bu0 ðtÞ þ Kf e0 ðt þ 1Þ  C e^xðt þ 1jtÞ  zk E0 u0 ðt þ 1Þ

 ðzI  A þ Kf C e AÞ^xðtjtÞ ¼ zk Bu0 ðtÞ þ Kf e0 ðt þ 1Þ  C e zk Bu0 ðtÞ  zk E0 u0 ðt þ 1Þ ^xðtjtÞ ¼ ðI  z1 ðA  Kf C e AÞÞ1 ðKf ðe0 ðtÞ þ E0 u0 ðt  kÞ  C e Bu0 ðt  k  1ÞÞ þ Bu0 ðt  k  1ÞÞ

ð8:27Þ The estimated states of the combined model can, therefore, be obtained using the Kalman filter, where the gain is computed from a Riccati equation solution (Chap. 13 and [9]). Simple manipulation of the above estimator equations reveals that the estimated states may be expressed in the form ^xðtjtÞ ¼ T f 1 ðz1 Þe0 ðtÞ þ T f 2 ðz1 Þu0 ðtÞ

ð8:28Þ

where the transfer operators, involving the filter sensitivity functions T f 1 ðz1 Þ ¼ ðI  z1 ðA  Kf C e AÞÞ1 Kf

ð8:29Þ

and
 T f 2 ðz1 Þ ¼ ðI  z1 ðA  Kf C e AÞÞ1 zk Kf E0 þ z1 ðI  Kf C e ÞBÞ

ð8:30Þ

Kalman predictor: It is important that the special structure of the system leads to a prediction equation, which is only dependent upon the linear disturbance and reference models. To derive the Kalman predictor equation note that the following signal does not depend on control action since it is cancelled by the last control term ^xdr ðtjtÞ ¼ ^xðtjtÞ  Uðz1 ÞBu0 ðt  kÞ
¼ ðzI  A þ Kf Ce AÞ1 Kf ðe0 ðt þ 1Þ þ zk E0 u0 ðt þ 1Þ  C e zk Bu0 ðtÞÞ þ zk Bu0 ðtÞ  Uzk Bu0 ðtÞ ¼ ðI  z1 ðA  Kf C e AÞÞ1 Kf ðe0 ðtÞ þ ðE0  Ce UBÞu0 ðt  kÞÞ

ð8:31Þ

8.2 System Description

355

Noting (8.23) this result confirms the fact that the filter state estimate, for just the component due to the stochastic signals, without control action, simply involves subtracting Uðz1 ÞBu0 ðt  kÞ from the state estimate. The predicted values of the states, due to the stochastic input signals to the system (minus the contribution of the control signal) may be defined to have the form Ak ^xdr ðtjtÞ. The total predicted state, k-steps ahead ðk [ 0Þ, only requires the predicted control action term adding. Thus we obtain ^xðt þ kjtÞ ¼ Ak ^xdr ðtjtÞ þ Uðz1 ÞBu0 ðtÞ
 ¼ Ak ^xðtjtÞ  Uðz1 ÞBu0 ðt  kÞ þ Uðz1 ÞBu0 ðtÞ ¼ Ak ^xðtjtÞ þ ðI  Ak zk ÞUðz1 ÞBu0 ðtÞ

ð8:32Þ

Kalman predictor: It follows that the k-steps ahead state estimates can be obtained as ^xðt þ kjtÞ ¼ Ak ^xðtjtÞ þ Ak1 Bu0 ðt  kÞ þ Ak2 Bu0 ðt  k þ 1Þ þ    þ ABu0 ðt  2Þ þ Bu0 ðt  1Þ or ^xðt þ kjtÞ ¼ Ak ^xðtjtÞ þ T0 ðk; z1 ÞBu0 ðtÞ

ð8:33Þ

where the transfer operator T0 ðk; z1 Þ is defined to have the following finite pulse-response form
 T0 ðk; z1 Þ ¼ ðI  Ak zk ÞUðz1 Þ ¼ z1 I þ z1 A þ z2 A2 þ    þ zk þ 1 Ak1 ð8:34Þ Observe from (8.28) for the filter to be unbiased the following relationship must be satisfied T f 2 ðz1 Þ  T f 1 ðz1 ÞW0k zk ¼ UBzk

ð8:35Þ

This may also be confirmed by substituting from (8.29) and (8.30).

8.3

Optimal Nonlinear GMV Control

The optimal NGMV control problem involves the minimization of the variance of the signal /0 ðtÞ in Fig. 8.2. This signal involves a dynamic cost-function weighting matrix Pc ðz1 Þ acting on the error, represented by a linear state-space subsystem, and a nonlinear dynamic control signal costing term ðF c uÞðtÞ: The choice of the

356

8 State-Space Approach to Nonlinear Optimal Control

φ0 = Pc e +

cu

ξd

+ + Error weighting

ω

Reference

Wr

Pc

e0

0 (.)

Disturbance model

c

Wd

Non-linear plant

Controller

r +

-

Control weighting

−k

u

z W0 k (

1k

u)

d + m

+ y

ξ

+ + v Fig. 8.2 Single degree of freedom closed-loop control system for the nonlinear plant (signal to be minimized /0 ðtÞ is dependent on the weightings and are shown dotted)

dynamic weightings is critical to the control system design. As noted in Chap. 5 the weighting Pc ðz1 Þ is typically a low-pass transfer function and the operator F c is a high-pass transfer function. The signal /0 ðtÞ ¼ Pc ðz1 ÞeðtÞ þ ðF c uÞðtÞ

ð8:36Þ

This signal will again be referred to as an inferred output since it is not a measured signal. This signal is to be minimized in a variance sense [10], so that the cost-index to be minimized may be defined as follows:      J ¼ E /T0 ðt þ kÞ/0 ðt þ kÞt ¼ E tracef/0 ðt þ kÞ/T0 ðt þ kÞtg

ð8:37Þ

where Efjtg denotes the conditional expectation. In the channels with explicit delay k the control signal affects the inferred output /0 ðtÞ at least k-steps later. Thence, the weighting on the control signal is defined to have the form ðF c uÞðtÞ ¼ zk ðF ck uÞðtÞ

ð8:38Þ

Before stating the main theorem, the assumptions required can be summarized as follows: 1. The nonlinear plant model W 1k is assumed finite gain stable but any linear plant subsystem W0k can be unstable. 2. The reference and disturbance models are assumed linear and the delays in the plant are the same in each channel of magnitude k (this assumption can be relaxed).

8.3 Optimal Nonlinear GMV Control

357

3. To ensure closed-loop stability the cost-function weightings are chosen so that the “generalized plant” operator ðPc W k  F ck Þ is finite gain m2 stable. 4. The control signal weighting operator F ck is assumed square, full-rank and invertible. Theorem 8.1: NGMV Optimal Control Signal The nonlinear operator of the generalized plant ðPc W k  F ck Þ is assumed to have a stable causal inverse, due to the choice of weighting operators Pc and F c . The NGMV optimal controller is required to minimize the variance of the weighted error and control signals   J ¼ E /T0 ðt þ k Þ/0 ðt þ kÞt In terms of the future predicted state the optimal state may be computed as
 uðtÞ ¼ F 1 xðt þ kjtÞ  E/ ðW 1k uÞðtÞ ck C/^

ð8:39Þ

In terms of the current state estimate the optimal control may be written as

 1
 C/ Ak ^xðtjtÞ uðtÞ ¼ F ck þ C/ T0 ðk; z1 ÞB þ E/ W 1k

ð8:40Þ

The NGMV optimal control may be written in the following alternative form

 k uðtÞ ¼ F 1 xðtjtÞ  C/ T0 ðk; z1 ÞB þ E/ ðW 1k uÞðtÞ ck C/ A ^

ð8:41Þ

where the finite pulse-response operator T0 ðk; z1 Þ ¼ ðI  Ak zk ÞUðz1 Þ

ð8:42Þ ■

Proof The derivation of the optimal controller and a discussion of stability properties is considered in the next section. Stability issues are discussed below in Sect. 8.3.5. ■

8.3.1

Solution for the NGMV Optimal Control

The solution of the optimal control problem may be obtained by expanding the expression for the inferred output /0 ðtÞ, and by then introducing a prediction equation. The analysis is similar to that used in Chap. 5, for polynomial based models. The expressions will first be obtained for the predicted values of the inferred output both in terms of the current state and the future state estimates.

358

8 State-Space Approach to Nonlinear Optimal Control

From Eq. (8.7) yp ðt þ kÞ ¼ Pc eðt þ kÞ and from (8.36) the inferred output being minimized, /0 ðt þ kÞ ¼ ðPc eÞðt þ kÞ þ ðF ck uÞðtÞ ¼ yp ðt þ kÞ þ ðF ck uÞðtÞ

ð8:43Þ

The, k-steps ahead prediction of /0 ðtÞ follows as ^ ðt þ kjtÞ ¼ ^yp ðt þ kjtÞ þ ðF ck uÞðtÞ / 0

ð8:44Þ

where from (8.22) and (8.33): ^yp ðt þ kjtÞ ¼ C/^xðt þ kjtÞ þ E/ u0 ðtÞ

ð8:45Þ


 ¼ C/ Ak ^xðtjtÞ þ C/ T0 ðk; z1 ÞB þ E/ u0 ðtÞ

ð8:46Þ

It follows that the k-steps ahead predicted inferred output may be written in terms of the current state estimate as
 ^ ðt þ kjtÞ ¼ C/ Ak ^xðtjtÞ þ C/ T0 ðk; z1 ÞB þ E/ u0 ðtÞ þ ðF ck uÞðtÞ / 0

ð8:47Þ

The inferred output can also be written in terms of the future state by substituting for yp ðtÞ from (8.22) in Eq. (8.43): /0 ðt þ kÞ ¼ C/ xðt þ kÞ þ E/ u0 ðtÞ þ ðF ck uÞðtÞ

ð8:48Þ

Also recall the output of the black-box input subsystem u0 ðtÞ ¼ ðW 1k uÞðtÞ, and hence the k-steps ahead predicted inferred output in terms of the future predicted state may be written as
 ^ ðt þ kjtÞ ¼ C/^xðt þ kjtÞ þ ðE/ W 1k þ F ck Þu ðtÞ / 0

ð8:49Þ

Cost-function: The cost-function involves the minimization of the weighted error and control signals. From (8.37), the variance to be minimized J ¼ Ef/0 ðt þ kÞT /0 ðt þ kÞjtg, and this may be written in terms of the prediction ^ ðt þ kjtÞ and the prediction error / ~ ðt þ kjtÞ. Since these signals that depend / 0 0 upon the Kalman filter state estimates are orthogonal [7], the cost-function simplifies as follows: o n n o ^ ðt þ kjtÞt þ E / ~ ðt þ kjtÞt ^ ðt þ kjtÞT / ~ ðt þ kjtÞT / J¼E / 0 0 0 0

ð8:50Þ

~ ðt þ kjtÞ does not depend upon Cost minimization: The prediction error / 0 control action and hence the cost-function is minimized by setting the k-steps ahead predicted values of the inferred output signal /0 ðtÞ to zero. From (8.49) the condition for optimality:

8.3 Optimal Nonlinear GMV Control

^ ðt þ kjtÞ ¼ C/^xðt þ kjtÞ þ ðE/ W 1k þ F ck ÞuðtÞ ¼ 0 / 0

359

ð8:51Þ

This provides the first expression for the optimal control signal in terms of the future predicted state as
 xðt þ kjtÞ  E/ ð W 1k uÞðtÞ uðtÞ ¼ F 1 ck C/^

ð8:52Þ

The optimal control using the current state estimate may be found by setting the ^ ðt þ kjtÞ to zero using (8.47). That is, predicted values of / 0
 ^ ðt þ kjtÞ ¼ C/ Ak ^xðtjtÞ þ C/ T0 ðk; z1 ÞB þ E/ u0 ðtÞ þ ðF ck uÞðtÞ / 0

  ¼ C/ Ak ^xðtjtÞ þ ðC/ T0 ðk; z1 ÞB þ E/ Þ W 1k þ F ck u ðtÞ ¼ 0 ð8:53Þ This provides an expression for the optimal control in terms of the current predicted state as

 1
 uðtÞ ¼ F ck þ C/ T0 ðk; z1 ÞB þ E/ W 1k C/ A k ^xðtjtÞ

ð8:54Þ

The optimal control may also be written from the condition for optimality (8.53) in a more useful form that may be compared with (8.52):

 k uðtÞ ¼ F 1 xðtjtÞ  C/ T0 ðk; z1 ÞB þ E/ ðW 1k uÞðtÞ ck C/ A ^ This completes the proof of the optimal control solution.

8.3.2

ð8:55Þ ■

Remarks on the NGMV Control Solution

The controller has a physically justifiable internal model control structure. Recall the nonlinear input plant subsystem W 1k was assumed stable, and forms part of the controller expressions, so that the controller is also nonlinear. The disturbance knowledge influences the design via the state equations of the stochastic disturbance model Wd ðz1 Þ. However, if rejecting a particular type of disturbance is important, its internal model can be included in the error weighting transfer Pc ðz1 Þ. To achieve good rejection of step load changes it can contain an integrator, whilst a sinusoidal disturbance of frequency f Hz requires a weighting filter of the form 1=ð1  2 cosð2pf Þz1 þ z2 Þ. See the example in Sect. 8.5. Stability is considered in Sect. 8.3.5 but note that if there exists a PID controller that will stabilize the nonlinear system, without transport delay elements, then a set of cost weightings can be defined as in Chap. 5 (Sect. 5.2) to guarantee the existence of this inverse, and ensure the stability of the closed-loop [10].

360

8 State-Space Approach to Nonlinear Optimal Control

Reference

r

+ -

e0

Cφ xˆ ( t + k | t )

Disturbance

− yˆ p (t + k t )

Controller structure

-

u0 (t − k )

−1 ck

u

Plant

d +

Output

+

y

1k

Eφ

u0 (t )

Fig. 8.3 NGMV control signal generation and controller modules

Remarks on the Controller Structure (i) The alternative expressions for the NGMV control signal (8.39)–(8.41) lead to the structures, shown in Figs. 8.3, 8.4 and 8.5, respectively. There are different benefits to the different structures, but the structure shown in Fig. 8.4 is not normally suitable for implementation but is of interest conceptually. Either Figs. 8.3 or 8.5 can be used as a basis for implementation. (ii) The first controller structure that is a good starting point for implementation is defined by (8.39) and depends upon the future predicted state as shown in Fig. 8.3. It provides a useful block diagram to understand behaviour. However, it may still require some additions to avoid the algebraic loop problem and to add anti-windup features (discussed below). The output of the Kalman estimator block involves the predicted state and only needs the through term E/ u0 ðtÞ adding to obtain the total predicted signal ^yp ðt þ k jtÞ ¼ C/^xðt þ kjtÞ þ E/ u0 ðtÞ in Fig. 8.3. This signal is indicated in the figure. ^ ðt þ k jtÞ ¼ Recall that in the proof above the optimal control sets / 0 ^yp ðt þ kjtÞ þ F ck uðtÞ to zero in (8.51) and hence F ck uðtÞ ¼ ^yp ðt þ kjtÞ. (iii) The second structure that is useful for implementation is defined by (8.41) and depends upon the current state estimate as shown in Fig. 8.5. The finite pulse-response block adds a term C/ T 0 ðz1 ÞB u0 ðtÞ that provides the con^ ðt þ k jtÞ of the input terms up to time t. The use of a finite tribution to / 0 pulse-response filter is common in signal processing but they do not arise so often in control problems. The signal ^yp ðt þ kjtÞ that satisfies (8.46) is also ^ ðt þ kjtÞ to zero. shown in this figure, and the optimal control again sets / 0

8.3 Optimal Nonlinear GMV Control

361 Disturbance

Controller Subsystem Reference r

+

e0

-

Kalman predictor

Cφ Ak xˆ(t t )

−(

ck

+ ( C φT 0 B + E φ )

d

Nonlinear plant

Nonlinear operator term k

)

−1

u

Output

+

y

+

u0 (t − k )

v

+ +

Fig. 8.4 NGMV controller structure for conceptual analysis Disturbance Controller structure

Reference

r

+ -

e0

k ∧

Cφ A x ( t | t ) u0 (t − k )

− yˆ p (t + k | t )

-

d

Plant

u

−1 ck

Output

+ +

1k

Cφ T 0 B + Eφ

y

Fig. 8.5 NGMV control signal generation and controller modules

8.3.3

Structure of Controller in Terms of Kalman Filter

The NGMV optimal feedback system, shown in Fig. 8.5, may be redrawn as in Fig. 8.6 and this has the advantage that the role of the Kalman filter is more obvious. If the controller includes integral action and the plant includes a saturation characteristic some form of anti-windup compensation will be needed, as described in Chap. 5. This may simply involve clamping appropriate states in the integral action terms that are present in the filter model (via the augmented system). In addition to anti-windup compensation the design of the controller should ensure the controller response should be stable when the system saturates, and the system is effectively open-loop. It is interesting that a “nonlinear” state estimator is not needed, because of the assumption of linearity on the disturbance and reference signal models. Observe that the state estimator used in the controller is driven by the input to the linear subsystem of the plant (computed from the output of the saturation nonlinearity u0(t)). Thus, the Kalman filter sees the input to the linear subsystem input so that the estimation problem is linear. Note that the explicit transport delay elements could be added as states in the augmented system model but then the filter model would include these states. This is avoided in the use of the predictor equations illustrated in Fig. 8.6. The channel delays do not, therefore, inflate the order of the estimator. It is valuable that the order of the Kalman filter depends only on the delay free linear subsystem, since this reduces computations and simplifies implementation.

362

8 State-Space Approach to Nonlinear Optimal Control

xˆ(t + 1 t )

Ce

e0 = r − z +

Kf Filter gain

+

xˆ(t t )

z −k I

+

xˆ(t + 1 t )

e0 = Reference-observations

Inverse control weighting

Prediction

Cφ Ak

A + +

Cφ Ak xˆ(t | t )

Kalman filter prediction Pure block-diagonal delay

Discrete-Time Kalman Filter

z −k B

+

−

+

−1 ck

Optimal control

u

−1

Cφ T0 (k , z ) B + Eφ Finite pulse response block

Nonlinear plant model

u0 1k

Fig. 8.6 NGMV optimal controller in state-space Kalman filtering form for systems with explicit transport delay elements

8.3.4

Avoiding Algebraic Loop Implementation Problem

When implementing the controller, note from (8.39) that the expression for the optimal control signal u(t) depends upon u(t) itself, as explained in Sect. 5.4 on the algebraic loop problem. A number of ways of avoiding this algebraic loop problem were mentioned, depending upon the nature of the black-box term W 1k . The most general method is probably that described in Sect. 5.4.5, which involves a modification to the cost-index (8.37). That is, the criterion is redefined as follows:      J ¼ E /T0 ðt þ mÞ/0 ðt þ mÞt ¼ E tracef/0 ðt þ mÞ/T0 ðt þ mÞtg

ð8:56Þ

and the control weighting term is slightly modified to become ðF c uÞðtÞ ¼ zm ðF cm uÞðtÞ for 1  m  k. The solution given above is then modified and (8.48) becomes /0 ðt þ mÞ ¼ C/ xðt þ mÞ þ E/ u0 ðt þ m  kÞ þ ðF cm uÞðtÞ The output of the black-box input subsystem u0 ðtÞ ¼ ðW 1 k uÞðtÞ and (8.56) is changed as /0 ðt þ mÞ ¼ C/ xðt þ mÞ þ E/ ð W 1k zmk uÞðtÞ þ ðF cm uÞðtÞ The condition for optimality now becomes ^ ðt þ mjtÞ ¼ C/^xðt þ mjtÞ þ ðE/ W 1k zmk þ F cm ÞuðtÞ ¼ 0 / 0 and the expression for the optimal control signal

8.3 Optimal Nonlinear GMV Control


 uðtÞ ¼ ðF cm Þ1 C/^xðt þ mjtÞ þ E/ ð W 1k uÞðt þ m  kÞ

363

ð8:57Þ

If m is defined as m = k – 1 these changes ensure the algebraic loop is not present in (8.57). This is suboptimal relative to the original cost minimization problem (8.37).

8.3.5

Stability of the Closed-Loop System

It was noted in Chap. 2 (Sect. 2.3.1) that the stability of the GMV controller with the “square of sums” cost-function is ensured for linear systems, when the combination of a control weighting and an error weighted plant model is strictly minimum phase. It is shown below that for nonlinear systems, a related operator equation, based on the state-space and operator models, must have a stable inverse. To investigate the stability results a different expression for the optimal control action is required in terms of the exogenous inputs. Consider just the effect of the reference signal input for the stability discussion. From Eqs. (8.33) and (8.51) the condition for optimality may be written as C/^xðt þ kjtÞ þ ðE/ W 1k þ F ck ÞuðtÞ ¼ 0 C/ Ak ^xðtjtÞ þ ðC/ T0 ðk; z1 ÞB þ E/ Þu0 ðtÞ þ F ck uðtÞ ¼ 0
 C/ A k T f 1 e0 ðtÞ þ T f 2 u0 ðtÞ þ ðC/ T0 ðk; z1 ÞB þ E/ Þu0 ðtÞ þ F ck uðtÞ ¼ 0
 C/ A k T f 1 ðrðtÞ  ðCxðtÞ þ E0 u0 ðt  kÞÞÞ þ T f 2 u0 ðtÞ
 þ C/ T0 ðk; z1 ÞB þ E/ u0 ðtÞ þ F ck uðtÞ ¼ 0 ð8:58Þ Recall the linear subsystem W0k ¼ ðE0 þ CUBÞ then the condition for optimality becomes C/ A k T f 1 rðtÞ þ F ck uðtÞ
 þ C/ ðA k T f 2  A k T f 1 W0k zk þ T0 BÞ þ E/ u0 ðtÞ ¼ 0

ð8:59Þ

A relationship is now required for the set of terms in control action on the right hand side of (8.59). These may be simplified by substituting for (8.34) and (8.35), namely T0 ðk; z1 Þ ¼ ðI  Ak zk ÞU and T f 2 ðz1 Þ  T f 1 ðz1 ÞW0k zk ¼ UBzk to obtain
 C/ Ak T f 2  Ak Tf 1 W0k zk þ T0 B þ E/ ¼ C/ U B þ E/ ¼ Pc W0k

ð8:60Þ

The terms in the condition for optimality can be simplified, using the result in (8.60), to obtain

364

8 State-Space Approach to Nonlinear Optimal Control

C/ A k T f 1 rðtÞ þ F ck uðtÞ  Pc W0k u0 ðtÞ ¼ 0 Closed-Loop Responses: The optimal control action for the desired closed-loop analysis follows as uðtÞ ¼ ðPc W k  F ck Þ1 C/ A k T f 1 rðtÞ

ð8:61Þ

ðWuÞðtÞ ¼ zk W k uðtÞ ¼ zk W k ðPc W k  F ck Þ1 C/ Ak T f 1 rðtÞ

ð8:62Þ

These are the desired expressions for the optimal control and the plant output, for just the reference signal input. The assumption must be made that the cost weightings are chosen, so that the so-called generalized plant operator ðPc W k  F ck Þ1 is finite gain m2 stable. This is a necessary condition for stability.

8.4

Relationship to the Smith Predictor and Robustness

The relationship of the NGMV controller in its polynomial form to the Smith Predictor was discussed in Chap. 5 (Sect. 5.5). The state-space version of the NGMV optimal controller can also be expressed in a similar form to that of a Smith Predictor. However, the use of this structure to implement the NGMV controller again limits its range of applications to plants that are open-loop stable. A nonlinear predictor form of controller may now be derived assuming the plant is open-loop stable. The controller structure shown in Fig. 8.6 involves the predicted states from the Kalman Filter. The two different paths from the observations and the control inputs can be separated as shown in Fig. 8.7, using (8.28). Changes may then be made to the linear subsystems by adding and subtracting equivalent terms, as shown in Fig. 8.8. Finally, the system may be simplified by combining the three “linear” inner-loop blocks, with the common input signal u0 ðtÞ, using (8.60), to obtain


 C/ Ak T f 2  C/ Ak T f 1 W 0 k zk þ C/ T 0 B þ E/ u0 ðtÞ ¼ Pc W 0k u0 ðtÞ

r

+ -

Disturbance d Output

Controller

Reference

e0 u0

Cφ A kT f 1 ( z −1 ) k

+ +

-

ck−1

Cφ A T f 2 ( z )

Kalman predictor

u



1k

−1

Cφ T 0 B + Eφ

ð8:63Þ

u0

Fig. 8.7 Feedback control signal generation and controller modules

Nonlinear plant

+ +

y

8.4 Relationship to the Smith Predictor and Robustness

365

Compensator Reference

r

e0

+ _

Cφ A kT f 1 ( z −1 )

u0

+ _

+ +

d

Nonlinear plant

_

u

ck−1

_

y

+ +

  1k

Cφ A kT f 2 ( z −1 ) Cφ T 0 B + Eφ

u0 Cφ A kT f 1 ( z −1 ) z − k W0 k

W0 k

z −k -

+

Fig. 8.8 Modifications to the controller structure (shown with dotted lines)

The controller then simplifies to the state-space version of what might be termed a nonlinear Smith predictor as shown in Fig. 8.9. Nonlinear Smith Predictor: Note the path in Fig. 8.9 from the control signal u to the feedback signal p is absent when the model zk W k ¼ zk W 0k W 1k matches the plant model. The effect of control action is cancelled out in the two parallel paths involving the plant and its model. The control signal that results from reference changes, is not, therefore, due to feedback from the output but involves the contribution of the open-loop stable Kalman predictor term C/ Ak T f 1 ðz1 Þ and the inner nonlinear feedback loop term involving the error weighting Pc ðz1 Þ . The lack of true feedback action is the reason this Smith Predictor form of implementation is not suitable for open-loop unstable systems. Disturbance

d

Compensator Reference

r

+ -

Cφ A kT f 1 ( z −1 )

ψ

Plant

+

ck

−1

Kalman predictor for disturbance signal

u

+ +

W 0k

Pc

1k

W0 k u0

z −k p

- +

Fig. 8.9 Nonlinear Smith predictor compensator and internal model control structure

Output

y

366

8 State-Space Approach to Nonlinear Optimal Control

Stability and Weighting Function Selection: The form of implementation of the NGMV controller shown in Fig. 8.8 indicates that the inner-loop has the weightings F 1 W k . The return-difference ck Pc acting on the open-loop (delay free) plant model
 operator for this loop within the controller is given by I  F 1 ck Pc W k . Clearly, the term F 1 ck Pc enters the return-difference equation rather like a feedback loop controller. This suggests, as discussed in Chap. 5, that if the plant already has a PID controller that stabilizes the delay free plant model, the ratio of the weightings can be chosen equal to the PID controller. This provides a starting point for weighting selection and a stable initial control design. The choice of the weightings to be equal to such a PID control law avoids the problem of finding the weightings to ensure the stability of the operator discussed in the assumptions preceding Theorem 8.1. In practice, the control weighting found by this design approach will normally require an additional lead term to ensure the controller rolls off, and it will also need some retuning because of the presence of the delay in the actual plant [11]. Under the given assumptions, the Smith predictor form of the resulting system is stable. This result follows because the plant was assumed stable, the inner-loop is stable (due to choice of weightings), and there are only stable terms due to the Kalman filter block.

8.4.1

Black-Box Nonlinearity and Robustness

Most of the control laws discussed in this text has the option of including a black-box subsystem in the plant model. It is interesting that this appears explicitly as part of the control law block diagram solution. However, it does not affect other calculations. If the black-box input subsystem is used to incorporate the uncertainty, a different approach can be taken. That is, the uncertainty can be estimated and included in the model directly, or the deviations can be estimated online. This might be done in various ways depending on the form of uncertainty and the access to signals. However, one way is to model the uncertain subsystem in linear state-space form and to estimate the unknown parameters using an extended Kalman filter.

8.5

Multivariable Control Design Example

The design of a multivariable NGMV optimal control system is considered for a system with static nonlinearities. The plant has a Hammerstein model form (see Chap. 1). The nonlinear input block W 1k represents the static nonlinear characteristics of the actuators [12], and the linear block W0k denotes the plant dynamics. The system has three inputs and three outputs, as shown in the schematic block diagram shown in Fig. 8.10.

8.5 Multivariable Control Design Example

367

1k

y(t)

u0 (t)

u(t) NL

0

0

0 0

NL 0

0 NL

z −k

W0k

Fig. 8.10 Multivariable Hammerstein plant model

The common transport delay of k samples are included in the model and the sample time Ts ¼ 1 s. The nonlinear actuator characteristics for the three input channels are shown in Fig. 8.11. They include a symmetric square root function, a multi-segment piecewise-linear characteristic with a dead-zone, and a shifted exponential function with a limited output range. The effect of this static nonlinear block is equivalent to a variable input gain, with a gain for the second channel that is null in the dead-zone area. In this example, a NGMV controller, that accounts for the nonlinearities and aims to minimize the output signal and control signal variances, will be found. The plant dynamics are represented by a linear 3  3 model W0k that is stable and is given in the state equation form x0 ðt þ 1Þ ¼ A0 x0 ðtÞ þ zk B0 u0 ðtÞ

ð8:64Þ

y0 ðtÞ ¼ C0 x0 ðtÞ þ zk E0 u0 ðtÞ

ð8:65Þ

Channel 1 nonlinearity

Channel 2 nonlinearity

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

-1

0

1

2

-2

Channel 3 nonlinearity 4 3 2 1 0 -1

-2

Fig. 8.11 Static actuator nonlinearities

-1

0

1

2

-2 -2

-1

0

1

2

368

8 State-Space Approach to Nonlinear Optimal Control

where the system matrices are defined as 2 6 6 6 6 A0 ¼ 6 6 6 6 4

0:5

0

0

0 0

0:1 0

0 0:8

0

0

0

0 0 0 0 2 1 0 6 C0 ¼ 4 0 0:24 0 0

0 0 0 0:8 0

0

0

0

0 0

0 0

0 0

3

7 7 7 7 7; 0:6 0 0 7 7 7 0 0:2 0 5 0 0 0:5 3 0 0:4 0 7 0 0 0 5 0:36 0 1

2

0:5

0

6 0:125 6 6 6 0 B0 ¼ 6 6 0 6 6 4 0

0

3

7 7 7 7 7 0 7 7 7 0:25 5

0 1

0 0

0:5

0 0 0 1 2 3 1 0 0:5 6 7 E0 ¼ 4 0:3 1 0 5 0 0:3 2

The time delay k is common to all three output channels and has been set to k = 6 samples. The diagonal linear disturbance and reference models have the following state-space forms. Disturbance model Wd: xd ðt þ 1Þ ¼ Ad xd ðtÞ þ Dd nd ðtÞ dðtÞ ¼ Cd xd ðtÞ where 2

0:5 Ad ¼ 4 0 0

0 0:5 0

3 0 0 5; 0:5

2

0:25 Dd ¼ 4 0 0

0 0:25 0

3 2 0 0:4 0 5; Cd ¼ 4 0 0:25 0

0 0:4 0

Reference model Wr: xr ðt þ 1Þ ¼ Ar xr ðtÞ þ Dr xðtÞ rðtÞ ¼ Cr xr ðtÞ where 2

0:999

6 Ar ¼ 4 0 0 2 0:4 6 Cr ¼ 4 0 0

0

0

0:999 0 0

0

3

7 0 5; 0:999 3

7 0:4 0 5 0 0:4

2

0:25

6 Dr ¼ 4 0 0

0 0:25 0

0

3

7 0 5; 0:25

3 0 0 5 0:4

8.5 Multivariable Control Design Example

369

The singular value plots of the plant, disturbance and reference signal models are shown in Fig. 8.12. A measurement noise model is not included, since the choice of the control weighting can ensure a fall-off in the controller gain at high frequencies. Open-loop Responses: Figure 8.13 shows the open-loop step responses of the plant (including the nonlinear block but without the delay) for a staircase reference input. The responses have been recorded for each input separately, while the other two inputs were maintained at zero. The plots clearly show different gains across the operating range and interactions between the channels. The dead-zone in the second channel, and the asymmetric characteristic in the third, are also visible. For this example, consider a uniform operating range [−2, 2] for all three input channels, corresponding approximately to the output operating range of [−2.5, 2.5]. The zero operating point is considered as the nominal. Multi-loop PID controller and nominal NGMV control: The plant is controlled by a nominal multi-loop PID controller denoted K0 with the tuning parameters collected in Table 8.1. The controller parameters have been tuned for good step response reference tracking. This nominal PID controller has been used as a starting point for NGMV weighting selection as in Chap. 5 (Sect. 5.5). The initial choice of weightings was obtained as Fck ¼ I and Pc ðz1 Þ ¼ K0 ðz1 Þ. In the state-space form, the output of Pc ðz1 Þ can be found as

Frequency responses of system models 60 Plant

50

Disturbance Reference

Singular Values (dB)

40 30 20 10 0 -10 -20 -30 -5 10

10

-4

10

-3

10

-2

Frequency (rad/s)

Fig. 8.12 Singular value plots of system models

10

-1

10

0

10

1

370

8 State-Space Approach to Nonlinear Optimal Control Output responses to steps in input 1

Output responses to steps in input 2

3

2 y1

2

1

y2

1

y3

0

0

-1 -1

-2 -3

0

50

100

150

200

250

-2

0

100

50

150

200

250

200

250

Input step sequence

Output responses to steps in input 3 1.5

2

1

1.5 1

0.5

0.5

0

0

-0.5

-0.5

-1

-1 0

50

100

150

200

250

-1.5

0

100

50

time (s)

150

time (s)

Fig. 8.13 Open-loop step responses to inputs in a channel with other inputs null

Table 8.1 Multi-loop PID controller parameters

Loop

K

Ti (s)

Td (s)

#1 #2 #3

0.1 0.5 0.1

3 10 1

1 1 0.4

xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp u0 ðtÞ yp ðtÞ ¼ Cp xp ðtÞ þ Ep u0 ðtÞ where 3 3 2 1:5 0:5 0 0 0 0 0:25 0 0 7 6 1 6 0 0 0 0 0 0 0 0 7 7 7 6 6 7 7 6 6 7 6 0 6 0 0 1:5 0:5 0 0 0:5 0 7 7; Bp ¼ 6 7 Ap ¼ 6 7 6 0 6 0 0 1 0 0 0 0 0 7 7 7 6 6 7 7 6 6 4 0 4 0 0 0 0 1:5 0:9999 5 0 0:5 5 0 0 0 0 0:5 0 0 0 0 3 2 2 0:067 0:16 0 0 0 0 0:22 0 7 6 6 Cp ¼ 4 0 0 0:4 0:45 0 0 1:05 5; Ep ¼ 4 0 0 0 0 0 0:1066 0:05333 0 0 2

; 3 0 7 0 5 0:1733

8.5 Multivariable Control Design Example

371

NGMV Controller Tuning and Transient Performance: Based on the initial design, the control weighting can be retuned as 2

q1 Fck ¼ 4 0 0

0 0:5 0

3 0 5 0 1 1  0:5z

Let the nominal value q1 ¼ 0:5. The frequency plots of the error and control weightings are as shown in Fig. 8.14. A sequence of reference step changes has been applied on all channels, and the tracking performance of both the original PID and the NGMV controllers is shown in Fig. 8.15. The effects of varying the parameter q1 are shown in Fig. 8.16. The influence of this parameter on the speed of response can be seen, and provides a simple controller tuning method, with the nominal choice q1 ¼ 0:5 being a good compromise. The responses for the other channels (not shown) remain almost unchanged, however, they may also be modified by introducing tuning parameters q2 and q3 . In all cases, although the designs were based on the original PID controller, the NGMV controller provides improved transient performance across the operating range. Sinusoidal disturbance rejection: In the following simulation, only two reference step changes have been applied, to channel 1 (at t = 10 and t = 100), as well as a disturbance step change of 0.2 at time 50 in channel 2 and also a sinusoidal disturbance of amplitude 0.2 and frequency 0.04 Hz again at the output of channel 1. As the nominal PID controller does not contain the model of the sinusoidal disturbance, it is not able to reject it effectively. However, NGMV control design can Frequency responses of weighting functions 70 Error weight

60

Control weight

Singular Values (dB)

50 40 30 20 10 0 -10 -20 -6 10

10

-4

10

-2

Frequency (rad/s)

Fig. 8.14 Singular value plots of the weightings

10

0

372

8 State-Space Approach to Nonlinear Optimal Control Output responses

0.5

0 -2 0

50

100

Control responses

1 NGMV PID

u1

y1

2

200

150

0 -0.5 0

4

50

100

150

200

0

50

100

150

200

0

50

100

150

200

2

0

u2

y2

2 0 -2 50

100

-2

200

150

1

1

0.5

u3

y3

0 2

0 -1

0

50

100

200

150

0 -0.5

time (s)

time (s)

Fig. 8.15 Tracking performance of PID and NGMV controllers

Output 1 3 =1

2

= 0.5

1

= 0.1

0 -1 -2

0

20

40

60

80

100

120

140

160

180

200

140

160

180

200

Control 1

2 1 0 -1 -2

0

20

40

60

80

100

time (s)

Fig. 8.16 NGMV controller tuning (channel 1)

120

8.5 Multivariable Control Design Example

373

address this problem by including the sinusoidal disturbance model in the error weighting Pc. The approach taken here was to consider the model: 1 1 Psin þ z2 Þ c ðz Þ ¼ ksin =ð1  2 cosð2pf Þz

ð8:66Þ

where f denotes the frequency in Hertz of the sinusoid [13]. This transfer function was added to each diagonal term of the original Pc weighting, with the gain ksin used as a tuning parameter. The control signal costing was also retuned slightly, and included the following elements Fck ð1; 1Þ ¼ 0:75 þ 0:075z1 Fck ð2; 2Þ ¼ 0:0125 þ 0:00125z1 Fck ð3; 3Þ ¼ 0:05 þ 0:03z1  0:0025z2 The results of the simulation are shown in Fig. 8.17. It can be seen that both the sinusoidal and step disturbances have been much reduced in all channels, particularly in steady state, by the NGMV controller. This is thanks to the combination of three features of this controller: the multivariable structure that is able to handle the system interactions; the design process that includes the internal models of the disturbances, and the nonlinear compensation properties of the controller.

Output responses

u

y

1

1

0.2

0

50

-0.2 100

150

200 1

0

0

2

0.2

u

2

0

y

0

PID NGMV

-1

0

50

100

150

200 0.05

0

0

u

3

0.02 3

0

50

100

150

200

0

50

100

150

200

0

50

100

150

200

-1

-0.2

y

Control responses

0.4

1

-0.02

-0.05 0

50

100

time (s)

150

200

time (s)

Fig. 8.17 PID and NGMV controller results for sinusoidal disturbance rejection

374

8.6

8 State-Space Approach to Nonlinear Optimal Control

Concluding Remarks

A relatively simple controller for nonlinear multivariable and possibly time-varying systems was described for systems represented in nonlinear operator and linear state-space form. The controller may be considered a generalization of the well-known minimum variance controller due to Åström [14], but for systems represented by state equations and also containing nonlinearities. The assumptions made in the definition of the system reference and disturbance models and the specification of the cost-index, were all aimed at generating a simple controller. The resulting controller is equivalent to the polynomial system version presented in Chap. 4. The structure of the problem ensured that the state-space matrix equations to be solved were all linear, and the controller is simple to compute and implement. The closed-loop stability of the system depends upon the existence of a stable inverse for a closed-loop operator. This involves the so-called generalized plant model and is dependent on the cost-function weighting definitions. A starting point for weighting selection that was described was through the relationship to a PID controller. The Smith Predictor form of solution was discussed that provides an intuitive understanding of the operation and properties of the NGMV controller. The controller structure shown in Fig. 8.9 that may be described as a Nonlinear Smith Predictor was particularly useful for design insights. Observe that all of the stochastic properties and most of the compensator structure are determined by the disturbance, reference and noise models. This is an advantage since the control law design, via the cost-function weighting choice, can focus upon performance, stability and robustness properties. There are many potential applications of this type of approach even for simple Wiener models that apply to problems such as continuous stirred tank reactors and pH neutralization processes [15]. There is also a potential for use in machinery based control problems like aircraft systems [16]. The nonlinear component of the plant model need not be available in a conventional equation form but can involve a fuzzy-neural network [17, 18]. The black-box term can, for example, include a neural network model to provide a form of adaptive control [17].

References 1. Grimble MJ, Majecki PM (2008) Nonlinear GMV control for unstable state dependent multivariable models. In: 47th IEEE conference on decision and control, Fiesta Americana Grand Coral Beach Hotel, Cancun, Mexico, pp 4767–4774 2. Grimble MJ (2006) Design of generalized minimum variance controllers for nonlinear systems. Int J Control Autom Syst 4(3):1–12 3. Luenberger DG (1963) Determining the state of a linear system with observers of low dynamic order. PhD dissertation, Dept. of Electrical Engineering, Stanford University, CA 4. Grimble MJ, Jukes KA, Goodall DP (1984) Nonlinear filters and operators and the constant gain extended Kalman filter. IMA J Math Control Inf 1:359–386

References

375

5. Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. J Basic Eng 83:95–108 6. Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82 (1):35–45 7. Anderson B, Moore J (1979) Optimal filtering. Prentice Hall, Englewood Cliffs 8. Grimble MJ, Johnson MA (1988) Optimal multivariable control and estimation theory: theory and applications, vols I and II. Wiley, London 9. Kleinman DL (1969) Optimal control of linear systems with time-delay and observation noise. IEEE Trans Autom Control 14:524–527 10. Grimble MJ (2007) GMV control of nonlinear continuous-time systems including common delays and state-space models. Int J Control 80(1):150–165 11. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 12. Atherton DP (1982) Nonlinear control engineering. Van Nostrand Reinhold, New York 13. Grimble MJ (2005) Nonlinear generalized minimum variance feedback, feedforward and tracking control. Automatica 41:957–969 14. Åström KJ (1979) Introduction to stochastic control theory. Academic Press, London 15. Kazemi M, Arefi MM (2017) Nonlinear generalized minimum variance control and control performance assessment of nonlinear systems based on a Wiener model. Trans Inst Meas Control 40(5):1538–1553 16. Dimogianopoulos DG, Hios JD, Fassois SD (2009) Nonlinear integral minimum variance-like control with application to an aircraft system. In: Mediterranean conference on control and automation (MED), Ancona, pp 1–6 17. Zhu Q, Ma Z, Warwick K (1999) Neural network enhanced generalised minimum variance self-tuning controller for nonlinear discrete-time systems. IEE Proc Control Theory Appl 146 (4):319–326 18. Hussain A, Zayed AS, Smith LS (2001) A new neural network and pole placement based adaptive composite self-tuning. In: IEEE multi-topic conference, Lahore, pp 267–271

Chapter 9

State-Space Nonlinear Predictive Optimal Control

Abstract The model predictive control problem is again considered but in this chapter, the linear subsystem models are represented by the state-equations. This is the most popular modelling approach and the solutions are, therefore, the most useful. The linear predictive control problem is first considered that requires the Kalman filter for state estimation. The nonlinear model predictive control problem is then discussed where an operator term is included in the plant model. Problems in predictive control system design and implementation are also explored. The control design example is concerned with automotive engine emissions control. The last section considers the so-called preview approach to a form of predictive control. This provides a slightly different perspective on the use of future information that can be useful in some applications.

9.1

Introduction

The Generalized Predictive Control (GPC) algorithm is one of the most popular predictive control algorithms that fall under the heading of Model Predictive Control (MPC). The GPC problem is derived in the first part of the chapter for the control of discrete-time multivariable systems represented in linear state-equation form. The GPC control problem was discussed in Chap. 7 for systems represented in polynomial matrix form. However, in this chapter, a Kalman filter is needed to estimate the states in the state-space model for the linear multivariable subsystem. The problem in this chapter is also more general than that considered in Chap. 7. This state-space version of a GPC problem was first solved by Ordys and Clarke [1]. The problem considered here is more general involving a cost-function that includes dynamic cost-function weightings on both the tracking error and control signals. It also includes a so-called control signal reference, which is a type of default feedforward control that is introduced when the control cost-function weighting is sufficiently large. An end-state or end point-error weighting term is also included in the cost-index with the aim of improving robustness and stability properties. The treatment of constraint handling can be the same as in Chap. 7 © Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_9

377

378

9 State-Space Nonlinear Predictive Optimal Control

(Sect. 7.6). The same vectors and matrices computed for the unconstrained problem are what is needed for use in the quadratic programming solution. The Nonlinear Predictive Generalized Minimum Variance (NPGMV) control problem is introduced in the second half of the chapter. The state-space form of the NPGMV controller is related to the predictive controller derived in Chap. 7 for polynomial systems. The plant model is again represented by a combination of a general nonlinear operator and a linear subsystem. In this chapter, the linear subsystem is represented in state-equation form and this can be open-loop unstable. A valuable feature of the NPGMV control law is that in the asymptotic case when the plant is linear, the controller reduces to a state-space version of the GPC controller [1, 2]. In the asymptotic case, when the plant is nonlinear, but the cost-function is single stage, the controller becomes equal to a state-space version of the Nonlinear Generalized Minimum Variance (NGMV) controller (Chap. 8).

9.1.1

Solution Philosophy

The solution for the GPC and NPGMV control laws is derived in the time-domain using a general operator representation of the process, similar to the solution in Chap. 7. The state observer required is linear because of the assumed structure of the system. A Kalman filter is, therefore, required but this is a little different from that in the previous chapter since the input is the measured outputs rather than the noisy error signal. The usual predictive controllers do, of course, have a configuration that is in a natural two degree of freedom form. The objectives here are similar to those in Chap. 7. That is, to first solve a GPC problem and then derive the NPGMV controller but in this case, using a state-space system model for the output subsystem. The plant model is assumed linear for the derivation of the state-space GPC law in the first part of the chapter. The derivation of the NPGMV controller in the second part of the chapter includes both linear and nonlinear subsystems and no structure needs to be known for the nonlinear operator or black-box input subsystem, but it must be assumed open-loop stable. As in the previous problems, this input subsystem can include hard nonlinearities, a state-space model, transfer operators or nonlinear function look-up tables. The multi-step predictive control cost-index to be minimized involves quadratic error and control signal costing terms for both control problems. The cost-function is quite general and can include dynamically weighted error and control signal costing terms, and end-state weighting terms [3, 4]. The NPGMV optimal control law also includes a nonlinear control signal cost weighting matrix. To ensure the existence of a stable nonlinear closed-loop system it is shown that an assumption must be made on the choice of the cost-function weightings.

9.2 System Description

9.2

379

System Description

The system can include the nonlinear plant model, together with the reference, measurement noise and disturbance signal models. It can also include a stochastic output disturbance represented by a linear time-invariant model driven by zero-mean white noise, as shown in Fig. 9.1. This is not restrictive since in many applications the disturbance model is taken as a linear time-invariant (LTI) approximation. The augmented system equations that will describe the total state-space system model will cover this and other more general plant configurations. The disturbance model will include a deterministic output disturbance component denoted d(t). The reference r(t) is also assumed to be deterministic and both of these signals r(t) and d(t) are assumed known for the prediction horizon to be used. If the system also has a deterministic input disturbance this can be accommodated as in Chap. 11. The white measurement noise v(t) is assumed to have a constant covariance matrix Rf ¼ RTf  0 and there is no loss of generality in assuming that the zero-mean white noise source nðtÞ has an identity covariance matrix. The signals v(t) and nðtÞ are vector zero-mean white noise signals. As in the predictive control problem in Chap. 7 there is no requirement to specify the distribution of the noise sources because the prediction equation is dependent upon the linear stochastic disturbance model. Nonlinear plant model: The plant input subsystem may have the following nonlinear operator form ðW 1 uÞðtÞ ¼ zk ðW 1k uÞðtÞ

r−y

φ0 = Pc e + c u + Fc0u0 Pc

Error weighting

Future reference Controller

Disturbance model

+ + +

ξ Control weighting

c

r

ð9:1Þ

Fc 0

u

0

1k

Predictive controller

Nonlinear operator subsystem

Input weighting

u0

Observations signal z

W0

Wd m

Linear statespace subsystem

+ + d + y

v

+ +

Fig. 9.1 NPGMV two degrees of freedom feedback control for a nonlinear plant with state-space based linear plant subsystem

380

9 State-Space Nonlinear Predictive Optimal Control

The delay term zk I denotes a diagonal matrix of the common delay elements in the output signal paths with k > 0. The output of the nonlinear subsystem W 1k will be denoted u0 ðtÞ ¼ ðW 1k uÞðtÞ: The nonlinear subsystem W 1 is assumed to be finite gain stable but the linear subsystem W0 ¼ zk W0k can contain any unstable modes. If the full plant model is to be included in the nonlinear subsystem then the linear output subsystem W0k can be defined as W0k ¼ I: Note that this does not mean that the linear augmented state-space equations for the system are non-dynamic, since the state-space model representing the stochastic disturbances will be present.

9.2.1

Linear State-Space Subsystem

The augmented state-space model includes the state-equations for the various subsystems, assembled in the same manner as in the previous chapter. That is, it contains the equations for the linear subsystem W0 in the plant model, the disturbance model and the dynamic cost-function weighting terms. The resulting augmented linear subsystem model can be represented in the following state-equation form, with equations for the states, output to be measured, observations and output to be controlled, respectively xðt þ 1Þ ¼ AxðtÞ þ Bu0 ðt  kÞ þ DnðtÞ

ð9:2Þ

yðtÞ ¼ dðtÞ þ CxðtÞ þ Eu0 ðt  kÞ

ð9:3Þ

zðtÞ ¼ dðtÞ þ CxðtÞ þ Eu0 ðt  kÞ þ vðtÞ

ð9:4Þ

yp ðtÞ ¼ dPD ðtÞ þ Cp xðtÞ þ Ep u0 ðt  kÞ

ð9:5Þ

where A, B, C, D, E, Cp ; Ep are constant matrices. This augmented system is shown in Fig. 9.2. As in Chap. 8 the augmented state-space model is assumed to be controllable and observable or as a lesser requirement stabilizable and detectable. Weighted output: The weighted output to be controlled yp(t) can include any stable dynamic cost-function weighting that will be denoted in the following polynomial matrix operator form yp ðtÞ ¼ Pc ðz1 ÞyðtÞ

ð9:6Þ

The last equation in the augmented system (9.5) defines the signal in the cost-function to be introduced. It includes the state-space equations for the dynamic output or error weighting function Pc ðz1 Þ: This weighted output for the augmented system, letting dPD ðtÞ ¼ Pc ðz1 ÞdðtÞ; may be written as

9.2 System Description

381 d PD Linear augmented subsystem dynamics

NL plant model

Cp

+

+ Weighted output

+

yp

Ep ξ

Control signal 1k (.,.)

C

(.,.)

u0 (t ) uc

−k

z I Delays

Disturbance

D B

d + +

u0 (t − k )

+

z

−1

C

A E

NL control weighting

Observations +

+

y

+

+ + Output

z

v

Measurement noise

Fig. 9.2 Nonlinear plant input and linear output and disturbance subsystem

yp ðtÞ ¼ Pc ðz1 ÞðdðtÞ þ CxðtÞ þ Eu0 ðt  kÞÞ ¼ dPD ðtÞ þ Pc ðz1 ÞðCxðtÞ þ Eu0 ðt  kÞÞ Delay free linear subsystem: The delay free transfer operator of the plant linear subsystem may be written in terms of these equations in the form W0k ¼ E þ CUB

ð9:7Þ

where the resolvent matrix Uðz1 Þ ¼ ðzI  AÞ1 (corresponding to the state-transition matrix UðtÞ). If all the explicit delays can be removed from the plant model the through term matrix E in (9.4) and (9.7) can be assumed full rank. However, in some applications, it is difficult to obtain the model in this form, and the through term is null. This only becomes a problem when the prediction horizon is short and in such a case note the discussion about the Discrete-Time Model for the rotational link in Chap. 11 (Sect. 11.9). Signal definitions: The input signal channels in the plant model are assumed to include a k-steps transport delay and the signals may be listed as follows: xðtÞ u0 ðtÞ uðtÞ yðtÞ zðtÞ rðtÞ yp ðtÞ rp ðtÞ

vector of n system states in the linear plant subsystem and the disturbance model. vector of m0 input signals to the linear subsystem. vector of m control signals applied to the nonlinear subsystem, representing the plant input. vector of r plant output signals that are measured. vector of r observations or measured plant output signals including measurement noise. vector of r plant setpoint or deterministic reference-signals. vector of m inferred output signals to be controlled including dynamic cost-function weightings. vector of r plant setpoint or reference-signals including cost weighting.

382

9 State-Space Nonlinear Predictive Optimal Control

9.2.2

Linear State Prediction Model

The future values of the states and outputs may be obtained by repeated use of the state-equation as follows: xðt þ 1Þ ¼ AxðtÞ þ Bu0 ðt  kÞ þ DnðtÞ xðt þ 2Þ ¼ A2 xðtÞ þ ABuðtÞ þ Bu0 ðt þ 1  kÞ þ ADnðtÞ þ Dnðt þ 1Þ xðt þ 3Þ ¼ A3 xðtÞ þ A2 Bu0 ðt  kÞ þ ABu0 ðt þ 1  kÞ þ Bu0 ðt þ 2  kÞ þ A2 DnðtÞ þ ADnðt þ 1Þ þ Dnðt þ 2Þ Generalizing this result, obtain the state at the future time t þ i; where i > 0, as xðt þ iÞ ¼ Ai xðtÞ þ

i X

Aij ðBu0 ðt þ j  1  kÞ þ Dnðt þ j  1ÞÞ

ð9:8Þ

j¼1

This equation is valid for i  0 if the summation term is defined as null when i = 0. This notation is adopted for the following. Note that the future states depend upon the future inputs and the state-vector at time t. The expression for the future states may also be modified by including the k-steps explicit transport-delay giving xðt þ i þ kÞ ¼ Ai xðt þ kÞ þ

i X

Aij ðBu0 ðt þ j  1Þ þ Dnðt þ j þ k  1ÞÞ

ð9:9Þ

j¼1

The weighted output signal yp ðtÞ to be regulated, noting (9.5), has the form (for i  1): yp ðt þ i þ kÞ ¼ dPD ðt þ i þ kÞ þ Cp Ai xðt þ kÞ þ

i X

Cp Aij ðBu0 ðt þ j  1Þ þ Dnðt þ j þ k  1ÞÞ þ Ep u0 ðt þ iÞ

j¼1

ð9:10Þ Introducing the Vector-Matrix Notation: The outputs to be computed are for controls in the interval s 2 ½t; t þ N: Introducing an obvious notation for the output signals they may be collected in the N + 1 vector form, where N  1; as:

9.2 System Description 2

3

2

383 2

3

32

3 u0 ðtÞ 76 6 6 y ðt þ 1 þ kÞ 7 6 C A 7 6 CB u0 ðt þ 1Þ 7 6 p 7 6 p 7 7 0 0 7 p 76 6 7 6 7 6 6 7 7 6 2 6 yp ðt þ 2 þ kÞ 7 6 Cp A 7 6 7 . 7 6 C AB C B .. p p 7xðt þ kÞ þ 6 6 7¼6 6 7 7 6 7 6 . 7 7 76 6 .. . . 6 7 6 . 7 6 7 7 6 .. .. . 4 5 4 . 5 4 5 0 5 4 N N1 N2 u ðt þ N  1Þ yp ðt þ N þ kÞ Cp A 0 Cp A B Cp A B    Cp B 3 2 3 3 2 3 2 0 0  0 2 nðt þ kÞ Ep u0 ðtÞ dPD ðt þ kÞ 7 6 .. 7 6 6 E u ðt þ 1Þ 7 6 d ðt þ 1 þ kÞ 7 6 Cp D 0 . 0 7 76 nðt þ 1 þ kÞ 7 7 6 PD 7 6 6 p 0 7 76 7 6 7 6 6 .. .. .. 76 7 6 Ep u0 ðt þ 2Þ 7 6 dPD ðt þ 2 þ kÞ 7 6 þ6 7 6 7þ6 7 þ 6 Cp AD . Cp D . . 7 7 6 7 7 7 6 6 6 . . . 7 76 7 6 7 6 6 . . . . . . . . 5 74 5 4 5 6 4 . . . . . . . 0 5 4 dPD ðt þ N þ kÞ Ep u0 ðt þ NÞ N1 N2 nðt þ N  1 þ kÞ Cp A D Cp A D    Cp D yp ðt þ kÞ

Cp I

0

0

 .. .

0

ð9:11Þ Future outputs: The vector of future outputs may be written as 0 Yt þ k;N ¼ CN AN xðt þ kÞ þ ðCN BN þ EN ÞUt;N þ Dt þ k;N þ CN DN Wt þ k;N

ð9:12Þ

where the following vectors and block-matrices may be defined for N  1: CN ¼ diagfCp ; Cp ; . . .; Cp g and EN ¼ diagfEp ; Ep ; . . .; Ep g 3

2

6 A 7 6 7 6 27 A 7 AN ¼ 6 6 7; 6 .. 7 4 . 5

6 6 6 6 BN ¼ 6 6 6 4

2

I

A 2 6 6 Wt;N ¼ 6 6 4

N

nðtÞ nðt þ 1Þ .. . nðt þ N  1Þ

0

0



0

 .. B . .. . AN1 B AN2 B    B 3 3 2 u0 ðtÞ 7 6 u0 ðt þ 1Þ 7 7 7 6 7; U 0 ¼ 6 7; .. t;N 7 7 6 5 5 4 . B .. .

0

u0 ðt þ NÞ

0

3

07 7 .. 7 7 . 7; 7 7 05 0

Rt;N

2

0 6 6 D 6 6 DN ¼ 6 AD 6 . 6 . 4 . AN1 D 3 2 rp ðtÞ 6 rp ðt þ 1Þ 7 7 6 7 ¼6 .. 7 6 5 4 .

ðN þ 1 squareÞ 0 0 D AN2 D

 .. . ..

. 

0

3

7 07 7 7 7 7 7 05 D

rp ðt þ NÞ

ð9:13Þ The transfer Wt;N denotes a vector of future white noise inputs and Ut;N denotes a block vector of future control signals. The block vector Rt;N denotes a vector of future weighted reference-signals rp ðtÞ ¼ Pc ðz1 ÞrðtÞ that is the same weighting as used to generate the output (9.10). Future end-state: The robustness of predictive control solutions can sometimes be improved by adding an end-state weighting term in the cost-function and an expression is, therefore, obtained below for this signal. From (9.8), the end-state

384

9 State-Space Nonlinear Predictive Optimal Control

xðt þ N þ kÞ ¼ AN xðt þ kÞ þ

N X

ANj ðBu0 ðt þ j  1Þ þ Dnðt þ j þ k  1ÞÞ

j¼1

 ¼ AN xðt þ kÞ þ AN1 B

2 6 6 6 6 ... B 6 6 6 4

AN2 B

3

u0 ðtÞ

7 7 7 7 7 7 7 5

u0 ðt þ 1Þ .. .

u0 ðt þ N  1Þ 3 nðt þ kÞ 6 nðt þ 1 þ kÞ 7 7 6 7 6 .. 7 6 D 6 7 . 7 6 .. 7 6 5 4 . nðt þ N  1 þ kÞ 2

 þ AN1 D

AN2 D



This can be written in the more concise form 0  t;N  t þ k;N xðt þ N þ kÞ ¼ AN xðt þ kÞ þ BU þ DW

ð9:14Þ

     ¼ AN1 D AN2 D    D :  ¼ AN1 B AN2 B . . . B 0 and D where B Tracking error: The k-steps ahead tracking error, that includes any dynamic error weighting, may be written in the following vector form as Et þ k;N ¼ Rt þ k;N  Yt þ k;N   ð9:15Þ 0 ¼ Rt þ k;N  CN AN xðt þ kÞ þ VN Ut;N þ Dt þ k;N þ CN DN Wt þ k;N The matrix VN used in (9.15) and defined below, for N  1; triangular form 2 0  0 Ep .. 6 6 Cp B Ep  . 6 6 . . .. .. VN ¼ CN BN þ EN ¼ 6 Cp B 6 6 . . .. E .. 4 C AN2 B p p N1 N2 Cp A B Cp A B    Cp B

is of block lower 0

3

7 7 7 7 7 7 7 0 5 Ep 0 .. .

ð9:16Þ

Special cases: For the special case of a single stage cost-function the horizon N = 0 and the system matrices may be defined as AN ¼ I; BN ¼ DN ¼ 0; CN ¼ Cp , EN ¼ Ep . The matrix (9.16) in this case should be defined as VN ¼ Ep , where Ep denotes the through term between the input signal u0 ðt  kÞ and the weighted output. This result is needed to demonstrate that the NPGMV controller is a special

9.2 System Description

385

case of the NGMV controller in this limiting case. For the case when the linear plant subsystem is not present define B = 0. Note this does not imply the linear state-space subsystem in Fig. 9.1 is totally absent, since the disturbance and weighting models form part of the system in (9.2).

9.2.3

Output Prediction Model

The i-steps ahead prediction of the output signal may be calculated by noting the above result (9.10) and assuming that the future values of the control action are known. The estimate of future weighted outputs can be found as ^yp ðt þ kjtÞ ¼ Efyp ðt þ kÞjtg, where the expectation is conditioned upon information available up to time t. Thus letting, ^yp ðt þ i þ kjtÞ ¼ Efyp ðt þ i þ kÞjtg, the predicted weighted output to be controlled ^yp ðt þ i þ kjtÞ ¼ Cp Ai^xðt þ kjtÞ þ

i X

Cp Aij Bu0 ðt þ j  1Þ þ Ep u0 ðt þ iÞ ð9:17Þ

j¼1

The signal ^xðt þ kjtÞ denotes a least squares state-estimate from a Kalman filter. Collecting the results for the case N > 0, the vector of predicted outputs Y^t þ k;N may be obtained in the following block matrix form 2

^yp ðt þ kjtÞ

3

2

dPD ðt þ kÞ

3

2

Cp I

3

6 ^y ðt þ 1 þ kjtÞ 7 6 d ðt þ 1 þ kÞ 7 6 C A 7 7 6 PD 7 6 p 7 6 p 7 7 6 7 6 6 6 ^yp ðt þ 2 þ kjtÞ 7 6 dPD ðt þ 2 þ kÞ 7 6 Cp A2 7 7 ^xðt þ kjtÞ 7¼6 7þ6 6 7 6 7 6 . 7 6 . . 7 6 7 6 . 7 6 .. .. 5 4 5 4 . 5 4 ^yp ðt þ N þ kjtÞ Cp AN dPD ðt þ N þ kÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} 2

Dt þ k;N

CN AN



3

3 2 u0 ðtÞ 7 7 6 Ep  0 7 7 6 u0 ðt þ 1Þ 7 76 7 7 .. 7 6 .. .. 7 6 . 7 Cp B . . 76 7 7 7 6 .. .. 74 5 5 . Ep . 0 u0 ðt þ NÞ Cp AN1 B Cp AN2 B    Cp B Ep |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} U0 Ep

6 6 C B p 6 6 6 .. þ6 . 6 6 6 4 Cp AN2 B

0

VN ¼CN BN þ EN

0 .. .

0

t;N

ð9:18Þ If m0 is the dimension of u0(t) the number of columns of VN is equal to m0  (N + 1).

386

9 State-Space Nonlinear Predictive Optimal Control

9.2.4

Kalman Estimator in Predictor Corrector Form

The Kalman filter was introduced in the previous chapter but the algorithm needs to be slightly different for this predicted control problem. The state-estimate ^xðtjtÞ found in this current chapter is from a Kalman filter where the estimator input is the observations signal z(t), rather than the error signal (as in the case of the NGMV controller). The predictive control approach is a natural two degrees of freedom control problem since the output measurement and the reference enter the control algorithm individually rather than via the error signal. The Kalman filter state-estimation algorithm is discussed in more detail in Chap. 13. It may be summarized for this particular problem as follows: ^xðt þ 1jtÞ ¼ A^xðtjtÞ þ Bu0 ðt  kÞ

ðPredictorÞ

^xðt þ 1jt þ 1Þ ¼ ^xðt þ 1jtÞ þ Kf ðzðt þ 1Þ  ^zðt þ 1jtÞÞ ðCorrectorÞ

ð9:19Þ ð9:20Þ

where ^zðt þ 1jtÞ ¼ dðt þ 1Þ þ C^xðt þ 1jtÞ þ Eu0 ðt þ 1  kÞ

ð9:21Þ

The Kalman gain matrix is chosen to minimize the state-estimation error ~xðt þ kjtÞ ¼ xðt þ kÞ  ^xðt þ kjtÞ

ð9:22Þ

Kalman predictor: The Kalman k-steps predictor equation is required for the predictive control computations and may be written as follows: ^xðt þ kjtÞ ¼ Ak ^xðtjtÞ þ T0 ðk; z1 ÞBu0 ðtÞ

ð9:23Þ

where T0 ðk; z1 Þ denotes a finite pulse-response block, T0 ð0; z1 Þ ¼ I and for k  1:   T0 ðk; z1 Þ ¼ ðI  Ak zk ÞðzI  AÞ1 ¼ z1 I þ z1 A þ z2 A2 þ . . . þ zk þ 1 Ak1 ð9:24Þ It is valuable that for the prediction form of the estimator, the number of states in the filter is not increased by the number of the synchronous delays k. Operator form: The optimal estimate may be written, using Eqs. (9.19) to (9.21), in the form ^xðt þ 1jt þ 1Þ ¼ A^xðtjtÞ þ Bu0 ðt  kÞ þ Kf ðzðt þ 1Þ  dðt þ 1Þ  CA^xðtjtÞ  CBu0 ðt  kÞ  Eu0 ðt þ 1  kÞÞ   ^xðtjtÞ ¼ ðI  z1 ðI  Kf CÞAÞ1 ðBz1  Kf CBz1  Kf EÞu0 ðt  kÞ þ Kf ðzðtÞ  dðtÞÞ

9.2 System Description

387

The estimate of current state follows as   ^xðtjtÞ ¼ ðI  z1 ðI  Kf CÞAÞ1 Kf ðzðtÞ  dðtÞÞ  zk ðKf E þ ðKf C  IÞBz1 Þu0 ðtÞ

Estimator: The above equation may. therefore. be written in the concise form ^xðtjtÞ ¼ Tf 1 ðz1 ÞðzðtÞ  dðtÞÞ þ Tf 2 ðz1 Þu0 ðtÞ

ð9:25Þ

where the linear operators Tf 1 ðz1 Þ ¼ ðI  z1 ðI  Kf CÞAÞ1 Kf   Tf 2 ðz1 Þ ¼ ðI  z1 ðI  Kf CÞAÞ1 zk Kf E þ ðKf C  IÞBz1

ð9:26Þ ð9:27Þ

Unbiased estimates property: Observe that for the Kalman filter to be unbiased the transfer between the control input u0(t) and estimated state in (9.25) must satisfy Tf 1 ðz1 ÞðCUðz1 ÞB þ EÞzk þ Tf 2 ðz1 Þ ¼ Uðz1 ÞBzk

ð9:28Þ

This is a useful result that may be verified using (9.26) and (9.27), as follows: Tf 1 ðz1 ÞðCUðz1 ÞB þ EÞzk þ Tf 2 ðz1 Þ ¼ ðI  z1 ðI  Kf CÞAÞ1 ½Kf C  ðKf C  IÞðI  z1 AÞðI  z1 AÞ1 Bzk1 ¼ ðI  z1 AÞ1 Bzk1 ¼ Uðz1 ÞBzk ð9:29Þ Output prediction: The (N + 1) step-ahead prediction vector in (9.18) can clearly be written in the form 0 Y^t þ k;N ¼ Dt þ k;N þ CN AN ^xðt þ kjtÞ þ VN Ut;N

ð9:30Þ

Output prediction error: Y~t þ k;N ¼ Yt þ k;N  Y^t þ k;N 0 0 ¼ CN AN xðt þ kÞ þ VN Ut;N þ CN DN Wt þ k;N  ðCN AN ^xðt þ kjtÞ þ VN Ut;N Þ ð9:31Þ

Recall the k-steps ahead state-estimation error from (9.22). Substituting into (9.31) the inferred output estimation error

388

9 State-Space Nonlinear Predictive Optimal Control

Y~t þ k;N ¼ CN AN ~xðt þ k jtÞ þ CN DN Wt þ k;N

ð9:32Þ

Note for later use that the state estimation error is independent of the choice of the control action. Also recall that the optimal ^xðt þ kjtÞ and ~xðt þ kjtÞ are orthogonal and the expectation of the product of the future values of the control action (assumed known in deriving the prediction equation) and the zero-mean white noise driving signals is null. It follows that the vector of predicted signals Y^t þ k;N in (9.30) and the prediction error Y~t;N are orthogonal.

9.3

Generalized Predictive Control

A brief review of the derivation of the “state-space” version of the GPC controller for linear systems is provided below. This is one of the most common forms of the Model Predictive Control (MPC) algorithm. It was introduced in Chap. 7 for polynomial system models. The nonlinearity is omitted in the derivation of the GPC law in the first part of this chapter. The input block nonlinearity W 1k will be replaced by the identity and the plant input will be the same as the linear subsystem with input u(t) = u0(t), shown in Fig. 9.2. This solution also provides results that are needed for introducing the more general nonlinear problem of interest in the latter part of the chapter. Only the main points in the solution are summarized. GPC cost-function: The GPC performance index has a more general form here than that covered in Chap. 7. It may be defined as follows: N   X J ¼ Ef ep ðt þ j þ kÞT ep ðt þ j þ kÞ þ u0 ðt þ jÞ   u0 ðt þ jÞÞT k2j ðu0 ðt þ jÞ   u0 ðt þ jÞÞ jtg j¼0

þ Efex ðt þ N þ kÞT Ps ex ðt þ N þ kÞjtg

ð9:33Þ

where Ef:jtg denotes the conditional expectation, conditioned on measurements up to time t, and kj denotes a control signal weighting factor that may be a scalar, or in the form of a diagonal matrix. Note that the control horizon does not need to be the same as the output prediction horizon. A connection matrix is introduced in Chap. 11 (Sect. 11.4.1) that enables different numbers of control moves, or time distributions of control actions, to be specified. The future optimal control signal is to be calculated over the interval s 2 ½t; t þ N: Dynamic weighting: The state-space model generating the weighted error signal ep(t) is defined in terms of the weighted reference rp(t) and output yp(t) signals. These depend upon the dynamic cost-function weighting matrix Pc ðz1 Þ that can introduce a low-pass filter to ensure low-frequency disturbances are penalized. Control reference: The cost-index includes dynamically weighted tracking errors at current and future times, and a term involving what might be termed a control reference u0 ðt þ jÞ; which is a known signal. The inclusion of the control reference signal is not common but can be useful if there are known desired values

9.3 Generalized Predictive Control

389

for the control action. It may, for example, be possible to compute the best control action in an ideal case when dynamics and disturbances are neglected that under high control costing will result in a type of feedforward control action. End-state weighting: The cost-function (9.33) also includes a final term representing an end-state weighting. The error in the desired end-state is represented by the signal ex ðt þ i þ kÞ ¼ rx ðt þ i þ kÞ  xðt þ i þ kÞ; where rx ðt þ i þ kÞ denotes a deterministic reference-signal. Such a term can be introduced to improve stability and robustness properties. Minimum cost-horizon: The prediction horizon should be sufficiently large for the cost-index (9.33) to be physically meaningful. For example, when N = 0 and k > 0 the single stage cost-function has the form J ¼ Efep ðt þ kÞT ep ðt þ kÞ þ k2j u0 ðtÞT u0 ðtÞÞjtg: If the system does not have a through term the control input affects the output at least (k + 1) steps later. In many real systems, the state-space plant model will not include a through term and it is not straightforward to rearrange the equations to pull out another delay term to obtain a model including a non-zero through term. For such systems, the minimum horizon can be taken as N = 1.

9.3.1

Vector Form of GPC Cost-Function

The multi-step cost-function (9.33) is expanded below to obtain the vector form of the criterion for the optimization stage in the next section. The criterion may first be written in the more concise vector-matrix form that follows: 0 0 T 2 0 0  t;N  t;N J ¼ EfJt g ¼ EfðRt þ k;N  Yt þ k;N ÞT ðRt þ k;N  Yt þ k;N Þ þ ðUt;N U Þ KN ðUt;N U Þjtg

þ Efex ðt þ N þ kÞT Ps ex ðt þ N þ kÞjtg

ð9:34Þ

0  t;N where the vector of future values of the control reference  u0 ðtÞ are denoted as U . The states are not normally available for feedback control and an optimal state estimator must be introduced. This is assumed here and the cost-function is expressed below in terms of the optimal state-estimate and the state-estimation error. The end-state weighting term can be used to tune robustness and stability properties and may first be simplified [5]. Substituting from (9.14):

ex ðt þ N þ kÞ ¼ rx ðt þ N þ kÞ  xðt þ N þ kÞ   0  t;N  t þ k;N þ DW ¼ rx ðt þ N þ kÞ  AN xðt þ kÞ þ BU If the Kalman filter is introduced for state-estimation and prediction, this equation can be written as

390

9 State-Space Nonlinear Predictive Optimal Control

  0  t;N  t þ k;N þ AN ~xðt þ kjtÞ ex ðt þ N þ kÞ ¼ rx ðt þ N þ kÞ  AN ^xðt þ kjtÞ  BU  DW To simplify the equations substitute rdd ðt þ kÞ ¼ rx ðt þ N þ kÞ  AN ^xðt þ kjtÞ

ð9:35Þ

and we obtain 0  t;N  t þ k;N  AN ~xðt þ kjtÞ ex ðt þ N þ kÞ ¼ rdd ðt þ kÞ  BU  DW

ð9:36Þ

Exploiting the orthogonality properties of the estimates ¼E



Efex ðt þ N þ kÞT Ps ex ðt þ N þ kÞjtg 0  t;N  t þ k;N  AN ~xðt þ kjtÞ rdd ðt þ kÞ  BU  DW

¼E



0  t;N rdd ðt þ kÞ  BU

T

T

 

0  t;N  t þ k;N  AN ~xðt þ kjtÞ jt Ps rdd ðt þ kÞ  BU  DW



0  Ps rdd ðt þ kÞ  BUt;N jt þ Js 

where the final term is independent of control action and is defined as Js ¼ E

n

   o  t þ k;N þ AN ~xðt þ kjtÞ jt  t þ k;N þ AN ~xðt þ kjtÞ T Ps DW DW

It follows that the cost-function end-state weighting term can be written as Efxðt þ N þ kÞT Ps xðt þ N þ kÞjtg  T  

0 0  t;N  t;N Ps rdd ðt þ kÞ  BU ¼ E rdd ðt þ kÞ  BU j t þ Js Vector form of GPC cost: The cost-function can now be written from (9.34) as J ¼ EfðRt þ k;N  ðY^t þ k;N þ Y~t þ k;N ÞÞT ðRt þ k;N  ðY^t þ k;N þ Y~t þ k;N ÞÞ 0 0 T 2 0 0  t;N  t;N þ ðUt;N U Þ KN ðUt;N U Þjtg  T  

0 0  t;N  t;N Ps rdd ðt þ kÞ  BU þ E rdd ðt þ kÞ  BU j t þ Js

ð9:37Þ

u0 ðtÞ are where the cost weighting on the future values of the inputs u0 ðtÞ   expressed as K2N ¼ diagfk20 ; k21 ; . . .; k2N g.

9.3.2

Orthogonality and GPC Solution

The terms in the cost-index (9.37) can now be simplified, by noting the optimal estimate Y^t þ k;N is orthogonal to the estimation error Y~t þ k;N . Also recall that the

9.3 Generalized Predictive Control

391

0  t;N future reference or setpoint trajectory Rt þ k;N , control reference U and measurable disturbance Dt þ k;N are assumed to be known signals over the N + 1 future steps. Using these results to simplify (9.37), the vector/matrix form of the cost-index may be obtained as 0T 2 0 0T 2 0 0T 2  0  t;N J ¼ ðRt þ k;N  Y^t þ k;N ÞT ðRt þ k;N  Y^t þ k;N Þ þ Ut;N KN Ut;N  U KN Ut;N  Ut;N KN Ut;N  T   0 0  t;N Ps rdd ðt þ kÞ  BU  t;N þ Js þ J0 þ rdd ðt þ kÞ  BU ð9:38Þ

The terms that are independent of control action may be collected together as 0T 2  0  t;N J0 ¼ EfY~tTþ k;N Y~t þ k;N þ U KN Ut;N jtg þ Js

To simplify the first term in (9.38) recall from Eq. (9.30) the expression for the vector of state-estimates 0 Rt þ k;N  Y^t þ k;N ¼ Rt þ k;N  Dt þ k;N  CN AN ^xðt þ kjtÞ  VN Ut;N

ð9:39Þ

Also, introduce the modified reference-signal vector as ~ t þ k;N ¼ Rt þ k;N  Dt þ k;N  CN AN ^xðt þ kjtÞ R

ð9:40Þ

where from (9.23) the state-estimate ^xðt þ kjtÞ depends only upon the past values of the control signal. The multi-step cost-function may now be written as 0 T ~ 0 0T 2 0 0T 2 0 0T 2  0 ~ t þ k;N  VN Ut;N  t;N J ¼ ðR Þ ðRt þ k;N  VN Ut;N Þ þ Ut;N KN Ut;N  U KN Ut;N  Ut;N KN Ut;N  T   0 0  t;N Ps rdd ðt þ kÞ  BU  t;N þ Js þ J0 þ rdd ðt þ kÞ  BU   0T 0 ~ Ttþ k;N R ~ t þ k;N  Ut;N ~ t þ k;N þ K2N U  t;N ¼R VNT R    T  0 0T 2 0 0T  t;N ~ Ttþ k;N VN þ U KN Ut;N þ Ut;N VN VN þ K2N Ut;N  R T T 0 0T  T 0T  T 0  t;N  t;N þ rdd ðt þ kÞPs rdd ðt þ kÞ  rdd ðt þ kÞPs BU  Ut;N þ Js þ J0 B Ps rdd ðt þ kÞ þ Ut;N B Ps BU

To simplify this expression introduce the constant matrix XN . This may be taken as a positive-definite real symmetric matrix, because of the weighting choice, and is defined as   T Ps B XN ¼ VNT VN þ K2N þ B

ð9:41Þ

392

9 State-Space Nonlinear Predictive Optimal Control

The cost-function may now be written as follows:   0T 0 0T 0 ~ t þ k;N þ K2N U  t;N  T Ps rdd ðt þ kÞ J ¼ Ut;N XN Ut;N  Ut;N VNT R þB   0T 2 T 0 ~ Ttþ k;N VN þ U  Ut;N  t;N  R KN þ rdd ðt þ kÞPs B

ð9:42Þ

T ~ t þ k;N þ rdd ~ Ttþ k;N R þR ðt þ kÞPs rdd ðt þ kÞ þ Js þ J0

GPC optimal control: The approach to minimize (9.42) is the same as when all the signals are deterministic. The gradient of the quadratic cost-function can be set to zero, to obtain the vector of future controls, as illustrated in Chap. 7 (Sect. 7.3). Observe that the J0 cost term is independent of the control action. Thus, from (9.42) by setting the gradient to zero, the vector of future optimal controls follows as   0 0 ~ t þ k;N þ K2N U  t;N  T Ps rdd ðt þ kÞ Ut;N ¼ XN1 VNT R þB

ð9:43Þ

~ t þ k;N is defined in (9.40). The GPC optimal control signal at time t is where R defined from this vector of future controls based on the receding-horizon principle introduced in Chap. 7 (Sect. 7.3.2). The optimal predictive control is taken as the 0 [6, 7, 8]. first element in the vector of future controls Ut;N The results can be summarized in the following theorem. Theorem 9.1: Generalized Predictive Control Consider the linear state-space system described in Sect. 9.2 with the input subsystem set to the identity ðW 1k ¼ I): The cost-function (9.34) is to be minimized for a given error weighting Pc ðz1 Þ; control weighting KN and end-state weighting Ps . Given the future state ^xðt þ kjtÞ; ~ t þ k;N from (9.40), and VNT and XN , from (9.16) and (9.41), the GPC reference R optimal control can be computed from the first element of the vector of future controls   0 0  t;N  T Ps rdd ðt þ kÞ Ut;N ¼ XN1 VNT ðRt þ k;N  Dt þ k;N  CN AN ^xðt þ kjtÞÞ þ K2N U þB ð9:44Þ where the cost-function weighting definitions ensure the matrix XN is non-singular. ■ Proof By collecting the above results.

■

Properties: These results can be generalized to a plant that has different delays in the various channels but this adds some complication [9]. In the limit as K2N ! 1  I the vector of future controls (9.43) tends to the control reference 0 0  t;N Ut;N !U , which may be null. This gives a way of forcing a predefined form of control action in this limiting case. It results in a type of feedforward action with a very rapid response that does not need to wait for the feedback control to change.

9.3 Generalized Predictive Control

393

Constrained optimization: The constrained solution by quadratic programming involves the same matrices as in the unconstrained solutions derived above, as well as the matrices defining the equality and inequality constraints [10]. Matrix structure: The matrix VNT has an important influence on the structure of the predictive control solution in (9.43). Note that VNT ¼ BTN CNT þ ENT has an upper-triangular block form 2

EpT 6 0 6 6 . T VN ¼ 6 6 .. 6 4 0 0

BT CpT EpT 0  0

 BT CpT .. . 0 

BT ATN2 CpT 0 .. . EpT 0

3 BT ATN1 CpT BT ATN2 CpT 7 7 7 .. 7 7 . 7 BT CpT 5 EpT

ð9:45Þ

0 The current control is the first element in the vector Ut;N and is affected by the first row in (9.45). It follows from (9.43) that even if the through term EpT is null the top row in (9.45) will contribute to the current control. Note that in the special and unusual case of a single stage cost-function where N = 0, the matrix VNT ¼ EpT .

9.3.3

Equivalent GPC Cost Optimisation Problem

The above GPC problem is equivalent to a special “square of sums” cost-minimisation control problem, which is needed later to motivate the criterion in the NPGMV problem in Sect. 9.4. To express the cost-function in an equivalent form first let the constant positive-definite, real symmetric matrix XN be factorized into the form   T Ps B Y T Y ¼ XN ¼ VNT VN þ K2N þ B

ð9:46Þ

Completing the squares: The cost-function in Eq. (9.42) may now be rewritten, using the factorization in (9.46) to complete the squares and we obtain     0T 0 0T 0 T 0T 2 0 ~ Ttþ k;N VN þ rdd  T Ps rdd ðt þ kÞ þ K2N U ~ t þ k;N þ B  t;N þU  t;N  R J ¼ Ut;N XN Ut;N  Ut;N VNT R ðt þ kÞPs B KN Ut;N T ~ Ttþ k;N R ~ t þ k;N þ rdd þR ðt þ kÞPs rdd ðt þ kÞ þ Js þ J0    T T 0T 2 1 ~   ¼ Rt þ k;N VN þ rdd ðt þ kÞPs B þ Ut;N KN Y  Ut0Tþ k;N Y T     0 0  T Ps rdd ðt þ kÞ þ K2N U ~ t þ k;N þ B  t;N  YUt;N  Y T VNT R T ~ Ttþ k;N R ~ t þ k;N þ rdd ðt þ kÞPs rdd ðt þ kÞ þ Js þ J0 þR    T 0T 2 0 ~ Ttþ k;N VN þ rdd  T Ps rdd ðt þ kÞ þ K2N U þU  t;N ~ t þ k;N þ B  t;N  R ðt þ kÞPs B KN Y 1 Y T VNT R



394

9 State-Space Nonlinear Predictive Optimal Control

Equivalent cost-index: The cost-function may, therefore, be written in an equivalent form as ^ ^T J¼U t þ k;N Ut þ k;N þ J10 ðtÞ

ð9:47Þ

where the predicted inferred signal   2 0 T 0 ^ t þ k;N ¼ Y T V T R ~  U þ B P r ðt þ kÞ þ K U t þ k;N s dd N t;N  YUt;N N

ð9:48Þ

The terms that are independent of the control action may be written as J10 ðtÞ ¼ Js þ J0 þ J1 ðtÞ where, T ~ Ttþ k;N R ~ t þ k;N þ rdd ðt þ kÞPs rdd ðt þ kÞ J1 ðtÞ ¼ R   0T 2  þU  t;N ~ Ttþ k;N VN þ rdd ðt þ kÞT Ps B KN  R   0 ~ t þ k;N þ B  t;N  T Ps rdd ðt þ kÞ þ K2N U  XN1 VNT R

ð9:49Þ

Manipulating the cost expression into a different form does not change the optimal GPC solution as discussed in Chap. 7. In fact the optimal control is easily ^ t þ k;N ¼ 0Þ; because the term found by setting the first term in (9.47) to zero ðU J10 ðtÞ in (9.47) does not depend upon control action. The resulting optimal control is the same as the GPC law derived in (9.43).

9.3.4

Modified Cost-Function Generating GPC Controller

The above discussion may be used to motivate the definition of a multi-step minimum variance cost minimization problem that is a special case of the nonlinear problem introduced in the next half of the chapter. This modified cost problem will also provide a GPC control law when the system is linear. There are some mathematical preliminaries and the optimal control is then derived. Consider first a new signal to be minimized, which is an extension of that in the basic NGMV cost-index involving a weighted sum of error and input signals, of the form / ¼ Pc e þ Fc0 u0 . Motivated by this NGMV cost minimization problem the vector of future values of the signal to be minimized, in a multi-step cost-index, is defined as 0 0 0  t;N  CN rdd ðt þ kÞ Ut þ k;N ¼ PCN Et þ k;N þ FCN Ut;N þ K2N U þP

ð9:50Þ

This includes the control reference and end-state weighting terms. The definition of the cost-function weighting terms depends upon the GPC cost weightings above.

9.3 Generalized Predictive Control

395

The weightings for this modified cost-optimization problem are defined to have the following constant matrix forms PCN ¼ VNT ;

0  T Ps B  FCN ¼ K2N  B

 T Ps :  CN ¼ B and P

ð9:51Þ

The minimum variance multi-step cost-index is defined in terms of the vector of signals in (9.50) as ~J ¼ Ef~Jt g ¼ EfUTtþ k;N Ut þ k;N jtg

ð9:52Þ

The choice of weightings in (9.51) can be justified by the results presented in Theorem 9.2. This reveals that the GPC law may be derived from this minimization problem which will be a limiting case of the more general nonlinear problem considered later in Sect. 9.4. The minimization of the criterion in (9.52) exploits the orthogonality properties of the optimal estimator. Prediction equation: Predicting forward k-steps 0 0 0  t;N  CN rdd ðt þ kÞ Ut þ k;N ¼ PCN ðRt þ k;N  Yt þ k;N Þ þ FCN Ut;N þ K2N U þP

ð9:53Þ

Substituting from Eq. (9.53), for the vector of outputs Yt þ k;N ¼ Y^t þ k;N þ Y~t þ k;N : 0 0 0  t;N  CN rdd ðt þ kÞ Ut;N þ K2N U þP Ut þ k;N ¼ PCN ðRt þ k;N  Y^t þ k;N Þ þ FCN  PCN Y~t þ k;N

ð9:54Þ

This expression may be written in terms of the estimate and the estimation error vector as ^ t þ k;N þ U ~ t þ k;N Ut þ k;N ¼ U

ð9:55Þ

where the prediction ^ t þ k;N ¼ PCN ðRt þ k;N  Y^t þ k;N Þ þ F 0 U 0 þ K2 U  0 U N t;N þ PCN rdd ðt þ kÞ CN t;N and the prediction error ~ t þ k;N ¼ PCN Y~t þ k;N U

ð9:56Þ

Multi-step performance index: The performance index (9.52) may, therefore, be simplified and written as ^ t þ k;N þ U ~ t þ k;N ÞT ðU ^ t þ k;N þ U ~ t þ k;N Þjtg ~J ¼ Ef~Jt g ¼ EfUTtþ k;N Ut þ k;N jtg ¼ EfðU The terms in the performance index (9.52) can again be simplified, recalling the optimal estimate Y^t þ k;N and the estimation error Y~t þ k;N are orthogonal, and the

396

9 State-Space Nonlinear Predictive Optimal Control

future reference or setpoint trajectory Rt þ k;N is a known signal. Noting these results, we obtain ^T ^T ~T ~T ^ ~ ^ ~ ~J ¼ EfU t þ k;N Ut þ k;N þ Ut þ k;N Ut þ k;N þ Ut þ k;N Ut þ k;N þ Ut þ k;N Ut þ k;N jtg ^T ~T ^ t þ k;N þ EfU ~ t þ k;N jtg ¼U U U t þ k;N

t þ k;N

ð9:57Þ Thence, the cost-function may be written as ^T ^ ~JðtÞ ¼ U ~ t þ k;N Ut þ k;N þ J1 ðtÞ

ð9:58Þ

The cost term that is independent of control action may be written, using (9.58), as: T ~T ~ ~J1 ðtÞ ¼ EfU ~ ~T t þ k;N Ut þ k;N jtg ¼ EfYt þ k;N PCN PCN Yt þ k;N jtg

ð9:59Þ

^ t þ k;N may be simplified, by substituting for The vector of predicted signals U Y^t þ k;N , from (9.30) and (9.46): ^ t þ k;N ¼ PCN ðRt þ k;N  Y^t þ k;N Þ þ F 0 U 0 þ K2 U 0  U N t;N þ PCN rdd ðt þ kÞ CN t;N 0 ¼ PCN ðRt þ k;N  Dt þ k;N Þ  PCN ðCN AN ^xðt þ kjtÞ þ VN Ut;N Þ 0 0 0  t;N  CN rdd ðt þ kÞ þ FCN Ut;N þ K2N U þP   ¼ PCN Rt þ k;N  Dt þ k;N  CN AN ^xðt þ kjtÞ 0 0  t;N  t;N  T Ps BÞU  CN rdd ðt þ kÞ þ K2N U þP  ðVNT VN þ K2N þ B

Noting (9.40) and (9.46) 0 0 ^ t þ k;N ¼ PCN R ~ t þ k;N  XN Ut;N  t;N  CN rdd ðt þ kÞ þ K2N U þP U

ð9:60Þ

From a similar argument to that in the previous section, the optimal multi-step minimum variance predictive control sets the first squared term in (9.58) to zero, ^ t þ k;N ¼ 0: namely U The optimal control to minimize the modified cost-function (9.52) follows from setting (9.60) to zero, giving   0 0 ~ t þ k;N þ K2N U  t;N  CN rdd ðt þ kÞ Ut;N ¼ XN1 VNT R þP

ð9:61Þ

This expression for the modified cost minimization problem is also the same as the vector of future GPC controls. The results are summarized in the theorem that follows:

9.3 Generalized Predictive Control

397

Theorem 9.2: Equivalent Minimum Variance Cost Minimization Problem Consider the minimization of the GPC cost-index (9.33), for the system and assumptions introduced in Sect. 5.2, where the input subsystem is the identity W 1k ¼ I: Redefine the cost-index to have a multi-step minimum variance form ~JðtÞ ¼ EfUTtþ k;N Ut þ k;N jtg

ð9:62Þ

where 0 0 0  t;N  CN rdd ðt þ kÞ Ut;N þ K2N U þP Ut þ k;N ¼ PCN Et þ k;N þ FCN

ð9:63Þ

Define the cost-function weightings by relating them to the GPC problem weightings as follows: PCN ¼ VNT ;

0   T Ps B; FCN ¼ K2N  B

 CN ¼ B  T Ps P

where VN ¼ CN BN þ EN . The vector of future optimal controls:   0 0  T Ps rdd ðt þ kÞ ~ t þ k;N þ K2N U  t;N Ut;N ¼ XN1 VNT R þB

ð9:64Þ

where ~ t þ k;N ¼ Rt þ k;N  Dt þ k;N  CN AN ^xðt þ kjtÞ: R This is the same as the vector of future GPC controls in (9.43).

■

Solution The proof follows by collecting results in the section above.

■

Note that the black-box nonlinear term was considered the identity in the above derivation. This solution, therefore, provides the GPC optimal control for a linear system with a plant input u0 ðtÞ (see Fig. 9.2). The more general nonlinear case, with input u(t) to the black-box term, is considered below in Sect. 9.4.

9.4

Nonlinear Predictive GMV Control Problem

The Nonlinear Predictive Generalized Minimum Variance (NPGMV) control law will now be derived which extends the linear GPC solution to nonlinear systems. The input nonlinearity W 1k is assumed to be included in the system model for the development of this algorithm. An important feature of the solution is that in the limiting case when the control weighting becomes zero and the system is linear the NPGMV controller becomes equal to the GPC solution. This desirable property is due to the definition of weighting terms in the results of Theorem 9.2.

398

9 State-Space Nonlinear Predictive Optimal Control

Recall that the actual input to the system defined in Sect. 9.2, and shown in Fig. 9.1, is the control signal u(t) rather than the intermediate signal input to the linear subsystem u0 ðtÞ: The cost-function for the nonlinear control problem of interest will, therefore, include an additional cost weighting term on the control action u(t). However, the cost-function weighting on the intermediate signal u0 ðtÞ; can be retained to examine limiting cases and to provide a possible cost weighting term on actuator outputs. If the smallest common delay in each output channel of the plant is of magnitude k-steps, then the control signal t affects the output at least k-steps later. The possibly nonlinear control signal costing will, therefore, be defined to have the form ðF c uÞðtÞ ¼ zk ðF ck ðu   uÞ Þ ð t Þ

ð9:65Þ

The control weighting operator F ck is assumed full rank and invertible. The signal uðtÞ denotes some known control reference that is often a null vector but can be used to introduce a form of feedforward action. This is related to the control signal reference u0 ðtÞ that was added into the GPC cost-function (9.33). If these terms are to be introduced into the criterion they must, of course, be consistent and hence define u0 ðtÞ ¼ ðW 1k uÞðtÞ: If these signals are not needed they can be set to zero, however, it provides a useful capability in some industrial applications. The solution can be sought by following similar steps to those in Chap. 5. In analogy with the GPC problem in the previous section, a multi-step cost-index may first be defined that is an extension of the cost-function in (9.62). Extended multi-step cost-index: 0 Jp ¼ EfU0T t þ k;N Ut þ k;N jtg

ð9:66Þ

The signal whose variance is to be minimized U0t þ k;N includes vectors of future weighted control and error terms   0 0 0  CN rdd ðt þ kÞ þ F ck;N Ut;N  U  t;N  t;N U0t þ k;N ¼ PCN Et þ k;N þ FCN Ut;N þ K2N U þP   0 0 0  t;N  CN rdd ðt þ kÞ þ F ck;N Ut;N  U  t;N Ut;N þ K2N U þP ¼ PCN ðRt þ k;N  Yt þ k;N Þ þ FCN

ð9:67Þ 0  P  CN ¼ B  T Ps B,  T Ps . By ¼ K2N  B where the weightings satisfy PCN ¼ VNT , FCN defining the criterion in this way the controller is in the limiting case, where F ck is null and the system is linear W 1k ¼ I, reverts to the GPC control design discussed above.

Remarks 0 • The problem simplifies for the single-stage case N = 0, defining FCN ¼ 0 and  CN ¼ 0: It then reduces to a single-stage non-predictive control problem, which P is a particular form of the NGMV control problem.

9.4 Nonlinear Predictive GMV Control Problem

399

• Noting the weighting PCN ¼ VNT the number of columns of VN is of dimension equal to (N + 1) times the dimension of u0(t). This must be equal to (N + 1) times the dimension of the control input for the terms to be added as in (9.67), which defines the signal to be minimized. This implies that the input subsystem is assumed square and that the weighted inferred output yp to be minimized must be defined to have the same dimension m0 as the input signal u0(t). • The cost-function (9.66) involves the use of the same horizons for the output and control terms but this is not essential. As mentioned above for the GPC algorithm different horizons may be used and a connection matrix may be used as in Chap. 11 to apply different patterns of future control action. 0  t;N  t;N in (9.67) denote some known control reference vec• The signals U and U tors, including future values of u0 ðtÞ and uðtÞ: These will often be defined as the null-vector but they can be used to introduce a type of feedforward action. Block-diagonal nonlinear weighting: The nonlinear weighting term F ck;N Ut;N will be defined to have the block-diagonal matrix structure    t;N ¼ diagfðF ck ðu   F ck;N Ut;N  U uÞÞðtÞ; ðF ck ðu  uÞÞðt þ 1Þ; . . .; ðF ck ðu   uÞÞðt þ N Þg

ð9:68Þ 0 ¼ W 1k;N Ut;N , where the operator W 1k;N also has a The vector of inputs Ut;N block-diagonal matrix form

ðW 1k;N Ut;N Þ ¼ diagfW 1k ; W 1k ; . . .; W 1k gUt;N ¼ ½ðW 1k uÞðtÞT ; . . .; ðW 1k uÞðt þ NÞT T

9.4.1

ð9:69Þ

Condition for Optimality

The solution now follows similar steps to those in Sect. 9.3.3. Observe U0t;N defined in (9.67) may be written as follows    t;N U0t;N ¼ Ut;N þ zk F ck;N Ut;N  U ^0 ~0 In terms of the prediction and prediction error U0t þ k;N ¼ U t þ k;N þ Ut þ k;N . It follows from (9.53) and (9.67)   ^ ^0  U t þ k;N ¼ Ut þ k;N þ F ck;N Ut;N  Ut;N

  0 0 0  CN rdd ðt þ kÞ þ F ck;N Ut;N  U  t;N  t;N ¼ PCN ðRt þ k;N  Y^t þ k;N Þ þ FCN Ut;N þ K2N U þP

ð9:70Þ

400

9 State-Space Nonlinear Predictive Optimal Control

The estimation error follows as T~ ~0 ~ U t þ k;N ¼ Ut þ k;N ¼ VN Yt þ k;N

ð9:71Þ

^0 The predicted values in the signal U t þ k;N involves the estimated vector of weighted outputs Y^t þ k;N , and these are orthogonal to Y~t þ k;N . The estimation error is zero-mean and the expected value of the product of a zero-mean signal with a deterministic signal is null. The cost-function may, therefore, be written as ^ 0T U ^0 ~JðtÞ ¼ U ~ t þ k;N t þ k;N þ J1 ðtÞ

ð9:72Þ

Condition for optimality: The condition for optimality for the NPGMV problem 0 ^0 is clearly U t þ k;N ¼ 0: Substituting in (9.70) for the vector of future controls Ut;N ¼ W 1k;N Ut;N we obtain 0 0  t;N W 1k;N Ut;N þ K2N U PCN ðRt þ k;N  Y^t þ k;N Þ þ FCN    CN rdd ðt þ kÞ þ F ck;N Ut;N  U  t;N ¼ 0 þP

9.4.2

ð9:73Þ

NPGMV Optimal Control Solution

The generalization involving the introduction of the control reference signal requires an assumption to be made in the solution of the NPGMV optimal control. Either of the following two assumptions can be made to simplify the expression for the condition for optimality (9.73). That is, assume 1. Either the control signal cost weighting is linear (F ck , F ck;N are linear operators) or assume,  t;N are null. 2. The reference control vector uðtÞ and future values U These assumptions will be valid for the vast majority of optimal predictive   t;N to be control problems considered and either enables the term F ck;N Ut;N  U  t;N . written as F ck;N Ut;N  F ck;N U NPGMV control signal: The vector of future optimal control signals, to minimize the cost-index (9.72), follows from the condition for optimality (9.73), and satisfies  1  0 0  t;N W 1k;N Ut;N ¼ F ck;N þ FCN  PCN ðRt þ k;N  Y^t þ k;N Þ  K2N U   CN rdd ðt þ kÞ þ F ck;N U  t;N P

ð9:74Þ

An alternative solution of Eq. (9.73) that is more suitable for implementation, gives

9.4 Nonlinear Predictive GMV Control Problem

   0 ^ Ut;N ¼ F 1 ck;N  PCN Rt þ k;N  Yt þ k;N  FCN W 1k;N Ut;N  0  t;N  CN rdd ðt þ kÞ þ U  t;N  K2N U P

401

ð9:75Þ

The optimal predictive control law is clearly nonlinear, since it involves the nonlinear model for the plant W 1k;N and the control signal cost-function weighting term F ck;N . These expressions may be simplified further to reveal the relationship to the estimated state. With the above assumptions and substituting from (9.30) in the condition for optimality (9.73), we obtain     0 0  CN rdd ðt þ kÞ  F ck;N U  t;N  t;N ¼ 0 PCN Rt þ k;N  Y^t þ k;N þ F ck;N þ FCN W 1k;N Ut;N þ K2N U þP     0 0 þ F ck;N þ FCN PCN Rt þ k;N  Dt þ k;N  CN AN ^xðt þ kjtÞ  VN Ut;N W 1k;N Ut;N 0  t;N  t;N ¼ 0  CN rdd ðt þ kÞ  F ck;N U þ K2N U þP

0 Note from (9.41), XN ¼ PCN VN  FCN and hence the condition for optimality now becomes

  0  t;N  CN rdd ðt þ kÞ PCN Rt þ k;N  Dt þ k;N  CN AN ^xðt þ kjtÞ þ K2N U þP    t;N ¼ 0 þ F ck;N  XN W 1k;N Ut;N  F ck;N U

ð9:76Þ

This condition for optimality is similar to that which follows from (9.61), but with the control weighting F ck;N terms added. To simplify (9.76) introduce the constant matrix C/ ¼ PCN CN AN ¼ VNT CN AN

ð9:77Þ

Optimal control: The two alternative solutions for the vector of future optimal controls, in terms of the estimate of the future predicted state, follow as Ut;N ¼ ðF ck;N  XN W 1k;N Þ1 ðPCN ðRt þ k;N  Dt þ k;N Þ þ C/^xðt þ kjtÞ 0  CN rdd ðt þ kÞ  K2N U  t;N  t;N Þ þ F ck;N U P

ð9:78Þ

Ut;N ¼ F 1 xðt þ kjtÞ þ XN W 1k;N Ut;N ck;N ðPCN ðRt þ k;N  Dt þ k;N Þ þ C/ ^ 2 0  CN rdd ðt þ kÞÞ þ U  t;N  KN Ut;N  P

ð9:79Þ

or

402

9 State-Space Nonlinear Predictive Optimal Control

Remarks on the solution • The control law in (9.78) or (9.79) includes an internal model for the nonlinear process. The control law is again to be implemented using a receding-horizon philosophy and it becomes equal to a GPC controller (9.43) in the limiting linear case when the control signal costing tends to zero (F ck;N ! 0, W 1k;N ¼ I). • In the unrealistic nonlinear case where the control weighting F ck;N ! 0 the vector of future controls will, from (9.78), introduce the inverse of the plant 0 will model W 1k;N (if one exists), and the resulting vector of future controls Ut;N then be the same as the GPC controls for the linear system that remains. • For integral wind-up protection, the same technique of exploiting the nonlinear control signal costing F ck may be used as in the NGMV design discussed in Chap. 5 (Sect. 5.3).

9.4.3

NPGMV Control Problem Results

The expression for the optimal control, in terms of the current state-estimate (rather than the predicted state), is also useful, and may be derived based on the assumptions at the start of the previous section. Substituting for the expression for the optimal predicted state in Eq. (9.23) the condition for optimality (9.76), may be obtained as   0  t;N PCN Rt þ k;N  Dt þ k;N  CN AN ðAk ^xðtjtÞ þ T0 ðk; z1 ÞBu0 ðtÞÞ þ K2N U    t;N ¼ 0  CN rdd ðt þ kÞ þ F ck;N  XN W 1k;N Ut;N  F ck;N U þP

ð9:80Þ

Recall u0 ðtÞ ¼ W 1k uðtÞ and write ~ 0t þ k;N ¼ Rt þ k;N  Dt þ k;N  CN AN Ak ^xðtjtÞ R

ð9:81Þ

Condition for optimality: An alternative form for the condition for optimality in terms of the current state follows from (9.77) and (9.80) as 0  CN rdd ðt þ kÞ ~ 0t þ k;N  C/ T0 ðk; z1 ÞBðW 1k uÞðtÞ þ K2N U  t;N þP PCN R  t;N þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0  F ck;N U

ð9:82Þ

Optimal control: The vector of future NPGMV optimal controls then becomes ~ 0t þ k;N þ C/ T0 ðk; z1 ÞBðW 1k uÞðtÞ Ut;N ¼ ðF ck;N  XN W 1k;N Þ1 ðPCN R 0  t;N  t;N Þ  CN rdd ðt þ kÞ  K2N U þ F ck;N U P

ð9:83Þ

9.4 Nonlinear Predictive GMV Control Problem

403

An alternative expression that is more useful for implementation follows from (9.82) as 1 ~0 Ut;N ¼ F 1 ck;N ððPCN Rt þ k;N  C/ T0 ðk; z ÞBðW 1k uÞðtÞÞ þ XN W 1k;N Ut;N 0  CN rdd ðt þ kÞÞ þ U  t;N  t;N  K2N U P

ð9:84Þ

The control at time t is computed for N > 0 from the vector of current and future controls using the matrix CI0 ¼ ½ I

0  ¼ ½I; 0; . . .; 0

ð9:85Þ

The control at time t can, therefore, be found as uðtÞ ¼ ½I; 0; . . .; 0Ut;N

ð9:86Þ

The dimensions are such that CI0 W 1k;N ¼ ½W 1k ; 0; . . .; 0: The vector of future NPGMV optimal controls (9.84) may now be written, in a form that provides a useful starting point for implementation, as 1 ~0 Ut;N ¼ F 1 ck;N ðPCN Rt þ k;N þ ðXN þ C/ T0 ðk; z ÞBCI0 ÞW 1k;N Ut;N 0  t;N  CN rdd ðt þ kÞÞ þ U  t;N  K2N U P

ð9:87Þ

Theorem 9.3: The NPGMV Optimal Control Law Consider the linear components of the plant, disturbance and output weighting models put in augmented state-equation form (9.2), (9.3), with input from the nonlinear finite gain stable plant dynamics W 1k . The multi-step predictive control cost-function to be minimized, with N > 0, involving a sum of future cost-function terms, is defined as 0 Jp ¼ EfU0T t þ k;N Ut þ k;N jtg

ð9:88Þ

where the signal U0t þ k;N depends upon future error, input and nonlinear control signal costing terms as follows:   0 0 0  CN rdd ðt þ kÞ þ F ck;N Ut;N  U  t;N  t;N U0t þ k;N ¼ PCN Et þ k;N þ FCN Ut;N þ K2N U þP ð9:89Þ The error and control input cost-function weightings are motivated by those in the GPC cost minimization problem (9.33). The weightings in (9.88) are defined as 0  T Ps B  T Ps . The control signal  and P  CN ¼ B PCN ¼ VNT , FCN ¼ K2N  B cost-function weighting F ck;N is also a block-diagonal control weighting matrix (9.68). The F ck and F ck;N are assumed full rank and invertible, and the assump t;N apply. tions listed on the term F ck;N U

404

9 State-Space Nonlinear Predictive Optimal Control

The NPGMV optimal control law to minimize, the criterion (9.88), in terms of the predicted state, is given as xðt þ kjtÞ þ XN W 1k;N Ut;N Ut;N ¼ F 1 ck;N ðPCN ðRt þ k;N  Dt þ k;N Þ þ C/ ^ 2 0  CN rdd ðt þ kÞÞ þ U  t;N  KN Ut;N  P

ð9:90Þ

 In terms of the current  T Ps B: where the constant matrix XN ¼ VNT VN þ K2N þ B state-estimate, the NPGMV control law may be expressed as 1 ~0 Ut;N ¼ F 1 ck;N ðPCN Rt þ k;N þ ðXN þ C/ T0 ðk; z ÞBCI0 ÞW 1k;N Ut;N

ð9:91Þ

0  t;N  CN rdd ðt þ kÞÞ þ U  t;N  K2N U P

~ 0t þ k;N ¼ PCN ðRt þ k;N  Dt þ k;N Þ  C/ Ak ^xðtjtÞ and C/ ¼ PCN CN AN , and where PCN R

T0 ðk; z1 Þ ¼ ðI  Ak zk ÞðzI  AÞ1 . The optimal control can be computed by invoking the receding-horizon principle from the first component in the vector of future controls Ut;N . ■

Solution The proof of the optimal control was given before the Theorem, which provides a summary of the results. The closed-loop stability analysis follows below. ■ Remarks (i) The two expressions for the NPGMV control signal (9.90) and (9.91) lead to two alternative structures for implementation of the nonlinear controller. They both include an algebraic-loop that can be avoided in implementation using similar steps to those described in Chap. 5 (Sect. 5.4). The second form of controller, based on the current state-estimate, is shown in Fig. 9.3. (ii) The controller involves a Kalman predictor stage and it is important to note that the order of the Kalman filter depends only on the delay free linear subsystems. Any channel delays do not, therefore, inflate the order of the estimator. NPGMV Controller Structure

Ut,N

Λ 2N U t0, N + P rdd (t + k ) CN

Rt + k , N − Dt + k , N

d

+

PCN

+ + +

Ut,N

+ +

−1

ck ,N

X N + Cφ T0 (k , z −1 ) BCI0

u

y

+ +

Output

U t0, N

Cφ Ak xˆ(t | t )

Noise v

Plant

Disturbance 1k , N

u0 (t − k )

Fig. 9.3 NPGMV feedback control using Kalman predictor

Observations signal

z

9.4 Nonlinear Predictive GMV Control Problem

405

(iii) If the output weighting Pc includes a near integrator, it appears both in the feedback and reference channels and it is desirable for implementation to move this integrator term into a common path that has the difference of signals. 0 (iv) Inspection of the form of the cost-function term (9.89) in the case when FCN , 0 0    PCN , Ut;N and Ut;N are null gives Ut þ k;N ¼ PCN Et þ k;N þ F ck;N Ut;N ¼ VNT Et þ k;N þ F ck;N Ut;N , and for a single stage cost VN ¼ Ep . In this case U0t þ k;N ¼ EpT Et þ k;N þ F ck;N Ut;N . The limiting case of the NPGMV controller is, therefore, related to a particular form of NGMV controller where the error weighting is scaled by the EpT term. This observation is useful when comparing behaviour and validating results in limiting cases.

9.4.4

Implementation of the Predictive Controller

The key to implementing the various MPC controllers, whether based on state-space or input–output models, is to take advantage of the simplifications that can be made in special cases. Some of the options are discussed later in Chap. 11. To compute the vector of future controls for t > 0 introduce the matrix C0I ¼ ½ 0 IN  and we obtain 2 f ¼ C0I Ut;N ¼ ½ 0 Ut;N

6 IN  4

uðtÞ .. .

uðt þ NÞ

3 uðt þ 1Þ 7 7 6 .. 5 5¼4 . 3

2

ð9:92Þ

uðt þ NÞ

1 1 From (9.85) note that CI0 F 1 ck;N ¼ ½F ck ; 0; . . .; 0 ¼ F ck CI0 . Optimal control: The optimal control at time t can now be computed, using (9.87), as 1 ~0 uðtÞ ¼ F 1 ck CI0 ðPCN Rt þ k;N þ ðXN þ C/ T0 ðk; z ÞBCI0 ÞW 1k;N Ut;N 0  t;N  CN rdd ðt þ kÞÞ þ uðtÞ  K2N U P

ð9:93Þ

The vector of future controls, computed at time t, may also be found as f 1 ~0 ¼ C0I F 1 Ut;N ck;N ðPCN Rt þ k;N þ ðXN þ C/ T0 ðk; z ÞBCI0 ÞW 1k;N Ut;N 0  CN rdd ðt þ kÞÞ þ C0I U  t;N  t;N  K2N U P

ð9:94Þ

where from (9.69), the vector W 1k;N Ut;N may be written (partitioning current and future terms) as

406

9 State-Space Nonlinear Predictive Optimal Control

ðW 1k;N Ut;N Þ ¼ ½ðW 1k uÞðtÞT ; . . .; ðW 1k uÞðt þ NÞT T ¼ ½ðW 1k uÞðtÞT ;

f T T ðW 1k;N1 Ut;N Þ 

ð9:95Þ

There are useful insights into the way the different Kalman filter blocks contribute to the NPGMV control law, obtained by considering an alternative expression for the optimal control. This follows from (9.23), (9.25) and (9.91), as follows:   k Ut;N ¼ F 1 xðtjtÞ ck;N ðPCN Rt þ k;N  Dt þ k;N þ C/ A ^   1 0  t;N  CN rdd ðt þ kÞÞ þ U  t;N P þ XN þ C/ T0 ðk; z ÞBCI0 W 1k;N Ut;N  K2N U k k ¼ F 1 ck;N ðPCN ðRt þ k;N  Dt þ k;N Þ þ C/ A T f 1 ðzðtÞ  dðtÞÞ þ C/ A T f 2 u0 ðtÞ   0  CN rdd ðt þ kÞÞ þ U  t;N  t;N P þ XN þ C/ T0 ðk; z1 ÞBCI0 W 1k;N Ut;N  K2N U

ð9:96Þ This form of implementation, which motivates the implementation of the controller, requires the two paths of the Kalman filter to be separated as shown in Fig. 9.4. The following observations may be useful concerning properties and implementation: • The optimal NPGMV controller includes an inner-loop, as shown in Fig. 9.3 or Fig. 9.4, giving rise to the so-called “algebraic-loop.” The steps that can be taken to avoid problems with the algebraic-loop in implementation are the same as discussed in Chap. 5. • The whole control sequence Ut;N is computed to obtain the optimal solution. • In the limiting case, when the nonlinear control weighting F ck tends to zero and the plant is linear, the control signal becomes equal to that of a GPC design. 0 ¼ K2N goes to • In the limiting case when the multi-step control weighting FCN T zero and the number of steps is unity (PCN ¼ Ep ) the controller reverts to a two degrees of freedom NGMV control design. This two degree of freedom NGMV solution may improve tracking accuracy in comparison with the one degree of freedom solution in some applications.

CN

-

z−d

+ u0

+ -

PCN

−

+ Cφ A T f 1 ( z ) + k

Noise v

Compensator structure

Λ 2N U t0, N + P rdd (t + k )

c

+

−1

Cφ Ak T f 2 ( z −1 )

N

+

φ 0

−1 ,

+ +

,

1k , N

u0

Fig. 9.4 Feedback control signal generation and controller modules

u

Plant

y

+ +

Output Disturbance

z

9.5 Systems Analysis and Stability

9.5

407

Systems Analysis and Stability

Useful insights into the stability of the NPGMV closed-loop designs and the relationship to cost-function weighting selection is explored in this section. The links to the Smith predictor structure are explored that also provides insights into the stability relationships. One of the merits of the NPGMV controller relative to the GPC solution is the highly intuitive block diagram structure available that is sometimes easier to justify than a black-box optimization algorithm.

9.5.1

System Stability

To investigate the stability of the closed-loop system an expression is required for the control and output signals in a closed-loop form. This analysis will reveal that a closed-loop operator representing a sensitivity function must have a stable inverse. This operator depends upon the choice of cost-function weightings, which implies that the cost-function weightings must be chosen to satisfy performance, stability and robustness requirements. An algebraic result is first required involving terms from the Kalman filter equations. From Eq. (9.28): C/ Ak T f 1 ðCUB þ EÞzk þ C/ Ak Tf 2 ¼ C/ Ak UBzk

ð9:97Þ

After substitution from (9.24), the desired expression follows as C/ T0 B þ C/ Ak T f 2 þ C/ Ak T f 1 zk W0k ¼ C/ UB

ð9:98Þ

where W0k ¼ E þ CUB: It may now be shown that a nonlinear operator must have a stable inverse for stability, where the measure of stability, such as “finite gain stability,” depends upon the assumption of stability on the nonlinear plant subsystem W 1k . Assume that the disturbance inputs and end-state weighting term are null. That is, the only input is due to the known future reference signal. The condition for ~ 0t þ k;N (from optimality in (9.82) may now be written using the expression for R (9.81)). Setting the aforementioned signals to zero, we obtain PCN ðRt þ k;N  CN AN Ak ^xðtjtÞÞ  C/ T0 BðW 1k uÞðtÞ þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0

408

9 State-Space Nonlinear Predictive Optimal Control

Noting the expression for the state-estimate (9.25) we obtain ðPCN Rt þ k;N  C/ Ak ðTf 1 ðzðtÞ  dðtÞÞ þ Tf 2 u0 ðtÞÞÞ  C/ T0 BðW 1k uÞðtÞ þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0 Also, note the expression for the observations signal (9.3) where in this case zðtÞ ! dðtÞ þ ðE þ CUBÞu0 ðt  kÞ, giving: ðPCN Rt þ k;N  C/ ðAk ðT f 1 W0k zk þ T f 2 Þ þ T0 BÞu0 ðtÞÞ þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0 By substituting from (9.98) we obtain ðPCN Rt þ k;N  C/ UBu0 ðtÞÞ þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0 The condition for optimality now follows as ðF ck;N  XN W 1k;N  C/ UBCI0 W 1k;N ÞUt;N þ PCN Rt þ k;N ¼ 0

ð9:99Þ

The desired expressions for the vectors of future optimal controls and the nonlinear plant outputs follow as Future optimal controls:  1 Ut;N ¼ ðXN þ C/ UBCI0 ÞW 1k;N  F ck;N PCN Rt þ k;N

ð9:100Þ

NL subsystem future outputs:  1 W 1k;N Ut;N ¼ W 1k;N ðXN þ C/ UBCI0 ÞW 1k;N  F ck;N PCN Rt þ k;N

ð9:101Þ

Total NL future plant outputs:  1 W k;N Ut;N ¼ W k;N ðXN þ C/ UBCI0 ÞW 1k;N  F ck;N PCN Rt þ k;N

ð9:102Þ

where from (9.69) the total future predicted plant block structure has the form ðW 1k;N Ut;N Þ ¼ diagfW 1k ; W 1k ; . . .; W 1k gUt;N ¼ ½ðW 1k uÞðtÞT ; . . .; ðW 1k uÞðt þ NÞT T The output signal involves this unstructured nonlinear input block, which is assumed stable, but the linear state-space output subsystem can be unstable. If it is assumed that the cost weightings are chosen so that the inverse of the operator ððXN þ C/ UBCI0 ÞW 1k;N  F ck;N Þ is finite gain m2 stable the expression for the

9.5 Systems Analysis and Stability

409

control signal (9.100) depends only on stable operators. The stability of the inverse of this operator is a necessary condition for stability. For a sufficient condition to be established further assumptions on the nature of the black-box input nonlinearity would be needed.

9.5.2

Cost-Function Weightings and Stability

It will be shown below that in a limiting case a set of cost weightings can be defined to guarantee the existence of the stable inverse of the operator ððXN þ C/ UBCI0 ÞW 1k;N  F ck;N Þ that appeared in the expressions for the control and output responses (9.100)–(9.102) in the previous section. As in Chap. 7 assume that a PID controller is known that will stabilize the nonlinear system (without transport-delay elements). Also assume that K2N ! 0 and that the control reference and endpoint weighting terms are null. Thus, only the error and control weighting terms are used and XN ¼ VNT VN þ K2N ! VNT VN . Then from (9.100)  1 Ut;N ¼ ðXN þ C/ UBCI0 ÞW 1k;N  F ck;N PCN Rt þ k;N  1 ! VNT ðVN þ CN AN UBCI0 ÞW 1k;N  F ck;N PCN Rt þ k;N For the single stage cost problem (where N = 0), the matrix VN ¼ Ep is square and non-singular. If the dynamic weighting is on the plant outputs yp ðtÞ ¼ Pc ðz1 ÞyðtÞ and Ep þ Cp UB ¼ Pc ðz1 ÞW0k . In this limiting case  1 uðtÞ ! EpT Pc W0k W 1k  F ck EpT rp ðtÞ

ð9:103Þ

One of the main problems in nonlinear control design is to obtain a stabilizing control law, and this depends on the selection of the cost-function weighting terms. A practical method of deriving such weightings is suggested by these results. The control weighting is normally assumed to be linear and hence denote F ck ¼ Fk . Then the term ðI þ Fk1 EpT Pc W0k W 1k Þ may be interpreted as the return-difference operator for a nonlinear system with delay free plant model W k ¼ W0k W 1k . If the plant has a PID controller that stabilizes this model, the ratio of weightings Fk1 EpT Pc can be chosen equal to this controller. As in previous chapters, the choice of the weightings to be equal to a PID control law is only suggested as a starting point for design. There are no guarantees when moving away from this special case but it provides a simple method of finding starting values of the weightings.

410

9 State-Space Nonlinear Predictive Optimal Control Compensator structure

Rt + k , N − Dt + k , N

d

+ -

+

-

Noise

Nonlinear plant

u

−1

v y

ck ,N

+

Output

+ +

z

1k , N

Kalman predictor

X N + Cφ ΦBCI 0

u0

Disturbance

z − kW0 k CI 0

p

-

+

Observations z

Fig. 9.5 Nonlinear smith predictor derived from NPGMV compensator structure

9.5.3

Relationship to the Smith Predictor

The NPGMV controller can be related to that of a Smith Predictor, but the use of this structure limits applications to open-loop stable systems. The controller structure, shown in Fig. 9.4, involves the predicted states from the Kalman Filter, with two different paths from the observations and control signal inputs. Changes can, therefore, be made to the linear subsystems by adding and subtracting equivalent terms. The three linear inner-loop blocks with the common input signal u0 ðtÞ ¼ ðW 1k uÞðtÞ can then be combined by using (9.98). The Smith Predictor style of controller that results is shown in Fig. 9.5. The path from the control signal u to the feedback signal p is null when the model zk W k ¼ zk W0k W 1k matches the plant model in Fig. 9.5. It follows that the control action, due to the reference-signal changes, is not due to the feedback but it involves the open-loop stable compensator, involving the block C/ Ak T f 1 ðz1 Þ, and the inner nonlinear feedback loop. This link is useful to explain the operation of the NPGMV algorithm but it is only, of course, suitable for stable open-loop systems.

9.6

Automotive Engine Control Example

A Spark Ignition (SI) engine is a combustion engine that is usually a petrol engine where the spark plug is used to ignite the air–fuel mixture. This is rather different to compression ignition based engines like diesel engines where the heat produced by compression together with the injection of fuel is all that is needed to initiate the ignition. There are increasing requirements in exhaust emission standards and fuel consumption, leading automotive companies to focus on the development of more advanced control laws. There is also an incentive to utilize engine models so that calibration effort is reduced and engine controls can be developed faster to cope with the increasing rate of new car developments.

9.6 Automotive Engine Control Example

411

The multivariable nature of the engine control problem is more challenging than in former years, because of the use of new actuator and measurement system configurations. For example, turbo chargers are now being applied extensively in spark ignition engines. The turbo uses the engine’s exhaust gas to drive a turbine, and this turbine drives a compressor that compresses the air entering the engine. The turbine’s variable geometry can be manipulated to optimize the engine’s behaviour, but it also makes the control problem more demanding. It is difficult to use advanced control in automotive systems, where the sampling periods are typically a few milliseconds and on-board computing power is limited. However, fortunately, the increase in processor speed and memory has enabled MPC to be applied in automotive applications [11, 12, 13]. An important task for an engine control system is to maintain the air–fuel ratio at the stoichiometric level k ¼ Air=Fuel ¼ aircharge=fuel charge Stoichiometric combustion defines the ideal conditions where fuel is burned completely and the ideal value of k is usually normalized to unity. There is a trade-off situation between the goals of good torque tracking and air–fuel ratio regulation. That is, fast transient responses and consequently, good torque performance, entail larger air–fuel ratio excursions.

9.6.1

Gasoline Engine Control

A petrol or SI-engine management system integrates the air and fuel paths, and the ignition controls. An Electronic Control Unit (ECU) uses sensors to monitor the engine parameters and it manages the throttle control, ignition timing and fuel injection. Calibration data is usually obtained via offline testing to generate maps covering engine load and speed, temperatures, and other parameters. Fuel and ignition, as well as cam timing settings, can be adjusted using these maps, to maximize performance from the engine. The torque of spark-ignition engines is primarily influenced by the throttle, controlling the mass airflow and the amount of air flowing into the combustion chamber. The engine management system continuously monitors the engine sensor outputs and calculates • The torque requested by the driver, through the accelerator pedal demands. • The air charge to be fed into the cylinders by the change in throttle angle actuator. • The injector actuator opening times to control the amount of fuel to provide a given air/fuel or stoichiometric mixture ratio, given the change in the air charge. • The ignition timing and possible variable valve timing and lift.

412

9 State-Space Nonlinear Predictive Optimal Control

Other variables influencing the variations of torque can include deactivation of the injection of given cylinders, boost pressure control for turbocharged engines, exhaust gas recirculation (EGR), and a variable manifold.

9.6.2

Air–Fuel Ratio Engine Predictive Control Design

The Nonlinear Predictive GMV (NPGMV) control design is illustrated using a nonlinear regression model of a spark-ignition engine. The engine diagram is shown in Fig. 9.6. The control problem involves regulating the air/fuel ratio (lambda, k) by adjusting the Fuel Pulse Width (FPW) input, subject to a varying Throttle Position Sensor (TPS) output (representing the current engine load), and varying engine speed in Revolutions per Minute (RPM). It can be. therefore. considered as a scalar control problem with measurable deterministic disturbances. Control Objective: The main objective in powertrain control is to respond to driver torque demands, whilst maintaining the Air/Fuel Ratio (AFR) at the nominal stoichiometric value, typically 14.7:1. Equivalently, the normalized ratio, denoted as k, should be kept near unity. Such a setting optimizes the combustion of fuel in the cylinders. It also makes the most of the catalytic converter located downstream of the exhaust manifold, which neutralizes harmful emissions, such as CO, NOx and hydrocarbons. Current emission level standards set the acceptable variations of k within 1–5% of the nominal value of unity. This requirement emphasizes the need for effective regulation of the variable. Typical throttle and engine speed data are shown in Fig. 9.7. NARX Model of the Engine: There exist very detailed and accurate models of Spark-Ignition (SI) engines developed from first principles and making use of mass, energy and momentum conservation laws, fluid dynamics equations, and some approximation of the combustion process. In this example, however, we use a simpler model, obtained by fitting a nonlinear regression model to experimental engine data. This model encapsulates the main features of the process and is adequate to illustrate the NPGMV control design solution. For the identification of Air-Fuel Ratio

Fuel Injector FPW (ms)

Intake Manifold

λ

Air MAF (g/s)

Throttle servo and setpoint SP (deg)

Catalyst

Exhaust Manifold

Valves

TPS (deg)

Exhaust MAP (kPa)

Tim (K)

Pexh (kPa)

Cylinder TQ (Nm) RPM

Fig. 9.6 Components of a spark-ignition engine

Texh (K)

9.6 Automotive Engine Control Example

413

TPS [V]

1.5

1

0.5

0

2

4

6

8

10

12 4

10

RPM [rpm]

3000

2000

1000

0

0

2

4

6

8

10

time (samples)

12 4

10

Fig. 9.7 Throttle input and engine speed data

linear systems, an autoregressive linear model (ARX model) may sometimes be used but for engine control, a nonlinear version of this class of algorithm was needed. The identified NARX model of the engine is defined to have one output, three inputs and eight states, and has a pseudo-linear state-space form xðt þ 1Þ ¼ AxðtÞ þ BuðtÞ yðtÞ ¼ CxðtÞ þ DuðtÞ

ð9:104Þ

where the system matrices are constant, but the input vector u(t) contains both linear and quadratic components  T uðtÞ ¼ TPSt ; FPWt ; Nt ; TPS2t ; FPWt2 ; Nt2

ð9:105Þ

The model, therefore, has a so-called Hammerstein model structure (Chap. 1). There are time delays in the fuel delivery path that are fixed, and in the lambda sensor path that are variable. For simplicity, these have been modelled by an average delay of k = 17 events. An additional constant input, representing a bias term, was added to the regression variables. Model Inputs and Output: To summarize, the model inputs and outputs are, respectively • • • •

Throttle position (Input TPS, in degrees) Fuel Pulse Width (Input FPW, in milliseconds) Engine Speed (Input N, in rpm) Air–Fuel Ratio (Output k, dimensionless and regulated to a magnitude of unity).

414

9 State-Space Nonlinear Predictive Optimal Control ⎡ TPSt ⎤ ⎢ RPM ⎥ t ⎥ utu = ⎢ ⎢ TPS 2 ⎥ t ⎢ ⎥ 2 ⎣⎢ RPM t ⎦⎥

Fig. 9.8 NARX model separation into the controlled and disturbance subsystems

Wu

⎡ FPWt ⎤ utc = ⎢ 2⎥ ⎣ FPWt ⎦

Wc

m

+

λ

+

Model Separation: The model needs to be separated into a nonlinear unstructured block and a linear state-space subsystem to make use of the predictive features of the NPGMV algorithm. This involves separating the control and disturbance components of the input vector u, as shown in Fig. 9.8. The inputs TPS and RPM are treated as measurable deterministic disturbances, and can be used to compute the feedforward control action. The control input signal uct , which includes the linear and quadratic FPW terms, requires special treatment. The approach utilized here was to obtain a SISO linear model by extracting a common factor from the 1  2 model Wc, as illustrated in Fig. 9.9. This linear model was used for the control design. NPGMV Design and Controller Structure: As with any classical predictive control algorithm, the NPGMV solution uses future reference information. Whilst ⎡ FPWt ⎤ utc = ⎢ 2⎥ ⎣ FPWt ⎦

m

Wc Wc = [WL

WNL ]

FPW

WL

(.)

2

+ +

WNL

m

WL = BL Bcom A−1 WNL = BNL Bcom A−1

FPW

BL (.)

2

BNL

+ +

u0

Bcom A−1 Linear model

Fig. 9.9 Common linear factor extraction from the NARX model control path

m

9.6 Automotive Engine Control Example

415

this is of no relevance for k regulation, where the setpoint remains constant, the predictive control still affects the disturbance rejection performance, and the dynamic responses. The controller robustness should also improve with increasing prediction horizon. The selection of the error and control signal dynamic weightings in this problem followed the general rules outlined in Chap. 5. The error weighting should contain an integrator to remove any steady-state lambda offset, whilst the control weighting may contain a lead term to penalize high-frequency control signal variations. The relative gain factors determine the trade-off between performance and stability/ robustness properties. The nominal error and control weights for the current design are chosen as Pc ðz1 Þ ¼ 13ð1  0:7685z1 Þ=ð1  0:999z1 Þ;

F ck ¼ 500

Note the positive sign of the control weighting, was motivated by the inverted system gain (the air–fuel ratio decreases with increasing fuel input). Figure 9.10 shows the internal structure of the NPGMV controller as implemented in a Simulink simulation. This may be compared with the more general block diagram of Fig. 9.4. In this example, the contributions from the deterministic disturbance inputs TPS and RPM were included in the estimator and in the controller subsystems. The signal dD is the model through term, which is removed from the Kalman filter input, and the signal distN contains the future disturbance values (assumed fixed at the current measurements). The block that contains the nonlinear part of the model is shown in the lower part of Fig. 9.9 (labelled Model_N), and is included in the block-diagonal operator. This subsystem represents the N-step “nonlinear prediction” into the future. Effect of Prediction Horizon: Simulation results. One of the properties of predictive control is that the robustness of the solution often improves with increasing prediction horizon. This is an intuitive property since the controller is able to “look” further into the future and predict the consequences of the control

Fig. 9.10 NPGMV controller structure for the engine control problem

416

9 State-Space Nonlinear Predictive Optimal Control 1.2

1.2 N=1 N=5 N=9

1.1

1.1

TPS (V)

lambda (-)

1.15

1.05

1 0.9

1 0.8

0.95 0.9 4000

4500

5000

5500

6000

0.7 4000

6500

4500

5000

5500

6000

6500

6000

6500

time (samples)

5

2000

4

1800

N (rpm)

CFC (mg)

time (samples)

3

1600

1400

2

1 4000

4500

5000

5500

6000

1200 4000

6500

4500

time (samples)

5000

5500

time (samples)

Fig. 9.11 NPGMV control of the air-fuel ratio (with N = 1, 5, 9)

Table 9.1 Integral square error for a varying prediction horizon N

1

5

9

ISE

10.21

3.98

2.59

action taken [14]. The receding-horizon control strategy accounts for uncertainties and disturbances in the system, by introducing the feedback control action. The improved robustness is apparent from the reduced overshoots in disturbance rejection responses. This is evident in Fig. 9.11, which shows a fragment of the driving cycle, and the control results for the prediction horizons of N = 1, 5 and 9. The rise time and settling time also reduce with the increasing horizon. The integral square error figures can confirm these conclusions. These were collected in Table 9.1 and computed for the full dataset.

9.7

Preview Control

The term preview control was used by Professor Masayoshi Tomizuka and co-workers at Berkeley [15, 16] to describe optimal systems that can use future setpoint or other information. The approach can be related to model-based

9.7 Preview Control

417

predictive control methods but they minimized a quadratic cost-function in an LQ or LQG sense. For linear systems, this resulted in controllers, which could use future reference information provided as a deterministic signal or modelled by a white noise signal driving a linear filter. The resulting theoretical solution was similar to the usual case for LQ state feedback or LQG state-estimate feedback control but including future reference or disturbance information. The relationship between preview control methods and MPC algorithms was explored by Ordys et al. [17]. To provide a brief introduction, assume for simplicity that the plant state model is given by (9.2) and the output to be controlled is given by (9.3) but with no explicit delay (k = 0) or through term (E = 0), and no deterministic disturbance (d(t) = 0). Also, assume the system is stabilizable and detectable. Reference-signal model: Future reference information is included in the preview control algorithms in a slightly different form to those in MPC problems. The model for the future reference is chosen to be deterministic within a preview horizon but is modelled as a random process thereafter. This latter assumption signifies that future reference knowledge is not available for more than the preview horizon of say N time steps. The approach is interesting because it enables the limiting case of infinite-horizon LQ optimal control problems to be considered, which is useful when determining stability properties. Denote the future reference as in (9.13), which is assumed known for N steps, as 2 6 6 Rt;N ¼ 6 4

rðtÞ rðt þ 1Þ .. .

3 7 7 7 5

ð9:106Þ

rðt þ N  1Þ The reference-signal is defined to be a known signal up to N, where N is referred to as the preview horizon. The reference can be defined, up to and beyond N, by a reference signal generator having the following reference state and output equations xR;N ðt þ 1Þ ¼ HN xR;N ðtÞ þ lN nr ðt þ NÞ

ð9:107Þ

rðtÞ ¼ CR xR;N ðtÞ

ð9:108Þ

where the vector xR;N ðtÞ includes the current and (N − 1) future values of the reference-signal, defining the signals in Rt;N from Eq. (9.106). The reference generator input matrix lN is a vector lN ¼ ½ 0   

0 1 T

where lN nr ðt þ NÞ ¼ nr ðtÞ and nt þ N is a Gaussian white noise signal of dimension ny. The reference-signal generator matrix HN is defined to specify the type of deterministic response required. For example, defining

418

9 State-Space Nonlinear Predictive Optimal Control

2

 . . .. 0  0

0 6 ... HN ¼ 6 4

I ..

0

3 0 .. 7 .7 I5

ð9:109Þ

I

ensures that the mean value of the reference-signal will remain constant after N steps, and letting 2

 . . .. 0  0

0 6 ... HN ¼ 6 4

I ..

0

3 0 .. 7 .7 I5

ð9:110Þ

0

ensures the mean value of the reference is set to zero after N steps. After the N steps into the future, the white noise introduces a stochastic component, which makes the reference trajectory uncertain. Thus, (9.109) provides a constant reference that will vary outside the interval due to the white noise driving signal, whilst (9.110) will provide a reference that becomes zero-mean white noise after N steps. At any given time t an arbitrary reference can be defined by setting the initial conditions in (9.107). This determines the response in the preview horizon. Augmented system: The combined plant and reference model has an extended state-vector of the form  vðtÞ ¼

xðtÞ xR;N ðtÞ



The state-equation, in terms of the extended state, may be written as  vðt þ 1Þ ¼

A

0

vðtÞ þ

 B

 uðtÞ þ

0 HN 0    ¼ AX ðtÞ þ BuðtÞ þ G1ðtÞ

D

0

0

lN



nðtÞ



nðt þ NÞ ð9:111Þ

The combined output and error equations 

yðtÞ rðtÞ



 ¼

C

0

0

CR



xðtÞ xR;N ðtÞ



eðtÞ ¼ rðtÞ  yðtÞ ¼ CR xR;N ðtÞ  CxðtÞ ¼ ½ C  ðtÞ ¼ CX

 xðtÞ CR  xR;N ðtÞ ð9:112Þ

9.7 Preview Control

9.7.1

419

Preview Cost-Function and Solution

The main difference between a preview type of LQ controller and an MPC solution lies in the form of the cost-function. The preview class of cost-functions appear similar to those in MPC, since they can include the same signals. However, the stochastic version of the preview control problem depends upon a cost-function employing an unconditional expectation. It is, therefore, related to the usual LQG control problem, which require Riccati equations to be solved. This is different to the MPC criteria, which involve conditional cost-functions, and where the solution is obtained by invoking the receding-horizon principle. For MPC there are only simple matrix manipulations (or quadratic programming in the constrained case). The definition of cost-functions was discussed in Chap. 2 (Sect. 2.3). The quadratic performance criterion to minimize to obtain the preview controller can be defined as follows: X M 1 Jt ¼ E fðyðt þ j þ 1Þ  rðt þ j þ 1ÞÞT ke ðyðt þ j þ 1Þ  rðt þ j þ 1ÞÞ M þ 1 j¼0

M X T þ uðt þ jÞ ku uðt þ jÞg ð9:113Þ j¼0

where the upper limit M of the summation is different to the preview horizon N. The design guidelines [17] suggest that the cost-horizon M can normally be much greater than N (the integer M may approach infinity). The preview time step N is normally chosen to be about 3 times the dominant time-constant of the closed-loop system. Thus, in a typical application the preview horizon N can be chosen from knowledge of the slowest closed-loop eigenvalue. The cost-function weightings ke  0 and ku [ 0 provide the tuning variables, and the performance-criterion (9.113) can written as ( ( )) M X 1 T T T  fX ðt þ j þ 1Þ C ke CX ðt þ j þ 1Þ þ uðt þ jÞ ku uðt þ jÞg Jt ¼ E M þ 1 j¼0 This is now in the form of a standard LQG control problem and the solution has the form  T   vðt þ jÞ  H  t þ j þ 1B  þ Ku 1 B T H  t þ j þ 1 A^ uðt þ jÞ ¼  B

ð9:114Þ

 k is determined by the following matrix Riccati difference equation that is where H solved backward in time  T  T H A T H   þA  T ke C k ¼ C  k þ 1A  k þ 1B  B  H  k þ 1B  þ ku 1 B T H  k þ 1A H  If M tends to  T ke C: t þ M þ 1 ¼ C and the final value equals the end-state term H infinity, the algebraic Riccati equation provides a solution.

420

9 State-Space Nonlinear Predictive Optimal Control

The preview control problem considered here is not as general as considered in the MPC solutions either above or in the following chapters. However, it does illustrate how future information can be included. Moreover, it provides the link to a further predictive control algorithm referred to as Linear Quadratic Gaussian Predictive Control (LQGPC). This can be related to MPC and LQG solutions in limiting cases [18] and it can be used with state-dependent or qLPV models for use in nonlinear and parameter varying systems. These algorithms are computationally more intensive but they may provide better stability and robustness than conventional MPC. This is an approach that deserves more attention, as discussed in Ordys et al. [17]. In more recent work Hazell produced a Ph.D. thesis on preview control [19], which minimizes H∞ or H2 cost-functions. He also developed a MATLAB Preview Control Toolbox with examples. The results are mainly for linear time-invariant systems. A nonlinear autonomous vehicles control system was designed, using a parameterized set of linear systems to represent the nonlinearity.

9.7.2

Merits of Preview and Possible Applications

The relative merits of the LQ based preview algorithms and the MPC algorithms may be summarized as follows: Advantages of preview based algorithms 1. Improved robustness and stability properties expected. 2. The criterion has a more physically realistic and meaningful value that can be used for benchmarking. 3. Lower cost in variance terms due to the time-averaged dynamic optimization. Disadvantages of the preview approach 1. Algorithms may be more intensive computationally than MPC. 2. Riccatti equations may be more daunting to classically trained engineers than simple matrix manipulations. 3. Theory and software has not reached the same level of development or generalization as MPC. Type of applications where preview might find a role 1. Problems where the variance related aspects of the cost-function are important. 2. Situations where the potential improvements in robustness and stability properties outweigh the increased computational burden. There are a number of applications for preview control such as robotic manipulator trajectory following and wind turbine control. Quite a number of applications have been considered for the automotive industry such as suspension systems to improve passenger comfort, guidance systems for autonomous vehicles; including

9.7 Preview Control

421

trajectory tracking for lane-keeping and changing. There are also examples of preview control to provide aid to the driver that can be classified under the heading of advanced driver assistance systems. Researchers working with Toyota and Nissan have described studies for path tracking and active suspension systems using preview control, respectively [20–22]. Tomizuka [23] also considered the use of preview in vehicle suspension systems. Cole et al. [24] compared both predictive control and linear quadratic optimal control theory to design path-following controllers for steering control with preview. It was shown that using their problem construction that for long preview lengths and control horizons, the predictive and LQ approaches gave identical results. Other automotive applications have included intelligent braking systems and adaptive cruise control.

9.8

Concluding Remarks

The aim here was to introduce a nonlinear predictive control law that is simple to implement in industry. It was also required to relate to some well-known and accepted control design approaches in various limiting cases. The initial model predictive control method introduced was a Generalized Predictive Control (GPC) algorithm for a linear plant model in state-equation form and a reasonably general cost-index. This was of interest since it is representative of the most popular MPC solution. It was also needed to motivate the nonlinear predictive control algorithm described in the second part of the chapter. The second predictive algorithm was termed Nonlinear Predictive Generalized Minimum Variance (NPGMV) control. It is a development of the Nonlinear Generalized Minimum Variance (NGMV) design method for state-space based models, which is easy to design and to implement. The NPGMV controller has the property that if the system is linear and the control weighting F ck tends to zero, the solution reverts to the GPC solution, which is a generalization of some common MPC algorithms. The dynamic weighting functions, control reference-signal and the end-state weighting terms in the GPC and NPGMV cost-functions are useful generalizations of the basic algorithms. If the predictive control cost-horizon is a single stage, then the NPGMV control law reverts to the NGMV control solution. This may also be a useful starting point for the design. It is well-known that as the horizon in a predictive controller, increases the responses normally become more damped and the solution is more robust. It, therefore, seems a practical step to obtain the limiting single stage form of NGMV controller, where a stabilizing set of weightings can easily be achieved, but where the system frequency and time responses may not be as good as for predictive control. The number of time steps in the predictive control horizon can then be increased until no further improvement is obtained. Clearly, if the responses are not improving sufficiently there is no need to increase the computations by increasing the horizon further.

422

9 State-Space Nonlinear Predictive Optimal Control

The NPGMV control law includes an internal model of the process but many of the computations, as in traditional predictive control, simply involve matrix multiplication. The controller can be implemented like a nonlinear version of a Smith-Predictor. However, in this form, the compensator cannot be used to stabilize an open-loop unstable process. The relationship to this well-known transport delay compensator provides confidence in the NPGMV solution. There are many applications of MPC and the latest trend is to use predictive controls in high-speed machinery control systems, like automotive and aerospace applications [25]. The MPC approach provides an attractive control design method for engine calibrators since the cost-function can be designed so that the cost-function weights (calibration parameters) correspond directly to the performance goals [26]. For a useful review of papers on MPC see Morari and Lee [6], Bemporad and Morari [5], Qin and Badgwell [27]. The robustness and stability of predictive controls are important for applications and these issues have been researched extensively [28–31].

References 1. Ordys AW, Clarke DW (1993) A state-space description for GPC controllers. Int J Syst Sci 24 (9):1727–1744 2. Ordys AW, Pike AW (1998) State-space generalized predictive control incorporating direct through terms. In: 37th IEEE control and decision conference. Tampa, Florida 3. Grimble MJ, Majecki P (2008) Nonlinear predictive GMV control. In: American control conference, Westin seattle hotel, Seattle, Washington 4. Grimble MJ, Majecki P (2010) State-space approach to nonlinear predictive generalized minimum variance control. Int J Control 83(8):1529–1547 5. Bemporad A, Morari M (2004) Robust model predictive control: a survey. In: Proceedings of European control conference, Porto, Portugal 6. Morari M, Lee J (1999) Model predictive control, past and future, in computer and chemical. Engineering 23:667–682 7. Kwon WH, Pearson AE (1977) A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Trans Autom Control 22(5):838–842 8. Mayne DQ, Michalska H (1990) Receding horizon control of nonlinear systems. IEEE Trans Autom Control 35:814–824 9. Marinescu B, Bourles H (1997) Robust predictive control for multi-input/output control systems with non-equal time delays. In: European control conference, Brussels 10. Lee YI, Kouvaritakis B, Cannon M (2003) Constrained receding horizon predictive control for nonlinear systems. Automatica 38(12):2093–2102 11. Hrovat DD, Di Cairano S, Tseng HE, Kolmanovsky IV (2012) The development of model predictive control in automotive industry: a survey. In: IEEE international conference on control applications. Dubrovnik, Croatia 12. Majecki P, van der Molen GM, Grimble MJ, Haskara I, Hu Y, Chang C-F (2015) Real-time predictive control for SI engines using linear parameter-varying models. In: 5th IFAC conference on nonlinear model predictive control, Seville, 2015 and IFAC-Papers On-Line, vol 48, no 23, pp 94–101 13. Majecki P, Grimble MJ, Haskara I, Hu Y, Chang CF (2017) Total engine optimization and control for SI engines using linear parameter-Varying Models. In: American control conference. Seattle, WA

References

423

14. Pike AW, Grimble MJ, Shakhour S, Ordys AW (1995) Predictive control, in the control handbook. In: Levine WS (ed) CRC Press Inc 15. Tomizuka M, Whitney DE (1975) Optimal discrete finite preview problems (why and how is future information important?). ASME J Dyn Syst Meas Control 97:319–325 16. Tomizuka M, Rosenthal DE (1979) On the optimal digital state vector feedback controller with integral and preview actions. Trans ASME 101:172–178 17. Ordys AW, Tomizuka M, Grimble MJ (2007) State-space dynamic performance preview predictive controller. ASME J Dyn Syst Meas Control 129(2):144–153 18. Grimble MJ, Ordys AW (2001) Non-linear predictive control for manufacturing and robotic applications. In: IEEE conference on methods and models in automation and robotics. Poland, pp 579–593 19. Hazell A (2008) Discrete-time optimal preview control, Ph.D. thesis, Imperial college, London 20. Boyali A, John V, Lyu Z, Swarn R, Mita S (2017) Self-scheduling robust preview controllers for path tracking and autonomous vehicles. In: Asian control conference 21. Tobata H, Fukuyama K, Kimura T, Fukushima N (1992) Advanced control methods of active suspension. International symposium on advanced vehicle control, Yokohama, Japan, pp 136–141 22. Tobata H, Fukuyama K, Kimura T, Fukushima N (1993) Advanced control methods of active suspension. Veh Syst Dyn 22(5):347–358 23. Tomizuka M (1976) Optimum linear preview control with application to vehicle suspension revisited. Trans ASME, J Dyn Syst, Meas Control 98(3):309–315 24. Cole DJ, Pick AJ, Odhams AMC (2006) Predictive and linear quadratic methods for potential application to modelling driver steering control. Veh Syst Dyn 44(3):259–284 25. Youssef A, Grimble MJ, Ordys A, Dutka A, Anderson D (2003) Robust nonlinear predictive flight control. In: European control conference. Cambridge, pp 1–4 26. Vermillion C, Butts K, Reidy K (2010) Model predictive engine torque control with real-time driver-in-the-loop simulation results. In: American control conference. Baltimore 27. Qin S, Badgwell T (2000) An overview of nonlinear model predictive control applications. In: Allgower F, Zheng A (eds) Nonlinear predictive control, Birkhauser, pp 369–393 28. Rossiter JA, Kouvaritakis B (1994) Numerical robustness and efficiency of generalised predictive control algorithms with guaranteed stability. Proc IEE, Pt-D 141:154–162 29. Rossiter JA, Rice MJ, Kouvaritakis B (1998) A numerically robust state-space approach to stable predictive control strategies. Automatica 34:65–73 30. Rossiter JA, Gossner JR, Kouvaritakis B (1996) Infinite horizon stable predictive control. IEEE Trans Autom Control 41:1522–1577 31. Kouvaritakis B, Rossiter JA, Chang AOT (1992) Stable generalised predictive control: an algorithm with guaranteed stability. IEE Proc, Pt D, vol 139, pp 349–362

Chapter 10

LPV and State-Dependent Nonlinear Optimal Control

Abstract In this chapter, a more general modelling paradigm is introduced. That is, the linear plant subsystem used previously is replaced by a linear parameter varying or a state-dependent state-space model. Taken together with a black-box operator subsystem very general nonlinear systems may be considered. The nonlinear generalized minimum variance controller provides an obvious starting point because of its simplicity, which is valuable when the problem becomes more complex. The most important message from this chapter is that the basic solution procedure is conceptually as simple as for the linear state-space system case, even if there are subtle differences in stability analysis and implementation. The automotive engine control example at the end of the chapter illustrates the value of the control approach and also considers the plant modelling and system identification problem.

10.1

Introduction

A Nonlinear Generalized Minimum Variance (NGMV) control law is derived in this chapter for systems represented by the combination of an unstructured input dynamic nonlinearity and a state-equation based output subsystem. The model of the plant used in this chapter is more general than in the two previous state-equation based chapters, where the state model was assumed linear time-invariant. The state model used here and introduced in Sect. 10.2 may represent a State-Dependent (SD), Linear Parameter Varying (LPV), Piecewise Affine (PA) or input-dependent system. As mentioned in Chap. 1, such systems are often referred to as quasi-LPV (qLPV) systems. The multivariable discrete-time qLPV model representing an output subsystem may be open-loop unstable. This is driven from an unstructured nonlinear input subsystem that can include an operator of a very general nonlinear form (black-box subsystem). The introduction of a quasi Linear Parameter Varying (qLPV) model for the output subsystem, that may be open-loop unstable, is needed for applications such as the continuous stirred tank reactor considered in Chap. 14 (Sect. 14.5).

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_10

425

426

10

LPV and State-Dependent Nonlinear Optimal Control

The LPV and qLPV system models have demonstrated their potential in aerospace, automotive, wind-turbine and marine control problems, and there are potential applications across the process industries. Changes in a process plant control loop can come about because of physical changes in the process itself or because of changes in the operating-point that introduce different nonlinear characteristics. To manage these nonlinear terms in a “classical control” approach may require configuration changes in the control scheme. For example, in classical control, the tuning parameters might be modified as a function of a process variable, such as the feed rate. An output or process variable might alternatively be fed back through a suitable static-function to enable a PID controller to manage a nonlinear process. Such a bespoke classical solution can provide a very effective and practical solution. However, to reduce design effort an optimal approach offers a standardized design procedure that also has the advantage that it applies to a range of complex multivariable and stochastic systems. The family of NGMV optimal controllers was introduced in the previous chapters for nonlinear model-based multivariable systems where the plant model could be decomposed into a linear subsystem and an input nonlinear subsystem. The advance in this chapter involves the introduction of a more general model structure, where the nonlinearities may be associated with either the inputs or outputs. The objective is to introduce the Nonlinear Generalized Minimum Variance (NGMV) controller for this more general plant model, where the criterion is chosen so that a relatively simple controller structure is obtained. It is a useful property that when the system is linear, the controller will become equal to the Generalized Minimum Variance (GMV) controller described in Chap. 2 that is now accepted as being useful in applications. Like other NGMV controllers, the solution can be manipulated into a similar form to the Smith Predictor for systems with explicit transport-delays. However, it has the advantage that in its basic form (before transforming into the Smith structure) it can stabilize open-loop unstable plants. There are several interesting generalizations in the solution obtained in this chapter. The outputs being measured can be different from those that are controlled. A two degrees of freedom structure is also used so that some limited future reference information may be incorporated. The total system model also includes both stochastic and deterministic reference and disturbance signal models. As in the previous two chapters the states of the augmented qLPV system must be estimated but in this case, a time-varying Kalman filter, rather than a constant gain Kalman filter, is required. A Kalman predictor stage is also needed in the implementation of the controller so that the predicted values of the weighted error and control signal to be minimized ð/0 ðtÞÞ may be obtained. The so-called state-dependent Riccati equation method is described briefly in the following section since this was one of the earliest feedback control solutions to systems represented in qLPV form.

10.1

Introduction

427

10.1.1 State-Dependent Riccati Equation Approach The use of a qLPV or state-dependent model for representing or approximating the behaviour of a nonlinear system was discussed in Chap. 1 (Sect. 1.7). The use of a qLPV model to generate a NGMV optimal controller is presented below. However, before considering this problem and solution a brief description will be given for one of the most popular methods for controlling such systems, namely the statedependent Riccati equation approach. The optimal control of nonlinear systems using a state-dependent approach was described in the work by Cloutier et al. [1] assessed on real applications. The first stage of the technique involved the use of a direct parameterization to bring the nonlinear system to a linear structure that had state-dependent coefficients. A simple Linear Quadratic (LQ), control strategy was then followed. This was called the state-dependent Riccati equation (SDRE) method, or alternatively the frozen Riccati equation (FRE) method (see [2–5]). The state-dependent Riccati equation optimal control approach involved a Linear Quadratic (LQ) optimal control problem with an infinite-time cost-horizon. The assumption was made in computing the current control action that the state-dependent matrices would remain constant from the current time. This is, of course, a gross approximation and an unrealistic assumption for nonlinear applications, where state-dependent plant models will change with the operating point. However, given this rather extreme assumption, the LQ optimal control law can be found simply by solving a steady-state or algebraic Riccati equation for the state-dependent system. For this computation, the model is assumed frozen in time from the current time t to 1. The optimal control law, therefore, requires the online solution of an algebraic Riccati equation, since at each time instant the model will change and the infinite-time LQ problem is solved assuming the model is linear and time-invariant (with state-dependent matrices frozen). The time-varying state-equation matrices depend upon the system states. It is an approximation to assume that these matrices are constant and to use the solution of the infinite-time LQ problem to compute the optimal gains. However, the time-varying nature is to some extent accounted for since the computation occurs at each sample instant. There are, therefore, parallels with the receding horizon principle used in predictive controls (Chaps. 7 and 9). The solution of the LQ optimal control problem does, of course, involve the solution of the steady-state or algebraic Riccati equation. The resulting State-Dependent Riccati Equation (SDRE) design method is a relatively simple idea but in its basic form, it requires the solution of the algebraic Riccati equation at each time instant. There are various ways the SDRE solution can be simplified for implementation (scheduling according to the nonlinear operating point is an obvious possibility). The early application trials on nonlinear systems were very promising. The technique has had some success in missile control applications [6]. The approach has also been shown to have the potential for industrial applications like hot strip mill controls (Chap. 14) and tandem cold mills [7]. The method is empirical even though considerable effort has been expended on trying to provide a full theoretical

428

10

LPV and State-Dependent Nonlinear Optimal Control

justification. Useful stability results were difficult to establish, but the exponential stability region for the so-called state-dependent Riccati equation based controllers has been established by Chang and Chung [8]. A useful survey of SDRE methods was presented by Cimen [9], who noted that issues still remained in establishing global asymptotic stability properties.

10.2

Nonlinear Operator and qLPV Plant Model

The plant model can be grossly nonlinear, dynamic and may have a very general structure but the output subsystem and disturbance model is assumed to be represented by a qLPV subsystem, which as mentioned includes state-dependent systems. This output subsystem can easily be specialized to a linear time-invariant model if required. The two degrees of freedom feedback system is shown in Fig. 10.1. It includes the nonlinear plant model and the reference and disturbance signals. The first plant subsystem is of a nonlinear operator form W 1 . Nonlinear input subsystem plant model: ðW 1 uÞðtÞ ¼ zk ðW 1k uÞðtÞ

ð10:1Þ

The zk I term denotes a diagonal matrix of the delay elements in the signal paths. Without delay terms the subsystem in Fig. 10.1 is denoted by W 1k . The output of this subsystem is denoted u0 ðtÞ ¼ ðW 1k uÞðtÞ, where W 1k is assumed to be finite-gain stable. The subscript k signifies that explicit delays have been removed from this unstructured nonlinear subsystem model. The input–output operator W 0 will denote the path from the input to the second subsystem to measured output. It will be assumed to have a qLPV form that will be denoted W 0k (without the k-steps delay). Disturbances

d 0d

0

rm

r0

vr

Nonlinear controller

C0

Nonlinear operator subsystem 1k

0 qLPV subsystem

Measurement or observations signal zm

d

yd Controlled output

y

+

u0

u

d0

+ +

ym

+ d 0m

+ +

vm Noise

dm

ydm

Fig. 10.1 Two-degrees of freedom feedback control system (nonlinear plant with operator and qLPV subsystems)

10.2

Nonlinear Operator and qLPV Plant Model

429

Nonlinear output subsystem plant model:
 ðW 0 u0 ÞðtÞ ¼ W 0k zk u0 ðtÞ

ð10:2Þ

The output of the unstructured subsystem W 1k acts as a delayed input to the qLPV block W 0k , and is denoted u0 ðt  kÞ: The total forward path to the system output that is measured ym ðtÞ can be written as
 ðW uÞðtÞ ¼ ðW 0k W 1 uÞðtÞ ¼ W 0k zk W 1k u ðtÞ

ð10:3Þ

The state-space form of the output subsystem W 0k is introduced in more detail below. This output subsystem qLPV model W 0k may be unstable.

10.2.1 Signal Definitions In some previous chapters, the outputs to be controlled were the same as the measured outputs but in this chapter, these signals will be assumed to be different. The output of the system to be controlled y(t) in Fig. 10.1 includes deterministic d(t) and stochastic yd ðtÞ components of the disturbance; the total disturbance d0 ðtÞ ¼ dðtÞ þ yd ðtÞ: The measured output ym ðtÞ, which may be different to y(t), also includes deterministic dm ðtÞ and stochastic ydm ðtÞ components of the disturbance signal, and the total disturbance d0m ðtÞ ¼ dm ðtÞ þ ydm ðtÞ: The zero-mean white measurement noise on the measured output is denoted vm ðtÞ and has a covariance matrix Rm : The plant model includes the driving white process noise f0 ðtÞ, to represent an input disturbance. The controlled output y(t) is required to follow a reference signal r0 ðtÞ, which is assumed to consist of a stochastic component rðtÞ and a known deterministic setpoint signal component rd ðtÞ: The total reference signal r0 ðtÞ ¼ rðtÞ þ rd ðtÞ may be corrupted by a zero-mean measurement noise vr ðtÞ that has a covariance matrix Rr . It will be assumed that the reference signal is known p steps into the future, where p  0: In most applications, the noise term vr ðtÞ will be null, but in some tracking problems, the target trajectory is not known precisely. The zero-mean, white noise signals f0 ðtÞ, xd ðtÞ and xr0 ðtÞ that feed the plant, disturbance and reference models, have the covariance matrices Q0, Qd and Qr0, respectively. The system structure is different to the one degree of freedom control approach in Chap. 8 (involving only a stochastic description of the reference signal), where the Kalman filter was driven by the noisy error, rather than the separate reference and observation signals used in the two degrees of freedom problem considered here. Signals: The signals may be listed as follows: x0 ðtÞ

Vector of n states in the qLPV plant subsystem and the input disturbance model.

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10

LPV and State-Dependent Nonlinear Optimal Control

u0 ðtÞ uðtÞ yðtÞ dðtÞ

Vector of m0 input signals to the qLPV subsystem. Vector of m control signals applied to the nonlinear subsystem. Vector of r plant output signals to be controlled. Vector of r deterministic disturbance signal values on the output to be controlled. yd ðtÞ Vector of r stochastic disturbance signal values on the output to be controlled. ym ðtÞ Vector of rm plant output signals that are measured. dm ðtÞ Vector of rm deterministic disturbance signal values on the measured output. ydm ðtÞ Vector of rm stochastic disturbance signal values on the measured output. zm ðtÞ Vector of rm observations on measured plant outputs. r0 ðtÞ Vector of r setpoint or reference signal values. d0 ðtÞ Vector of r known output disturbance signal values. d0d ðtÞ Vector of q known input disturbance signal values.

10.2.2 Quasi-LPV Model Dynamics The so-called state-dependent state-space model form has been used very successfully with state-dependent Riccati equation (SDRE) optimal control solutions [4, 5], and this helps to motivate the use of this type of structure for the output subsystem. As described in Chap. 1, the term quasi-LPV can be used to cover the various models that are similar to state-dependent including Piecewise Affine (PWA), Linear Parameter Varying (LPV) and even a class of hybrid-system models. The acronym qLPV is used here to cover all these various models but note that this terminology is not adopted universally. This generalization to qLPV models does not add much to the complexity of the problem, since it is similar to assuming the output subsystem state-equation matrices are linear and time-varying. Looking deeper into the models there is an important difference between state-dependent models that can represent a nonlinear system and LPV models that are for the linear state-equations. However, superficially there is not a great difference to the solution of the optimal control problems. There are more profound differences when considering questions of stability. The advantages of the qLPV modelling approach may be listed as follows: 1. Engineers have an intuitive understanding of the control of linear systems and the linear structure of qLPV provides confidence that they can use a similar design and tuning procedures. 2. Once a qLPV model is obtained there are many options for linear control laws that might be applied and these are generally better understood than say black-box nonlinear optimization algorithms.

10.2

Nonlinear Operator and qLPV Plant Model

431

3. A lot of effort has been invested in using MPC with qLPV models in different applications and results have been very promising. The main disadvantage with the use of the qLPV modelling approach is that such models are not unique. The choice of structure for the model is not obvious unless it follows directly from the physical plant equations. When the model is only an approximation there may not be a close link with the physical plant structure. It is also unfortunate that the choice of qLPV model can lead to very different robustness characteristics for a feedback system. The best choice of qLPV model seems to be when it is close to the physical system plant model, and scaled appropriately. The system to be controlled includes the qLPV model for the output subsystem of the plant and the models for the disturbance, reference and cost-function error weighting subsystems. The resulting augmented qLPV subsystem model is similar to a time-varying linear system and is assumed to be pointwise stabilizable and detectable. The input signal channels in the qLPV plant model are assumed to include a common explicit k-steps transport delay. The block diagram of the combined qLPV system is shown in Fig. 10.2. For further discussion on qLPV models see Chap. 1, Sect. 1.7.3. Plant Model: The output subsystem plant matrices can be functions of signals in this case, including the model states, the input at time (t – k), or a vector of parameters. This vector denoted p0(t) can represent parameter variations due to known signals such as measured disturbances. The nonlinear plant output subsystem model therefore has the following qLPV model form x0 ðt þ 1Þ ¼ A0 ðx0 ; u0 ; p0 Þx0 ðtÞ þ B0 ðx0 ; u0 ; p0 Þ u0 ðt  kÞ

ð10:4Þ

þ D0 ðx0 ; u0 ; p0 Þf0 ðtÞ þ G0 ðx0 ; u0 ; p0 Þd0d ðtÞ

The controlled output and the noise free measured output have the output equations

State-dependent plant dynamics

+

0

u0 1k (.,.)

u(t)

(.,.)

0m

+

+ +

z -k

0

uc

x0

z

+

ym + +

+

1

0

+

+

0

u0 (t k )

d0 +

+

r0

r0

Reference model

+

z

1

r0

y

zm Error weighting subsystem

e

_

+

+ +

r0 +

0

r0

Measured output

Controlled output

0

NL control weighting

vm

k)

+

0

Delay k c

0m u 0 (t

0

NL subsystem dynamics

Control

Noise

d0m

d0d

r

e p (t )

rm Measured reference

vr

+

rd

Fig. 10.2 Nonlinear unstructured and state-dependent/qLPV plant model (including disturbance and reference models u0 ðtÞ ¼ W 1k ð:; :ÞuðtÞ

432

10

LPV and State-Dependent Nonlinear Optimal Control

yðtÞ ¼ d0 ðtÞ þ C0 ðx0 ; u0 ; p0 Þx0 ðtÞ þ E 0 ðx0 ; u0 ; p0 Þ u0 ðt  kÞ

ð10:5Þ

ym ðtÞ ¼ d0m ðtÞ þ C0m ðx0 ; u0 ; p0 Þ x0 ðtÞ þ E 0m ðx0 ; u0 ; p0 Þ u0 ðt  kÞ

ð10:6Þ

where x0 ðtÞ is a vector of subsystem states, u0 ðtÞ is a vector of the subsystem inputs, p0 ðtÞ is a vector of parameters and yðtÞ is a vector of possibly unmeasured output signals. The deterministic component of the input disturbance is denoted d0d ðtÞ: Notation: The qLPV model is a function of plant states, inputs and the parameters ðx0 ðtÞ; u0 ðt  kÞ; p0 ðtÞÞ: To simplify the notation in Eqs. (10.4), (10.5) write A0 ðtÞ ¼ A0 ðx0 ðtÞ; u0 ðt  kÞ; p0 ðtÞÞ, and similarly for the matrices B0 ; C0 and E 0 . Explicit delay: The explicit transport delay k is often assumed greater than zero and in this case even if E 0 is a non-zero full rank term, the input cannot affect the output without at least a unit-step delay. Note that for many real systems the state-dependent/qLPV model is in a form where the system has no through term E 0 . This can be treated as a special case since it is assumed full rank here. However, treating the different possible cases complicates the results and if this case arises, it may be possible to assume that E 0 is just full rank and consider the limiting case where E 0 ¼ eI and the scalar e [ 0 is small. The results will be suboptimal but this approximation may not be significant from a practical viewpoint. Disturbance on output: The stochastic output disturbance model, driven by zero-mean unity-variance white noise xd ðtÞ, may be implemented in state-equation form as follows: xd ðt þ 1Þ ¼ Ad xd ðtÞ þ Dd xd ðtÞ;

xd ðtÞ 2 Rnd

ð10:7Þ

yd ðtÞ ¼ Cd xd ðtÞ

ð10:8Þ

ydm ðtÞ ¼ Cdm xd ðtÞ

ð10:9Þ

The two output signals represent the two potential stochastic disturbance terms added to the controlled plant output and the measured plant outputs, respectively.

10.2.3 Tracking Control and Reference Generation The optimal control solution will apply to the tracking of either stochastic or deterministic reference or setpoint signals. It will also enable some limited future reference signal knowledge to be included. This feature is more associated with predictive control problems, but in fact, it is limited since the amount of preview or future information that can be used is less than the size of the transport-delay k. However, the NGMV solution is simpler than predictive control solutions that involve a multi-step cost-function. As noted the reference signal is assumed to have both stochastic and deterministic components r0 ðtÞ ¼ rðtÞ þ rd ðtÞ: It is assumed that

10.2

Nonlinear Operator and qLPV Plant Model

433

the future values of the reference signal are known for p time steps ahead. The noise free signal r0 ðt þ pÞ denotes the desired future values of the reference signal p steps ahead. The model for the stochastic component of the reference r(t + p) is driven by zero-mean white noise xr ðtÞ, and is determined by the state-equation modelled subsystem Reference state model: xr0 ðt þ 1Þ ¼ Ar0 xr0 ðtÞ þ Dr0 xr0 ðtÞ;

xr0 ðtÞ 2 Rnr0

ð10:10Þ

Stochastic component of reference: rðt þ pÞ ¼ Cr0 xr0 ðtÞ

ð10:11Þ

In some tracking problems the reference r0 ðtÞ may be corrupted by a measurement noise signal vr(t). For example, a target may be followed that is not known precisely. Then the measured reference rm ðt þ pÞ ¼ vr ðtÞ þ r0 ðt þ pÞ ¼ vr ðtÞ þ Cr0 xr0 ðtÞ þ rd ðt þ pÞ

ð10:12Þ

If the noise on the reference vr(t) is zero, the reference will be just the known signal r0(t). Models are also required to represent the future values of the stochastic component of the reference signal r(t + p), r(t + p − 1), …, r(t), where p > 0. State variables can be defined as the delayed values of the future reference signal xr1 ðtÞ ¼ r ðt þ p  1Þ ¼ Cr0 xr0 ðt  1Þ xr2 ðtÞ ¼ r ðt þ p  2Þ ¼ xr1 ðt  1Þ ¼ Cr0 xr0 ðt  2Þ .. .. .. . . . xrp ðtÞ ¼ r ðtÞ ¼ xrðp1Þ ðt  1Þ ¼ Cr0 xr0 ðt  pÞ

ð10:13Þ

This model represents the reference at time t and at all future times up to time t + p. The state-equation model for the future stochastic reference signal variations becomes

434

2

10

2

3

Ar0 7 6 Cr0 6 7 6 6 7 6 0 6 7 6 . 6 7¼6 . 6 7 6 . 6 7 6 6 4 xr p1 ðt þ 1Þ 5 4 ... xrp ðt þ 1Þ 0 xr0 ðt þ 1Þ xr1 ðt þ 1Þ xr2 ðt þ 1Þ .. .

LPV and State-Dependent Nonlinear Optimal Control

  0 .. . .. 0 .  0 0 0 I .. .

0 0 0 0  0 .. . .. . 0 0 I

32 3 3 2 Dr0 xr0 ðtÞ 0 xr0 ðtÞ 7 7 6 6 0 07 76 xr1 ðtÞ 7 6 7 .. 7 6 xr2 ðtÞ 7 6 07 . 7 7 7 6 6 .. 76 7 7þ6 .. . . 76 . 7 7 6 . . 7 7 6 6 .. 7 5 5 5 4 4 xr p1 ðtÞ . 0 xrp ðtÞ 0 0

The reference model matrices and vectors may now be defined as 2

Ar0 0    0 6 6 Cr0 0    0 6 6 0 I 0  6 . . . Ar ¼ 6 .. .. 6 .. 6 6 . .. .. 6 . . 0 . 4 . 0  0 0 3 2 Dr0 xr0 ðtÞ 7 6 0 7 6 7 6 . 7 6 . Dr xr ðtÞ ¼ 6 . 7 7 6 .. 7 6 5 4 .

0 0 0 .. . 0 I

3 0 7 07 7 07 7 .. 7 ; .7 7 .. 7 7 .5 0

Cr ¼ ½ Cr0

0

   0 ;

0 and the future values of the reference can now be collected in the vector xr ðtÞ, where h iT xr ðtÞ ¼ xTr0 ðtÞ; ; xTr1 ðtÞ ; ::; xTrp ðtÞ The future stochastic components of the reference model may be collected in the model Future reference model: xr ðt þ 1Þ ¼ Ar xr ðtÞ þ Dr xr ðtÞ;

xr0 ðtÞ 2 Rnr0

rðt þ pÞ ¼ Cr0 xr0 ðtÞ ¼ Cr xr ðtÞ

ð10:14Þ ð10:15Þ

The noise-corrupted measurement of the stochastic and deterministic components of the future reference signal may be written as rm ðt þ pÞ ¼ vr ðtÞ þ Cr0 xr0 ðtÞ þ rd ðt þ pÞ ¼ vr ðtÞ þ Cr xr ðtÞ þ rd ðt þ pÞ

ð10:16Þ

10.2

Nonlinear Operator and qLPV Plant Model

435

10.2.4 Error and Observation Signals The total qLPV model for the system, shown in Fig. 10.2, distinguishes between the measured and the weighted outputs. The signals of interest also include the error and the observations signal (measurements). Tracking error: eðtÞ ¼ r0 ðtÞ  yðtÞ ¼ xrp ðtÞ þ rd ðtÞ  yðtÞ ¼ Crp xr ðtÞ þ rd ðtÞ  yðtÞ where the matrix to Crp ¼ ½ 0    0 I :

pick

out

the

reference

at

ð10:17Þ

the

current

time

Plant observations signal: zm ðtÞ ¼ ym ðtÞ þ vm ðtÞ

ð10:18Þ

Error weighting: The weighted tracking error will involve the signal to be controlled in the system and this weighting may be implemented in state-equation form as follows: xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp eðtÞ;

xp ðtÞ 2 Rnp

ep ðtÞ ¼ Cp xp ðtÞ þ E p eðtÞ

ð10:19Þ ð10:20Þ

Remarks The time-functions are contained in extensions of the discrete Marcinkiewicz space. The structure in Fig. 10.1 is quite general but if the measured and controlled outputs are the same, the solution may be obtained by defining C0m ¼ C0 , E 0m ¼ E 0 , d0m ¼ d0 .

10.2.5 Total Augmented System The augmented system equations may now be introduced for the total r output m input multivariable system shown in Fig. 10.2. The derivation of a suitable state-space qLPV plant model for design is a difficult problem as explained in Chap. 1. However, the remaining subsystem models will often be linear time-invariant and be chosen by the designer based on analysis or experience. The augmented state-space qLPV model will be assumed to be well behaved, in the

436

10

LPV and State-Dependent Nonlinear Optimal Control

sense that it is controllable and observable, or at least stabilizable and detectable. The vector of the augmented state-space system model states for the qLPV model is defined to be of the form  xðtÞ ¼ xT0 ðtÞ xTd ðtÞ

xTr ðtÞ

xTp ðtÞ

T

The combined state model for the qLPV subsystems in the previous section, and the disturbance inputs are summarized below. The notation used for the time-varying augmented system matrices and the relationship to the subsystem models is established in the next section. Augmented system: xðt þ 1Þ ¼ At xðtÞ þ Bt u0 ðt  kÞ þ Dt nðtÞ þ dd ðtÞ;

xðtÞ 2 Rn

ð10:21Þ

yðtÞ ¼ dðtÞ þ Ct xðtÞ þ E t u0 ðt  kÞ

ð10:22Þ

m ym ðtÞ ¼ dm ðtÞ þ Cm t xðtÞ þ E t u0 ðt  kÞ

ð10:23Þ

m zm ðtÞ ¼ vm ðtÞ þ dm ðtÞ þ Cm t xðtÞ þ E t u0 ðt  kÞ

ð10:24Þ

The weighted error or output to be controlled may be written similarly in terms of the augmented qLPV model ep ðtÞ ¼ dp ðtÞ þ Cp t xðtÞ þ E p t u0 ðt  kÞ

ð10:25Þ

When the controlled outputs are the same as the measured outputs Ct ¼ Cm t and Et ¼ Em t , dðtÞ ¼ dm ðtÞ.

10.2.6 Definition of the Augmented System Matrices The relationship between the augmented system matrices in equations in (10.21)– (10.25), for the plant, disturbance, reference and weightings is now established. Equation (10.17) can be used to expand the expression for the dynamic error weighting states (using Eqs. (10.19) and (10.20)) as
 xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp eðtÞ ¼ Ap xp ðtÞ þ Bp Crp xr ðtÞ þ rd ðtÞ  yðtÞ

ð10:26Þ

From (10.5), noting d0 ðtÞ ¼ dðtÞ þ yd ðtÞ, the controlled output yðtÞ ¼ d0 ðtÞ þ C0 x0 ðtÞ þ E 0 u0 ðt  kÞ ¼ dðtÞ þ yd ðtÞ þ C0 x0 ðtÞ þ E 0 u0 ðt  kÞ

10.2

Nonlinear Operator and qLPV Plant Model

437

Noting (10.8), (10.11), and (10.26), we obtain
 xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp Crp xr ðtÞ þ rd ðtÞ  Bp ðdðtÞ þ yd ðtÞ þ C0 x0 ðtÞ þ E 0 u0 ðt  kÞÞ ¼ Ap xp ðtÞ þ Bp Crp xr ðtÞ þ Bp rd ðtÞ  Bp dðtÞ  Bp Cd xd ðtÞ  Bp C0 x0 ðtÞ  Bp E 0 u0 ðt  kÞ

The state-equation for the augmented system may now be obtained from the above equations as 3 2 x0 ðt þ 1Þ A0 6 x ðt þ 1Þ 7 6 0 6 d 7 6 6 7¼6 4 xr ðt þ 1Þ 5 4 0 2

xp ðt þ 1Þ

0 Ad

0 0

0

Ar

32 3 2 0 x0 ðtÞ 6 7 6 0 7 76 xd ðtÞ 7 6 76 7þ6 0 54 xr ðtÞ 5 4

B0 0 0

3 7 7 7u0 ðt  kÞ 5

Bp E 0 Bp C0 Bp Cd Bp Crp Ap xp ðtÞ 2 3 2 3 3 G0 0 D0 0 0 2 f0 ðtÞ  6 7 6 0 D 7 d0d ðtÞ 0 76 d 6 7 60 0 7 þ6 7 74 xd ðtÞ 5 þ 6 4 0 0 5 ðrd ðtÞ  dðtÞÞ 4 0 0 Dr 5 xr ðtÞ 0 Bp 0 0 0 ð10:27Þ

Augmented System: This equation may clearly be written in the form xðt þ 1Þ ¼ At xðtÞ þ Bt u0 ðt  kÞ þ Dt nðtÞ þ dd ðtÞ 2

ð10:28Þ

3 2 3 B0 0 6 0 7 0 7 7, 7 where Bt ¼ 6 4 0 5, 0 5 Bp E 0 A 2 p 3 2 G0 0 D0 60 0 7 6 0 7 Dt ¼ 6 and Gt ¼ 6 40 0 5 4 0 0 Bp 0 2 3 G0 d0d ðtÞ   6 7 0 d0d ðtÞ 7 dd ðtÞ ¼ Gt ¼6 4 5 0 ðrd ðtÞ  dðtÞÞ Bp ðrd ðtÞ  dðtÞÞ Output to be Controlled: The output to be controlled, noting d0 ðtÞ ¼ dðtÞ þ yd ðtÞ and Eqs. (10.5) and (10.8), may be expressed as A0 6 0 At ¼ 6 4 0 B C 3 p 0 0 0 Dd 0 7 7, 0 Dr 5 0 0

0 Ad 0 Bp Cd

0 0 Ar Bp Crp

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10

LPV and State-Dependent Nonlinear Optimal Control

yðtÞ ¼ dðtÞ þ Cd xd ðtÞ þ C0 x0 ðtÞ þ E 0 u0 ðt  kÞ 2 3 x0 ðtÞ 6 x ðtÞ 7 6 d 7 ¼ dðtÞ þ ½ C0 Cd 0 0 6 7 þ E 0 u0 ðt  kÞ 4 xr ðtÞ 5

ð10:29Þ

xp ðtÞ This may be written in terms of the augmented system model in (10.22) as yðtÞ ¼ dðtÞ þ Ct xðtÞ þ E t u0 ðt  kÞ where Ct ¼ ½ C0

Cd

0

0

and

Et ¼ E0

ð10:30Þ

Measured output model: The measured output, noting Eqs. (10.6) and (10.9), and d0m ðtÞ ¼ dm ðtÞ þ ydm ðtÞ ym ðtÞ ¼ dm ðtÞ þ ydm ðtÞ þ C0m x0 ðtÞ þ E 0m u0 ðt  kÞ 2 3 x0 ðtÞ 6 x ðtÞ 7 6 d 7 ¼ dm ðtÞ þ ½ C0m Cdm 0 0 6 7 þ E 0m u0 ðt  kÞ 4 xr ðtÞ 5

ð10:31Þ

xp ðtÞ This equation may be written in the augmented system form of Eq. (10.23) as m ym ðtÞ ¼ dm ðtÞ þ Cm t xðtÞ þ E t u0 ðt  kÞ

where Cm t ¼ ½ C 0m

Cdm

0

0

and E m t ¼ E 0m

ð10:32Þ

Weighted tracking error: To compute the augmented matrices for the weighted error, from (10.20) ep ðtÞ ¼ Cp xp ðtÞ þ E p ðr0 ðtÞ  yðtÞÞ ¼ Cp xp ðtÞ þ E p ðr0 ðtÞ  ðdðtÞ þ Cd xd ðtÞ þ C0 x0 ðtÞ þ E 0 u0 ðt  kÞÞÞ 2 3 x0 ðtÞ 6 x ðtÞ 7 6 d 7 ¼ E p ðrd ðtÞ  dðtÞÞ þ ½ E p C0 E p Cd E p Crp Cp 6 7  E p E 0 u0 ðt  kÞ 4 xr ðtÞ 5 xp ðtÞ

ð10:33Þ

10.2

Nonlinear Operator and qLPV Plant Model

439

This equation is in the form of the augmented system Eq. (10.25) ep ðtÞ ¼ dp ðtÞ þ Cp t xðtÞ þ E p t u0 ðt  kÞ where Cp t ¼ ½ E p C0

E p Cd

E p Crp

Cp  and

E p t ¼ E p E 0

ð10:34Þ

and the through “deterministic” term dp ðtÞ ¼ E p ðrd ðtÞ  dðtÞÞ. Summary of the Augmented System Matrices: Collecting the above results the matrices that define the augmented system are defined as 2

A0 6 0 6 At ¼ 4 0 Bp C0

0 Ad 0 Bp Cd

Cp t

0 0 Ar Bp Crp

3 0 0 7 7; 0 5 Ap

2

3 B0 6 0 7 7 Bt ¼ 6 4 0 5; Bp E 0

2

D0 6 0 6 Dt ¼ 4 0 0

0 Dd 0 0

3 0 0 7 7 Dr 5 0

Ct ¼ ½ C0 Cd 0 0 ; E t ¼ E 0 Cm 0 0 ; E m t ¼ ½ C0m Cdm t ¼ E 0m ¼ ½ E p C0 E p Cd E p Crp Cp ; E p t ¼ E p E 0

ð10:35Þ Recall the component of the reference signal rd ðtÞ, and the known disturbance signal dðtÞ, are both deterministic signals. Thence combining the deterministic signals 2

3 G0 d0d ðtÞ 6 7 0 7 dd ðtÞ ¼ 6 4 5 0 Bp ðrd ðtÞ  dðtÞÞ

and

The white noise driving terms  T nT ðtÞ ¼ fT0 ðtÞ xTd ðtÞ xTr ðtÞ .

dp ðtÞ ¼ E p ðrd ðtÞ  dðtÞÞ

are

combined

in

ð10:36Þ

the

vector

Remarks The subscript t on the state matrices is used for the augmented system and in a slight abuse of notation it indicates that these matrices are evaluated at time t, so that the augmented system matrix at t + 1 is written as At þ 1 and so on.

10.3

Predicted Plant Outputs and States

Assume for the present that the future values of the control signal and the deterministic disturbance components are known, so that the future values of the system matrices may be estimated. Then the future values of the states and outputs may be obtained as

440

10

LPV and State-Dependent Nonlinear Optimal Control

xðt þ 1Þ ¼ At xðtÞ þ Bt u0 ðt  kÞ þ Dt nðtÞ þ dd ðtÞ

ð10:37Þ

Similarly, xðt þ 2Þ ¼ At þ 1 ðAt xðtÞ þ Bt u0 ðt  kÞ þ Dt nðtÞ þ dd ðtÞÞ þ Bt þ 1 u0 ðt þ 1  kÞ þ Dt þ 1 nðt þ 1Þ þ dd ðt þ 1Þ ¼ At þ 1 At xðtÞ þ At þ 1 Bt u0 ðt  kÞ þ Bt þ 1 u0 ðt þ 1  kÞ

ð10:38Þ

þ At Dt nðtÞ þ Dt þ 1 nðt þ 1Þ þ At þ 1 dd ðtÞ þ dd ðt þ 1Þ and xðt þ 3Þ ¼ At þ 2 At þ 1 At xðtÞ þ At þ 2 At þ 1 Bt u0 ðt  kÞ þ At þ 2 Bt þ 1 u0 ðt þ 1  kÞ þ Bt þ 2 u0 ðt þ 2  kÞ þ At þ 2 At þ 1 Dt nðtÞ þ At þ 2 Dt þ 1 nðt þ 1Þ þ Dt þ 2 nðt þ 2Þ þ At þ 2 At þ 1 dd ðtÞ þ At þ 2 dd ðt þ 1Þ þ dd ðt þ 2Þ

ð10:39Þ Future states: The expression for a i-steps ahead state-vector may be obtained by generalizing the above result to obtain (for i > 1): xðt þ iÞ ¼ At þ i1 At þ i2 . . .At xðtÞ þ At þ i1 At þ i2 . . .At þ 1 Bt u0 ðt  kÞ

þ    þ At þ i1 Bt þ i2 u0 ðt þ i  k  2Þ þ Bt þ i1 u0 ðt þ i  k  1Þ þ At þ i1 At þ i2 . . .At þ 1 Dt nðtÞ

þ    þ At þ i1 Dt þ i2 nðt þ i  2Þ þ Dt þ i1 nðt þ i  1Þ þ At þ i1 At þ i2 . . .At þ 1 dd ðtÞ

þ    þ At þ i1 dd ðt þ i  2Þ þ dd ðt þ i  1Þ

ð10:40Þ

It is assumed that the future values of the reference signal are assumed known for p time steps ahead. It is also assumed the future values of the system matrices that are dependent on future controls can be computed based on prediction. Note from (10.28) the computation of the estimate of x(t + k) involves knowledge of the system matrices up to t + k − 1 and controls up to t − 1, which are all known at time t. Notation: This equation may now be written in a more compact form, by introducing the notation Aim t þ m ¼ At þ i1 At þ i2 . . .At þ m

for i [ m;

where A0t þ m ¼ I

for i ¼ m ð10:41Þ

and Ait ¼ At þ i1 At þ i2 . . .At

for i [ 0; where A0t ¼ I

for i ¼ 0

ð10:42Þ

10.3

Predicted Plant Outputs and States

441

Using this notation and a summation write (10.40) as

þ

i P

xðt þ iÞ ¼ At þ i1 At þ i2 . . .:At xðtÞ
 At þ i1 At þ i2 . . .:At þ j Bt þ j1 u0 ðt þ j  1  kÞ þ Dt þ j1 nðt þ j  1Þ þ dd ðt þ j  1Þ

j¼1

where when j = i the term At þ i1 At þ i2 . . .:At þ j is defined to be the identity. This equation is valid for i  0 if the summation term is defined to be null when i = 0. This may now be written as xðt þ iÞ ¼ Ait xðtÞ þ

i X


 Aij t þ j Bt þ j1 u0 ðt þ j  1  kÞ þ Dt þ j1 nðt þ j  1Þ þ ddd ðt þ i  1Þ

j¼1

ð10:43Þ Predicted known disturbance: The predicted known disturbance term in this equation ddd ðt þ i  1Þ ¼

i X

Aij t þ j dd ðt þ j  1Þ

ð10:44Þ

j¼1

These Eqs. (10.43) and (10.44) are valid for i  0 if the summation terms are defined as null for i = 0. That is, the predicted known disturbance term ddd ðt þ i  1Þ ¼ ddd ðt  1Þ ¼ 0;

i P j¼1

Aij t þ j dd ðt þ j  1Þ for i [ 0 for i ¼ 0

9 = ;

ð10:45Þ

Future weighted errors: Noting (10.34) the weighted error or output signal ep ðtÞ to be regulated at future times (for i  0), has the form ep ðt þ iÞ ¼ dp ðt þ iÞ þ Cp t þ i xðt þ iÞ þ E p t þ i u0 ðt þ i  kÞ i  P ¼ dp ðt þ iÞ þ Cp t þ i Ait xðtÞ þ Cp t þ i Aij B u ðt þ j  1  kÞ t þ j t þ j1 0 þ Cp t þ i

i  P j¼1

or



j¼1

Aij t þ j Dt þ j1 nðt þ j  1Þ þ Cp t þ i ddd ðt þ i  1Þ þ E p t þ i u0 ðt þ i  kÞ

442

10

LPV and State-Dependent Nonlinear Optimal Control

ep ðt þ iÞ ¼ dpd ðt þ iÞ þ Cp t þ i Ait xðtÞ þ Cp t þ i

i  X

Aij t þ j Bt þ j1 u0 ðt þ j  1  kÞ



j¼1

þ Cp t þ i

i  X

Aij D nðt þ j  1Þ þ E p t þ i u0 ðt þ i  kÞ t þ j1 tþj

j¼1

ð10:46Þ The deterministic signals in this equation are combined as dpd ðt þ iÞ ¼ dp ðt þ iÞ þ Cp t þ i ddd ðt þ i  1Þ

ð10:47Þ

10.3.1 Expression for Predicted Plant Outputs and States Expressions are required for the predicted state and the predicted weighted error terms. Noting Eq. (10.43) the i-steps ahead predictor of the state may now be introduced. State-prediction: The i-steps predicted state, noting the stochastic disturbance is zero-mean, can be expressed as ^xðt þ ijtÞ ¼ Ait ^xðtjtÞ þ

i X

Aij t þ j Bt þ j1 u0 ðt þ j  1  kÞ þ ddd ðt þ i  1Þ ð10:48Þ

j¼1

where Aij t þ j ¼ At þ i1 At þ i2 . . .At þ j and ddd ðt þ i  1Þ ¼

i P j¼1

Aij t þ j dd ðt þ j  1Þ, but if

i = 0 define ddd ðt  1Þ ¼ 0 Error prediction: The i-steps predicted weighted error follows from (10.34) and the predicted state as ^ep ðt þ ijtÞ ¼ dp ðt þ iÞ þ Cp t þ i^xðt þ ijtÞ þ E p t þ i u0 ðt þ i  kÞ

ð10:49Þ

Alternative form: These equations may be simplified by introducing a finite pulse response model (noting (10.48)), of the form T ði; z1 Þ ¼

i X

j1k Aij t þ j Bt þ j1 z

j¼1

The state-prediction equation above may be written as

ð10:50Þ

10.3

Predicted Plant Outputs and States

443

^xðt þ ijtÞ ¼ Ait ^xðtjtÞ þ T ði; z1 Þu0 ðtÞ þ ddd ðt þ i  1Þ

ð10:51Þ

Substituting from this equation into (10.49)
 ^ep ðt þ ijtÞ ¼ dp ðt þ iÞ þ Cp t þ i Ait ^xðtjtÞ þ T ði; z1 Þu0 ðtÞ þ ddd ðt þ i  1Þ þ E p t þ i u0 ðt þ i  kÞ ¼ dpd ðt þ iÞ þ Cp t þ i Ait ^xðtjtÞ þ Cp t þ i T ði; z1 Þu0 ðtÞ þ E p t þ i u0 ðt þ i  kÞ ð10:52Þ where dpd ðt þ iÞ is defined by (10.47).

10.3.2 Kalman Predictor qLPV Model The Kalman filter is required for this linear time-varying system and two degrees of freedom control problem. The result below is extended in an obvious way to accommodate the delays on input channels and the use of through terms for the observations signal model. The Kalman filter equations can also allow for the presence of a known bias or disturbance on the observations [10]. Such changes do not affect the basic stochastic relationships or the gain of the optimal filter. The Kalman filter input involves the noisy plant observations or measurements zm ðtÞ and the measured reference signal rm ðtÞ. For the design of the filter the combined observations signal (from (10.16))  zðtÞ ¼

   m zm ðtÞ v ðtÞ þ dm ðtÞ þ Cm t xðtÞ þ E t u0 ðt  kÞ ¼ m rm ðt þ pÞ vr ðtÞ þ rd ðt þ pÞ þ Cr xr ðtÞ

ð10:53Þ

The total observations input to the filter may now be written as zðtÞ ¼ vðtÞ þ df ðtÞ þ Ctf xðtÞ þ E tf u0 ðt  kÞ

ð10:54Þ

where 

 vm ðtÞ ; vðtÞ ¼ vr ðtÞ



 dm ðtÞ df ðtÞ ¼ ; rd ðtÞ



Ctf

C0m ¼ 0

Cdm 0

0 Cr

 0 ; 0



E tf

Em t ¼ 0



ð10:55Þ The process and measurement noise covariance matrices, have the form Qtf ¼ diagfQ0 ; Qd ; Qr g

and

Rtf ¼ diagfRm ; Rr g

Remarks If the stochastic component of the reference is zero the corresponding covariance Qr can be set to zero. Similarly, if there is no measurement noise on the

444

10

LPV and State-Dependent Nonlinear Optimal Control

reference the covariance Rr can be set to zero. However, a small measurement noise covariance for the reference may help the numerical solution for the filter gain. In practical applications, the covariance matrices Qd ; Qr ; Rr may often be treated as tuning variables (depending on the application involved). Augmented system for Kalman filter design: The computation of the Kalman filter state-estimation is also based on a model in qLPV model form. The calculations are similar to those for a known time-varying linear system if the states determining the state matrices are assumed known. Augmented plant model: xðt þ 1Þ ¼ At xðtÞ þ Bt u0 ðt  kÞ þ Dt nðtÞ þ dd ðtÞ

ð10:56Þ

Observations signal: zðtÞ ¼ vðtÞ þ df ðtÞ þ Ctf xðtÞ þ E tf u0 ðt  kÞ

ð10:57Þ

where the signals dd(t) and df(t) were defined by (10.36) and (10.55), respectively. The controller is in a two degrees of freedom form, since the Kalman filter that is part of the control system is driven from the measured output and the measured reference in different input channels. The deterministic component of the reference signal affects both the signals dd(t) and df(t).

10.3.3 Kalman Predictor Corrector Form The time-varying Kalman filter is to be applied to the combined qLPV model in Eqs. (10.56) and (10.57) that was assumed pointwise observable and controllable. The qLPV model matrices may, of course, be a function of state variables, parameters and possibly inputs but for computing the Kalman filter the time-varying system matrices are assumed known. Recall that the parameters p0 ðtÞ are known and the plant includes a transport-delay term so that the previous control actions are known. The states are not, of course, known and if the system matrices are state-dependent they must be computed using state-estimates. It will be assumed that the system matrices are computable up to time t. This is a sound assumption since experience reveals that in the majority of applications the Kalman estimator provides reasonable estimates, even when the uncertainties are present. The discrete-time Kalman filter equations for a time-varying linear state-space model in predictor/corrector form is suitable for this qLPV estimation problem. Assume the exogenous signals have known means, then the estimation equations follow as

10.3

Predicted Plant Outputs and States

^xðt þ 1jtÞ ¼ At ^xðtjtÞ þ Bt u0 ðt  kÞ þ dd ðtÞ

445

ðPredictor Þ ð10:58Þ

^xðt þ 1jt þ 1Þ ¼ ^xðt þ 1jtÞ þ Kf t þ 1 ðzðt þ 1Þ  ^zðt þ 1jtÞÞ ðCorrector Þ ð10:59Þ where ^zðt þ 1jtÞ ¼ df ðt þ 1Þ þ Ctfþ 1^xðt þ 1jtÞ þ E tfþ 1 u0 ðt þ 1  kÞ

ð10:60Þ

The process and measurement noise covariance matrices are denoted Qft and Rft . The Kalman filter gain and the Riccati equation becomes Kalman gain: Kf t þ 1 ¼ Pðt þ 1jtÞCft þT 1 ½Ctfþ 1 Pðt þ 1jtÞCft þT 1 þ Rtfþ 1 1

ð10:61Þ

A priori covariance: Pðt þ 1jtÞ ¼ At PðtjtÞATt þ Dt Qtf DTt

ð10:62Þ

A posteriori covariance: Pðt þ 1jt þ 1Þ ¼ Pðt þ 1jtÞ  Kðt þ 1jtÞCtfþ 1 Pðt þ 1jtÞ

ð10:63Þ

Initial conditions: ^xð0j0Þ ¼ m0 and Pð0j0Þ ¼ P0 : The Kalman filtering problem is discussed in more detail in Chap. 13, in Anderson and Moore [11] and Grimble and Johnson [10]. It is useful that the order of the Kalman filter depends only on the delay free subsystems. The channel delays do not increase the order of the filter. An operator form of the estimator equations is derived below since it is useful for the later analysis that explores the structure of the controller to be implemented and the stability of the closed-loop design. Estimates in closed-loop operator form: From the Kalman filter equations, substituting (10.58) and (10.60) into (10.59) obtain ^xðt þ 1jt þ 1Þ ¼ At ^xðtjtÞ þ Bt u0 ðt  kÞ þ dd ðtÞ  
þ Kf t þ 1 zðt þ 1Þ  df ðt þ 1Þ þ Ctfþ 1^xðt þ 1jtÞ þ E tfþ 1 u0 ðt þ 1  kÞ 

¼ At ^xðtjtÞ þ Bt u0 ðt  kÞ þ dd ðtÞ

þ Kf t þ 1 zðt þ 1Þ  df ðt þ 1Þ  Ctfþ 1 ðAt ^xðtjtÞ þ Bt u0 ðt  kÞ þ dd ðtÞÞ  E tfþ 1 u0 ðt þ 1  kÞ

After manipulation and multiplying both sides by z−1 obtain



446




10

LPV and State-Dependent Nonlinear Optimal Control

 I  ðI  Kf t Ctf Þz1 At ^xðtjtÞ ¼ ðKf t zðtÞ þ ððI  Kf t Ctf Þz1 Bt  Kf t E tf Þu0 ðt  kÞ þ dd ðt  1Þ  Kf t ðdf ðtÞ þ Ctf dd ðt  1ÞÞÞ The state estimate may, therefore, be expressed as

 1 ^xðtjtÞ ¼ I  I  Kf t Ctf z1 At ðKf t ðzðtÞ  df ðtÞÞ

 þ ðI  Kf t Ctf Þz1 Bt  Kf t E tf u0 ðt  kÞ þ I  Kf t Ctf dd ðt  1ÞÞ ð10:64Þ

The state-estimate (10.64) may now be expressed in the operator form
 ^xðtjtÞ ¼ Tf 1 ðz1 Þ zðtÞ  df ðtÞ þ Tf 2 ðz1 Þu0 ðtÞ þ df ðtÞ

ð10:65Þ

It follows by comparison that the operators in (10.65) may be defined as follows:

 1 Tf 1 ðz1 Þ ¼ I  I  Kf t Ctf z1 At Kf t

ð10:66Þ



 1
 Tf 2 ðz1 Þ ¼ I  I  Kf t Ctf z1 At ðI  Kf t Ctf Þz1 Bt  Kf t E tf zk ð10:67Þ

 1
 df ðtÞ ¼ I  I  Kf t Ctf z1 At I  Kf t Ctf dd ðt  1Þ

+ Control weighting

0

ep

+

ep

c

ð10:68Þ

u c

e

Error weighting

+

r0 -

c

Noise

y

Disturbances

Reference model

vm

Controller r

r

r + r0 + rm + +

0

(t, z 1 )

u

z

k 0k

1k

ym

+ +

Non-linear plant

rd

vr

Plant observations

zm

Fig. 10.3 Closed-loop control system for the nonlinear plant (output signal to be minimized is shown dotted)

10.3

Predicted Plant Outputs and States

447

The following is a useful result that can be verified using (10.66) and (10.67), as follows:  1 m k Tf 1 ðz1 Þ Cm þ Tf 2 ðz1 Þ t ðzI  At Þ Bt þ E t z ¼ ðI  z1 At Þ1 z1 Bt zk ¼ Uðz1 ÞBt zk

10.4

ð10:69Þ

Nonlinear Generalized Minimum Variance Control

The NGMV cost-minimisation problem may now be introduced, for the system, which is shown in concise operator form in Fig. 10.3. The optimal NGMV control problem involves the minimisation of the variance of the signal /0 ðtÞ, where the signal to be minimized and the cost-function, are respectively /0 ðtÞ ¼ ep ðtÞ þ ðF c uÞðtÞ

ð10:70Þ

and J ¼ Ef/T0 ðt þ kÞ/0 ðt þ kÞjtg ¼ Eftracef/0 ðt þ kÞ/T0 ðt þ kÞgjtg

ð10:71Þ

and E fjtg denotes the conditional expectation operator. The first term in the inferred output signal /0 ðtÞ to be minimized in this problem involves the weighted error between the reference and controlled outputs. This may be denoted in the operator form ep ðtÞ ¼ ðP c eÞðtÞ and is represented by the state-equation model (10.19) and (10.20), described in Sect. 10.2. The final term in the criterion is the nonlinear dynamic control signal weighting operator ðF c uÞðtÞ expressed as ðF c uÞðtÞ ¼ zk ðF ck uÞðtÞ

ð10:72Þ

where F ck will often be a linear operator that is assumed full rank and minimum-phase. It may also be chosen to be a nonlinear dynamic operator that is full rank and has a stable inverse. The choice of the dynamic cost-function weightings influences both stability and performance.

10.4.1 Minimization of NGMV Control Problem The solution of the optimal control problem is straightforward and follows the minimum variance strategy established in previous chapters. The first step is to obtain a prediction equation by expanding the expression for the inferred output signal to be minimized. This is not normally a physical signal but is constructed as

448

10

LPV and State-Dependent Nonlinear Optimal Control

/0 ðtÞ ¼ ðP c eÞðtÞ þ ðF c uÞðtÞ ¼ ep ðtÞ þ ðF c uÞðtÞ

ð10:73Þ

As in the polynomial based NGMV designs in Chap. 5 the weighting P c in (10.73) is typically a low-pass filter and F c is a high-pass filter. This problem is related to GMV control for linear systems [12], and is a qLPV model version of NGMV control [13–17]. To minimize the variance of (10.70) the output to be minimized may be written, substituting from (10.34) as /0 ðtÞ ¼ ep ðtÞ þ ðF c uÞðtÞ ¼ dp ðtÞ þ Cp t xðtÞ þ E p t u0 ðt  kÞ þ ðF c uÞðtÞ

ð10:74Þ

Recall that the signal u0 ðtÞ ¼ W 1k uðtÞ and that the control signal weighting ðF c uÞðtÞ ¼ zk ðF ck uÞðtÞ. Substituting in (10.74)
 /0 ðtÞ ¼ dp ðtÞ þ Cp t xðtÞ þ ðE p t W 1k þ F ck Þu ðt  kÞ

ð10:75Þ

Thence, the future values of the inferred output: /0 ðt þ kÞ ¼ dp ðt þ kÞ þ Cp t þ k xðt þ kÞ þ ðE p t þ k W 1k þ F ck ÞuðtÞ Recalling the first term dp ðtÞ ¼ E p ðrd ðtÞ  dðtÞÞ is assumed a known signal, the condition for optimality becomes ^ ðt þ kjtÞ ¼ dp ðt þ kÞ þ Cp t þ k ^xðt þ kjtÞ þ ðE p t þ k W 1k þ F ck ÞuðtÞ ¼ 0 ð10:76Þ / 0 The cost-function to be minimized (10.71) involves the minimization of the variance of the weighted error and control signals. This criterion may now be ^ ðt þ kjtÞ and the prediction error written in terms of the predicted value / 0 ~ ðt þ kjtÞ, noting these signals are orthogonal / 0 J ¼ Ef/0 ðt þ kÞT /0 ðt þ kÞjtg ^ ðt þ kjtÞjtg þ Ef/ ~ ðt þ kjtÞjtg ~ ðt þ kjtÞT / ^ ðt þ kjtÞT / ¼ Ef/ 0 0 0 0

ð10:77Þ

~ ðt þ kjtÞ does not depend upon control action and hence the The prediction error / 0 cost is clearly minimized by setting the k-steps-ahead predicted values of the signal /0 ðtÞ to zero.

10.4.2 Solution of the NGMV Optimal Control Problem The optimal control law is chosen to set the predicted values of the signal /0 ðtÞ to zero and thereby minimize its variance. Thus, setting the prediction (10.76) to zero provides two possible expressions for the optimal control in terms of the predicted state estimate

10.4

Nonlinear Generalized Minimum Variance Control


 uðtÞ ¼ ðE p t þ k W 1k þ F ck Þ1 dp ðt þ kÞ  Cp t þ k ^xðt þ kjtÞ

449

ð10:78Þ

The following expression is more useful for implementation
 xðt þ k jtÞ  ðE p t þ k W 1k uÞðtÞ uðtÞ ¼ F 1 ck dp ðt þ kÞ  Cp t þ k ^

ð10:79Þ

Alternative prediction equation: To derive the expression for the optimal control signal, in terms of the current state-estimate, recall the prediction equation (including the finite pulse response term), is obtained from equation (10.51): ^xðt þ kjtÞ ¼ Akt ^xðtjtÞ þ T ðk; z1 Þu0 ðtÞ þ ddd ðt þ k  1Þ where from (10.42) Akt ¼ At þ k1 At þ k2 . . .At and the operator T ðk; z1 Þ was defined by (10.50). The predicted inferred output may now be found by substituting in (10.76) and using (10.47):
 ^ ðt þ kjtÞ ¼ dp ðt þ kÞ þ Cp t þ k ^xðt þ k jtÞ þ ðE p t þ k W 1k þ F ck Þu ðtÞ / 0 ¼ dp ðt þ kÞ þ Cp t þ k Akt ^xðtjtÞ
þ Cp t þ k T ðk; z1 Þu0ðtÞ þ Cp t þ k ddd ðt þ k  1Þ þ ðE p t þ k W 1k þ F ck Þu ðtÞ
 ¼ dpd ðt þ kÞ þ Cp t þ k Akt ^xðtjtÞ þ ððE p t þ k þ Cp t þ k T ðk; z1 ÞÞW 1k þ F ck Þu ðtÞ ð10:80Þ Alternative control solution: If the Kalman predictor is defined in terms of the current state estimate using the finite pulse response model then setting the predicted values of (10.80) to zero also provides two alternative expressions for the optimal control uðtÞ ¼





1
  E p t þ k þ Cp t þ k T ðk; z1 Þ W 1k þ F ck dpd ðt þ kÞ  Cp t þ k Akt ^xðtjtÞ ð10:81Þ

k xðtjtÞ  ððE p t þ k þ Cp t þ k T ðk; z1 ÞÞW 1k uÞðtÞÞ uðtÞ ¼ F 1 ck ðdpd ðt þ kÞ  Cp t þ k At ^

ð10:82Þ The most practical forms for implementation are either (10.79) or (10.82). Reviewing the assumptions made earlier observe that (10.82) includes the Kalman filter state-estimate and the prediction blocks, and both depend upon the qLPV output subsystem plant model. The nonlinear input plant subsystem W 1k was assumed finite-gain stable but no structure was assumed. However, the qLPV nonlinear model W 0k may be unstable. For optimal state estimation, the assumption

450

10

LPV and State-Dependent Nonlinear Optimal Control

was made that the future values of the qLPV models could be computed. Finally note that the control cost-function weighting was defined so that the inverse of F ck exists, and thence the inverse of the operators in (10.79) or (10.82) may be found. The main theorem for qLPV model-based NGMV control now follows. Theorem 10.1: NGMV Controller for qLPV and Nonlinear Systems Under the above assumptions, the NGMV optimal control to minimize the variance of the weighted error and control signals may be computed in terms of the predicted state-estimate as follows:
 uðtÞ ¼ F 1 xðt þ k jtÞ  E p t þ k ðW 1k uÞðtÞ ck dp ðt þ kÞ  Cp t þ k ^

ð10:83Þ

where dp ðtÞ ¼ E p ðrd ðtÞ  dðtÞÞ, Cp t ¼ ½ E p C0 E p Cd E p Crp Cp  and E p t ¼ E p E 0 . Alternatively, in terms of the current state-estimate, the optimal control

  k uðtÞ ¼ F 1 xðtjtÞ  ðE p t þ k þ Cp t þ k T ÞW 1k u ðtÞ ck dpd ðt þ kÞ  C p t þ k At ^ ð10:84Þ dpd ðt þ iÞ ¼ dp ðt þ iÞ þ Cp t þ i ddd ðt þ i  1Þ

where T ðk; z1 Þ ¼

k P j¼1

and

j1k Akj . t þ j Bt þ j1 z



Proof The proof of optimality involves collecting the above results, subject to the assumptions.  Remarks Some of the properties of the system and solution can be listed as • The input plant subsystem and the control weighting can both be nonlinear and no particular structure needs to be assumed for the former other than open-loop stability. The stability of the closed-loop system is explored in Sect. 10.6 below.

Plant

Controller structure d p (t k)

pt k

xˆ(t

dd , d f

+ +

-

y

u (t )

1

ym +

ck

-

Controlled output

+

k | t) pt k

u0 (t k )

zm

1k

Disturbances

vm

u0 (t ) Measured output or observations signal

Fig. 10.4 Implementation of the NGMV qLPV controller in terms of the predicted state

10.4

Nonlinear Generalized Minimum Variance Control

451 Plant

Controller structure

d pd ( t

k)

+ +

Kalman predictor k pt k

t

-

u

1

+

ck

-

ym + 1k

xˆ(t | t ) pt k

dd , d f

vm

y

pt k

Disturbances

(k , z 1 )

u0

zm

Observations

Fig. 10.5 Implementation of the NGMV qLPV controller in terms of the current state

y Controller subsystems

Reference

rm

d pd (t k ) k pt k

t

xˆ(t | t )

Kalman predictor

ck

(

pt k

+

pt k

Disturbances

(k , z 1 ))

Nonlinear operator term

1

u

+ +

1k

Measured output

ym

Nonlinear plant

zm

vm

+ +

Fig. 10.6 Conceptual qLPV nonlinear controller structure

• The qLPV model matrices depend on the future values of the states, but these states are assumed to be measured or can be estimated using (10.51). The qLPV output subsystem may also depend on future control inputs. However, because of the plant delay, the control signal u(t) is known k-steps ahead. It follows that the qLPV output subsystem W 0k can be treated as known over the prediction interval. • The expressions for the NGMV optimal control signal (10.83) and (10.84), involving the predicted state or the current state estimate, respectively, lead to the two alternative control structures, shown in Fig. 10.4 and Fig. 10.5, respectively. The conceptual solution in terms of the current state estimate (10.81) has the block diagram form shown in Fig. 10.6, but this is not so useful for implementation. • The inferred output signal to be costed can involve the deviation from a known signal uðtÞ, by defining /0 ðtÞ ¼ ðP c eÞðtÞ þ ðF c ðu   uÞÞðtÞ. A review of the solution for this case reveals that the minimization of the deviation of the control signal u(t) has only a minor influence on the solution. That is, it provides a modification that acts as a feedforward term, as follows:

452

10

LPV and State-Dependent Nonlinear Optimal Control


 xðt þ k jtÞ  E p t þ k ðW 1k uÞðtÞ þ  uðtÞ ¼ F 1 uðtÞ ck dp ðt þ kÞ  C p t þ k ^

10.4.3 Advantages and Disadvantages of NGMV Control The advantages and disadvantages of NGMV control may now be summarized. Most of these also apply to the polynomial form of the NGMV controller that was introduced earlier in Chap. 4. Advantages of NGMV Control • Experience suggests often works easily out of the box. • Simple controller structure in a block form that is easy to understand. • Simple rule exists for choosing the cost-function by reference to a known PID solution. • Limited computational requirements, since do not involve a long prediction horizon and simple recursive solution of Kalman filter gains. • Suitable for multivariable systems and can accommodate black-box input nonlinearities and an output subsystem that can include a qLPV subsystem. Disadvantages of NGMV Control • The NGMV controller does not use the full future setpoint information as in the model-based predictive control methods. • There are some ranges of cost-function weightings and system descriptions, which can result in instability, and the weightings, therefore, have to be chosen following good practice. • The version for quasi-LPV and related systems is more computationally demanding than the basic state-space NGMV version in Chap. 8, which only uses an output system model that is linear time-invariant.

10.5

Controller Structure and Implementation

Two aspects of implementing the qLPV version of the NGMV controller are discussed in this section. The first involves a more detailed view of the use of the Kalman filter in the controller structure. The second is the simplified structure for the controller that can be derived when the nonlinear input subsystem can be decomposed into a non-dynamic through term and other terms involving at least a unit-step delay W 1k ð:Þ ¼ W 1k0 þ z1 W 1k1 ðz1 ; :Þ.

10.5

Controller Structure and Implementation f t 1 0

u (t 1 k )

+ z (t 1) d f (t 1)

453

Time-Varying Kalman Filter/Predictor Filter gain

-

ft 1

f t 1

+ +

xˆ(t 1| t 1)

z

1

d pd (t k )

xˆ(t 1| t)

+ dd (t )

0

d0 d + +

k t

xˆ ( t | t )

-

k pt k

t

z (t 1) zˆ(t 1| t)

k

xˆ(t | t)

pt k

+

-

pt k

1 ck

Delay elements

u(t)

(k , z 1 )

t

z kI

Optimal control

Control weighting inverse

Prediction

u 0 (t )

NL model of plant subsystem 1k

Finite pulse response block

Fig. 10.7 NGMV Optimal Controller in a state-space Kalman filtering form

10.5.1 Kalman Filter-Based Controller Structure The controller involves a time-varying Kalman filter discussed in Sect. 10.3.2 with a predictor stage [18, 10]. This is part of the more physically intuitive structure of the controller shown in Fig. 10.7. To justify this structure, in terms of the current state estimate, recall (10.84) and (10.57), so that the estimated observations signal becomes: ^zðt þ 1jtÞ ¼ Ctfþ 1^xðt þ 1jtÞ þ Ctfþ 1 u0 ðt þ 1  kÞ þ df ðt þ 1Þ

ð10:85Þ

Introducing the Kalman filter structure leads to the realisation of the controller in Fig. 10.7, which is easy to implement. The state-estimates may also be useful for condition monitoring purposes as discussed in Chap. 13. Note the attempt to show the time dependence of matrices for the filter in this diagram. Extended Kalman Filter: The controllers in this and the following chapter depend upon the time-varying Kalman filter solution. However, if the qLPV model is only an approximation to a nonlinear process there is an alternative approach that may be taken. That is, this estimator may be replaced with an Extended Kalman Filter (EKF) or one of the other nonlinear state-estimators described in Chap. 13. The two possibilities are as follows: • Using a qLPV model for control design and a Kalman filter with qLPV model for state estimation that has the advantage of being consistent but both involve approximations to the underlying nonlinear system. • Using the qLPV model for control design but an EKF or nonlinear filter for estimation that may reduce estimation errors but where there is a mismatch with the model used in the control solution.

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LPV and State-Dependent Nonlinear Optimal Control

It is not clear which would be the most successful approach for a particular application in which case a comparison simulation may be needed.

10.5.2 Simplified Solution for a Special Case The controller (10.83), shown in Fig. 10.4, is simpler to compute if the nonlinear control signal cost-weighting operator F ck ð:Þ is defined to have a non-dynamic first component (full rank function), where F ck can be written in the form F ck ð:Þ ¼ F ck0 þ z1 F ck1 ð z1 ; :Þ and also if it is assumed the nonlinear plant input subsystem is decomposed as W 1k ð:Þ ¼ W 1k0 þ z1 W 1k1 ðz1 ; :Þ where the term W 1k0 is a non-dynamic matrix term. Substituting into the condition for optimality (10.76) obtain
 dp ðt þ kÞ þ Cp t þ k ^xðt þ k jtÞ þ E p t þ k W 1k0 þ F ck0 uðtÞ þ E p t þ k z1 W 1k1 þ z1 F ck1 ¼0 The NGMV optimal control solution then follows as



1 uðtÞ ¼ E p t þ k W 1k0 þ F ck0 dp ðt þ kÞ  Cp t þ k ^xðt þ kjtÞ  ðE p t þ k z1 W 1k1 þ z1 F ck1 ÞuðtÞ

pt k

xˆ(t

-

+ +

1 ck 0

pt k

k)

1k

y

u (t )

+

1k 0

ym +

k | t)

d d , d f u0 (t

pt k

ð10:86Þ

Controlled output

Plant

Controller Structure

d p (t k)



z

1 1k 1

z

1 ck1

Disturbances

zm vm

u (t k ) Measured output or observations signal

Fig. 10.8 Implementation of NGMV state-dependent/qLPV controller structure

10.5

Controller Structure and Implementation

455

Remarks • The controller structure that results from (10.86) is shown in Fig. 10.8. • The matrix W 1k0 must be known explicitly but W 1k1 ðz1 ; :Þ is a nonlinear operators that can be in a black-box model form. • The implementation of the family of NGMV controllers in (10.86) is straightforward but care must be taken about the algebraic-loop that is involved, as explained in Chap. 5 (Sect. 5.4). The assumptions made in this “simplified” case enables the algebraic-loop problem to be avoided, since the inner feedback-loop in Fig. 10.8 includes at least a unit-step delay. Further simplification: A further simplification is possible if the plant can be totally represented by the qLPV model. If the input subsystem is not needed in the plant model define W 1k ¼ I, and if it is assumed F ck is linear let F ck ðz1 Þ ¼ Fck0 þ z1 Fck1 ðz1 Þ. In this case, the expression for the optimal control (10.83) simplifies as 
1
uðtÞ ¼ Fck0 þ E p t þ k dp ðt þ kÞ  Cp t þ k ^xðt þ kjtÞ  Fck1 ðz1 Þuðt  1Þ ð10:87Þ

10.6

Properties and Stability of the Closed-Loop

For linear systems it was noted in Chap. 2 that stability is ensured for Generalized Minimum Variance control designs when the combination of a control weighting and an error weighted plant model is strictly minimum-phase [14]. For nonlinear systems, it is shown below that a related operator equation must have a stable inverse. First assume the stochastic inputs are null, so that xðt þ 1Þ ¼ At xðtÞ þ Bt u0 ðt  kÞ þ dd ðtÞ The state-vector may, therefore, be written as xðtÞ ¼ ðzI  At Þ1 ðBt u0 ðt  kÞ þ dd ðtÞÞ

ð10:88Þ

where the resolvent operator for the augmented system Ut ¼ ðzI  At Þ1 ¼ ðI  z1 At Þ1 z1 The state can now be represented as xðtÞ ¼ Ut ðBt u0 ðt  kÞ þ dd ðtÞÞ

ð10:89Þ

456

10

LPV and State-Dependent Nonlinear Optimal Control

It also follows that the expression for the future state, predicted k-steps ahead, becomes: ^xðt þ kjtÞ ¼ xðt þ kÞ ¼ Ut þ k ðBt þ k u0 ðtÞ þ dd ðt þ kÞÞ

ð10:90Þ

The condition for optimality follows from setting the predicted /0 ðtÞ in (10.76) to zero ^ ðt þ kjtÞ ¼ dp ðt þ kÞ þ Cp t þ k ^xðt þ kjtÞ þ ðE p t þ k W 1k þ F ck ÞuðtÞ ¼ 0 / 0 Thus, in this special case of the stochastic inputs being null dp ðt þ kÞ þ Cp t þ k ðUt þ k ðBt þ k u0 ðtÞ þ dd ðt þ kÞÞÞ þ ðE p t þ k W 1k þ F ck ÞuðtÞ ¼ 0 The condition for optimality may be expressed as


 ðE p t þ k þ Cp t þ k Ut þ k Bt þ k ÞW 1k þ F ck uðtÞ ¼ dp ðt þ kÞ  Cp t þ k Ut þ k dd ðt þ kÞ ð10:91Þ This enables the following expressions to be obtained. Optimal control:


1
uðtÞ ¼ ðCp t þ k Ut þ k Bt þ k þ E p t þ k ÞW 1k þ F ck dp ðt þ kÞ  Cp t þ k Ut þ k dd ðt þ kÞ

ð10:92Þ Nonlinear subsystem output: 
1
ðW 1k uÞðtÞ ¼ W 1k ðCp t þ k Ut þ k Bt þ k þ E p t þ k ÞW 1k þ F ck dp ðt þ kÞ  Cp t þ k Ut þ k dd ðt þ kÞ

ð10:93Þ System plant outputs:
1
 ðWuÞðtÞ ¼ W ðCp t þ k Ut þ k Bt þ k þ E p t þ k ÞW 1k þ F ck dp ðt þ kÞ  Cp t þ k Ut þ k dd ðt þ kÞ

ð10:94Þ Stability of the closed-loop: From these equations, a necessary condition for stability is that the operator ððCp t þ k Ut þ k Bt þ k þ E p t þ k ÞW 1k þ F ck Þ should be finite-gain m2 stable. For the sufficient condition for optimality some further structure on the nature of the nonlinearity needs to be assumed (see Chap. 4 and [15]). The work of Safonov and Athans [16] might be revisited for analyzing stability and robustness of this type of system. They described an approach to simplify a feedback controller that used LQ control gains and an extended fixed gain Kalman filter, and analyzed the stability and robustness properties.

10.6

Properties and Stability of the Closed-Loop

457

10.6.1 Cost-Function Weighting Choice The two main approaches to weighting selection, namely free choice or PID inspired, have been used in many of the examples and applications described. Both methods are relatively simple and should be tried since it is difficult to say which will be best for particular applications. The free choice method of cost-function weighting selection was described in Chap. 5 (Sect. 5.2.1), based upon the use of parameterized frequency-dependent weighting functions. The PID inspired approach for the qLPV state-equation model in this chapter will now be considered. Recall that a method was described in Chap. 5 (Sect. 5.2.2) for obtaining the starting values of the NGMV cost-function weightings. The same result is described briefly below but using a different argument in the derivation. The problem here is more general, being for a qLPV plant model and a two degrees of freedom controller. However, the philosophy is similar to that described in Chaps. 5 and 8. From Eqs. (10.34) and (10.89), the predicted weighted error ep ðt þ kÞ ¼ dp ðt þ kÞ þ Cp t þ k Ut þ k ðBt þ k u0 ðtÞ þ dd ðt þ kÞÞ þ E p t þ k u0 ðtÞ
 ¼ dp ðt þ kÞ þ Cp t þ k Ut þ k dd ðt þ kÞ þ Cp t þ k Ut þ k Bt þ k þ E p t þ k u0 ðtÞ ð10:95Þ Assume that the measured outputs and those to be controlled are the same in this discussion, and the error weighting is to be applied to the output tracking error. Then, from Eq. (10.25) the error weighting term can be written as P c W 0k ¼ Cp t þ k Ut þ k Bt þ k þ E p t þ k

ð10:96Þ

and from (10.95) the predicted weighted error ep ðt þ kÞ ¼ dp ðt þ kÞ þ Cp t þ k Ut þ k dd ðt þ kÞ þ P c ðW 0k u0 ÞðtÞ This result (10.96) may be used to simplify the operator in the control and output Eqs. (10.92)–(10.94): ðCp t þ k Ut þ k Bt þ k þ E p t þ k ÞW 1k þ F ck ¼ ðP c W 0k W 1k þ F ck Þ

ð10:97Þ

As noted in the previous section, the inverse of this operator must be finite-gain stable to satisfy the necessary condition for stability of the closed-loop system. A simple method of finding initial choices for the weightings is to use the method described in Chap. 5 (Sect. 5.2.2). It was argued that if there exists a PID controller KPID that will stabilize the nonlinear system, without transport-delay elements, then cost-weightings could be defined to guarantee the existence of a stable inverse for the operator that determines the necessary condition for stability (equivalent to (10.97) for this chapter). This operator ðP c W 0k W 1k þ F ck Þ was referred to earlier as the generalized plant model. In most cases the control

458

10

LPV and State-Dependent Nonlinear Optimal Control

weighting may be assumed linear and the ratio of weightings can be chosen as F 1 ck P c ¼ KPID . Here KPID denotes a controller that stabilizes the feedback-loop for a plant with model W 0k W 1k . This definition of the cost-function weightings to be equal to a PID control law provides a starting point for the selection of cost-function weightings. To summarize, the operator in (10.97) must have a stable inverse for the NGMV control law to be stabilizing. A stabilizing PID controller may be available if the design is to improve an existing process, or it may be found by simulation. There is no guarantee of stability when moving away from this cost-function weighting choice, but it provides a simple method of finding the starting values of the weightings. There is an approximation involved in the argument when the PID controller is obtained based upon a real working system since the result (10.97) applies to the model without the delay term. However, in practice, if a controller stabilizes a plant with delays it may well stabilize the plant when the delay is null. The problem does not arise if a suitable PID controller is found by simulation where the delay can be omitted. Note that any low-order controller that stabilizes the system can be used for this purpose and not only a PID controller, however, the cost-weightings should not be high-order since this increases the order of the controller.

10.6.2 Alternative Expression for Optimal Control The controller can be represented in an alternative form where the paths through the Kalman filter are separated. This is useful for the analysis in the next section. First recall from Eq. (10.80)
 ^ ðt þ kjtÞ ¼ dpd ðt þ kÞ þ Cp t þ k Ak ^xðtjtÞ þ ðE p t þ k þ Cp t þ k T ðk; z1 ÞÞW 1k þ F ck uðtÞ / 0 t

Substituting from (10.65)
 ^ ðt þ kjtÞ ¼ dpd ðt þ kÞ þ Cp t þ k Ak Tf 1 ðzm ðtÞ  dm ðtÞÞ þ Tf 2 u0 ðtÞ þ df ðtÞ / 0 t
 þ ðE p t þ k þ Cp t þ k T ÞW 1k þ F ck uðtÞ ^ ðt þ kjtÞ ¼ 0), as in the Optimal control: Setting the predicted value to zero (/ 0 previous analysis, the condition for optimality yields the optimal control as
k k uðtÞ ¼ F 1 ck dpd ðt þ kÞ  Cp t þ k At df ðtÞ  Cp t þ k At Tf 1 ðzm ðtÞ  dm ðtÞÞ ð10:98Þ  Cp t þ k Akt Tf 2 u0 ðtÞ  ðE p t þ k þ Cp t þ k T ÞW 1k uðtÞ The controller structure shown in Fig. 10.5 involves the predictor acting on the current state estimates from the Kalman Filter. However, it is useful to separate the

10.6

Properties and Stability of the Closed-Loop

459

Controller

-

zm (t ) d m (t ) k pt k

d m (t ) +

u0

t

k pt k

t

Tf 1 Tf 2

+

-

+

-

k)

ck

u

1

+

1k

( k pt k

Disturbance and noise

+

+ +

Kalman predictor

d pd (t

Nonlinear plant

t

pt k

+

pt k

u0

)

d f (t )

Observations zm (t )

Fig. 10.9 Feedback control signal generation and controller modules

two different paths from the observations and the control inputs as shown in Fig. 10.9, based on Eq. (10.98).

10.6.3 Use of Future Reference Information The inferred output and the condition for optimality in Eq. (10.76) depend upon the deterministic and estimated stochastic components of the reference signal k-steps ahead. This may be demonstrated since the signal dd ðtÞ in (10.36) and the signal dp ðt þ kÞ ¼ E p ðrd ðt þ kÞ  dðt þ kÞÞ both depend upon the deterministic component of the reference rd ðtÞ. Moreover, the predicted state-estimate will depend upon the stochastic component of the reference, since 2

Cp t þ k ^xðt þ k jtÞ ¼ ½ E p C0

E p Cd

E p Crp

3 ^x0 ðt þ k jtÞ 6 ^xd ðt þ kjtÞ 7 7 C p jt þ k 6 4 ^xr ðt þ kjtÞ 5 ^xp ðt þ k jtÞ

where E p Crp^xr ðt þ kjtÞ ¼ E p^xrp ðt þ k jtÞ ¼ E p^r ðt þ kjtÞ. Unlike predictive controllers the optimal control solution involving the k-steps ahead prediction can only utilize k-steps of future reference information. That is, unlike the predictive controls the number of steps of future reference knowledge p that is required can be set equal to k. It follows that if a system has large process delays, relative to the dominant plant time-constant, then future reference or setpoint changes may be valuable in NGMV control, but if the transport-delay is small, the introduction of limited future reference knowledge may have little impact.

460

10

LPV and State-Dependent Nonlinear Optimal Control

10.6.4 Enhanced NGMV Control The basic NGMV algorithm may be enhanced to provide a future setpoint tracking capability in an Extended Nonlinear Generalized Minimum Variance (ENGMV) algorithm [19]. This requires the criterion to include at least one future tracking error component. The ENGMV controller is an attempt to obtain some of the properties of the NGMV solution combined with the benefits of predictive control. The solution can be related to a basic NGMV controller but the computations allow future setpoint information and end-state weightings to be introduced. However, the future information is only built in a rather restricted way so that the computations remain simple. It is, therefore, a compromise, which provides some future setpoint tracking capability and possibly improved robustness relative to NGMV, but with a little additional complication.

10.7

Links to the Smith Predictor

As in previous chapters, the optimal controller can be expressed in a similar form to that of a Smith Predictor to provide some confidence in the design, but this limits the applications to open-loop stable systems. A Nonlinear Smith Predictor form of this controller can be derived using similar arguments to those in Chap. 5 (Sect. 5.5). Only the main points will, therefore, be summarized. First note the measured output from (10.23) and (10.21): m ym ðtÞ ¼ dm ðtÞ þ Cm t xðtÞ þ E t u0 ðt  kÞ  1 1 m m ¼ dm ðtÞ þ Cm t ðzI  At Þ Bt þ E t u0 ðt  kÞ þ Ct ðzI  At Þ ðDt nðtÞ þ dd ðtÞÞ

Recall the resolvent operator is defined as Ut ¼ ðzI  At Þ1 and note from this result that the transfer-operator from the signal u0 ðt  kÞ to the measured output may be written as 1 m m m W 0k ¼ Cm t ðzI  At Þ Bt þ E t ¼ Ct Ut Bt þ E t

ð10:99Þ

The expression for the measured output in terms of this plant model follows as m ym ðtÞ ¼ dm ðtÞ þ W 0k u0 ðt  kÞ þ Cm t Ut Dt nðtÞ þ Ct Ut dd ðtÞ

ð10:100Þ

Operator relationships: Some useful operator relationships are now obtained that will be needed in deriving an interesting form of the NGMV controller. From (10.69):

10.7

Links to the Smith Predictor

461

 1 m k Tf 1 C m ðzI  A Þ B þ E þ Tf 2 ¼ Uðz1 ÞBt zk t t t t z Substituting from (10.99): Tf 1 W 0k zk þ Tf 2 ¼ Uðz1 ÞBt zk

ð10:101Þ

From the definition of the finite pulse response term in (10.50) note T ¼ Ut þ k Bt þ k  At þ k1 At þ k2 . . .At Ut zk Bt þ k

ð10:102Þ

where Akt ¼ At þ k1 At þ k2 . . . At . Thence, from (10.101) we obtain for the sum of the following terms, the result

Compensator dm

-

k

+

pt k

t

pt k

t

u0

k

Tf 1

+ +

Tf 2

+ +

+ +

ck

-

pt k pt k

u

1

+

zm

+ 1k

k

d pd (t k )

Disturbance and noise

Nonlinear plant

t

+

1

pt k

d f (t )

k pt k

t

Tf 1

(k,z ) 0k

z

u0

k

0k

z

-

k

+

Fig. 10.10 Modifications to the controller structure shown dotted Disturbance and noise

Compensator

dm

+

k pt k

t

-

Tf1

u

1

+

Kalman predictor

k t

Plant

1k

d pd (t k ) pt k

+

ck

+ + pt k

+

pt k

t k

u0

t k

d f (t ) 0k

p

-

z

k

+

Fig. 10.11 Nonlinear Smith Predictor compensator and internal model

Output

z

462

10

LPV and State-Dependent Nonlinear Optimal Control

E p t þ k þ Cp t þ k T þ Cp t þ k Akt Tf 1 W 0k zk þ Cp t þ k Akt Tf 2
 ¼ E p t þ k þ Cp t þ k T þ Akt Uðz1 ÞBt zk ¼ E p t þ k þ C p t þ k Ut þ k B t þ k

ð10:103Þ

Smith predictor form: Changes may be made to the qLPV subsystems in the controller shown in Fig. 10.9, by adding and subtracting equivalent terms to obtain the system shown in Fig. 10.10. The three inner-loop blocks, with the input signal u0 ðtÞ in Fig. 10.10, may then be combined using the result in (10.103). The controller can, therefore, be expressed in the Smith predictor structure shown in Fig. 10.11. If the NGMV controller is implemented in this form it cannot, of course, stabilize an open-loop unstable system. However, it does provide a useful intuitive compensator structure for nonlinear systems with transport delays. It is interesting that the transfer-operator from the control signal u to the feedback signal p in Fig. 10.11 is null when the model W ¼ zk W k ¼ zk W 0k W 1k matches the plant model. The control action, due to the reference signal r(t) is not, therefore, due to feedback but involves the open-loop stable compensator Cp t þ k Akt Tf 1 that has an input from the disturbance estimate, and the contribution of the inner “nonlinear” feedback-loop. This inner-loop feedback path transfer includes
 E p t þ k þ Cp t þ k Ut þ k Bt þ k ¼ P c W 0k ðt þ kÞ The open-loop transfer-operator for the inner-loop, therefore, includes the weightings F 1 ck P c acting like an inner-loop controller. If these weightings are chosen by the PID motivated approach in Sect. 10.6.1, this inner-loop will replicate a stabilizing PID controlled response behaviour. This follows because the open-loop plant is stable, the inner-loop is stable (due to choice of weightings) and there are only stable terms in the input block. Thus, for this initial choice of weightings and for an open-loop stable system, stability will be ensured.

10.8

SI Automotive Engine Multivariable Control

The control problem considered in the following is the multivariable extension of the example used in Chap. 9 on Spark Ignition (SI) engines. The engine torque is added as the second controlled output. Simultaneous air–fuel ratio and torque control in spark ignition engines have been the subject of many investigations. Authors mostly use physical mean-value models to describe engine dynamics; however, in this example, a simpler multivariable nonlinear regression model identified from driving cycle data is again considered. The model is assumed to have two outputs: torque (TRQ) and air–fuel ratio or lambda (k), and three inputs: throttle position (TPS), fuel pulse width (FPW) and the engine speed (RPM). Other

10.8

SI Automotive Engine Multivariable Control

463

Ambient conditions

SP

Throttle flow

Torque production

Intake manifold

TQ

Ratio

FP Fuel path

RPM

Lambda sensor

Fig. 10.12 Black-Box model of a spark ignition engine

engine variables are considered external disturbances or internal states of the model, and are not modelled explicitly. The “outside” view of the system with the basic internal structure captured by the regression is shown in Fig. 10.12. In a conventional “old-fashioned” SI engine control scheme, the throttle plate is directly linked to the acceleration pedal and it regulates the amount of air entering the engine. The torque produced, therefore, depends upon the throttle position determined by the driver. The fuel flow is usually controlled by a single-loop feedforward plus feedback controller that aims to maintain a steady stoichiometric ratio of the air–fuel mixture. The controller considered here uses an Electronic Throttle Control (ETC), which decouples the throttle from the pedal (drive-by-wire). In this case, the optimal torque setpoint is determined by the pedal position, together with other measurements, design specifications and nonlinear calibration mappings. The setpoint angle for the throttle servo is driven from the ETC. This enables multivariable optimal control to be implemented for the control of torque and lambda.

10.8.1 NARX Model Identification The Nonlinear Autoregressive Exogenous (NARX) model of the engine was identified from Federal Test Procedure (FTP) driving cycle data (Environmental Protection Agency Federal Test Procedure). The data were used to identify a multi-input, multi-output (MIMO) NARX model. Consider a model with a control input vector ut, a disturbance input vector dt, and an output vector yt, as shown in Fig. 10.13. The objective is to fit the assumed NARX model to the measurement data. The NARX model structure considered here includes linear and quadratic inputs, as well as a constant input, followed by a Linear Time-Invariant (LTI) subsystem model. The NARX model may be represented in a state-space form as

464

10

LPV and State-Dependent Nonlinear Optimal Control ut

Fig. 10.13 System model in a NARX structure

NARX model ut dt 2 2 ut (.) 2 d t 2 (.)

Fig. 10.14 NARX model structure for identification (explicit time-delays shown)

yt

System

dt

yˆ t

LTI Model

1

1 TPS FPW RPM

NARX model

z z

xt þ 1 ¼ Uxt þ Gut þ E yt ¼ Hxt þ Aut þ F

kTRQ

TR

k

ð10:104Þ

where  T ut ¼ TPSt ; FPWt ; Nt ; TPS2t ; FPWt2 ; Nt2 and yt ¼ ½TRQ; kT The MIMO transfer-function model was first identified from the first half of the data (identification set), and the parameters were estimated using least squares. The resulting model was then converted to a state-space form. The output time-delays kTRQ and kk were chosen to minimize the difference between the data and model output, in the mean square error sense, and their optimal values were found as kTRQ ¼ 4 and kk ¼ 11 events. The input-delay on Fuel Pulse Width (FPW) was assumed incorporated in the model. The resulting model structure is shown in Fig. 10.14. qLPV engine model: The special structure of the model allows the qLPV matrices to be defined using xt þ 1 ¼ Uxt þ ðG1 þ G4 TPSt ÞTPSt þ ðG2 þ G5 FPWt ÞFPWt þ ðG3 þ G6 Nt ÞNt þ E yt ¼ Hxt þ ðA1 þ A4 TPSt ÞTPSt þ ðA2 þ A5 FPWt ÞFPWt þ ðA3 þ A6 Nt ÞNt þ F ð10:105Þ By redefining u ¼ ½TPS; FPW T and d ¼ ½N; 1 T this can be rewritten as

10.8

SI Automotive Engine Multivariable Control

xt þ 1 ¼ Uxt þ ½ ðG1 þ G4 TPSt Þ yt ¼ Hxt þ ½ ðA1 þ A4 TPSt Þ

465

ðG2 þ G5 FPWt Þ ut þ ½ ðG3 þ G6 Nt Þ ðA2 þ A5 FPWt Þ ut þ ½ ðA3 þ A6 Nt Þ

E dt E dt ð10:106Þ

with parameter and input-dependent matrices. For practical control design, the matrices involving TPSt and FPWt must be evaluated using the control signal values as applied in the previous step.

10.8.2 Control Design and Simulation The frequency responses for the dynamic design weightings are shown in Fig. 10.15. Simple PID motivated error weighting design was used and a constant control weighting. The reference signals were modelled as integrators and the disturbance model was driven by a white noise source. Since all system-dynamics are included in the qLPV subsystem, the nonlinear model W 1k was set to the identity. Figure 10.16 shows the results of the multivariable NGMV control of the qLPV modelled engine with the actual driving cycle data used as the input to the simulation. The measured torque signal was used as the torque demand, and the air–fuel ratio setpoint was set to a constant value (unity). The results are identical to those for a “black-box” NGMV control design, where the entire plant model is included in the black-box term. This was confirmed by a separate simulation. This gives some confidence in the implementation of the controller.

Singular Values 100 Pc Fck

Singular Values (dB)

80

60

40

20

0

-20 -6 10

10

-4

10

-2

10

0

Frequency (rad/s)

Fig. 10.15 Frequency responses: magnitude of the reference model, error weighting and control weighting

466

10 Torque

350

Air-fuel ratio

1.1 set-point TQ

300

1.05

250 200

[-]

TRQ [Nm]

LPV and State-Dependent Nonlinear Optimal Control

150 100

1 0.95

50 0

1

1.2

1.4

1.6

1.8

2

time [events]

0.9

2.2 10

1.6

1.8

2

2.2 4 10

2

2.2

10

25

FPW [ms]

TPS [deg]

1.4

Fuel Pulse Width

12

30

20 15 10

8 6 4 2

5 0

1.2

time [events]

Throttle Position

35

1

4

1

1.2

1.4

1.6

1.8

2

time [events]

0

2.2 10

4

1

1.2

1.4

1.6

1.8

time [events]

10

4

Fig. 10.16 NGMV control of the qLPV modelled engine (fragment of dataset)

The qLPV approach which involves the separate input-operator and qLPV models is more general than in previous chapters. Since the plant model was stable in this example both modelling approaches are equivalent. In the case of open-loop unstable systems, the model for the unstable section must be in the output subsystem model. The assumption that the black-box term is stable would be violated by placing the full open-loop unstable plant model in this input subsystem operator term.

10.9

Concluding Remarks

An optimal controller for nonlinear multivariable systems was described that extends the family of NGMV controllers to more general plant models and to a two degrees of freedom controller structures. The solution applies to systems represented by a combination of nonlinear operator and qLPV model forms. The inclusion of the qLPV subsystem model provides the main generalization. It may be used to represent open-loop unstable plants with input or output nonlinearities [20, 21]. The qLPV model covers state-dependent structures, linear parameter varying systems and even some hybrid control systems. The closed-loop stability of the system depends upon the existence of a stable inverse for a particular loop operator. This operator depends upon the cost-function weighting definitions. A starting point for the cost-function weighting selection was provided by the relationship of the weightings to an existing stabilising controller. The assumption made in the specification of the cost-index (sum of error and

10.9

Concluding Remarks

467

control terms) leads to a relatively simple control solution. The structure of the problem means that most of the equations to be solved are similar to those for a linear system. The NGMV controller, even for qLPV systems, is therefore, simple to compute, understand and implement. A qLPV predictive controller can also be developed using an extension of this NGMV philosophy [22] and a receding horizon strategy [23]. This is described in the next chapter. The stability of NGMV controlled systems, in the presence of modelling uncertainties, requires further analysis [16]. It may be possible to introduce learning features in the controller by using a neural network in the black-box input subsystem [24, 25]. A related NGMV control approach was developed for piecewise affine systems by Pang and Grimble [26], that applied to some classes of hybrid-system. Links to the work of James Taylor and co-workers [27–30] can also be established. The class of NGMV controllers, therefore, applies to a wide class of systems and problems [31–33] but further applications experience is needed.

References 1. Cloutier JR, D’ Souza CN, Mracek CP (1996) Nonlinear regulation and nonlinear H∞ control via the state-dependent Riccati equation technique. In: International conference on nonlinear problems in aviation and aerospace, Daytona Beach, Florida, pp 117–130 2. Cloutier JR (1997) State-dependent Riccati equation techniques: an overview. In: American control conference, Albuquerque, New Mexico, pp 1072–1073 3. Sznair M, Cloutier J, Jacques D, Mracek C (1998) A receding horizon state dependent Riccati equation approach to sub optimal regulation of nonlinear systems. In: 37th IEEE conference on decision and control, Tampa, Florida, pp 1792–1797 4. Hammett KD (1997) Control of non-linear systems via state-feedback state-dependent Riccati equation techniques. PhD dissertation, Air Force Institute of Technology, Dayton, Ohio 5. Hammett KD, Hall CD, Ridgely DB (1998) Controllability issues in nonlinear state-dependent Riccati equation control. AIAA J Guid Control Dyn 21(5):767–773 6. Cloutier JR, Stansbery DT (2001) Nonlinear hybrid bank-to-turn/skid-to-turn autopilot design. In: Proceedings of the AIAA guidance, navigation and control conference, Montreal, Canada 7. Pittner J, Simaan MA (2012) Advanced technique for control of the threading of a tandem hot-metal-strip rolling mill. IEEE Trans Ind Appl 48(5):1683–1691 8. Chang I, Chung SJ (2009) Exponential stability region estimates for the state-dependent Riccati equation controllers. In: Joint 48th IEEE conference on decision and control and 28th Chinese control conference, Shanghai, China 9. Cimen T (2008) State-dependent Riccati equation (SDRE) control: a survey. In: 17th IFAC world congress, Seoul, Korea, pp 3761–3775 10. Grimble MJ, Johnson MA (1988) Optimal multivariable control and estimation theory: theory and applications, vol I and II. Wiley, London 11. Anderson B, Moore J (1979) Optimal filtering. Prentice Hall, Englewood Cliffs 12. Clarke DW, Hastings-James R (1971) Design of digital controllers for randomly disturbed systems. Proc IEE 118(10):1502–1506 13. Grimble MJ (2004) GMV control of nonlinear multivariable systems. In: UKACC conference control, University of Bath, 6–9 September

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14. Grimble MJ (2005) Non-linear generalised minimum variance feedback. Feed Track Control Autom 41:957–969 15. Grimble MJ (2005) Robust industrial control. Wiley, Chichester 16. Safonov MG, Athans M (1978) Robustness and computational aspects of nonlinear stochastic estimators and regulators. IEEE Trans Autom Control 23:717–725 17. Grimble MJ, Majecki P (2005) Nonlinear generalised minimum variance control under actuator saturation. In: IFAC world congress, Prague 18. Åström KJ (1979) Introduction to stochastic control theory. Academic Press, London 19. Grimble MJ, Majecki P, Katebi R (2017) Extended NGMV predictive control of quasi-LPV systems. In: 20th IFAC world congress, Toulouse, France, pp 4162–4168 20. Grimble MJ, Pang Y (2007) NGMV control of state-dependent multivariable systems. In: 46th IEEE conference on decision and control, pp 1628–1633 21. Pang Y, Grimble MJ (2010) NGMV control of delayed piecewise affine systems. IEEE Trans Autom Control 55(12):2817–2821 22. Grimble MJ, Majecki P, Giovanini L (2007) Polynomial approach to nonlinear predictive GMV control. In: European control conference, Koss, Greece 23. Mayne DQ, Michalska H (1990) Receding horizon control of constrained non-linear systems. IEEE Trans Autom Control 35(7):814–824 24. Lewis FL, Ge SS (2006) Neural networks in feedback control systems. In: Kutz M (ed) Mechanical engineer’s handbook, instrumentation, systems, controls, Chapter 19. Wiley, New York 25. Zhu Q, Ma Z, Warwick K (1999) Neural network enhanced generalised minimum variance self-tuning controller for nonlinear discrete-time systems. IEE Proc Control Theory Appl 146(4):319–326 26. Pang Y, Grimble MJ (2009) State dependent NGMV control of delayed piecewise affine systems. In: 48th IEEE conference on decision and control, joint with 28th Chinese control conference, pp 7192–7197 27. Taylor CJ, Chotai A, Burnham KJ (2011) Controllable forms for stabilising pole assignment design of generalised bilinear systems. Electron Lett 47(7):437–439 28. Taylor CJ, Chotai A, Young PC (2009) Nonlinear control by input-output state variable feedback pole assignment. Int J Control 82(6):1029–1044 29. Taylor CJ, Shaban EM, Stables MA, Ako S (2007) Proportional-integral-plus (PIP) control applications of state dependent parameter models. IMECHE Proc Part I J Syst Control Eng 221(17):1019–1032 30. Taylor CJ, Pedregal DJ, Young PC, Tych W (2007) Environmental time series analysis and forecasting with the captain toolbox. Environ Model Softw 22(6):797–814 31. Grimble MJ, Majecki P (2008) Nonlinear GMV control for unstable state-dependent multivariable models. In: 47th IEEE conference on decision and control, Cancun, pp 4767–4774 32. Grimble MJ, Majecki P (2006) H∞ control of nonlinear systems with common multi-channel delays. In: American control conference, Minneapolis, pp 5626–5631 33. Grimble MJ (2007) GMV control of non-linear continuous-time systems including common delays and state-space models. Int J Control 80(1):150–165

Chapter 11

LPV/State-Dependent Nonlinear Predictive Optimal Control

Abstract This is the second chapter where the nonlinear plant is represented by a more general LPV or state-dependent model combined with a black-box operator term. However, in this case the model predictive control problem is considered with some enhancements including, for example, a connection matrix defining the pattern of control moves allowed in the prediction horizon. The generalized predictive controller is first considered for the plant without the operator term since this provides a simple model predictive control solution. In the later part of the chapter, the full plant model is considered and a nonlinear control weighting term is also included in the cost-function. The rotational link example considered has a natural state-dependent form and it illustrates the modelling and control design options. The final section considers the problem of restricted structure control where the aim is to use a low-order controller within the feedback loop to simplify retuning. There are several lessons since the approach is unusual, particularly regarding the improvement in robustness properties that may occur, and the way the gains vary similar to an adaptive system.

11.1

Introduction

A similar model structure is used for the plant that was employed in the previous chapter. The system has an output subsystem, which is a linear parameter varying or state-dependent model. The term quasi-LPV or qLPV will again be used to describe this type of subsystem. The input subsystem can be black-box or an unstructured nonlinearity, but the initial analysis concerning the Nonlinear Generalized Predictive Control (NGPC) problem will assume the plant is represented by only the output subsystem involving the qLPV model. In the previous chapters on predictive control, the Generalized Predictive Control (GPC) problem was introduced first. A similar strategy is followed in this chapter but since the plant model is initially treated as being qLPV, it is termed a Nonlinear GPC or NGPC problem.

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_11

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The main Model Predictive Control (MPC) problem to be considered is that discussed in the latter part of the chapter, which is a form of Nonlinear Predictive Generalized Minimum Variance (NPGMV) control. In this case, the plant involves both the nonlinear black-box subsystem and the qLPV output subsystem model. The objective is to design an industrial controller, which has the advantages of the popular MPC or Generalized Predictive Control (GPC) algorithms, but for quite general nonlinear systems. This is similar to the approach in Chap. 9 but is for a much more general system description. Nonlinear systems have more complex behaviour than linear systems, including limit cycle responses and even chaotic behaviour. There are of course some existing simple nonlinear control design methods, like gain scheduling [1], that rely on linearization, but a linearized model may not be representative or controllable at certain operating points. The proposed controller does not rely on this type of local linearization and instead provides a global solution for a flexible choice of the system model. This is important because the certification of systems such as flight controls at a large number of linearized operating points is an expensive process and a controller that is valid over a much wider range provides a significant benefit. The control strategy builds upon the previous results on Nonlinear Generalized Minimum Variance (NGMV) control, for nonlinear model-based multivariable systems (as described in Chaps. 4, 5, 8 and 10 and Grimble [2]). Two nonlinear predictive control solutions are obtained in this chapter. The first Nonlinear Generalized Predictive Control (NGPC) is of value in its own right but is also a step in the solution of the second more general control problem (Nonlinear Generalized Minimum Variance Control). In both cases a multi-step predictive control cost-function is to be minimized, which can be used with different error and control signal cost horizons. The NGPC solution is quite simple and applies to a reasonably large class of nonlinear control problems. It is therefore considered a practical option for some applications. To improve the generality of the results, two alternative types of control signal input to the plant model are considered. The first control term is the traditional absolute control signal input and this signal is penalized in the predictive control criterion in the usual way. However, it is sometimes desirable to augment the plant model with an integrator to provide a simple way of introducing integral action. In the augmented system, the new system input is then the change (or increment) in control action, and this signal is also therefore penalized in the performance criterion. The results obtained in the following will apply to both cases, and a parameter change between b = 0 and b = 1 will determine the type of control action that is included in the criterion. The criterion includes dynamic cost-function weightings on both error and control signal terms that must be chosen to satisfy both performance and stability/robustness requirements. A simple method is proposed in Sect. 11.6.1 for obtaining starting values for this choice of weightings, in a limiting situation. There is a rich history of research on nonlinear predictive control [3–16], but the approach taken here is somewhat different, since it is closer in spirit to that of a model-based fixed-structure control design for a time-varying system. An

11.1

Introduction

471

advantage of this predictive control approach is that the plant model can be in a rather general nonlinear operator form. This can involve hard nonlinearities, transfer operators or nonlinear function look-up tables. No structure needs to be assumed for the input nonlinear subsystem, but this subsystem must be open-loop stable. As described in Chap. 2 for the equivalent linear system designs, stability is ensured when the combination of a control weighting function and an error weighted plant model is strictly minimum phase. For qLPV nonlinear systems, a similar result is described later in Sect. 11.6.

11.2

Nonlinear Operator and qLPV System Model

The assumption is made that the plant model can be decomposed into a set of delay terms, a general input black-box nonlinear subsystem (that has to be stable) and a qLPV output subsystem that can include unstable modes. The plant description is assumed similar to that for the NGMV control problem discussed in Chap. 10, which can involve qLPV models [17]. The black-box operator input subsystem model is not so important when the qLPV model is included, since the qLPV model may also represent some nonlinear elements. These nonlinear elements are of a more restricted nature but the NGPC solution that is presented first and only includes the qLPV model is simple and practical to use. Consider the feedback control system shown in Fig. 11.1, where the output plant subsystem is represented in qLPV form that can include the state-dependent and LPV models [18]. The plant input subsystem is a general nonlinear operator denoted as W 1 . Nonlinear Input Subsystem Plant Model: ðW 1 uÞðtÞ ¼ zk ðW 1k uÞðtÞ

ð11:1Þ

where u0 ðtÞ ¼ ðW 1k uÞðtÞ. The output subsystem may also be nonlinear and the operator W 0 will denote the path from the second subsystem input to measured output, and without the explicit delay term, it is denoted by W 0k . Disturbances



r

d0d

Nonlinear operator subsystem

Nonlinear controller

u

d 0  d  yd +

u0

+ +

y

Controlled Output

ym

+ Measurement or observations signal

zm

qLPV subsystem

Fig. 11.1 Two degrees-of-freedom feedback control system

d0m  dm  ydm

+ +

Noise

vm

472

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LPV/State-Dependent Nonlinear Predictive Optimal Control

Nonlinear output subsystem plant model:
 ðW 0 u0 ÞðtÞ ¼ W 0k zk u0 ðtÞ

ð11:2Þ

A generalization to different explicit delays in different signal paths complicates the solution [19]. In this case, it may be better to absorb any delay terms into the subsystem models.

11.2.1 Signal Definitions The output of the system to be controlled y(t) in Fig. 11.1 includes the output of the qLPV (possibly state-dependent) block dynamics W 0 and both the deterministic d(t) and stochastic yd ðtÞ components of the disturbances. The measured output ym ðtÞ also includes deterministic dm ðtÞ and stochastic ydm ðtÞ components of the disturbance signals. The stochastic component is modelled by a qLPV disturbance model, driven by a white noise signal f0 ðtÞ. The measurement noise signal is denoted by vm ðtÞ and is assumed to be white and zero-mean, and to have a constant covariance matrix Rf ¼ RTf  0. There is no loss of generality in assuming that the zero-mean white noise signal f0 ðtÞ has an identity covariance matrix. The structure of the system again leads to a prediction equation that is only dependent on the qLPV model. The controlled output should follow the reference signal rðtÞ that is assumed deterministic and is shown in Fig. 11.1. This is a known signal for a given period into the future, determined by the predictive control cost-function horizon. The approach is slightly different from that used in Chap. 8, where a stochastic description of the reference signal was used. Signals: The signals may be listed as follows: x0 ðtÞ

Vector of n states in the qLPV plant subsystem and the input disturbance model. u0 ðtÞ Vector of m0 input signals to the qLPV subsystem. uðtÞ Vector of m control signals applied to the nonlinear subsystem. yðtÞ Vector of r plant output signals to be controlled. dðtÞ Vector of r deterministic disturbance signal values on the output to be controlled. yd ðtÞ Vector of r stochastic disturbance signal values on the output to be controlled. ym ðtÞ Vector of rm plant output signals that are measured. dm ðtÞ Vector of rm deterministic disturbance signal values on the measured output. ydm ðtÞ Vector of rm stochastic disturbance signal values on the measured output. zm ðtÞ Vector of rm observations or measured outputs.

11.2

Nonlinear Operator and qLPV System Model

473

rðtÞ ep ðtÞ

Vector of r known setpoint or reference signal values. Vector of m tracking output or error signals to be controlled including dynamic cost-weightings. d0 ðtÞ Vector of r output-disturbance signal values. d0m ðtÞ Vector of rm output-disturbance signal values. d0d ðtÞ Vector of q known input-disturbance signal values.

11.2.2 qLPV Subsystem Models The second subsystem that is assumed to have a quasi-LPV model form may now be introduced. It includes the plant output subsystem, disturbance model and the error weighting model. The system model is similar (leaving aside the black-box term), to those used for state-dependent Riccati equation optimal control solutions described in Chap. 10 [20–23]. The qLPV modelling approach was introduced in Chap. 1, and was employed in the previous chapter. Plant model: The nonlinear plant output subsystem is again assumed to have the following qLPV model form: x0 ðt þ 1Þ ¼ A0 ðx0 ; u0 ; pÞx0 ðtÞ þ B0 ðx0 ; u0 ; pÞu0 ðt  kÞ þ D0 ðx0 ; u0 ; pÞf0 ðtÞ þ G0 ðx0 ; u0 ; pÞd0d ðtÞ

ð11:3Þ

where the input signal channels include a common k-steps transport delay. The plant outputs to be controlled (not including weightings), and measured outputs (without measurement noise) may be written as follows: yðtÞ ¼ d0 ðtÞ þ C0 ðx0 ; u0 ; pÞx0 ðtÞ þ E 0 ðx0 ; u0 ; pÞu0 ðt  kÞ

ð11:4Þ

ym ðtÞ ¼ d0m ðtÞ þ C0m ðx0 ; u0 ; pÞx0 ðtÞ þ E 0m ðx0 ; u0 ; pÞu0 ðt  kÞ

ð11:5Þ

where x0 ðtÞ 2 Rn0 . The model matrices are functions of the states, inputs and parameters ðxðtÞ; u0 ðt  kÞ; pðtÞÞ. The deterministic component of the input disturbance is denoted by d0d ðtÞ. Disturbance: The total disturbance on the output to be controlled is composed of two terms d0 ðtÞ ¼ dðtÞ þ yd ðtÞ. That is, a known deterministic component dðtÞ and a stochastic component yd ðtÞ, which is determined by a subsystem defined below. Similarly the total disturbance on the measured output d0m ðtÞ ¼ dm ðtÞ þ ydm ðtÞ, where dm ðtÞ is a known deterministic disturbance signal. The stochastic disturbance component ydm ðtÞ is defined below. Disturbance on output model: The plant (11.3) often includes process noise f0 ðtÞ, which can be considered an input disturbance, and also a disturbance model contributing directly to the output. This output disturbance model, driven by zero-mean white noise x(t), may be implemented in a state-equation form, as follows:

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xd ðt þ 1Þ ¼ Ad xd ðtÞ þ Dd xðtÞ;

xd ðtÞ 2 Rnd

ð11:6Þ

yd ðtÞ ¼ Cd xd ðtÞ ydm ðtÞ ¼ Cdm xd ðtÞ These two output signals represent the two potential stochastic disturbance terms added to the controlled plant output and measured plant outputs, respectively. Error and observations: The error (involving the output to be controlled) and the observation signals (involving the measured output) are defined as follows: Error signal: eðtÞ ¼ rðtÞ  yðtÞ

ð11:7Þ

zm ðtÞ ¼ ym ðtÞ þ vm ðtÞ

ð11:8Þ

Observation signal:

Error weighting: The signal to be controlled will include the weighted tracking error in the system and this weighting may be implemented in state-equation form as follows: xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp ðrðtÞ  yðtÞÞ;

xp ðtÞ 2 Rnp

ye ðtÞ ¼ Cp xp ðtÞ þ E p ðrðtÞ  yðtÞÞ

ð11:9Þ ð11:10Þ

Ideal response model: If an ideal response model is required, this can be introduced into the solution relatively easily. The predictive control problem is a two degrees of freedom solution and the reference only enters the plant equations in (11.9) and (11.10). Thus, if the reference is denoted by rs ðtÞ, then the ideal response r(t) can be assumed to be driven by an ideal response model rðtÞ ¼ Wi ðz1 Þrs ðtÞ. The following results need not, therefore, include this model since it only involves adding a filter on the signal r(t). Integral action: The traditional method of introducing integral action in predictive controls is to augment the system input by adding an integrator. To include this option define an additional input subsystem, which becomes part of the control law: xi ðt þ 1Þ ¼ bxi ðtÞ þ Du0 ðt  kÞ;

xi ðtÞ 2 Rni

u0 ðt  kÞ ¼ bxi ðtÞ þ Du0 ðt  kÞ

ð11:11Þ ð11:12Þ

11.2

Nonlinear Operator and qLPV System Model

475

The input–output transfer-operator for this model (11.11), (11.12) is as follows: u0 ðt  kÞ ¼ bðz  bÞ1 Du0 ðt  kÞ þ Du0 ðt  kÞ ¼ ð1  bz1 Þ1 Du0 ðt  kÞ

ð11:13Þ

The Δ can be considered an operator in unit-delay terms D ¼ ð1  bz1 Þ. For b ¼ 1, this transfer is an integrator without additional delay and if b ¼ 0, then u0 ðt  kÞ ¼ Du0 ðt  kÞ which is the absolute control action normally used. The results can, therefore, apply to both cases: using the absolute control input ðb ¼ 0Þ, or using an augmented system with a rate of change of control input ðb ¼ 1Þ. It will be assumed that the plant model is augmented by this subsystem, making the input to the subsystem W 0k the signal Du0 ðt  kÞ. The case b ¼ 1 is the traditional method of augmenting the plant by an integrator to force integral action into a predictive controller. Dynamic control weighting: The predictive control will include the usual constant control weighting terms and for the NPGMV solution, a dynamic weighting that can be included like the F c ðz1 Þ control cost term in the previous NGMV controls. However, this latter term is not included in the NGPC control solution that is discussed in the first part of this chapter, and it may not be required when using the simplified NPGMV control solution discussed later in the chapter in Sect. 11.8.1 . Thus, it is useful to add a dynamic input signal costing subsystem that can be used in these cases. Define the input weighting subsystem as xu ðt þ 1Þ ¼ Au xu ðtÞ þ Bu Du0 ðt  kÞ;

xu ðtÞ 2 Rnu

yu ðtÞ ¼ Cu xu ðtÞ þ E u Du0 ðt  kÞ

ð11:14Þ ð11:15Þ

Remarks • The structure shown in Fig. 11.1 is quite general and it can describe the situation where the controlled and measured outputs are different or the same. • The incremental and absolute forms of control apply to the NGPC results presented in Sect. 11.4 that follows. • However, in the NPGMV control problem in Sect. 11.5 there is a problem in the incremental control case ðb ¼ 1Þ. That is, the input subsystem must be assumed absent (W 1k ¼ I), or linear, square and diagonal. This assumption is needed so that the integrator can easily be moved from the augmented system into the controller block. There is no such assumption needed in the absolute control case ðb ¼ 0Þ.

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11.2.3 Augmented System The qLPV state-space system model, for the augmented r  m multivariable system, shown in Fig. 11.2, may now be defined. Combining the plant, disturbance, integral, weighted error and weighted control equations, the state vector may be defined as h iT xðtÞ ¼ xT0 ðtÞ xTd ðtÞ xTi ðtÞ xTp ðtÞ xTu ðtÞ ð11:16Þ As in the previous chapter, to simplify the notation, the state or parameter dependence will not be shown explicitly. Thus, write At ¼ AðxðtÞ; u0 ðt  kÞ; pðtÞÞ and similarly for the other time-varying matrices. The use of the subscript t on the augmented system matrices is a slight abuse of notation but it simplifies the presentation. Thus the matrix at time t is denoted by At and at t + 1 is At þ 1 and so on. The augmented system state-space model will again be assumed controllable and observable, or stabilizable and detectable. The system equations and the weighted error or output to be controlled may be written, in terms of the augmented qLPV model, as follows: Augmented state model: xðt þ 1Þ ¼ At xðtÞ þ Bt Du0 ðt  kÞ þ Dt nðtÞ þ dd ðtÞ

ð11:17Þ

Controlled output: yðtÞ ¼ dðtÞ þ Ct xðtÞ þ E t Du0 ðt  kÞ

ð11:18Þ

ym ðtÞ ¼ dm ðtÞ þ Cm t xðtÞ þ E m t Du0 ðt  kÞ

ð11:19Þ

Measured output:

Observations output: zm ðtÞ ¼ vm ðtÞ þ dm ðtÞ þ Cm t xðtÞ þ E m t Du0 ðt  kÞ

ð11:20Þ

Weighted error: ep ðtÞ ¼ dp ðtÞ þ Cp t xðtÞ þ E p t Du0 ðt  kÞ

ð11:21Þ

with state xðtÞ 2 Rn . In most cases note that Ct ¼ Ct m and E t ¼ E t m . The relationship between the augmented system matrices and the plant, disturbance and weighting subsystems is established in the section below and this is summarized in the following section. The augmented system state-equation model in Eqs. (11.17)–(11.21) may be represented as in Fig. 11.2. Note that the augmented qLPV model has an input Du0 ðtÞ and the corresponding change in control input is denoted by DuðtÞ.

11.2

Nonlinear Operator and qLPV System Model

477 dp

qLPV nonlinear dynamics pt

NL plant model



Control signal

u

dd

u0 1k (.,.)

(.,.)

z I

uc

Explicit delays

NL control weighting

t

+

+

+

+

Weighted error

+

ep

pt

+ t

k

+ +

Disturbances dm

x z

mt

+

+

t

u0 (t  k )

Observations

ym +

+

1

mt

Output

zm

+

vm

Measurement noise

Fig. 11.2 Nonlinear and qLPV plant and disturbance model ðDu0 ðtÞ ¼ W 1k ð:; :ÞDuðtÞÞ

11.2.4 Definition of Augmented System Matrices The augmented system model may be obtained by combining the various subsystem models. The relationship between the augmented system matrices including the plant, disturbance and weighting subsystems is now established (see Eqs. (11.17)– (11.21)). From Eqs. (11.3)–(11.20):
 xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp rðtÞ  Bp yðtÞ but d0 ðtÞ ¼ dðtÞ þ yd ðtÞ, and from (11.3), (11.4) and (11.12): yðtÞ ¼ d0 ðtÞ þ C0 x0 ðtÞ þ E 0 u0 ðt  kÞ ¼ dðtÞ þ C0 x0 ðtÞ þ Cd xd ðtÞ þ E 0 bxi ðtÞ þ E 0 Du0 ðt  kÞ

ð11:22Þ

Thence we obtain xp ðt þ 1Þ ¼ Ap xp ðtÞ þ Bp rðtÞ  Bp dðtÞ  Bp C0 x0 ðtÞ  Bp Cd xd ðtÞ  Bp E 0 bxi ðtÞ  Bp E 0 Du0 ðt  kÞ

ð11:23Þ

Now from (11.3), (11.12) and u0 ðt  kÞ ¼ bxi ðtÞ þ Du0 ðt  kÞ, we obtain x0 ðt þ 1Þ ¼ A0 x0 ðtÞ þ B0 u0 ðt  kÞ þ D0 f0 ðtÞ þ G0 d0d ðtÞ ¼ A0 x0 ðtÞ þ bB0 xi ðtÞ þ B0 Du0 ðt  kÞ þ D0 f0 ðtÞ þ G0 d0d ðtÞ: The augmented system matrices may now be identified from these equations and (11.3)–(11.13) and (11.23).

478

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

Augmented state matrices: The augmented state-equation model becomes 2

x0 ðt þ 1Þ

3

2

A0

bB0

0

0

0

32

x0 ðtÞ

3

2

B0

3

76 7 7 6 7 6 6 Ad 0 0 0 76 xd ðtÞ 7 6 0 7 6 xd ðt þ 1Þ 7 6 0 76 7 7 6 7 6 6 7 7 6 6 xi ðt þ 1Þ 7 ¼ 6 0 6 0 bI 0 0 7 76 xi ðtÞ 7 þ 6 I 7Du0 ðt  kÞ 7 6 6 76 7 7 6 7 6 6 4 xp ðt þ 1Þ 5 4 Bp C0 Bp Cd bBp E 0 Ap 0 54 xp ðtÞ 5 4 Bp E 0 5 xu ðt þ 1Þ 0 0 0 0 Au xu ðtÞ Bu 3 3 2 2 G0 0 D0 0 7 7 6 6 6 0 Dd 7  6 0 0 7  7 f0 ðtÞ 7 6 6 d0d ðtÞ 7 60 0 7 þ 0 0 þ6 7 xðtÞ 7 ðrðtÞ  dðtÞÞ 6 6 7 7 6 6 0 5 4 0 4 0 Bp 5 0 0 0 0

ð11:24Þ This equation may clearly be written in terms of the augmented system matrices and vectors as xðt þ 1Þ ¼ At xðtÞ þ Bt Du0 ðt  kÞ þ Dt nðtÞ þ dd ðtÞ where 2

A0 6 0 6 At ¼ 6 6 0 4 Bp C0 20 D0 6 0 6 Dt ¼ 6 6 0 4 0 0

0 bB0 Ad 0 0 bI Bp Cd bBp E 0 30 20 0 G0 60 Dd 7 7 6 6 0 7 7; Gt ¼ 6 0 40 0 5 0 0

0 0 0 Ap 03 0 0 7 7 0 7 7 Bp 5 0

3 0 0 7 7 0 7 7; 0 5 Au

2

3 B0 6 0 7 6 7 7 Bt ¼ 6 6 I 7; 4 Bp E 0 5 Bu

and 2

G0   60 6 d0d ðtÞ dd ðtÞ ¼ Gt ¼6 60 ðrðtÞ  dðtÞÞ 40 0

3 3 2 0 G0 d0d ðtÞ 7   6 0 7 0 7 7 6 d0d ðtÞ 7 7 6 0 7 0 ¼6 7 ðrðtÞ  dðtÞÞ 5 4 Bp Bp ðrðtÞ  dðtÞÞ 5 0 0 ð11:25Þ

11.2

Nonlinear Operator and qLPV System Model

479

Output to be controlled: Recall from (11.22) the plant output to be controlled: yðtÞ ¼ dðtÞ þ C0 x0 ðtÞ þ Cd xd ðtÞ þ E 0 bxi ðtÞ þ E 0 Du0 ðt  kÞ This equation may be written in terms of the augmented system model in (11.18). That is, yðtÞ ¼ dðtÞ þ Ct xðtÞ þ E t Du0 ðt  kÞ where Ct ¼ ½ C0 Cd E 0 b 0 0  and E t ¼ E 0 . Measured output model: From (11.3) and (11.5), noting d0m ðtÞ ¼ dm ðtÞ þ ydm ðtÞ and u0 ðt  kÞ ¼ bxi ðtÞ þ Du0 ðt  kÞ We obtain ym ðtÞ ¼ d0m ðtÞ þ C0m x0 ðtÞ þ E 0m u0 ðt  kÞ ¼ dm ðtÞ þ Cdm xd ðtÞ þ C0m x0 ðtÞ þ E 0m bxi ðtÞ þ E 0m Du0 ðt  kÞ 3 2 x0 ðtÞ 6 x ðtÞ 7 6 d 7 7 6 7 ¼ dm ðtÞ þ ½ C0m Cdm E 0m b 0 0 6 6 xi ðtÞ 7 þ E 0m Du0 ðt  kÞ 7 6 4 xp ðtÞ 5

ð11:26Þ

xu ðtÞ This equation may be written in the augmented system form of Eq. (11.19): ym ðtÞ ¼ dm ðtÞ þ Cm t xðtÞ þ E m t Du0 ðt  kÞ

ð11:27Þ

where Cm t ¼ ½ C0m

Cdm

E 0m b

0

0  and

E m t ¼ E 0m

Weighted tracking error: The cost-function to be introduced later will include dynamically weighted system error and control action terms. The weighted tracking error will be denoted as ep ðtÞ. To compute the future weighted tracking error, note (11.10), (11.15) and (11.22), and define the weighted error to be minimized as     y ðtÞ Cp xp ðtÞ þ E p ðrðtÞ  yðtÞÞ ep ðtÞ ¼ e ¼ yu ðtÞ Cu xu ðtÞ þ E u Du0 ðt  kÞ but rðtÞ  yðtÞ ¼ rðtÞ  dðtÞ  C0 x0 ðtÞ  Cd xd ðtÞ  E 0 bxi ðtÞ  E 0 Du0 ðt  kÞ

480

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

On substituting, we get 

Cp xp ðtÞ þ E p ðrðtÞ  dðtÞ  C0 x0 ðtÞ  Cd xd ðtÞ  E 0 bxi ðtÞ  E 0 Du0 ðt  kÞÞ Cu xu ðtÞ þ E u Du0 ðt  kÞ 2 3 x0 ðtÞ    6 xd ðtÞ 7 7 E p C0 E p Cd bE p E 0 Cp 0 6 E p ðrðtÞ  dðtÞÞ 6 xi ðtÞ 7 þ ¼ 6 7 0 0 0 0 0 Cu 4 xp ðtÞ 5 xu ðtÞ   E p E 0 þ Du0 ðt  kÞ Eu



ep ðtÞ ¼

ð11:28Þ This equation may be written in terms of the augmented system model (11.21) as ep ðtÞ ¼ dp ðtÞ þ Cp t xðtÞ þ E p t Du0 ðt  kÞ where the vector and matrices are  dp ðtÞ ¼



 E p ðrðtÞ  dðtÞÞ ; 0

Cp t ¼ Ep t ¼

and

E p C0 E p Cd 0 0   E p E 0

bE p E 0 0

Cp 0

0 Cu



ð11:29Þ

Eu

The above results justify the definition of the augmented system matrices in Eqs. (11.17)–(11.21), where the state vector is defined in (11.16). Summary of Augmented System Matrices: The system matrices, defined in terms of the plant, disturbance and weighting subsystems may be collected as 2

0

bB0

0

Ad

0

0

0

bI

0

Bp Cd

bBp E 0

Ap

0

0

0

A0

6 6 0 6 6 At ¼ 6 0 6 6 B C 4 p 0 0

" Cp t ¼

Cm t ¼ ½ C0m

Cdm

0

3

2

7 0 7 7 7 0 7; 7 0 7 5

3

7 6 6 0 7 7 6 7 6 B t ¼ 6 I 7; 7 6 6 B E 7 4 p 05

2

0 0 ;

E p C0

E p Cd

bE p E 0

Cp

0

0

0

0

0

Cu

E m t ¼ E 0m ; "

# ;

Ep t ¼

D0

6 6 0 6 6 Dt ¼ 6 0 6 6 0 4

Bu

Au

E 0m b

B0

0 E p E 0

0

3

7 Dd 7 7 7 0 7 7 0 7 5 0

#

Eu

ð11:30Þ Recall that the reference r(t) and disturbance d(t) are both deterministic signals. Thus, define the deterministic signal vectors:

11.2

Nonlinear Operator and qLPV System Model

481

2

G0 d0d ðtÞ

3

7 6 0 7  6 7 6 d0d ðtÞ 7 6 0 dd ðtÞ ¼ Gt ¼6 7 ðrðtÞ  dðtÞÞ 7 6 4 Bp ðrðtÞ  dðtÞÞ 5 0   E p ðrðtÞ  dðtÞÞ and dp ðtÞ ¼ 0 

The white noise driving  T nT ðtÞ ¼ fT0 ðtÞ xT ðtÞ .

11.3

sources

can

be

combined

ð11:31Þ

in

the

vector

qLPV Model Future State and Error Predictions

The prediction model may now be obtained for the qLPV augmented system based on the results in Chap. 10. There is only a minor difference in the “delta” form of the control action used in this chapter. The expression for the future values of the states and outputs may be obtained by repeated use of the state equation (11.17) to obtain xðt þ iÞ ¼ Ait xðtÞ þ

i X

Aij t þ j ðB t þ j1 Du0 ðt þ j  1  kÞ

j¼1

þ Dt þ j1 nðt þ j  1ÞÞ þ ddd ðt þ i  1Þ

ð11:32Þ

It is assumed that the future values of the disturbance up to t þ i  1 are known. As in the previous chapter, assume the notation: 0 Aim t þ m ¼ At þ i1 At þ i2 . . .At þ m for i [ m; where At þ m ¼ I for i ¼ m

ð11:33Þ

and Ait ¼ At þ i1 At þ i2 . . .At for i [ 0; where A0t ¼ I for i ¼ 0 where when j = i the term At þ i1 At þ i2 . . .:At þ j is defined as the identity. The future known disturbance term is ddd ðt þ i  1Þ ¼ ddd ðt  1Þ ¼ 0;

i P j¼1

Aij t þ j dd ðt þ j  1Þ

for i [ 0 for i ¼ 0

9 = ;

ð11:34Þ

Equations (11.32) and (11.34) are valid for i  0 if the summation terms are defined as null for i = 0. Noting (11.21) and again following the approach in Chap. 10

482

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

(Sect. 10.3), the weighted error or output signal ep(t) to be regulated at future times has the form: ep ðt þ iÞ ¼ dpd ðt þ iÞ þ Cp t þ i Ait xðtÞ þ Cp t þ i

i X

Aij t þ j B t þ j1 Du0 ðt þ j  1  kÞ

j¼1

þ Cp t þ i

i X

Aij t þ j Dt þ j1 nðt þ j  1Þ þ E p t þ i Du0 ðt þ i  kÞ

j¼1

ð11:35Þ where i  0. The combined deterministic signals in this equation are defined as dpd ðt þ iÞ ¼ dp ðt þ iÞ þ Cp t þ i ddd ðt þ i  1Þ

ð11:36Þ

dp ðtÞ ¼ E p ðrðtÞ  dðtÞÞ

ð11:37Þ

where

For later manipulations note that the disturbance input nðtÞ affects this equation in a similar manner to the control input Du0 ðt  kÞ in (11.35).

11.3.1 State Estimates for qLPV Prediction Models The prediction equations are required in the derivation of the predictive controller. The i-steps prediction i  0 of the state and output signals may now be obtained. The predicted state, noting (11.32), may be expressed as ^xðt þ ijtÞ ¼ Ait ^xðtjtÞ þ

i X

Aij t þ j Bt þ j1 Du0 ðt þ j  1  kÞ þ ddd ðt þ i  1Þ

j¼1

ð11:38Þ The predicted output follows from (11.18), and the predicted state follows as ^yðt þ ijtÞ ¼ dðt þ iÞ þ Ct þ i^xðt þ ijtÞ þ E t þ i Du0 ðt  k þ iÞ

ð11:39Þ

The weighted prediction error in terms of the estimated state, for i  0 is ^ep ðt þ ijtÞ ¼ dp ðt þ iÞ þ Cp t þ i^xðt þ ijtÞ þ E p t þ i Du0 ðt þ i  kÞ

ð11:40Þ

k-steps ahead prediction: The k-steps prediction follows from (11.38), and is obtained as (i = k, k  0):

11.3

qLPV Model Future State and Error Predictions

^xðt þ kjtÞ ¼ Akt ^xðtjtÞ þ

k X

483

Akj t þ j Bt þ j1 u0 ðt þ j  1  kÞ þ ddd ðt þ k  1Þ

j¼1

ð11:41Þ k + i steps prediction: The expression for the future predicted states and error signals for the predictive controller may be obtained by changing the prediction time in (11.38) from t ! t þ k. For i  0: ^xðt þ k þ ijtÞ ¼ Ait þ k ^xðt þ kjtÞ þ

i X

Aij t þ k þ j Bt þ k þ j1 Du0 ðt þ j  1Þ

j¼1

þ ddd ðt þ k þ i  1Þ

ð11:42Þ

The expression also applies for the case i = 0 if the summation term in (11.42) is assumed null. Predicted weighted output error: The expression for the future states and outputs may be obtained by substituting in (11.40), for i ! i þ k, to obtain ^ep ðt þ i þ kjtÞ ¼ dp ðt þ i þ kÞ þ E p t þ i þ k Du0 ðt þ iÞ þ Cp t þ i þ k ^xðt þ i þ kjtÞ ð11:43Þ Substituting in (11.43) from the expression for the predicted states in (11.42), we obtain: ^ep ðt þ i þ kjtÞ ¼ dpd ðt þ i þ kÞ þ E p t þ i þ k Du0 ðt þ iÞ þ Cp t þ i þ k Ait þ k ^xðt þ kjtÞ þ Cp t þ i þ k

i X

Aij t þ k þ j B t þ k þ j1 u0 ðt þ j  1Þ

ð11:44Þ

j¼1

where the deterministic signals dpd ðt þ i þ kÞ in this equation are combined as in (11.36). That is, for i  0: dpd ðt þ i þ kÞ ¼ dp ðt þ i þ kÞ þ

i X

Cp t þ i þ k Aij t þ k þ j dd ðt þ k þ j  1Þ

ð11:45Þ

j¼1

where the summation is taken to be null for i = 0.

11.3.2 Vector Matrix Notation The predicted errors or outputs may be computed for controls in a future interval s 2 ½t; t þ N for N  1, and the expression (11.44) may be used to compute the

484

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

vector of future inferred weighted error signals. These are needed in the cost-function introduced below. These weighted error signals for k  1 may be collected in the following N + 1 vector form: 3 3 2 3 2 3 2 Cp t þ k I ^ep ðt þ kÞ E p t þ k Du0 ðtÞ dpd ðt þ kÞ 1 6 ^ep ðt þ 1 þ kÞ 7 6 dpd ðt þ k þ 1Þ 7 6 Cp t þ 1 þ k At þ k 7 6 E p t þ k þ 1 Du0 ðt þ 1Þ 7 7 7 6 7 6 7 6 6 7 6 ^ep ðt þ 2 þ kÞ 7 6 dpd ðt þ k þ 2Þ 7 6 Cp t þ 2 þ k A2 7 6 xðt þ kjtÞ þ 6 E p t þ k þ 2 Du0 ðt þ 2Þ 7 t þ k 7^ 7¼6 7þ6 6 7 7 6 7 6 7 6 6 .. .. .. . 5 5 5 5 4 4 4 4 . . . . . N ^ep ðt þ N þ kÞ dpd ðt þ k þ NÞ E Du ðt þ NÞ C A p t þ k þ N 0 ptþN þk tþk 3 2 3 0 0  0 0 2 Du0 ðtÞ .. 7 6 Cp t þ k þ 1 Bt þ k 0 . 0 0 76 Du0 ðt þ 1Þ 7 6 76 7 6 7 6 6 .. ... 7 þ 6 Cp t þ k þ 2 A1t þ k þ 1 Bt þ k 76 7 Cp t þ k þ 2 Bt þ k þ 1 0  . 76 7 6 74 Du0 ðt þ N  1Þ 5 6 .. .. .. 5 4 . 0 0 . . Du0 ðt þ NÞ Cp t þ k þ N AN1 Cp t þ k þ N AN2    Cp t þ k þ N Bt þ k þ N1 0 t þ k þ 1 Bt þ k t þ k þ 2 Bt þ k þ 1 2

ð11:46Þ Future error and predicted error: With an obvious definition of terms, this equation may be written as 0 b P t þ k;N ¼ DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ ðCP t þ k;N Bt þ k;N þ E P t þ k;N ÞDUt;N E

ð11:47Þ To simplify this equation define the non-dynamic time-varying matrix: V P t þ k;N ¼ CP t þ k;N Bt þ k;N þ E P t þ k;N

ð11:48Þ

so that 0 b P t þ k;N ¼ DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ V P t þ k;N DUt;N E

ð11:49Þ

Noting that the control input and disturbances enter the equations in a similar way, the expression for the weighted future errors may, therefore, be written, including the stochastic disturbance inputs Nt þ k;N , as 0 EP t þ k;N ¼ DP t þ k;N þ CP t þ k;N At þ k;N xðt þ kÞ þ V P t þ k;N DUt;N þ CP t þ k;N Dt þ k;N Nt þ k;N

ð11:50Þ Block matrices: Noting Eq. (11.46) the vectors and block matrices, for the general case of N  1, may be defined. The N + 1 square block output and through term matrices may be defined as CP t þ k;N ¼ diagfCp t þ k ; Cp t þ 1 þ k ; Cp t þ 2 þ k ; . . .; Cp t þ N þ k g

11.3

qLPV Model Future State and Error Predictions

485

E P t þ k;N ¼ diagfE p t þ k ; E p t þ 1 þ k ; . . .; E p t þ N þ k g

ð11:51Þ

Also, define the block system matrices as 2

At þ k;N

2

3

6 A1 7 6 tþk 7 6 2 7 7 6 ¼ 6 At þ k 7 ; 6 . 7 6 . 7 4 . 5 2

Dt þ k;N

I

Bt þ k;N

ANtþ k

0 6 6 Dt þ k 6 6 6 ¼ 6 At þ k þ 1 D t þ k 6 6 .. 6 . 4 AN1 t þ k þ 1 Dt þ k

0 6 6 Bt þ k 6 6 6 ¼ 6 A1t þ k þ 1 Bt þ k 6 6 .. 6 . 4 AN1 t þ k þ 1 Bt þ k 0



0

 .. . .. . 

Dt þ k þ 1 .. . AN2 t þ k þ 2 Dt þ k þ 1

0



0

 .. . .. . 0    Bt þ k þ N1

0 .. .

0

3

7 07 7 .. 7 7 . 7; 7 7 07 5 0

Bt þ k þ 1 .. . AtN2 þ k þ 2 Bt þ k þ 1 3 0 2 3 7 .. nðtÞ 7 . 7 6 nðt þ 1Þ 7 7 6 7 7 7; 7; Nt;N ¼ 6 .. 6 7 7 4 5 . 7 7 0 5 nðt þ N  1Þ Dt þ k þ N1

ð11:52Þ 3 ^ep ðt þ kÞ 6 ^ep ðt þ 1 þ kÞ 7 7 6 7 6 ¼ 6 ^ep ðt þ 2 þ kÞ 7; 7 6 .. 5 4 . ^ep ðt þ N þ kÞ 2

b P t þ k;N E

3 Du0 ðtÞ 6 Du0 ðt þ 1Þ 7 7 6 7 6 ¼ 6 Du0 ðt þ 2Þ 7; 7 6 .. 5 4 . 2

0 Ut;N

Du0 ðt þ NÞ

3 dpd ðtÞ 6 dpd ðt þ 1Þ 7 7 6 7 6 ¼ 6 dpd ðt þ 2Þ 7 7 6 .. 5 4 . dpd ðt þ NÞ 2

DP t;N

ð11:53Þ 0 The signal DUt;N denotes a block vector of future input signals. The block vector DP t;N denotes a vector of future reference minus known disturbance signal components. The above system matrices At þ k;N ; Bt þ k;N ; Dt þ k;N are assumed functions of future states (and possibly parameters and inputs), and the following assumption must therefore be introduced. For the special case of a single-step cost-function N = 0 and the above matrices simplify to become At þ k;N ¼ I, Bt þ k;N ¼ 0, Dt þ k;N ¼ 0, CP t þ k;N ¼ Cp t þ k , E P t þ k;N ¼ E p t þ k (as in Chap. 9). Assumptions: The assumption is made that the future state is calculable. This is valid if, for example, it is assumed that the stochastic disturbance nðtÞ is null and future inputs within the prediction horizon are known (so that future states can be calculated from the model). In practice, the exact prediction is not, of course, possible and an approximation is introduced. Also from (11.48), the matrix V P t þ k;N ¼ CP t þ k;N Bt þ k;N þ E P t þ k;N is assumed full rank. Recall the matrices CP t þ k;N and E P t þ k;N are specified by the designer, and depend on the cost-function weightings chosen.

486

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

11.3.3 Tracking Error The k-steps ahead tracking error, including any dynamic error weighting, may now be written in the following vector-form, noting (11.50) as: 0 EP t þ k;N ¼ DP t þ k;N þ CP t þ k;N At þ k;N xðt þ kÞ þ V P t þ k;N DUt;N þ CP t þ k;N Dt þ k;N Nt þ k;N

ð11:54Þ The weighted inferred output to be minimized in the predictive control problem is assumed to have the same dimension as the control signal. The matrix V P t þ k;N in (11.54) is square and of a block lower-triangular form, and given below, for N  1, as 2

V P t;N

Ep t

0



0 .. .

0

3

7 6 6 Cp t þ 1 Bt Ep t þ 1  0 7 7 6 .. 7 .. 6 ¼ 6 Cp t þ 2 A1 Bt 7 . . C B p t þ 2 t þ 1 tþ1 7 6 7 6 . . . .. .. .. 4 E p t þ N1 0 5 Cp t þ N AN1 Cp t þ N AN2 t þ 1 Bt t þ 2 Bt þ 1    Cp t þ N Bt þ N1 E p t þ N |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} V P t;N ¼ CP t;N Bt;N þ E P t;N

ð11:55Þ Special case: This matrix (11.55) is important since it relates the vector of future controls to the error being minimized. For the special case of a single-step cost N = 0, the matrix (11.55) can be defined as V P t;N ¼ E p t . Here E p t ¼ E p E 0 denotes the through term between the input signal u0 ðt  kÞ and the weighted output. This special case is of interest when demonstrating that the NPGMV controller becomes a type of NGMV controller for the limiting case of single-stage criterion. Note that Cp t þ 1 Bt is null if C0 B0 is null and for examples like that considered later in the chapter, this implies the number of steps chosen for cost-index which should be N [ 1. Output prediction error: Based on (11.47) and (11.50) obtain the expression for the prediction error, representing the difference between the vector of future errors e P t þ k;N ¼ EP t þ k;N  E b P t þ k;N Þ: and estimates ð E 0 e P t þ k;N ¼ DP t þ k;N þ CP t þ k;N At þ k;N xðt þ kÞ þ V P t þ k;N DUt;N þ CP t þ k;N Dt þ k;N Nt þ k;N E 0  ðDP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ V P t þ k;N DUt;N Þ

ð11:56Þ Thence, the inferred output estimation error:

11.3

qLPV Model Future State and Error Predictions

487

e P t þ k;N ¼ CP t þ k;N At þ k;N ~xðt þ kjtÞ þ CP t þ k;N Dt þ k;N Nt þ k;N E

ð11:57Þ

where the k-steps ahead state-estimation error ~xðt þ kjtÞ ¼ xðt þ kÞ  ^xðt þ kjtÞ. Note for later use that the estimation error is independent of the choice of control action. Also recall that the optimal estimate ^xðt þ kjtÞ and estimation error ~xðt þ kjtÞ are orthogonal, and the expectation of the product of the future values of the control action (assumed known in deriving the prediction equation) and the zero-mean white noise signals are null. It follows that the vector of predicted weighted error b P t þ k;N in (11.47) and the prediction error E e P t þ k;N are orthogonal. signals E

11.3.4 Kalman Filter Predictor–Corrector Form If it is assumed that the states are not available for feedback, then an optimal state estimator must be introduced, and the cost-function must be expressed in terms of the optimal state estimate and the state-estimation error. The Kalman filter equations are introduced below for a system containing delays on input channels, through terms and an input Du0 ðt  kÞ. The estimates are to be generated for the time-varying known augmented system, and the algorithm may be summarized as ^xðt þ 1jtÞ ¼ At ^xðtjtÞ þ Bt Du0 ðt  kÞ þ dd ðtÞ

ðPredictor Þ

^xðt þ 1jt þ 1Þ ¼ ^xðt þ 1jtÞ þ Kf t þ 1 ðzm ðt þ 1Þ  ^zm ðt þ 1jtÞÞ

ð11:58Þ

ðCorrector Þ ð11:59Þ

where ^zm ðt þ 1jtÞ ¼ dm ðt þ 1Þ þ Cm xðt þ 1jtÞ þ E m t þ 1^ t þ 1 Du0 ðt þ 1  kÞ

ð11:60Þ

The Kalman filter has been so successful in applications that it is likely to be the most reliable part of the predictive control solutions obtained [19, 24, 25]. Predicted state: The state estimate ^xðt þ kjtÞ may be obtained, k-steps ahead, in a computationally efficient form from a time-varying Kalman filter (see the previous chapter and [24]). In this form of the estimator, the number of states in the filter is not increased by the number of the synchronous delays k. Prediction model: The k-steps prediction may be written in a more concise form. Introduce a finite pulse-response block into (11.41): ^xðt þ kjtÞ ¼ Akt ^xðtjtÞ þ T ðk; z1 ÞDu0 ðtÞ þ ddd ðt þ k  1Þ where the finite pulse-response model term is defined as

ð11:61Þ

488

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

T ðk; z1 Þ ¼

k X

j1k Akj t þ j Bt þ j1 z

ð11:62Þ

j¼1

and the combination of known disturbance terms ddd ðtÞ in (11.61) was defined by (11.34) with i = k. As in (11.42) the summation terms in (11.62) will be assumed to be null for k = 0, then T ð0; z1 Þ ¼ 0 and ddd ðt  1Þ ¼ 0. The controller will be found to include the Kalman predictor equation. It is valuable that the order of the Kalman filter depends only on the delay-free qLPV subsystems and the channel delays do not add to the order of the controller. Note for implementation that the integrator introduced in the augmented system equations will appear in the Kalman estimator equation. In fact whether integral action is introduced via a dynamic error cost-function weighting or whether it be introduced explicitly by using a delta form of the weighted control action, in either event, the integrator becomes part of the augmented system and it is therefore included within the Kalman filter.

11.4

Nonlinear Generalized Predictive Control

The Generalized Predictive Controller (GPC) was introduced in Chap. 7. This is an approach, which is representative of many Model Predictive Control (MPC) algorithms for linear systems. The main advantage is the simplicity of the solution but it does not apply directly to nonlinear systems. However, by combining this method with a qLPV model representation, a simple and effective LPV/ state-dependent approach is obtained that can be used in many linear and nonlinear applications. The derivation of the Nonlinear Generalized Predictive Control (NGPC) algorithm, for a qLPV system, is provided first below. The steps in the solution are similar to those in Chap. 7, for the linear case, which is an attractive feature. The solution will be summarized only briefly. In this first part of the chapter, the black-box input subsystem is not used and is set to the identity. This means the input to the plant will be taken to be that of the qLPV subsystem, denoted by u0 ðtÞ in Fig. 11.1. This provides a solution to the NGPC problem that is similar superficially to the GPC solution for linear time-varying systems. The NGPC solution for a qLPV model is a slight extension of linear MPC. It is good for industrial applications since it is simple conceptually and hard constraints may be applied using quadratic programming in a similar way as for the linear case. The NGPC problem also helps to motivate results that are required for the more general NPGMV problem considered later in Sect. 11.5. For this more general problem, the black-box input subsystem in Fig. 11.1 will be reintroduced and the control input will then be taken as the signal u(t). NGPC Performance Index: The NGPC performance index to be minimized may be defined as

11.4

Nonlinear Generalized Predictive Control

( J¼E

N X j¼0

T

ep ðt þ j þ kÞ ep ðt þ j þ kÞ þ

489 Nu X

) ðDu0 ðt þ jÞÞT k2j Du0 ðt þ jÞjt

j¼0

ð11:63Þ where Ef:jtg denotes the conditional expectation, conditioned on measurements up to time t and k2j denotes a diagonal control signal weighting matrix for the MIMO case. In this definition of the cost-function to be minimized, the weighted error term is k-steps ahead of the control signal since u0 ðtÞ affects ep ðt þ kÞ after k steps. This cost-index is more general than it first appears. The error signal being penalized in the first term has dynamic error costing and by suitable definition of the augmented system, it can include a dynamic input costing and state-costing terms. The second term in the criterion can represent absolute control action or control increments. Also, note the future control action is to be calculated for the interval s 2 ½t; t þ Nu . This depends on the number of steps ðNu þ 1Þ in the control costing term in (11.63). Vector form NGPC Cost-Index: The multi-step cost-function may be written in a more concise form by introducing the vectors defined in Sect. 11.3.2. n o 2 0T 0 J ¼ EfJt g ¼ E EPT t þ k;N EP t þ k;N þ DUt;N K DU jt Nu t;Nu u

ð11:64Þ

To expand the cost-function terms, recall from (11.57) that the weighted tracking e P t þ k;N þ E b P t þ k;N . Assume the Kalman filter is introerror satisfies EP t þ k;N ¼ E duced for state estimation and prediction, and use (11.53) and (11.64) to obtain 0T 0 e P t þ k;N ÞT ð E b P t þ k;N þ E e P t þ k;N Þ þ DUt;N b P t þ k;N þ E K2Nu DUt;N jtg J ¼ Efð E u u

ð11:65Þ The cost-function weightings on inputs Du0 ðtÞ at future times are written as K2Nu ¼ diagfk20 ; k21 ; . . .; k2Nu g. The terms in the cost-index can therefore be simplib P t þ k;N is orthogonal to the estimation fied, by noting that the optimal estimate E e error E P t þ k;N . The vector/matrix form of the cost-function follows as 0T 0 b P t þ k;N þ DUt;N b PT t þ k;N E K2Nu DUt;N þ J0 J¼E u u

ð11:66Þ

where the final cost term is independent of control action and is defined as e P t þ k;N jtg e PT t þ k;N E J0 ¼ Ef E To solve the resulting optimal control problem, note that the vector of future controls can be a different length for the future error and the control terms. These depend upon the cost-function horizons employed. To deal with this and more general problems, the concept of a connection matrix is now introduced.

490

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

11.4.1 Connection Matrix and Control Profile It is common practice in predictive control to use a control horizon of length Nu that is less than the error or output horizon N. The reason is that the size of the main matrix to be inverted is Nu square. The future controls, in this case, are assumed to change initially but then become constant for times greater than Nu. In the following a more general solution is provided that enables the control action to be distributed throughout the prediction horizon N. This requires the introduction of a control profile Pu, defined to be of the form: rowfPu g ¼ ½ðlength of control interval; in samplesÞ

ðnumber of repetitionsÞ

The solution to the NGPC class of problems may be obtained with different control and error horizons, or the problem can be considered where the horizons are the same, but the control changes are not allowed at each sample instant but at fewer times as the future time increases (‘moves blocking’). The example that follows illustrates the approach. Example 11.1: Connection Matrix for the Absolute Control Case Letting 2

1 Pu ¼ 4 2 3

3 3 25 1

The first row signifies the control action is first allowed to change at every sampling instant and this occurs for three time steps. The second row signifies that the control then becomes constant over two steps and this occurs twice. Finally, the third row signifies that the control action remains constant for three steps for just one final stage. The total number of steps is the sum of the row products 1  3 + 2  2 + 3  1 = 10, whilst the number of controls to be computed is the sum of the second column 3 + 2 + 1 = 6. By spreading out the future control activity throughout the prediction interval, control that is more effective may be achieved, whilst keeping the computational burden low. Given a control profile, a transformation or “connection” matrix Tu can be defined, relating the control moves to be optimized (vector V), to the full horizon control vector (U). The relationship may be expressed as U = Tu  V. For the above example, the relationship between the full and reduced set of future controls is as follows:

11.4

Nonlinear Generalized Predictive Control

491

3 3 2 uðtÞ uðtÞ 7 6 uðt þ 1Þ 7 6 uðt þ 1Þ 7 7 6 6 7 6 uðt þ 2Þ 7 6 uðt þ 2Þ 7 7 6 6 7 6 uðt þ 3Þ 7 6 uðt þ 3Þ 7 7 6 6 6 uðt þ 4Þ 7 6 uðt þ 4Þ ¼ uðt þ 3Þ 7 7 7¼6 )6 7 6 uðt þ 5Þ 7 6 uðt þ 5Þ 7 7 6 6 6 uðt þ 6Þ 7 6 uðt þ 6Þ ¼ uðt þ 5Þ 7 7 7 6 6 7 6 uðt þ 7Þ 7 6 uðt þ 7Þ 7 7 6 6 4 uðt þ 8Þ 5 4 uðt þ 8Þ ¼ uðt þ 7Þ 5 uðt þ 9Þ 3 uðt þ 9Þ ¼ uðt þ 7Þ 0 0 0 0 0 1 0 0 0 07 72 3 0 1 0 0 07 7 v1 6 7 0 0 1 0 07 7 6 v2 7 7 7 0 0 1 0 0 76 6 v3 7 7 6 0 0 0 1 0 7 6 v4 7 7 4 5 0 0 0 1 07 7 v5 0 0 0 0 17 7 v6 0 0 0 0 15 0 0 0 0 1 2

U ¼ Tu V

2

1 60 6 60 6 60 6 60 ¼6 60 6 60 6 6 60 40 0

The connection matrix Tu on the right of this equation clearly sets the pattern of controls. Note that in the multivariable case, where the control signal is a vector, the unity entries in these matrices must be replaced by identity matrices of a corresponding dimension. Delta control NGPC problem: The control horizon may be less than the error horizon, and in this case, we may define the full vector of future control changes 0 as DUt;N 0 0 ¼ Tu DUt;N DUt;N u

ð11:67Þ

Example 11.2: Connection Matrix for Incremental Control Case Consider again the previous example but for the case of the incremental control formulation. The relationship between the full and reduced set of future incremental controls is as follows:

492

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

3 3 2 DuðtÞ DuðtÞ 6 Duðt þ 1Þ 7 6 Duðt þ 1Þ 7 7 7 6 6 6 Duðt þ 2Þ 7 6 Duðt þ 2Þ 7 7 7 6 6 6 Duðt þ 3Þ 7 6 Duðt þ 3Þ 7 7 7 6 6 6 Duðt þ 4Þ 7 6 Duðt þ 4Þ ¼ 0 7 7 7¼6 DU ¼ TDu DV ) 6 6 Duðt þ 5Þ 7 6 Duðt þ 5Þ 7 7 7 6 6 6 Duðt þ 6Þ 7 6 Duðt þ 6Þ ¼ 0 7 7 7 6 6 6 Duðt þ 7Þ 7 6 Duðt þ 7Þ 7 7 7 6 6 4 Duðt þ 8Þ 5 4 Duðt þ 8Þ ¼ 0 5 Duðt þ 9Þ ¼ 0 Duðt þ 9Þ3 2 1 0 0 0 0 0 60 1 0 0 0 07 6 72 3 6 0 0 1 0 0 0 7 Dv1 6 7 6 0 0 0 1 0 0 76 Dv2 7 6 7 76 6 0 0 0 0 0 0 76 Dv3 7 7 76 ¼6 6 0 0 0 0 1 0 76 Dv4 7 6 7 76 6 0 0 0 0 0 0 74 Dv5 5 7 6 6 0 0 0 0 0 1 7 Dv6 7 6 40 0 0 0 0 05 0 0 0 0 0 0 2

Clearly, this again represents a situation with Nu = 3 + 2 + 1 = 6 control moves and involves a total of N = 10 sample points. In this example, four control moves will not be computed, representing a substantial computational saving in the predictive control law calculation. Example 11.3: Connection Matrix for Incremental Control and Traditional Case The traditional approach is to assume future control changes that are null after the control horizon Nu and in this case, the connection matrix can be defined to have N + 1 rows and Nu columns. The form of the T matrix, when N = 6 and Nu 0 0 = 2 (recalling DUt;N has N + 1 rows and DUt;N has Nu + 1 rows): u 2

1 60 6 60 6 T¼6 60 60 6 40 0

0 1 0 0 0 0 0

3 0 07 7 17 7 07 7 07 7 05 0

3 Du0 ðtÞ 6 Du0 ðt þ 1Þ 7 7 6 6 Du0 ðt þ 2Þ 7 7 6 7 0 ¼6 7 6 7 6 0 7 6 5 4 0 0 2

and

0 0 DUt;N ¼ Tu DUt;N u

For simplicity the same symbol will be used in the following to represent the connection matrix for the control and the incremental control cases ðTu Þ but it should be recalled that different definitions apply in the two cases.

11.4

Nonlinear Generalized Predictive Control

493

11.4.2 NGPC State-Dependent/qLPV Solution The NGPC controller for qLPV systems is derived below. The controller is important in its own right since it is a useful extension of linear GPC control. It is also a stage in the derivation of the more general nonlinear predictive controller presented in the later sections. To compute the vector of future weighted error signals first note 0 0 V P t þ k;N DUt;N ¼ V P t þ k;N Tu DUt;N u

ð11:68Þ

Then from (11.49) and (11.68), the vector of future predicted errors: 0 b P t þ k;N ¼ DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ V P t þ k;N DUt;N E 0 ¼ DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ V P t þ k;N Tu DUt;N u

or e P t þ k;N þ V P t þ k;N Tu DU 0 b P t þ k;N ¼ D E t;Nu

ð11:69Þ

e P t þ k;N ¼ DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ D

ð11:70Þ

where

Expanded cost-index: Noting (11.48) and substituting from Eq. (11.47) for the vector of state estimates: e P t þ k;N þ V P t þ k;N Tu DU 0 ÞT ð D e P t þ k;N þ V P t þ k;N Tu DU 0 Þ þ DU 0T K2 DU 0 þ J0 J ¼ ðD t;Nu t;Nu t;Nu Nu t;Nu 0T T T 0 eT eT e e ¼D P t þ k;N D P t þ k;N þ DUt;Nu Tu V P t þ k;N D P t þ k;N þ D P t þ k;N V P t þ k;N Tu DUt;Nu 0T 0 þ DUt;N X t þ k;Nu DUt;N þ J0 u u

ð11:71Þ where X t þ k;Nu ¼ TuT V TP t þ k;N V P t þ k;N Tu þ K2Nu

ð11:72Þ

Optimization: The procedure for minimizing this cost term, if the signals are deterministic, is almost identical to that when the conditional cost-function is considered. That is, the gradient of the cost-function must be set to zero, to obtain

494

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

the vector of future optimal control signals. From a similar perturbation and gradient calculation in [24], or the argument in Sect. 7.3.1, the vector of NGPC future optimal control signals for this problem becomes 0 T T e DUt;N ¼ X 1 t þ k;Nu Tu V P t þ k;N D P t þ k;N u

ð11:73Þ

The NGPC optimal control signal at time t is defined from this vector based on the receding horizon principle mentioned previously [26]. The optimal control is 0 . therefore taken as the first element in the vector of future control increments DUt;N u The main results for the NGMV controller are summarized in the theorem that follows. Theorem 11.1: Nonlinear Generalized Predictive Control Consider the linear qLPV state-space system described in Sect. 11.2 but with the input subsystem set to the identity ðW 1k ¼ IÞ. The cost-function (11.64) is to be minimized for a given dynamic error weighting defined by (11.9) and (11.10), and a control weighting K2Nu ¼ diagfk20 ; k21 ; . . .; k2Nu g. Given the future state ^xðt þ kjtÞ, and the known future disturbance and reference vector DP t þ k;N from (11.53), and the matrices V TP t þ k;N and X t þ k;Nu , from (11.55) and (11.72) the NGPC optimal control can be computed from the first element of the vector of future controls as
 0 T T DUt;N ¼ X 1 xðt þ kjtÞ t þ k;Nu Tu V P t þ k;N DP t þ k;N þ CP t þ k;N At þ k;N ^ u

ð11:74Þ

where the cost-function weighting definitions ensure the matrix XN is non-singular. ■ Proof The proof follows by collecting the results preceding the theorem.

■

Notice that the reference signal affects the vector DP t;N of future controls in (11.74) since DP t;N is given by (11.53):  T T ðtÞ dpd ðt þ 1Þ    DP t;N ¼ dpd

T dpd ðt þ NÞ

T

and this is a vector that depends upon both the future disturbance and reference signals since from (11.36) the dpd ðt þ iÞ depend upon (11.31) and (11.37), where   T dp ðtÞ ¼ E p ðrðtÞ  dðtÞÞ and dd ðtÞ ¼ Gt d0d ðtÞ ðrðtÞ  dðtÞÞT . The reference, therefore, affects the computed controls via both a through term and the model dynamics.

11.4

Nonlinear Generalized Predictive Control

495

11.4.3 NGPC Equivalent Cost Optimization Problem It is now shown that the above problem is equivalent to a special cost-minimisation control problem. This is needed to motivate the cost-function in the NPGMV problem defined later. First let the positive-definite, real symmetric matrix X t þ k;Nu (defined in (11.72)), that enters the above solution (11.74), be factorized into the form: Y Ttþ k;Nu Y t þ k;Nu ¼ X t þ k;Nu ¼ TuT V TP t þ k;N V P t þ k;N Tu þ K2Nu

ð11:75Þ

Then observe that by completing the squares in (11.71), the cost-function may be written as 0T T T 0 eT eT e e J¼D P t þ k;N D P t þ k;N þ DUt;Nu Tu V P t þ k;N D P t þ k;N þ D P t þ k;N V P t þ k;N Tu DUt;Nu 0T 0 þ DUt;N Y Ttþ k;Nu Y t þ k;Nu DUt;N þ J0 u u



T 1 T T T 0T e e P t þ k;N þ Y t þ k;N DU 0 Y t þ k;Nu TuT V P t þ k;N D ¼ D u t;Nu P t þ k;N V P t þ k;N Tu Y t þ k;Nu þ DUt;Nu Y t þ k;Nu



1 T T T e eT þD P t þ k;N ðI  V P t þ k;N Tu Y t þ k;Nu Y t þ k;Nu Tu V P t þ k;N Þ D P t þ k;N þ J0

ð11:76Þ Equivalent cost-index: By comparison with (11.76), the cost-function may be written in an equivalent form: b 0T b0 J¼W t þ k;Nu W t þ k;Nu þ J10 ðtÞ

ð11:77Þ

where the signal within the “squared” term in (11.76):

¼

T T 0 e b0 Y T W t þ k;Nu ¼
t þ k;Nu Tu V P t þ k;N D P t þ k;N þ Y t þ k;Nu DUt;Nu T T T 0 Y t þ k;Nu Tu V P t þ k;N DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ Y t þ k;Nu DUt;N u

ð11:78Þ The cost terms that are independent of the control action may be written as J10 ðtÞ ¼ J0 þ J1 ðtÞ where 1 T T T eT e J1 ðtÞ ¼ D P t þ k;N ðI  V P t þ k;N Tu Y t þ k;Nu Y t þ k;Nu Tu V P t þ k;N Þ D P t þ k;N

ð11:79Þ

Optimization: Since the last term J10 ðtÞ in Eq. (11.77) does not depend upon control action, the optimal control is found by setting the first term to zero ^0 W t þ k;Nu ¼ 0. This result leads to the same optimal control as in (11.74). Thus, the NGPC optimal controller for the above system is the same as the controller to b0 minimize the Euclidean norm of the signal W t þ k;Nu , defined in (11.78). That is, the vector of optimal future controls:

496

11

¼

LPV/State-Dependent Nonlinear Predictive Optimal Control

0 T T e DUt;N ¼ X 1 t
þ k;Nu Tu V P t þ k;N D P t þ k;N u  1 T T X t þ k;Nu Tu V P t þ k;N DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ

ð11:80Þ

In the spirit of receding horizon optimal control, the current change in control follows from the first element of this vector and the current control is given by (11.13): u0 ðtÞ ¼ Du0 ðtÞ þ bu0 ðt  1Þ

ð11:81Þ

11.4.4 NGPC-Modified Cost-Function and Solution The above discussion motivates the definition of a new multi-step minimum variance cost problem that is similar to the above cost-minimization problem (11.77) but is in a form where the link to NGMV design can be established. There are some mathematical preliminaries and the result is then presented. Recall that the signal to be minimized, in the NGMV problem, involves a weighted sum of error and input signals that have the form /ðt þ kÞ ¼ Pc eðt þ kÞ þ Fc 0 uðtÞ. Extending this cost expression, a multi-step cost-index, can be defined as 0 0 Ut þ k;N ¼ PCN; t EP t þ k;N þ FCN; t DUt;Nu

ð11:82Þ

The cost-function weightings will be defined to have the following constant matrix forms: PCN;t ¼ TuT V TP t þ k;N

and

0 FCN;t ¼ K2Nu

ð11:83Þ

The weightings in (11.83) are clearly dependent on the NGPC weightings in (11.64). These definitions of the weightings use foresight and will be justified using the property established in Theorem 11.2 established below. The error weighting in (11.83) depends upon the matrix V TP t þ k;N , and from (11.55) this is of an upper-triangular form: 3 T T  BTt ATt þN1 E Tp t BTt CTp t þ 1 BTt AT1 t þ 1 Cp t þ 2 1 Cp t þ N 7 6 T E Tp t þ 1 BTtþ 1 CTp t þ 2  BTtþ 1 ATt þN2 6 0 2 Cp t þ N 7 7 6 .. .. .. 7 6 ¼6 0 7 . . 0 . 7 6 . .. .. 7 6 . T T T . E p t þ N1 . Bt þ N1 Cp t þ N 5 4 . 0 0  0 E Tp t þ N |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2

V TP t;N

V TP t;N

P t;N

¼BTt;N CTP t;N þ E TP t;N

11.4

Nonlinear Generalized Predictive Control

497

Special case: In the special case of N = 0, the matrix V TP t;N ¼ E Tp t . As in the related case discussed in Sect. 9.3, if the through term E p t is null the horizon N must be chosen to be greater than zero to be physically realistic. Related cost-function: To develop the results of this theorem, define a minimum variance multi-step cost-function, using a vector of signals: e J ðtÞ ¼ EfUTtþ k;N Ut þ k;N jtg

ð11:84Þ

0 0 Ut þ k;N ¼ PCN;t EP t þ k;N þ FCN;t DUt;N u

ð11:85Þ

Predicting forward k steps:

Consider the signal Ut þ k;Nu and substitute for the vector of outputs b P t þ k;N þ E e P t þ k;N . Then from (11.85), we obtain EP t þ k;N ¼ E 0 0 b P t þ k;N þ E e P t þ k;N Þ þ FCN;t Ut þ k;N ¼ PCN;t ð E DUt;N u

0 0 b P t þ k;N þ FCN;t e P t þ k;N ¼ PCN;t E DUt;N þ PCN;t E u

ð11:86Þ

This expression may be written in terms of the estimate and the estimation error vectors as b t þ k;N þ U e t þ k;N Ut þ k;N ¼ U

ð11:87Þ



0 0 b t þ k;N ¼ PCN;t E b P t þ k;N þ FCN;t DUt;N U u

ð11:88Þ

The predicted signal:

The prediction error: e t þ k;N ¼ PCN;t E e P t þ k;N U

ð11:89Þ

Multi-step performance index: The performance index (11.84) may be simplified and written as b t þ k;N þ U e t þ k;N ÞT ð U b t þ k;N þ U e t þ k;N Þjtg e J ¼ EfUTtþ k;N Ut þ k;N jtg ¼ Efð U The terms in the performance index (11.84) can again be simplified, recalling the e P t þ k;N are orthogonal. b P t þ k;N , and the estimation error E optimal estimate E Invoking these orthogonality properties, we obtain bT eT eT b e b e bT e J ¼ Ef U t þ k;N U t þ k;N jtg þ Ef U t þ k;N U t þ k;N jtg þ Ef U t þ k;N U t þ k;N jtg þ Ef U t þ k;N U t þ k;N jtg T T e b e b ð11:90Þ ¼U t þ k;N U t þ k;N þ Ef U t þ k;N U t þ k;N jtg

498

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

The cost-function may be written as bT b e e J ðtÞ ¼ U t þ k;N U t þ k;N þ J 1 ðtÞ

ð11:91Þ

The last cost-function term is independent of control action and may be written, using (11.89), as T e eT e e eT J 1 ðtÞ ¼ Ef U t þ k;N U t þ k;N jtg ¼ Ef E P t þ k;N PCN;t PCN;t E P t þ k;N jtg

ð11:92Þ

b t þ k;N ¼ 0. Condition for optimality: The condition for optimality is clearly U b The vector of predicted signals U t þ k;N may be simplified by substituting from (11.69) into (11.88): 0 0 b t þ k;N ¼ PCN;t E b P t þ k;N þ FCN;t U DUt;N u e P t þ k;N þ V P t þ k;N Tu DU 0 Þ þ F 0 DU 0 ¼ PCN;t ð D t;Nu CN;t t;Nu

Further simplification is possible by substituting from (11.75) and (11.83), and noting 2 0 T T PCN;t V P t þ k;N Tu þ FCN; t ¼ Tu V P t þ k;N V P t þ k;N Tu þ KNu ¼ X t þ k;Nu

ð11:93Þ

Using (11.93) now we obtain

2 0 e P t þ k;N þ T T V T b t þ k;N ¼ PCN;t D U u P t þ k;N V P t þ k;N Tu þ KNu DUt;Nu e P t þ k;N þ X t þ k;N DU 0 ¼ PCN;t D u t;Nu

ð11:94Þ

The cost-function weightings are chosen so that the time-varying matrix X t þ k;Nu is non-singular. From a similar argument to that in the previous section, the optimal multi-step minimum variance predictive control sets the first squared term in (11.91) to zero. The optimal control, therefore, follows from setting (11.94) to zero, giving
 0 T T DUt;N ¼ X 1 xðt þ kjtÞ t þ k;Nu Tu V P t þ k;N DP t þ k;N þ CP t þ k;N At þ k;N ^ u

ð11:95Þ

This expression is the same as the vector of future NGPC controls. The results are summarized in the theorem below. Theorem 11.2: Equivalent NGPC Cost Optimization Problem Consider the minimization of the NGPC cost-index (11.63) for the system and assumptions introduced in Sect. 11.2, where the input subsystem is absent ðW 1k ¼ IÞ. Assume that the cost-index is redefined to have a multi-step minimum variance form (11.84):

11.4

Nonlinear Generalized Predictive Control

499

e J ðtÞ ¼ EfUTtþ k;N Ut þ k;N jtg

ð11:96Þ

0 0 where Ut þ k;Nu ¼ PCN;t EP t þ k;N þ FCN;t DUt;N and the cost weightings are defined u relative to the NGPC cost-index as

PCN;t ¼ TuT V TP t þ k;N

and

0 FCN;t ¼ K2Nu

ð11:97Þ

The vector of future optimal controls that minimizes (11.96) follows as
 0 T T DUt;N ¼ X 1 xðt þ kjtÞ t þ k;Nu Tu V P t þ k;N DP t þ k;N þ CP t þ k;N At þ k;N ^ u

ð11:98Þ

where X t þ k;Nu ¼ TuT V TP t þ k;N V P t þ k;N Tu þ K2Nu . It follows that the optimal control (11.98) is identical to the vector of NGPC future controls in (11.74). ■ Solution The proof follows by collecting results in the section above.

■

Remarks There is no restriction on the number of inputs and outputs being controlled in this solution. That is, the outputs being controlled can be different in number to the number of inputs. The above results can clearly be related to the polynomial-based predictive controls in Chap. 7 [27].

11.5

Nonlinear Predictive GMV Optimal Control

A more general nonlinear control design approach will now be considered for systems described by qLPV models. The Nonlinear Predictive Generalized Minimum Variance (NPGMV) controller is obtained below for a system where the input nonlinear dynamic term W 1k is present. The signals u(t) and u0(t) are therefore different in this problem. The actual input to the system is the control signal u(t), shown in Fig. 11.1, rather than the input to the qLPV subsystem, that was denoted by u0(t). The cost-function for the NPGMV control problem must therefore include an additional control signal cost-function weighting term on u(t). The costing on the intermediate signal u0(t) can be retained to examine limiting cases and to provide what may be a useful actuator cost-function weighting term. For the NPGMV control problem (that may include an incremental cost term), the control signal cost weighting may be defined to have the form:
 ðF c DuÞðtÞ ¼ F ck zk Du ðtÞ

ð11:99Þ

This weighting on the nonlinear subsystem input will typically be a linear dynamic operator but it may also be chosen to be a nonlinear operator if required. It may, for example, be chosen to compensate for plant input nonlinearities or may be used to introduce an anti-windup capability [28]. The control weighting operator F ck will be assumed full rank and invertible. In analogy with the previous NGPC problem,

500

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

a multi-step cost-index may now be defined that is an extension of the cost-function in (11.84). Extended Multi-step Cost-Index: 0 Jp ¼ EfU0T t þ k;N Ut þ k;N jtg

ð11:100Þ

Now consider a new signal whose variance is to be minimized, involving a weighted sum of error, subsystem input and control-input signals. The signal U0t þ k;N is defined to include the future control signal costing terms: 0 0 U0t þ k;N ¼ PCN;t EP t þ k;N þ FCN;t DUt;N þ F ck;Nu DUt;Nu u

ð11:101Þ

The nonlinear function F ck;Nu DUt;Nu will normally be defined to have a simple block diagonal matrix form: ðF ck;Nu DUt;Nu Þ ¼ diagfðF ck DuÞðtÞ; ðF ck DuÞðt þ 1Þ; . . .; ðF ck DuÞðt þ Nu Þg ð11:102Þ Note the vector of changes at the input of the qLPV subsystem: 0 ¼ ðW 1k;Nu DUt;Nu Þ DUt;N u

ð11:103Þ

This represents the output of the nonlinear input subsystem W 1k;Nu , which also has a block diagonal matrix form: ðW 1k;Nu DUt;Nu Þ ¼ diagfW 1k ; W 1k ; . . .; W 1k gDUt;Nu ¼ ½ðW 1k DuÞðtÞT ; . . .; ðW 1k DuÞðt þ Nu ÞT T

ð11:104Þ

11.5.1 NPGMV Cost-Index Note for later use that the state-estimation error is independent of the choice of control action. Also recall that the optimal ^xðt þ kjtÞ and ~xðt þ kjtÞ are orthogonal and the expectation of the product of the future values of the control action (assumed known in deriving the prediction equation), and the zero-mean white noise b P t þ k;N and driving signals, is null. It follows that the vector of predicted signals E e the prediction error E P t þ k;N are orthogonal. The solution of the NPGMV control problem follows from very similar ideas to those in Chap. 9 (Sect. 9.4) and is therefore only summarized. The signal U0t þ k;N to be minimized can be written, noting (11.85) in the form:

11.5

Nonlinear Predictive GMV Optimal Control

501

0 0 U0t þ k;N ¼ PCN;t EP t þ k;N þ FCN;t DUt;N þ F ck;Nu DUt;Nu ¼ Ut þ k;N þ F ck;Nu DUt;Nu u

b0 e0 and in terms of the prediction and prediction error U0t þ k;N ¼ U t þ k;N þ U t þ k;N . It b0 may be written as follows that the predicted signal U t þ k;N

b b0 U t þ k;N ¼ U t þ k;N þ ðF ck;Nu DUt;Nu Þ 0 0 b P t þ k;N þ FCN;t DUt;N þ ðF ck;Nu DUt;Nu Þ ¼ PCN;t E u

ð11:105Þ

From (11.89) the estimation error: T T e0 e e e U t þ k;N ¼ U t þ k;N ¼ PCN;t E P t þ k;N ¼ Tu V P t þ k;N E P t þ k;N

ð11:106Þ

b0 The future predicted values of the signal U t þ k;N involve the weighted predicted b e P t þ k;N . The estimation error is errors PCN;t E P t þ k;N , which are orthogonal to PCN;t E zero-mean and the expected value of the product with any known signal is null. Extended multi-step cost-index: The cost-function may, therefore, be written as b0 b 0T U e e J ðtÞ ¼ U t þ k;N t þ k;N þ J 1 ðtÞ

ð11:107Þ

b0 The condition for optimality U t þ k;N ¼ 0, that determines the optimal solution, therefore has the form: 0 0 b P t þ k;N þ FCN;t PCN;t E DUt;N þ F ck;Nu DUt ;Nu ¼ 0 u

ð11:108Þ

This equation is obviously related to the condition for optimality for the previous b t þ k;N ¼ 0, but with the control signal weighting F ck;Nu added. NGPC problem U

11.5.2 NPGMV Optimal Control Solution The vector of future optimal control signals, to minimize the cost-function (11.107), follows from the condition for optimality in Eq. (11.108):

0 0 b E DUt;Nu ¼ F 1 P  F DU CN;t P t þ k;N ck;Nu CN;t t;Nu Optimal control: The optimal control signal becomes 2 T T b DUt;Nu ¼ F 1 ck;Nu ðTu V P t þ k;N E P t þ k;N  KNu W 1k;Nu DUt;Nu Þ

ð11:109Þ

The optimal predictive control law is nonlinear since it involves the nonlinear control signal costing term F ck;Nu and the nonlinear model for the plant W 1k;Nu . This expression can be modified to avoid the algebraic-loop problem. This is

502

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

discussed in the following sections on implementation and was also considered in Chap. 5 (Sect. 5.4). Further simplification is possible by again using the condition for optimality. This may be written from (11.69), (11.75), (11.97) and (11.105) as 0 0 b P t þ k;N þ FCN;t PCN;t E DUt;N þ ðF ck;Nu DUt;Nu Þ u T T 0 e ¼ PCN;t D P t þ k;N þ ðTu V P t þ k;N V P t þ k;N Tu þ K2Nu ÞDUt;N þ ðF ck;Nu DUt;Nu Þ u 0 e ¼ PCN;t D P t þ k;N þ X t þ k;Nu DUt;Nu þ ðF ck;Nu DUt;Nu Þ ¼ 0

ð11:110Þ b0 This condition for optimality U t þ k;N ¼ 0 may be written, noting (11.70) and (11.103), as 

PCN;t DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ X t þ k;Nu W 1k;Nu þ F ck;Nu DUt;Nu ¼ 0 ð11:111Þ An expression for the optimal control that is not so valuable for implementation follows as
1
 DUt;Nu ¼ X t þ k;Nu W 1k;Nu þ F ck;Nu PCN;t DP t þ k0 ;N  C/t ^xðt þ kjtÞ ð11:112Þ Optimal control: The solution for the vector of future optimal controls, in terms of the estimate of the future predicted state and useful for implementation follows from (11.110) as

e DUt;Nu ¼ F 1 P  X W DU D CN;t P t þ k;N t þ k;Nu 1k;Nu t;Nu ck;Nu
 ^ ð11:113Þ P D  C W DU x ðt þ kjtÞ  X ¼ F 1 CN;t P t þ k;N /t t þ k;Nu 1k;Nu t;Nu ck;Nu where PCN;t ¼ TuT V TP t þ k;N and hence, C/t ¼ PCN;t CP t þ k;N At þ k;N ¼ TuT V TP t þ k;N CP t þ k;N At þ k;N

ð11:114Þ

Remarks • The optimal control law in Eq. (11.113) is clearly model-based and includes an internal model for the nonlinear process. • The control law must be implemented using a receding horizon philosophy, as explained in Chap. 7 (Sect. 7.3.2).

11.5

Nonlinear Predictive GMV Optimal Control

503

• It is clear from (11.112) that the optimal control is identical to the NGPC controller (11.74) in the limiting case when the control signal costing tends to zero ðF ck;Nu ! 0Þ and W 1k;N ¼ I. • The problem construction enables a useful property to be predicted, which can be confirmed from (11.112). That is, if the control weighting F ck;Nu ! 0 then DUt;Nu will introduce the inverse of the plant model W 1k;Nu (if one exists). The resulting vector of future control changes DUt;Nu will then be the same as that for the GPC controller for the qLPV system that remains. This is not a practical solution but it does provide some confidence in the control philosophy in this limiting case.

11.5.3 Nonlinear Predictive Control Summary These results for the Nonlinear Predictive Generalized Minimum Variance controller for qLPV systems may be summarized in the following theorem. Theorem 11.3: NPGMV State-Dependent/qLPV Optimal Control Law Consider the components of the qLPV plant, disturbance and output weighting models put in the augmented state-equation form (11.17), with input from the nonlinear finite-gain stable plant dynamics W 1k . Assume that the multi-step predictive control cost-function to be minimized involves a sum of future cost terms, defined in vector form as: 0 Jp ¼ EfU0T t þ k;N Ut þ k;N jtg

ð11:115Þ

where the signal U0t þ k0 ;N depends upon future error, input and nonlinear control signal costing terms: 0 0 U0t þ k;N ¼ PCN;t EP t þ k;N þ FCN;t DUt;N þ F ck;Nu DUt;Nu u

ð11:116Þ

Assume the error and input cost-function weightings are defined similar to those in an NGPC problem (11.63) and define the related block matrix cost-weightings 0 ¼ K2Nu . Assume that the control signal cost-weighting PCN;t ¼ TuT V TP t þ k;N and FCN;t is nonlinear and is of the form ðF c DuÞðtÞ ¼ ðF ck DuÞðt  kÞ, where F ck is a full rank and invertible operator. Then the NPGMV optimal control law to minimize the variance (11.115) is given as
 DUt;N ¼ F 1 xðt þ kjtÞ  X t þ k;Nu W 1k;Nu DUt;Nu ck;Nu PCN;t DP t þ k;N  C/t ^ ð11:117Þ

504

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

where X t þ k;Nu ¼ TuT V TP t þ k;N V P t þ k;N Tu þ K2Nu and C/ t ¼ PCN;t CP t þ k;N At þ k;N . The current control can be computed using the receding horizon principle from the first component in the vector of future optimal controls. ■ Solution The proof of the optimal control was given before the theorem. The assumption needed for closed-loop stability is explained in the stability analysis that follows below. ■ Remarks (i) The expression for the NPGMV optimal control signal in (11.117), in terms of the predicted state estimate, is best suited for implementation. This is shown in the controller structure of Fig. 11.3. Unfortunately, it includes a so-called algebraic loop (constant through term) in the inner feedback loop but there are various ways to deal with this problem as discussed in Chap. 5. (ii) Inspection of the form of the cost-function term (11.116) in the case when the 0 input costing FCN is null gives U0t þ k;N ¼ PCN;t EP t þ k;N þ F ck;N Ut;N . For a single-stage cost-function N = 0, PCN;t ¼ TuT V TP t þ k;N and if TuT ¼ I then PCN;t ¼ V TP t þ k;N where V P t;0 ¼ E p t ¼ E p E 0 . The limiting case of the NPGMV controller is therefore related to an NGMV controller, where the error weighting is scaled by the PCN;t ¼ V TP t þ k;N ¼ E Tp t þ k term. (iii) It is useful to separate the vector of control changes into that to be applied at time t and that in the future. This can be achieved by introducing the matrices CI0 ¼ ½I; 0; . . .; 0 and C0I ¼ ½ 0 IN  so that the current and future control f ¼ C0I DUt;N , changes can be found as DuðtÞ ¼ ½I; 0; . . .; 0DUt;N and DUt;N respectively.

P t k , N

PCN , t

+ +

-

1

U t , Nu

c k , Nu

-

State estimator ˆ(t t x

dd , dm

u

ym +



1k , Nu



Controlled output

+ Disturbances

 k | t)

u0 (t  k )

y

Plant

Controller Structure

zm vm

 Measured output

Fig. 11.3 Implementation of NPGMV controller for qLPV modelled systems

11.5

Nonlinear Predictive GMV Optimal Control

505

11.5.4 Including Anti-windup Protection in Predictive Control The windup phenomenon was described in Sect. 5.3 regarding NGMV control. It arises when a controller includes integral action and the plant involves input limits or a saturation type of characteristic. In the optimal and predictive control philosophies described previously, integral action was normally introduced via a dynamic cost-function weighting term Pc ðz1 Þ, on the system error signal. An augmented state-equation-based system model was then formed including the error weighting. This resulted in an integrator being present within the Kalman filtering or observer estimation block that provided the state estimates for the subsystem Pc ðz1 Þ. Unfortunately, this integrator can be subject to the same wind-up phenomena that occur in classical feedback control systems. The nonlinear optimal and predictive control methods will take account of the effect of a plant input limit or saturation. If, for example, the nonlinear system model for the plant includes a saturation nonlinearity on the input channel, or if it contains control signal maximum and minimum limits, the resulting optimal system is aware that the nonlinearity is present. This type of nonlinearity can conveniently be included in the black-box input subsystem term in the NGMV or NPGMV solutions. It may also be introduced in these and other predictive algorithms by using a qLPV plant model, where the input B matrix (that can be a function of the input) could be modified to approximate the saturation or limiting behaviour. Unfortunately, it is not clear that simply including the nonlinearity in the plant models will provide anti-windup in the predictive control that is as effective as traditional measures. This is because the vector of future controls that are computed does not match the actual controls applied using the receding horizon principle. Including the saturation or limits in the plant model ensures accurate state estimation. This is because the control signal that enters the Kalman filter or observer will be subject to the same constraint (or limit), that the plant observes. Thus, the estimator should give accurate state estimates, since it has been informed of the presence of the input nonlinearity. It is, of course, a general rule that any known signals, which enter the plant, should also be input to the Kalman filter model via the same input map. There is another factor which affects windup and that is the second part of the problem definition which involves the cost-function weightings. Assume that the error-weighting model P c ðz1 Þ includes integrators for reducing steady-state errors in the multivariable system. This will result in integrators being present in the Kalman filter, via the augmented system model, which provides the controller integral action. Thus there is a direct link between the error weighting choice and the windup problem. It suggests that one approach to mitigate the effects of windup is to change the performance requirements when entering the windup situation. This can be achieved by adopting a qLPV or time-varying model for this error weighting, and modifying the requirements if the system enters saturation or exceeds hard constraints.

506

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

A more pragmatic way to consider this problem is that the integrator must not continue to integrate if windup is to be avoided in saturation conditions. In the scalar case, this suggests freezing the error-weighting integrator within the filter until the error reverses. However, for the multivariable case it is not so obvious what action is required. The error weighting P c ðz1 Þ is normally a diagonal matrix. If the system has the same number of inputs as outputs being controlled (not necessarily a square plant) and is only weakly interactive, then the scalar approach may be applied to the respective channels. However, if a particular input has a significant effect on more than one error channel then the integrators for these channels should be frozen. Thus, a determination must be made at the design stage regarding the couplings involved. This approach can be related to changing the performance requirements when the saturation makes it impossible to achieve a low error. It is a recognition that further integration of an error signal does not reflect the control action that is needed for the “nonlinear” system. The judgement about the degree of interaction and thereby which integrator to freeze should not be critical but some experimentation may be useful to optimize performance. The resulting controller will account for the saturation in the plant both within the Kalman filter and within the control law calculation. It will introduce a form of anti-windup behaviour without ad hoc changes. Finally, note that the anti-windup approach for the NGMV control previously proposed in Chap. 5 [28] also attempts to change cost-function performance requirements to cope with the windup problem, but in this case it modifies the control weighting matrix F ck ðz1 Þ. This is assumed invertible and a hard limit is replaced by a piecewise linear function with small slope for large amplitude inputs, rather than no slope. The above procedure applied to the error-weighting terms is probably easier to implement by changes to the software, which simply clamp the integrator output signals.

11.6

Stability of the Closed-Loop and Robust Design Issues

To consider stability properties, an expression is required for the control and output signals in a closed-loop form. For the proposed nonlinear predictive control (11.117), it is shown below that a nonlinear operator:



  C/t Ut þ k Bt þ k CI0 þ X t þ k;Nu W 1k;Nu þ F ck;Nu

must have a stable inverse. The measure of stability used, such as finite-gain stability, depends upon the type of stability assumed for the nonlinear plant subsystem W 1k . Assume that the stochastic external inputs are null so that the only inputs are those due to the known (deterministic) reference and output disturbance

11.6

Stability of the Closed-Loop and Robust Design Issues

507

signals. Also recall the resolvent operator Ut ¼ ðI  z1 At Þ1 z1 , so that the state (11.17) can be represented as xðtÞ ¼ ðI  z1 At Þ1 z1 ðBt Du0 ðt  kÞ þ dd ðtÞÞ ¼ Ut ðBt Du0 ðt  kÞ þ dd ðtÞÞ The expression for the future state in this deterministic case follows as xðt þ kÞ ¼ Ut þ k ðBt þ k Du0 ðtÞ þ dd ðt þ kÞÞ

ð11:118Þ

and the predicted state ^xðt þ kjtÞ ¼ xðt þ kÞ. From (11.111) and recalling C/t ¼ PCN;t CP t þ k;N At þ k;N , the condition for optimality follows as



 PCN;t DP t þ k;N þ C/t ^xðt þ kjtÞ þ X t þ k;Nu W 1k;Nu þ F ck;Nu DUt;Nu ¼ 0

Substituting from (11.118): PCN;t DP t þ k;N þ C/t Ut þ k dd ðt þ kÞ 
þ ðC/t Ut þ k Bt þ k CI0 þ X t þ k;Nu ÞW 1k;Nu þ F ck;Nu DUt;Nu ¼ 0

ð11:119Þ

where the vector W 1k;Nu DUt;Nu may be written as ðW 1k;Nu DUt;Nu Þ ¼ ½ðW 1k DuÞðtÞT ; . . .; ðW 1k DuÞðt þ Nu ÞT T . The future controls and outputs in closed-loop operator form satisfy the following: Future controls:

 1 DUt;Nu ¼ C/t Ut þ k Bt þ k CI0 þ X t þ k;Nu W 1k;Nu þ F ck;Nu
  PCN;t DP t þ k;N  C/t Ut þ k dd ðt þ kÞ ð11:120Þ NL subsystem future outputs:

 1 W 1k;Nu DUt;Nu ¼ W 1k;Nu C/t Ut þ k Bt þ k CI0 þ X t þ k;Nu W 1k;Nu þ F ck;Nu
  PCN;t DP t þ k;N  C/t Ut þ k dd ðt þ kÞ ð11:121Þ Future plant outputs:



 1 W k;Nu DUt;Nu ¼ W k;Nu C/t Ut þ k Bt þ k CI0 þ X t þ k;Nu W 1k;Nu þ F ck;Nu
  PCN;t DP t þ k;N  C/t Ut þ k dd ðt þ kÞ ð11:122Þ

Stability of the closed loop: From (11.120)–(11.122) a necessary condition for stability is that the inverse of the operator:



  C/t Ut þ k Bt þ k CI0 þ X t þ k;Nu W 1k;Nu þ F ck;Nu

is finite-gain m2 stable. For the sufficient condition for stability, some further structure on the nature of the nonlinearity must be assumed (see Chap. 7 and [29]).

508

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

11.6.1 Cost-Function Weightings and Relationship to Stability Similar to the results in previous chapters in a limiting case, if there exists say a controller that will stabilize the nonlinear system, without transport delay elements, then a set of cost-weightings can be defined to guarantee the existence of this inverse, and ensure the stability of the closed loop. One of the problems in nonlinear control design is to obtain a stabilizing control law and this depends on the selection of the cost-function weightings. To demonstrate the result for this type of problem, first assume that only the error and control weightings are used, and the input weighting K2Nu ! 0. Then, from (11.75) X t þ k;Nu ! TuT V TP t þ k;N V P t þ k;N Tu , and from Eq. (11.120): 

 1
C/t Ut þ k Bt þ k CI0 þ X t þ k;Nu W 1k;Nu þ F ck;Nu PCN;t DP t þ k;N  C/t Ut þ k dd ðt þ kÞ

1
PCN;t DP t þ k;N ! C/t Ut þ k Bt þ k CI0 þ TuT V TP t þ k;N V P t þ k;N Tu W 1k;Nu þ F ck;Nu  C/t Ut þ k dd ðt þ kÞ

DUt;Nu ¼

In the case of a single-stage cost problem with a through term (N = 0, Tu ¼ I and 0 ¼ k2 IÞ, the matrix V P t þ k;N ¼ E t þ k;N is assumed non-singular. The block FCN;t matrices, from (11.52), are At þ k;N ¼ I, Bt þ k;N ¼ 0, Dt þ k;N ¼ 0, CP t þ k;N ¼ Cp t þ k , E P t þ k;N ¼ E p t þ k , and hence V P t;N ¼ E p t and PCN;t ¼ TuT V TP t þ k;N ¼ E Tp t þ k , C/t ¼ E Tp t þ k Cp t þ k . Thence we obtain

1
 DuðtÞ ! E Tp t þ k Cp t þ k Ut þ k Bt þ k þ E p t þ k W 1k þ F ck;Nu E Tp t þ k DP t þ k;N

E Tp t þ k Cp t þ k Ut þ k dd ðt þ kÞ If the dynamic weighting is on the plant outputs yp ðtÞ ¼ P c ðz1 ÞyðtÞ then E p t þ k þ Cp t þ k Ut þ k Bt þ k ¼ P c W 0k . From these results, it follows that in this limiting case:



1 DuðtÞ ! E Tp t þ k P c W 0k W 1k þ F ck E Tp t þ k DP t þ k;N  E Tp t þ k Cp t þ k Ut þ k dd ðt þ kÞ

ð11:123Þ A possible method of deriving cost-weightings that will stabilize the system is suggested from these limiting results. That is, the term ðE Tp t þ k P c W 0k W 1k þ F ck Þ may be interpreted as the return-difference operator for a nonlinear system with delay-free plant model W k ¼ W 0k W 1k . Thus, if the plant already has a PID controller that stabilizes this model, the ratio of weightings in this limiting case can be T chosen as F 1 ck E p t þ k P c ¼ KPID .

11.6

Stability of the Closed-Loop and Robust Design Issues

509

11.6.2 MPC Robustness, Design and Guidelines In the early days of linear Model Predictive Control (MPC), there was not much attention to stability questions and even less to achieving robust control design solutions. One of the main current research areas in linear and nonlinear predictive control lies in this area of robust predictive control systems design. There are of course different types of robustness. The most obvious robustness requirement is that disturbances are rejected efficiently, even if there is some uncertainty regarding their nature. The second very critical requirement for robustness is that stability properties should not be compromised by gain, time constant or other variations within the system. Finally, there is the robustness of the control law with respect to tuning parameters. This latter type of robustness is more of a practical requirement than a theoretical challenge. A well-behaved control system will be one where the tuning of the controller parameters will not be too critical to the stability or the performance properties, and this is an unusual but important type of robustness. This last measure of system robustness has not received so much attention. Clearly if the tuning parameters of a controller (like PID gains, or optimal control cost-weightings), are such that when small changes are made they completely change the transient behaviour of the system, then this is not robust in a tuning sense. This is related to the so-called controller fragility problem. Controllers can be classified as being either fragile, non-fragile or resilient. A controller is fragile if very small perturbations in the coefficients of the controller destabilize the closed-loop control system. To assess controller fragility an index can be defined which relates the loss of robustness of the control loop, when controller parameters change to the nominal robustness of the control loop. The robust tuning problem is therefore related to the fragility of the controller. Several practical design steps can influence robustness [30, 31]. For example: • The model used for control design may be obtained from a nonlinear plant model derived by testing or from physical equations. This model will often require approximations or simplifications to be made. The quasi-LPV or state-dependent model is not unique and the choices taken should try to maintain the link with the physical system and as the assumptions demand be pointwise stabilizable and detectable. • Increasing the prediction horizon for linear time-invariant systems will normally improve robustness in the sense that transient response overshoots will decrease suggesting that sensitivities are improved. Note that this behaviour does assume weightings might be modified to maintain similar speeds of response. However, for qLPV systems it may be that using a shorter prediction horizon actually improves robustness. The beneficial results of increasing the prediction horizon are sometimes countered by the increased inaccuracies in the prediction errors. There will, therefore, be a mismatch in the future control signal and state trajectories, which depend upon the previously calculated control sequence. This can be reduced by using iteration to improve the future control sequence but

510

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

experience reveals this is not often very effective, whilst adding to the computational burden. • Stability can be guaranteed in linear MPC, with a finite prediction horizon, by adding a zero-terminal constraint at the end of the prediction horizon (adding the constraint x(t + N) = 0). Alternatively, an end-state weighting may be introduced in the cost-function. These cost-function terms have an influence on robustness and may provide a valuable tuning capability for nonlinear systems.

11.6.3 Modelling the Parameter Variations The model for qLPV systems introduced in Chap. 1 (Sect. 1.7) can allow for stochastic variations in the parameters. If say the parameter vector is qðtÞ, then the parameter variations hðtÞ may be modelled by a known component hd ðtÞ and a stochastic term as follows: hðt þ 1Þ ¼ Ah hðtÞ þ Dh nh ðtÞ qðtÞ ¼ Ch hðtÞ þ hd ðtÞ where nh ðtÞ is white noise of zero-mean and given covariance. The augmented system will include this model and the Kalman filter will provide an estimate of the states b hðtÞ. The resulting controller will, therefore, accommodate these expected variations, which may improve robustness when there is uncertainty in the knowledge of the parameters of the model. This is justified in the same way as for disturbance models that are often a crude approximation but still a useful way to allow for uncertainties.

11.6.4 Formal Robust Predictive Control Approaches The main disadvantage of the traditional linear predictive control solutions is that they do not allow for plant model nonlinearities or uncertainties in a formal way. The design methods presented here provide a formal approach to deal with nonlinearities, but not to uncertainties. The Robust Model Predictive Control (Robust MPC) is a class of model-based control methods that can formally and explicitly account for the presence of modelling uncertainties in the plant. The extension of traditional MPC strategies to robust control solutions can involve improvements in the state/output predictions, and to the closed-loop stability properties (see [7, 32– 38]). To allow for uncertainty in predictions, a number of uncertainty descriptions can be used to obtain future state/output expressions. Uncertainty might be represented by an additive unknown but bounded disturbance, or by say a multiplicative uncertainty. This might be caused by a number of factors such as imprecise plant

11.6

Stability of the Closed-Loop and Robust Design Issues

511

knowledge, or the time-varying nature of the model parameters, which are unknown but belong to a bounded set. Models for uncertain systems were discussed in Chap. 6 (Sect. 6.5.3). The GPC type of on-line constrained minimization problem can be changed into a min-max problem to deal with uncertainties. This can involve minimizing the worst-case value of the cost-function, where the worst case is taken over the set of uncertain models. Over recent years, min-max optimization has been used to analyse robust MPC problems with work from the late 1990s. The cost-function is usually defined using a weighted two-norm summation. This is maximized in terms of system uncertainties and an upper bound is minimized with respect to the control inputs. There are various approaches in current predictive control systems to improve robustness by allowing for uncertainty explicitly. The main difficulty is the increase in complexity that results, detracting from a major advantage of predictive control, which is its relative simplicity. Rigorous formal solutions to the robust MPC problem have been produced by, for example, Kouvaritakis et al. [38] and the value in applications was demonstrated by Nagy and Braatz [39] for a batch crystallization process. They considered the effect of parameter uncertainty in both the Extended Kalman Filter (EKF) and a nonlinear MPC algorithm. Linear Matrix Inequality (LMI) methods were discussed briefly in Chap. 5 (Sect. 5.6.4). Robust predictive control algorithms have been proposed using LMI-based algorithms [40]. The basic idea is to interpret the MPC problem as a semi-definite programming problem [35]. This may involve an optimization problem with linear objective-function and positive-definite constraints, involving symmetric matrices related to the decision variables. The infinite-horizon MPC problem, with input and output constraints, and plant uncertainty, was formulated as a convex optimization problem involving LMI’s by Kothare et al. [7]. The LMI optimization for a predictive control must normally be solved online at each time step, which can be computationally intensive. However, it represents a formal approach to handling this difficult problem and is growing in popularity.

11.7

NPGMV Simplifications for Implementation

Two simplifications will be introduced that considerably simplify the computation of the controller. The first is to assume the nonlinear plant model can be represented using only the qLPV state-equation model [41]. In this case the black-box model W 1k can be set equal to the identity ðW 1k ¼ I so that W 1k;Nu ¼ IN Þ, and the signals u(t) = u0(t). From the condition for optimality, in (11.111):
 PCN;t ðDP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞÞ þ X t þ k;Nu þ F ck;Nu DUt;Nu ¼ 0 ð11:124Þ

512

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

The vector of future optimal control signals, in terms of the predicted state, for this system that only contains the qLPV subsystem, becomes
1 DUt;N ¼ X t þ k;Nu þ F ck;Nu ðPCN;t DP t þ k;N  PCN;t CP t þ k;N At þ k;N ^xðt þ kjtÞÞ
1 ¼ X t þ k;Nu þ F ck;Nu ðPCN;t DP t þ k;N  C/t ^xðt þ kjtÞÞ ð11:125Þ Recall matrices X t þ k;Nu ¼ TuT V TP t þ k;N V P t þ k;N Tu þ K2Nu and C/t ¼ PCN;t CP t þ k;N At þ k;N , and note the control weighting involves both the constant K2N and the dynamic F ck;N weighting terms. These determine the inverse matrix in (11.125).

11.7.1 NPGMV Special Weighting Case The algorithm that follows is the simplest predictive control solution for nonlinear systems that avoids the difficulty with algebraic loops. Assume that the dynamic control signal weighting F ck ðz1 Þ is linear, or has a nonlinear decomposition into a non-dynamic or constant term F ack and a dynamic operator term F bck ðz1 Þ, including at least a unit-step delay. This can, of course, be arranged by the designer who specifies the form of the control weighting function. That is, we write F ck ðz1 Þ ¼ F ack þ F bck ðz1 Þ The block-diagonal version of these functions involves the decomposition of F ck;Nu into the terms F ack;Nu and F bck;Nu ðz1 Þ, so that F ck;Nu ðz1 Þ ¼ F ack;Nu þ F bck;Nu ðz1 Þ. Then, from Eq. (11.124):

ðPCN;t DP t þ k;N þ C/t ^xðt þ kjtÞÞ þ X t þ k;Nu þ F ack;Nu þ F bck;Nu ðz1 Þ DUt;Nu ¼ 0 or

PCN;t DP t þ k;N þ C/t ^xðt þ kjtÞ þ F bck;Nu ðz1 ÞDUt;Nu þ X t þ k;Nu þ F ack;Nu DUt;Nu ¼ 0 Thence, we obtain for a linear control signal cost term:

1

DUt;Nu ¼ X t þ k;Nu þ F ack;Nu PCN;t DP t þ k;N  C/t ^xðt þ kjtÞ  F bck;Nu ðz1 ÞDUt;Nu

ð11:126Þ where C/t ¼ TuT V TP t þ k;N CP t þ k;N At þ k;N and PCN;t ¼ TuT V TP t þ k;N .

11.7

NPGMV Simplifications for Implementation

513

Controller Structure

P t k ,N

P

CN , t

Kalman predictor

ˆ(t t x dd d

+ +

-

Plant

U t , Nu (

 k | t)

t  k , Nu

b



a c k , Nu

c k , Nu ( z

1

u ( t  k )

)

CI0

1

)

(1   z )

1 1

u

Controlled output

y

+ ym +

Disturbances

zm

vm

z k Measurements or observations signal



Fig. 11.4 Simplified NPGMV controller structure uðtÞ ¼ ð1  bz1 Þ1 DuðtÞ

Remarks on special weighting case: • The above solution involved a number of simplifications and fast and efficient algorithm results since there is no algebraic loop in the control law computations. • Similar simple results can be obtained, as discussed in Chap. 15 (Sect. 15.2.3), when the input subsystem W 1k ðsÞ can be decomposed as ðW 1k uÞðtÞ ¼ G0 uðtÞ þ G1 ðuðtÞÞ. This is, of course, more general than assuming the input subsystem is not needed (replaced by the identity matrix). However, this is a property of the plant description and may not be achievable without serious approximations. • The implementation of the controller, in terms of the predicted state, is shown in Fig. 11.4. The estimator input includes the control term and the deterministic output disturbance d(t) and the reference and input disturbance term dd ðtÞ. • The matrix inverse required here ðX t þ k;Nu þ F ack;Nu Þ1 is time-varying but non-dynamic, which simplifies the computations.

11.8

NPGMV in Terms of Finite Pulse Response

The NPGMV controller can be expressed in terms of the current state, rather than the predicted state. This is illustrated for the general problem discussed in Sect. 11.5 above, and the special simplified problem considered in Sect. 11.7. Note that this approach of using the current state was exploited in previous chapters to demonstrate the structural similarities to Smith Predictors and the Internal Model-Based Control methods. The predicted state may be expressed in terms of the current state in (11.61), using the finite pulse-response operator T ðk; z1 Þ, as follows:

514

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

^xðt þ kjtÞ ¼ Akt ^xðtjtÞ þ T ðk; z1 ÞDu0 ðtÞ þ ddd ðt þ k  1Þ The condition for optimality (11.111) may now be expanded using the predicted state as

 PCN;t DP t þ k;N þ CP t þ k;N At þ k;N ^xðt þ kjtÞ þ X t þ k;Nu W 1k;Nu þ F ck;Nu DUt;Nu ¼ 0 Since C/t ¼ PCN;t CP t þ k;N At þ k;N , we obtain
k  PCN;t DP t þ k;N
þ C/t At ^xðtjtÞ þ T Du0ðtÞ þ ddd ðt þ k  1Þ þ X t þ k;Nu W 1k;Nu þ F ck;Nu DUt;Nu ¼ 0 This equation may be written as PCN;t DP t þ k;N þ C/t Akt ^xðtjtÞ þ C/t ddd ðt þ k  1Þ
 þ C/t T CI0 W 1k;Nu þ X t þ k;Nu W 1k;Nu þ F ck;Nu DUt;Nu ¼ 0

ð11:127Þ

In terms of the current state estimate and the finite pulse response T ðk; z1 Þ, the vector of future controls: k DUt;Nu ¼ F 1 xðtjtÞ þ ddd ðt þ k  1ÞÞ ck;Nu ðPCN;t DP t þ k;N  C/t ðAt ^
  X t þ k;Nu þ C/t T CI0 W 1k;Nu DUt;Nu Þ

ð11:128Þ

This controller structure is illustrated in block diagram form in Fig. 11.5.

Controller Structure P t k ,N

P

CN , t

d

+

C t (

k t

+ +

1

c k , Nu

-

U t , Nu

u

u0 (t  k )

t  k , Nu



t

CI0

Noise vm

ym

(1   z 1 )1 CI0

xˆ (t | t )

 d dd (t  k  1)) d d , d dd

-

Plant

1k , Nu

Disturbances

U t0, Nu

Fig. 11.5 Implementation of general NPGMV and qLPV model controller structure

+ +

zm

11.8

NPGMV in Terms of Finite Pulse Response

515

11.8.1 Simplified Controller Structure and Special Weighting Case The simplified controller derived above in Sect. 11.7 may also be derived using the same assumptions but expressed in terms of the current state estimates. That is, with the assumption that the input nonlinear block is set to the identity, recall from (11.127) that the condition for optimality is PCN;t DP t þ k;N þ C/t Akt ^xðtjtÞ þ C/t ddd ðt þ k  1Þ
 þ C/t T CI0 þ X t þ k;Nu þ F ck;Nu DUt;Nu ¼ 0 The equation simplifies as



 PCN;t DP t þ k;N þ C/t Akt ^xðtjtÞ þ T DuðtÞ þ ddd ðt þ k  1Þ þ ðF ck;Nu þ X t þ k;Nu ÞDUt;Nu ¼ 0

ð11:129Þ

The vector of future optimal control signal changes therefore become
1 DUt;Nu ¼ F ck;Nu þ X t þ k;Nu ðPCN;t DP t þ k;N  C/t Akt ^xðtjtÞ C/t ðT ðk; z1 ÞDuðtÞ þ ddd ðt þ k  1ÞÞÞ

ð11:130Þ

Control weighting structure: If the dynamic control signal weighting F ck or F ck;Nu is linear (or has a nonlinear decomposition) and can be represented in terms of a non-dynamic or constant term F 0ck or F 0ck;Nu , and an operator term that includes at least a unit delay F 1ck ðz1 Þ or F 1ck;Nu ðz1 Þ, then the condition for optimality (11.129) becomes


PCN;t DP t þ k;N þ C/t ðAkt ^xðtjtÞ þ T DuðtÞ þ ddd ðt þ k  1ÞÞ

þ F 0ck;Nu þ F 1ck;Nu ðz1 Þ þ X t þ k;Nu DUt;Nu ¼ 0



It follows that the vector of future optimal controls, when there is no black-box plant model term, and the control weighting is of a linear form:

1 DUt;Nu ¼ F 0ck;Nu þ X t þ k;Nu Þ  ðPCN;t DP t þ k;N þ C/t ðAkt ^xðtjtÞ

þ T DuðtÞ þ ddd ðt þ k  1ÞÞÞ  F 1ck;N ðz1 ÞDUt;Nu

ð11:131Þ

516

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

Controller Structure P t k ,N

d

+

PCN ,t

Kalman predictor

C t dd

k

xˆ(t | t )

-

+ +

( c0k , Nu 

+ +

+ +

) 1

(1   z 1 )1 CI0

1 c k , Nu ( z )

d

t

 t dd

z

t  k , Nu

(k , z 1 )

vm

Plant

U t , Nu

u

ym

+ +

zm

Disturbance u (t  k )

k

Observations signal

Fig. 11.6 Implementation of simplified NPGMV controller structure

Remarks • The structure of the controller in Fig. 11.6 avoids the algebraic-loop problem. To demonstrate this result, recall that F 1ck ðz1 Þ is an operator, which includes a k P j1k Akj unit delay. Also from (11.62), the function T ðk; z1 Þ ¼ t þ j Bt þ j1 z j¼1

includes a unit delay. It follows that the current control does not depend through the inner loop on itself at time t, and the algebraic-loop problem is avoided. • The matrix inverse ðF 0ck;Nu þ X t þ k;Nu Þ1 is time-varying but non-dynamic which also simplifies the computations.

11.9

Rotational Link Control Design

The NPGMV controller described above is now applied to the control of the rotational link shown in Fig. 11.7. This is a common problem in the control of mechanisms. It may, for example, be viewed as a simplified model of a robotic manipulator with flexible joints. The motor torque must be controlled so that the motor rotates through a specified angle, whilst stabilizing the vibration of the robot or mechanism arm. Different control solutions to this problem were summarized by Zhang [42], who also described a predictive control solution. The rotational link is a highly nonlinear system, where a nonlinear controller is required. A Direct Current (DC) motor is used to rotate the link in the vertical plane. The equilibrium condition is defined to be the angle h = p, where the arm is hanging straight down. The objective is to control the motor such that the link is stabilized at some chosen reference angle. That is, the torque T(t) is applied to the rotational link so that the angular position h follows the desired trajectory, denoted

11.9

Rotational Link Control Design

517

Fig. 11.7 Rotational link motor and arm components

m, J

θ

g L

T

by href . This system has one control input uðtÞ ¼ TðtÞ. This is the torque demand to the motor which accelerates the link, through the torque produced [43]. Plant dynamics: The nonlinear continuous-time dynamics of the system can be described as follows:     _ d hðtÞ hðtÞ   ¼ _ _ hðtÞ _  þ uðtÞ mgL sinðhðtÞÞ=J  chðtÞ dt hðtÞ

ð11:132Þ

Let the angular position h be taken as the first system state. The continuous-time _ state variables can then be defined as x1 ðtÞ ¼ hðtÞ and x2 ðtÞ ¼ hðtÞ, where  T T T _ . The equilibrium conditions of the system xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ  ¼ hðtÞ hðtÞ may be expressed as     x s xeq ¼ 1 ¼ and ueq ðtÞ ¼ mgL sin x1 ðtÞ=J x2 0 where h ¼ x1 ¼ s is the scheduling variable. The state-dependent or qLPV model that arises naturally in this problem has the form:  x_ ðtÞ ¼

0 mgL sin x1 Jx1

   1 0 xðtÞ þ uðtÞ cjx2 j 1

ð11:133Þ

The angle output or noise-free measurement yðtÞ ¼ ½ 1 0 xðtÞ. Oscillator: For the nonlinear system, evaluated at one operating point, the system matrix may be approximated (crudely) in the form: 

Alin

0 ¼ a

 1 ; b



sI  Alin

s ¼ a

 1 ; sþb

detðsI  Alin Þ ¼ s2 þ sb þ a

518

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

It can be noted that near the steady-state operating condition the parameter b = 0 and the system is likely to behave like an oscillator. Discrete-Time Model: The discrete-time equivalent of the model can be approximated by replacing the continuous-time model as follows:  xðt þ 1Þ  xðtÞ ¼

0 Ts mgL sin x1 Jx1

   Ts 0 uðtÞ xðtÞ þ Ts cjx2 j Ts

or  xðt þ 1Þ ¼

1 Ts mgL sin x1 Jx1

   Ts 0 uðtÞ xðtÞ þ 1  Ts cjx2 j Ts

A high sample rate is assumed which ensures the discrete model is more appropriate. That is, the sample period Ts ¼ 0:001. If it, that a single-step delay is included, then the model to be used for design can be written as  xðt þ 1Þ ¼

1 Ts mgL sin x1 Jx1

   Ts 0 uðt  1Þ xðtÞ þ 1  Ts cjx2 j Ts

yðtÞ ¼ ½ 1 0 xðtÞ þ quðt  1Þ The output equation includes the scalar q, which represents the through term. In the derivation of the control law the assumption was that all explicit delay terms are taken out the model into the z−k term, which implies the delay k > 0. The discrete-time model is, of course, an approximation to the underlying continuous-time model, so rather arbitrarily the through term can be set to a small number q ¼ 0:0001, for control design purposes. The following values for the parameters may be assumed m = 1, g = 9.81, L = 0.5, J = 0.25, c = 5. For simplicity, the same predictive control prediction horizon will be used for each solution of N = 15. To allow for computing delays a minimum one-step explicit delay can be assumed (k = 1). The open-loop response of the system to step reference changes is shown in Fig. 11.8, which reveals the system is oscillatory at the stable operating point. PID Control: For the closed-loop simulations, a PID controller was defined for comparison as Cpid ¼ ð55  104:9z1 þ 50z2 Þ=ð1  z1 Þ ¼ 55ð1  0:9718z1 Þð1  0:9354z1 Þ=ð1  z1 Þ Predictive Controllers Error Weighting: The input weighting K2Nu ¼ 0:05 and the error weighting for the predictive controllers can use PID inspired weightings (Chap. 5), and is defined in the s-domain as

11.9

Rotational Link Control Design

519 Link position (deg)

200 150 100 50 0

5

10

15

10

15

10

15

Input torque (Nm) 0 -5 -10 0

5 Link velocity (deg/s)

100 50 0 -50 -100 0

5

time (s)

Fig. 11.8 Open-loop output time response

Pc ¼ Kp ð1 þ 1=ðTi sÞ þ Td s=ðss þ 1ÞÞ where Kp ¼ 5; Ti ¼ 0:5; Td ¼ 0:1; s ¼ 0:01 and discretized gives Pc ¼ 6:6643

ðz  0:9876Þðz  0:9802Þ ðz  1Þðz  0:9672Þ

The additional control weighting Fck ¼ ð0:2  0:16z1 Þ=ð1  0:99z1 Þ was also used for the NPGMV design, which provides the user with more freedom to shape the responses with respect to the basic NGPC control law. NGMV Optimal Control: In NGMV control the signal, whose variance is to be minimized, is defined as /0 ðtÞ ¼ Pc eðtÞ þ Fc0 u0 ðtÞ þ ðFck uÞðtÞ

ð11:134Þ

The NGMV control costing is linear in this case Fck ¼ 1 and the error weighting:

520

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

Pc ¼ 38:3833

ð1  0:9722z1 Þð1  0:9375z1 Þ ð1  z1 Þð1  0:3333z1 Þ

This initial NGMV design is based on the PID controller and gives a broadly similar type of response. Simulations: The sampling time was taken as 10 ms for the simulations ðTs ¼ 0:01 sÞ. The simulation time T ¼ 15 s and the reference angular position signal was defined as 8 180 > > < 150 rðtÞ ¼ 180 > > : 140

0t2 2\t  7 7\t  12 12\t  15

The starting angle was first set at a stable equilibrium (arm hanging down), requiring zero initial torque. A comparison of the responses is shown in Fig. 11.9. The future setpoint knowledge helps the NGPC and NPGMV predictive controllers. The retuned NGMV controller improves upon the PID responses, but cannot utilize much future setpoint information (as noted in Chaps. 4 and 10, the NGMV controller can only use the future setpoint information up to time k). For this example,

Link position (deg)

190 180 170

PID NGMV NGPC NPGMV

160 150 140 6.5

7

7.5

8

8.5

8

8.5

8

8.5

Input torque (Nm) 100 50 0 -50 -100 6.5

7

7.5 Link velocity (deg/s)

300 200 100 0 -100 6.5

7

7.5

time (s)

Fig. 11.9 Comparison of PID, NGMV, NGPC and NPGMV responses

11.9

Rotational Link Control Design

521

there is only a one-step delay, and there is little benefit to the NGMV control from future reference or setpoint knowledge. Initial Unstable Equilibrium: If the starting initial condition is changed to 90° and the same weights are employed, the NPGMV and NGMV controllers have the responses shown in Fig. 11.10. The responses for the predictive controller are clearly sharper. The starting angle from the horizontal position is not a stable equilibrium, such that the initial torque is nonzero. Detailed Comparison: The responses can be seen more clearly for a shorter time horizon, as in Fig. 11.11. If the reference trajectory is changed to a sine wave, the responses are as shown in Fig. 11.12. Note that the reference and NPGMV responses are similar for the scale used. If a disturbance is added at the system output (on angle position), the results are as shown in Fig. 11.13. The disturbance is an integrator with a gain 2  180=p and white unity variance driving noise. The responses are less than ideal in this case but the predictive solution is clearly better than NGMV. The effect of the choice of the prediction horizon is illustrated in Fig. 11.14, where the control weighting Fck has been set to zero for sharper short-horizon responses. It is clear there is some initial improvement for longer horizons (in particular, note its stabilizing effect) but little benefit is gained for horizons greater than N = 10.

Link position (deg)

100

NGMV NPGMV

80 60 40 0

5

10

15

10

15

Input torque (Nm) 100 50 0 -50 -100 0

5

Link velocity (deg/s)

300 200 100 0 -100 -200 0

5

10

time (s)

Fig. 11.10 Comparison of NGMV and NPGMV responses for 90° initial condition

15

522

11

LPV/State-Dependent Nonlinear Predictive Optimal Control Link position (deg)

100 90 80

NGMV NPGMV

70 60 6.8

7

7.2

7.4

7.6

7.8

8

7.6

7.8

8

7.8

8

Input torque (Nm) 100 50 0 -50 -100 6.8

7

7.2

7.4

Link velocity (deg/s)

300 200 100 0 -100 -200 6.8

7

7.2

7.4

7.6

time (s)

Fig. 11.11 Comparison of NGMV and NPGMV responses for a portion of time response Link position (deg) 120 100 80

NGMV NPGMV

60 0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.2

2.4

2.6

2.2

2.4

2.6

Input torque (Nm) 100 50 0 -50 -100 0.8

1

1.2

1.4

300 200 100 0 -100 -200 0.8

1.6

1.8

2

Link velocity (deg/s)

1

1.2

1.4

1.6

1.8

2

time (s)

Fig. 11.12 Comparison of NGMV and NPGMV responses for sine wave tracking

11.9

Rotational Link Control Design

523 Link position (deg)

120 100 80

NGMV NPGMV

60 0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.2

2.4

2.6

2.2

2.4

2.6

Input torque (Nm) 100 50 0 -50 -100 0.8

1

1.2

1.4

1.8

2

Link velocity (deg/s)

300 200 100 0 -100 -200 0.8

1.6

1

1.2

1.4

1.6

1.8

2

time (s)

Fig. 11.13 Comparison of NGMV and NPGMV responses for sine wave tracking including a significant output disturbance Link position (deg)

190 180 170 160

N=4 N=7 N=10 N=30

150 140 6.8

6.9

7

7.1

7.2

7.3

7.4

7.2

7.3

7.4

7.3

7.4

Input torque (Nm) 100 50 0 -50 -100 6.8

6.9

7

300 200 100 0 -100 -200 6.8

7.1

Link velocity (deg/s)

6.9

7

7.1

7.2

time (s)

Fig. 11.14 Comparison NPGMV responses for different tracking horizons without output disturbances

524

11

LPV/State-Dependent Nonlinear Predictive Optimal Control Link position (deg)

190 180 170

NGPC abs u(t) NPGMV abs u(t) NGPC inc u(t) NPGMV inc u(t)

160 150 140 6.7

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5

7.3

7.4

7.5

7.3

7.4

7.5

Input torque (Nm)

100 50 0 -50 -100 6.7

6.8

6.9

7

7.1

7.2

Link velocity (deg/s)

300 200 100 0 -100 6.7

6.8

6.9

7

7.1

7.2

time (s)

Fig. 11.15 Comparison NPGMV and NGPC responses for the absolute control and incremental control cases Link position (deg)

200

PID NPGMV abs u(t) NPGMV inc u(t)

150

100 1

2

3

4

5

6

7

8

9

10

7

8

9

10

8

9

10

Input torque (Nm)

50 0 -50 -100 1

2

3

4

5

6

Link velocity (deg/s)

100 0 -100 -200 -300 1

2

3

4

5

6

7

time (s)

Fig. 11.16 Comparison NPGMV responses for the absolute control and incremental control cases and PID for a series of steps

11.9

Rotational Link Control Design

525 Link position (deg)

180 160 140 120 0

NGMV NPGMV

5

10

15

10

15

10

15

Input torque (Nm) 100 50 0 -50 -100 0

5 Link velocity (deg/s)

400 200 0 -200 -400 0

5

time (s)

Fig. 11.17 Comparison of NGMV and NPGMV when no future setpoint information used in predictive control

Incremental Controls: A comparison of the use of incremental controls DuðtÞ and the absolute controls u(t) is shown in Fig. 11.15, where very similar results are obtained. The cost-function weightings employed in this example follow: NPGMV and NGPC Input Weighting: K2Nu ¼ 0:2 NPGMV Error Weighting: Pc ðz1 Þ ¼ 2:1311ðz  0:9692Þ=ðz  0:9672Þ NGPC Error Weighting: Pc ðz1 Þ ¼ 1 Observe that no integrator term is needed in the error weighting since in the incremental form, the integrator is already augmented with the plant model. The step responses of an NPGMV control design are shown in Fig. 11.16, together with the PID control design responses. These include both the absolute and incremental control cases and cover a wide range of the nonlinear system operating points. As might be expected, both of the predictive controls give superior performance. Importance of Future Setpoint Knowledge: A comparison of NGMV and NPGMV designs when there is no future setpoint information used in the NPGMV predictive control is shown in Fig. 11.17. This again indicates the importance of future reference or setpoint knowledge. In addition to the benefit obtained from the early movement of actuators, it is also clear the dynamic responses compare more favourably when future information is available. The conclusion is therefore that at least part of the improvement in responses from the use of a predictive controller is due to the presence of future setpoint knowledge.

526

11.10

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

Restricted Structure Generalized Predictive Control

To provide a more efficient online algorithm for implementing a model-based control a fixed low-order controller structure can be used. The NGPC approach can be used to select the gains in such a structure and this is a so-called Restricted Structure (RS) design approach. The Restricted Structure-Generalized Predictive Control (RS-GPC) that results is quite different from the solutions described in Chap. 2 where linear time-invariant controllers were assumed. The restricted structure controller introduced here involves a predictive control cost-index and a plant model that has a similar structure to that of the NGPC algorithm described in the first half of the chapter. The difference in this section is that a low-order controller structure is assumed to be predefined. For example, in a scalar problem a one degree-of-freedom PID structure, or a lead–lag transfer-function, may be chosen where the gains are to be optimized at each sample instant. For the traditional GPC or NGPC algorithms, matrix computations provide a vector of future controls and the first element of this vector is chosen for implementation. For the RS-GPC algorithm, a fixed controller structure is defined and the controller gains are computed to minimize a GPC cost-index. To a classically trained control engineer, this first appears like a PID controller with time-varying gains. However, the solution can involve much more general structures including multivariable systems. This RS approach provides the benefits of model-based control and also a simple tuning interface (such as the PID tuning parameters). The controller structure can include functions like a time-constant term or double integrator, to obtain an extended PID control law. The functions can represent transfer-functions with frequency characteristics that introduce gain in frequency ranges where improved control is desirable. The classical PID controller structure does, of course, have an integral function, which introduces gain at low frequencies and a derivative term, which provides phase lead at high frequencies. Clearly, there may be other functions, which would be important in mid–frequency regions that might also be added. The structure used can be of a more general transfer-function controller structure that may be of low-order where the gains and time constants are optimized. The RSGPC optimization algorithm can compute the controller gains and unknown parameters online. If there are no signal changes like setpoint changes or disturbances, then the gains will be found to settle down at constant values. If changes in the disturbance occur, the gains will be modified accordingly. This behaviour is rather like adaptation but it is not adaptive in a traditional sense. The gains change when the reference or disturbance signals change since the gains are obtained by minimizing a predictive control cost-index at each time instant. A similar qLPV plant model is assumed to that defined earlier in Sect. 11.3. If the plant model parameters change then the gains will also change, since the algorithm is updated using both the qLPV model and the plant measurements. Benefits: One of the attractive features of the approach is that a fixed structure, like PID, can allow technicians to use their existing knowledge and skills for

11.10

Restricted Structure Generalized Predictive Control

527

returning the gains. That is, the gains computed by the model-based optimization algorithm can be tweaked to accommodate some practical requirements of plant technicians. Moreover, all of the benefits of model-based control are available with this algorithm. A less obvious advantage but one that is very important is that the RS-GPC solution often seems to inherit the good natural robustness properties of both low-order controllers, and those of predictive controllers. At the same time, a simple controller structure is available where the gains can be modified by staff without requiring advanced control knowledge. Note that there are different ways in which this retuning can be achieved either ad hoc (multiplying the computed gains by scalar tuning parameters) or as part of an optimal solution (varying the cost-weightings related to the controller terms).

11.10.1

Restricted Structure GPC Criterion

A brief introduction to the RS-GPC controller is provided below. The plant is assumed qLPV and is the same as that described in Sect. 11.3, except that the unstructured black-box input subsystem is not included ðW 1k ¼ IÞ. Such a term can be added using a very similar argument to that in Sect. 11.5. For simplicity, only the absolute control case is considered ðb ¼ 0Þ. The GPC performance index motivates the criterion to be minimized. This may be defined with a dynamic cost-function weighting on the error signal, as ( J¼E

N X

) ep ðt þ j þ kÞ

T

ep ðt þ j þ kÞ þ u0 ðt þ jÞT k2j u0 ðt þ jÞÞjt

j¼0

where Ef:jtg denotes the conditional expectation, conditioned on measurements up to time t and k2j denotes a scalar or diagonal control signal weighting matrix. The future optimal control signal is to be calculated for the interval s 2 ½t; t þ N. The state-space models generating the signals rp ðtÞ and yp ðtÞ may include a dynamic cost-function weighting P c ðz1 Þ that may be in a qLPV form but is often just an integrator. The usual GPC criterion [16, 30, 31], that is to be extended, may be written in a vector-matrix form using the definitions in Sect. 11.3 as follows: n o 0T 2 0 J ¼ EfJt g ¼ E EPT t þ k;N EP t þ k;N þ Ut;N KN Ut;N jt

ð11:135Þ

The criterion (11.135) used in GPC and NGPC control will now be generalized for the restricted structure control problem. The RS-GPC cost-function is defined to have a similar form but with some enhancements. A term to limit the gains of the controller (denoted by kc ðtÞÞ may be added into the cost-index so that large gains are penalized. The rate of gain variations Dkc ðtÞ involving the difference of the gains may also be costed, where

528

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

Dkc ðtÞ ¼ kc ðtÞ  kc ðt  1Þ

ð11:136Þ

An end-state cost term, which represents the error in the desired end-state, is also included, defined as ex ðt þ i þ kÞ ¼ rx ðt þ i þ kÞ  xðt þ i þ kÞ where rx ðt þ i þ kÞ denotes a given reference end-state. RS-GPC cost-function definition: The multi-step RS-GPC cost-function of interest may now be written in the extended vector-matrix form: 0T 2 0 J ¼ EfJt g ¼ EfEPT t þ k;N EP t þ k;N þ Ut;N KN Ut;N þ kcT ðtÞK2K kc ðtÞ þ DkcT ðtÞK2D Dkc ðtÞ

þ ex ðt þ N þ kÞT Ps ex ðt þ N þ kÞjtg ð11:137Þ The number of steps in the prediction horizon is denoted by N and the number of functions employed in parameterizing the controller (explained in the next section) is denoted by Ne. The control gain weighting matrices in (11.137) may be defined to have the following diagonal forms: • The cost-weightings on the future inputs u0 are defined as K2N ¼ diagfk20 ; k21 ; . . .; k2Ne g. • The cost-weightings on the controller gains are defined as K2K ¼ diagfq20 ; q21 ; . . .; q2Ne g. • The cost-weightings on the difference of the gains is denoted by K2D ¼ diagfc20 ; c21 ; . . .; c2Ne g. The end-state weighting term may now be written as ex ðt þ N þ kÞ ¼ rx ðt þ N þ kÞ  xðt þ N þ kÞ ¼ rx ðt þ N þ kÞ  ddd ðt þ N þ k  1Þ  ANtþ k xðt þ kÞ 0  Dt þ k;N Nt þ k;N  Bt þ k;N Ut;N

 where Bt þ k;N ¼ AN1 t þ 1 þ k B t þ k ; .. .  N1 Dt þ k;N ¼ At þ 1 þ k Dt þ k ; . . . ; Dt þ N þ k1 .

11.10.2

ð11:138Þ ; Bt þ N þ k1

0



and

Parameterizing a Restricted Structure Controller

A three degree-of-freedom controller in a restricted structure form can be introduced, so that the controller has separate feedback, tracking and feedforward terms. The functions fj ðz1 ; kj ðtÞÞ that determine the controller structure are chosen by the

11.10

Restricted Structure Generalized Predictive Control

529

designer before the optimization is performed. These might be the constant, integrator and derivative functions found in PID control, or they might be the delay operator terms in a general multivariable controller structure. A total of Ne linear dynamic functions (frequency sensitive) can be chosen to have well-defined frequency response characteristics. The control signal can then be realized as follows: uðtÞ ¼

Ne X

fj ðz1 ; kj ðtÞÞe0 ðtÞ

ð11:139Þ

j¼1

The controller structure involves a sum of vector functions that form the control signal. In a one degree-of-freedom (DOF) case, these might be the sum of proportional, integral and filtered derivative terms for each of the channels. The functions fj ðz1 ; kj ðtÞÞ and the gains kj ðtÞ determine the restricted structure controller dynamics and the controller gains, respectively. Expanding terms in (11.139): uðtÞ ¼ f1 ðz1 ; k1 ðtÞÞe0 ðtÞ þ f2 ðz1 ; k2 ðtÞÞe0 ðtÞ þ    þ fNe ðz1 ; kNe ðtÞÞe0 ðtÞ ð11:140Þ The functions fj ðz1 ; kj ðtÞÞ in this equation are dynamic terms that are frequency sensitive and specified by the designer. The gains kj ðtÞ represent a set of time-varying non-dynamic gain vectors for the multivariable controller, with a total of Ne function block terms. There are clear links to explicit predictive control algorithms (Chap. 7 and [44]). Example 11.4: Filtered PID Controller The term f1 ðz1 Þk1 ðtÞe0 ðtÞ in a one DOF PID controller may represent a proportional term, and the second term f2 ðz1 Þk2 ðtÞe0 ðtÞ represents an integral term and third term a filtered derivative term ðNe ¼ 3Þ. For a scalar system, let the functions be chosen as f1 ðz1 Þ ¼ 1, f2 ðz1 Þ ¼ 1 ð1  z1 Þ and f3 ðz1 Þ ¼ ð1  z1 Þ ð1  az1 Þ, and the controller has the familiar form: uðtÞ ¼ k1 e0 ðtÞ þ

1 ð1  z1 Þ k k3 e0 ðtÞ e ðtÞ þ 2 0 ð1  z1 Þ ð1  az1 Þ ■

Degrees of freedom: The definition of the signal e0(t) is different for the one, two and three DOF cases [46]. That is, • For a one DOF regulation problem the signal e0(t) represents the noisy tracking error e0 ðtÞ ¼ rðtÞ  zðtÞ.  T • For the two DOF regulation and tracking problems e0 ðtÞ ¼ zT ðtÞ r T ðtÞ , representing a vector of observations and tracking reference signals.

530

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

• For three DOF regulation, tracking and feedforward problems  T e0 ðtÞ ¼ zT ðtÞ r T ðtÞ d T ðtÞ , representing a vector of observations, reference and measured disturbance signals. Matrix form of the controller functions: The function fj ðz1 ; kj ðtÞÞ and gains kj ðtÞ can be defined to have the following matrix forms: 2

j f11j ðz1 Þk11 ðtÞ 6 . ..
 6 6 fj z1 ; kj ðtÞ ¼ 6 6 4 j j fm1 ðz1 Þkm1 ðtÞ

2

3 j f1rj ðz1 Þk1q ðtÞ 7 .. 7 j 7 f22j ðz1 Þk22 ðtÞ . 7 ð11:141Þ .. 7 5 . j 1 j  fmr ðz Þkmq ðtÞ 

j k11

6 6 j 6k kj ¼ 6 21 6 .. 4 .

j km1

j k12 j k22



 ..

.

3 j k1q .. 7 7 . 7 7 7 5 kmj q

ð11:142Þ

where the number of columns is a function of the controller configuration (1, 2, 3  T DOF or others) and the length of the vectors in e0 ðtÞ ¼ zT ðtÞ r T ðtÞ d T ðtÞ . In the classical one DOF SISO problem solution q = r and in the simple multi-loop multivariable case, the function matrix fj ðz1 ; kj ðtÞÞ, and the gain matrix kj can be diagonal matrices.

11.10.3

Parametrizing a General RS Controller

The RS controller need not be in an extended PID form but can be in a general transfer-function form of low order. This type of classical controller can also be expressed as a product of the gains to be computed with known signals. Assume a classical controller is expressed, in unit-delay operator form, as follows: uðtÞ ¼ C0 ðz1 ; tÞe0 ðtÞ ¼ ðI þ C1d z1 þ C2d z2 þ    þ C2d znd Þ1 ðC0n þ C1n z1 þ    þ C2n znm Þe0 ðtÞ

where the parameter matrices are time-varying. This is referred to here as the “general” control structure relative to the PID structure (assuming the RS controller is linear). Clearly, the control signal may be expressed in the form:

11.10

Restricted Structure Generalized Predictive Control

531

uðtÞ ¼ ðC0n þ C1n z1 þ    þ C2n znm Þe0 ðtÞ  ðC1d z1 þ C2d z2 þ    þ C2d znd ÞuðtÞ

ð11:143Þ Example 11.5 For a scalar system, the “general” second-order control structure can be written as uðtÞ¼ðI þ C1d ðtÞz1 þ C2d ðtÞz2 Þ1 ðC0n ðtÞ þ C1n ðtÞz1 þ C2n ðtÞz2 Þe0 ðtÞ If the denominator matrix has known coefficients, then only the numerator coefficient matrices are to be computed which is similar to the PID problem. However, if all the controller numerator and denominator coefficients are to be computed Eq. (11.143) can be used. That is, the controller coefficients (or scalar gains to be computed) determine the control action u(t) as follows: uðtÞ ¼ C0n ðtÞe0 ðtÞ þ C1n ðtÞe0 ðt  1Þ þ C2n ðtÞe0 ðt  2Þ  C1d ðtÞuðt  1Þ  C2d ðtÞuðt  2Þ ¼ k1 ðtÞe0 ðtÞ þ k2 ðtÞe0 ðt  1Þ þ k3 ðtÞe0 ðt  2Þ  k4 ðtÞuðt  1Þ  k5 ðtÞuðt  2Þ 

General controller: Similar results apply to a general multivariable structure, where the feedback controller may be represented as C0 ðz1 ; tÞ ¼ C0d ðz1 ; tÞ1 C0n ðz1 ; tÞ, and the controller is time or parameter varying. The controller can be parameterized to cover this more general controller structure (11.143), by modifying (11.139) to  T include the vector c0 ðtÞ ¼ eT0 ðtÞ uT ðtÞ . Thus, let the controller have the restricted structure: uðtÞ ¼

Ne X

fj ðz1 ; kj ðtÞÞc0 ðtÞ

ð11:144Þ

j¼1

This applies to the general controller structure, where the controller denominator terms must also be optimized. To summarize, the two main RS cases involve different definitions for c0 ðtÞ. • Extended PID structures: In this case c0 ðtÞ ¼ e0 ðtÞ, where e0 ðtÞ depends upon the degrees of freedom listed above. • General classical controller structure: For the more general structure, the signal  T c0 ðtÞ must also include control action c0 ðtÞ ¼ eT0 ðtÞ uT ðtÞ where again e0 ðtÞ depends upon the assumed degrees of freedom. Example 11.6 For a general controller structure the two output and two input plant, with a one degree-of-freedom controller has the form:

532

11 "

e0 ðtÞ

1

fj ðz ; kj ðtÞÞ

#

uðtÞ

" ¼

LPV/State-Dependent Nonlinear Predictive Optimal Control

j j j j f11j ðz1 Þk11 e01 ðtÞ þ f12j ðz1 Þk12 e02 ðtÞ  f13j ðz1 Þk13 u1 ðtÞ  f14j ðz1 Þk14 u2 ðtÞ

#

j j j j e01 ðtÞ þ f22j ðz1 Þk22 e02 ðtÞ  f23j ðz1 Þk23 u1 ðtÞ  f24j ðz1 Þk24 u2 ðtÞ f21j ðz1 Þk21

For a single output system with two inputs, with three degrees of freedom, the controller has the form: 2

zðtÞ

3

# 6 7 " j 1 j j j j 6 rðtÞ 7 rðtÞ þ f13j ðz1 Þk13 dðtÞ  f14j ðz1 Þk14 uðtÞ f11 ðz Þk11 zðtÞ þ f12j ðz1 Þk12 7¼ fj ðz1 ; kj ðtÞÞ6 6 dðtÞ 7 j j j j zðtÞ þ f22j ðz1 Þk22 rðtÞ þ f23j ðz1 Þk23 dðtÞ  f24j ðz1 Þk24 uðtÞ f21j ðz1 Þk21 4 5 uðtÞ

For the general controller structure and the 3-DOF case the signals e0 ðtÞ and c0 ðtÞ:  e0 ðtÞ ¼ zT ðtÞ

r T ðtÞ

d T ðtÞ

T

and  c0 ðtÞ ¼ zT ðtÞ r T ðtÞ d T ðtÞ

uT ðtÞ

T ■

Generalizing the results: Considering this more general problem, the vector c0 ðtÞ may be written in terms of scalar signals for each error channel as: c0 ðtÞ ¼ ½ c01 ðtÞ

c02 ðtÞ   

c0q ðtÞ T

ð11:145Þ

Each
of theterms in the summation for the control signal in (11.144) has the form fj z1 ; kj ðtÞ c0 ðtÞ. From (11.141) and (11.145), the contribution in the general case of only the jth function term in each channel can be written as 3 j j ðtÞ  f1qj ðz1 Þk1q ðtÞ 2 c01 ðtÞ 3 f11j ðz1 Þk11 7 6 .. .. 76 c02 ðtÞ 7 6 j
1  76 7 6 ðtÞ . . f22j ðz1 Þk22 fj z ; kj ðtÞ c0 ðtÞ ¼ 6 76 .. 7 .. 74 . 5 6 5 4 . c0q ðtÞ j j j j ðz1 Þkm1 ðtÞ  fmq ðz1 Þkmq ðtÞ fm1 3 2 j 1 j j j f11 ðz Þk11 c01 ðtÞ þ f12j ðz1 Þk12 c02 ðtÞ þ . . . þ f1qj ðz1 Þk1q c0q ðtÞ 7 6 j 1 j j j j j 6 f21 ðz Þk21 c01 ðtÞ þ f22 ðz1 Þk22 c02 ðtÞ þ . . . þ f2q ðz1 Þk2q c0q ðtÞ 7 7 6 ¼6 .. 7 5 4 . j j 1 j 1 j j 1 j fm1 ðz Þkm1 c01 ðtÞ þ fm2 ðz Þkm2 c02 ðtÞ þ . . . þ fmq ðz Þkmq c0q ðtÞ 2

ð11:146Þ

11.10

Restricted Structure Generalized Predictive Control

533

The expression for the control signal, in terms of the general parameterized controller, therefore becomes uðtÞ ¼

Ne X
 fj z1 ; kj ðtÞ c0 ðtÞ j¼1

2

Ne n P

j j j f11j ðz1 Þk11 c01 ðtÞ þ f12j ðz1 Þk12 c02 ðtÞ þ . . . þ f1qj ðz1 Þk1q c0q ðtÞ

o 3

7 6 j¼1 7 6 6 P n o 7 7 6 Ne j j j 1 j 1 j 1 j f21 ðz Þk21 c01 ðtÞ þ f22 ðz Þk22 c02 ðtÞ þ . . . þ f2q ðz Þk2q c0q ðtÞ 7 6 7 ¼6 7 6 j¼1 7 6 . .. 7 6 7 6 Ne n o 5 4 P j 1 j j j j j fm1 ðz Þkm1 c01 ðtÞ þ fm2 ðz1 Þkm2 c02 ðtÞ þ . . . þ fmq ðz1 Þkmq c0q ðtÞ j¼1

ð11:147Þ Example 11.7: One DOF and Two-Input and Two-Output Plant For a two-input and two-output plant and a one DOF controller T c0 ðtÞ ¼ e0 ðtÞ ¼ ½ e01 ðtÞ e02 ðtÞ  , with three control functions ðNe ¼ 3Þ, the related  j j  k11 k12 gains kj ¼ . The function terms become j j k21 k22   j 1 j j   c01 ðtÞ f11 ðz Þk11 f12j ðz1 Þk12 fj z ; kj ðtÞ c0 ðtÞ ¼ j j 1 j c02 ðtÞ f22j ðz1 Þk22  f21j ðz1 Þk21  j j 1 j f11 ðz Þk11 c01 ðtÞ þ f12 ðz Þk12 c02 ðtÞ ¼ j j f21j ðz1 Þk21 c01 ðtÞ þ f22j ðz1 Þk22 c02 ðtÞ


1

Thence, the sum of terms that determines the current control action: 

 X Ne


1   u1 ðtÞ fj z ; kj ðtÞ c0 ðtÞ ¼ u2 ðtÞ 2j¼1 3 3 3 P P j j j j 1 1 f ðz Þc ðtÞk þ f ðz Þc ðtÞk 01 02 11 11 12 12 7 6 j¼1 6 j¼1 7 ¼6 3 7 3 P 4 P j 1 j j j 5 1 f21 ðz Þc01 ðtÞk21 þ f22 ðz Þc02 ðtÞk22 j¼1

or expanding

j¼1

ð11:148Þ

534

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

1 1 1 2 1 2 3 1 3 u1 ðtÞ ¼ f11 ðz Þc01 k11 þ f11 ðz Þc01 ðtÞk11 þ f11 ðz Þc01 ðtÞk11 1 1 1 2 1 2 3 1 3 þ f12 ðz Þc02 ðtÞk12 þ f12 ðz Þc02 ðtÞk12 þ f12 ðz Þc02 ðtÞk12 1 1 1 2 1 2 3 1 3 u2 ðtÞ ¼ f21 ðz Þc01 ðtÞk21 þ f21 ðz Þc01 ðtÞk21 þ f21 ðz Þc01 ðtÞk21 1 1 1 2 1 2 3 1 3 þ f22 ðz Þc02 ðtÞk22 þ f22 ðz Þc02 ðtÞk22 þ f22 ðz Þc02 ðtÞk22

Note that (11.148) may be written in an alternative and more useful block-diagonal form for computing the gains: 2 j 3


6 k11 j 7 f12j ðz1 Þc02 ðtÞ 0 0 6 k12 7 6 j j j 7 5 0 f21 ðz1 Þc01 ðtÞ f22 ðz1 Þc02 ðtÞ 4 k21 j k22





f j ðz1 Þc01 ðtÞ fj z ; kj ðtÞ c0 ðtÞ ¼ 11 0 1

ð11:149Þ The block-diagonal form for the functions in (11.149) is more convenient for gain calculation. For the three function block terms in the example, the control signal may also be expressed as

uðtÞ ¼

3  X j¼1

f11j ðz1 Þc01 ðtÞ f12j ðz1 Þc02 ðtÞ 0 0

0 f21j ðz1 Þc01 ðtÞ

2 j 3 6 k11 j 7 0 6 k12 7 6 j 7 5 f22j ðz1 Þc02 ðtÞ 4 k21 j k22

Introducing the following row vectors of functions and column vectors of gains: 1 1 2 1 3 1 fe11 ¼ ½f11 ðz Þc01 ðtÞ f11 ðz Þc01 ðtÞ f11 ðz Þc01 ðtÞ 1 1 2 1 3 1 ðz Þc02 ðtÞ f12 ðz Þc02 ðtÞ f12 ðz Þc02 ðtÞ fe12 ¼ ½f12 1 1 2 1 3 1 ðz Þc01 ðtÞ f21 ðz Þc01 ðtÞ f21 ðz Þc01 ðtÞ fe21 ¼ ½f21 1 1 2 1 3 1 ðz Þc02 ðtÞ f22 ðz Þc02 ðtÞ f22 ðz Þc02 ðtÞ fe22 ¼ ½f22

2

kc11

3 1 k11 2 5; ¼ 4 k11 3 k11

2

kc12

3 1 k12 2 5; ¼ 4 k12 3 k12

2

kc21

3 1 k21 2 5; ¼ 4 k21 3 k21

2

kc22

3 1 k22 2 5 ¼ 4 k22 3 k22

The resulting control signal may then be written as

uðtÞ ¼

Ne X j¼1




1   f 11 fj z ; kj ðtÞ c0 ðtÞ ¼ e 0

fe12 0

0 fe21

2 11 3  kc12 7 0 6 6 kc 7 22 4 21 5 fe kc kc22 ■

11.10

Restricted Structure Generalized Predictive Control

535

Example 11.8: Two-DOF Controller and Two-Input and Two-Output System Again, consider the above example with Ne ¼ 3, but with a two DOF controller  T c0 ðtÞ ¼ zT ðtÞ r T ðtÞ , so that Eq. (11.148) becomes 3 2 c01 ðtÞ " j #6 7 j j j j 6 c02 ðtÞ 7
1  f11 ðz1 Þk11 f12j ðz1 Þk12 f13j ðz1 Þk13 f14j ðz1 Þk14 7 6 fj z ; kj ðtÞ c0 ðtÞ ¼ 6 c ðtÞ 7 j j j j f21j ðz1 Þk21 f22j ðz1 Þk22 f21j ðz1 Þk23 f21j ðz1 Þk24 4 03 5 c04 ðtÞ " j # j j j j f11 ðz1 Þk11 c01 ðtÞ þ f12j ðz1 Þk12 c02 ðtÞ þ f13j ðz1 Þk13 c03 ðtÞ þ f14j ðz1 Þk14 c04 ðtÞ ¼ j j j j f21j ðz1 Þk21 c01 ðtÞ þ f22j ðz1 Þk22 c02 ðtÞ þ f13j ðz1 Þk13 c03 ðtÞ þ f24j ðz1 Þk24 c04 ðtÞ ¼ diagff11j ðz1 Þc01 ðtÞ f12j ðz1 Þc02 ðtÞ f13j ðz1 Þc03 ðtÞ f14j ðz1 Þc04 ðtÞ; f21j ðz1 Þc01 ðtÞ f22j ðz1 Þc02 ðtÞ f23j ðz1 Þc03 ðtÞ f24j ðz1 Þc04 ðtÞg  j  j j j j j j j T  k11 k12 k13 k14 k21 k22 k23 k24 This equation has a similar structure to (11.149), and following similar steps to those in the previous example, the control signal can be expressed in the form: 3 kc11 6 k 12 7 6 c 7 6 13 7 6k 7 6 c14 7 7 0 6 6 kc 7 21 7 fe24 6 6 kc 7 6 22 7 6 kc 7 6 23 7 4 kc 5 kc24 2

 uðtÞ ¼

fe11 0

fe12 0

fe13 0

fe14 0

0 fe21

0 fe22

0 fe23

■ The expression (11.147) provides a valid parameterization of the controller but as the above example demonstrated, it can be expressed in a more convenient form for the optimization of the gains. The matrix expressions in the example and in (11.149) enable the gains to be collected in one vector.

11.10.4

Block Diagonal Parameterization Matrix

The general case will now be considered. Motivated by the summation terms in (11.147) define the function term: feis ¼ ½fis1 ðz1 Þc0s ðtÞ fis2 ðz1 Þc0s ðtÞ . . .: fisNe ðz1 Þc0s ðtÞ

ð11:150Þ

536

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

and the gain vector:  kcis ¼ kis1T

kis2T

   kisNe T

T

ð11:151Þ

Thus, from (11.150) and (11.151) we obtain 2 feis kcis

¼

½fis1 ðz1 Þc0s ðtÞ fis2 ðz1 Þc0s ðtÞ

. . .:

kis1

6 2 6 kis 6 .. 4 .

6 fisNe ðz1 Þc0s ðtÞ6

3 7 7 7 7 7 5

kisNe Observe that kcis is a vector containing gains corresponding to each function used in the RS controller. The contribution to the control signal in channel i, corresponding to the controller input in channel s, can be obtained as feis kcis ¼

Ne X

fisj ðz1 Þc0s ðtÞkisj



ð11:152Þ

j¼1

From (11.147) and (11.152) the RS parameterized control can be computed as 2

fe11 kc11 þ fe12 kc12 þ . . . þ fe1q kc1q

6 21 21 6 fe kc þ fe22 kc22 þ . . . þ fe2q kc2q 6 uðtÞ ¼ 6 .. 6 . 4

3 7 7 7 7 7 5

ð11:153Þ

fem1 kcm1 þ fem2 kcm2 þ . . . þ femq kcmq But from (11.152) the term feis kcis is the contribution to controller output i due to controller input s, and involving the contributions from all Ne functions. Block-diagonal structure: By introducing a block-diagonal matrix structure, it is easier to see the contributions of the different channel signals to the control action. Define the block-diagonal matrix Fe ðtÞ as 2

fe11 6 6 0 6 6 Fe ðtÞ ¼ 6 6 0 6 6 0 4 0

fe12



fe1q

0







0



0

fe21

fe22



fe2q

0

0



0

0

0



0

0

0

0



0 ..

.

 .. .

 fem11



 fem12

   fem1 q 

0













0





fem1

fem2



0

3

7 0 7 7 7 7 0 7 7 0 7 5 fem q

ð11:154Þ

11.10

Restricted Structure Generalized Predictive Control

537

Collect the gains in the following vectors by defining 2 1 3 3 2 1 3 1 k1q k11 k12 6 k2 7 2 7 2 7 6 k11 6 k12 6 1q 7 6 7 6 7 7 ¼ 6 . 7; kc12 ¼ 6 . 7; . . . ; kc1q ¼ 6 6 .. 7; 4 .. 5 4 .. 5 4 . 5 Ne Ne Ne k11 k12 k1q 3 2 2 1 3 2 1 3 1 k2q k21 k22 6 k2 7 2 7 6 k21 6 2 7 6 2q 7 6 7 22 6 k22 7 2q 7 ¼ 6 . 7; kc ¼ 6 . 7; . . .: ; kc ¼ 6 6 .. 7; 4 .. 5 4 .. 5 4 . 5 Ne Ne k21 k22 k Ne 2 12q 3 kmq 6 k2 7 6 mq 7 7 . . . ; kcmq ¼ 6 6 .. 7: 4 . 5 Ne kmq 2

kc11

kc21

The components of the gain vector kci j are numbered according to the controller row number (i) and controller input number (q), and their dimension depends on the number of functions. For a one DOF solution, the number of controller inputs equals the number of measurement channels q = r. In terms of the vector of all the gains kc ðtÞ, the expression for the parameterized control has the form: 2 6 6 uðtÞ ¼ Fe ðtÞkc ðtÞ ¼ 6 4

fe11 kc11 þ fe12 kc12 þ . . . þ fe1q kc1q fe21 kc21 þ fe22 kc22 þ . . . þ fe2q kc2q .. .

3 7 7 7 5

ð11:155Þ

fem1 kcm1 þ fem2 kcm2 þ . . . þ femq kcmq The block-diagonal matrix Fe ðtÞ in (11.154) can clearly be represented in the form: Fe ðtÞ ¼ diagf ef 1 ðtÞ ef 2 ðtÞ

   ef m ðtÞ g

ð11:156Þ

where the contribution to each controller output s depends on the row vector s:  efs ðtÞ ¼ fes1

fes2



fesq



for s 2 ½1; m:

The various channel gains in (11.155) may be grouped according to the dimensions of the row vectors in (11.156) to obtain the total gain vector in the form:  T kc ðtÞ ¼ kc1

T kc2

T    kcm

T

ð11:157Þ

538

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

 T where kc i ðtÞ ¼ kci 1 T kci2 T    kci q T and the total vector of gains is ordered in terms of the channel numbers. Control in terms of functional gains: To summarize, the control signal (11.147), in terms of the pre-specified functional controller structure, and the controller gain vector, follows as 2

uðtÞ ¼ Fe ðtÞkc ðtÞ ¼ diagf ef 1 ðtÞ

3 kc1 6 kc2 7 6 7    ef m ðtÞ g6 . 7 4 .. 5

ef 2 ðtÞ

ð11:158Þ

kcm where the total gain vector kc includes the functional controller gains for each channel. These gains may be written as kc ðtÞ ¼



T kcTm ¼

T kc1

T kc2



1 ¼ k11

2 k11

Ne    k11

h

"

kc11T kc12T    kc1qT |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

kc21T kc22T    kc2qT |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

channel 1 gains

channel 2 gains

1 k12

2 k12



Ne k12



km1 q



kcm1T kcm2T    kcm qT |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

#T

channel m gains

km2 q



kmNeq

iT

ð11:159Þ where kc 2 RrmNe . The gain vector contains groups of gains, each of which involves Ne functions, grouped for row one, two and so on. Example 11.9: PI Controller for m-Square Diagonal Control Structure Consider an m-square single DOF system with m inputs and m outputs. In the special case of a diagonal control structure, the matrix in (11.141) must be diagonal and the matrix Fe ðtÞ will have the following form: 2 6 Fe ðtÞ ¼ 4

1 1 f11 ðz Þc01 ðtÞ

2 1 f11 ðz Þc01 ðtÞ

0 0



Ne 1 f11 ðz Þc01 ðtÞ



0 .. 0

. 

1 fmm ðz1 Þc0m ðtÞ

0 .. . 

3

Ne 1 fmm ðz Þc0m ðtÞ

For this diagonal structure, the control signal to be implemented in (11.147) simplifies as 3 j j 1 f ðz Þc ðtÞk 01 11 7 6 j¼1 11 7 2 e ðtÞk 3 6 7 6 P f1 c1 Ne 6 j 7 e ðtÞk f22j ðz1 Þc02 ðtÞk22 7 6 6 f 2 c2 7 7 7¼6 uðtÞ ¼ 6 j¼1 6 7 ¼ Fe ðtÞkc . 7 4 6 . 5 . 7 6 . .. 7 6 ef m ðtÞkc m 7 6 Ne 5 4 P j 1 j fmm ðz Þc0m ðtÞkmm 2

Ne P

j¼1

ð11:160Þ



7 5

11.10

Restricted Structure Generalized Predictive Control

539

where " kc ¼

Ne 1 2 k11    k11 k11 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

1 2 k22 k22    k Ne |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl22ffl}

channel 1 gains

channel 2 gains



1 2 Ne kmm    kmm kmm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

#T

channel m gains

■ Example 11.10: PI Diagonal Controller for 2  2 Systems If say, a PI controller is used in each channel Ne ¼ 2. For the 2-square single DOF system considered in the previous diagonal controller example, the RS controller matrices will have the following form: 2  Fe ðtÞkc ¼

1 k11

3

6 2 7 6 k11 7 6 7  1 1 2 1 1 7 0 f22 ðz Þc02 ðtÞ f22 ðz Þc02 ðtÞ 6 4 k22 5 2 k22 "
#  1 1 1 2 1 2 f11 ðz Þk11 þ f11 ðz Þk11 c01 ðtÞ  ¼
1 1 1 2 1 2 f22 ðz Þk22 þ f22 ðz Þk22 c02 ðtÞ

1 1 ðz Þc01 ðtÞ f11

2 1 f11 ðz Þc01 ðtÞ





0

■

11.10.5

Parameterizing the Vector of Future Controls

In traditional predictive control such as GPC, the vector of future controls is computed to minimize the cost-index over the prediction horizon N. Recall that a GPC optimal control is found at time t is based on the receding horizon principle [26]. The optimal control is taken as the first element in the vector of future controls 0 Ut;N . The optimal control is therefore computed for the full prediction horizon N, but only the value at time t is actually used. The RS control approach is unusual because the predicted control law is implemented using the chosen fixed controller structure. However, the approach is related to the receding horizon predictive control philosophy. The gains are computed to minimize the future tracking and control signal variances over the prediction interval, based on the assumption that the gains over the prediction horizon N are constant. These gains are used to compute the vector of future controls that may be needed for evaluating the future values of the qLPV model. In the spirit of the receding horizon philosophy, only the control signal at the current time t is implemented. The computation of the RS-GPC controller gains, based on a predictive control philosophy, provides the gains in (11.158) in a very simple manner. The procedure is repeated at each time instant and the new gains are found and used. That is, the

540

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

controller gains kc ðtÞ are assumed constant over the prediction interval [0, N] and are chosen to minimize the cost-function. The computed value kc ðtÞ is then used to compute the optimal control for time t. Clearly, this process may involve varying gains if either the reference or disturbance signals are changing. The future controls are only needed if the qLPV model matrices are control dependent. The result is a fixed structure controller where the gain vector is computed at each sample time. For implementation in very fast systems, some form of pre-computation of the gains, followed by scheduling, may be needed. Vector of future controls: Under the receding horizon type of assumption discussed above, the vector of future restricted structure controls Ut;N can be computed. Using (11.160) and the predicted future values of Fe ðtÞ, we obtain 2

Ut;N

3 2 3 Fe ðtÞ uðtÞ 6 uðt þ 1Þ 7 6 F 7 b 6 7 6 e ðt þ 1Þ 7 ¼6 7¼6 7kc ðtÞ .. .. 4 5 4 5 . . b uðt þ NÞ F e ðt þ NÞ

ð11:161Þ

Philosophy: In the RS-GPC solution the structure of the controller is fixed and it is only the gains that are to be determined. These are assumed constant over the prediction interval N and once found, for time t the optimal control is computed and implemented. The optimal solution is in terms of the gains that minimize the cost-function for the chosen fixed controller structure. This RS controller structure that may be one DOF is different from conventional predictive control. Recall that in traditional MPC (like GPC), that the future control is computed using a two degrees-of-freedom control structure. This arises because the observations and future reference are input at different points in the controller structure. Even if a two degrees-of-freedom controller structure is chosen (with separate reference and feedback signal inputs) for the RS controller, the solution is still likely to be suboptimal, since a low-order solution is normally required. However, the benefit of the RS design approach lies in using a low-order controller that can be re-tuned by relatively unskilled staff (without advanced control design engineering expertise), for a solution that has the potential to be more robust. Prediction control matrix: The matrix in (11.161) will be denoted by Ufe and is defined as follows:  Ufe ðtÞ ¼ FeT ðtÞ

b eT ðt þ 1Þ F



b eT ðt þ NÞ F

T

ð11:162Þ

This matrix has the ith (i > 1) block row in (11.162): b e ðt þ iÞ ¼ diagf ^ef 1 ðt þ iÞ ^ef 2 ðt þ iÞ F

   ^ef m ðt þ iÞ g

The predicted values of the i step-ahead control, given the predicted values of error, follow as

11.10

Restricted Structure Generalized Predictive Control

541

b e ðt þ iÞkc ðtÞ ¼ diagf ^ef 1 ðt þ iÞ ^ef 2 ðt þ iÞ uðt þ iÞ ¼ F

   ^ef m ðt þ iÞ gkc ðtÞ

Parameterized predictive control: The vector of future controls may now be written in terms of the parameterized predictive controller as Ut;N ¼ Uf e ðtÞkc ðtÞ

ð11:163Þ

where 2 6 6 Ufe ðtÞ ¼ 6 4

diagf ef 1 ðtÞ ef 2 ðtÞ    ef m ðtÞ g diagf ^ef 1 ðt þ 1Þ ^ef 2 ðt þ 1Þ    ^ef m ðt þ 1Þ g .. .

diagf ^ef 1 ðt þ NÞ ^ef 2 ðt þ NÞ

3 7 7 7 5

   ^ef m ðt þ NÞ g

  and ef i ðtÞ ¼ fei1 fei2    fei r . Note that the matrix Ufe in (11.161) has (N + 1)  m rows and q  mNe columns. For a reasonable prediction horizon, there will normally be more rows than columns since the number of functions Ne will be much less than the prediction horizon. A potential benefit of RS control is therefore that the computations will be less onerous than standard MPC (depending on the choice and dimensions of the RS controller structure).

Background Gain Controller Computations Function error terms Ne

Kalman filter stage and gain computation

u (t )   f j ( z , k j (t ))c0 (t ) 1

e fs (t )  f es1

f es 2

f esq

Fe (t )  diag e f 1 (t ), e f 2 (t ),

d (t )

c0 (t )

, e f m (t )

Fe (t )

d

xˆ(t | t )

z(t)

j 1

kc (t )  

1 t k ,N

kc (t )



0 P t k ,N

dd d p

  2D kc (t  1)



Gain computations Disturbance  (t )

 z (t )

r (t ) Reference

c0 (t ) 

r (t )

c0 (t )

Output

u (t )  Fe (t )kc (t )

calculation

d (t )  u (t )

Fe (t ) kc (t )

y (t )

Restricted structure controller Plant

u(t)

+

Observations z(t)

+

vd

Fig. 11.18 RS-GPC three DOF state-space controller structure

542

11

11.10.6

LPV/State-Dependent Nonlinear Predictive Optimal Control

RS-GPC Theorem

The main results for the Restricted Structure-Generalized Predictive Control (RS-GPC), may now be summarized in the following theorem. The block diagram implementation of the controller is shown in Fig. 11.18. The background computations in this solution (at the top of the figure) are similar in complexity to the NGPC solution discussed in the first part of the chapter. The main difference lies in the controller structure chosen within the feedback loop. Theorem 11.4: Restricted Structure-Generalized Predictive Controller Consider the system and assumptions introduced in Sect. 11.2 where the system is represented by the qLPV model and W 1k ¼ I. The Restricted Structure-Generalized Predictive Controller is required to minimize the following criterion: 0T 2 0 J ¼ EfJt g ¼ EfEPT t þ k;N EP t þ k;N þ Ut;N KN Ut;N þ kcT ðtÞK2K kc ðtÞ þ DkcT ðtÞK2D Dkc ðtÞ þ ex ðt þ N þ kÞT Ps ex ðt þ N þ kÞjtg

ð11:164Þ The RS-GPC control can be implemented as uðtÞ ¼

Ne X

fj ðz1 ; kj ðtÞÞc0 ðtÞ ¼ Fe ðtÞkc ðtÞ

ð11:165Þ

j¼1

where the block-diagonal matrix: Fe ðtÞ ¼ diagf ef 1 ðtÞ

ef 2 ðtÞ

   efm ðtÞ g

ð11:166Þ


 The functions fj z1 ; kj ðtÞ for j 2 ½1; Ne  are pre-specified and determine the   controller structure. For each i, the matrix ef i ðtÞ ¼ fei1 fei2    fei q , and the   row vector functions feis ¼ fis1 ðz1 Þc0s ðtÞ    fisNe ðz1 Þc0s ðtÞ , where fpgj ðz1 Þ for p 2 ½1; m and g 2 ½1; q. Optimal gains: The vector of optimal Restricted Structure Controller gains which follows by invoking the receding horizon type of philosophy and is given as kc ðtÞ ¼ X 1 xðt þ kjtÞ  P C t þ k;N rdd ðt þ kÞ t þ k;N ðP C t þ k;N DP t þ k;N þ C / t þ k;N ^ 2  KD kc ðt  1ÞÞ ð11:167Þ where the non-singular matrix: T

X t þ k;N ¼ UfeT ðtÞðV TP t þ k;N V P t þ k;N þ K2N þ Bt þ k;N Ps Bt þ k;N ÞUf e ðtÞ þ K2K þ K2D

11.10

Restricted Structure Generalized Predictive Control

C/ t þ k;N ¼ UfeT V TP t þ k;N CP t þ k;N At þ k;N ;

543

P C t þ k;N ¼ UfeT V TP t þ k;N ;

T

P C t þ k;N ¼ UfeT Bt þ k;N Ps  and Bt þ k;N ¼ AN1 t þ 1 þ k Bt þ k ;    multivariable system has the form:

; Bt þ N þ k1

 T T T kc ðtÞ ¼ kc1 kc2    kcTm " kc11T kc12T    kc1qT kc21T kc22T    kc2qT |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ¼ |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} channel 1 gains

 0 . The gain vector for a

   kcm1T kcm2T    kcm qT |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

channel 2 gains

#T

channel m gains

ð11:168Þ The optimal feedback control to be implemented may then be computed using the RS-GPC control law (11.165), as shown in Fig. 11.18. The parameterized vector of future controls may also be obtained in the form: Ut;N ¼ Ufe ðtÞkc ðtÞ  and Ufe ðtÞ ¼ FeT ðtÞ

b eT ðt þ 1Þ F



b eT ðt þ NÞ F

ð11:169Þ T

:

■

Solution The proof is similar to that in Sect. 11.4, but optimizing the gains rather than the vector of future controls. The substitution is made as Ut;N ¼ Ufe ðtÞkc ðtÞ, and the following analysis is then similar to that in Sect. 11.4.3 and presented in Grimble and Majecki [45] and Grimble [46]. ■ Comments on the form of the solution: Observe that the “denominator matrix” in the RS-GPC controller gain expression (11.167) is full rank because of the cost-weighting definitions and the system description. The expression for the gain vector here seems rather similar in form to the usual GPC solution but the denominator matrix in (11.167) is time-varying even for linear time-invariant systems, and it can be of much lower dimension than arises in traditional MPC or GPC control. The size of this matrix depends upon the number of unknown gains in the parameterized controller and this is usually less than the prediction and control horizons in GPC control design. If the weightings K2K and K2D on the controller gain and the gain difference terms are not present, the denominator matrix can become singular. It is a valuable feature to be able to avoid this numerical problem and to be able to penalize the magnitudes of the gains and the rate of change of gains. Optimization and varying gains: The optimization process provides a vector of future predictive controls, from (11.169), that is needed for the qLPV prediction models. The controller structure is parameterized so that it may have a one, two or

544

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

three DOF form with given functional blocks and time-varying gains. The gains are found to minimize predictive control cost-function (11.64). When the system reaches a steady-state, the signals will normally settle down at almost constant values and the resulting feedback controller gains should become nearly constant. However, when there are large disturbances or reference changes then the type of behaviour will depend on the RS controller chosen. For example, in a three DOF controller the disturbance feedforward or tracking controller gains, respectively, are likely to change substantially. In a one DOF controller, when there are either reference or disturbance signal changes, all the feedback controller gains may change. Constrained RS-GPC: Soft constraints can be applied to the controller gains or to the rate of change of gains by variation of the cost-weighting terms in (11.164), namely K2K and K2D . More unusually, hard constraints may be applied to the magnitude of the controller gains, or to their rate of change, by use of quadratic programming. Hard constraints may also be applied to the control signals and to the system outputs [47]. The ability to bind the gain or gain variations may be very useful practically. Limiting the magnitude of controller gains has a direct effect on actuator power demands and an indirect effect on nonlinear behaviour and stability. Limiting the rate of change of gains by increasing the weighting will lead to performance similar to that of a fixed gain controller. By changing this weighting, the control design priorities may be changed so that either high performance is achieved or predictable behaviour is ensured. The ability to control the maximum levels of gain, or the variation in controller gains, should be a welcome addition to the design engineer’s armoury. Implementation: The full solution shown in Fig. 11.18 can be used for implementation, however, for fast systems a numerically simpler solution may be essential. For example, the calibration of automotive engines is a very expensive process and the controllers have to be relatively easy to implement online. A possibility for such an application is to use the full RS-solution in simulations using a government-approved driving cycle, such as the US06 Supplemental Federal Test Procedure, and to record the gains over a range of operating points. A solution might then be implemented online using the stored gains in a form of scheduling.

11.10.7

Adaptive Behaviour

The RS-GPC controller is not adaptive in the usual sense of self-tuning control, or model reference adaptive control. However, there are some interesting links and differences: • The controller is fixed in the sense that given the plant signals and any known varying parameters, the controller has a known fixed structure and gains that minimize a criterion. • The gains change with disturbances, setpoint changes or parameter variations.

11.10

Restricted Structure Generalized Predictive Control

545

• If there is a modelling error, the signals will be different from the ideal case when there is no plant model mismatch. The gains depend upon these signals so the controller itself will change. This is different from the usual case of a fixed controller where control action only changes when the controller error input changes. • There is, therefore, a degree of adaptation but not the same risk as in some adaptive systems where if the model identified is in error it has serious consequences for stability. • In RS-GPC when a mismatch arises, the optimal gains do their best to minimize the criterion given the measurements and the controller structure imposed. Much of the gain calculation also depends upon the measurements, although the assumed plant model is still of course involved.

11.11

Concluding Remarks

The tracking results in this chapter are more general than in previous chapters because of the general model choice and the cost-functions that involved additional terms. The cost-function involved either incremental control or control signal costing terms and the control profile could be specified using a connection matrix, adding to the generality of the results. The plant output subsystem was assumed to be a qLPV state-space model that includes the class of state-dependent models. Both of the predictive control design problems discussed can, therefore, include nonlinearities. The Nonlinear Generalized Predictive Control (NGPC) design method was introduced first because it is relatively easy to design and to implement. The proposed NGPC approach has the nice property that if the system is linear then the controller reduces to the Generalized Predictive Controller that may be considered a general version of MPC. The Nonlinear Predictive Generalized Minimum Variance (NPGMV) control design method was considered in the second half of the chapter and the black-box input subsystem was assumed to be included in this case. If the predictive control cost horizon becomes only a single-stage cost-function, the control law reverts to a special form of the so-called NGMV solution. This may be used as a useful starting point for design since it is well known that as the number of steps in a predictive controller increases, the responses normally become more damped and the solution more robust. It therefore seems a practical step to develop a predictive form of NGMV controller, where a stabilizing set of weightings can easily be achieved but where the time responses may not be as good as for NPGMV control. The number of steps in the predictive control horizon can then be increased until no further improvement is obtained. Clearly, if the responses are not improving by using further steps there is no need to increase the complexity of the computations.

546

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

The simplified NPGMV problem and the control structure provided in the later sections (Sect. 11.8) is particularly valuable for applications and avoids the algebraic-loop problem. This is important when dealing with qLPV (state-dependent or LPV) models where results are necessarily more complicated than for LTI state models. The NPGMV predictive control law includes an internal model of the process and many of the computations only involve matrix multiplication. There are many design, stability and robustness issues still to explore but experience reveals that better results can normally be achieved than with NGMV control. Some current research is on techniques for reducing the sensitivity of the closed-loop system to modelling errors and disturbances. There is also research on algorithms to improve computational methods to reduce computing demands. The principal shortcoming of many design techniques for MPC is their inability to deal explicitly with plant model uncertainty. There is more to do in the development of truly robust nonlinear MPC solutions, particularly for fast processes [47, 48]. The natural robustness of low-order controllers can be exploited in some cases. A Restricted Structure Controller, like the RS-GPC solution described in the previous section, offers a way of obtaining a low-order controller that still provides most of the performance benefits of multivariable model-based predictive control [45]. The true effectiveness of this approach is still to be assessed but trials on different automotive problems have been promising. There is an interesting link with the Quantitative Feedback Theory (QFT) design approach, which enables low-order controllers to be designed, whilst limiting the controller gains [49]. Horowitz [50] who developed the QFT approach, referred to the gain as representing the “cost of control.”

11.11.1

Pros and Cons of Some Model-Based Control Methods

The advantages and disadvantages of some of the model-based control methods discussed are summarized in the following Table 11.1.

11.11

Concluding Remarks

547

Table 11.1 Advantages and disadvantages of some model-based control methods Method

Advantages

Disadvantages

NGMV

• Relatively simple and can be related to the Smith Predictor which provides some confidence in the approach • Includes a black-box nonlinear input subsystem, which can be beneficial if models are not available in traditional equation for • Cost-function and plant model can be made reasonable general • Usually works with relatively little design effort • Full predictive control capabilities • Simple to understand • Relates to well-known GPC algorithms for linear systems • Can easily be made more general both in system type and cost-function description • Well-proven track record in process applications • Includes explicit constraint handling • The most general algorithm • Can be specialized to NGMV or NGPC if required • Often provides the best performance because of its range of tuning options

• Does not have a predictive capability beyond the number of delay steps k • Performance may not be as good as achieved with the longer range predictive controls • Hard constraints cannot be handled explicitly (“soft constraints” can be applied through the cost function)

NGPC

NPGMV

RS-GPC

• Provides a simple structure for the controller, which can be manually tuned • Seems to have an inherent robustness due to the low-order controller involved • The gain calculations need not be at the same sample rate as the PID controller providing economies for implementation • Has a predictive capability and lower complexity than NGPC • Constraints may be placed on controller gains

• A little more restricted in the type of system description and cost-function it employs • Medium level of complexity algorithm • If constraints are included, it is more complex

• Possibly the most confusing algorithm because of its generality • The most difficult to implement in a numerically efficient form and to implement constraints in the general case • This is the newest algorithm in its current form, which is very promising but whose properties are still being investigated • The performance cannot be as good as a full predictive control because the controller structure is restricted, but it may of course get close to best performance • Use of QP methods and indirect constraint handling for control signals and states

548

11

LPV/State-Dependent Nonlinear Predictive Optimal Control

References 1. Chisci L, Falugi P, Zappa G (2003) Gain-scheduling MPC of nonlinear systems. Int J Robust Nonlinear Control 13:3–4 2. Grimble MJ (2004) GMV control of nonlinear multivariable systems. In: UKACC control 2004 conference, University of Bath 3. Cannon M, Kouvaritakis B (2001) Open loop and closed-loop optimality in interpolation MPC. In: Nonlinear predictive control: theory and practice, IEE Press, London, pp 131–149 4. Cannon M, Kouvaritakis B (2002) Efficient constrained model predictive control with asymptotic optimality. Siam J Control Optim 41(1):60–82 5. Cannon M, Kouvaritakis B, Lee YI, Brooms AC (2001) Efficient nonlinear predictive control. Int J Control 74(4):361–372 6. Cannon M, Deshmukh V, Kouvaritakis B (2003) Nonlinear model predictive control with polytopic invariant sets. Automatica 39(8):1487–1494 7. Kothare MV, Balakrishnan V, Morari M (1996) Robust constrained model predictive control using linear matrix inequalities. Automatica 32(10):1361–1379 8. Michalska H, Mayne DQ (1993) Robust receding horizon control of constrained nonlinear systems. IEEE Trans Autom Control 38:1623–1633 9. Shamma JS, Athans M (1990) Analysis of gain scheduled control for nonlinear plants. IEEE Trans Autom Control 35:898–907 10. Kouvaritakis B, Cannon M, Rossiter JA (1999) Nonlinear model based predictive control. Int J Control 72(10):919–928 11. Lee YI, Kouvaritakis B, Cannon M (2003) Constrained receding horizon predictive control for nonlinear systems. Automatica 38(12):2093–2102 12. Mayne DQ, Rawlings JB, Rao CV, Scokaert POM (2000) Constrained model predictive control: stability and optimality. Automatica 36(6):789–814 13. Scokaert POM, Mayne DQ, Rawlings JB (1999) Suboptimal model predictive control (feasibility implies stability). IEEE Trans Autom Control 44(3):648–654 14. Brooms AC, Kouvaritakis B (2000) Successive constrained optimisation and interpolation in nonlinear model based predictive control. Int J Control 73(4):312–316 15. Allgower F,, Findeisen R (1998) Nonlinear predictive control of a distillation column. In: International symposium on nonlinear model predictive control, Ascona, Switzerland 16. Camacho EF (1993) Constrained generalized predictive control. IEEE Trans Autom Control 38:327–332 17. Grimble MJ, Pang Y (2007) NGMV control of state dependent multivariable systems. In: 46th IEEE conference on decision and control, New Orleans, pp 1628–1633 18. Casavola A, Famularo D, Franze G (2002) A feedback min-max MPC algorithm for LPV systems subject to bounded rates of change of parameters. IEEE Trans Autom Control 47 (7):1147–1153 19. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 20. Hammett KD (1997) Control of nonlinear systems via state-feedback state-dependent riccati equation techniques, PhD, Dissertation, Air Force Institute of Technology, Dayton, Ohio 21. Hammett KD, Hall CD, Ridgely DB (1998) Controllability issues in nonlinear state-dependent Riccati equation control. AIAA J Guid, Control Dyn 21(5):767–773 22. Cloutier JR (1997) State-dependent Riccati equation techniques: an overview. In: American control conference, Albuquerque, New Mexico, vol 2 23. Sznair M, Cloutier J, Jacques D, Mracek C (1998) A receding horizon state dependent Riccati equation approach to sub-optimal regulation of nonlinear systems. In: 37th IEEE conference on decision and control, Tampa, Florida, pp 1792–1797 24. Grimble MJ, Johnson MA (1988) Optimal control and stochastic estimation, vols I and II. Wiley, Chichester 25. Grimble MJ (2006) Robust industrial control. Wiley, Chichester

References

549

26. Kwon WH, Pearson AE (1977) A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Trans Autom Control 22(5):838–842 27. Grimble MJ, Majecki P, Giovanini L (2007) Polynomial approach to nonlinear predictive GMV control. In: European control conference, Koss, Greece 28. Grimble MJ, Majecki P (2005) Nonlinear generalised minimum variance control under actuator saturation. In: IFAC world congress, Prague 29. Grimble MJ (2005) Nonlinear generalised minimum variance feedback, feedforward and tracking control. Automatica 41:957–969 30. Ordys AW (1993) Model-system parameters mismatch in GPC control. Int J Adapt Control Signal Process 7(4):239–253 31. Ordys AW (1999) Steady state offset in predictive control. In: American control conference, San Diego 32. Wang Y (2002) Robust model predictive control, PhD thesis, Department of Chemical Engineering, University of Wisconsin-Madison 33. Scokaert POM, Mayne DQ (1998) Min-max feedback model predictive control of constrained linear systems. IEEE Trans Autom Control 43:1136–1142 34. Bemporad A, Morari M (2001) Robust model predictive control: a survey. In: European control conference, Porto, Portugal, pp 939–944 35. Boyd S, Ghaoui LEl, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. Studies in applied mathematics, vol 15. SIAM, Philadelphia 36. Bemporad A, Morari M (1999) Robust model predictive control: a survey. In: Garulli, Tesi (Eds) Robustness in identification and control, vol 245. Springer, London 37. Camacho EF, Bordons C (2003) Model predictive control. Springer, Berlin 38. Kouvaritakis B, Rossiter JA, Schuurmans J (2000) Efficient robust predictive control. IEEE Trans Autom Control 45(8):1545–1549 39. Nagy ZK, Braatz RD (2003) Robust nonlinear model predictive control of batch processes. AIChE J 49(7):1776–1786 40. Gahinet P, Nemirovski A, Lamb AJ, Chilali M (1995) LMI control toolbox: for use with MATLAB®. The Mathworks Inc, Natick, MA 41. Grimble MJ, Majecki P (2015) Nonlinear predictive generalised minimum variance state-dependent control. IET Control Theory Appl 9(16):2438–2450 42. Zhang Z (2008) Predictive function control of the single-link manipulator with flexible joint. In: IEEE international conference on automation and logistics, Qingdao, China 43. Shawky A, Ordys A, Grimble MJ (2002) End-point control of a flexible-link manipulator using H∞ nonlinear control via a state-dependent Riccati equation. In: IEEE conference on control applications, vol 1, pp 501–506 44. Besselmann T, Lofberg J, Morari M (2012) Explicit MPC for LPV systems: stability and optimality. IEEE Trans Autom Control 57(9):2322–2332 45. Grimble MJ, Majecki P (2018) Restricted structure predictive control for linear and nonlinear systems. Int J Control. 46. Grimble MJ (2018) Three degrees of freedom restricted structure optimal control for quasi-LPV systems. In: 57th IEEE conference on decision and control, Fontainebleau Hotel, Miami Beach 47. Casavola A, Famularo D, Franzè G (2003) Predictive control of constrained nonlinear systems via LPV linear embeddings. Int J Robust Nonlinear Control 13(3–4):281–294 48. Li D, Xi Y (2010) The feedback robust MPC for LPV systems with bounded rates of parameter changes. IEEE Trans Autom Control 55(2):503–507 49. Garcia-Sanz M, Elso J (2009) Beyond the linear limitations by combining switching and QFT: application to wind turbines pitch control systems. Int J Robust Nonlinear Control 19 (1):40–58 50. Horowitz I (2001) Survey of quantitative feedback theory (QFT). Int J Robust Nonlinear Control 11(10):887–921

Part IV

Estimation, Condition Monitoring and Fault Detection for Nonlinear Systems

Chapter 12

Nonlinear Estimation Methods: Polynomial Systems Approach

Abstract Attention now turns to nonlinear filtering problems that are related to some of the minimum variance control laws discussed previously. Two forms of the estimator are described that have a nonlinear minimum variance form. The first has the virtue of simplicity and the second is useful since it can be related to Wiener or Kalman filtering when the system is linear. They can both include nonlinear communication channel dynamics in the problem construction. This is the most useful learning point and is a feature not usually considered. The channel equalization design example illustrates the useful structure of the polynomial-based solution and the computations involved.

12.1

Introduction

Signal processing and condition monitoring can be as important as control loop design to the safe operation of an industrial process. For example, nuclear reactor core condition monitoring is essential to the everyday operation of a reactor, and to the lifetime of safe operation that may be achieved [1]. There are also important signal processing problems in non-industrial areas such as medical applications like heart and lung sound estimation [2–4]. Linear estimation methods [5] may be used for many applications, even when they are inherently nonlinear. However, there are increasing demands on estimation accuracy, which suggests there will be a growing need for simple and effective nonlinear filtering solutions. This is the problem that is now considered. There are two main approaches to signal estimation found in control problems: namely Bayesian and Least Squares methods. The Bayesian methods use prior knowledge of the distribution of the states (and noise/disturbances) to find the most probable estimates of the states. The Least Squares (LS) approaches use no prior assumptions on the distributions to find these estimates. They involve minimizing a quadratic cost-function and for Gaussian noise and disturbances, and linear models, the Bayesian and Least Squares approaches lead to the same solution. The emphasis

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_12

553

554

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Nonlinear Estimation Methods: Polynomial Systems Approach

in this chapter is on the LS-based estimation methods but for nonlinear applications in areas such as robotics where nonlinearities are dominant. Classical techniques of signal processing for linear systems are well-established, including transfer-function based filtering methods that depend upon the frequency response properties of systems. The solution of linear filtering problems using optimal least squares estimation methods is also a popular approach. The Wiener filter [6], based on spectral-factorization and involving partial fraction expansion of equations, became influential in the 1940s. The reign of the Wiener filter in control related applications was relatively short. In the 1960s the Kalman filter [7, 8], was introduced and this is now much preferred for control applications. Recall the Kalman filter was introduced in Sect. 8.2 for linear time-invariant systems, and in Sect. 10.5 for time-varying systems, including quasi Linear Parameter Varying (qLPV) systems. The Kalman filter is an optimal state-estimator that is an essential part of the state-equation based control solution presented in previous chapters. These all involved what might be termed (rather imprecisely) a separation principle type of structure. Interesting links between the frequency response and state-space methods of estimation, that provided valuable insights into the relationships, were explored by Shaked [9]. The development of practical nonlinear filtering methods has been slower than for the linear case but the nonlinear problem is much more complex because of the wide range of different types. There are of course many potential applications for nonlinear estimation methods in areas such as signal processing, control systems, fault detection, condition monitoring, sensor fusion and communications. Unfortunately, many nonlinear filtering techniques involve advanced mathematics and are not so simple to implement, or they involve approximations and ad hoc solutions. A nonlinear Luenberger observer does provide an obvious and intuitive approach, at least as regards the structure of the observer, but methods for computing the gains are problematic. There are of course more elegant possibilities that include sliding mode observers and adaptive observers and some of these options are discussed briefly in the following sections and in Chap. 13.

12.1.1 Nonlinear Minimum Variance Options This chapter focuses on a polynomial systems approach to optimal nonlinear filtering, smoothing and prediction problems. Two types of the nonlinear optimal estimator are to be introduced. These might loosely be called the duals of two of the nonlinear control problems, discussed in Chaps. 4 and 6, respectively. As in the development of the NGMV optimal control solutions, the intention is to introduce nonlinear estimators that are simple to implement and yet have a formal theoretical basis. The first is an estimation problem with a construction and optimization argument that might be termed the dual of the NGMV control problem of Chap. 4. The second that is a little more complicated, but usually has improved properties,

12.1

Introduction

555

and is the dual of the NQGMV problem of Chap. 6. The duality is not as precise as in the well-known duality between Linear Quadratic optimal control and Kalman filtering problems [7] but there is a clear relationship between the mathematics involved in the feedback control and estimation problems. The two estimation problems considered appear similar superficially but the definitions of the noise sources ensure rather different results are obtained. The first estimator is the Nonlinear Minimum Variance (NMV) solution and its main virtue is the simplicity of the approach. The second estimation approach, which is referred to as the Wiener Nonlinear Minimum Variance (WNMV) solution, has a useful special case. That is, when the system is linear and the system model is the same as in Wiener or Kalman filtering problems, then the estimator reduces to one of these optimal filters [10]. The cost of this useful property, relative to NMV estimators, is some additional complexity in both the solution and the off-line calculations. However, both of these estimators are relatively simple to implement.

12.1.2 Nonlinear Minimum Variance Estimation The solution of a special class of Nonlinear Minimum Variance (NMV) estimation problems is considered first. The signal and noise models of interest involve both linear and nonlinear subsystems. By restricting the generality of the problem a simple nonlinear estimation algorithm can be derived. The cost-function to be minimized involves the variance of the estimation error and a simple optimisation procedure is applied [11, 12]. This may be thought of informally as the dual to the NGMV optimal control problem. It is not a strict mathematical dual. An advantage of this estimation problem is that a simple theoretical solution is available. Most nonlinear filtering problems involve a complex mathematical solution but the results obtained here involve only a least squares type of analysis. A simple solution follows because of the assumption of linearity for the signal generating models. The nonlinearities are assumed to be in the signal channel or possibly in a noise channel representing the uncertainty. These nonlinearities may be important in communications, radar and speech processing problems amongst others [13]. The filtering problem considered is to estimate an output signal given a noisy measurement in a system that may be multivariable. That is, it may have multiple inputs and several output measurements possibly obtained via a communication channel. The multichannel estimation problem involves a signal that is modelled by a colouring filter driven by white noise. The signal is then assumed to enter a signal channel, which is stable but may involve linear and nonlinear subsystems. The nonlinear channel dynamics can be represented by a stable nonlinear operator (black-box model). This might involve a set of nonlinear equations or could contain computer code and look-up tables, or a fuzzy-neural network. The signal channel can include different delays in different channels before the output is measured.

556

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Nonlinear Estimation Methods: Polynomial Systems Approach

To allow for uncertainty the measurements may also include an interference signal, that may enter through a parallel channel. This channel will probably not be present in the actual system but may be considered a fictitious model used to represent uncertainties. It can be included for design purposes to shape the measurement noise attenuation characteristics of the estimator. The solution requires an assumption to be made that a particular nonlinear operator has a stable inverse. This operator depends upon the parallel nonlinear channel interference model which is specified by the designer.

12.2

Nonlinear Multichannel Estimation Problem

The nonlinear inferential estimation problem considered involves the estimation of a signal, which enters a communication channel including nonlinearities and transport-delay elements [14]. The measurements are assumed to be corrupted by a noise signal that is correlated with the signal to be estimated and is provided through a parallel signal channel. This same channel can be used to represent uncertainty in the system model. The signal and noise models are assumed to have Linear Time-Invariant (LTI) model representations, which is rather restrictive, but in many applications, the models for the signal and noise signals are only LTI approximations [15, 16]. The signal processing problem is illustrated in Fig. 12.1 and includes the nonlinear signal channel model and the linear measurement noise and signal generation models. The signal to be estimated (for greater generality) is taken as the output of a linear block Wc that in most cases will represent the linear part of the communications path Wc0 (in this case Wc ¼ Wc0 ). The message signal to be estimated is denoted sðtÞ ¼ Wc ðz1 ÞyðtÞ. This is a so-called inferential estimation problem since the signal being estimated is not directly linked to the measurements obtained. That is, it is the weighted signal that enters the communication channel that is to be estimated and not the signal at the output of the channel itself. To further generalize the results a dynamic cost-function weighting term is introduced and the weighted estimation error is to be minimized, where the weighted signal sq ðtÞ ¼ Wq ðz1 Þ sðtÞ. There is no loss of generality in assuming that the signal model is driven by a zero-mean white noise source eðtÞ with an identity covariance matrix. As in the chapters on the control problems, there is no need to specify the distribution of the noise sources since the special structure of the system leads to a prediction equation that only depends upon the linear subsystems. System models: The r  r linear multivariable signal generation and noise models, in polynomial matrix form, may now be introduced. The assumption the multivariable system is square can be relaxed somewhat but this would complicate the analysis. The input signals generation model Ws and the noise model Wn have a left-coprime polynomial matrix representation

Wc ( z 1 )

y

+

+

n



Wn ( z 1 )

Message signal : s  Wc y Weighted message : sq  Wq s

Ws ( z 1 )

f

s0

Channel delays

z 0

Message signal : s  Wc 0 y ( special case : Wc  Wc 0 )

Coloured noise n

Wc 0 ( z 1 )

Linear model

z 0 sd

nd

Fig. 12.1 Canonical nonlinear filtering problem with noise sources and channel interference



Signal source

Nonlinear noise design channel

)

)

1

1

Nonlinear main channel

c1 ( z

c0 ( z

sc

+

Nonlinear estimator

( z 1 )

sˆ f

+ z

Estimated signal

nc

12.2 Nonlinear Multichannel Estimation Problem 557

558

12

Nonlinear Estimation Methods: Polynomial Systems Approach

1 1 1 ½Ws ðz1 Þ; Wn ðz1 Þ ¼ A1 0 ðz Þ½Cs ðz Þ; Cn ðz Þ

ð12:1Þ

where the unit-delay operator is defined as z1 xðtÞ ¼ xðt  1Þ. The polynomial matrix system models, for the system, may, therefore, be listed as Signal model: 1 1 Ws ðz1 Þ ¼ A1 0 ðz ÞCs ðz Þ

ð12:2Þ

1 1 Wn ðz1 Þ ¼ A1 0 ðz ÞCn ðz Þ

ð12:3Þ

Noise model:

These models can be taken to be asymptotically stable and without loss of generality they may be scaled to have a common denominator polynomial matrix A0 ðz1 Þ with identity first term and numerator signal Cs ðz1 Þ and noise Cn ðz1 Þ polynomial matrices of the following form A0 ðz1 Þ ¼ Ir þ A1 z1 þ    þ Ana zna Cs ðz1 Þ ¼ Cs0 þ Cs1 z1 þ    þ Csncs zncs Cn ðz1 Þ ¼ Cn0 þ Cn1 z1 þ    þ Cnnn znn The arguments of the polynomial matrices are often omitted for notational simplicity. There are many signal processing problems where polynomial methods of system representation are very suitable [17–20].

12.2.1 Nonlinear Channel Models The signal channel model can include both a linear transfer-function Wc0 and nonlinear dynamics. The latter is represented by a general stable nonlinear operator W c1 that in the equivalent control problems was referred to as a black-box model. The total nonlinear signal channel dynamics with different delays in different channels may be denoted as W c ¼ W c1 zK0 Wc0

ð12:4Þ

where zK0 denotes a diagonal matrix K0  I of the common delay elements in the respective output paths zK0 ¼ diagfzk1 ; zk2 ; . . . ; zkr g

ð12:5Þ

12.2

Nonlinear Multichannel Estimation Problem

559

The nonlinear channel model output (without noise and uncertainty)

 sc ðtÞ ¼ W c1 zK0 s0 ðtÞ ¼ W c1 zK0 Wc0 f ðtÞ

ð12:6Þ

where the output of the linear channel subsystem W c0 is denoted as s0 ðtÞ ¼ ðWc0 f ÞðtÞ

ð12:7Þ

and s0 ðt  K0 Þ ¼ ðzK0 s0 ÞðtÞ denotes the signal delayed by different amounts in different channels. The nonlinear subsystem W c1 can be assumed to be finite gain stable [17]. There are many possible applications where the nonlinear channel dynamics are static functions and simple to identify, or where they are dynamic functions and can be represented by well-established physical equations. Some of the previous work covering different modelling structures includes Pinter and Fernando [21], Gomez and Baeyens [22, 23], Celka et al. [24–26]. Uncertainties: There are often significant uncertainties due to noise and interference. To provide a design facility to compensate for uncertainties and noise, a nonlinear parallel channel that includes dynamics is introduced. For simplicity, this is assumed to contain the same channel transport delays. This shaping operator will be referred to as an uncertainty tuning function since it can be chosen to try to allow for noise or model uncertainties. It will normally be linear and stable but may be nonlinear and it is defined to have the form F c ðz1 Þ ¼ F c0 ðz1 ÞzK0

ð12:8Þ

The parallel path dynamics in Fig. 12.1 are shown using a dotted line. This is because this channel will probably not exist physically. It provides a term that enters the cost-function to be introduced and enables the noise attenuation to be frequency shaped. The F c0 ðz1 Þ function is specified by the designer and it will be assumed to have a stable inverse. To draw parallels with the “dual” NGMV control problem it uses a similar notation to the nonlinear control signal weighting function in the NGMV control cost-function.

12.2.2 Signals in the Signal Processing Problem The infinite-time estimation problem, where the filter is assumed to be in operation from an initial time t0 ¼ 1, is considered [27, 28]. The input and noise generating processes have an innovations signal model with white noise signal input eðtÞ 2 Rr . This signal may be assumed to be zero-mean with a covariance matrix cov½eðtÞ; eðsÞ ¼ Idts , where dts denotes the Kronecker delta function. The signal source output and coloured noise signal f ðtÞ 2 Rr may be written as

560

12

Nonlinear Estimation Methods: Polynomial Systems Approach

f ðtÞ ¼ yðtÞ þ nðtÞ

ð12:9Þ

The signals shown in the closed-loop system model of Fig. 12.1 may be listed as follows: Noise: nð t Þ ¼ W n e ð t Þ

ð12:10Þ

yð t Þ ¼ W s e ð t Þ

ð12:11Þ

f ð t Þ ¼ y ð t Þ þ nð t Þ

ð12:12Þ

s0 ðtÞ ¼ ðWc0 f ÞðtÞ

ð12:13Þ

nc ðtÞ ¼ ðF c eÞðtÞ

ð12:14Þ

Input signal:

Channel input:

Linear channel subsystem:

Channel interference:

Nonlinear channel subsystem: sc ðtÞ ¼ ðW c1 sd ÞðtÞ

ð12:15Þ

sd ðtÞ ¼ zK0 s0 ðtÞ ¼ s0 ðt  K0 Þ

ð12:16Þ

z ð t Þ ¼ nc ð t Þ þ s c ð t Þ

ð12:17Þ

Nonlinear channel input:

Observations signal:

Message signal to be estimated: sðtÞ ¼ Wc yðtÞ ¼ Wc Ws eðtÞ

ð12:18Þ

Weighted message signal: sq ðtÞ ¼ Wq Wc yðtÞ

ð12:19Þ

12.2

Nonlinear Multichannel Estimation Problem

561

Estimation error signal: ~sðtjtÞ ¼ sðtÞ  ^sðtjtÞ

ð12:20Þ

The effect of any zero-frequency or DC bias signal can be treated in a similar manner to that in the equivalent control problem [17].

12.2.3 Spectral Factorization The power spectrum for the combined linear signal and noise models can be computed, noting Uff ¼ ðWs þ Wn ÞðWs þ Wn Þ. Recall the notation for the adjoint of Ws . That is, Ws ðz1 Þ ¼ WsT ðzÞ, wherein this case the symbol z denotes the complex number in the z-domain. The generalized spectral-factor Yf satisfies Yf Yf ¼ Uff , and this may be written in the left-coprime polynomial matrix form Yf ¼ A1 0 Df 0

ð12:21Þ

From the above equations and this spectral-factor result observe that the observations signal may be expressed as a nonlinear function of the driving noise: zðtÞ ¼ nc ðtÞ þ sc ðtÞ ¼ ðF c eÞðtÞ þ ðW c1 sd ÞðtÞ ¼ ðF c eÞðtÞ þ ðW c1 s0 Þðt  K0 Þ

 ¼ ðF c eÞðtÞ þ W c1 Wc0 Yf e ðt  K0 Þ ¼ F c0 þ W c1 Wc0 Yf eðt  K0 Þ ð12:22Þ The system models are assumed such that Df 0 in (12.21) is a strictly Schur polynomial matrix [19], Shaked [9] that satisfies Df 0 Df 0 ¼ ðCs þ Cn ÞðCs þ Cn Þ

ð12:23Þ

The right-coprime polynomial matrix model may also be defined as 

   Cf 1 Wq Wc Ws A ¼ Df Yf

ð12:24Þ

where Aðz1 Þ is normalized so that Að0Þ ¼ I, and the weighted signal model Wq Wc Ws ¼ Cf A1

ð12:25Þ

The spectral-factor may now be written in either the left-coprime polynomial matrix form Yf ¼ A1 0 Df 0 or in the common denominator polynomial matrix form Yf ¼ Df A1 . Observe that the signal and noise models are correlated. However, the signal model Ws is normally defined to be a low-pass transfer-function and the noise

562

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Nonlinear Estimation Methods: Polynomial Systems Approach

model Wn to be a high-pass transfer-function. In this case, it may be shown that the estimation results for the special case of a linear problem, with no signal channel dynamics, will be similar to those for the Wiener or coloured noise Kalman filtering problems [6, 7].

12.3

Nonlinear Estimation Problem

The estimation problem is concerned with finding the best estimate of the signal sðtÞ in the presence of an interference noise term. Recall the minimum variance optimal deconvolution problem [29] involves the minimization of the estimation error ~sðtjt  ‘Þ ¼ sðtÞ  ^sðtjt  ‘Þ

ð12:26Þ

where^sðtjt  ‘Þ denotes the optimal linear estimate of the signal s(t) at time t, given the observations signal z(s) over the semi-infinite internal s 2 ð1; t  ‘. That is, the estimate given measurements up to time ðt  ‘Þ. The scalar ‘ may be positive or negative (for filtering ‘ ¼ 0, for fixed lag smoothing ‘ \ 0 and for prediction ‘ [ 0), [30]. Cost-function: A weighted estimation error cost-function is to be minimized in the current problem. The weighted estimation error criterion to be minimized has the form J ¼ tracefEfWq~sðtjt  ‘ÞðWq~sðtjt  ‘ÞÞT gg ¼ tracefEf~sq ðtjt  ‘Þ~sTq ðtjt  ‘Þgg ð12:27Þ where Ef:g denotes the expectation operator and Wq denotes a linear strictly minimum phase dynamic cost-function weighting function matrix. This weighting is assumed to be strictly minimum phase, square and invertible. The estimate ^sðtjt  ‘Þ is assumed to be generated from a nonlinear estimator of the form Estimator: ^sðtjt  ‘Þ ¼ Hf ðt; z1 Þzðt  ‘Þ

ð12:28Þ

where Hf ðt; z1 Þ denotes a minimal realization of the optimal nonlinear estimator. Since an infinite-time estimation problem is of interest no initial condition term is required (the estimator is assumed to have been in operation from an initial time t0 ! −∞).

12.3

Nonlinear Estimation Problem

563

12.3.1 Solution of the Nonlinear Estimation Problem An expression for the weighted estimation error may now be obtained. This is required for the optimization process. This signal may be defined as follows: sq ðtÞ ¼ Wq~sðtjt  ‘Þ

ð12:29Þ

From (12.9) to (12.11) the signal source and noise input can be represented as 1 f ðtÞ ¼ A1 0 Cn eðtÞ þ A0 Cs eðtÞ

ð12:30Þ

A realization of f ðtÞ can be obtained using the spectral-factor defined above as f ¼ A1 0 Df 0 e

ð12:31Þ

where Df 0 satisfies (12.23). Also, recall the weighted message signal is obtained from (12.19) as sq ¼ Wq Wc Ws e

ð12:32Þ

Estimation error: From equations: (12.2), (12.24), (12.32) obtain the weighted estimation error as
 ~sq ðtjt  ‘Þ ¼ sq ðtÞ  ^sq ðtjt  ‘Þ ¼ Wq Wc Ws eðtÞ  Hf zðt  ‘Þ ¼ Cf A1 eðtÞ  Wq Hf zðt  ‘Þ

ð12:33Þ

Recall from (12.17) that the observations z ¼ nc þ sc and substituting from Eq. (12.33) ~sq ðtjt  ‘Þ ¼ Cf A1 eðtÞ  Wq Hf ðnc ðt  ‘Þ þ sc ðt  ‘ÞÞ From (12.12), (12.14), (12.15) and (12.16) sc ðtÞ ¼ ðW c1 sd ÞðtÞ and nc ðtÞ ¼ ðF c eÞðtÞ we obtain ~sq ðtjt  ‘Þ ¼ Cf A1 eðtÞ  Wq Hf ððF c eÞðt  ‘Þ þ ðW c1 sd Þðt  ‘ÞÞ
 ¼ Cf A1 eðtÞ  Wq Hf ðF c0 zK0 eÞðt  ‘Þ þ ðW c1 zK0 s0 Þðt  ‘Þ ð12:34Þ Also using equations: (12.13), (12.18), (12.31)
 ~sq ðtjt  ‘Þ ¼ Cf A1 eðtÞ  Wq Hf ðF c0 zK0 eÞðt  ‘Þ þ ðW c1 zK0 Wc0 Yf eÞðt  ‘Þ

564

12

Nonlinear Estimation Methods: Polynomial Systems Approach

Assumption: The assumption will be made that (Wc0 Yf and zK0 ) commute, and hence we obtain
 ~sq ðtjt  ‘Þ ¼ Cf A1 eðtÞ  Wq Hf ðF c0 zK0 eÞðt  ‘Þ þ ðW c1 Wc0 Yf zK0 eÞðt  ‘Þ ¼ Cf A1 eðtÞ  Wq Hf ðF c0 þ W c1 Wc0 Yf Þeðt  K0  ‘Þ

ð12:35Þ

There is some abuse of notation in this equation. In the multichannel case, from (12.5) the K0 matrix is diagonal, and hence eðt  K0  ‘Þ ¼ zK0 ‘I eðtÞ ¼ diagfzk1 ‘ ; zk2 ‘ ; . . .; zkr‘ geðtÞ It follows an element of row j in the vector eðt  K0  ‘Þ is defined as eðt  kj  ‘Þ. Now introduce the following Diophantine equation where degðF0 Þ\K0 þ ‘ I F0 A þ G0 zK0 ‘ ¼ Cf

ð12:36Þ

Weighted estimation error: The first term in (12.35) may be simplified using the Diophantine equation. This enables the term to be broken into a sum of white and coloured noise components as follows: ~sq ðtjt  ‘Þ ¼ ðF0 þ G0 A1 zK0 ‘ ÞeðtÞ  Wq Hf ðF c0 þ W c1 Wc0 Yf Þeðt  K0  ‘Þ ¼ F0 eðtÞ þ G0 A1 eðt  K0  ‘Þ  Wq Hf ðF c0 þ W c1 Wc0 Yf Þeðt  K0  ‘Þ

ð12:37Þ where it is assumed that the “denominator” polynomial A matrix and the delay matrix zK0 commute.

12.3.2 Optimization and Solution The solution for the optimal linear estimator may now be obtained. First, inspect the form of the weighted estimation error in Eq. (12.37), and advance the signal by ðt þ ‘ÞI þ K0 to obtain ~sq ðt þ ‘ þ K0 jt þ K0 Þ ¼ F0 eðt þ ‘ þ K0 Þ þ ½G0 A1  Wq Hf ðF c0 þ W c1 Wc0 Yf ÞeðtÞ ¼ F0 eðt þ ‘ þ K0 Þ þ ½G0  Wq Hf ðF c0 A þ W c1 Wc0 Df ÞA1 eðtÞ

ð12:38Þ The equation may also be written in a form that provides the block diagram matrix form of the estimator

12.3

Nonlinear Estimation Problem

~sq ðt þ ‘ þ K0 jt þ K0 Þ ¼ F0 eðt þ ‘ þ K0 Þ þ ½G0 A1  Wq Hf ðI þ W c1 Wc0 Yf F 1 c0 ÞF c0 eðtÞ

565

ð12:39Þ

The polynomial matrix term F0 in (12.39) may be expanded as a sequence of terms in the unit advanced operator fz; z2 ; z3 ; z4 ; . . .; z‘ þ K0 g. It follows that the first term in (12.39) is dependent upon the future values of the white noise signal components eðt þ 1Þ; eðt þ 2Þ; eðt þ 3Þ; . . .; eðt þ ‘ þ K0 Þ. The second group of terms in the square brackets are all dependent upon past values of the white noise signals, whereas the first term depends upon future values. It follows that these two groups of terms are statistically uncorrelated and the expected value of the cross-terms is null. The first terms on the right-hand side of the estimation error expressions (12.38) or (12.39) are independent of the choice of estimator. It follows that the smallest variance is achieved when the remaining square bracketed terms in (12.38) or (12.39) are set to zero. Assuming the existence of a finite gain stable causal inverse to the nonlinear operator and noting Yf ¼ Df A1 the optimal estimator follows as: 1 1 Hf ¼ Wq1 G0 ðF c0 A þ W c1 Wc0 Df Þ1 ¼ Wq1 G0 A1 F 1 c0 ðI þ W c1 Wc0 Yf F c0 Þ

ð12:40Þ Before summarizing the results recall the assumptions that Wc0 Yf and zK0 commute which is satisfied if the delays are the same in the signal paths zK0 ¼ zk I, or if these models are diagonal. The assumption was also made that the operator ðF c0 þ W c1 Wc0 Yf Þ has a stable inverse. This is a reasonable assumption since there is a free choice of the design function F c0 discussed later. Under these assumptions, the following theorem applies. Theorem 12.1: Optimal Estimator for Nonlinear Systems The nonlinear deconvolution filter ð‘ ¼ 0Þ, predictor ð‘ [ 0Þ, or smoother ð‘\0Þ to minimize the variance of the estimation error (12.27), for the system in Sect. 12.2, can be calculated from the following spectral-factor, Diophantine and operator equations. 1 The spectral-factor Yf ¼ A1 0 Df 0 , where Df 0 is asymptotically stable and is defined from the polynomial factorization Df 0 Df 0 ¼ ðCs þ Cn ÞðCs þ Cn Þ

ð12:41Þ

The right-coprime polynomial matrices ðCf ; Df Þ and A satisfy 

   Wq Wc Ws Cf 1 A ¼ Df Yf

ð12:42Þ

The polynomials G0 and F0 may be obtained from the minimal degree solution ðG0 ; F0 Þ, with respect to F0 , of the Diophantine equation

566

12

Nonlinear Estimation Methods: Polynomial Systems Approach

F0 A þ G0 zK0 ‘ ¼ Cf

ð12:43Þ

NMV estimator: The causal linear estimate ^sðtjt  ‘Þ ¼ Hf ðt; z1 Þzðt  ‘Þ to minimize the variance of the estimation error (12.27), may be found as Hf ¼ Wq1 G0 ðF c0 A þ W c1 Wc0 Df Þ1

ð12:44Þ

Alternatively, the estimator may be computed in the form 1 1 Hf ¼ Wq1 G0 A1 F 1 c0 ðI þ W c1 Wc0 Yf F c0 Þ

ð12:45Þ

Minimum cost: The minimum variance of the estimation error may be computed using 9 8 > > < 1 I dz= ð12:46Þ J ¼ trace fF0 F0 g > z> ; :2pj jzj¼1

Proof The proof follows by collecting the above results.



Remarks • The polynomial block structure for the estimator defined in (12.45) clearly has the structure shown in Fig. 12.2. • To analyze the stability of the estimator, note that Wq is assumed strictly minimum phase and the operator ðF c0 þ W c1 Wc0 Yf Þ is also assumed to have a finite gain stable inverse, by suitable choice of uncertainty tuning function or shaping operator F c0 , and hence the estimator Hf is stable. • The degree of F0 is determined by the solution of F0 A þ G0 zK0 ‘ ¼ Cf and as ‘ increases the degree of F0 also increases, which increases the order of the estimator. • The expression (12.46) may be used to test the optimal estimator by comparing the actual performance against the ideal. This result provides a possible method of benchmarking the nonlinear estimator. • When the signal model Ws ðz1 Þ and main channel model W c1 ðz1 Þ are low-pass and both the noise model Wn ðz1 Þ and the parallel channel shaping operator F c0 ðz1 Þ are high-pass, the estimator also provides an approximation to the solution of the more conventional filtering problem, where the signal and noise are statistically uncorrelated. This is discussed further in the next section.

12.3

Nonlinear Estimation Problem

567

Nonlinear estimator

f

Estimate sˆ(t t  )

Observations z(t) +

1

G0 A 1

c0

-

Wc 0

c1

Wq1

Yf

Transmission Signal/noise channel model Spectral model

Fig. 12.2 Nonlinear estimator structure including channel model

12.3.3 Significance of Parallel Path Dynamics The main strength of the NMV estimation approach is that for a restricted class of systems, where the technique is applicable, it provides a very simple solution to a difficult nonlinear estimation problem. In certain cases, the results are also valuable more generally. For example, consider the case where a system has correlated noise and signal models but where the noise model has basically a high frequency characteristic and the signal model is dominantly low frequency. If the results are applied to this type of system, it will provide an approximate solution to some common estimation problems (for the linear case with no channel dynamics these correspond to Wiener/Kalman filtering problems). That is, the fact that the signals in the assumed model are correlated will have little effect on the results at both low frequencies and high frequencies (because of the choice of the signal and noise models). It has already been noted in Sect. 12.2.1 that the parallel path dynamics shown in Fig. 12.1 do not normally represent a real physical system and there is, therefore, design freedom in the specification of the dynamical uncertainty tuning function or shaping operator term F c0 , which defines the coloured noise parallel channel output term. This operator that is normally high-pass may be used to shape the frequency response to noise or to uncertainties which may be difficult to model accurately. The signal models are correlated but since W c1 is normally low-pass and F c0 is high-pass the results for the special case of the linear problem will be similar to those for the coloured noise Wiener filtering problem [6], where the measurement and signal noise sources are independent. The parallel path shaping operator dynamics F c0 is therefore considered a design weighting term that is tuned to obtain the desired estimator characteristics (see the example in Sect. 12.3.5). Note that there is the complication that the tuning should also ensure a stable inverse of the nonlinear operator N c ¼ ðF c0 þ W c1 Wc0 Yf Þ exists.

568

12

Nonlinear Estimation Methods: Polynomial Systems Approach

12.3.4 Design and Implementation Issues The computation of the estimator is relatively straightforward. The selection of the shaping operator or uncertainty tuning function F c0 is a dual problem to the selection of the NGMV optimal control cost-function weightings in Chap. 5. The requirement on the nonlinear operator to have a stable inverse is, of course, mandatory. A simple starting point is, therefore, to assume the uncertainty model F c0 is a constant and of a small magnitude. This corresponds to the situation where the uncertainty is simply white noise added at the output of the communications channel (before it enters the estimator). Also, observe that the estimator includes a model for the channel dynamics which was assumed to be known. In fact, the nonlinear subsystem dynamics W c1 can represent all the channel dynamics (absorbing the linear subsystem as well) and it is useful that the dynamical equations need not be known in traditional equation form. It is clear from (12.47) that the equations for the black-box model are not required (just the ability to compute an output from a given input). This suggests that an adaptive estimator could be developed where the black-box model is computed from a neural network for systems with changing signal channel dynamics. For implementation note that the output of the nonlinear operator in the estimator expression (12.44) may be written as mðtÞ ¼ ðF c0 A þ W c1 Wc0 Df Þ1 zðtÞ and the computation is straightforward when the shaping operator or uncertainty design term F c0 is chosen to be linear F c0 ðz1 Þ ¼ Fc0 ðz1 Þ and has a stable inverse. Without loss of generality, the plant polynomial matrix Aðz1 Þ may be written in the form of a through term and polynomial terms Aðz1 Þ ¼ I þ z1 A1 ðz1 Þ. Thus write, mðtÞ ¼ ðFc0 þ Fc0 z1 A1 þ W c1 Wc0 Df Þ1 zðtÞ After manipulation
 1 zðtÞ  Fc0 A1 mðt  1Þ  W c1 Wc0 Df mðtÞ mðtÞ ¼ Fc0

ð12:47Þ

A so-called algebraic-loop is present (since m(t) depends on itself) but the equation can usually be reordered to avoid this problem as for the NGMV control problem discussed in Chap. 5 (Sect. 5.4).

12.3

Nonlinear Estimation Problem

569

12.3.5 Nonlinear Channel Equalization Problem The design of an equalizer for a communication channel is a possible applications area. Channel equalisation is important in digital communications and in this case, the signal y(t) represents a digital information sequence. The noise is due to the signal nðtÞ and the interference between channel signals. The error is defined in terms of the difference between the equalizer output ^sðtjt  ‘Þ and the delayed values of the sequence transmitted y(t). Recall ~sðt þ ‘jtÞ ¼ yðt þ ‘Þ  ^sðt þ ‘jtÞ and if the delay ‘ ¼ k then the estimation error becomes ~sðt  kjtÞ ¼ yðt  kÞ  ^sðt  kjtÞ. In this equalization problem, the purpose is to design an equalizer Hf which estimates ^sðt  kjtÞ from the observations signal zðtÞ, to minimize the estimation error ~sðt  kjtÞ between the equalizer output and the delayed signal of the transmitted sequence. In the special case when Wc ðz1 Þ is defined as Wc ðz1 Þ ¼ zk Ir this becomes a so-called fixed-lag smoothing problem. Such problems are considered further in Sect. 12.7.

12.4

Automotive Nonlinear Filtering Problem

There follows an automotive simulation example for a feedback-loop controlling the air–fuel ratio using a lambda sensor. The sensor measures the residual oxygen in the exhaust gas and passes this information to the engine control unit (ECU), which adjusts the optimum air–fuel mixture. The lambda sensor is located at some distance from the point of interest and there is, therefore, a considerable delay in the measurement, as illustrated in Fig. 12.3. In this example let the sample-time Ts = 0.025 s and the discrete delay (zn ) be n = 7 events. Plant linear subsystem model: The linear subsystem may be represented by a discrete-time transfer-function Wc0 ðz1 Þ ¼

0:259z1 1  0:7408z1

Engine Control Unit

Air Fuel

Air-Fuel Mixture

Lambda Sensor

Engine

Exhaust

Delay

Fig. 12.3 Location of the Lambda sensor

570

12

Nonlinear Estimation Methods: Polynomial Systems Approach UEGO Sensor Characteristic

EGO Sensor Characteristic 3 2.8

1

Sensor Output (Volts)

Sensor Output (Volts)

1.2

0.8 0.6 0.4 0.2 0 0.85

2.6 2.4 2.2 2 1.8 1.6 1.4

0.9

0.95

1

1.05

1.1

1.15

1.2

1.2 0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Lambda Ratio

Lambda Ratio

Fig. 12.4 Two types of lambda sensor characteristics

Sensor and nonlinear subsystem: There are two types of lambda or Air/Fuel ratio sensor that are used in the automotive industry. The Universal Exhaust Gas Oxygen (UEGO) sensor uses the changes of voltage that are representative of the oxygen levels in the reference and test chambers of the sensor to feed an electronic control unit for feedback control. The second type is the Exhaust Gas Oxygen (EGO) sensor which is an electrochemical cell used to measure the oxygen content of an engine’s exhaust. In closed-loop operation, this signal is used to monitor the fuel mixture. The two different static nonlinear sensor characteristics [31], are shown in Fig. 12.4.

12.4.1 Design Problem and Simulation Results The NMV estimation algorithm can be compared below for three different cases. In the first case, a single EGO sensor is used and the results obtained are as shown in Table 12.1 and Fig. 12.5. In the second case, a single UEGO sensor was tested and the results obtained are as shown in Table 12.2 and in Fig. 12.6. In the last case, both EGO and UEGO sensors are assumed to be present and the results obtained in this case are as shown in Table 12.3 and shown in Fig. 12.7.

Table 12.1 Using a single EGO sensor Tuning filter (F c0 )

Theoretical minimum variance

Minimum variance based on simulation

1

ð1  0:7686z1 Þ=ð50ð1  z1 ÞÞ

1.08e−5

1.65e−5

2

ð1  0:7686z1 Þ=ð60ð1  z1 ÞÞ

1.08e−5

1.56e−5

3

ð1  0:7686z1 Þ=ð70ð1  z1 ÞÞ

1.08e−5

1.49e−5

12.4

Automotive Nonlinear Filtering Problem

571 Tuning Filter Response Used For EGO Sensor

EGO SENSOR

1.015

40 35

1.005 1 0.995 0.99 0.985 0.98 0

30

Magnitude (dB)

Lambda Ratio

1.01

25 20 15 10 5 0

Actual Signal Estimated Signal

5

10

-5 15

-10 -1 10

0

1

10

10

2

10

3

10

Frequency (rad/s)

Time(Sec)

Fig. 12.5 Left: EGO comparison between actual frequency and estimated Lambda; right: the EGO tuning filter F 1 c0 response Table 12.2 Using single UEGO sensor Tuning filter (F c0 )

Theoretical minimum variance

Minimum variance based on simulation

1

ð1  0:2865z1 Þ=ð50ð1  z1 ÞÞ

1.08e−5

1.36e−4

2

ð1  0:2865z1 Þ=ð60ð1  z1 ÞÞ

1.08e−5

1.42e−4

3

ð1  0:2865z1 Þ=ð70ð1  z1 ÞÞ

1.08e−5

1.57e−4

Tunning Filter Response Used For UEGO Sensor

UEGO SENSOR 1.015

40 35 30

1.005

Magnitude (dB)

Lambda Ratio

1.01

1 0.995 0.99 0.985 0.98 0

10

Time(Sec)

20 15 10 5 0

Actual Signal Estimated Signal

5

25

-5 15

-10 -1 10

0

10

1

10

2

10

3

10

Frequency (rad/s)

Fig. 12.6 Left: UEGO comparison between actual frequency response and estimated Lambda; right: the UEGO tuning filter F 1 c0 response

As explained above the estimator depends on the design of the weighting or shaping filter F c0 . This is chosen to ensure a stable inverse of the operator F c0 A þ W c1 Wc0 Df . The “DC gain” and the cut-off frequency of F c0 play a major role in the design of the NMV estimation algorithm. In the UEGO case, the

572

12

Nonlinear Estimation Methods: Polynomial Systems Approach

Table 12.3 Using EGO and UEGO sensors together Tuning filters (F c0 )

Theoretical minimum variance

Minimum variance based on simulation

1

ð1  0:7686z1 Þ=ð70ð1  z1 ÞÞ and ð1  0:2865z1 Þ=ð50ð1  z1 ÞÞ

1.08e−5

5.51e−5

2

ð1  0:7686z1 Þ=ð50ð1  z1 ÞÞ and ð1  0:2865z1 Þ=ð40ð1  z1 ÞÞ

1.08e−5

5.78e−5

3

ð1  0:7686z1 Þ=ð80ð1  z1 ÞÞ and ð1  0:2865z1 Þ=ð60ð1  z1 ÞÞ

1.08e−5

5.43e−5

estimation results are close to the actual signal due to the design of F c0 but in the other two cases, the results are also encouraging. The use of the two sensors and filters leads to worse results in this case but this is purely dependent on the choice of the different tuning functions and scaling.

12.5

Introduction to the Wiener NMV Estimation Problem

In this second half of the chapter, attention turns to a slightly more sophisticated estimator which has useful asymptotic properties and has a more general problem construction. It will be shown that a similar solution strategy may be followed as above but with some subtleties. The estimator is almost as simple to implement as the NMV filter. The off-line computations are a little more complicated but it has improved properties and it deals with the more common situation where the noise and signal models are uncorrelated. It will be referred to as the Wiener Nonlinear Minimum Variance (WNMV) estimation problem. The signal and noise channels may again be nonlinear and can be represented in a general black-box nonlinear operator form. The WNMV estimation algorithm to be derived will be shown to be relatively simple to use and to implement. It is more complicated than the NMV estimator and can be implemented as a recursive algorithm using a discrete-time nonlinear difference equation. A very useful property is that in the limiting case of a linear system, without channel dynamics, the estimator reduces to a Wiener filter in a discrete polynomial matrix form [32]. In the previous case of the NMV filter, this type of behaviour was only an approximation when the system frequency responses were of a special type.

12.5.1 Nonlinear Multichannel System Description A multichannel estimation problem is again considered [33], and the signal to be estimated is assumed to be generated by feeding white noise into a colouring filter. The resulting signal then enters a linear subsystem representing part of the channel

12.5

Introduction to the Wiener NMV Estimation Problem UEGO Plus EGO SENSOR

573

Tuning Filters Response

1.015

40

1.01

Magnitude (dB)

Lambda Ratio

30 1.005 1 0.995 0.99 0.985 0.98 0

10

Time(Sec)

10 0 -10

Actual Signal Estimated Signal

5

20

15

-20 -1 10

EGO Tuning Filter Response UEGO Tuning Filter Response 0

1

10

10

10

2

3

10

Frequency (rad/s)

Fig. 12.7 Left: UEGO plus EGO comparison between actual frequency response and estimated Lambda; right: the UEGO plus EGO tuning filter F 1 c0 response

dynamics which can be non-minimum phase. The input to the nonlinear channel dynamics includes an additive coloured noise signal representing airborne noise or pick up for example. Unlike the previous approach, the signal and noise terms are not assumed correlated. The signal enters a nonlinear subsystem which is assumed to be stable but is otherwise quite general. It is one of the strengths of the technique that the nonlinear channel dynamics can again be represented by a general nonlinear operator (including a set of nonlinear equations or a black-box model containing with computer code that is not known). The problem is illustrated in Fig. 12.8, which shows the nonlinear signal channel model and linear measurement noise and signal models. The signal generation process is assumed to be represented by white noise feeding a linear time-invariant model. The objective is to estimate the signal sðtÞ at some point in the signal channel path sðtÞ ¼ Wc yðtÞ. In the usual case where this involves the output of the channel Wc0 ðz1 Þ, let Wc ðz1 Þ ¼ Wc0 ðz1 Þ. In the channel equalisation or deconvolution estimation problems, the signal input to the channel is to be found and Wc ðz1 Þ should be defined as Wc ðz1 Þ ¼ zq I, where q [ 0. The steady-state, or so-called infinite-time estimation problem, where the filter is assumed to be in operation from the initial time t0 ! 1, is considered.

12.5.2 Signals in the Signal Processing System The nonlinear system representing the optimal estimation problem is illustrated in Fig. 12.8. The white noise sources in the system are assumed to be zero-mean and to have an identity covariance matrix. The polynomial matrix system models and the r  m multichannel system may now be introduced. The signal Wc0 Ws and noise Wn models have a stable left-coprime polynomial matrix representation

Ws ( z )

1

y

Wc

s0

Signal s

Wc 0 ( z )

1

Linear channel +

+

n

f

z

0

Channel delays

sd

Fig. 12.8 Nonlinear filtering problem with coloured signal and noise and channel dynamics



Signal model



1 c1 ( z )

Nonlinear channel

z

f

( z 1 )

Nonlinear estimator

Estimated signal

sˆ

12

Wn ( z 1 )

Coloured noise

574 Nonlinear Estimation Methods: Polynomial Systems Approach

12.5

Introduction to the Wiener NMV Estimation Problem 1 1 1 ½Wc0 ðz1 Þ Ws ðz1 Þ; Wn ðz1 Þ ¼ A1 0 ðz Þ½Cs ðz Þ; Cn ðz Þ

575

ð12:48Þ

These system models may be listed as Channel and signal model: 1 1 Wc0 ðz1 ÞWs ðz1 Þ ¼ A1 0 ðz ÞCs ðz Þ

ð12:49Þ

1 1 Wn ðz1 Þ ¼ A1 0 ðz ÞCn ðz Þ

ð12:50Þ

Noise model:

Signal to be minimized: The signal to be estimated will be assumed to be given by sðtÞ ¼ Wc ðz1 ÞyðtÞ, where Wc ðz1 Þ will be equal to the signal path Wc ¼ Wc0 in most problems. The cost-index may involve a dynamic cost-function weighting Wq where the weighted signal sq ðtÞ ¼ Wq ðz1 ÞsðtÞ ¼ Wq ðz1 ÞWc ðz1 ÞyðtÞ

ð12:51Þ

Nonlinear channel model: After separating the signal channel dynamics into a linear input subsystem and a nonlinear output subsystem model the nonlinear channel model output, with different delays in different channels, may be expressed as
 sc ðtÞ ¼ W c1 zK0 f ðtÞ

ð12:52Þ

where the diagonal delay matrix zK0 ¼ diagfzk1 ; zk2 ; . . .; zkr g. As before, with some abuse of notation, the delayed signals sðt  K0 Þ ¼ zK0 sðtÞ may involve different delays in different communication channels. The nonlinear subsystem W c1 is assumed to be finite gain stable. Linear model: The linear time-invariant part of the input channel subsystem dynamics is denoted by Wc0 ðz1 Þ. The signal model is denoted y ¼ Ws n, and the signal output from the linear channel dynamics Wc0 ðz1 Þ is assumed to be corrupted by coloured measurement noise. The combination of these signals is denoted f ðtÞ 2 Rr . The white noise signals nðtÞ 2 Rqs and xðtÞ 2 Rqn are mutually independent and trend free and the covariance matrices are defined as cov½nðtÞ; nðsÞ ¼ Qs dts and cov½xðtÞ; xðsÞ ¼ Qn dts , respectively. The signals in the signal and communication channel model shown in Fig. 12.8, may be listed as follows: Noise: nðtÞ ¼ Wn xðtÞ

ð12:53Þ

576

12

Nonlinear Estimation Methods: Polynomial Systems Approach

Message signal: yðtÞ ¼ Ws nðtÞ

ð12:54Þ

s0 ðtÞ ¼ Wc0 ðz1 ÞyðtÞ

ð12:55Þ

f ðtÞ ¼ Wc0 ðz1 ÞyðtÞ þ nðtÞ

ð12:56Þ

Linear channel signal output:

Output of linear channel:

Nonlinear channel input: sd ðtÞ ¼ zK0 f ðtÞ ¼ f ðt  K0 Þ

ð12:57Þ

The power spectrum for the combined signal f can be computed, noting these are linear subsystems, using  Uff ¼ Us0 s0 þ Unn ¼ Wc0 Ws Qs Ws Wc0 þ Wn Qn Wn

ð12:58Þ

where the notation for the adjoint operator of Ws implies Ws ðz1 Þ ¼ WsT ðzÞ. The generalized spectral-factor Yf may be computed using Yf Yf ¼ Uff , which by reference to (12.2) may be written in the left-coprime polynomial matrix form Yf ¼ A1 0 Df 0

ð12:59Þ

The system models are assumed to be such that Df 0 is a strictly Schur polynomial matrix that satisfies Df 0 Df 0 ¼ Cs Qs Cs þ Cn Qn Cn

ð12:60Þ

It follows that by using the spectral-factor result (12.23), a realization of f ðtÞ can be obtained as f ðtÞ ¼ A1 0 Df 0 eðtÞ

ð12:61Þ

where eðtÞ denotes a unity covariance zero-mean white noise source, and where Df 0 satisfies (12.23). The signals associated with the nonlinear system elements and estimator may now be listed as Nonlinear channel output: sc ðtÞ ¼ ðW c1 sd ÞðtÞ ¼ ðW c1 zK0 f ÞðtÞ

ð12:62Þ

12.5

Introduction to the Wiener NMV Estimation Problem

577

Channel noise and interference: nc ðtÞ ¼ ðF c0 zK0 eÞðtÞ

ð12:63Þ

z ð t Þ ¼ nc ð t Þ þ s c ð t Þ

ð12:64Þ

Observations signal:

Message signal to be estimated: sðtÞ ¼ Wc yðtÞ

ð12:65Þ

~sðtjtÞ ¼ sðtÞ  ^sðtjtÞ

ð12:66Þ

Wq ~sðtjtÞ ¼ Wq ðsðtÞ  ^sðtjtÞÞ

ð12:67Þ

Estimation error signal:

Weighted estimation error:

12.5.3 Design and Weighting Functions To obtain a solution to this estimation problem an assumption must be made that a particular nonlinear operator has a stable inverse. To ensure this condition is satisfied requires the specification of the nonlinear noise or interference model, which is now introduced. This function can be considered a design variable accounting for the noise and channel uncertainties. The system of interest can be represented, as shown in Fig. 12.9, which involves a number of generalizations. As in the NMV estimator derived above a nonlinear parallel channel with delays including a stable uncertainty tuning function or shaping operator may be introduced of the form F c ðz1 Þ ¼ F c0 ðz1 ÞzK0

ð12:68Þ

The measured output is assumed to include this nonlinear noise or distortion term representing measurement inaccuracies and uncertainties, and its choice becomes part of the design process. The parallel path dynamics are shown in Fig. 12.9 using a dotted line because the channel may not exist in practice. It is included as in the NMV filtering problem to provide additional design freedom and to allow for uncertainties in models or noise. If for example, it has a high gain at high frequency the output will represent high frequency noise and the filter will increase the measurement noise attenuation at high frequencies. In the NQGMV optimal control problem described in Chap. 6 this term is equivalent to the control signal costing term used for shaping the controller frequency responses [34]. This may be called the dual problem, although it is not a strict duality.

WsQsCs* D*f 01

y +

+

f

Linear channel dynamics Weighted signal Wq Wc sq

Wc0

s0

n

*1 f0

z 0

Channel delays model

sd

1

)

Nonlinear signal channel dynamics

1 c (z ) 1

c0 ( z

+

+

sc

nc

z

f

( z1 )

Nonlinear estimator

Fig. 12.9 Nonlinear filtering problem with innovations signal and noise models and dynamic estimation error signal costing



Signal model

* n

z 0

Nonlinear noise shaping channel

Wq

sˆq

sˆ

Estimated signal

12

Wn Qn C D



Coloured noise

578 Nonlinear Estimation Methods: Polynomial Systems Approach

12.5

Introduction to the Wiener NMV Estimation Problem

579

The noise sources used in Fig. 12.9 have also been modified, relative to the problem shown in Fig. 12.8. That is, the signal and noise models are represented by the innovations signal model shown in Fig. 12.9. It is well-known that the traditional Wiener [35] or Kalman filtering [7] problem for a linear system can also be solved by using an innovations signal model. The justification for modifying the noise sources to generate the signal f (t) is provided in Sect. 12.6.1 that follows. For later use define a nonlinear dynamic operator Combined Nonlinear Operator:
 N c ¼ F c0 þ W c1 Yf

ð12:69Þ

The nonlinear channel dynamics are assumed to be r input and r output and the operator N c is assumed to be invertible with an inverse that is finite gain stable. As mentioned F c0 can be chosen by the designer and a causal inverse of N c can be achieved by ensuring the first term in F c0 ð0Þ is full rank and invertible.

12.6

Nonlinear Estimation Problem and Solution

The output estimation problem is similar to that defined previously and involves finding the best estimate of the signal sðtÞ in the presence of the uncertainty, as shown in Fig. 12.9. Recall the minimum variance optimal deconvolution problem involves the minimization of the estimation error: ~sðtjt  ‘Þ ¼ sðtÞ  ^sðtjt  ‘Þ

ð12:70Þ

where ^sðtjt  ‘Þ denotes the optimal estimate of the signal sðtÞ at time t, given observations zðsÞ over the semi-infinite internal s 2 ð1; t  ‘ up to time ð t  ‘Þ. The scalar ‘ may be positive or negative depending on the type of the estimation problem [29, 30, 36, 37]. The smoothing problem has a solution for the cases where the smoothing lag satisfies K0 þ ‘I  I, signifying the combination of the smoothing lag and the channel delay must be greater than unity for each channel. The weighted estimation error cost-function is again to be minimized so that the estimation error can be penalized in particular frequency ranges. The cost-index may be expressed in the form J ¼ tracefEfðWq~sðtjt  ‘ÞÞðWq~sðtjt  ‘ÞÞT gg

ð12:71Þ

where Ef:g denotes the expectation operator and Wq denotes a linear dynamic weighting function matrix, which may be represented in polynomial matrix form as Wq ¼ A1 q Bq . The weighting matrices Wq and Bq are assumed to be strictly minimum phase, square and invertible [38]. The estimate ^sðtjt  ‘Þ is assumed to be generated from a causal estimator of the form

580

12

Nonlinear Estimation Methods: Polynomial Systems Approach

^sðtjt  ‘Þ ¼ Hf ðt; z1 Þzðt  ‘Þ

ð12:72Þ

where Hf ðt; z1 Þ again denotes a minimal realization of the optimal nonlinear estimator.

12.6.1 Solution of the WNMV Estimation Problem To obtain an expression for the estimation error ~sðtÞ, the noise sources in Fig. 12.8 may be modified, as shown in Fig. 12.9. The new noise model has the same spectrum but is not physically realizable. That is, the combined signal and noise defined by (12.56) and represented in Fig. 12.8 is equivalent to the signal f ðtÞ shown in Fig. 12.9. This may be confirmed by noting (12.49), (12.59), (12.60), and obtaining the following expression for the signal f ðtÞ in Fig. 12.9, as follows:  1 f ðtÞ ¼ Wc0 Ws Qs Cs D1 f 0 eðtÞ þ Wn Qn Cn Df 0 eðtÞ
   1 1 ¼ A1 0 Cs Qs Cs e þ Cn Qn Cn Df 0 eðtÞ ¼ A0 Df 0 eðtÞ

ð12:73Þ

Clearly, the signal f ðtÞ has the same spectrum in the systems shown in Figs. 12.8 and 12.9 as the innovations signal model f ðtÞ ¼ A1 0 Df 0 eðtÞ, where eðtÞ denotes a unity covariance white noise signal source. This is a similar argument to the method used in Chap. 6 to solve the NQGMV control problem, where the cost-function weightings did not have to be physically realizable models. Introduce the left-coprime polynomial matrices for the weighted signal model A1 cs Ccs ¼ Bq Wc Ws

ð12:74Þ

The weighted message signal, for the modified system shown in Fig. 12.9, may now be written as 1 1  1 sq ðtÞ ¼ Wq Wc Ws Qs Cs D1 f 0 eðtÞ ¼ Aq Acs Ccs Qs Cs Df 0 eðtÞ

ð12:75Þ

From this equation and (12.72) obtain the expression for the estimation error 1  1 ~sq ðtjt  ‘Þ ¼ Wq ðsðtÞ  ^sðtjt  ‘ÞÞ ¼ A1 q Acs Ccs Qs Cs Df 0 eðtÞ  Wq Hf zðt  ‘Þ

ð12:76Þ Recall from (12.17) that the observations z ¼ nc þ sc and hence we obtain 1  1 ~sq ðtjt  ‘Þ ¼ A1 q Acs Ccs Qs Cs Df 0 eðtÞ  Wq Hf ðnc ðt  ‘Þ þ sc ðt  ‘ÞÞ

12.6

Nonlinear Estimation Problem and Solution

581

From (12.62) and noting nc ðtÞ ¼ ðF c eÞðtÞ and sc ðtÞ ¼ ðW c1 sd ÞðtÞ, after substitution 1  1 ~sq ðtjt  ‘Þ ¼ A1  Wq Hf ððF c eÞðt  ‘Þ þ ðW c1 sd Þðt  ‘ÞÞ q Acs Ccs Qs Cs Df 0 eðtÞ
 1 1  1 ¼ Aq Acs Ccs Qs Cs Df 0 eðtÞ  Wq Hf ðF c0 zK0 eÞðt  ‘Þ þ ðW c1 zK0 f Þðt  ‘Þ

ð12:77Þ Substituting in (12.77), noting f ðtÞ ¼ A1 0 Df 0 eðtÞ ¼ Yf eðtÞ, we obtain
 1  1 K0 ~sq ðtjt  ‘Þ ¼ A1 þ W c1 zK0 Yf Þeðt  ‘Þ q Acs Ccs Qs Cs Df 0 eðtÞ  Wq Hf ðF c0 z Nonlinear operator: To simplify notation let the nonlinear operator
 N c ¼ F c0 þ W cl Yf . Weighted estimation error: Now assume the delay term commutes with Yf and we obtain 1  1 ~sq ðtjt  ‘Þ ¼ A1 q Acs Ccs Qs Cs Df 0 eðtÞ  Wq Hf N c eðt  K0  ‘Þ

where the same abuse of notation in expressing the delay terms referred to earlier arises. Substituting obtain, 1  1 K0 þ ‘I ~sq ðt þ K0 þ ‘jt þ K0 Þ ¼ A1 eðtÞ  Wq Hf N c eðtÞ ð12:78Þ q Acs Ccs Qs Cs Df 0 z

Diophantine equation: This Eq. (12.78) may be simplified using a Diophantine equation [20], to break up the first term in (12.78) into causal and non-causal terms. The solution ðF0 ; G0 Þ is required where the degree of F0 is less than some positive integer g. This integer is the smallest number that ensures the following Diophantine equation involves only powers of z1 Acs Aq F0 þ G0 Df 0 zg ¼ Ccs Qs Cs zK0 þ ‘IgI

ð12:79Þ

The equation may, therefore, be solved by letting degðF0 Þ ¼ g  1, so that after division, g 1 1 1 1  1 K0 þ ‘I F0 D1 f 0 z þ Aq Acs G0 ¼ Aq Acs Ccs Qs Cs Df 0 z

ð12:80Þ

Hence (12.78) becomes 1  1 K0 þ ‘I ~sq ðt þ K0 þ ‘jt þ K0 Þ ¼ A1 eðtÞ  Wq Hf N c eðtÞ q Acs Ccs Qs Cs Df 0 z g 1 1 ¼ ðF0 D1 f 0 z þ Aq Acs G0 ÞeðtÞ  Wq Hf N c eðtÞ  1  1 ¼ F0 D1 f 0 eðt þ gÞ þ Aq Acs G0  Bq Hf N c eðtÞ  1
 1 ¼ F0 D1 f 0 eðt þ gÞ þ Aq Acs G0  Bq Hf F c0 þ W c1 Yf eðtÞ

ð12:81Þ

582

12

Nonlinear Estimation Methods: Polynomial Systems Approach

Note that F0 zg is a polynomial matrix in positive powers of z. Also, note that the smoothing cases must be restricted to the cases where K0 þ ‘I  I.

12.6.2 Optimization to Compute the Estimator The solution of the optimization problem for the optimal estimator now follows. First, inspect the form of the first term in the weighted estimation error Eq. (12.81). g  1 The signal F0 D1 f 0 eðt þ gÞ ¼ F0 ðz Df 0 Þ eðtÞ is multiplied by the matrix g  1 L1 c ¼ ðz Df 0 Þ . Recall the matrix Lc is strictly non-Schur and the degree of F0 is

less than the integer g. The first polynomial matrix term F0 ðzg Df 0 Þ1 in Eq. (12.81) may, therefore, be expanded as a convergent sequence of terms in the unit advance operator fz; z2 ; z3 ; z4 ; . . .:g. It follows that the first term in (12.81) is dependent upon the future values of the white noise signal components {eðt þ 1Þ, eðt þ 2Þ, eðt þ 3Þ….}. The final group of terms in (12.81) (in the square brackets) are all dependent upon past values of the white noise signal. The two groups of terms in the weighted estimation error cost are therefore statistically uncorrelated, and the expected value of the cross-terms is null. Also, note that the first term on the right-hand side of (12.81) is independent of the estimator action. It follows that the smallest variance is achieved when the remaining terms in the square brackets in (12.81) are set to zero. Assuming the existence of a finite gain stable causal inverse to the operator N c the optimal estimate is obtained by setting this second group of terms in (12.81) to zero. This relationship may be simplified by defining the following right-coprime polynomial matrices as 

  1 1  Gf 1 Bq Acs G0 A ¼ Df Yf

ð12:82Þ

The optimal estimator therefore becomes 1

1

1 1 Hf ¼ B1 q Acs G0 N c ¼ Gf A N c

ð12:83Þ

The Wiener Nonlinear Minimum Variance (WNMV) estimator may, therefore, be written as
1
1 Hf ¼ Gf A1 F c0 þ W c1 Yf ¼ Gf F c0 A þ W c1 Df

ð12:84Þ

12.6

Nonlinear Estimation Problem and Solution

583

12.6.3 WNMV Optimal Estimator Before quoting the theorem which summarises these results recall the assumptions. It was assumed that Yf and zK0 commute and that the system models are such that Df 0 is a strictly Schur polynomial matrix. Also through the appropriate definition of F c0 the operator N c is assumed to have a finite gain stable causal inverse. Theorem 12.2: Optimal Wiener Estimator for Nonlinear Multichannel Systems The deconvolution estimator to minimize the variance of the estimation error (12.27), for the system described in Sect. 12.5, can be calculated following the solution of spectral-factorization,, Diophantine and operator equations where for the filter ð‘ ¼ 0Þ, predictor ð‘ [ 0Þ, or smoother ð‘\ 0Þ. The signal Wc0 Ws and noise Wn models have the left-coprime polynomial matrix representation 1 1 1 ½Wc0 ðz1 Þ Ws ðz1 Þ; Wn ðz1 Þ ¼ A1 0 ðz Þ½Cs ðz Þ; Cn ðz Þ 1 The spectral-factor Yf ¼ A1 0 Df 0 , where the operator Df 0 is asymptotically stable and satisfies

Df 0 Df 0 ¼ Cs Qs Cs þ Cn Qn Cn

ð12:85Þ

and the left-coprime polynomial matrices Acs and Ccs satisfy A1 cs Ccs ¼ Bq Wc Ws . The minimal degree solution ðG0 ; F0 Þ, with respect to F0 , is also required of the Diophantine equation Acs Aq F0 þ G0 Df 0 zg ¼ Ccs Qs Cs zK0 þ ‘IgI

ð12:86Þ

It is assumed that the weighting F c0 is chosen to ensure the following nonlinear operator has a finite gain stable inverse
 N c ¼ F c0 þ W c1 Yf

ð12:87Þ

The optimal causal WNMV estimate ^sðtjt  ‘Þ ¼ Hf ðt; z1 Þzðt  ‘Þ to minimize 1 1 the variance of the estimation error (12.27), is given by Hf ¼ B1 q Acs G0 N c , and may be written as Estimator:
1
1 Hf ¼ Gf A1 F c0 þ W c1 Yf ¼ Gf F c0 A þ W c1 Df

ð12:88Þ

584

12

Nonlinear Estimation Methods: Polynomial Systems Approach

where the following right-coprime polynomial matrices satisfy 

  1 1  Gf 1 Bq Acs G0 A ¼ Df Yf

ð12:89Þ

Minimum cost: The minimum variance of the cost-index may be computed as 9 8 > > = < 1 I  1 dz fD1 F F D g J ¼ trace f 0 0 f > z> ; :2pj

ð12:90Þ

jzj¼1

Proof The proof follows by collecting the results in the above section. Remarks • A valuable property of the above estimator is that when the system is linear the estimator reverts to well-known solutions. In the limiting case when the nonlinear channel is replaced by the identity matrix the results correspond to the traditional linear Wiener deconvolution filter. If the linear channel dynamics are also reduced to the identity it becomes equal to the Wiener or Kalman filters. • The existence of the stable inverse to the nonlinear operator N c ¼
 F c0 þ W cl Yf was assumed which depends upon the design function F c0 . This can be used to compensate for uncertainty and measurement noise at the nonlinear channel output. • The expression (12.90) may be used to test the optimal estimator by comparing the actual performance against the ideal. This, therefore, provides a possible method of benchmarking nonlinear estimators [39]. • The degree of F0 is determined by the solution of (12.86) and as ‘ increases the degree of F0 and the order of the estimator increases. The estimator may be expressed in the useful block diagram form shown in Fig. 12.10, by noting that if the signal pðtÞ ¼ ðF c0 A þ W c1 Df Þ1 zðtÞ it may also be written as:
 pðtÞ ¼ ðF c0 AÞ1 zðtÞ  ðW c1 Df ÞðpðtÞÞ ¼ ðF c0 AÞ1 eðtÞ

ð12:91Þ

where eðtÞ ¼ zðtÞ  ðW c1 Df ÞðpðtÞÞ. The signal e(t) in Fig. 12.10 feeds the uncertainty tuning function F c0 and may be expressed in terms of a sensitivity function as eðtÞ ¼ ðI þ W c1 Df ðF c0 AÞ1 Þ1 zðtÞ

ð12:92Þ

12.6

Nonlinear Estimation Problem and Solution

585

Nonlinear Estimator

Observations + z(t)

e(t)

( c 0 A) 1

-

c1

p(t)

f

Gf

Estimate

sˆ(t t  )

Df

Transmission channel model

Fig. 12.10 Wiener nonlinear estimator for filtering, prediction and smoothing

12.6.4 Limiting Features of WNMV and NMV Estimators It is useful to consider the limiting linear form of the WNMV estimator so that it may be related to existing filtering solutions. As the nonlinear channel dynamics tend to the identity and the uncertainty weighting F c0 tends to zero, the estimator becomes equivalent to a Wiener deconvolution estimator in a linear polynomial matrix form. This may be shown by considering the linear filtering problem, where the channel dynamics are absent W c1 ðz1 Þ ¼ I and the channel has a unit-delay K0 ¼ I. If the uncertainty and output noise tend to zero because F c0 ðz1 Þ ! 0 and a single-step smoothing problem is considered (‘ ¼ I), then the problem reverts to a Wiener filtering problem. The resulting estimator follows from the above theorem and becomes 1 1 1 Hf ¼ Gf D1 f ¼ Bq Acs G0 Yf

ð12:93Þ

where G0 satisfies Acs Aq F0 þ G0 Df 0 zg ¼ Ccs Qs Cs zg , A1 cs Ccs ¼ Bq Ws and Df 0 satisfies the spectral-factor equation Df 0 Df 0 ¼ Cs Qs Cs þ Cn Qn Cn . The estimator is therefore in this special case equivalent to a Wiener filter, or equivalently a frequency-domain version of the Kalman filter [19]. Thus, the Wiener version of the nonlinear minimum variance estimator has the useful property that if the channel dynamics are absent and the noise is white the estimator reduces to that of the Kalman or Wiener filter. Unfortunately, this does not apply to the NMV filter discussed in the first part of the chapter. However, the NMV estimator is the simplest to compute and implement. The estimation problem in the NMV case assumes that there is a correlation between the measurement noise and the signal models which leads to the simplifications achieved. The WNMV estimator is slightly more complicated but applies when the measurement noise and signals are uncorrelated. The WNMV estimator has a noise and disturbance structure which more closely parallels the situation in the usual Wiener or Kalman filtering

586

12

Nonlinear Estimation Methods: Polynomial Systems Approach

problems. In fact, the noise and signal descriptions in the WNMV problem were constructed to ensure this limiting result was obtained.

12.6.5 Parallel Path Dynamics and Uncertainty As in the case of the NMV estimator described in the first part of the chapter, the parallel path dynamics can represent a real physical system path, and in this case, there is no freedom in the specification of the dynamical nonlinear term F c0 , however, this is unlikely. It may also represent a noise channel to represent uncertainty, and, therefore, provide an uncertainty tuning capability. In fact, the parallel path dynamics represented by F c0 normally acts like a design weighting term that may be used to ensure the inverse of the nonlinear operator N c exists when the channel dynamics W c1 are not stably invertible. There is freedom in the choice of F c0 , since uncertainty is by definition difficult to describe precisely. The signal, noise and channel models may include other types of probabilistic uncertainty, which might involve parameters that have known means and variances. Alternatively, the models might be subject to additive or multiplicative uncertainties. The aim is of course to construct an estimation problem that leads to robust estimators that are not influenced significantly by uncertainties. A design can be based not only on nominal design models but also on specified sets of deviations between the models and the true systems. These are sometimes referred to as error models, or uncertainty models [40, 41]. However, the approach taken here has been to try to improve robustness of the filter by the choice of weighting functions. It may be possible to optimize robustness properties by using appropriate uncertainty descriptions but this does introduce an additional level of complication into a nonlinear estimation problem which was deliberately constructed to ensure simple results are obtained.

12.7

Example of Design Issues and Channel Equalization

A nonlinear channel equalization problem, introduced in Sect. 12.3.5, is considered for the design example. If a signal is passed through a channel and the effects of the channel on the signal are removed by making an inverse channel filter, this is referred to as equalisation problem. An equalisation filter attempts to restore the frequency and phase characteristics of the signal to their values prior to the transmission. This technique is used widely in telecommunications to maximize the bandwidth for transmission and to reduce errors caused by the channel characteristics. It is often desirable to equalize or remove the effect that channel distortion has on a signal. The problem considered here is a particular equalisation problem

12.7

Example of Design Issues and Channel Equalization

587

for systems that may involve nonlinear distortion. Equalisation is needed in telephone modems, fax machines and other communications applications. The channel equalization problem is based on that considered by Choi et al. [42] that was solved using recurrent neural network (RNN). It will enable the performance of the optimal estimator to be demonstrated. Channel equalization can be accomplished through a number of techniques, such as high-pass filtering the data at the transmitter and using tunable impedance matching networks. The problem considered is based on a decision feedback equalization problem. A general model of a digital communications system with an equalizer involves a signal source and a signal channel possibly including both linear and nonlinear types of distortion [42]. A sequence s(t) is extracted from a source of information and is then transmitted but the transmitted signals are often corrupted by channel distortion and noise. The channel containing the memoryless nonlinear distortion is modelled below using the tanh function. The linear finite impulse response of the channel is denoted by Hðz1 Þ. Such a nonlinear model can be encountered in digital satellite communications and in magnetic recording [43, 44]. The WNMV filter is computed below for a typical application and a simulation is used to verify the results. A decision feedback equalization is characterized by the three integers: feedforward order, feedback order, and decision delay, respectively. The decision delay d ¼ ‘ was selected as 2 steps. The signal and noise models have the following delay operator forms, respectively: Ws ¼ 2 ð0:61  1:203 z1 þ 0:6 z2 Þ=ð1  2:496 z1 þ 1:998 z2  0:5 z3 Þ Wn ¼ ð0:08562z1 Þ=ð1  0:2865 z1 Þ The cost-weighting Wq ¼ 1 and Wc ¼ Wc0 ¼ 1, and the channel delay ¼ z1 . The linear non-minimum phase channel characteristics are defined as Hðz1 Þ ¼ 0:3482 þ 0:8704z1 þ 0:3482z2

ð12:94Þ

where the roots are at z = −1.9996 and z = −0.5001. The model is often used in channel equalization case studies. The nonlinearity in the signal channel is modelled as zðtÞ ¼ tanhðsd ðtÞÞ ¼ tanhðf ðt  K0 ÞÞ

ð12:95Þ

and the inverse cothð xÞ ¼ 1=tanhð xÞ. The nonlinearity is a function of the signal output of the linear channel dynamics Hðz1 Þ, and can take into account saturation effects in the transmitting amplifier. The dc gain and changes in the cut-off frequency of the weighting filter F c0 influence the accuracy of estimation. The uncertainty tuning function or shaping operator F c0 can be optimized for the particular problem and has the discrete-time operator model F c0 ¼ ðz  0:9835Þ=ð3:333 z  3:333Þ. The logic of this choice is that the channel interference is likely to be more serious at low frequencies and

588

12

Nonlinear Estimation Methods: Polynomial Systems Approach

hence the gain is higher at low frequencies. The overall system and the resulting WNMV filter are shown in Fig. 12.11. In this case Wc ¼ Wc0 ¼ 1 and the signal y(t) = s(t) is corrupted by the noise signal n(t) with the relative magnitudes shown in Fig. 12.12. Because of the tanh(x) nonlinear function, the signal entering the communications channel is limited in magnitude, as illustrated in Fig. 12.13. The resulting optimal estimate may be compared with the signal before transmission through the channel as in Fig. 12.14. The performance of the filter can be seen from the comparison of the clipped output of the channel and the estimates from the filter as shown in Fig. 12.15. The filter is clearly effective at reconstructing the underlying signal even when the measurement has a clipped output due to the tanh function effects (involves a saturation characteristic). The estimation error is shown in Fig. 12.16 and the value of the minimum variance is obtained from the simulation (in this case it is 0.0320 for the system with one step delay). If the channel has 10 sample steps more than for the design the variance increases to 0.2048 and the results are as shown in Fig. 12.17. This clearly reveals the effect of the increased delay. If the filter is designed for this long delay case the estimates improve, as shown in Fig. 12.18. The variance falls in this optimal case to 0.0420, since the signals are now more in phase. The use of prediction in the 11 step channel delay optimal case is shown in Fig. 12.19, where the prediction interval is 5 steps ð‘ ¼ 5Þ and the variance 0.0483. The equivalent smoothing problem for a 5 step smoothing lag ð‘ ¼ 5Þ has a lower variance of 0.0373 and is shown in Fig. 12.20. As expected the variance in the optimal prediction case is increased, whilst the variance in the optimal smoothing case is lower. If the uncertainty is made constant and very small (F c0 ¼ 0:001) the results are the best. This is reflected in the estimation error variance of 0.0216. In this case, since little uncertainty is assumed the estimator attempts to introduce an inverse of the nonlinear channel dynamics. However, it cannot cancel the linear subsystem dynamics that are non-minimum phase and hence the inverse includes the minimum phase image of Hðz1 Þ.

 (t )

Noise model

Wn ξ(t)

Ws

s(t)  

n(t)

WNMV filter

Channel dynamics

f (t )

z (t )

tanh  f (t   0 ) 



sˆ(t  )

1

G f A1

c0



Signal model

Yf



tanh fˆ (t   0 )



fˆ (t )

Fig. 12.11 WNMV estimator and signal generation model along with channel dynamics

12.7

Example of Design Issues and Channel Equalization

589

Message and Noise at NL Channel Input 2.5 Output Wco with input Message Noise Signal

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Fig. 12.12 The message and noise at input of nonlinear communications channel

Total Input to NL Channel and Measurement of Output of NL Channel 3 Input to NL Channel Output from NL Channel

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Fig. 12.13 Input and output from the communications channel

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Nonlinear Estimation Methods: Polynomial Systems Approach Estimate and Signal

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Fig. 12.14 Signal and its estimate from WNMV filter

Estimate and Measured Output of Channel

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Fig. 12.15 Channel output signal and signal estimate

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Example of Design Issues and Channel Equalization

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Fig. 12.16 Estimation error time response

Estimate and Signal

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Fig. 12.17 Signal and estimate when channel contains 11 steps (designed for unity delay)

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Nonlinear Estimation Methods: Polynomial Systems Approach Estimate and Signal

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Fig. 12.18 Signal and optimal estimate when channel contains 11 steps delay

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Fig. 12.19 Signal and optimal prediction when channel contains 11 steps delay

10

12.8

Concluding Remarks

593 Estimate and Signal

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Fig. 12.20 Signal and optimal smoothed signal when channel contains 11 steps delay

12.8

Concluding Remarks

The proposed nonlinear filters are applicable to some more specialized estimation problems than found in traditional Wiener or Kalman filtering situations since they include nonlinear channel dynamics. However, in the limiting case when the dynamics are linear and the channel is just a gain, the WNMV estimator becomes equal to a Wiener or Kalman filter in a polynomial system description form. The solutions are relatively easy to understand and to implement. Other nonlinear filtering techniques like the extended Kalman filter involve approximations and are not optimal. The estimators proposed here, if the system model fits the assumptions (not common), do not involve such an approximation. The full applications potential of this class of estimators has not been explored. They were derived for applications in control, fault monitoring and detection problems. However, there are opportunities for simple computationally efficient estimators in communication systems and wireless sensor networking. The key to their potential lies in the simplicity of the online computations and in the ability to handle uncertain, nonlinear communication channels. Leaving aside the question of optimality the results may also be applicable to systems that include time-varying channel dynamics. In fact, one of the advantages of the approach is that the signal channels can be represented by either a set of known nonlinear equations or can be replaced by say a neural network to introduce learning or adaptive features. A simple H1 nonlinear filter can also be produced

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following similar ideas [45–49]. An alternative approach that might improve robustness is to represent any poorly known parameters probabilistically [50]. Tools from the so-called Nonlinear Minimum Variance family of controllers and estimators provide a possible condition monitoring solution for a wide variety of nonlinear system types and structures [51]. There are also obvious applications in fault detection since the communications channel can represent the path between the fault and the signal [52]. Restricted structure filter designs may be found following similar strategies to the linear case [53], and can be related to the restricted structure control problems in Chap. 2.

References 1. Bonivento C, Grimble MJ, Giovanini L, Monatri M, Paoli A (2007) State and parameter estimation approach to monitoring AGR nuclear core. In: Picci G, Valcher M (eds) A tribute to Antonio Lepschy. University of Padova, Chapter 13, pp 31–56 2. Charleston S, Azimi-Sadjadi MR (1996) Reduced order Kalman filtering for the enhancement of respiratory sounds. IEEE Trans Biomed Eng 43(4): 421–424 3. Charleston-Villalobos S, Dominguez-Robert LF, Gonzalez-Camarena R, Aljama-Corrales AT (2006) Heart sounds interference cancellation in lungs sounds. In: International conference of the IEEE engineering medicine and biology society, vol 1, New York, pp 1694–1697 4. Cortes S, Jane R, Torres A, Fiz JA, Morera J (2006) Detection and adaptive cancellation of heart sound interference in tracheal sounds. IEEE Eng Med Biol Soc N Y 1:2860–2863 5. Kailath T (1974) A view of three decades of linear filtering theory. IEEE Trans Inf Theory 20 (2):146–181 6. Wiener N (1949) Extrapolation, interpolation and smoothing of stationary time series, with engineering applications. Technology Press and Wiley (issued in February 1942 as a classified US National Defence Research Council Report), New York 7. Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 35–45 8. Kalman RE (1961) New methods in Wiener filtering theory. In: Symposium on engineering applications of random function theory and probability, pp 270–388 9. Shaked U (1979) A transfer function approach to the linear discrete stationary filtering and the steady-state discrete optimal control problems. Int J Control 29(2):279–291 10. Åström KJ (1979) Introduction to stochastic control theory. Academic Press, London 11. Grimble MJ (2007) NMV optimal estimation for nonlinear discrete-time multi-channel systems. In: 46th IEEE conference on decision and control, New Orleans, pp 4281–4286 12. Grimble MJ, Naz SA (2010) Optimal minimum variance estimation for nonlinear discrete-time multichannel systems. IET Signal Process J 4(6): 618–629 13. Ali-Naz S, Grimble MJ (2009) Design and implementation of nonlinear minimum variance filters. Int J Adv Mechatron Syst Interscience 1(4) 14. Haykin S (2001) Communication systems, 4th edn. Wiley 15. Grimble MJ (1995) Multichannel optimal linear deconvolution filters and strip thickness estimation from gauge measurements. ASME J Dyn Syst Meas Control 117:165–174 16. Grimble MJ (2006) Robust industrial control: optimal design approach for polynomial systems. Wiley, Chichester 17. Grimble MJ (2005) Nonlinear generalised minimum variance feedback, feedforward and tracking control. Automatica 41:957–969 18. Grimble MJ, Kucera V (1996) Polynomial methods for control systems design. Springer, London

References

595

19. Grimble MJ (1985) Polynomial systems approach to optimal linear filtering and prediction. IJC 41(6):1545–1564 20. Kucera V (1979) Discrete linear control. Wiley, Chichester 21. Pinter S, Fernando XN (2007) Equalization of multiuser wireless CDMA downlink considering transmitter nonlinearity using Walsh codes. EURASIP J Wireless Commun Netw:1–9 22. Gomez JC, Baeyens E (2003) Hammerstein and Wiener model identification using rational orthonormal bases. Lat Am Appl Res 33(4):449–456 23. Gómez JC, Baeyens E (2004) Identification of block-oriented nonlinear systems using orthonormal bases. J Process Control 14(6):685–697 24. Celka P, Bershad NJ, Vesin JM (2001) Stochastic gradient identification of polynomial Wiener systems: analysis and application. IEEE Trans Sig Proc 49(2):301–313 25. Bershad NJ, Celka P, McLaughlin S (2001) Analysis of stochastic gradient identification of Wiener-Hammerstein systems for nonlinearities with Hermite polynomial expansions. IEEE Trans Signal Process 49(5):1060–1072 26. Celka P, Bershad NJ, Vesin JM (2000) Fluctuation analysis of stochastic gradient identification of polynomial Wiener systems. IEEE Trans Signal Proc 48(6):1820–1825 27. Grimble MJ, Jukes KA, Goodall DP (1984) Nonlinear filters and operators and the constant gain extended Kalman filter. IMA J Math Cont Inf 1:359–386 28. Kwakernaak H, Sivan R (1991) Modern signals and systems. Prentice Hall 29. Moir TJ (1986) Optimal deconvolution smoother. IEE Proc Pt D 133(1):13–18 30. Grimble MJ (1996) H∞ optimal multichannel linear deconvolution filters, predictors and smoothers. Int J Control 63(3):519–533 31. Berggren P, Perkovic A (1996) Cylinder individual lambda feedback control in an SI engine. M.Sc. Thesis, Linkoping 32. Grimble MJ, Ali Naz S (2009) Nonlinear minimum variance estimation for discrete-time multi-channel systems. IEEE Trans Signal Proc 57(7):2437–2444 33. Liu X, Sun JW, Liu D (2006) Nonlinear multifunctional sensor signal reconstruction based on total least squares. J Phys Conf Ser 48:281–286 34. Grimble MJ (1984) LQG multivariable controllers: minimum variance interpretation for use in self-tuning systems. Int J Control 40(4):831–842 35. Ahlen A, Sternad M (1991) Wiener filter design using polynomial equations. IEEE Trans Signal Process 39:2387–2399 36. Anderson BDO, Moore JB (1979) Optimal filtering. Prentice Hall Inc., New Jersey 37. Chi CY, Mendel JM (1984) Performance of minimum variance deconvolution filter. IEEE Trans Acoust Speed Signal Process 32(60):1145–1152 38. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 39. Grimble MJ (2004) Data driven weighted estimation error benchmarking for estimators and condition monitoring systems. IEE Proc Control Theory Appl 151(4):511–521 40. Sternad M, Ahlén A (1996) H2 design of model-based nominal and robust discrete time filters. In: Grimble MJ, Kucera V (eds) A polynomial approach to H2 and H∞ robust control design, Chapter 5. Springer, London 41. Grimble MJ (1996) Robust filter design for uncertain systems defined by both hard and soft bounds. IEEE Trans Signal Process 44(5):1063–1071 42. Choi J, Bouchard M, Yeap TH, Kwon O (2004) A derivative-free Kalman filter for parameter estimation of recurrent neural networks and its applications to nonlinear channel equalization. In: Fourth ISCS symposium on engineering of intelligent systems, Madeira, Portugal 43. Chen S, Gibson GJ, Mulgrew B, McLaughlin S (1990) Adaptive equalization of finite nonlinear channels using multilayer perceptrons. Signal Process 20:107–119 44. Ataslar B (2004) Robust flow control for data communication networks. Ph.D. Dissertation, Anadolu University, Eskisehir, Turkey 45. Grimble MJ (1987) H∞ design of optimal linear filters. In: Byrnes CI, Martin CF, Saeks RE (eds) MTNS conference, Phoenix, Arizona, 1988. North Holland, Amsterdam, pp 553–540

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46. Grimble MJ, ElSayed A (1990) Solution of the H∞ optimal linear filtering problem for discrete-time systems. IEEE Trans Signal Process 38(7):1092–1104 47. Yaesh I, Shaked U (1992) Transfer function approach to the problems of discrete-time systems: H∞ optimal linear control and filtering. IEEE Trans Autom Control 36(11):1264–1271 48. Shaked U (1990) H∞ minimum error state estimation of linear stationary processes. IEEE Trans Autom Control 35(5):554–558 49. Aliyu MDS, Boukas EK (2008) Discrete-time mixed H2/H∞ nonlinear filtering. American control conference, June 11–13, Seattle, Washington 50. Grimble MJ (1984) Wiener and Kalman filters for systems with random parameters. IEEE Trans 29(6):552–554 51. Alkaya A, Grimble MJ (2015) Non-linear minimum variance estimation for fault detection systems. Trans Inst Meas Control 37(6):805–812 52. Chen J, Patton RJ, Zhang HY (1996) Design of unknown input observers and robust fault detection filters. Int J Control 63(1):85–105 53. Ali Naz S, Grimble MJ, Majecki P (2011) Multi-channel restricted structure estimators for linear and nonlinear systems. IET J Signal Process 5(4):407–417

Chapter 13

Nonlinear Estimation and Condition Monitoring: State-Space Approach

Abstract This second chapter on nonlinear filtering involves systems that use state-space models to represent the linear subsystems. The main estimation problem considered is the nonlinear minimum variance filtering problem. However, there is also an introduction to other linear and nonlinear estimation problems. The Kalman filter was used in many of the control solutions considered earlier and this is described in more detail. The algorithm for the computation of the Kalman gains is presented. The extended Kalman filter for systems represented by nonlinear state-equation-based models is also described. A brief introduction to alternative nonlinear estimators is included beginning with particle filters and covering the unscented transformation and Unscented Kalman Filter. The nonlinear minimum variance filtering problem is unusual because it includes a possible nonlinear communications channel. To illustrate applications of this filter for state models the automotive air–fuel ratio estimation problem is again considered. A final example illustrates its use in condition monitoring, fault monitoring and detection problems. The main learning point is the natural structure the problem has for such applications, where the signal to be estimated lies deep within a system that includes uncertainties and nonlinearities before the measurements are obtained.

13.1

Introduction

A state-space version of the Nonlinear Minimum Variance (NMV) estimator is introduced in this chapter. In many estimation problems, the process dynamics and the observations equation can be “nonlinear” even within the normal operating region. There is, therefore, a need for effective nonlinear estimators. Potential application areas include control systems but also encompass weather forecasting, economics, radar tracking and navigation systems. The use of the estimator in condition monitoring, fault-monitoring, fault-detection and fault-isolation problems is explored in later sections. Before turning attention to the NMV estimator, a brief introduction to other linear and nonlinear estimation methods is included.

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_13

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The state-space approach to optimal filtering for “linear” time-invariant and time-varying systems is well known. The optimal state-equation-based observer or filter is commonly referred to as a Kalman filter [1, 2]. This was introduced in Chap. 8 and was used in Chaps. 9, 10 and 11 employing different types of state-equation and state-dependent models. It is a discrete-time linear filter that is optimal in a mean square sense. The Kalman–Bucy filter is the term that is often used to describe the continuous-time version of the filter, in honour of Kalman and Bucy [3] who made this seminal contribution. The main advantage of the discrete-time Kalman filtering approach over the older Wiener transfer-function-based optimal filtering methods [4] is the convenient form for digital implementation. The Kalman filter is arguably the most successful contribution that has been made to “modern control and estimation theory.” The system is assumed linear in the Kalman filtering problem and a statistical criterion is minimized so that the estimate is optimal in an average sense [5]. The Kalman filter provides an estimate of the mean and the covariance in real time. Unfortunately, real systems may involve a significant nonlinearity that will have a material effect on filter performance in some ranges of operation. The distribution of the signals will then deviate from the Gaussian distribution that is often assumed. The estimation, control and condition monitoring problems are closely linked. The larger the number of measurements available, the more effective a control loop can become. Unfortunately the system can also become less reliable, since the addition of more hardware may introduce other potential causes of failure. The “soft-sensing,” or so-called “inferential estimation” approach can provide alternative estimates of key signals without the need for additional measurements. That is, other measurements can be used together with models to infer a particular signal. This can be a cheaper alternative to provide a measure of redundancy than the duplication of sensors or measurement systems.

13.1.1 Kalman Filter The state-estimate update equation was introduced in previous chapters for different cases but the computation of the Kalman filter gain was not covered, and hence this process will be described briefly. The process (or plant) is assumed to be stabilizable and detectable. The process and measurement noise signals fðtÞ and vðtÞ are assumed to be independent white noise signals and have covariance matrices QðtÞ and RðtÞ, respectively. These signals are usually assumed to be zero-mean but in the algorithm below, they can contain non-zero bias terms (a minor modification to the results). An explicit k-steps delay is also included in the control channel. Process model: xðt þ 1Þ ¼ AðtÞxðtÞ þ BðtÞuðt  kÞ þ DðtÞfðtÞ Observations model: zðtÞ ¼ CðtÞxðtÞ þ vðtÞ

13.1

Introduction

599

The Kalman filter for this system only utilizes the first two moments of the state (the mean and covariance) in its recursive algorithm. The innovations signal is required in the following, which is the difference between the observations signal z (t) and the predicted observations. Kalman filter algorithm in predictor–corrector form Predictor update: ^xðt þ 1jtÞ ¼ AðtÞ^xðtjtÞ þ BðtÞuðt  kÞ þ DðtÞfðtÞ Estimate update: ^xðt þ 1jt þ 1Þ ¼ ^xðt þ 1jtÞ þ Kðt þ 1Þ~yðt þ 1jtÞ Innovations signal: ~yðt þ 1jtÞ ¼ zðt þ 1Þ  Cðt þ 1Þ^xðt þ 1jtÞ  vðtÞ A priori covariance: Pðt þ 1jtÞ ¼ AðtÞPðtjtÞAT ðtÞ þ DðtÞQðtÞDT ðtÞ Kalman gain matrix:
1 Kðt þ 1Þ ¼ Pðt þ 1jtÞC T ðt þ 1Þ Cðt þ 1ÞPðt þ 1jtÞC T ðt þ 1Þ þ Rðt þ 1Þ A posteriori covariance: Pðt þ 1jt þ 1Þ ¼ Pðt þ 1jtÞ  Kðt þ 1ÞCðt þ 1ÞPðt þ 1jtÞ Initial conditions: xð0j0Þ ¼ m0

and

Pð0j0Þ ¼ P0

Bias terms: fðtÞ ¼ E ffðtÞg;

vðtÞ ¼ EfvðtÞg:

Tuning and enhancing the Kalman filter There are some practical measures to improve control when using the Kalman filter for observer-based control applications. The covariance matrices of the process and the measurement noise are sometimes considered the tuning variables even though in theory they are determined by the physical properties of the process. In the case of the disturbance signals, the covariance elements are usually easy to determine based on previous operating records. However, there is much more uncertainty in the covariances related to other plant states and hence these covariances can be treated as design variables.

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Note that all the plant states should be considered to be excited by non-zero white noise, which implies that none of the covariances for the plant noise inputs can be null. This will improve the numerical properties of the Kalman filter gain computations and possibly the robustness of the feedback control. The robustness improvement is not guaranteed but often occurs because increased noise on particular plant states can in some measure compensate for uncertainty in plant model knowledge. It is often the case that low-frequency disturbances, or unknown bias signals on measurements, can be estimated very effectively by adding an integrator driven by white noise in the system model at inputs or outputs, respectively. The idea of adding another subsystem, either on plant input states or at the plant outputs, is also empirical but is useful in applications (Sect. 5.6). Such additions can compensate for unknown bias signals or other uncertainties.

13.1.2 Extended Kalman Filter There are many techniques for nonlinear estimation, and as mentioned in Chap. 8, one of the best known for control applications is the extended Kalman filter (EKF). An EKF linearizes the distribution around the mean of the current state estimate, and then uses these results for prediction and to update the estimated states. The EKF model has a similar structure to the Kalman filter, but it contains a nonlinear state-equation model of the process [6]. To compute the equivalent of the Kalman filter gain matrix for the EKF, the system is linearized about the current state-estimate and current control signal. Only the first-order terms in the Taylor series expansion of the nonlinear plant function are retained. The extended Kalman gain matrix in the innovations signal channel is then computed on-line, rather like a time-varying Kalman filter. The EKF gain computations are intuitively reasonable but they involve an approximation and there is not a formal proof to show they minimize the estimation error in general cases. To accommodate the nonlinearity, the Kalman gain for the EKF is computed at each time step by first computing the Jacobian to obtain a linearized model. This model is used to update the estimated covariance matrix and the Kalman filter gain matrix. The system nonlinearity is taken into account in a way that is intuitively justifiable which is useful to engineers when deploying the estimator. It is probably the most popular approach used in nonlinear control applications, ranging from feedback control to sensor fusion. The noise covariance matrix elements are often used to tune the filter and it is usually very reliable for systems that are not too nonlinear. It is not very reliable for cases where the EKF is used for simultaneous state and parameter estimation [6]. The EKF algorithm obtained below can be motivated by the Kalman filter algorithm given in the previous section. The plant model involves a differentiable nonlinear function for the states f ð:Þ and the observations hð:Þ, and is of the form:

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Introduction

601

Process model: xðt þ 1Þ ¼ f ðxðtÞ; uðt  kÞÞ þ DðtÞfðtÞ Observations model: zðtÞ ¼ hðxðtÞÞ þ vðtÞ The process and measurement of white noise signals fðtÞ and vðtÞ are not assumed zero-mean in this case. The means are denoted by fðtÞ and vðtÞ, and the covariance matrices are denoted by QðtÞ and RðtÞ, respectively. Extended Kalman filter algorithm in predictor–corrector form Predictor update: ^xðt þ 1jtÞ ¼ f ðxðtÞ; uðt  kÞÞ þ DðtÞfðtÞ State-estimate update: ^xðt þ 1jt þ 1Þ ¼ ^xðt þ 1jtÞ þ Kðt þ 1Þ~yðt þ 1jtÞ Innovations signal: ~yðt þ 1jtÞ ¼ zðt þ 1Þ  hð^xðt þ 1jtÞÞ  vðtÞ A priori covariance: Pðt þ 1jtÞ ¼ AðtÞPðtjtÞAT ðtÞ þ DðtÞQðtÞDT ðtÞ Innovations covariance: Sðt þ 1Þ ¼ Cðt þ 1ÞPðt þ 1jtÞC T ðt þ 1Þ þ Rðt þ 1Þ Kalman gain matrix: Kðt þ 1Þ ¼ Pðt þ 1jtÞCT ðt þ 1ÞSðt þ 1Þ1 A posteriori covariance: Pðt þ 1jt þ 1Þ ¼ ðI  Kðt þ 1ÞCðt þ 1ÞÞPðt þ 1jtÞ Initial conditions: xð0j0Þ ¼ m0

and Pð0j0Þ ¼ P0

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Bias terms: fðtÞ ¼ E ffðtÞg and vðtÞ ¼ EfvðtÞg where the system or state-transition matrix AðtÞ and the observations matrix CðtÞ are defined by the Jacobians:  @f  AðtÞ ¼  @x ^xðtjt1Þ

and

 @h CðtÞ ¼  @x ^xðtjt1Þ

The flow diagram illustrating the computations for the EKF is shown in Fig. 13.1. Limitations of the EKF: The EKF can be valuable in some applications but it has limitations: • The gain matrix cannot be computed offline as in the Kalman filter for linear time-invariant systems. • The gain calculation, based on linearization, involves a logical procedure but not one that can be proven from first principles. • The approximation involved in the first-order truncation of the Taylor series can make the EKF difficult to tune (via the noise covariance matrices), and they can lead to divergence when there are significant nonlinearities. • The plant model must be differentiable or can be approximated by a model that is differentiable. • For complex systems, it may be difficult to obtain the Jacobians if they are to be obtained analytically. If the Jacobians are to be computed numerically, the computational cost may be very large. Relationship between EKF and Kalman filter: The state estimates computed using the Kalman filter and the EKF are both affected by the estimation error covariance matrix. However, the covariance estimate in the Kalman filter is not

Kalman Gain Matrix Computation xˆ0 , P0

S (t  1)  C  t  1 P  t  1 t  C T  t  1  R  t  1

K  t  1  P  t  1 t  C T  t  1 S (t  1) 1

Prediction stage

xˆ  t  1 t   f  x  t  , u  t  k    D  t    t  P  t  1 t   A  t  P  t t  AT  t 

 D  t  Q  t  DT  t 

State estimation stage

xˆ  t  1| t  1  xˆ  t  1| t   K  t  1 y (t  1 t ) y (t  1 t )  z  t  1  h  xˆ  t  1| t    v  t  P  t  1| t  1   I  K  t  1 C  t  1  P  t  1| t 

Fig. 13.1 Extended Kalman filter algorithm for a discrete-time system

13.1

Introduction

603

affected by the state estimates or the observations. In fact the estimation error covariance matrix may be computed before receiving any observed data. This is different from the EKF, where the covariance matrix estimate depends upon both the state estimate and the observations. The linearized transformations in the EKF are reliable if the error propagation can be approximated with reasonable accuracy by a linear function. The estimates will diverge when the approximation is poor. The Kalman filter in applications is normally very reliable if the system model is reasonably accurate. This is not the case for the EKF that is much more sensitive to the type of problem. Related estimators: It may be possible to simplify the EKF by fixing its gain matrix and this can help in the stability and robustness analysis [7, 8]. A second-order EKF can also be proposed, which uses the Hessian matrix (second term of the Taylor series). This may reduce the possibility of the estimates diverging. However, the estimation accuracy may not improve significantly unless the process and observation functions are quadratic. The increased complexity may also prohibit its use. The problems with the EKF resulted in the development of a set of filtering methods that offered improved accuracy and consistency of the estimates. These employ deterministic sampling techniques that avoid the need to calculate the derivatives analytically (as for the Jacobians in the EKF algorithm). A deterministic sampling approach is employed using algorithms referred to as Sigma-Point Kalman Filters. Rather than linearizing a nonlinear function using a truncated Taylor-series expansion (evaluated at a single point equal to the mean value of the random variable), the function can be linearized using a weighted statistical linear regression. This requires points drawn from the prior distribution of the random variable, and the true nonlinear function evaluated at these points. This statistical approximation should reduce the errors in linearization that arise in the EKF. The Unscented Kalman Filter (UKF) is a so-called particle filter, where the sample points are chosen using a “sigma point approach.” These are discussed briefly below. The UKF may be considered a derivative-free approach to extended Kalman filtering, which provides improved performance for a similar computational complexity.

13.1.3 Particle Filters Monte Carlo methods are a class of numerical algorithms that involve repeated random sampling to compute results. They employ randomness to solve problems that might be deterministic but are difficult to solve by more analytical techniques. Monte Carlo methods have existed since the 1950s [9], but their development was limited by the computing power available. The particle filtering approach is a particular Monte Carlo technique for performing inference in state-space models, where the state of the system is changing with time and noisy measurements are available.

604

13

Nonlinear Estimation and Condition Monitoring: …

Recursive Bayesian estimation is a probabilistic approach to estimation for estimating the probability density function (pdf) recursively using a process model and measurements. The pdf includes all available statistical information that can provide a complete solution to the estimation problem. However, if the integrals required for a Bayesian recursive filter cannot be solved analytically, a possibility is to represent the posterior probability density function by a set of randomly chosen weighted samples. The solution to the Bayesian filtering problem may then be obtained by estimating the posterior density of the state using a set of weighted particles. The particle filtering approach was introduced in the 1990s [10] and is now a well-established method for use with discrete-time state-equation models, and is often the preferred filtering approach for highly nonlinear and non-Gaussian problems. A particle filter is a sequential Monte Carlo Bayesian estimator that can provide an approximate solution to discrete-time recursive filtering problems by updating an approximate description of the posterior density function. It uses a discrete approximation of the continuous probability density function and recursive computation (approximating the probability distribution using discrete random measures). The “particle” or point mass weights are the probability masses computed using Bayesian theory. The approach may be applied even when the posterior density has a multimodal shape or when the distribution of the noise is non-Gaussian. The main drawback with the particle filtering is that it is computationally demanding. It involves an approximation and increasing the number of particles can not only improve accuracy but also increase computations. Nevertheless, the particle filters or sequential Monte Carlo filtering methods are promising candidates for a wide range of applications including signal processing, computer vision, mobile robots, target tracking, neuroscience, biomedical and biochemical networks [11–14].

13.1.4 Unscented Kalman Filter The main problem in the use of the EKF lies in the approximations that are involved in its derivation, particularly the use of linearization [15, 16]. An alternative approach to obtain a nonlinear version of a Kalman filter (that also involves an approximation) is to use an unscented transformation (UT). This can be used to propagate the mean and covariance through nonlinear transformations, and gives rise to the so-called Unscented Kalman Filter (UKF). The unscented transformation is a technique for calculating the statistics of a random variable that is subject to a nonlinear transformation. It is used in this type of application because it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function. The Unscented Kalman Filter (UKF) belongs to a larger class of filters, which use statistical linearization techniques. It has properties that lie between the low

13.1

Introduction

605

computational effort of the Kalman filter and the high performance of the particle filter. The particle filter is related to a UKF in the sense that it transforms a set of points via known nonlinear equations and combines the results to estimate the mean and covariance of the state. The number of points needed in a particle filter is usually much larger than those required for the UKF. It has been shown that the estimation error in a UKF does not converge to zero in any sense, whereas the estimation error in a particle filter converges to zero as the number of particles approaches infinity [17]. Particle filtering algorithms can, therefore, provide guaranteed convergence and will outperform suboptimal estimators if enough samples are computed at each time step. However, filters based on linearization are still popular in the control community. The UKF uses the maps in the state equations f(.) and h(.) directly and is derivative-free, which simplifies implementation. The UKF has the benefit that it involves the same order of calculations as the EKF needs for the linearization computations. The particle filter and the Unscented Kalman Filter provide useful options but are not as accessible as the EKF. That is, the mathematical techniques may not be so familiar to process engineers, whereas the EKF can be explained by treating it as a rather obvious extension of the Kalman filter. The nonlinear estimation method described in the remainder of the chapter is the state-equation-related version of the filter described in the previous chapter (Chap. 12), which is quite different from those mentioned above. It has the benefit of being simple to understand and implement. It also considers a rather special problem where there are nonlinear channel dynamics between the signal source and the measurements.

13.1.5 Nonlinear Minimum Variance Filters A state-equation and nonlinear operator-based approach to Nonlinear Minimum Variance (NMV) estimation is now introduced for discrete-time multi-channel systems that can be compared with the polynomial solution in the previous chapter. This is a deconvolution or inferential estimation problem, where a signal enters a communication channel that contains both nonlinearities and transport delays. The measurements are assumed to be corrupted by a coloured noise signal that is correlated with the signal to be estimated both at the inputs and the outputs of the channel. The communications channel may include either static or dynamic nonlinearities represented in a general nonlinear operator form. The main difference with the NMV estimation results in the previous chapter is that the signal, noise, disturbance and input channel models are all in a state-space, rather than a polynomial system, model form. The solution of the NMV estimation problem is again obtained using a least squares optimization argument. The optimal nonlinear estimator is derived in terms of the state equations and the nonlinear operator that describes the system. The algorithm is relatively simple to derive and to implement in a recursive form.

606

13

Nonlinear Estimation and Condition Monitoring: …

The main advantage of the approach is the simplicity of the nonlinear estimation theory and the straightforward structure of the resulting solution. As in the Nonlinear Generalized Minimum Variance (NGMV) optimal control approach, the problem is restricted in generality so that a simple nonlinear estimation algorithm is derived. The results in this chapter parallel the NGMV state-space optimal control approach in Chap. 8 and in Grimble [18] and Grimble and Majecki [19]. As in the NGMV control problem, the solution requires an assumption that a particular nonlinear operator has a stable inverse. This operator in the filtering problem depends upon the nonlinear channel interference noise or “uncertainty tuning function,” which is included for design purposes. The uncertainty and noise model is in a parallel communications path that contains the channel uncertainty model F c 0 ðz1 Þ. This is chosen by the designer and plays a similar role to the control weighting function in the NGMV control design problem.

13.2

Nonlinear Multi-channel Estimation Problem

The nonlinear filtering problem involves the estimation of a signal, which enters a communication channel including nonlinearities and transport-delay elements [20]. The measurements are assumed to be corrupted by noise, which is correlated with the signal to be estimated. The signal and noise models are assumed to have linear time-invariant state-space representations. The system is shown in Fig. 13.2 and includes the nonlinear signal channel model and the linear signal and measurement noise models. The message or signal to be estimated is at the output of a linear block denoted by s ¼ Wc y. A cost-function weighting is introduced to provide design flexibility and the weighted signal sq ¼ Wq s is to be estimated. It is assumed that the zero-mean white noise innovations signal eðtÞ that feeds the system as shown in Fig. 13.2 has an identity covariance matrix [21]. The system description for the ðr  rÞ linear multivariable signal and noise models may now be introduced. The input signal generation model is denoted by Ws and the coloured noise model by Wn , and these are functions of the unit-delay operator z1 .

13.2.1 Operator Forms of Nonlinear Channel Models The signal goes through a channel which is assumed to be stable, and the nonlinear signal channel dynamics are represented as W c ðz1 Þ ¼ W c1 ðz1 ÞzK0

ð13:1Þ

s  Wc y

Message signal

n

Coloured Noise n



Wn ( z 1 )

+

Signal y +

Wc ( z 1 )

Ws ( z 1 )

Signal source

f

z  0 Channel delays

Wc 0 ( z 1 ) Linear sub-system for main channel

s0

z

 0

sd

nd

1

1

)

)

Nonlinear main channel

c1 ( z

c0 ( z

sc

+

+

nc

z

( z 1 ) Nonlinear estimator

f

sˆ

Estimated signal

Fig. 13.2 Canonical nonlinear filtering problem with noise sources and channel interference (weighted message sq ¼ Wq s and channel signal s0 ¼ Wc0 f )



Nonlinear noise design channel

13.2 Nonlinear Multi-channel Estimation Problem 607

608

13

Nonlinear Estimation and Condition Monitoring: …

where zK0 denotes a diagonal matrix of the k-step delay elements in the signal paths and K0 ¼ kI. The uncertainties due to noise and interference may be modelled by the parallel nonlinear channel dynamics and time delays. After separating the communications channel into a linear input subsystem and a nonlinear output model, the model for the channel output (without noise and uncertainty) may be defined as follows:     sc ðtÞ ¼ W c1 zk s0 ðtÞ ¼ W c1 zk Wc0 f ðtÞ

ð13:2Þ

The output of the linear subsystem W c0 , with any explicit channel delays, is denoted as s0 ðtÞ ¼ ðWc0 f ÞðtÞ

ð13:3Þ

and s0 ðt  kÞ ¼ ðzk s0 ÞðtÞ denotes the signal delayed by different amounts in different channels. The nonlinear subsystem W c1 is assumed to be finite-gain stable. As in the previous chapter, the uncertainty tuning function or shaping operator is defined to be stable and have the form: F c ðz1 Þ ¼ F c 0 ðz1 Þzk

ð13:4Þ

The parallel path dynamics in Fig. 13.2 are shown using a dotted line to emphasize it is included as an additional tuning model to limit uncertainties and noise. It provides a signal that enters the cost-function and enables the noise attenuation characteristics to be frequency shaped.

13.2.2 Signals in the Signal Processing System The optimal nonlinear estimation problem is illustrated in Fig. 13.2. The input and noise generating processes have an innovations signal model with a zero-mean white noise signal eðtÞ 2 Rr and covariance matrix cov½eðtÞ; eðsÞ ¼ Idts , where dts denotes the Kronecker delta function. The signals in the system may be listed as follows: Noise: nðtÞ ¼ Wn eðtÞ

ð13:5Þ

yðtÞ ¼ Ws eðtÞ

ð13:6Þ

Input signal:

13.2

Nonlinear Multi-channel Estimation Problem

609

Channel input: f ðtÞ ¼ yðtÞ þ nðtÞ

ð13:7Þ

s0 ðtÞ ¼ ðWc0 f ÞðtÞ

ð13:8Þ

Linear channel subsystem:

Weighted channel interference: nc ðtÞ ¼ ðF c eÞðtÞ

ð13:9Þ

Nonlinear channel subsystem: sc ðtÞ ¼ ðW c1 sd ÞðtÞ

ð13:10Þ

sd ðtÞ ¼ zk s0 ðtÞ ¼ s0 ðt  kÞ

ð13:11Þ

zðtÞ ¼ nc ðtÞ þ sc ðtÞ

ð13:12Þ

Nonlinear channel input:

Observations signal:

Message signal to be estimated: sðtÞ ¼ Wc yðtÞ ¼ Wc Ws eðtÞ

ð13:13Þ

Weighted message signal: sq ðtÞ ¼ Wq Wc yðtÞ

ð13:14Þ

~sðtjtÞ ¼ sðtÞ  ^sðtjtÞ

ð13:15Þ

Estimation error signal:

The time sequences for stability analysis can be considered to be contained in extensions of the discrete Marcinkiewicz space m2 ðR þ ; Rn Þ, [8]. The infinite-time estimation problem, where the filter is assumed to be in operation from an initial time t0 ! 1, is considered. The noise and signal models are correlated. However, Ws is normally low-pass and Wn is normally high-pass and the results for the special case of the linear problem, with no channel dynamics, are similar to those for the Wiener or Kalman filtering problems [1, 4]. The coloured noise model may represent output disturbances such as eccentricity models in steel rolling mill processes [22].

610

13

Nonlinear Estimation and Condition Monitoring: …

13.2.3 Signal Analysis and Noise Models The linear subsystems, illustrated in Fig. 13.3, are assumed to be represented in state-equation form and to be observable, or at least detectable. The observations signal, representing the output of the communications channel, has the form: zðtÞ ¼ nc ðtÞ þ sc ðtÞ ¼ ðF c eÞðtÞ þ ðW c1 sd ÞðtÞ ¼ ðF c eÞðtÞ þ ðW c1 s0 Þðt  kÞ The signal and noise entering the communications channel: f ðtÞ ¼ yðtÞ þ nðtÞ ¼ Cs xs ðtÞ þ Cn xn ðtÞ þ En eðtÞ

ð13:16Þ

The state-space matrix form of the system models may be listed as follows: Signal generator: xs ðt þ 1Þ ¼ As xs ðtÞ þ Ds eðtÞ yðtÞ ¼ Cs xs ðtÞ and let Ws ðz1 Þ ¼ Cs ðzI  As Þ1 Ds

ð13:17Þ

Coloured noise: xn ðt þ 1Þ ¼ An xn ðtÞ þ Dn eðtÞ nðtÞ ¼ Cn xn ðtÞ þ En eðtÞ

and Wn ðz1 Þ ¼ Cn ðzI  An Þ1 Dn þ En

ð13:18Þ

Linear part channel: x0 ðt þ 1Þ ¼ A0 x0 ðtÞ þ B0 f ðtÞ ¼ A0 x0 ðtÞ þ B0 Cs xs ðtÞ þ B0 Cn xn ðtÞ þ B0 En eðtÞ s0 ðtÞ ¼ C0 x0 ðtÞ þ E0 f ðtÞ ¼ C0 x0 ðtÞ þ E0 Cs xs ðtÞ þ E0 Cn xn ðtÞ þ E0 En eðtÞ ð13:19Þ and Wc0 ðz1 Þ ¼ C0 ðzI  A0 Þ1 D0 þ E0

ð13:20Þ

Inferential signal path: xc ðt þ 1Þ ¼ Ac xc ðtÞ þ Bc yðtÞ ¼ Ac xc ðtÞ þ Bc Cs xs ðtÞ sðtÞ ¼ Cc xc ðtÞ þ Ec yðtÞ ¼ Cc xc ðtÞ þ Ec Cs xs ðtÞ

ð13:21Þ

Ds

+

An

z 1

+

+

As

z 1

xn

xs

Linear message dynamics

Dn

Noise model

+

Cs

En

Cn

y

yn

+

+

+

+

n f +

+

A0

z 1

+

Ac

z 1

Inferential model dynamics +

Bc

B0

Linear channel dynamics

xc

x0

Ec

Cc

E0

C0

Fig. 13.3 Linear message and noise models and linear and nonlinear channel dynamics





+

+

+

+ Delays

z 0

s

Unmeasured signal

s0 sd

NL channel dynamics

c1(.,.)

+

+

z

Observations

sc

nc

( z 1 ) Nonlinear estimator

f

sˆ

Estimated signal

13.2 Nonlinear Multi-channel Estimation Problem 611

612

13

Nonlinear Estimation and Condition Monitoring: …

and Wc ðz1 Þ ¼ Cc ðzI  Ac Þ1 Dc þ Ec

ð13:22Þ

Weighted message: xq ðt þ 1Þ ¼ Aq xq ðtÞ þ Bq sðtÞ ¼ Aq xq ðtÞ þ Bq Cc xc ðtÞ þ Bq Ec Cs xs ðtÞ sq ðtÞ ¼ Cq xq ðtÞ þ Eq sðtÞ ¼ Cq xq ðtÞ þ Eq Cc xc ðtÞ þ Eq Ec Cs xs ðtÞ

ð13:23Þ

where Wq ðz1 Þ ¼ Cq ðzI  Aq Þ1 Dq þ Eq

ð13:24Þ

13.2.4 Augmented State Model To obtain a total linear subsystem model, let the state vector of the linear subsystems that contribute to the weighted error be denoted as h iT x ¼ xTs xTn xTc xTq . Then the above equations may be expressed in a combined state-equation form: Augmented state equation: xðt þ 1Þ ¼ A xðtÞ þ D eðtÞ

ð13:25Þ

f ðtÞ ¼ Cf xðtÞ þ Ef eðtÞ

ð13:26Þ

sq ðtÞ ¼ CxðtÞ þ EeðtÞ

ð13:27Þ

Signal plus noise:

Weighted message output:

Also, define the total resolvent matrix as Uðz1 Þ ¼ ðzI  AÞ1 and Wf ðz1 Þ ¼ Cf Uðz1 ÞD. The augmented system matrices follow from the above subsystem equations as 2

As 6 0 A¼6 4 Bc Cs Bq Ec Cs

0 An 0 0

0 0 Ac Bq Cc

3 2 3 0 Ds 6 Dn 7 0 7 7 and D ¼ 6 7 4 0 5 0 5 Aq 0

13.2

Nonlinear Multi-channel Estimation Problem

C ¼ ½ Eq Ec Cs Cf ¼ ½ Cs

13.3

0 Cn

Eq Cc 0

0

613

Cq  and

and E ¼ 0 Ef ¼ En

ð13:28Þ

Discrete-Time Kalman Filter

The results for the Kalman filter are summarized in Lemma 13.1, for the case where the process and measurement noise terms (fðtÞ and vðtÞ) are correlated. The following results on Kalman filtering and spectral-factorization are required for later use and depend upon the linear model subsystems defined above. Note that it is assumed the noise covariances and system ensure a strictly Schur spectral-factor exists or equivalently that the filter is asymptotically stable [23, 24]. Lemma 13.1: Computation of the Spectral-Factor from State-Equation Models If the signal f ðtÞ is measured, the Kalman filter may be computed using the results that follow. Consider an innovations signal model for the system, where the covariance matrix for the noise eðtÞ is given as cov½eðtÞ; eðrÞ ¼ Qdtr and   DQDT DQEfT  0: The Kalman one-step ahead predictor: Ef QDT Ef QEfT   ^xðt þ 1jtÞ ¼ A^xðtjt  1Þ þ Kf f ðtÞ  ^f ðtjt  1Þ ^f ðt þ 1jtÞ ¼ Cf ^xðt þ 1jtÞ

ð13:29Þ

The Kalman filter gain matrix in the estimator equation: Kf ¼ ðAPCfT þ DQEfT ÞR1 f where the a priori covariance P matrix satisfies the Riccati equation: P ¼ APAT þ DQDT  Kf Rf KfT and Rf ¼ ðEf QEfT þ Cf PCfT Þ

ð13:30Þ

The spectral-factor of the combined channel input and noise satisfies Yf ðz1 ÞYfT ðzÞ ¼ ðWs ðz1 Þ þ Wn ðz1 ÞÞQðWsT ðzÞ þ WnT ðzÞÞ This may be computed directly from the state models as

ð13:31Þ

614

13

Nonlinear Estimation and Condition Monitoring: …

Yf ðz1 Þ ¼ ðIr þ Cf Uðz1 ÞKf ÞRf

1=2

ð13:32Þ

A realization of the signal f(t) may, therefore, be obtained as f ðtÞ ¼ Yf ðz1 ÞeðtÞ. Proof The proof of this lemma follows in the next section.

13.3.1 Return Difference and Spectral-Factorization Some of the results for the linear Kalman filtering problem will first be reviewed and this will provide the basis for deriving the useful filter return-difference and spectral-factorization relationships. The system includes constant coefficient state equations and is assumed to be stabilizable and detectable. The LTI system equations that proceed the nonlinear channel dynamics block are assumed to be of the form: xðt þ 1Þ ¼ AxðtÞ þ DeðtÞ

ð13:33Þ

yðtÞ ¼ Cf xðtÞ

ð13:34Þ

f ðtÞ ¼ Cf xðtÞ þ Ef vðtÞ

ð13:35Þ

and

where xðtÞ 2 Rn ; eðtÞ 2 Rq and f ðtÞ 2 Rr . The process noise signal eðtÞ and the measurement noise sequence v(t) are assumed to be zero-mean, white and stationary with the following noise covariances: 9   cov½eðtÞ; eðrÞ ¼ Qdtr = DGEfT DQDT cov½vðtÞ; vðrÞ ¼ Rdtr where Q  0 and R [ 0; and 0 Ef GT DT Ef REfT ; cov½eðtÞ; vðrÞ ¼ Gdtr The dtr denotes the Kronecker delta function and the resolvent and transfer-function matrices are defined as Uðz1 Þ ¼ ðzI  AÞ1 and W0 ðz1 Þ ¼ Cf Uðz1 ÞD, respectively. For the current estimation problem vðtÞ ¼ eðtÞ and R = G = Q. In this case, the state and the output equations: xðtÞ ¼ ðzI  AÞ1 DeðtÞ ¼ Uðz1 ÞDeðtÞ and

13.3

Discrete-Time Kalman Filter

  f ðtÞ ¼ Cf Uðz1 ÞD þ Ef eðtÞ

615

ð13:36Þ

This is a correlated signal and noise problem. The Kalman filtering solution is summarized in the above lemma when a single-step delay is involved in computing the estimates. This is sometimes referred to as the single-stage predictor problem. For a stable solution of the linear filtering problem and for stability in limiting cases of the nonlinear problem the transfer in (13.36) must be minimum phase. To ensure this minimum phase condition holds a realization of the signal f(t) is obtained using the spectral-factorization result (13.32), as described below. Spectral-factorization: The estimator derived in this chapter should only use state-equation-based calculations but a spectral-factorization calculation is needed in the derivation. Spectral-factorization is normally performed using polynomial matrix-based calculations but can be achieved by exploiting the relationship between spectral-factorization and the return-difference matrix for the Kalman filter. This is now illustrated for the Kalman single-stage predictor defined in the above lemma [25]. The solution of the spectral-factorization problem, in terms of the state matrices, follows below and the state-space version of the NMV estimator is then derived in the next section (Sect. 13.4). First, let P be assumed to satisfy the steady-state Riccati equation: P ¼ APAT þ DQDT  Kf Rf KfT Manipulating this Riccati equation by adding and subtracting the same terms we obtain DQDT ¼ P  APAT þ Kf Rf KfT þ zðPAT  PAT Þ þ z1 ðAP  APÞ ¼ ðzI  AÞPðz1 I  AT Þ þ APðz1 I  AT Þ þ ðzI  AÞPAT þ Kf Rf KfT where z1 denotes the inverse of the z-transform complex number in this section. Pre- and postmultiplying both sides of this equation we obtain Cf Uðz1 ÞDQDT UT ðzÞCfT ¼ Cf PCfT þ Cf PAT UðzÞT CfT þ Cf Uðz1 ÞAPCfT þ Cf Uðz1 ÞKf Rf KfT UðzÞT CfT ð13:37Þ The Kalman filter gain expression may be written as Kf Rf ¼ APCfT þ DQEfT and hence (13.37) becomes Cf Uðz1 ÞDQDT UT ðzÞCfT ¼ Cf PCfT þ ðKf Rf  DQEfT ÞT UðzÞT CfT

þ Cf Uðz1 Þ Kf Rf  DQEfT þ Cf Uðz1 ÞKf Rf KfT UðzÞT CfT

616

13

Nonlinear Estimation and Condition Monitoring: …

or reordering Cf Uðz1 ÞDQDT UT ðzÞCfT þ Ef QDT UðzÞT CfT þ Cf Uðz1 ÞDQEfT ¼ Cf PCfT þ ðKf Rf ÞT UðzÞT CfT þ Cf Uðz1 ÞKf Rf þ Cf Uðz1 ÞKf Rf KfT UðzÞT CfT ð13:38Þ Observe from (13.36) that the combined signal and noise source model: Wf ¼ Cf Uðz1 ÞD þ Ef ¼ W0 ðz1 Þ þ Ef has the spectrum: Uff ðz1 Þ ¼ W0 ðz1 ÞQW0T ðzÞ þ Ef QEfT þ Ef QW0 ðzÞT þ W0 ðz1 ÞQEfT Adding the term Ef QEfT to both sides of (13.38) and noting this last result and (13.30), Uff ðz1 Þ ¼ Ef QEfT þ Cf PCfT þ ðKf Rf ÞT UðzÞT CfT þ Cf Uðz1 ÞKf Rf þ Cf Uðz1 ÞKf Rf KfT UðzÞT CfT T T T ¼ Rf þ ðKf Rf Þ UðzÞ Cf þ Cf Uðz1 ÞKf Rf þ Cf Uðz1 ÞKf Rf KfT UðzÞT CfT ð13:39Þ This may be related to the return-difference matrix for the Kalman filter loop that is defined as Fðz1 Þ ¼ Ir þ Cf Uðz1 ÞKf

ð13:40Þ

The desired return-difference matrix and spectral density result follows by first noting Fðz1 ÞRf F T ðzÞ ¼ Uff ðz1 Þ ¼ ðIr þ Cf Uðz1 ÞKf ÞRf ðIr þ KfT UðzÞT CfT Þ ¼ Rf þ Rf KfT UðzÞT CfT þ Cf Uðz1 ÞKf Rf þ Cf Uðz1 ÞKf Rf KfT UðzÞT CfT Then from (13.39) the return-difference matrix can be related to the spectral factor Yf ðz1 Þ, since ¼ W0 ðz

1

Fðz1 ÞRf F T ðzÞ ¼ Uff ðz1 Þ T ÞQW0 ðzÞ þ Ef QEfT þ Ef QW0 ðzÞT

þ W0 ðz1 ÞQEfT

ð13:41Þ

13.3

Discrete-Time Kalman Filter

617

and the filter spectral-factor may, therefore, be computed as Yf ðz1 Þ ¼ Fðz1 ÞRf

1=2

¼ ðIr þ Cf Uðz1 ÞKf ÞRf

1=2

ð13:42Þ

A realization of the signal f(t) may now be obtained as f ðtÞ ¼ Yf ðz1 ÞeðtÞ

ð13:43Þ

The signal f ðtÞ is not measured and the Kalman single-step predictor defined above is not actually to be implemented for state estimation. The relationships of the Kalman predictor are needed in the following analysis.

13.4

Nonlinear Channel Estimation Problem

The filtering or prediction problem is concerned with finding the best estimate of the signal sðtÞ, for the system in Fig. 13.3, in the presence of an interference noise term. The minimum variance optimal deconvolution problem involves the minimization of the estimation error: ~sðtjt  ‘Þ ¼ sðtÞ  ^sðtjt  ‘Þ

ð13:44Þ

where ^sðtj t  ‘Þ denotes the optimal linear estimate of the signal s(t) at time t, given the observations zðsÞ over s 2 ð1 ; t  ‘ up to time ðt  ‘Þ. As in the previous chapter, the scalar ‘ may be defined according to the type of problem. This scalar is null for Filtering ð‘ ¼ 0Þ, positive for Prediction ‘ [ 0 and negative for Smoothing ‘\ 0 [26, 27]. Cost-function: The weighted estimation error cost-function to be minimized has the form: J ¼ tracefEfWq~sðtjt  ‘ÞðWq~sðtjt  ‘ÞÞT gg ¼ tracefEf~sq ðtjt  ‘Þ~sTq ðtjt  ‘Þgg

ð13:45Þ

where Ef:g denotes the expectation, sq ¼ Wq s, ~sq ¼ Wq~s, and Wq denotes a linear dynamic cost-function weighting matrix. This weighting matrix is assumed to be strictly minimum-phase, square and invertible. It may be used to penalize the low-frequency estimation errors if the gain is increased at low frequencies [28]. The signal estimate ^sðt j t  ‘Þ is assumed to be generated from a nonlinear estimator of the form: Estimator equation: ^sðtjt  ‘Þ ¼ Hf ðt; z1 Þzðt  ‘Þ

ð13:46Þ

618

13

Nonlinear Estimation and Condition Monitoring: …

where Hf ðt; z1 Þ denotes a minimal realization of the optimal nonlinear estimator. No initial condition term needs to be specified since an infinite-time estimation problem is of interest ðs0 ! 1Þ.

13.4.1 Solution of the Nonlinear Estimation Problem An expression for the weighted estimation error ~sq ðtjt  ‘Þ ¼ Wq ~sðtjt  ‘Þ is required. Recall the weighted message signal is given from (13.14) as sq ðtÞ ¼ Wq Wc Ws eðtÞ

ð13:47Þ

From Eq. (13.47), we obtain ~sq ðtjt  ‘Þ ¼ sq ðtÞ  ^sq ðtjt  ‘Þ ¼ Wq Wc Ws eðtÞ  Wq Hf zðt  ‘Þ

ð13:48Þ

Also from (13.12) the observations z ¼ nc þ sc . Thus, substituting in Eq. (13.48): ~sq ðtjt  ‘Þ ¼ Wq Wc Ws eðtÞ  Wq Hf ðnc ðt  ‘Þ þ sc ðt  ‘ÞÞ From (13.7), (13.9), (13.10) and (13.11), noting sc ðtÞ ¼ ðW c1 sd ÞðtÞ and nc ðtÞ ¼ ðF c eÞðtÞ we obtain ~sq ðtjt  ‘Þ ¼ Wq Wc Ws eðtÞ  Wq Hf ððF c eÞðt  ‘Þ þ ðW c1 sd Þðt  ‘ÞÞ ¼ Wq Wc Ws eðtÞ  Wq Hf ððF c0 eÞðt  k  ‘Þ þ ðW c1 Wc0 f Þðt  k  ‘ÞÞ

ð13:49Þ

The desired expression for the estimation error follows as ~sq ðt þ k þ ‘jt þ kÞ ¼ Wq Wc Ws eðt þ k þ ‘Þ  Wq Hf ððF c0 eÞðtÞ þ ðW c1 Wc0 f ÞðtÞÞ

ð13:50Þ

13.4.2 Prediction Equation As in the previous chapter, there is not a requirement to specify the distribution of the noise sources since the structure of the system leads to a prediction equation, which is only dependent upon the linear models. For the estimator solution the first term on the right of (13.50) must be split into terms which depend upon the future, and terms which depend upon the past white noise signal eðtÞ. The prediction equation is first required. The future values of the states and outputs follow from repeated use of (13.25) and (13.27), as follows:

13.4

Nonlinear Channel Estimation Problem

619

xðt þ 2Þ ¼ AðAxðtÞ þ DeðtÞÞ þ Deðt þ 1Þ ¼ A2 xðtÞ þ ADeðtÞ þ Deðt þ 1Þ sq ðt þ 1Þ ¼ Cxðt þ 1Þ þ Eeðt þ 1Þ ¼ CAxðtÞ þ CDeðtÞ þ Eeðt þ 1Þ Similarly, xðt þ 3Þ ¼ Axðt þ 2Þ þ Deðt þ 2Þ ¼ A3 xðtÞ þ A2 DeðtÞ þ ADeðt þ 1Þ þ Deðt þ 2Þ sq ðt þ 2Þ ¼ Cxðt þ 2Þ þ Eeðt þ 2Þ ¼ CA2 xðtÞ þ CADeðtÞ þ CDeðt þ 1Þ þ Eeðt þ 2Þ Generalzing this result, we obtain the state at the future time t þ i, as xðt þ iÞ ¼ Ai xðtÞ þ

i X

Aij Deðt þ j  1Þ

j¼1

sq ðt þ iÞ ¼ CAi xðtÞ þ

i X

CAij Deðt þ j  1Þ þ Eeðt þ iÞ

ð13:51Þ

j¼1

Thence, we obtain xðt þ k þ ‘Þ ¼ Ak þ ‘ xðtÞ þ

kþ‘ X

Ak þ ‘j Deðt þ j  1Þ

j¼1

¼ Ak þ ‘ xðtÞ þ Ak þ ‘1 DeðtÞ þ Ak þ ‘2 Deðt þ 1Þ þ . . . þ Deðt þ k þ ‘  1Þ

ð13:52Þ Similarly, sq ðt þ k þ ‘Þ ¼ Cxðt þ k þ ‘Þ þ Eeðt þ k þ ‘Þ ¼ CAk þ ‘ xðtÞ þ CAk þ ‘1 DeðtÞ þ CAk þ ‘2 Deðt þ 1Þ þ       þ CDeðt þ k þ ‘  1Þ þ Eeðt þ k þ ‘Þ or sq ðt þ k þ ‘Þ ¼ CAk þ ‘ xðtÞ þ

kþ‘ X j¼1

CAk þ ‘j Deðt þ j  1Þ þ Eeðt þ k þ ‘Þ ð13:53Þ

620

13

Nonlinear Estimation and Condition Monitoring: …

13.4.3 Separation into Future and Past Terms Returning to the solution procedure, an expression for the weighted message output can be obtained. First recall the weighted message output was defined as sq ðt þ k þ ‘Þ ¼ Wq Wc Ws eðt þ k þ ‘Þ where k þ ‘  1. Using (13.53) the following results may be obtained for the system of interest, sq ðt þ k þ ‘Þ ¼ CAk þ ‘ xðtÞ þ

kþ‘ X

CAk þ ‘j Deðt þ j  1Þ þ Eeðt þ k þ ‘Þ

j¼1

  ¼ CAk þ ‘ xðtÞ þ CAk þ ‘1 DeðtÞ þ ðCAk þ ‘2 Deðt þ 1Þ    þ CDeðt þ k þ ‘  1Þ þ Eeðt þ k þ ‘ÞÞ

ð13:54Þ

The weighted signal may, therefore, be written, from (13.54), in the form: sq ðt þ k þ ‘Þ ¼ ðCAk þ ‘ UD þ CAk þ ‘1 DÞeðtÞ þ F0 eðt þ k þ ‘Þ

ð13:55Þ

The finite pulse-response estimation term F0 ðz1 Þ in this equation may be identified comparing (13.54) and (13.55) (assuming the first term is null, when k þ ‘ ¼ 1) as 1

F0 ðz Þ ¼

kþ‘ X

! k þ ‘j

CA

Dz

k‘ þ j1

þE

ð13:56Þ

j¼2

The first term in Eq. (13.55) may be simplified by noting: ðCAk þ ‘ UD þ CAk þ ‘1 DÞ ¼ CAk þ ‘1 ðAU þ IÞD ¼ CAk þ ‘1 ðI  z1 AÞ1 D It follows that the weighted message signal to be estimated, for future times k þ ‘  1, may be expressed as sq ðt þ k þ ‘Þ ¼ H0 ðz1 ÞeðtÞ þ F0 ðz1 Þeðt þ k þ ‘Þ

ð13:57Þ

where the stable filter H0 ðz1 Þ is defined as H0 ðz1 Þ ¼ CAk þ ‘1 ðI  z1 AÞ1 D

ð13:58Þ

Thus, for the following solution note the signal sq ðt þ k þ ‘Þ may be represented in terms of the signals H0 ðz1 ÞeðtÞ and F0 eðt þ k þ ‘Þ. These involve terms dependent on past and future values of the white noise signal, and are statistically

13.4

Nonlinear Channel Estimation Problem

621

independent. Also, observe that the subsystem H0 ðz1 Þ ¼ CAk þ ‘1 ðI  z1 AÞ1 D in (13.58) may be implemented in state-space form using the state-equation model: xh ðt þ 1Þ ¼ Axh ðtÞ þ DmðtÞ

ð13:59Þ

pðtÞ ¼ CAk þ ‘1 xh ðt þ 1Þ

ð13:60Þ

13.4.4 Optimization After establishing these properties, we may return to the solution of the optimal nonlinear estimation problem. The weighted estimation error in Eq. (13.50) may be advanced by t þ k þ ‘, to obtain ~sq ðt þ k þ ‘jt þ kÞ ¼ Wq Wc Ws eðt þ k þ ‘Þ  Wq Hf ððF c0 eÞðtÞ þ ðW c1 Wc0 f ÞðtÞÞ Now substitute for the result in (13.57) to obtain   ~sq ðt þ k þ ‘jt þ kÞ ¼ F0 eðt þ k þ ‘Þ þ H0 eðtÞ  Wq Hf ððF c0 eÞðtÞ þ ðW c1 Wc0 f ÞðtÞÞ ð13:61Þ As discussed above the signal f ðtÞ may be realized as f ðtÞ ¼ Yf ðz1 ÞeðtÞ, where Yf ðz1 Þ is defined to be strictly minimum phase, by using the spectral-factor relationship (13.32). This is necessary for the stability of the estimator when the uncertainty tuning function or shaping operator F c0 gain is small. Substituting in (13.61):   ~sq ðt þ k þ ‘jt þ k Þ ¼ F0 eðt þ k þ ‘Þ þ H0  Wq Hf ðF c0 þ W c1 Wc0 Yf Þ eðtÞ ð13:62Þ This equation may be written in a form which is useful when defining the block diagram of the estimator:   ~sq ðt þ k þ ‘jt þ kÞ ¼ F0 eðt þ k þ ‘Þ þ H0  Wq Hf ðI þ W c1 Wc0 Yf F 1 c0 ÞF c0 eðtÞ ð13:63Þ The case where k þ ‘  1 or ‘  1  k will be assumed. The first term in (13.62) or (13.63) is dependent upon the future values of the white noise signal components eðt þ 1Þ, eðt þ 2Þ, …, eðt þ k þ ‘Þ. The second group of terms in (13.62) or (13.63) is all dependent upon past values of the white noise signals. It follows that these two groups of terms are statistically independent and the expected values of any cross-terms are null. Also, note that the first terms on the right-hand sides of (13.62)

622

13

Nonlinear Estimation and Condition Monitoring: …

or (13.63) are independent of the choice of estimator. The smallest variance is achieved when the remaining terms are set to zero [22]. Assuming the existence of a finite-gain stable causal inverse to the nonlinear operator, the optimal estimator is obtained by setting this second group of terms to zero, giving Hf ¼ Wq1 H0 ðF c0 þ W c1 Wc0 Yf Þ1

ð13:64Þ

1 1 Hf ¼ Wq1 H0 F 1 c0 ðI þ W c1 Wc0 Yf F c0 Þ

ð13:65Þ

or

The optimal estimation error is defined by the terms that remain in (13.62) or (13.63), and may be written as ~sqmin ðt þ k þ ‘jt þ kÞ ¼ F0 eðt þ k þ ‘Þ

ð13:66Þ

The assumption was made that the operator ðF c0 þ W c1 Wc0 Yf Þ has a stable inverse. There is a free choice of the uncertainty tuning function F c0 that is often chosen to be linear, minimum phase and stable. It is also assumed that the smoothing delay satisfies ‘  1  k. The following theorem applies under these assumptions. Theorem 13.1: Optimal State Equation Based Estimator for Nonlinear Systems The nonlinear deconvolution filter ð‘ ¼ 0Þ or predictor ð‘ [ 0Þ to minimize the variance of the estimation error (13.45), for the system described in Sect. 13.2, can be calculated from spectral-factor and operator equations. The spectral-factor Yf can be computed from the Kalman filter Eq. (13.32): Yf ðz1 Þ ¼ ðIr þ Cf Uðz1 ÞKf ÞRf

1=2

The optimal casual estimate ^sðtjt  ‘Þ ¼ Hf ðt; z1 Þzðt  ‘Þ, to minimize the variance of the estimation error (13.45) is given as Hf ¼ Wq1 H0 ðF c0 þ W c1 Wc0 Yf Þ1

ð13:67Þ

where H0 ðz1 Þ ¼ CAk þ ‘1 ðI  z1 AÞ1 D. Alternatively, assuming the definition of the uncertainty tuning function F c0 has a stable causal inverse, the estimator may be computed in the form: 1 1 Hf ¼ Wq1 H0 F 1 c0 ðI þ W c1 Wc0 Yf F c0 Þ

ð13:68Þ

where the filter H0 may be implemented in state-space model form (13.59), (13.60). ■ Proof The proof follows by collecting the above results.

■

13.4

Nonlinear Channel Estimation Problem

623 Nonlinear estimator

Fig. 13.4 Nonlinear estimator structure including a model of nonlinear channel Observations z(t) + -

m(t) 1

c0

f

p(t)

H0

Wq1

Estimate sˆ(t t  )

c1 Wc 0 Y f

Transmission path model

Remarks • To consider the stability of the estimator, note that the estimation error weighting Wq was assumed to be strictly minimum phase and the operator ðF c0 þ W c1 Wc0 Yf Þ was assumed to have a finite-gain stable inverse. In this case, the estimator Hf is stable. • For systems where the signal model Ws ðz1 Þ and main channel model W c1 ðz1 Þ are low-pass and both the noise model Wn ðz1 Þ and F c 0 ðz1 Þ are high-pass, the estimator provides an approximation to the solution of the more conventional problem (where the noise and signal are statistically independent). • The expression for the estimator (13.68) gives rise to the structure shown in Fig. 13.4, which suggests the simplest way to implement the estimator. The algebraic loop that is present can be avoided for implementation in much the same ways as in Chap. 5.

13.4.5 Benchmarking the Estimator The minimum cost of the optimal estimator may easily be computed using the results below and this can be used to compare with the performance of alternative “non-optimal” estimators. Lemma 13.2: Minimum of the Cost-Function The minimum variance of the cost may be computed, noting (13.55), as 9 8 > > = < 1 I  dz ð13:69Þ J ¼ trace tracefF0 QF0 g > z> ; :2pj jzj¼1

where

624

13

1

F0 ðz Þ ¼

kþ‘ X

Nonlinear Estimation and Condition Monitoring: …

! k þ ‘j

CA

Dz

k‘ þ j1

þE

j¼2

■ Proof From (13.66) the optimal estimation error becomes ~sqmin ðt þ k þ ‘j t þ kÞ ¼ F0 eðt þ k þ ‘Þ

ð13:70Þ

where this term involves the white noise signals eðt þ k þ ‘  1 Þ; eðt þ k þ ‘  2 Þ; . . .; eðt þ 1Þ, and (13.69) follows. ■ These expressions may be used to test the optimal estimator by comparing the actual performance against the ideal. This provides a possible method of benchmarking optimal nonlinear estimators.

13.5

Design and Implementation Issues

The design and implementation of the estimator may now be considered. Notice that the performance of the estimator depends upon the choice of the uncertainty tuning function F c0 , which also determines the estimator stability (noting the presence of the inverse operator in (13.67)). The selection of the uncertainty tuning function F c0 is a related problem to the selection of the optimal control cost-function weightings described in earlier chapters.

13.5.1 Parallel Path Dynamics and Uncertainty The linear models in this system are represented in state-equation form but it is still useful to consider frequency-domain characteristics. As in the previous chapter, the parallel path dynamics do not normally represent a real physical system. This path is included to add design freedom via the specification of the frequency response characteristics of the term F c0 that can be nonlinear and include dynamics. This function enters the equations like a nonlinear signal channel that corrupts the signal from the communication channel with noise nc. It may be used to shape the frequency response to noise and uncertainty. Reference back to Fig. 13.2 reveals that when F c0 is a constant matrix the output of this channel is white noise that corrupts the observations into the estimator. The fictitious parallel path dynamics, including the uncertainty tuning function F c0 , must be chosen so that a stable inverse of the operator N c ¼ ðF c0 þ W c1 Wc0 Yf Þ can be computed. Note the channel dynamics W c1 Wc0 may not be stably invertible.

13.5

Design and Implementation Issues

625

The dynamic response of F c0 should reflect the type of uncertainty expected. If it is chosen as a simple linear system with transfer-function representation, the dc-gain and the cut-off frequency of F c0 provide tuning variables for the design of the NMV estimator. Uncertainty is of course often associated with high-frequency behaviour and hence a lead term might be used to represent the frequency response of F c0 . However, there may also be situations where the function can be defined to include a static nonlinearity to help counter nonlinearities in the signal channel. There are many applications in the process industries for such estimators [29].

13.5.2 Implementation Issues The computation of the output of the nonlinear operator mðtÞ ¼ 1 1 ðI þ F 1 c0 W c1 Wc0 Yf Þ F c0 zðtÞ can be performed easily when the uncertainty tuning function term F c0 is chosen to be linear and of the form: F c0 ðz1 Þ ¼ Fc0 ðz1 Þ ¼ F0 þ F1 ðz1 Þz1 , where F0 is a constant invertible matrix. In this case we obtain mðtÞ ¼ ðF0 þ F1 z1 þ W c1 Wc0 Yf Þ1 zðtÞ or ðF0 þ F1 z1 þ W c1 Wc0 Yf ÞmðtÞ ¼ zðtÞ so that,   mðtÞ ¼ F01 zðtÞ  F1 ðz1 Þmðt  1Þ  W c1 Wc0 Yf mðtÞ

ð13:71Þ

Since the signal mðtÞ arises on both sides of this equation, an algebraic loop is present, but if the through term in W c1 Wc0 Yf is computed, the equation can be rearranged to avoid this problem. It is clear from (13.71) that the equations for the channel model are not required, just the ability to compute an output from a given input. Thus the nonlinear subsystem dynamics W c1 , that can represent all the channel dynamics (absorbing the linear subsystem as well), need not be known in equation form. Such a model may, therefore, be used with a neural network to generate an adaptive estimator for systems with slowly changing channel dynamics.

13.6

Automotive Nonlinear Filtering Problem

Consider the example of a lambda or air–fuel ratio sensor to monitor the fuel mixture in an automobile engine, as discussed in Chap. 12. The sensor measures the residual oxygen in the exhaust gas and passes the information to the engine control unit, which adjusts the optimum air–fuel mixture for feedback control [30]. The problem described is a little simplistic but it illustrates the main idea. Let the sample

626

13

Nonlinear Estimation and Condition Monitoring: …

time 0.025 s and the channel delay k = 7 sample instants. The linear part of the channel dynamics and the signal and noise models are, respectively Wco ðz1 Þ ¼ ð0:36  0:1008z1 Þ=ð1  0:7408z1 Þ Ws ðz1 Þ ¼ 0:001499=ð1  0:9985z1 Þ Wn ðz1 Þ ¼ ð0:042  0:04199z1 Þ=ð1  0:8825z1 Þ The frequency response characteristics of the signal, noise and linear channel models are shown in Fig. 13.5. The static nonlinear characteristic of the EGO sensor is shown in Fig. 13.6, which defines the nonlinear channel subsystem W c1 . The spectral-factor computation provides the frequency response of Yf and this is compared in Fig. 13.7 with the noise and signal model responses. The message signal path transfer Wc and the estimation error dynamic weighting Wq will first be assumed unity, and the uncertainty tuning function F c0 will be assumed to be a small constant: F c0 ¼ 0:001. The minimum value of the cost for the system, which includes the uncertainty channel model, then follows from (13.70) and may be computed as J ¼ 2:2466  106 . The EGO sensor is used with the uncertainty tuning function but the fictitious parallel uncertainty model is not included in the simulation, so the computed cost J ¼ 8:49  106 is not be expected to coincide with this theoretical minimum value. The time responses are shown in Fig. 13.8 and Table 13.1, where the effect of the channel delay of k  Ts ¼ 0:1750 seconds is evident.

Bode Frequency Responses of Ws, Wn and Wco Magnitude (dB)

0

Ws Wn Wco

-20 -40 -60 -80

Phase (deg)

90 45 0 -45 -90 10-4

10-2

100

Frequency (rad/s)

Fig. 13.5 Signal, noise and channel frequency responses

102

13.6

Automotive Nonlinear Filtering Problem

627

EGO Sensor Characteristic 1 0.9

System Output (volts)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.85

0.9

0.95

1

1.05

1.1

1.15

Value of Lambda Ratio Input

Fig. 13.6 EGO lambda sensor behaviour

Magnitude (dB)

Bode Diagram Frequency Responses of Yf, Ws and Wn Yf Ws Wn

0 -20 -40 -60 -80

Phase (deg)

90 45 0 -45 -90 10-4

10-2

100

102

Frequency (rad/s)

Fig. 13.7 Signal, noise and spectral-factor noise and spectral-factor frequency responses

The estimation error is reduced when the uncertainty tuning gain decreases so long as an optimal filter actually exists, which explains much of the behaviour in the table of values. The second case in Table 13.1 has a reasonably low cost

628

13

Nonlinear Estimation and Condition Monitoring: …

Signal and Signal Estimates 0.99 Actual Signal Estimated Signal

0.985

Lambda Ratio

0.98 0.975 0.97 0.965 0.96 0.955 0.95

0

1

2

3

4

5

6

7

8

9

Time (s)

Fig. 13.8 Signal and estimated signal time responses Table 13.1 Effect of uncertainty model in parallel fictitious path Uncertainty tuning function F c0 1

10

Variance from simulation

3

J ¼ 8:4902  106 1

1

2

0:001ð3  2:472z Þ=ð1  0:4724z Þ

J ¼ 8:5173  106

3

0:01ð1  0:9779z1 Þ=ð1  0:7788z1 Þ

J ¼ 9:2878  106

4 5

1

1

ð1  0:9991z Þ=ð1  0:08208z Þ 1

1

0:1ð3  2:995z Þ=ð1  0:4724z Þ

J ¼ 12:743  106 J ¼ 14:355  106

corresponding to the dynamic uncertainty tuning function F c0 shown in Fig. 13.9. The estimated signal and the actual signal are as shown in Fig. 13.10 for this second dynamic uncertainty case. Comparing the observations (measurement from the channel) and the estimate in Fig. 13.11, this solution is clearly very effective. Considering the level of noise on the signal and the gain change due to the channel the estimates are good. The main advantage of the estimator is that the computations involved are straightforward.

13.6

Automotive Nonlinear Filtering Problem

629

Uncertainty Weighting Function Frequency Response -48

Magnitude (dB)

-50

-52

-54

-56

-58

-60 10 -2

10 -1

10 0

10 1

10 2

Frequency (rad/s)

Fig. 13.9 Uncertainty tuning function F c0

Signal and Signal Estimates 0.99 Actual Signal Estimated Signal

0.985

Lambda Ratio

0.98 0.975 0.97 0.965 0.96 0.955 0.95

0

1

2

3

4

5

6

7

8

Time (s)

Fig. 13.10 Actual and estimated lambda time responses using the EGO sensor

9

630

13

Nonlinear Estimation and Condition Monitoring: …

Observations and Signal Estimates 1 0.9 0.8

Lambda Ratio

0.7 0.6 0.5 0.4 0.3 0.2 Observations Signal Estimated Signal

0.1 0

0

1

2

3

4

5

6

7

8

9

Time (s)

Fig. 13.11 Measured channel output and estimate

13.7

Condition Monitoring and Fault Detection

There are close links between the various signal monitoring topics: • Condition monitoring: use of system measurements and signal processing methods to check normal operation of the plant, to assess performance, predict when maintenance is needed, or predict possible failure of components or machines. • Performance monitoring: the methods used to assess the quality product or determine the status of process variables (outputs) to assess if targets or performance goals are being met. • Fault detection and isolation: the methods used to diagnose system faults or malfunctions and to determine the type of fault and location. • Process monitoring: often refers to Statistical Process Control (SPC) which is a method of quality control that employs statistical methods to monitor and control a process. The signal processing problem considered below involves channel dynamics with both linear and nonlinear subsystems. The problem construction can, therefore, be used to solve a range of condition monitoring problems [31–37]. To illustrate the type of problem, consider a fault estimation problem in rotating machinery. In many cases this machinery may be subject to random disturbances, causing additional loading on different mechanical components. Shaft vibration might provide an

13.7

Condition Monitoring and Fault Detection

631

indication of possible future bearing failure and this could be measured on the casing of the machinery. The variations may of course also result from other signal paths through different bearings. The main signal channel can, therefore, represent the path of the fault to the measurement, and the possible parallel paths can be represented by the uncertainty model F c 0 ðz1 Þ: The Fault Detection (FD) methods can be classified as being model-based or data-driven. The model-based Fault Detection Isolation (FDI) approaches include parity space, parameter estimation and observer-based approaches. Data-driven classification techniques that are used include support vector machines and artificial neural networks. The focus here is on model-based FDI methods and amongst these the most effective are the observer-based methods. The model-based approaches typically involve two steps: • Residual generation: the procedure of extracting fault symptoms from the process, and • Residual evaluation: the procedure for decision-making. The residuals may be generated using either an observer for deterministic models or an optimal filter for stochastic models. Observe-based FD methods use measurements of the actual signal and estimates of the signal to generate a residual. To avoid false alarms, the residual must remain below a threshold when no fault exists [6]. The residual should become large when a fault occurs, but remain small when not in fault conditions. The key problem is to decide whether the residual is large enough to indicate a fault, or if the increase in the residual is due to uncertainties such as disturbances and modelling errors. There is a rich history of work in model-based fault monitoring, detection and isolation, mostly using linear system models (see [31–58]).

13.7.1 NMV Estimator-Based Fault Detection The Nonlinear Minimum Variance (NMV) estimator can be used to generate a residual signal for fault detection applications. A valuable feature of this approach is that the general nonlinear operator may be used to represent the nonlinearity in the signal channel or in the sensor system. The nonlinearities can be present both in the signal channel and in the noise channel representing the uncertainty. The fault detection estimator is required to generate residuals which have to be sensitive to faults and independent of disturbances. This involves comparing the behaviour of the actual signal and an estimated signal. In the absence of a fault, the observer residual ideally approaches zero. When a fault exists, the residual should be relatively large so that it may be monitored and serves as a fault indicator. The block diagram of the proposed nonlinear minimum variance filter-based residual generation for fault detection is shown in Fig. 13.12. An advantage of the NMV estimator approach is that no on-line linearization is required, as required in the extended Kalman filter, and implementation is also straightforward.

632

13

Nonlinear Estimation and Condition Monitoring: …

Weighted channel interference

nc

sc Measured signal

sˆ

+ +

f

Wc1

yˆ  fˆ

Wc 0 ( z 1 ) Linear subsystem

NMV Estimator

c1 ( z

1

)

Nonlinear main channel

Residual generation

r Fig. 13.12 Model of an NMV-based fault detection system

Residual: Residuals are normally obtained by using a mathematical model of the system and measurements from sensors and actuators. A residual signal for this problem can be generated using the measured signal sc ðtÞ and its filtered estimate ^sc ðtÞ, (see the analysis in Alkaya and Grimble [59]) as rðtÞ ¼ sc ðtÞ  ^sc ðtÞ In the ideal case, the residual is zero if there is no fault and changes significantly when a fault arises. NMV Residual calculation: The inferential subsystem Wc is assumed linear, minimum phase and invertible. The NMV filtering algorithm estimates the signal ^sðtj tÞ so that ^sc ðtÞ ¼ ^sc ðtj tÞ might be defined in term of the signal ^sðtÞ ¼ ^sðtj tÞ (simplifying notation), as follows: ^yðtÞ ¼ Wc1^sðtÞ and ^f ðtÞ ¼ ^yðtÞ ¼ Wc1^s

ð13:72Þ

Thence, we obtain ^sc ðtÞ ¼ W c1 Wc0^f ðtÞ ¼ W c1 Wc0 Wc1^sðtÞ and the residual: rðtÞ ¼ sc ðtÞ  ^sc ðtÞ ¼ sc ðtÞ  W c1 Wc0 Wc1^sðtÞ

ð13:73Þ

This residual signal rðtÞ can be monitored using a threshold device to detect when a fault has occurred. When there is a fault at the signal estimation point, the residual becomes rðtÞ ¼ sc ðtÞ  W c1 Wc0 Wc1^sðtÞ þ /f ðtÞ ¼ W c1 Wc0 Wc1 sðtÞ  W c1 Wc0 Wc1^sðtÞ þ /f ðtÞ

ð13:74Þ

where /f 6¼ 0 when there are fault conditions. The residual is related to the estimation error. In the special case when the system is linear, (13.74) simplifies as

13.7

Condition Monitoring and Fault Detection

633

rðtÞ ¼ Wc1 Wc0 Wc1 ð~sðtjtÞÞ þ /f ðtÞ where ~sðtÞ ¼ ~sðtjtÞ denotes the estimation error. The fault can only be detected if the residual term is significant compared with the estimation errors and the signal noise. Threshold computation: Fault detection can be based on the residual signal, but a threshold is needed to distinguish the faults from the disturbances and uncertainties. The threshold determining what is a “significant” change in residual often depends on engineers previous experience. Let Tmin \0 and Tmax [ 0 denote the minimum and maximum values of the threshold values. The residual r(t) may then be monitored and the limits may be expressed as follows: • If r\Tmin or r [ Tmax , an alarm can be indicated since a fault has been detected. • If Tmin  r  Tmax , no alarm is warranted since the system is fault-free.

13.7.2 Fault Detection Example The selection of the uncertainty tuning function F c0 is a dual problem to the selection of the NGMV optimal control cost-function weighting. A starting point is to assume the uncertainty model F c0 is a constant and is of a small magnitude. This corresponds to the situation where the uncertainty is represented by white noise added at the output of the communications channel. To illustrate the NMV filter-based fault detection system, a nonlinear SISO problem will be considered. Consider a system having the following signal and noise models: Ws ¼ 0:2=(1  0:9999z1 ) and Wn ¼ 0:4=(1  0:1z1 ) The linear channel dynamics are defined as Wc0 ¼ 0:2=(1  0:1z1 ) and the channel delay k ¼ z1 , so that K0 ¼ k ¼ 1. The static nonlinear characteristic of the system is defined as shown in Fig. 13.13. Also, let the weighting Wq ¼ 1 and Wc ¼ 0:2=(1  0:2z1 ). The dc-gain and frequency response of the weighting filter F c 0 influences the accuracy of estimation. The tuning function for this example has the following model: F c 0 ¼ 0:02ð1  0:1z1 Þ=ð1  0:9z1 Þ Results: Under normal operation conditions (fault-free), the measured signal and estimated signal are as illustrated in Fig. 13.14. The estimation is not too good because the tuning is aimed at fault detection. The tuning function and model frequency responses are as shown in Fig. 13.15. The calculated residual signal and the confidence level thresholds are shown in Fig. 13.16. As shown in Fig. 13.16, the residual signal is well under the threshold when the system is operating normally.

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System Characteristics 5 4 3 2

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Fig. 13.13 Nonlinear behaviour of the output subsystem

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Condition Monitoring and Fault Detection

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Frequency Response of Signal, Noise, Linear Channel and Weighting 80 Signal Model Ws Noise Model Wn Linear Channel Wc0 Weighting Function Fc0

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Fig. 13.15 System model and tuning function responses

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Fig. 13.16 Residual signal with thresholds (no fault)

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Two types of faults were applied to demonstrate the effectiveness of the proposed NMV estimator in fault detection applications. A sensor fault is represented by a signal applied at the output of the signal channel where the measurement is taken. An actuator fault is represented by a signal that appears at the output of the signal source that will appear at the input to the signal channel. Sensor Fault: Assume a sensor fault is applied at the output of the signal channel, represented by a pulse of magnitude 2.2 and acting from time 300 to 450 s. After a sensor fault is applied, the actual and estimated signals are as illustrated in Fig. 13.17. The calculated residual signal and confidence level threshold are shown in Fig. 13.18. The fault in this rather artificial situation can clearly be detected successfully. In practice, a fault isolation algorithm capable of distinguishing between sensor and actuator faults would of course also be needed. Actuator Fault: In this case assume an actuator fault is applied at the input to the signal channel, represented by a pulse of magnitude −2.2 and acting from time 180 to 280 s. After the actuator fault is applied, the measured and estimated signal are as illustrated in Fig. 13.19. The calculated residual signal and confidence level thresholds are as shown in Fig. 13.20. The fault can clearly be detected successfully with an accurate estimate of the time, but again the problem is rather artificial. Remarks on the uncertainty: The parallel path in Fig. 13.2, shown as dotted, is used as a way of modelling uncertainty and ensuring estimator stability. The freedom and ability to tune the estimated signal, using the F c 0 tuning function, is important in fault detection. However, there are some physical problems, as in fault detection, where the parallel channel might actually exist and in this case F c 0 cannot be treated as a freely chosen design function but is determined by the physical plant equations. 2.5 2

Amplitude

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Fig. 13.17 Measured and estimated signal (sensor fault)

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Condition Monitoring and Fault Detection

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Fig. 13.18 Residual signal with thresholds (sensor fault)

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Fig. 13.20 Residual signal with thresholds (actuator fault)

13.8

Conclusions

The filtering problem may be posed in either the polynomial matrix form of the previous chapter or in the state-equation-based framework utilized here. The state-space approach to modelling the linear subsystems is valuable since the resulting computations mostly involve the manipulation of constant coefficient matrix equations which are well-conditioned numerically. As mentioned in the previous chapter the NMV estimator may be applied to the solution of channel equalization problems in communications or to fault detection problems in control applications, amongst others. The estimator is very simple to implement and design, as the examples in both chapters demonstrated. The NMV estimator solution can be thought of as the “dual” of the solution to the basic NGMV state-space control problem although they are not strictly the mathematical duals. The main strength of the NMV nonlinear filtering approach is that it provides a very simple solution to a difficult nonlinear estimation problem. The benefits of implementation were explored in Ali Naz and Grimble [60]. Competing techniques like the extended Kalman filter involve approximations resulting from the linearization and they are not strictly optimal estimators. The solution proposed here does not involve any linearization, however, it is for a rather special class of problems, which is the price of the simplicity achieved. An advantage of the approach described in this and the previous chapter is the flexibility the nonlinear black-box subsystem in the signal channels provides. There are many options for use of this type of approach in related problems. For example, a NMV estimator for quasi-Linear Parameter-Varying (qLPV) systems may be

13.8

Conclusions

639

developed that includes time-varying, Linear Parameter Varying and/or state-dependent systems. A one-block H∞ estimator may also be produced that is related to the linear case [61, 62]. The application of the approach in industrial systems also requires experience to be gained and comparison with traditional estimation methods [63, 64].

References 1. Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 35–45 2. Kalman RE (1961) New methods in Wiener filtering theory. In: Proceedings of the symposium on engineering applications of random function theory and probability, pp 270– 388 3. Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. J Basic Eng Trans ASME 95–108 4. Wiener NE (1949) Extrapolation, interpolation, and smoothing of stationary time series, with engineering applications. MIT Press, Cambridge, MA (Issued in February 1942 as a classified US National Defence Research Council Report) 5. Kwakernaak H, Sivan R (1991) Modern signals and systems. Prentice Hall 6. Gelb A (1974) Applied optimal estimation. The Analytic Sciences Corporation. MIT Press, Cambridge, MA 7. Athans M, Safonov MG (1978) Robustness and computational aspects of nonlinear stochastic estimators and regulators. IEEE Trans Autom Control 23(4):717–725 8. Grimble MJ, Jukes KA, Goodall DP (1984) Nonlinear filters and operators and the constant gain extended Kalman filter. IMA J Math Control Inf 1:359–386 9. Hammersley JM, Morton KW (1954) Poor man’s Monte Carlo. J R Stat Soc 16(1):23–38 10. Gordon NJ, Salmond DJ, Smith AFM (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc Pt F 140(2):107–113 11. Zhe C (2003) Bayesian filtering: from Kalman filters to particle filters, and beyond. Technical report, Adaptive Systems Laboratory, McMaster University 12. Hutter F, Dearden R (2003) The Gaussian particle filter for diagnosis of non-linear systems. In: 14th international conference on principles of diagnosis, Washington, DC, pp 65–70 13. Cadini F, Zio E (2008) Application of particle filtering for estimating the dynamics of nuclear systems. IEEE Trans Nucl Sci 55(2):748–757 14. Djuric PM, Kotecha JH, Zhang J, Huang Y, Ghirmai T, Bugallo MF, Miguez J (2003) Particle filtering. IEEE Signal Proc Mag 20(5):19–37 15. Julier SJ, Uhlmann JK (2004) Unscented filtering and nonlinear estimation. Proc IEEE 92 (3):401–422 16. Almeida J, Oliveira P, Silvestre C, Pascoal A (2015) State estimation of nonlinear systems using the unscented Kalman filter. In: TENCON 2015 conference, Macao, China 17. Simon D (2006) Optimal state estimation: Kalman, H∞, and nonlinear approaches. Wiley 18. Grimble MJ (2005) Nonlinear generalised minimum variance feedback, feedforward and tracking control. Automatica 41:957–969 19. Grimble MJ, Majecki P (2005) Nonlinear generalised minimum variance control under actuator saturation. In: IFAC world congress, Prague 20. Grimble MJ (2007) NMV optimal estimation for nonlinear discrete-time multi-channel systems. In: 46th IEEE conference on decision and control, New Orleans, pp 4281–4286 21. Åström KJ (1979) Introduction to stochastic control theory. Academic Press, London 22. Grimble MJ (2006) Robust industrial control: optimal design approach for polynomial systems. Wiley, Chichester

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13

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23. Kailath T (1969) Fredholm resolvents, Wiener-Hopf equations, and Riccati differential equations. IEEE Trans Inf Theory 15(6):665–672 24. Anderson BDO, Moore JB, Loo SG (1969) Spectral factorization of time-varying covariance functions. IEEE Trans Inf Theory 15(5):550–557 25. Grimble MJ, Johnson MA (1988) Optimal multivariable control and estimation theory, vol I and II. Wiley, London 26. Moir TJ (1986) Optimal deconvolution smoother. IEE Proc Pt D 133(1):13–18 27. Jiang SS, Sawchuk A (1986) Noise updating repeated Wiener filter and other adaptive noise smoothing filters using local image statistics. Appl Opt 25:2326–2337 28. Grimble MJ (2001) Industrial control systems design. Wiley, Chichester 29. Grimble MJ (1995) Multichannel optimal linear deconvolution filters and strip thickness estimation from gauge measurements. ASME J Dyn Syst Meas Control 117:165–174 30. Berggren P, Perkovic A (1996) Cylinder individual lambda feedback control in an SI engine. M.Sc. thesis, Linkoping 31. Zhang X, Parisini T, Polycarpou MM (2001) Integrated design of fault diagnosis and accommodation schemes for a class of nonlinear systems. In: 40th IEEE conference on decision and control, Orlando, FL, pp 1448–1453 32. Isermann R, Balle P (1997) Trends in the application of model-based fault detection and diagnosis of technical processes. Control Eng Pract 5(5):709–719 33. Chen J, Patton RJ (1999) Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publisher 34. Gertler J (1998) Fault detection and diagnosis in engineering systems. Marcel Dekker, Basel, Switzerland 35. Forsman K, Stattin A (1999) A new criterion for detecting oscillations in control loops. In: European control conference, Karlsruhe, Germany 36. Hagglund T (1995) A control-loop performance monitor. Control Eng Pract 3:1543–1551 37. Horch A (1999) A simple method for detection of stiction in control valves. Control Eng Pract 7:1221–1231 38. Willsky AS (1976) A survey of design methods for failure detection in dynamic systems. Automatica 12:601–611 39. Frank PM (1987) Advanced fault detection and isolation schemes using non-linear and robust observers (survey paper). In: Proceedings of the 10th IFAC world congress Munich, pp 63–68 40. Frank PM (1990) Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy. Automatica 26:459–474 41. Frank PM, Ding X (1994) Frequency domain approach to optimally robust residual generation and evaluation for model-based fault diagnosis. Automatica 30(4):789–804 42. Frank PM (1994) Enhancement of robustness in observer-based fault detection. Int J Control 59(4):955–981 43. Patton RJ, Frank PM, Clark RN (eds) (2000) Issues of fault diagnosis for dynamic systems. Springer, London 44. Patton RJ, Chen J (1993) A survey of robustness in quantitative model-based fault diagnosis. Appl Math Comp Sci 3(3):399–416 45. Patton RJ, Chen J (1994) A review of parity space approaches to fault diagnosis for aerospace systems. J Guid Control Dyn 17(2):278–285 46. Patton RJ, Chen J (1996) Robust fault detection and isolation (FDI) systems, control and dynamic systems. In: Leondes C (ed). Mita Press, pp 171–224 47. Patton RJ, Chen J, Nielsen SB (1995) Model-based methods for fault diagnosis: some guide-lines. Trans Inst Meas Control 17(2):73–83 48. Basseville M, Nikiforov IV (1993) Detection of abrupt changes: theory and application. Prentice-Hall Inc., Englewood Cliffs, N.J 49. Chen J, Patton RJ (1999) Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, Dordrecht 50. Isermann R (2005) Fault-diagnosis systems: an introduction from fault detection to fault tolerance. Springer, Berlin

References

641

51. Markovsky I, De Moor B (2005) Linear dynamic filtering with noisy input and output. Automatica 41(1):167–171 52. Simani S, Fantuzzi C, Beghelli S (2000) Diagnosis techniques for sensor faults of industrial processes. IEEE Trans Control Syst Technol 8(5):848–855 53. Chen J, Patton RJ (1998) Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers 54. Chow EY, Millsky AS (1988) Analytical redundancy and the design of robust failure detection systems. IEEE Trans Autom Control 29:603–614 55. Clark RN, Fosth DC, Walton WM (1975) Detecting instrument malfunctions in control systems. IEEE Trans Aerosp Electron Syst 11:465–473 56. Thumati B, Jagannathan S (2010) A model based fault detection and prediction scheme for nonlinear multivariable discrete-time systems with asymptotic stability guarantees. IEEE Trans Neural Netw 21(3):404–423 57. Ferdowsi H, Jagannathan S (2013) A unified model-based fault diagnosis scheme for nonlinear discrete-time systems with additive and multiplicative faults. Trans Inst Meas Control 35(6):452–462 58. Zhang X, Polycarpou MM, Parisini T (2000) Abrupt and incipient fault isolation of nonlinear uncertain systems. In: American control conference, Chicago, IL, pp 3713–3717 59. Alkaya A, Grimble MJ (2015) Non-linear minimum variance estimation for fault detection systems. Trans Inst Meas Control 37(6):805–812 60. Ali Naz S, Grimble MJ (2008) Design and real time implementation of nonlinear minimum variance filter. In: UKACC conference, Manchester 61. Grimble MJ (1987) H∞ design of optimal linear filters. In: MTNS conference, Phoenix, Arizona. Byrnes CI, Martin CF, Saeks RE (1988) North Holland, Amsterdam, pp 540–553 62. Grimble MJ, ElSayed A (1990) Solution of the H∞ optimal linear filtering problem for discrete-time systems. IEEE Trans Signal Proc 38(7):1092–1104 63. Moore JB, Anderson BDO (1967) Solution of a time-varying Wiener filtering problem. Electron Lett 3:562–563 64. Grimble MJ (2011) Nonlinear minimum variance state based estimation for discrete-time multi-channel systems. IET J Sig Proc 5(4):365–378

Part V

Industrial Applications

Chapter 14

Nonlinear Industrial Process and Power Control Applications

Abstract The remaining two chapters consider applications of nonlinear multivariable control design methods. This chapter covers the topics of power generation using wind energy, and metal processing involving rolling processes. It also includes design studies on process control applications involving a heavy oil fractionator, control of stirred tank reactors and finally predictive control for evaporators. The main message from the results in this chapter is that it is practical to apply the advanced nonlinear control design methods proposed and it is not unduly difficult to do so. The valuable role that LPV and state-dependent models have for industrial process modelling is also demonstrated. The results suggest performance benefits should be obtained by applying advanced controls, but whether these are really significant will depend upon the application and the time spent on system design and tuning.

14.1

Introduction

The drivers in the industry are to consistently attain high product quality, to use energy more efficiently and to reduce the impact on the environment. Often these requirements all combine to impose greater demands on control systems than can be met by traditional control methods. The case for using advanced control is intimately related to the need for high performance since reasonable control can often be designed, using single-loop techniques, for even large nonlinear multivariable systems. However, the demands on performance can often be translated into a need for higher gains and for constraint handling, and both inexorably lead to the need for advanced control solutions. The chapter begins with a wind turbine power regulation problem where a qLPV model is used. This is followed by a looper control design, for a hot strip finishing mill, where the looper dynamics are very nonlinear. Three process control applications are then considered, beginning with a petrochemical fractionator process, which has many inputs and outputs and includes constraints. A chemical reactor control and benchmarking problem are then described involving a continuous © Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_14

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stirred tank reactor. Finally, an evaporator control problem is considered, where an LPV/qLPV model is employed. The corresponding plant models are introduced and the use of the model-based control methods described in previous chapters is then discussed. The focus is on the control law synthesis and design questions that arise. In process control applications, it may not be necessary to follow setpoints tightly on all outputs, but zone or range control may be needed so that outputs are maintained between high and low limits. The early applications of Model Predictive Control (MPC) were in the process industries and the ability to handle constraints was one of its most valuable features. Nonlinearities due to actuators or process dynamics were dealt with using scheduling. This was and is used routinely for processes that contain significant nonlinearities. Nonlinearities can affect the efficient operation or more seriously can lead to unstable behaviour tripping safety systems and destroying high value product. Process operations that are nonlinear can often be related to a set of operating regions based on the values of key process variables. The process model parameters can then be determined or identified, and the controller tuned for reasonable performance in each region. When the process operations change, moving the process from one region of operation to another, an MPC law can be updated using scheduling [1]. However, in general, the selection of the operating regions for scheduling is a non-trivial task. That is, the process operating space must be divided into a sufficient number of regions to provide adequate performance but not so many that they increase the complexity of the controller significantly. Unfortunately switching between linear controllers can introduce transients and result in additional wear and tear on system components such as actuators. Bumpless transfer methods must, therefore, be used when implementing a scheduled solution. The model-based control methods described enable scheduling to be either avoided or implemented in a more planned regime. The solutions may, for example, be obtained by using qLPV models and these will apply across the range of operation (the range where the qLPV model is valid). In some cases to simplify implementation on limited computing devices, a form of scheduling may be used but this will be easier to design given the qLPV solution.

14.2

Wind Energy Systems

Wind energy devices provide one of the most important source of “renewable” energy at the present time which is secure for the future. It is an environmentally friendly source of electrical power with reasonable costs so long as serious breakdowns and failures are limited. Wind turbine control algorithms have to deal with stringent requirements on reliability, the cost of energy and the extreme operating conditions of offshore installations [2, 3]. Both fixed and variable-speed wind turbines are used with different operating characteristics and problems. For example, fixed-speed wind turbines can produce variations in power due to the

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Wind Energy Systems

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so-called tower shadow effects. Variable-speed wind turbines eliminate the flicker effect but harmonic currents are introduced by the inverters employed. The advanced control of wind turbines can be used to improve energy capture and to reduce dynamic loading, and wear and tear [4]. It is particularly important for high reliability to be maintained in offshore wind turbines, where the costs of maintenance and repairs can make operation uneconomic. A good controller for offshore turbines is one which provides adequate performance and limits extreme stresses. Many different control philosophies have been proposed for use with individual wind turbines, ranging from LQG/H2 and H∞ robust control design methods to fuzzy control design approaches [5–7]. Wind Farms and Network: Unfortunately, good wind resources are often in remote areas, where the electrical system is weak. Early wind power installations were limited to a few hundred kilowatts but individual wind turbines now have megawatt capacities and wind farms can have a capacity of several thousand MWs. Unfortunately, the increased power of wind farms in areas of good wind resources can interfere with the power system and there is the need to carefully integrate large wind energy resources into the grid. For power systems, where the proportion of wind power is significant, power quality and stability problems may arise and the line frequency may vary. Large wind farms have to share some of the responsibilities of conventional power plants, such as regulating active and reactive power and performing frequency and voltage control. More severe problems may arise in the future due to the complexity of networks and the distributed nature of the power generation devices and users. The control of very large systems has been problematic for many years but as described earlier model-based predictive control techniques have been applied very successfully to large systems in the process industries. Wind turbines have much faster dynamics and significant nonlinearities, but predictive control methods also appear very suitable for wind farm control. A model-based predictive controller requires a model of the system and some knowledge or assumptions regarding future setpoint and disturbance variations. Optimal setpoints for power, rotational speed and pitch angle can be found offline as functions of the wind speed [8], and wind speed prediction can be based on LIDAR. Mechanical limits on the rate of change or on the movement of actuators can be accommodated using a predictive controller, and there also is the possibility of using the constraint handling features to deal with faults. For example, the loss of a power line can be considered a special form of hard constraint. Offshore Wind Farms: To deal with the severe environmental conditions in offshore wind farms, good condition monitoring is needed in addition to safe and reliable control methods. Intelligent sensor systems are valuable to monitor stresses, wear and tear, and loading on the wind turbine components. These problems were investigated in a wind energy project supported by the European Commission called AEOLUS, which employed supervisory predictive control for offshore wind farm control. The aim was to “harvest” the wind efficiently and to minimize fatigue loading and wear. Both the actuators for wind turbines and the functions that describe fatigue are nonlinear. There are problems that stem from the large scale,

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complex and distributed nature of the wind farm. These problems are exacerbated by the location of wind farms that are normally remote from the users of power. A model predictive supervisory control solution was proposed because of the relatively simple computations, which is an important factor when a large number of wind turbines are involved.

14.2.1 Introduction to Individual Wind Turbine Control The control of individual wind turbines will be the focus of the remainder of this discussion on wind energy. The following section is based on contributions by our colleague Dr. Petros Savvidis and further details may be found in his Ph.D. thesis [9]. A wind turbine converts the kinetic energy of the wind into electrical power. The wind power driving the rotor is first translated into mechanical power and subsequently into electrical power via the transmission system and the generator. The amount of power which can be extracted from the wind is determined by the area swept out by the turbine rotor and is limited by a factor which varies with the pitch angle of the blades and the tip-speed ratio. The tip-speed ratio is normally denoted by k and is the ratio between the tangential speed of the tip of a blade and the velocity of the wind. The wind turbine model considered below was developed by the US National Renewable Energy Laboratory (NREL) for a 5 MW offshore wind turbine. The four regions of wind turbine operation are illustrated in Fig. 14.1. The objectives of the control law depend on the regions of operation, and may be summarised as follows: Available power as a function of wind speed for a 5MW NREL turbine 5

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• Region 1: (0–4 m/s). Start-up/Low wind speed: Objective is to maximize power constrained by minimum rotor speed. • Region 2: (4–10.9 m/s). Below rated wind speed: Objective is to maximize power. • Region 21/2: (10.9–11.5 m/s). Below rated wind speed and rotor speed: Objective is to maximize the power generated constrained by the nominal/ maximum rotor speed. • Region 3: (11.5–25 m/s). Above rated wind speed: Objective is to maintain rated power and nominal/maximum rotor speed.

14.2.2 Controller Structure Consider the control of a large single wind turbine. The aim is to regulate the electrical power produced and to compensate for the turbulent wind speed variations and gusts. When the wind turbine operates below rated wind speed, the control strategy is to maximize the electrical power produced. The pitch of the blades is therefore set to zero to capture as much of the energy available in the wind as possible. The optimal torque reference to the generator can be derived from look-up tables, designed to optimize the power generated. In the above rated wind speed conditions, both the generator torque and the blade angle control, can be used. The main control objective in this region is the regulation of the electrical power to achieve its rated value. The control strategies that may be applied involve two possible configurations (see Fig. 14.2), either scalar Single Input Single Output (SISO) or Multi Input Multi Output (MIMO): 1. Fixed Torque/Variable Pitch: The generator torque is maintained at the rated value, whilst the pitch is manipulated to regulate the power to its rated value, compensating for the wind speed variations. 2. Variable Torque/Variable Pitch: Both the generator torque and the generator pitch may be manipulated to regulate the generator speed and power at the rated values, during wind speed variations. For simplicity, attention will mostly focus on the SISO option although the final results are presented for the multivariable case. The main emphasis will be on the use of the Nonlinear Predictive Generalized Minimum Variance (NPGMV) controller to account for the process nonlinearities (Chaps. 9 and 11). Generator torque reference (Nm) Pitch angle reference (deg.)

Input 1 Input 2

Output 1

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Generator speed (rad/s) Power produced (W)

Fig. 14.2 Multivariable Underlying Control Problem (The SISO configuration involves Input 2/ Output 2)

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Inputs and Outputs: The main output or controlled variable in this problem is the electrical power produced. The input or manipulated variable is the turbine pitch-angle reference. The disturbance variable is the effective wind speed as experienced by the blades of the turbine. Control Architecture: The control architecture involves a feedback and a feedforward component, as illustrated in Fig. 14.3. It is assumed that a wind speed measurement or estimate is available. Feedforward control depends on the optimal trajectories for the pitch angle, rotor/generator speed, power and generator torque. The aim is to keep power production at the rated value in the above rated wind speed zone. The optimal reference curves, stored in the form of look-up tables, are used to generate the power and speed reference signals. For Tthe above rated wind speed operation, the optimal pitch angle is increased, whilst the power, generator speed and torque characteristics remain fixed at the rated values. The optimal curves for the wind turbine considered in this example are shown in Fig. 14.4. Feedback control is used to minimize power variations around the rated value and to compensate for model uncertainties and nonlinearities. These nonlinearities are often accommodated by gain scheduling [10, 11].

14.2.3 Wind Turbine Model Description The wind turbine model may be separated into the following subsystems: • Blade Pitch Actuator (linear dynamic) • Rotor Aerodynamics (nonlinear static) • Transmission System or Drivetrain (linear dynamic)

τg_0 τg LUT

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δβ_c

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Fig. 14.3 SISO control system feedback and feedforward structure

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14.2

Wind Energy Systems Optimal power curve

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v [m/s]

Fig. 14.4 Optimal reference curves for the 5 MW NREL wind turbine

• Generator and Converter System (nonlinear dynamic) • Tower Bending Dynamics (linear dynamic). The main source of nonlinearities is in rotor aerodynamics, i.e. generation of rotor torque caused by the wind exerting force on the blades. The model components are described in more detail below. Energy in Wind: The total available power from the wind at the turbine rotor is given by the relation Ptotal ¼ qAv3 =2 ¼ qpR2 v3 =2, where A is the area swept by the rotor (m2), v is the wind speed (m/s), q is the air density (kg/m3), and R is the rotor radius (m). The Betz law is informative and states that less than 16/27 (or 59%) of the total wind power can be converted to mechanical power and this is known as the Betz limit or Betz efficiency. Tip-Speed Ratio: The tip-speed ratio is the ratio of the speed of the rotating blade tip to the speed of the free stream wind, and it is denoted as k ¼ xr R=v, where xr denotes the rotor speed (rad/s) and R is the blade length (m). There is an optimum angle of attack, which creates the highest lift to drag ratio. The angle of attack is dependent on the wind speed, and there is therefore an optimum target tip-speed ratio. Wind turbines are designed to achieve the optimal tip-speed ratios to extract as much power from the wind as possible. Pitch Actuator: The pitch actuator is controlled to provide rotation of the blade around the pitch axis to change the angle of attack. In some large wind turbines, the blades can be controlled individually but a common actuator is assumed here for all

652

14

Nonlinear Industrial Process and Power Control Applications

three blades. The pitch actuator servo will be represented by a second-order system bðsÞ=bref ðsÞ ¼ x2n =ðs2 þ 2nxn s þ x2n Þ, where b is the collective pitch angle in degrees, xn is the natural frequency in radians/second and n is the damping-ratio. Rotor Aerodynamics: The main nonlinear component in the wind turbine involves the aerodynamic conversion from the wind energy captured by the rotor, to the resulting drive torque. The mechanical extractable power can be represented as Pext ¼ Cp ðk; bÞPtotal ¼ 0:5Cp ðk; bÞqpR2 v3 , where Cp ðk; bÞ is the power coefficient that varies with the tip-speed ratio k and the blade pitch angle b. The optimum value Cp ðk; bÞ is approximately 0.4. The aerodynamic torque applied to the rotor shaft follows as sr ¼ Pext =xr ¼ Cp ðk;bÞ0:5qpR2 v3 =xr ¼ Cq ðk; bÞ0:5qpR3 v2

ð14:1Þ

where the torque-coefficient Cq ðk;bÞ ¼ Cp ðk;bÞ=k. The thrust force exerted by the wind on the rotor Ft ¼ Ct ðk; bÞ0:5qpR2 v2 will produce a motion of the wind turbine tower in the fore-aft direction. The thrust coefficient Ct and the coefficients Cp, Cq, are given by look-up tables, parameterised by the pitch angle and the tip-speed ratio. The Cp characteristics for the NREL wind turbine are shown in Fig. 14.5. Transmission System, Tower Bending and Generator: The transmission system or drivetrain transfers the mechanical power generated at the turbine rotor to the electrical generator. It consists of two shafts (low and high-speed), which are connected via a gearbox. The gearbox introduces an increase of speed from the low-speed rotor to the high-speed generator shaft. This is modelled by a flexible shaft, subject to torsional torque, and may be modelled by a third-order system. The drivetrain configuration involves the aerodynamic torque sr generated at the rotor, which translates into torque on the generator side sg. The high-speed shaft stiffness Khs can be related to the low shaft stiffness Kls via the gear ratio N. This also applies to the angular positions, expressed in terms of the low shaft torsion-angle hr (just h for simplicity), and can be taken further by expressing Dg in terms of the low shaft friction damping Dr (or Dls). Power coefficient (Cp) parameterized by pitch angle ( ) 0.5

0.5 = 0 deg = 4 deg = 6 deg = 8 deg = 12 deg

0.45 0.4

0.4 0.35

0.3 0.25

0.3 0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05 0

2

4

=3 =6 =8 = 14

0.45

Cp [-]

Cp [-]

0.35

0

Power coefficient (Cp) parameterized by tip speed ratio ( )

6

8

10

[-]

12

14

16

18

0

0

5

10

15

20

[deg]

Fig. 14.5 The 5 MW NREL Cp curve with respect to different values of b and k

25

30

14.2

Wind Energy Systems

653

The equations that describe the transmission system follow as 1 1 ðsr  Kls h  Dls xr  Dls xg Þ Jr N

x_ r ¼

ð14:2Þ


 1 1 1 1 sg þ Kls h  Dls xr  2 Dls xg x_ g ¼ Jg N N N

ð14:3Þ

1 h_ ¼ xr  xg N

ð14:4Þ

Only the longitudinal displacement (fore-aft) tower motion dynamics are considered. These are due to the thrust force Ft on the turbine structure when the wind passes through the rotor. The state-space system representing the tower bending: 

v_ FA x_ FA





FA D mFA ¼ 1

 mKFA FA 0 

vFA ¼ ½ 1



v 0  FA xFA

 vFA þ Ft xFA

ð14:5Þ

 ð14:6Þ

where vFA = tower fore-aft velocity (m/s), xFA = tower fore-aft displacement (m), KFA = tower fore-aft stiffness (N/m), DFA = tower fore-aft damping (N/(m/s)), mFA = tower fore-aft mass (kg) and Ft = thrust force (N). The active power control at the turbine is considered. It is assumed that the power controller provides the electrical generator with a torque reference. The generator dynamics is modelled by a first-order system: Generator Torque: sg ðsÞ ¼ sg;ref ðsÞagc =ðs þ agc Þ

ð14:7Þ

PðtÞ ¼ gg xg ðtÞsg ðtÞ

ð14:8Þ

Convertor Power:

where sg = generator torque, sg, ref = generator torque reference, agc = first-order term coefficient and ηg = generator efficiency. Figure 14.6 reveals how the subsystems are combined to provide the total model of the wind turbine. The system involves significant interactions, disturbances and is nonlinear. Care must also be taken with both control design and implementation to avoid excessive loading and wear [12].

654

Pitch angle ref. (deg.)

14

Nonlinear Industrial Process and Power Control Applications

Rotor speed (rad/s)

Rotor torque (Nm)

Pitch Actuator

Wind speed (m/s)

Pitch angle (deg)

Rotor Aerodynamics

Thrust force (N)

Effective wind speed (m/s) Tower displacement (m/s)

Drive Train Load torque (Nm)

Generator speed (rad/s)

Tower Dynamics

Generator torque ref. (Nm)

Generator

Power produced (W)

Fig. 14.6 Wind turbine subsystems and interconnections

14.2.4 LPV Wind Turbine Model for Control Design Linear Parameter-Varying (LPV) models are becoming popular for wind turbine controls and other applications [9, 13, 14]. An analytical derivation and a discretization were performed to obtain the LPV model for the computation of the NPGMV controller. The model reflects small deviations along the optimal trajectory and involves the subsystems listed above. The wind turbine system can be put into an LPV structure since the torque and thrust coefficient look-up tables can be replaced by linearized versions parameterised by wind speed. It is assumed that there are only three output measurements available for control and these are the generator speed, pitch angle and the electric power output of the generator. State vector: • Blade pitch angle, b (deg) • Blade pitch angle, b_ (deg/s) • Low-speed shaft torsional angle, h (rad) • Generator speed, xg (rad/s) • Rotor speed, xr (rad/s) • Generator load torque, sg (Nm) • Tower fore-aft displacement, xFA (m) • Tower fore-aft velocity, vFA (m/s) Input vector: • Blade effective wind speed, veff (m/s) • Generator load torque reference, sg,ref (Nm) • Pitch angle reference, br (deg) Output vector: • Generator speed, xg (rad/s) • Blade pitch angle, b (deg) • Produced electrical power, Pel (W)

3 2 3 b x1 6 x2 7 6 b_ 7 7 6 7 6 6 x3 7 6 h 7 7 6 7 6 6 x4 7 6 xg 7 7 6 7 x¼6 6 x5 7 ¼ 6 xr 7 7 6 7 6 6 x6 7 6 sg 7 7 6 7 6 4 x7 5 4 xFA 5 x8 vFA 2

2

3 2 3 veff u1 4 5 4 u ¼ u2 ¼ sg ref 5 u3 bref 2

3 2 3 xg y1 y ¼ 4 y2 5 ¼ 4 b 5 y3 Pel

14.2

Wind Energy Systems

655

The state and output equations for the linearized model are written in the vector-matrix form as follows: 2 3 0 b 6 x2n _ 6 b 7 6 6 7 6 6 h 7 6 0 6 7 6 0 6 6 d 6 xg 7 7¼6 6 fs ;b 7 x dt 6 6 r 7 6 Jrr 6 sg 7 6 0 6 7 6 4 xFA 5 6 0 4 fFt ;b vFA 2

mFA

1 2nxn 0 0

0 0 0 Kls NJg

0 0  N1  ND2lsJg

0 0 0 0

 KJrls 0 0 0

 DJrls 0 0 0

2

0 6 0 6 6 0 6 6 0 þ6 6 fsrJ;veff 6 r 6 0 6 4 0

fFt ;veff mFA

0 0 1 Dls  NJ g



ðDls þ fsr ;xr Þ NJr

0 0 

fFt ;xr mFA

0 0 0  J1g

0 0 0 0

0 agc 0 0

0 0 0  mKFA FA

3 2 3 7 b _ 76 b 7 76 7 76 h 7 76 7 76 xg 7 7 6 7 f 76 xr 7  srJ;vreff 76 7 7 6 7 0 76 sg 7 7 4 5 x FA 1 5 ðDFA þ fFt ;veff Þ v FA  0 0 0 0

mFA

3

0 0 0 0 0 agc 0 0

0 x2n 7 7 2 3 0 7 7 veff 0 7 74 sg;ref 5 0 7 7 bref 0 7 7 0 5 0

0 0

3 b 6 b_ 7 7 6 36 h 7 2 7 0 0 6 0 0 6 xg 7 7þ40 0 0 0 56 6 xr 7 7 0 0 6 0 0 6 sg 7 7 6 4 xFA 5 vFA

ð14:9Þ

2

2

3 2 0 0 xg 4 b 5¼4 p 0 180 0 0 Pel

0 0 0

1 0 sg;op

0 0 0

xg;op

32 3 0 veff 0 54 sg;ref 5 0 bref

ð14:10Þ where the coefficients fsr and fFt were obtained from the Cq and Ct coefficients look-up tables by linearizing, interpolating and parameterising with the wind speed. The combined model was then discretized using the Euler method with a sample period of 50 ms.

14.2.5 Wind Turbine Simulation and Performance The nonlinear NGMV and predictive NPGMV controllers were used for wind turbine control by Savvidis [9] and Savvidis et al. [14], with promising results. The three controllers are compared in this section: a simple PID controller, an NGMV controller and an LPV-based NPGMV control design. They are evaluated below for various operating scenarios in the above rated wind speed operating region. The

656

14

Nonlinear Industrial Process and Power Control Applications

performance is quantified below using the Root Mean Square Error (RMSE) metric. The scenarios include different types of wind speed variations, due to gusts and turbulence, to illustrate the disturbance rejection properties. The future reference changes are assumed known when demonstrating the performance of the NPGMV controller. Disturbance Rejection: The disturbance rejection properties are determined by the disturbance models and the cost-function weighting choices. For example, the power tracking weighting includes integral action at low frequencies and rolls off at mid-frequencies so as to remove steady-state tracking errors. Wind Gust Scenarios: Consider first Scenario 1A for small gust variations of 14–16 m/s (nominal wind speed at 15 m/s). The model-based NGMV and NPGMV controls, shown in Fig. 14.7, clearly provide improved power regulation with minimum changes to pitch control (very small at 0:5°). Note that small variations in pitch give relatively significant changes in the power output. For this region of operation, generator torque is not controlled. Table 14.1 shows the computed mean square error values for this scenario. The regulation performance depends on controller tuning, and the predictive controller actually has worse performance than the NGMV controller in this scenario. The performance of predictive controllers generally improves with the prediction horizon. Electric Power [MW]

5.1

0.3

5.05

0.28

5

0.26

4.95

0.24

4.9

15

20

25

30

Pitch Angle [deg]

20

0.22 15 42

PID NGMV NPGMV (N=5) NPGMV (N=10)

15

Shaft Torsion [deg]

20

25

30

Generator Torque [kNm]

41.5 41 40.5

10

40 5

15

20

25

30

Wind Speed [m/s]

20

5.7

20

25

30

Tower Deflection [m]

5.6

18

5.5

16

5.4

14 12

39.5 15

5.3 15

20

25

30

5.2 15

time [sec]

Fig. 14.7 Small wind gust variations of 14–16 m/s (Scenario 1A)

20

25

time [sec]

30

14.2

Wind Energy Systems

657

Table 14.1 Power root mean square error values for different controllers and scenarios (in kW) Scenario

PID

NGMV

NPGMV, N = 5

NPGMV, N = 10

1A (small gust) 1B (large gust) 2 (turbulent)

17.87 25.20 26.61

11.02 15.76 24.15

12.86 18.89 26.45

11.49 16.47 22.61

Scenario 1B: Now consider the effect of large gust deviations in Scenario 1B, as shown in Fig. 14.8. In the case of large gust variations of 12–18 m/s all the mean square error values are of course larger, but as shown in Table 14.1 the pattern of results is the same as for the small wind gust behaviour. Scenario 2: This scenario is for turbulent wind speed variations between 12 and 21 m/s, as shown in Fig. 14.9. The results of Table 14.1 reveal that the long horizon predictive control provides the best performance in this case, but the NGMV control is also good and is simpler to implement and understand. Reference Tracking: The reference tracking Scenario 3 involves a step change in power reference, due to fluctuations in power demand. The results for a small down-step change in power demand are shown in Fig. 14.10 and the benefit of future reference knowledge is clearly evident. Electric Power [MW]

Shaft Torsion [deg]

5.15

0.28

5.1

0.27

5.05

0.26

5

0.25

4.95

0.24

4.9

15

20

25

30

0.23

15

20

25

30

Generator Torque [kNm]

Pitch Angle [deg] 42

25 PID NGMV NPGMV (N=5) NPGMV (N=10)

20 15

41.5 41 40.5

10

40

5

39.5

15

20

25

30

15

Wind Speed [m/s]

20

25

30

Tower Deflection [m]

22

5.8

20 5.6

18 16

5.4

14 12

15

20

25

30

5.2

15

time [sec]

Fig. 14.8 Large wind gust variations of 12–18 m/s (Scenario 1B)

20

25

time [sec]

30

658

14

Nonlinear Industrial Process and Power Control Applications

Electric Power [MW]

5.1

Shaft Torsion [deg]

0.35

5.05

0.3

5 0.25

4.95 4.9 30

35

40

45

Pitch Angle [deg]

18

0.2 30

35

40

45

Generator Torque [kNm]

42 41.5

16

41

14

40.5

PID NGMV NPGMV (N=5) NPGMV (N=10)

12 10 30

35

40

40 39.5 45

Wind Speed [m/s]

18

30

35

40

45

Tower Deflection [m]

6.5 6

17

5.5 16 15

5 30

35

40

45

4.5

30

time [sec]

35

40

45

time [sec]

Fig. 14.9 Turbulent wind variations of 12–21 m/s (Scenario 2)

Reducing Torsional Shaft Vibrations Using MIMO Control: The previous SISO solutions involved the use of blade pitch angle to control power output. To illustrate the benefit of MIMO control, a two-input and two-output NPGMV controller is now used for the small gust Scenario 1A. The generator torque is used as a second control input with a requirement to limit shaft torsion as well as regulate shaft power output to the desired setpoint. The transient response results are shown in Fig. 14.11, and the second input involving generator torque is now active. The MIMO MPC not only has slightly worse power regulation performance than the dedicated SISO MPC but it also reduces oscillations in the shaft torsion, as required. The avoidance of excessive loading is an important objective in both the design and the implementation of wind turbine controllers, particularly for offshore wind farms. The problems in implementing wind turbine controllers have been considered by Leith and Leithead [12].

14.3

Tension Control in Hot Strip Finishing Mills Electric Power [MW]

5.1

0.3

5

659 Shaft Torsion [deg]

0.25

4.9 0.2

4.8 4.7 14.5

15

15.5

16

16.5

17

0.15 14.5

Pitch Angle [deg] 42

15 PID NGMV NPGMV (N=5) NPGMV (N=10)

14 13

16

16.5

17

Generator Torque [kNm]

41 40.5 40

15

15.5

16

16.5

17

39.5 14.5

Wind Speed [m/s] 16

5.48

15.5

5.46

15

15.5

16

16.5

17

Tower Deflection [m]

5.44

15

5.42

14.5 14 14.5

15.5

41.5

12 11 14.5

15

5.4 15

15.5

16

time [sec]

16.5

17

5.38 14.5

15

15.5

16

16.5

17

time [sec]

Fig. 14.10 Response to electric power reference small down-step change (Scenario 3)

14.3

Tension Control in Hot Strip Finishing Mills

The hot rolling mill is one of the main processes in steel strip mill production. The process involves reheating steel slabs close to their melting point, then rolling them through successive rolling mill stands until they are thinner and longer. A stored slab is first heated to the required rolling temperature in a reheat furnace and it is then reduced in thickness in roughing mill stands. The resulting “transfer bars” are at high temperature (typically 1100 °C for mild steel) and these enter the finishing mill, where the thickness is reduced further. The tandem hot strip finishing mill normally reduces the thickness by a factor of about 10. Because of friction in the roll gap, the width of the strip does not change and there will, therefore, be a corresponding increase in the length of the bar by the same factor. The tandem hot strip finishing mill is where the more advanced controls are needed. Each of the mill stands has a pair of independently driven work rolls supported by backup rolls that are of a larger diameter. The backup rolls provide rigid support and limit the bending of the work rolls. They are needed to reduce the convexity of the loaded roll gap. At the exit of the finishing mill, the strip is cooled in a run-out table before being coiled. The hot strip mill is a major part of the

660

14

Nonlinear Industrial Process and Power Control Applications

Electric Power [MW]

5.05

Shaft Torsion [deg]

0.3 0.28

5

0.26 0.24

4.95

15

20

25

30

Pitch Angle [deg]

20

15

20

25

30

Generator Torque [kNm]

40.9 SISO NPGMV MIMO NPGMV

15

0.22

40.8 40.7 40.6

10

40.5 5

15

20

25

30

Wind Speed [m/s]

20

15

20

25

30

Tower Deflection [m]

5.7 5.6

18

5.5

16

5.4

14 12

40.4

5.3 15

20

25

time [sec]

30

5.2

15

20

25

30

time [sec]

Fig. 14.11 Multivariable control solution for small wind gust variations (Scenario 4)

processing of flat steel products. After the steel is rolled into a continuous strip, the hot rolled coil is usually transported, stored and is then processed in a multi-stand cold rolling mill. The strip is thinner and harder when it enters the cold rolling mill but further reduction in the strip thickness occurs. The cold-rolled strip may be processed further in tinning or powder coating lines. The finished product is then delivered to the manufacturers of vehicles, food and drink cans, building materials and kitchen appliances. The control of tandem hot and cold rolling mills is a complex multivariable control problem [15, 16]. The control of tension is a problem in a hot rolling mill and loopers (described below) must be introduced between the mill stands. However, in the hot rolling mill process flow stresses are low, hence forces and power requirements are relatively low. The ductility is also high and large deformations can be achieved with less force and power. There are many quantities to control in rolling mills including roll speeds, strip tension, strip thickness, strip profile and so-called strip shape. This latter quantity refers to the tension stress distribution and the ability of the material to lie flat on a flat surface. Since there is the transport of material involved, there are transport-delays between each stand and these depend on the line speed [17, 18].

14.3

Tension Control in Hot Strip Finishing Mills

661

The control design must accommodate these varying delays which are destabilizing and also influence the exit temperature of the strip [19]. The main difference between a cold mill and a hot rolling mill lies in the use of the loopers that are placed between the hot rolling mill stands to control the strip tension. The yield strength of the strip in a hot mill is much lower than in a cold mill and the loopers are therefore needed to control the product quality and to reduce tension transients. A looper includes a roller on an arm and a pivot that is activated by hydraulics. This enables a varying force to be applied to the moving strip for disturbance rejection. A looper reduces tension variations by changing its angle, to improve the quality of the product. It also maintains stable operation by absorbing any increased loop of the strip that arises from a mass flow unbalance [20]. The control of a looper for a particular mill stand zone is the problem considered in Sect. 14.3.2.

14.3.1 Hot Mill Control A typical tandem hot strip rolling mill with loopers between stands is illustrated in Fig. 14.12. This is a highly interactive process and there are therefore significant advantages from using a true multivariable control design approach for tension, thickness and flatness control [17]. The state-dependent or qLPV modelling approach is very suitable for this application, as noted by Pittner and Simaan [21]. They employed a state-dependent Riccati equation approach for tandem hot rolling mill control (see Chaps. 10 and 11 and [22]). The hot rolling mill control system should enable operators to maximize the production of strip at a minimum cost whilst maintaining the product quality. The quality measures include the strip thickness (gauge), temperature, profile (also

Co-ordinated Supervisory Control ----------Local regulator level -----------

Load

C1

C2 L

C3 L

C4 L

C5 L

L

Material flow

Temperature and thickness sensor

Thickness, profile and flatness measurements Looper Roll speed Looper angle

Fig. 14.12 Hot strip finishing mill train with local and supervisory controls and looper for the control of the looper angle and strip tension (Based on the thesis and by courtesy Hearns, 2000 [23])

662

14

Nonlinear Industrial Process and Power Control Applications

known as crown), shape (also known as flatness), width and surface finish. The production costs involve the cost of equipment, energy consumption, raw materials, human resources and mill maintenance. The time to roll a coil must also be minimized if the throughput of the mill is to be maximized. Throughput may be maximized by starting to thread a coil during the tail out of the previous coil to maximize the mill throughput.

14.3.2 Looper Control Systems If the tension becomes too high in an interstand zone of a hot mill, the strip can break since the yield strength will be much lower than in a cold mill, and this results in a long down time that will be expensive. A looper is therefore placed midway between the stands of a tandem mill (the figures were based on the thesis by Hearns, 2000 [23]). It is pivoted on a short arm so that it can be raised above the pass-line, and form a “loop” in the strip between the stands. The zone between the stands stores a small amount of strip so that the looper angle can change and ensure disturbances that do not cause the strip to become either too tight or too loose. An ideal looper maintains the strip tension constant, and this has the very desirable effect of effectively decoupling the upstream and downstream stands. The looper is shown in Fig. 14.13 for the zone between the two stands, and in Fig. 14.12 for the loopers placed between the different stands of a tandem mill. The variation in the interstand strip tension has an important impact on the dimensional accuracy and the threading performance of hot strip mills. It is desirable that the interstand strip tension and the looper angle be kept approximately constant, but this conflicts with the need to regulate against the effects of disturbances. The range of looper operation is therefore limited, and the aim is to maintain the looper angle at a value that is approximately constant. The interstand strip tension and the looper angle are controlled using both the looper motor and the mill stand drive motors. If the interaction with thickness control is neglected, the looper control system has two inputs (looper torque and drive speed reference) and two outputs (looper angle and strip tension). Fig. 14.13 Hot strip mill looper roll angle control geometry (Based on the thesis and by courtesy Hearns, 2000 [23])

14.3

Tension Control in Hot Strip Finishing Mills

663

When choosing a control philosophy, the looper nonlinearity is important and NGMV control seems a natural design strategy. This is a regulating problem and predictive control does not seem so appropriate. However, predictive control still has a role in the wider hot strip mill control problem. The relationship between the looper angle and the torque (or equivalently looper motor current and the strip tension) is a nonlinear function of the looper arm angle. To keep the looper angle approximately constant, any changes in looper motor current must be accompanied by changes in the stand motor drive speeds. The looper is usually controlled by two servo loops responsible for the tension and the looper height. The strip tension control in a classical design involves regulation of the looper motor, whilst the looper angle is mainly controlled by adjusting the rotational speed of the upstream stand main drive motor. Poor control results in irregular up and down motions of the looper, which reduces the product quality and can lead to strip breakage. This is likely to result in serious down time whilst the strip is cleared from between the stands and the mill rolls are changed. Process Model: Consider the multivariable control of a single zone between two stands as shown in Fig. 14.13. Assume the two variables to be regulated are the strip tension r and the looper angle h. These are to be regulated by adjusting the looper torque reference M and the drive speed reference xr. The model is very nonlinear and subject to strip velocity perturbations. The schematic block diagram of the system is shown in Fig. 14.14 [23]. The open-loop responses to pulse changes in the torque and drive speed reference reveal a wide range of nonlinearities, as illustrated in Fig. 14.15.

Drive speed Reference ωr

Looper and Strip Tension i Strip Exit Velocity i

+

Strip velocity disturbance

–

Young’s Modulus

Strip Exit Velocity i+1 Strip exit thickness i

Strip Tension σ

– + σ

+ +

h Load Torque

Load Torque

Geometric Loop Length Looper arm and roll

Torque reference M

Looper Angle θ

+ – – Looper drive

θ Load Torque

Fig. 14.14 Hot strip mill looper schematic for the looper tension and angle model (Based on the thesis and by courtesy Hearns, 2000 [23])

664

14

Nonlinear Industrial Process and Power Control Applications 2

Looper angle [deg]

Strip tension [MN/m ] 15

25 20

10

15 10

5 0

20

40

60

0

Looper torque reference [kNm]

20

40

60

10 -3 Drive speed reference [Hz] 0

14 12

-5

10 -10 0

20

40

60

0

time [s]

20

40

60

time [s]

Fig. 14.15 Open-loop step responses of the looper and tension control system

14.3.3 Looper Control System Design The looper model in this example is represented by a black-box input subsystem W 1k , which becomes part of the NGMV controller structure, as shown in Fig. 14.16. For the design and implementation of the controller, the model was discretized using the Euler algorithm with a sample interval of 10 ms. Note that the system model is open-loop stable and that there are no explicit transport-delays. The parameter k will, therefore, be set to unity to allow for possible computing delays. There is no separate linear plant output subsystem in this example ðW0k ¼ IÞ. However, the output of the plant model block W 1k includes a linear disturbance model signal. The reference and disturbance models are both represented by near integrators: 

1 1 Wd ðz Þ ¼ Wr ðz Þ ¼ diag ; ð1  0:9999z1 Þ ð1  0:9999z1 Þ 1

1

 ð14:11Þ

It is assumed that there exists a stabilizing PID controller, which can be used as the starting point for tuning the NGMV error weightings, as discussed in Chap. 5 (Sect. 5.2.2). The NGMV controller was first computed based on the PID controller

14.3

Tension Control in Hot Strip Finishing Mills

665

Fig. 14.16 NGMV Controller structure for the looper and tension control application

motivated weightings and was then retuned. The error weighting, using a common denominator, was defined as follows: Pc ðz1 Þ ¼ diag

n

ð0:040760:04054z1 Þ ; ð1z1 Þ

Looper angle [deg]

15.5

ð0:001450:0024z1 þ 0:001z2 Þ ð1z1 Þ

2

10 7

o

ð14:12Þ

Strip tension [N]

15 1.5

14.5 14

1

13.5 PID NGMV

13 12.5

54

56

58

60

62

64

0.5 0

54

56

Speed reference [rad/s]

0.08

1.6

0.06

1.4

0.04

1.2

0.02

1

0

0.8

-0.02

0.6

-0.04

54

56

58

60

Time [sec]

58

60

62

64

Time [sec]

Time [sec]

62

64

0.4

10 4 Torque reference [Nm]

54

56

58

60

Time [sec]

Fig. 14.17 Regulation of responses to step disturbance change for PID and NGMV

62

64

666

14

Nonlinear Industrial Process and Power Control Applications

The controller synthesis returns the two linear compensator blocks in the NGMV controller structure shown in Fig. 14.16. Simulation Results: Simulations were performed with both the PID and NGMV controllers in the loop. In terms of disturbance rejection, the PID and NGMV controllers give comparable regulating performance as seen in Fig. 14.17. This confirms the assertion that the PID-based NGMV weightings selection is a useful starting point for controller tuning. However, the PID performance degrades when away from the operating point. This can be illustrated by changing the looper angle setpoint, as shown in Fig. 14.18. The NGMV controller compensates for the nonlinearities and reduces oscillations as compared with the original (fixed gain) PID control.

Looper angle [deg]

16

3

14

2

12

1

10

8

56

58

Strip tension [N]

0

PID NGMV

54

10 7

60

62

64

-1

54

56

Time [sec]

Speed reference [rad/s]

0.2

58

60

62

64

Time [sec] 2.5

10 4

Torque reference [Nm]

2 0 1.5 1

-0.2

0.5 -0.4 0 -0.6

54

56

58

60

Time [sec]

62

64

-0.5

54

56

58

60

62

64

Time [sec]

Fig. 14.18 Disturbance rejection and looper angle tracking responses for PID and NGMV Designs

14.4

14.4

Control of a Heavy Oil Fractionator

667

Control of a Heavy Oil Fractionator

The first of the process control design studies involves a heavy oil fractionator that has large transport-delays and is a multivariable system with a high degree of interaction. The NGMV control design method will be applied but first, the general fractionator control problem will be introduced. Fractionators are also known as fractional distillation columns. They are used in the downstream oil and gas sector for the separation of crude oil into its constituent parts. This is a process known as fractional distillation. Fractionation systems have different objectives, such as removing a light component from a heavy product, removing a heavy component from a light product, making two products, or making more than two products. These categories are referred to as stripping, rectification, fractionation and complex fractionation. The Shell heavy oil fractionator control problem was the outcome of a workshop [24]. The fractionation involved a separation process in which a quantity of a mixture (solid, liquid, suspension or isotope) was divided into a number of smaller quantities. The control design problem required a multivariable controller to track setpoints, handle constraints, reject disturbances and meet economic objectives. The control challenges involve the presence of long deadtimes, gain uncertainty, multiple constraints and mixed fast and slow responses. The fractionator problem had seven measured outputs, five inputs, and the product streams were divided into three parts from the top, side and bottom of the fractionator [25]. Three controlled inputs were present and two disturbance inputs were included. The system is interesting since it contains strong cross-coupled loop interactions. Figure 14.19 shows a schematic diagram of the heavy oil fractionator. There are three circulating loops (or reflux) located at the top, middle (intermediate reflux) and the bottom of the fractionator. These three reflux loops are used to remove heat from the system through heat exchangers. The heat originates from the gaseous feed stream whilst the reflux loops act like heat exchangers. These are responsible for removing some heat from the fractionator, and they can also be used to supply heat to other processes. The outputs of the multivariable problem are from the three product draws (top draw product composition, side draw product composition, and bottom-draw reflux temperature). The control inputs or manipulated variables involve valves controlling the top and side product-flow rates. The third manipulated variable involves control of the bottom reflux loop valve. Model predictive control is an obvious candidate for such systems [26, 27]. The heat required enters into the system, along with the feed, through the bottom of the fractionator. Fractionators are set up in series with other fractionating columns and heat is taken out from one column through the heat exchangers for the top two circulating reflux loops (upper reflux duty and intermediate reflux duty). This is used to provide the heat for steam generation processes or other fractionating columns. There is a significant difference between the three circulating, or reflux loops, and their effects on the system. The intermediate reflux loop can be

668

14

Nonlinear Industrial Process and Power Control Applications

PC

y3 T LC

Upper reflux

FC u1

d1 y4 T

Top draw

y1 A

Intermediate reflux d2

T

y6 Q(F,T) control

Side stripper

T y 5

FC u2

LC

Bottoms reflux u3

T

Side draw

y2 A T

F

y7

LC

Feed

Bottoms

Fig. 14.19 Shell heavy oil fractionator (Prett and Morari, 1987 [24]). Note Adapted from the text “Shell Process Control Workshop,” Butterworth, by Prett, D. M. and Morari, M., page 356, 1987, with permission from Elsevier

considered a measured disturbance that may be used for feedforward control, whilst the upper reflux loop must be treated as an unmeasured disturbance. Increasing the steam produced implies reducing the heat duty at the bottom loop making the process run more economically. The bottom reflux duty can be used to manipulate the product draws from the fractionator, and is used as a control input or manipulated variable.

14.4.1 Modelling the Fractionator System The model for the system may now be introduced. The top and side draws are the flow rates of products drawn from the top and sides of the fractionator. The controlled outputs (i.e. top end-point, y1 and side end-point, y2), are the output compositions. Prett and Morari [24] modelled the fractionator system as a 5  7 multivariable system of first-order transfer-functions with deadtime. Table 14.2

14.4

Control of a Heavy Oil Fractionator

669

Table 14.2 Shell heavy oil fractionator inputs and outputs Parameter

Signal

Symbol

Top draw Side draw Bottom reflux duty Intermediate reflux duty Upper reflux duty Top end-point composition Side end-point composition Top temperature Upper reflux temperature Side draw temperature Intermediate reflux temperature Bottoms reflux temperature

Control input Control input Control input Measured disturbance Unmeasured disturbance Controlled/measured output Controlled/measured output Measured output Measured output Measured output Measured output Controlled/measured output

u1 u2 u3 d2 = dmd d1 = dumd y1 y2 y3 y4 y5 y6 y7

Table 14.3 Transfer-function model of Shell heavy oil fractionator (deadtime and time-constants given in minutes) u1

u2

u3

d2

d1

y1

4:05e27s 50s þ 1

1:77e28s 60s þ 1

5:88e27s 50s þ 1

1:20e27s 45s þ 1

1:44e27s 40s þ 1

y2

5:39e18s 50s þ 1

5:72e14s 60s þ 1

6:90e15s 40s þ 1

y3

3:66e2s 9s þ 1

1:65e20s 30s þ 1

5:53e2s 40s þ 1

1:52e15s 25s þ 1 1:16 11s þ 1

1:83e15s 20s þ 1 1:27 6s þ 1

y4

5:95e11s 12s þ 1

2:54e12s 27s þ 1

8:10e2s 20s þ 1

1:73 5s þ 1

1:79 19s þ 1

2s

1:31 2s þ 1

1:26 22s þ 1

1:19 19s þ 1

1:17 24s þ 1

1:14 27s þ 1

1:26 32s þ 1

y5

7s

5s

2:38e 19s þ 1

6:23e 10s þ 1

8s

4s

1s

4:13e 8s þ 1

y6

4:06e 13s þ 1

4:18e 33s þ 1

y7

4:38e20s 33s þ 1

4:42e22s 44s þ 1

6:53e 9s þ 1 7:20 19s þ 1

shows the input and output signals of the fractionator. Table 14.3 shows the transfer-function relationships between each control and disturbance input to their corresponding outputs.

14.4.2 Fractionator System Objectives and Constraints A controller is required to meet the following requirements: (1) To keep the top and side draw end-point product compositions, y1 and y2, within the range 0.0 ± 0.005, at the desired steady-state levels. (2) To maximize steam production which involves minimizing the heat duty in the bottom circulating reflux loop (bottom reflux temperature, y7). To maximize the

670

14

Nonlinear Industrial Process and Power Control Applications

heat removal and save cost; this should be kept at a minimum permissible bottom heat duty of not less than –0.5. (3) To compensate for the unmeasured d1 = dumd and measured d2 = dmd disturbances entering the system, from upper reflux and intermediate loops, due to changes that may occur in the heat duty requirements of other fractionating columns. The disturbance signals (unmeasured and measured) due to the upper and intermediate reflux duties should stay within the range −0.5 to 0.5. (4) To maintain the closed-loop speed of response between 0.8 and 1.25 of the open-loop process bandwidth. Constraints: The following constraints should be satisfied by the proposed control action: (1) The fastest sampling time should not be less than 1 min and the manipulated variables are restricted to a maximum move size of 0.05 per minute. (2) The top and side product draw rates (u1 and u2) and the bottoms reflux duty (u3) should be kept within hard maximum and minimum transient limits of 0.5 and – 0.5 ð0:5  ui  0:5Þ. (3) The minimum value of the bottom reflux-draw temperature is –0.5 ð0:5  y7 Þ. (4) The top end-point composition should not exceed the maximum and minimum values of 0.5 and –0.5 ð0:5  y1  0:5Þ. (5) The side end-point composition y2 and bottoms reflux temperature y7 should not be out with the maximum and minimum values of 0.5 and –0.5 (0:5  y2  0:5 and 0:5  y7  0:5).

14.4.3 System Model and Controller Design The Shell fractionator model has different transport-delays in the output channels. Note that a sample time of 4 min was assumed and hence for a maximum input move of 0.05/min the slew rate limits can be set a ±0.2. For an NGMV controller design, a common delay term is normally extracted, equal to the largest common delay in the controlled outputs and control inputs. In the following, the delay is assumed to be the minimum of just one step, which is represented by a common delay term of k = 1 (corresponding to 4 min). The NGMV optimal control approach is very suitable for process control applications that include troublesome delays because of its relationship to the Smith Predictor that is widely used in process control. The fractionator system is large, multivariable and interacting, but the only nonlinearities considered in this example are those due to constraints, which are treated here as saturation style hard limits. To illustrate the design of NGMV controllers, both of the cases with and without constraints will be considered. Both a classical discrete-time PID controller and an NGMV controller are considered below.

14.4

Control of a Heavy Oil Fractionator

671

Cost weightings: The system and weighting models used to compute the NGMV controller were as follows: Reference model: Wr ðz1 Þ ¼

1 I3 ð1  0:999z1 Þ

ð14:13Þ

Control Weighting:   0:4ð1  0:01z1 Þ 1  0:3z1 1  0:3z1 ;  ;  F ck ðz1 Þ ¼ diag  ð1  0:001z1 Þ ð1  0:01z1 Þ ð1  0:01z1 Þ ð14:14Þ Error Weighting: 

0:12ð1  0:8z1 Þ 0:15ð1  0:91z1 Þ 0:3ð1  0:91z1 Þ Pc ðz Þ ¼ diag ; ; ð1  z1 Þ ð1  z1 Þ ð1  0:999z1 Þ 1



ð14:15Þ Simulation Results: The NGMV controller was designed for different simulation scenarios: • With or without nonlinearities or constraints in the controllers or plant. • In the absence or presence of the unmeasured disturbances (upper reflux duty). Open-Loop Responses: Figure 14.20 shows the open-loop responses of the heavy oil fractionator. It is a reasonably well-behaved system but with very different transport delays in each channel. The two responses shown in this figure are the continuous-time open-loop responses and the discrete-time approximation. The common delay of k = 1 samples (due to the u3 ! y7 path) was used in the NGMV controller design, and the additional delays were included in the W0k model, adding to the system order. Closed-Loop Responses: The closed-loop responses of the fractionator system were generated using both the NGMV and PID controllers for both the unconstrained and constrained cases. The results in Fig. 14.21 are for the case where there are no saturation constraints or nonlinearities. Both the NGMV and the PID controllers can meet the output end-point specification of 0.0 ± 0.005, but the PID

672

14 From u1

To y 1

4

To y 2

From u2

2

From u3

6 4

2

1 2

0

0

50

100

150

200

0

6

6

4

4

2

2

0

50

100

150

200

0

0

50

100

150

200

0

50

100

150

200

10

5

0

To y 7

Nonlinear Industrial Process and Power Control Applications

0

50

100

150

200

0

6

6

4

4

2

2

0

50

100

150

200

0

10

5

0

0

50

100

150

200

0

continuous-time model discretized model

0

50

100

150

200

0

0

50

100

150

200

Time (minutes)

Fig. 14.20 Open-loop step responses of the fractionator for outputs y1, y2 and y7

controller has a more sluggish response compared to the NGMV controller. The bottom reflux temperature was minimized to almost the minimum set value (most economic level). The controllers do not satisfy the transient constraint on the top draw but the NGMV design has the least overshoot. For the case shown in Fig. 14.22, where the plant includes saturation terms representing constraints, the NGMV controller includes the saturation subsystem within its structure, but neither controller keeps the top end point composition (y1) within the specified composition limit of 0.0 ± 0.005. The NGMV was better in this respect but the input-one control signals u1 for both controllers drifted off due to windup effects. This can be improved in the PID design by introducing anti-windup or a limit on controller outputs, which is not, of course, the best solution but is sufficient for current purposes. However, the NGMV design has a neater solution, discussed in Chap. 5, whose results are shown in Fig. 14.23. In this case, anti-windup is achieved by defining the inverse of the control signal costing term F 1 ck to include the 0:5 limits.

14.4

Control of a Heavy Oil Fractionator Output y1

0.1

0.4 0.2

-0.1 NGMV PID

-0.2

0

500

1000

1500

2000

0

2500

Output y2

0.2

-0.2

0

-0.2

-0.1

0

500

1000

1500

2000

2500

Output y7

0

0

500

-0.2

1000

1500

2000

2500

2000

2500

2000

2500

Control u2

0.1

0

-0.4

Control u1

0.6

0

-0.3

673

0

500

1000

1500

Control u3

0 -0.1

-0.2

-0.2 -0.4 -0.6

-0.3

0

500

1000

1500

time (min)

2000

2500

-0.4

0

500

1000

1500

time (min)

Fig. 14.21 System responses in the absence of input constraints comparing PID and NGMV control and no output disturbances

Unmeasured Disturbance Rejection: In a similar scenario considering first the unconstrained case, unmeasured disturbances, driven by alternating step signals between −0.5 to 0.5, were fed into the system within a runtime interval from 1148 to 1448 min. The system response with NGMV and PID controllers is shown in Fig. 14.24. In the presence of the unmeasured disturbances, the system oscillates during the time period for which the disturbance lasts. The NGMV controller has the worst transient disturbance rejection response for the top end-point y1 compared

674

14

Nonlinear Industrial Process and Power Control Applications

Output y1

0

Control u1

1.5 1

-0.1

0.5 NGMV PID

-0.2 -0.3

0

500

1000

1500

2000

0

2500

Output y2

0.2

-0.5

0

500

1000

1500

2000

2500

2000

2500

2000

2500

Control u2

0.1 0.05

0

0 -0.2 -0.4

-0.05

0

500

1000

1500

2000

2500

Output y7

0

-0.1

0

500

1000

1500

Control u3

0 -0.1

-0.2

-0.2 -0.4 -0.6

-0.3

0

500

1000

1500

time (min)

2000

2500

-0.4

0

500

1000

1500

time (min)

Fig. 14.22 System responses with plant input constraints for PID and NGMV controllers with no disturbances (note drift on control signal one)

with the PID, but the latter has more difficulty restoring the top end-point composition to its initial state before the effect of the disturbance. However, for the side end-point y2 and bottom reflux temperature y7, both controllers show good steady-state disturbance rejection capabilities (returning the system to setpoint after the effect of the disturbance). The control signal u1 does exceed the transient response limits for both controllers but the same approach may be taken as mentioned above to limit the windup effect. The results when the constraints are applied and disturbances are present are shown in Fig. 14.25.

14.4

Control of a Heavy Oil Fractionator Output y1

0

675 Control u1

0.6 0.4

-0.1

0.2 -0.2 -0.3

0

0

500

1000

1500

2000

2500

Output y2

0.2

-0.2

0

500

1000

1500

2000

2500

2000

2500

2000

2500

Control u2

0.1 0.05

0

0 -0.2 -0.4

-0.05

0

500

1000

1500

2000

2500

Output y7

0

-0.1

0

500

1000

1500

Control u3

0 -0.1

-0.2

-0.2 -0.4 -0.6

-0.3

0

500

1000

1500

time (min)

2000

2500

-0.4

0

500

1000

1500

time (min)

Fig. 14.23 System responses with plant input constraints for NGMV controller with no disturbances (NGMV windup limits applied through control signal costing)

Benefits of the NGMV control approach: The nonlinear generalized minimum variance control methodology is well suited for systems with strong nonlinear characteristics. The oil fractionator model is a system that can be approximated by multivariable linear first-order transfer-functions with deadtimes. Although the presence of constraints introduces a nonlinearity, the system is still predominantly linear. The model contains different delay terms at the outputs and use of the common delay term k for the design of the NGMV controller was only an

676

14

Nonlinear Industrial Process and Power Control Applications

Output y1

0.4

NGMV PID

0.2

Control u1

1

0

0.5

-0.2 -0.4

0

500

1000

1500

2000

2500

Output y2

1

0

0

500

1500

2000

2500

2000

2500

2000

2500

Control u2

0.2

0.5

1000

0

0 -0.2

-0.5 -1

0

500

1000

1500

2000

2500

Output y7

0

-0.4

0

500

1500

Control u3

0

-0.2

1000

-0.2

-0.4 -0.4

-0.6 -0.8

0

500

1000

1500

time (min)

2000

2500

-0.6

0

500

1000

1500

time (min)

Fig. 14.24 System responses to an unmeasured disturbance for the PID and NGMV controllers without plant or controller constraints

approximation. The controller was also designed based on parameters estimated which involves some degree of approximation. Real fractionators will probably have an additional significant nonlinearity. In this case the NGMV controller would give better responses than those shown, assuming this nonlinearity is included in the NGMV control design.

14.5

Control of a CSTR Process

677

Output y1

0.4

NGMV PID

0.2

0.4

0

0.2

-0.2

0

-0.4

0

500

1000

1500

2000

2500

-0.2

Output y2

1

Control u1

0.6

0

500

1500

2000

2500

2000

2500

2000

2500

Control u2

0.2

0.5

1000

0

0 -0.2

-0.5 -1

0

500

1000

1500

2000

2500

-0.4

Output y7

0

-0.1

-0.4

-0.2

-0.6

-0.3

0

500

1000

1500

500

2000

2500

-0.4

time (min)

1000

1500

Control u3

0

-0.2

-0.8

0

0

500

1000

1500

time (min)

Fig. 14.25 System responses to an unmeasured disturbance for the PID and NGMV controllers with plant constraints and controller compensation (NGMV limits applied through control signal costing)

14.5

Control of a CSTR Process

The second process control example involves a continuous stirred tank reactor (CSTR). The NGMV control design strategy will again be applied but first, the general problem will be introduced. A reactor is a major element of a chemical production plant, where the starting materials, or reactants, react together to form a new substance or product [28]. A reactor should be controlled so that the conditions for an optimal reaction are maintained. Nonlinearities arise in all process plants and a spherical tank reactor has a particular nonlinear behaviour. The amount of liquid required to change the level by one centimetre varies according to the height of liquid because of the changing tank diameter. To control the level, the nonlinearity must be addressed and an obvious classical approach is to use gain scheduling.

678 Fig. 14.26 Continuous stirred tank reactor

14

Nonlinear Industrial Process and Power Control Applications

q, T0, Ca0

A  B + heat

qc, Tc0

T, Ca q, T, Ca

The process to be considered below is an irreversible exothermic first-order reaction, which takes place in a CSTR. Figure 14.26 shows a graphical representation of the process and the input and output process variables. A batch reactor includes an impeller or other mixing device to ensure efficient mixing. The various quantities and parameters are collected in Table 14.4. It will be assumed that • The liquid in the reactor is perfectly mixed so that the temperature and concentration of the product stream and the bulk reactor fluid are the same. • The feed flow is equal to the product outflow so that the liquid volume in the reactor is constant.

Table 14.4 Reactor input and output variables Symbol

Unit

Nominal value

Product concentration Reactor temperature Coolant flow rate Process flow rate Feed concentration Feed temperature Inlet coolant temperature CSTR volume Heat transfer term Reaction rate constant

Ca T qc q Ca0 T0 Tc0 V hA k0

mol/l K l/min l/min mol/l K K l cal/min/K

0.1 438.54 103.41 100 1 350 350 100 7.105

min1

Activation energy term

E=R

Heat of reaction

DH

K1 cal/mol

7:2  1010 1.104

Feed density Coolant density Feed-specific heat Coolant-specific heat

r rc Cp Cpc

g/l g/l cal/g/K cal/g/K

2  105 1.103 1.103 1 1

14.5

Control of a CSTR Process

679

Liu and Lewis [29] note that this problem exhibits significant nonlinearities and extreme parametric sensitivity requiring a robust control solution. Under the assumptions above, the CSTR process can be modelled by the following equations:
 qðtÞ E=R C_ a ðtÞ ¼ ðCa0  Ca ðtÞÞ  k0 Ca ðtÞexp  V T ðt Þ
 qð t Þ DH E=R T_ ðtÞ ¼ ðT0  T ðtÞÞ þ k0 Ca ðtÞexp  V qCp T ðt Þ 
 q Cpc hA qc ðtÞ 1  exp þ c ðTc0  T ðtÞÞ qc Cpc qc ðtÞ qCp V

ð14:16Þ

ð14:17Þ

Equation (14.16) results from the material balance. The second term contains the reaction rate for the first-order reaction. Equation (14.17) follows from the energy balance, which involves the heat produced by the reaction and the heat removed from the reactor. The problematic behaviour of the CSTR process is considered about a setpoint where the concentration of the product stream is 0.1 mol/l. The coolant flow rate qc (l/min) is regarded as the input to the process and the product concentration Ca (mol/l) is regarded as the output. The CSTR process exhibits some rich nonlinear behaviour. It has multiple steady-state solutions and it has a clear nonlinear dynamic response, as seen in Fig. 14.27. These are widely used in batch process manufacturing involving low-volume and high-value products where difficulties in control are very costly.

CSTR open-loop responses

0.15

110

0.1

0.05

100

0

20

40

60

time [min]

Fig. 14.27 Open-loop responses of the CSTR

80

100

90 120

qc [l/min]

Ca [mol/l]

control input output

680

14

Nonlinear Industrial Process and Power Control Applications

14.5.1 NGMV Control of a CSTR A linearized model was obtained for the CSTR and the NGMV controller was computed for the resulting linear system [30]. The NGMV controller provides a GMV control solution when the linearized model is used. Two choices of dynamic weightings were employed to conduct a comparison between the four resulting NGMV and GMV controllers. For the NGMV nonlinear controllers, the nonlinear plant model was included in the input operator subsystem model, or so-called black-box term of the NGMV controller. For the GMV linear controllers, the black box term W 1k was set to the identity. The differential Eqs. (14.16) and (14.17) can be linearized about the nominal operating point. Using the sampling period Ts = 0.1 min, the linearized system is obtained as Wlin ¼ z5

104 ð0:01571 þ 1:863z1 þ 0:4614z2  1:14z3 Þ 1  2:338z1 þ 1:88z2  0:5098z3

The eigenvalues of this system are located inside the unit-circle and hence the nominal working point is stable. The open-loop model verification for “small” and “large” deviations from the operating point is shown in Fig. 14.28. It can be seen that the linear model is only applicable to relatively small excursions about the working point.

Small deviations

Ca [mol/l]

0.11

Linear model Nonlinear model

0.105 0.1 0.095 0.09

0

10

20

30

40

50

60

70

80

90

Ca [mol/l]

Large deviations Linear model Nonlinear model

0.15

0.1

0.05

0

10

20

30

40

50

time [min]

Fig. 14.28 Verification of the linear model

60

70

80

90

14.5

Control of a CSTR Process

681

The disturbance model was chosen as Wd ¼ 0:001=ð1  0:95z1 Þ. The reference model is assumed zero, but integral action will be introduced to the controller through an integral error cost-weighting. Design of GMV and NGMV Cost-Function Weightings: Two choices of dynamic weightings were considered for the controller design: (1) Derived directly from an existing PI controller design: P1c ¼

14ð1  0:5z1 Þ ; ð1  z1 Þ

1 Fck ¼ 1

(2) Retuned for faster tracking: P2c ¼

1  0:85z1 ; ð1  z1 Þ

2 Fck ¼ 0:02ð1  0:1z1 Þ

These weightings were used to design the two pairs of controllers: GMV1/ NGMV1 and GMV2/NGMV2. The simulation results in Fig. 14.29 show the transient responses with these controllers in the feedback loop (without noise) for “large” deviations from the nominal setpoint. The results confirm a good performance of the nonlinear GMV controller over the whole operating range. Whilst the linear GMV control is adequate for small deviations from the nominal operating point, its performance degrades away from it Ca (mol/L)

0.14 0.12 0.1 0.08 0.06

0

10

20

30

40

50

60

70

90

PI GMV1 GMV2 NGMV1 NGMV2

qc (L/min)

110

80

105

100

95

0

10

20

30

40

50

60

time (s)

Fig. 14.29 CSTR step responses with GMV and NGMV control

70

80

90

682

14

Nonlinear Industrial Process and Power Control Applications

(compared with the nonlinear control). This is especially true for higher values of the product concentration Ca where the nonlinearity becomes significant. It is less noticeable for smaller values where the main difference between the use of linear and nonlinear models lies in the steady-state offset, which does not affect dynamic performance. From Fig. 14.29 it can be seen that the controllers GMV2 and NGMV2 result in faster transients, due to a greater bandwidth, and that the response for the linear controller GMV2 is starting to become unstable in the higher operating region. The PI-based controller weights for GMV1 and NGMV1 lead to responses that are close to the PI design, as might be expected.

14.5.2 Using State-Dependent NGMV Control for the CSTR A different approach will now be taken to the NGMV controller design for an unstable CSTR model problem [31]. In this case, a state-dependent or qLPV model of a CSTR will be employed. The cooling jacket temperature Tc is the input to the process and the product concentration Ca is again the output. The CSTR process model has nonlinear behaviour that involves multiple steady-state solutions, and both stable and unstable equilibrium points. The normalized dimensionless model given in Hernandez and Arkun [32] will be used for the following example, where the values of the system parameters are as listed in Table 14.5. The state equations for the model may be written as x_ 1 ¼ x1 þ Da ð1  x1 Þexpðx2 ð1 þ x2 /ÞÞ x_ 2 ¼ x2 þ BDa ð1  x1 Þexpðx2 ð1 þ x2 /ÞÞ þ bðu  x2 Þ y ¼ x1

ð14:18Þ ð14:19Þ ð14:20Þ

Defining a new-scaled input us ¼ ðu þ 4Þ=8, both the input and output signals are contained in the range (0, 1). The model for control design must often be found using system identification methods. In this second CSTR problem, a polynomial NARMAX model was determined using the System Identification Toolbox for MATLAB. A dimensionless sample time of 0.5 units was used to generate the data for identification. The CSTR model was found to include two stable regions, at the two ends of the output range, Table 14.5 CSTR reactor parameters for dimensionless model

Parameter

Meaning

Value

Da / B b

Damköhler number Dimensionless activation energy Heat of reaction coefficient Heat transfer coefficient

0.072 20.0 8.0 0.3

14.5

Control of a CSTR Process

683

separated by an unstable region [31]. The following model was found to provide a good balance between model accuracy and complexity: yðtÞ ¼ h0 þ h1 yðt  1Þ þ h2 yðt  2Þ þ h3 us ðt  1Þ þ h4 us ðt  2Þ þ h5 y2 ðt  1Þ þ h6 yðt  1Þyðt  2Þ þ h7 yðt  1Þus ðt  1Þ þ h8 yðt  1Þus ðt  2Þ þ h9 y3 ðt  1Þ þ h10 y2 ðt  1Þyðt  2Þ ð14:21Þ with the estimated parameter values: h0 ¼ 0:0129;

h1 ¼ 0:0390;

h5 ¼ 7:8027; h6 ¼ 6:5853; h10 ¼ 7:0073

h2 ¼ 0:5112; h7 ¼ 0:1232;

h3 ¼ 0:0219; h8 ¼ 0:0877;

h4 ¼ 0:0290; h9 ¼ 7:9623;

The model verification is shown in Fig. 14.30 and a good match is confirmed between the plant and model responses for both the estimation and the validation data. A comparison between the continuous-time system and the NARX model static characteristics is shown in Fig. 14.31. The curves were plotted by considering a range of steady-state outputs and computing the corresponding inputs from Eqs. (14.20) and (14.21). Multiple equilibrium points are evident and the unstable equilibria have been highlighted in green. These correspond to the local negative

Output y

1

data estimate

0.5

0

0

50

100

150

200

250

300

350

400

250

300

350

400

Input us

1

0.5

0

0

50

100

150

200

Time

Fig. 14.30 NARX model validation: plant (solid), model (dashed) (the estimation data used for identification: up to time 200)

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1 0.9

steady-state output

0.8 0.7 0.6 0.5 0.4 0.3 0.2

System NARX model Unstable region

0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

steady-state input

Fig. 14.31 Static characteristic of the reactor system

gain (slope) of the model. For the NGMV control, the NARX model was converted to a state-dependent or a qLPV model by defining the following states: x1 ðtÞ ¼ yðtÞ;

x2 ðtÞ ¼ yðt  1Þ;

x3 ðtÞ ¼ us ðt  1Þ

The qLPV model follows as 3 2 2 x1 ðt þ 1Þ x1 þ x1 þ 1 6 x2 ðt þ 1Þ 7 6 1 6 7 6 4 x3 ðt þ 1Þ 5 ¼ 4 0 x4 ðt þ 1Þ 0 2

x21 þ x1 þ 1 0 0 0

x1 þ 1 0 0 0

32 3 3 2 x1 ðtÞ x1 þ 1 1 7 6 7 6 07 76 x2 ðtÞ 7 þ 6 0 7u ðtÞ 0 54 x3 ðtÞ 5 4 1 5 s 0 x4 ðtÞ 1 ð14:22Þ

yðtÞ ¼ x1 ðtÞ

ð14:23Þ

14.5.3 Control Design and Simulation Results A major challenge is to control the system around an unstable equilibrium point, i.e. for the mid-range concentration values. As an example, a control input uðtÞ ¼ 0:5 was selected. Based on the model equations, the corresponding output steady-state values can be computed as y01 ¼ 0:1437, y02 ¼ 0:3652 and y03 ¼ 0:7658. In terms of these three solutions, y02 is the unstable equilibrium, and the control objective will be assumed to regulate the system around this level. The NGMV controller was designed for this problem using the following models:

14.5

Control of a CSTR Process

685

Reference model: Wr ðz1 Þ ¼

0:05 ð1  0:95z1 Þ

Error weighting: We ðz1 Þ ¼

ð1  0:1z1 Þ2 ð1  0:97z1 Þ2

Control weighting: Wu ðz1 Þ ¼ 0:55

ð1  0:3z1 Þ2 ð1  0:9z1 Þ2

The control problem is defined to move the process from the stable equilibrium y01 ¼ 0:1437 to the unstable operating point y02 ¼ 0:3652 and keep it there in the presence of step output disturbances. The comparison of the PID and NGMV control results for this scenario is shown in Fig. 14.32. The NGMV controller clearly gives a much smoother output for the tracking performance. Switching dynamic weightings: The final test consisted of tracking the concentration setpoint across the operating range. The approach used was to define a separate set of the control weighting parameters for each operating region (four Output

0.5 0.4 0.3

PID StDep NGMV

0.2 0.1

0

50

100

150

200

250

150

200

250

Control

1

0.5

0

-0.5

0

50

100

time (s)

Fig. 14.32 Output tracking and regulation about an unstable operating point

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14

Nonlinear Industrial Process and Power Control Applications

regions were specified). A simple switching scheme was then used when moving from one region to another. The NGMV control weighting function was parameterized in the following form: Fck ðz1 Þ ¼ q 

ð1  bz1 Þ2

ð14:24Þ

ð1  az1 Þ2

where the parameters q, b and a were tuned separately for each region. Table 14.6 contains the values of the parameters. The control signal costing values were collected in look-up tables and switched using gain scheduling. NGMV control results: The nominal simulation results (with the NARX model as the “plant”) are shown in Fig. 14.33, and good tracking is achieved across the operating range. On the other hand, the simulations with the original model of the system, shown in Fig. 14.34, result in some deterioration in performance in the Table 14.6 Control weighting parameters

Region centre (Ca)

b

a

q

0.1 0.4 0.55 0.8

−0.3 −0.3 −0.3 −0.5

0.92 0.92 0.92 0.9

0.4 0.65 0.3 1.2

Output 0.8 0.6 Unstable Region 0.4 0.2 0

0

50

100

150

200

250

300

350

400

450

500

300

350

400

450

500

Control

1

0.5

0

-0.5

0

50

100

150

200

250

time (s)

Fig. 14.33 NGMV control of output tracking across the operating range (with a gain-scheduled control weighting and nominal model)

14.5

Control of a CSTR Process

687 Output

0.8 0.6 Unstable Region 0.4 0.2 0

0

50

100

150

200

250

300

350

400

450

500

300

350

400

450

500

Control 1.5 1 0.5 0 -0.5 0

50

100

150

200

250

time (s)

Fig. 14.34 NGMV Control output tracking across the operating range (with a gain-scheduled control weighting for the original model)

unstable region. This is the result of the plant model mismatch and can be corrected by identifying a better model, possibly by including more regression terms, but not necessarily polynomial ones.

14.6

Nonlinear Predictive Control of an Evaporator Process

The third process control example considered in this chapter involves the multivariable control of a forced circulation evaporation process. In this case, a predictive control solution is used. The general control problem and background will first be introduced. Evaporators are used in a number of industries such as chemicals and pharmaceuticals, pulp and paper, beverages and sugar production, and many others. An evaporator is often used in the food processing industry to remove some of the water from food products. This may be needed for further processing or to increase the solids content. It can help to preserve the food being produced, whilst often improving the colour and adding to the flavour. An evaporator is also used in many industrial processes to evaporate the solvent from a feed stream. A somewhat different but common application is in refrigeration systems, where a compressed chemical is able to perform cooling by absorbing heat in the evaporation process. Evaporation is not the same as distillation since there is no attempt to separate the vapours into individual components.

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Nonlinear Industrial Process and Power Control Applications

14.6.1 Process Model and Evaporator Control Problem An evaporator in a process can turn a liquid into a vapour. This is the part of a refrigeration system where the refrigerant absorbs heat, which results in the change from a liquid into a gas. In the following forced circulation evaporator example, described in Newell and Lee [25], the concentration of the liquor is increased by evaporating solvent from a feed stream through a vertical heat exchanger. This model has often been used to compare designs and was a problem considered in Maciejowski [26] to illustrate the performance of model predictive control laws. The process considered here is essentially a heat exchanger, where the steam comes in at the top of the evaporator and condensate leaves at the bottom. This is represented schematically in Fig. 14.35 (figure based on the problem described in Chap. 2 of Newell and Lee [25]. As the recirculation liquor boils in the evaporator, Disturbances F1

F3

X1 L2

F2 Inputs

P100

Evaporator

F200

X2 P2

F4, T3

F200, T200

Vertical heat exchanger Separator

Heater steam jacket input

Steam

P100 F100

L2

P2

Cooling water Condenser F5 Condensate

LC

Level control loop for separation vessel

T100

Forced circulation evaporator

Product composition is main variable to be controlled to minimize variations

Condensate

Liquid feed

T201

Outputs

F1, X1, T1

F3 Pump

Recirculation Liquid drawn off

F2, X2, T2

Product

Fig. 14.35 Evaporator system with steam, cooling and feed inputs, and product and condensate outputs. The economic objective is to minimize operational cost related to steam, cooling water and pump effort (Based on the system described in the text by Newell and Lee, 1989, Chapter 2 [25])

14.6

Nonlinear Predictive Control of an Evaporator Process

689

a two-phase mixture of liquid and vapour flows to the separator unit. The liquid and vapour are then separated from each other. The resulting liquid is, of course, more concentrated than when it enters the evaporator. It is then pumped around again whilst some of it is drawn off as a product. The vapour flows to a condenser (another type of heat exchanger), where in this case the heat is exchanged with cooling water. The condensate is drawn off as a product of the process. Evaporator Control Problem: The feed stream is assumed to enter the process with a concentration X1, temperature T1 and with a flow rate F1. The control problem is to concentrate the liquid fed into the process by heating it with steam and then separating the vapour. A part of the concentrated liquid is retrieved as the product, whilst the rest is recirculated back into the evaporator by a pump. It is important to keep the variations in the composition as small as possible since this maximizes profitability. Out of specification, the product cannot be sold at the best price and it is important to maintain the product composition X2 better than the lower limit. The control of the evaporator must also be safe, avoiding harm to operators and damage to the plant. This requires the pressure in the evaporator P2 and the liquid level in the separator L2 to be controlled accurately. Note that if a separator overflows the condenser will be damaged, and if a pump runs dry then the pump may be damaged [33]. Disturbances: The evaporator performance is affected by disturbances and some of these stem from interactions with the other variables that are controlled which may drift away from their reference levels. This applies to the circulating flow rate F3, the feed flow rate F1, the feed composition X1, the feed temperature T1 and the cooling water inlet temperature T200. In this problem, the disturbances are taken to be the variables F3, F1 and X1. Control inputs: The control inputs (manipulated variables) can be considered the mass flow rate of the product drawn off F2, the pressure of steam entering the evaporator P100 and the mass flow rate of the cooling water entering the condenser F200. Controlled outputs: There are three outputs (controlled variables), namely the liquid level in the separator L2, the concentration X2 that determines product quality, and the evaporator pressure P2. The product composition must be controlled to reduce off specification material and safe operation requires the evaporator pressure P2 and the separator level L2 to be controlled. The process is assumed to operate around the operating point (L20, X20, P20) = (1 m, 30%, 50.5 kPa). States and state equation: The process model has three states: separator level L2 (m), product concentration X2 (%) and operating pressure P2 (kPa), which are also the controlled outputs. The dynamic equations of the system can be written as follows: qA

dL2 ¼ F1  F4  F2 dt

ð14:25Þ

M

dX2 ¼ F1 X1  F2 X2 dt

ð14:26Þ

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Nonlinear Industrial Process and Power Control Applications

C

dP2 ¼ F4  F5 dt

ð14:27Þ

where F4 ¼ ½0:16ðF1 þ F3 Þð0:3126X2  0:5616P2 þ 0:1538P100 þ 41:57Þ  F1 Cp ð0:3126X2 þ 0:5616P2 þ 48:43  T1 Þ=ks1

ð14:28Þ

and   F5 ¼ 2UA2 ð0:507P2 þ 55  T200 ÞCp F200 = ks2 UA2 þ 2Cp F200

ð14:29Þ

All the variables with their nominal values are collected in Table 14.7. The process [34] is subject to the following constraints, related to product specification, safety and design limits: X2  25:5%

ð14:30Þ

40 kPa  P2  80 kPa

ð14:31Þ

Table 14.7 Evaporator variables [25] Variable

Description

Value

Unit

F1 F2 F3 F4 F5 X1 X2 T1 T2 T3 L2 P2 F100 T100 P100 Q100 F200 T200 T201 Q200

Feed flow rate Product flow rate Circulating flow rate Vapour flow rate Condensate flow rate Feed composition Product composition Feed temperature Product temperature Vapour temperature Separator level Operating pressure Steam flow rate Steam temperature Steam pressure Heater duty Cooling water flow rate Cooling water inlet temperature Cooling water outlet temperature Condenser duty

10.0 20.0 50.0 8.0 8.0 5.0 25.0 40.0 84.6 80.6 1.0 50.5 9.3 119.9 194.7 339.0 208.0 25.0 46.1 307.9

kg/min kg/min kg/min kg/min kg/min % % °C °C °C m kPa kg/min °C kPa kW kg/min °C °C kW

14.6

Nonlinear Predictive Control of an Evaporator Process

691

P100  400 kPa

ð14:32Þ

F200  400 kg/min

ð14:33Þ

0 kg/min  F3  100 kg/min

ð14:34Þ

The process is open- loop unstable due to the integrating behaviour of the tank (Eq. 14.25). Thus, the level control loop must always be closed by using feedback to control the flow F2.

14.6.2 LPV Modelling of the Evaporator An LPV model of the evaporator was derived based on analytical linearization of the nonlinear model. A modified Jacobian LPV model with an affine term was then used to represent system local dynamics about a general operating point: d_xðtÞ ¼ f ðxt ; ut ; dt Þ þ





@f

@f

@f

dx þ du þ ddt t t @x op @u op @d op

ð14:35Þ

or d_xðtÞ ¼ ct þ Aðpt Þdxt þ Bðpt Þdut þ Gðpt Þddt

ð14:36Þ

with the function f (∙) described by Eqs. (14.25)–(14.29). At any given time t, the LPV system matrices Aðpt Þ, Bðpt Þ and Gðpt Þ can be evaluated by substituting for the vector of scheduling parameters pt (assumed measured or estimated). These include all of the following variables: L2, X2, P2, F2, P100, F200, F3, F1, X1, T1 and T200. The presence of the state and control variables in the parameter vector pt makes the model quasi-LPV. The model (14.36) is defined for deviations from the current operating point (xt, ut, dt), which is not necessarily an equilibrium point, and hence the presence of the “free” term ct in the equation. This more accurate model formulation was found to provide improved results relative to the classical steady-state linearization approach. It is more computationally intensive, but this can be justified for process control applications with relatively low controller sampling rates.

14.6.3 Simulation Scenarios and Results Nonlinear predictive controllers were designed based on the LPV model of the evaporator, and the results for several different scenarios are presented below. Scenario 1: This is a scenario used by Maciejowski [26] and involves ramp profiles on both the product demand X2 and evaporator pressure P2. The separator

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Nonlinear Industrial Process and Power Control Applications

1 50

60

70

80

90

10

100

4 3.5 3 2.5 2 30

40

50

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70

80

90

F1 [kg/min]

51 50.5 50 49.5 49 30

40

50

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90

100

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210 200 190 180 30

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F200SP [kg/min]]

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100

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100

30

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100

4 30

40

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250 200 150 100 50 0

11

6

10.5

5.5

X1 [%]

30

P100SP [kPa]

F2SP [kg/min]

20 15

1.5

F3 [kg/min]

25

P2 [kPa]

1.02

0.98

75

30 PID NGMV NGPC

1.04

X2 [%]

L2 [m]

1.06

10 9.5 9

30

40

50

60

70

80

90

100

5 4.5

Fig. 14.36 Control of evaporator: results of Scenario 1 (PID, NGMV and NGPC)

level is to be regulated but its variations are not important as long as the level is maintained within limits. Simulation results are shown for the following three controllers in Fig. 14.36, including • Well-tuned classical PID control, • Basic NGMV control, • The NGPC control with a prediction horizon N = 15. Note that the NGPC control includes a dynamic control cost-weighting Wu term and it is, therefore, similar but not the same as the NPGMV form of the solution, which involves a control weighting F ck . The NGMV controller provides an improved tracking performance over the PID design, and the NGPC provides the best tracking of the ramp setpoint. Scenario 2: This scenario includes a step change in the product concentration setpoint and also a step change in the feed flow rate disturbance. The results for the same three controllers as in Scenario 1 are shown Fig. 14.37, and the improved performance of the NGMV control of the pressure P2 relative to PID can again be noted. The NGPC controller is superior of all the three controllers, which is particularly evident in the disturbance rejection. Scenario 3: The compensation of the process nonlinearities by the NGPC controller can be compared with the linear GPC controller in Fig. 14.38. Both controllers were tuned using the same design parameters (horizons and weights), and as expected the nonlinear version provides improved performance. Scenario 4: In process control applications, especially operating in a batch mode, the setpoint profiles are often prespecified and therefore the future reference knowledge can be assumed and exploited by the predictive controllers. This is illustrated in Fig. 14.39, where the pre-emptive control action leads to better tracking of the ramp profiles. This is at the cost of a slightly bigger deviation from the level setpoint, which is acceptable in this application.

Nonlinear Predictive Control of an Evaporator Process 52

32

51

F2SP [kg/min]

0.85

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26 24

100

P100SP [kPa]

2.5 2 1.5 1 0.5 0

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F1 [kg/min]

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F200SP [kg/min]]

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X1 [%]

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P2 [kPa]

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F3 [kg/min]

693

1.05

X2 [%]

L2 [m]

14.6

5 4.5 4

100

1

25

GPC NGPC

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P100SP [kPa]

4 3.5 3 2.5 2 1.5

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F1 [kg/min]

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6

10

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250

11

X1 [%]

0.995

75

P2 [kPa]

30

F200SP [kg/min]

1.005

X2 [%]

L2 [m]

Fig. 14.37 Control of evaporator: results of Scenario 2 (PID, NGMV and NGPC)

5 4.5 4

Fig. 14.38 Control of evaporator: results of Scenario 3 (GPC and NGPC)

Scenario 5: The final set of results demonstrate the effectiveness of the solution to the constrained MPC problem. In this example, the control rate limits were set to (±1, ±50, ±50), for the F2 flow, P100 pressure and F200 flow, respectively. These may correspond to the actuator physical constraints. The constraints are satisfied as can be verified by inspecting the plots of the control signals in Fig. 14.40.

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Nonlinear Industrial Process and Power Control Applications 30

0.995 40

50

60

70

20 15 30

80

4 3.5 3 2.5 2 1.5 30

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F1 [kg/min]

F3 [kg/min]

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P100SP [kPa]

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P2 [kPa]

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X1 [%]

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F200SP [kg/min]

without preview with preview

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1.015

5 4.5

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0.96

2 1.5 1 0.5 30 40 50 60 70 80 90 100

51 50.5 50 49.5 49

30 40 50 60 70 80 90 100

26 24

30 40 50 60 70 80 90 100

2.5

0

F3 [kg/min]

unconstrained constrained

P100SP [kPa]

F2SP [kg/min]

0.94

28

300 250 200 150 100

30 40 50 60 70 80 90 100

51 50.5 50

30 40 50 60 70 80 90 100

300 200 100 0

6

9.5

5.5

9 8.5 8

30 40 50 60 70 80 90 100

30 40 50 60 70 80 90 100

400

10

X1 [%]

1 0.98

51.5

P2 [kPa]

30

F200SP [kg/min]

32

1.02

X2 [%]

1.04

F1 [kg/min]

L2 [m]

Fig. 14.39 NGPC control of evaporator: results of Scenario 4 (without and with future reference knowledge)

30 40 50 60 70 80 90 100

5 4.5 4 30 40 50 60 70 80 90 100

Fig. 14.40 NGPC control of evaporator: results of Scenario 5 (unconstrained and constrained solutions)

14.7

Conclusions

The model-based predictive and non-predictive nonlinear control methods have been illustrated on various industrial applications, which can challenge conventional controls. The system models included difficult dynamics like interactions in multivariable models, transport-delays and/or several nonlinearities. The different advanced control methods have revealed rather different benefits depending on the application; however, there are some common features. They are clearly able to

14.7

Conclusions

695

exploit model knowledge by taking account of the nonlinearities. The methods were based on optimal control designs with cost-function weightings that provided the tuning parameters. This type of approach provides the basis for a “formalized design procedure” and a way to benchmark performance. There are of course concerns whether the nonlinear optimal control design methods will deliver the required performance and be practical for installation. Predictive control is one of the success stories for modern control methods. Many potential applications for model-based predictive control have been described in the process, power and other industries (see [27, 34–37]). The success of linear MPC methods suggests that there is no real barrier to their use. However, there is the question as to whether nonlinear MPC methods will provide a robust and reliable performance needed. Unlike the use of linear MPC, there are many more options and approaches for true nonlinear MPC methods. The key to success is in choosing the right approach and design philosophy for the particular application. There are also questions whether more advanced control design methods will be easy to set up and tune. Process operators and plant control specialists should have an intuitive understanding of the behaviour of an advanced process control system, and ideally have a simple mechanism to adjust such controllers. The tuning procedures have to be tailored to the application [38]. Software interfaces can be provided to give intuitive tuning inputs that vary parameterized cost-function weightings. However, for a large multivariable industrial system non-expert changes to the solution provided by an original equipment manufacturer (OEM) might involve a high risk. It may be more practical to provide limited tuning options by storing alternative cost-weightings and then computing controllers, or alternatively just storing alternative solutions with different gains for critical loops. There are many ways to provide some operator tuning capability for model-based optimal controls that is in a more familiar form, but there is not yet an accepted general approach. A further opportunity for research!

References 1. Forbes MG, Patwardhan RS, Hamadah H, Gopaluni RB (2015) Model predictive control in industry: challenges and opportunities. In: 9th IFAC international symposium on advanced control of chemical processes, Whistler, British Columbia, Canada 2. De LaSalle SA, Reardon D, Leithead WE, Grimble MJ (1990) Review of wind turbine control. Int J Control 52(6):1295–1310 3. Bianchi FD, de Battista H, Mantz RJ (2007) Wind turbine control systems: principles, modelling and gain scheduling design. Springer 4. Leith DJ, Leithead WE, Hardan F, Markou H (1999) Direct regulation of large speed excursions for variable speed wind turbines. In: European wind energy conference, Nice, France, pp 857–860 5. Leithead WE, De la Salle SA, Reardon D, Grimble MJ (1991) Control system design for horizontal axis wind turbines (HAWT). In: American control conference. The Boston Park Plaza Hotel, Boston, Massachusetts

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6. Connor B, Iyer SN, Leithead WE, Grimble MJ (1992) Control of a horizontal axis wind turbine using H∞ control. In: First IEEE conference on control applications, Dayton, OH, pp 117–122 7. Connor B, Leithead WE, Grimble MJ (1994) LQG control of a constant speed horizontal axis wind turbine. In: Third IEEE conference on control applications, pp 251–252 8. Odgaard PF, Hovgaard TG (2015) Selection of references in wind turbine model predictive control design. IFAC-PapersOnLine 48(30):333–338 9. Savvidis P (2017) Nonlinear control: an LPV nonlinear predictive generalised minimum variance perspective. PhD thesis, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow 10. Leith DJ, Leithead WE (1996) Appropriate realisation of gain-scheduled controllers with application of wind turbine regulation. Int J Control 65(2):223–248 11. Leith DJ, Leithead WE (1999) Global gain-scheduling control for variable speed wind turbines. In: European wind energy conference, EWEC’99, Nice, France 12. Leith DJ, Leithead WE (1997) Implementation of wind turbine controllers. Int J Control 66 (3):349–380 13. Østergaard K, Brath P, Stoustrup J (2009) Linear parameter varying control of wind turbines covering both partial load and full load conditions. Int J Robust Nonlinear Control 19:92–116 14. Savvidis P, Grimble M, Majecki P, Pang Y (2016) Nonlinear predictive generalized minimum variance LPV control of wind turbines. In: 5th IET international conference on renewable power generation (RPG). IET London, vol 6, pp 48–54 15. Bryant GF (1973) Automation of tandem mills. Published by the Iron and Steel Institute 16. Pittner J, Simaan MA (2010) Tandem cold metal rolling mill control: using practical advanced methods. Springer, AIC Series 17. Hearns G, van der Molen GM, Grimble MJ (1996) Hot strip mill mass flow control. In: UWCC international conference on control, pp 23–28 18. Grimble MJ, Hearns G (1998) LQG controller for state-space systems with pure transport delays: application to hot strip mills. Automatica 34(10):1169–1184 19. Hearns G, Grimble M (2010) Temperature control in transport delay systems. In: 2010 American control conference, Baltimore, MD., pp 6089–6094 20. Choi IS, Rossiter JA, Fleming PJ (2007) Looper and tension control in hot rolling mills: a survey. J Process Control 17:509–521 21. Pittner J, Simaan MA (2018) Streamlining the tandem hot-metal-strip mill. IEEE Ind Appl Mag 1077–2618 22. Pittner J, Simaan MA (2012) Control of tandem hot metal strip rolling processes using an improvement to the state dependent Riccati equation technique. In: IEEE conference on decision and control, Maui, pp 6358–6363 23. Hearns G (2000) Advanced control design for hot strip finishing mills. PhD, Thesis, University of Strathclyde, Glasgow, Scotland 24. Prett DM, Morari M (1987) Shell process control workshop, butterworth, pp 335–360 25. Newell RB, Lee PL (1989) Applied process control, a case study. Prentice-Hall, Australia 26. Maciejowski JM (2002) Predictive control: with constraints. Pearson Education Ltd., Harlow, Essex 27. Qin S, Badgwell T (2003) A survey of industrial model predictive control technology. Control Eng Pract 11:733–764 28. Ravi R, Vinu R, Gummadi SN (2017) Coulson and Richardson’s chemical engineering, vol. 3A: chemical and biochemical reactors and reaction engineering. Butterworth-Heinemann 29. Liu K, Lewis EL (1994) Robust control of a continuous stirred-tank reactor. In: Proceedings of ACC, Baltimore, Maryland, pp 2350–2354 30. Alpbaz M, Hapolu H, Güresinli C (1998) Application of nonlinear generalized minimum variance control to a tubular flow reactor. Comput Chem Eng 22(15):839–842 31. Grimble MJ, Majecki P (2013) Non-linear generalised minimum variance control using unstable state-dependent multivariable models. IET Control Theory Appl 7(4):551–564

References

697

32. Hernandez E, Arkun Y (1993) Control of nonlinear systems using polynomial ARMA models. AIChE J 39:446–460 33. Ahmed S, Petrov M, Taneva A, Todorov Y (2012) Nonlinear model predictive control of an evaporator system using fuzzy-neural model. In: Workshop on dynamics and control in agriculture and food processing, Plovdiv, Bulgaria 34. Atuonwu JC, Cao Y, Rangaiah GP, Tade´ MO (2010) Identification and predictive control of a multistage evaporator. Control Eng Pract 18:1418–1428 35. Bulut B, Katebi MR, Grimble MJ (2002) Co-ordinated control of profile and shape in hot strip finishing mills with nonlinear dynamics. IEE Proc Control Theory Appl 149(5):471–480 36. Ordys AW, Grimble MJ (1996) A multivariable dynamic performance predictive control with application to power generation plants. IFAC World Congress, San Francisco 37. Cannon M, Kouvaritakis B, Grimble MJ, Bulut B (2003) Nonlinear predictive control of hot strip rolling mill. Int J Robust Nonlinear Control 13(3–4):365–380 38. Grimble MJ (1998) 3½ DOF polynomial solution of the inferential H2/H∞ control problem with application to metal rolling. ASME J Dyn Syst Meas Control 120:445–455

Chapter 15

Nonlinear Automotive, Aerospace, Marine and Robotics Applications

Abstract The nonlinear control applications in this chapter are mainly for faster systems that provide new challenges when limited computational power is available. The need for nonlinear control in robotics is demonstrated in a robot manipulator control design. This is followed by two application problems in marine systems involving ship roll stabilization and ship positioning, where computational power is less of an issue but where nonlinearities and constraints are significant. The diesel engine and the sightline servo-system control problems involve both significant nonlinearities and have greater restrictions on the computational power available. The chapter illustrates the importance of the state-dependent and LPV modelling philosophies, and the value of simple nonlinear controller structures, when processing speed and complexity is an issue.

15.1

Introduction

This final chapter provides further examples of the application of the nonlinear control design methods in industrial situations. It covers robotic manipulators, ship fin and rudder roll-stabilization systems, dynamic ship positioning, automotive diesel engine air path control, a high accuracy gimbal-based servo-system for defence systems, and flight controls. With the exception of the robotics problem, all the other examples build on the experience gained on industrial applications projects. Automotive systems provide an exemplar of future likely trends in a range of applications. The performance must be optimized throughout the range of operation to accommodate tighter emissions, fuel economy and ensure safe operation. Significant nonlinearities are a common feature in automotive engine dynamic systems. For example, the air mass flow passing through a throttle valve is modelled by Bernoulli’s fluid equation. Applications in the automotive industry are constantly providing new opportunities for advanced controls that can optimize the system response, whilst satisfying constraints [1].

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8_15

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Another potential area for future advanced control applications lies in the aerospace industry. The servo problem considered later involves a very fast acting and high accuracy servo-system. In this case, predictive control is not so appropriate since the reference is quickly changing and future behaviour is uncertain. However, there are opportunities in aircraft flight control systems where predictive control may be very suitable. Fighter aircraft have particularly challenging dynamics, are nonlinear, aerodynamics are uncertain, and the controls have position and rate limits. There are also stringent performance requirements on the flight control system. Stability is important and there is a wide range of operating conditions where high performance is demanded. For good manoeuvrability, the controls must be able to exploit the full range of actuator control capabilities. A major difficulty in applying Model Predictive Control (MPC) in many applications is that a constrained optimization problem must be solvable on-line for a system which is intrinsically nonlinear. Producing a control law that is practical for implementation is often problematic. A feedback linearizing control law that holds globally was used in Deori et al. [2] to make the system dynamics linear with respect to the new state and input variables. After the use of feedback linearization, the constraints were approximated in the new state and control variables to make them convex, leading to a simpler MPC solution online. The application problems are somewhat less challenging in marine systems where the weight and dimensions of processing devices are not important and the safety requirements are much less regulated. In fact, ship operators will evaluate and accept new advanced controls with little fuss, and this is understandable since there is time to correct errors in the ships heading by human intervention. This is not the case in aircraft flight control systems or even in lane-changing for autonomous automotive systems. There are more problems in applications where the constrained versions of predictive control algorithms are needed. This lies in the verification and validation of a control law that involves a black-box numerical optimization. Such algorithms also demand significantly more computing resources, and are slow in comparison with unconstrained versions. If unconstrained predictive control can provide a reasonable performance and soft constraints are effective, then such algorithms are preferable, particularly for fast applications. Explicit MPC methods can provide a more efficient alternative to on-line constrained optimization algorithms, as described in Chap. 7 (Sect. 7.1.4). If a more intuitive controller structure is needed than predictive control, then one of the class of Nonlinear Generalized Minimum Variance (NGMV) controllers may be suitable. The industrial application trials and simulations demonstrate that there is no best-recommended control solution. It is important to choose the right tool or design method for such a problem but there are often alternative solutions all of which can be tuned to give similar performance.

15.2

15.2

Multivariable Control of a Two-Link Robot Manipulator

701

Multivariable Control of a Two-Link Robot Manipulator

An obvious application area for nonlinear predictive control lies in the many control problems in industrial robotics. These include areas such as welding or paint spraying robots, where the reference trajectory is often planned well in advance. Robot manipulators have coupling between joints and nonlinearities due to actuators and axes transformations that can make control difficult. Flexible manipulators require less material, and are lighter in weight and consume less power. They require smaller actuators, are lower cost and have a higher payload to robot weight ratio. Unfortunately, flexible links have more compliance and resonant modes making good control design more challenging. Robotics will provide a continuous source of advanced control problems. Advances will affect everyone’s lives with applications including manufacturing plants, pick-and-place operations, microsurgery, and space robotics, movement of radioactive or bio-hazardous materials, the provision of home help and companionship and many others. Consider a planar manipulator with two rigid links, as shown in Fig. 15.1. The objective is to control the vector of joint angular positions q(t) with the vector of torques s(t), applied at the manipulator joints, so that they follow a desired reference trajectory qd(t). This problem was analysed in Slotine and Li [3], and it was shown that a multi-loop PD controller could be used to control the links to desired fixed positions. Other more advanced controllers may be applied, such as Model Predictive Controls [4, 5], when the benefits of model-based control are required. The dynamics of the two-link system are clearly nonlinear, and are described by the continuous-time differential equations [3] that follow:

Fig. 15.1 Two-link robot manipulator

e

m2, I2 q2 2

lc2 l1 lc1

m1, I1 q1

1

702




H11 H21

15

H12 H22




Nonlinear Automotive, Aerospace, Marine and Robotics …


€ h q_ 2 þ d1 q1 ðtÞ þ € h q_ 1 q2 ðtÞ

hðq_ 1 þ q_ 2 Þ d2






 g1 ðtÞ s1 ðtÞ q_ 1 ðtÞ þ ¼ g2 ðtÞ s2 ðtÞ q_ 2 ðtÞ

ð15:1Þ This equation may be written as follows: _ q_ þ gðqÞ ¼ s HðqÞ€q þ Cðq; qÞ

ð15:2Þ

_ q_ is the vector of centripetal and Coriolis where HðqÞ is the inertia matrix, Cðq; qÞ torques, and gðqÞ is a vector of torque components due to gravity. The parameters d1 and d2 represent the system damping due to friction (in the undamped nominal case: d1 ¼ d2 ¼ 0Þ. Assume the manipulator is operating in the horizontal plane so that gðqÞ ¼ 0. By reference to the geometry of the manipulator arm in Fig. 15.1, the matrix elements for HðqÞ, in (15.1) and (15.2), may be obtained as in the following equations.

Fig. 15.2 Open-loop responses with and without damping term (including torque pulse inputs)

15.2

Multivariable Control of a Two-Link Robot Manipulator

703

H11 ¼ a1 þ 2a3 cos q2 þ 2a4 sin q2 H12 ¼ H21 ¼ a2 þ a3 cos q2 þ a4 sin q2 ;

H22 ¼ a2

The parameters are defined as h ¼ a3 sin q2  a4 cos q2 and a1 ¼ I1 þ m1 l2c1 þ I2 þ m2 l2c2 þ m2 l21 a2 ¼ I2 þ m2 l2c2 ; a3 ¼ m2 l1 lc2 cosde and a4 ¼ m2 l1 lc2 sinde The following numerical values of parameters were used for the simulations: m1 ¼ 1; I1 ¼ 0:12; l1 ¼ 1; lc1 ¼ 0:5; m2 ¼ 2; I2 ¼ 0:25; lc2 ¼ 0:6; de ¼ 30 : Open-loop responses: The open-loop responses of the system, subject to torque pulses on both of the links, are shown in Fig. 15.2. The ideal undamped responses are shown in green, and are compared against the damped responses, in red (defined as d1 ¼ d2 ¼ 1). It is evident that closed-loop control is needed for this system.

15.2.1 State-Dependent Solution There are two different ways of modelling this system and designing NPGMV predictive controllers that are considered in the following. The first approach uses a state-dependent or quasi-Linear Parameter-Varying (qLPV) model, which is the more natural solution and was described in Chap. 11. In this first case, the input subsystem can be replaced by the identity and the total nonlinear plant and disturbance model can be represented by a state-dependent output subsystem. The NPGMV solution using this type of model was also presented in Chap. 11 (Sect. 11.7). The second approach that is described in the next section is rather contrived, but it illustrates how the “black-box” input model can be used to represent the full nonlinear plant model. The two-link robot manipulator Eq. (15.3) is nearly in the required state-dependent or quasi-Linear Parameter-Varying (qLPV) form. This may be confirmed by rewriting Eq. (15.2). The matrix H may be assumed invertible for this application and hence the plant equations may be written as follows: 



_ qðtÞ 0 qðtÞ 0 I x_ q ðtÞ ¼ þ sðtÞ ¼ _ _ €qðtÞ qðtÞ H 1 ðqÞ 0 H 1 ðqÞCðqÞ ð15:3Þ
   qðtÞ 1 ðqÞsðtÞ þ H yðtÞ ¼ €qðtÞ ¼ 0 H 1 ðqÞCðqÞ _ _ qðtÞ The continuous-time Eq. (15.3) can be discretized numerically using a Runge– Kutta algorithm to obtain the output subsystem model W 0k . A nominal one-step delay (k ¼ 1) may then be assumed to account for computing and actuator delays.

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The control signal costing term is defined as a linear transfer-function term in this problem and the NPGMV solution may, therefore, be found from Eq. (11.131) in Chap. 11 (Sect. 11.8.1). A comparison of the system responses using NGPC, NPGMV and PID controllers is as shown in Fig. 15.3, which reveals that the NGPC and NPGMV designs are not too different. The NPGMV responses have less overshoot but the NGPC design provides better decoupling. The PID controlled system has greater overshoots but control action is not so aggressive. It is difficult to fairly compare the performance of NGPC and NPGMV controlled systems because results depend upon the cost-weighting choices, and the NPGMV design has more tuning variables available. Note that the error weightings were chosen to have a PID form to obtain these results. The absolute and incremental control choices (see Chap. 11) lead to different results that are shown in Fig. 15.4, and in a detailed view in Fig. 15.5. It is not clear which method is preferable since the results depend upon dynamic weighting choices. For example, the incremental control case has a better plant output on channel 1 but worse on channel 2, but changing the relative size of the two output

Fig. 15.3 NGPC and NPGMV designs compared with PID control using absolute control action and PID-inspired error weighting

15.2

Multivariable Control of a Two-Link Robot Manipulator

705

Fig. 15.4 NPGMV designs compared for absolute control and incremental control action cases for a free error weighting choice

weightings can improve output 2 at the expense of output 1. Thus, the best choice will depend upon the application and the way that integral action is introduced. If the same case is considered as that used for the results in Fig. 15.4, but a stochastic disturbance is also added to the plant outputs, the results are as shown in Fig. 15.6. The predictive control designs have sharp responses in the noise free case and this is maintained when stochastic disturbances are present. They also result in a smaller variance for the control action as compared with the simple PID. Design questions: These two predictive control options illustrate one of the dilemmas the control designer must face, which was much less important when linear control methods were mostly employed. That is, there is great freedom in the choice of the nonlinear model structure and the cost-minimization problem description. These design decisions will affect the difficulty in tuning the system and the performance and robustness that is achievable. Robot control systems must provide fast and accurate tracking in the presence of variations of inertia and the gravitational load of the manipulator. There is also the question of how best to account for the model uncertainties that are present in robotics applications due to the changing payload and friction.

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Fig. 15.5 NPGMV designs compared for absolute control and incremental control action cases for free error weighting choice (expanded time scale)

Fig. 15.6 Comparison NPGMV, NGPC and PID for absolute control input, PD-inspired error weighting (including deterministic and stochastic disturbances on all outputs)

15.2

Multivariable Control of a Two-Link Robot Manipulator

Fig. 15.7 Separation of the two-link robot model into nonlinear and linear terms

(t )

NL robot dynamics

q(t )

707

1 s

1 s

W0

1 s

1 s

q(t )

1k

15.2.2 LTI Output Block Solution For the second modelling approach, the open-loop plant model may be separated into nonlinear and linear components. The main nonlinear plant dynamics are to be included in the “black-box” model term W 1k . However, anything within this block is assumed stable so the double integrator involved must be removed, as shown in Fig. 15.7. The separation was performed by defining the accelerations of the links as the output of the black-box model W 1k . From Eq. (15.3), this subsystem will have a direct feed-through term. The two double integrators associated with a link can be extracted to define the output linear subsystem W0 . This is rather artificial and not recommended but it is useful to demonstrate the option of including most of the discretized plant model within the black-box term. It is a general rule that to improve the robustness of the Kalman filter design, all the plant and disturbance model states in a system should be assumed to be driven by white process noise, even if this does not arise from the physical system description. In this application, it is particularly important that the states corresponding to the integrators should be assumed to be driven by white noise.

15.2.3 Simplifying Implementation The input subsystem model W 1k in this problem has known equations and parameters. This knowledge of the model can be exploited to simplify the implementation of the controller as described in Chap. 11 (Sect. 11.7.2). The model given in the state-equation form (15.3) may be written in terms of a matrix G0 ðsÞ ¼ H 1 ðqÞs, which is a static through term, and a term G1 ðsÞ ¼ _ q_ that has a unit transport delay. The state vector in these expressions H 1 ðqÞCðqÞ  T _ T xq ðtÞ ¼ qðtÞT qðtÞ was updated numerically using a 4th-order Runge–Kutta discretization algorithm. The discrete version of the model W 1k can, therefore, be separated into a component due to the current control (the through term) and a part dependent on the past values of control. That is, using the same notation for the discretized models:

708

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Nonlinear Automotive, Aerospace, Marine and Robotics …

W 1k ðsðtÞÞ ¼ G0 sðtÞ þ G1 ðsðtÞÞ

ð15:4Þ

The latter term G1 ðsðtÞ contains at least a unit-step transport-delay matrix z−1I. The special structure of the model in this second approach can be exploited to avoid the algebraic-loop problem discussed in Chap. 5, and thereby can simplify  CN ¼ 0 and K2N ¼ 0, and implementation. Consider the case where the weightings P  t;N ¼ 0. Also assume that the control signal costing term, the control reference U with a vector of torque s(t) input, may be written as F ck ðsðtÞÞ ¼ F 0 sðtÞ þ F 1 ðsðtÞÞ

ð15:5Þ

In each of (15.4) and (15.5), the first term is non-dynamic and the last term includes at least a unit-step transport delay. Note the condition for optimality in Chap. 9 (Sect. 9.4.3). Employing a similar block matrix notation for the block-diagonal structures of F ck;N and W 1k;N , Eq. (9.82) may be written as 

~ 0t þ k;N  C/ T0 BðW 1k uÞðtÞ þ ðF ck;N  XN W 1k;N ÞUt;N ¼ 0 PCN R   ~ 0t þ k;N þ ðXN þ C/ T0 BCI0 ÞðG0;N þ G1;N ÞUt;N F 0;N þ F 1;N Ut;N ¼ PCN R

The expression can now be written as 

  ~ 0t þ k;N þ XN G0;N Ut;N þ XN G1;N Ut;N þ C/ T0 BCI0 ðG0;N þ G1;N ÞUt;N F 0;N þ F 1;N Ut;N ¼ PCN R

Regrouping these terms, we obtain 

  ~ 0t þ k;N ðF 1;N  XN G1;N ÞUt;N þ C/ T0 BCI0 W 1k;N Ut;N F 0;N  XN G0;N Ut;N ¼ PCN R

The NPGMV vector of optimal controls Ut;N follows, since the static terms on the left of this expression will be invertible:  1  ~ 0t þ k;N ðF 1;N  XN G1;N ÞUt;N þ C/ T0 BCI0 W 1k;N Ut;N Ut;N ¼ F 0;N  XN G0;N PCN R

ð15:6Þ ~ 0t þ k;N ¼ Rt þ k;N  Dt þ k;N  CN AN Ak ^xðtjtÞ. where R Implementation: The terms F 1;N , G1;N ðsÞ and T0 ðk; z1 Þ contain at least a unit-step delay, and the right-hand side of (15.6) therefore involves the vector of controls computed in the previous step. The current control is therefore only dependent on past controls and inputs, and this solution does not include an algebraic loop. For the inversion of the operator ðF 0;N  XN G0;N Þ, note that at each sample instant the operators G0;N ðq; sÞ and F 0;N are non-dynamic matrix terms.

15.2

Multivariable Control of a Two-Link Robot Manipulator

709

The structure of the controller is shown in Fig. 15.8, where ½  N denotes an N-block diagonal model and the Dt þ k;N term is null in this case. These results are related to those in Chap. 11 (Sect. 11.7). Design results: Following the tuning guidelines given in Sect. 5.2, the NPGMV controller design for this second modelling approach was initially based on a PD controller, denoted by CPD ðz1 Þ, with the tuning gains selected as Kp ¼ 2000I and Kd ¼ 100I. Derivative filters with a time constant 0.001 s were added to the differential terms. For demonstration purposes, the dynamic weightings were defined as Pc ðz1 Þ ¼ CPD ðz1 Þ and Fck ¼ VN , where VN was defined in Chap. 9. The input weighting K2N was initially set to zero. As discussed in Chap. 11, these weightings correspond to the limiting case of N ¼ 0 that is equivalent to NGMV control. The horizon N [ 0 corresponds to the predictive control case. Multivariable control: The nominal simulation results for this limiting case and the sample time Ts = 0.001 s are shown in Fig. 15.9. The torque saturation limits in this study were set to ±200 Nm for both links. The links were subject to reference step changes, and a step load disturbance was applied to link 2 at time t = 5 s. With the nominal PID inspired weightings, the NPGMV control performance was very close to that of PD control. This method of selecting weightings is often a useful practical way of starting a design. Note that for the rather unrealistic case of an open-loop undamped system, the assumption on the strict stability of the black-box model W 1k;N does not hold, and it is expected that the accumulation of the modelling errors with an increasing prediction horizon N would lead to an unstable design. Damped system responses: The above problem does not arise if damping is present in the system. If the weightings are chosen carefully, the responses remain stable. They are also improved with increasing N. In the following modified design, a lead term has been included in the control weighting, with the error weighting

C N AN Ak

–

–

PCN

+

0,N

+

XN

0,N

(xq )

(t )

Ut,N

1

CI 0 Torque demand

Prediction

Rt

+

k ,N

Future reference

Rt

0

– 1,N

+

k, N

xq , N (t )

xq

Aq ( xq ) xq xq , N (t )

XN

Bq ( xq )

N

Runge-Kutta discretization

1, N

1k N

q (t )

xˆ(t t )

C T0 BC I 0

Estimation Kalman Filter

z

k

0, N

1, N

CI0

Fig. 15.8 Detailed implementation of the NPGMV controller (avoiding the algebraic loop)

710

15 Position of link 1

80

Position of link 2

50

PD NPGMV, N = 0

40 30

q 2 [deg]

60

q 1 [deg]

Nonlinear Automotive, Aerospace, Marine and Robotics …

40 20

20 10

0

0 -10

-20 0

1

2

3

4

5

0

6

2

3

4

5

6

Torque applied to link 2

Torque applied to link 1 200

100

100

[Nm]

200

0

2

0

1

[Nm]

1

-100

-100

-200

-200 0

1

2

3

4

5

0

6

1

2

3

4

5

6

time (sec)

time (sec)

Fig. 15.9 Robot position control showing nominal results for PD-inspired weighting

Pc ðz1 Þ retuned but still based on a PD controller. A small weighting is also included on the input signal (penalizing excessive acceleration of the links). These cost-function weightings may be listed as F ck ¼ Fck ¼ 0:408  ð1  0:9z1 Þ  I
2 1

1

Pc ðz Þ ¼ 52000ð1  0:9615z ÞI2

and

0:5 K ¼ 0 2

0 0:5



The step responses for the NPGMV controller and varying prediction horizons from N = 0 to 10 (with the sample time set to Ts ¼ 0:002 s in this case), are shown in Fig. 15.10. All the responses are stable improve with increasing N (although there is a limited improvement beyond N = 5). In the final design using this approach, the sample time Ts was increased to 0.03 s and the weightings were retuned to obtain

 10 0 0:1 0 2 1 F ck ¼ Fck ¼ ð1  0:92z Þ  ; K ¼ ; 0 30 0 0:1

 5925  5363z1 6225  5100z1 Pc ðz1 Þ ¼ diag 1 þ 0:875z1 1 þ 0:875z1

15.2

Multivariable Control of a Two-Link Robot Manipulator

711

Fig. 15.10 Damped system responses for varying N (Ts = 0.002 s)

The responses for N = 0 are somewhat oscillatory for these weightings but again there is rapid improvement with increasing N. With the increased sample time, only a few steps in the horizon were needed to obtain a satisfactory design. The results for N = 1, 5, 10 are shown in Fig. 15.11. The control effort decreases with increasing horizon N, which is typical of predictive controls. Design comments: A predictive controller has the advantage that the control action can begin before the changes in the reference signal occur, thanks to the future setpoint knowledge. This reduces control effort and the tracking error at the same time. The NPGMV method does, of course, have its limitations; that is, the black-box nonlinear part of the model is assumed stable. Thus to apply to the undamped open-loop robot system in the second design approach, some damping in the model was needed. The predictions, in this case, depended upon the linear subsystem model. Based on the simulation results it was found that as expected, the NPGMV controller offered advantages relative to the NGMV control design but at the expense of some additional complexity in implementation.

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Fig. 15.11 Damped system responses for varying N (N = 1, 5, 10 and Ts = 0.03 s)

15.3

Ship Rudder and Fin Roll Stabilization

The stabilization of the roll angle of a vessel is important for effective operation in heavy seas. The rolling motion of a ship due to sea wave disturbances in bad sea-state conditions has an adverse effect on passenger comfort, crew efficiency and the safety of cargo. It is particularly undesirable for cruise liners, container ships and naval vessels. Roll-stabilization systems are important for offshore wind turbine support vessels and oil rig support vessels since it is impossible to transfer maintenance staff or stores when there is too much motion. At the other end of the scale, aircraft carriers must provide a stable platform if operations are to continue in adverse weather conditions. Rolling motions can be reduced by using a roll-stabilization control system. There are several passive and active solutions of differing effectiveness. As an example of the former, ballast tanks are good at correcting steady-state heel angles and counteracting low-frequency sea wave disturbances. Active fin roll-stabilization control systems can be used to stabilize the ships roll angle, particularly at the natural roll frequency. The force on a fin varies in proportion to the square of the ships speed relative to the water flow. A gyro is normally used to measure the

15.3

Ship Rudder and Fin Roll Stabilization

713

rolling motions and the feedback controller varies the fin angles to produce lift forces to oppose the rolling motion. The fin actuators are electrohydraulic systems with constraints on the maximum angles and the rate of change of fin position. The fins are only effective when the speed is above about 10 knots. Classical fin roll-stabilization control systems are often used [6–8], which mostly involve linear control strategies such as PID control with some nonlinear compensation. It is interesting that the minimization of the sensitivity function that determines the rolling motions is a natural H1 control problem that has also been tested successfully [9].

15.3.1 Ship Rudder Roll-Stabilization Problem The roll of a vessel can be controlled by the combined use of fin roll stabilizers and a ship’s rudder, as discussed in Chap. 7. However, the problem considered here is rather different from the multivariable ship roll-stabilization and heading control problem of Chap. 7 where predictive control was employed. In this section, the heading control (autopilot loop), and the fin roll-stabilization controllers are assumed to be designed separately using the NGMV control approach. The design problem considered therefore only involves the control of the rudder to improve the roll-stabilization action over and above the stabilization provided by the fins [10]. This situation would arise when rudder roll stabilization is added to an existing ship autopilot and fin roll-stabilization system. In fact, this separation of control-loop designs is the more likely approach following current design practice. The rudder roll stabilization (RRS) systems are a more recent development than fin roll-stabilization systems. They have mainly been used on naval vessels and have the advantage that RRS systems require less investment. For ships that have fin roll stabilizers already fitted, a logical extension is to combine RRS with the control of the fins to provide an integrated control scheme. By combining all the aspects of ship motion control, namely an autopilot controlling the rudder, RRS and fin stabilizers, a total ship control methodology can be established. This offers potential benefits to the total system behaviour by coordinating control loops so that they are operating in harmony. However, in this example, the design of a two-layer integrated fin-rudder controller is described. The lower level of the hierarchy consists of the autopilot and fin controller loops, which in this study are assumed tuned and fixed. An NGMV controller is proposed here for the higher rudder roll-stabilization level. Ship model: The basic dynamics of the ship rolling motion [11], with respect to the fin and rudder, and the sea wave disturbances (wa, wd) are shown in block diagram form in Fig. 15.12. The rolling motion /ðtÞ is strongly influenced by the yaw motion wðtÞ and the ships speed. The other signals in this figure represent controller output terms or actuator inputs. The interaction Gd/ has

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Nonlinear Automotive, Aerospace, Marine and Robotics …

15

Total roll controller

C w e

+ +

Roll angle

G

–

Roll model

G

c

Fins

Fin controller

C

G – +

c

w

G

G

Rudder

Yaw model

Yaw

+ +

– +

C

Heading reference

Autopilot controller

r

Fig. 15.12 The integrated roll and yaw model

non-minimum-phase dynamics, which makes the control design more difficult. The ship rolling motion model G/ exhibits resonant behaviour, whilst the yaw model Gw contains an integrator. A polynomial-based linear ship model will be used so that the design results in Chaps. 4 and 5 may be invoked. The nonlinearities are due to the hard limits on actuator movements. The fin and rudder actuator models Ga and Gd , and the roll G/ , yaw Gw and rudder-to-roll model Gd/ are the same as used in Chap. 7 and shown in the block diagram of Fig. 7.8. Control problem: The rolling motion is at high frequencies compared with the yawing motion, and this frequency separation enables the roll and yaw controllers (Ca and Cd ), to be decoupled and designed independently. In this example, it is assumed that the fin-stabilization controller and the autopilot have been designed previously, and the controllers have an adaptation to account for ship speed variations. The total rolling motion controller Cad is required to minimize the rolling motions given the limitations imposed by this structural assumption. Actuator model: The servo-systems have magnitude and rate constraints, which limit the available bandwidth and performance. In this study, we focus on the rudder constraints, with typical angle and rate limits: dmax 2 ð25; 35Þ deg

and

d_ max 2 ð2; 7Þ deg/s

ð15:7Þ

For efficient roll reduction, the rudder rates required are of the order of 15–20 °/s, which is attainable by modern steering machines.

15.3

Ship Rudder and Fin Roll Stabilization

Rudder to roll ship model

-G

G

Yaw CL model (with autopilot)

Rudder to roll interaction

715

w + +

e

G

w

Roll CL model (with fin controller)

C Total roll controller Fig. 15.13 Combined closed-loop system model

Wave model: The wave disturbance is modelled using a second-order linear approximation to the Pierson–Moskowitz sea wave spectrum [20]: G ¼ kw s=ðs2 þ 2fw xw s þ x2w Þ

ð15:8Þ

where the typical frequencies xw of the wave motion lie in the band 0.5–1.2 rad/s, and the damping-factor is of the order 0.1–0.2. Combined system model: After some simple block diagram manipulation, the system of Fig. 15.12 can be reduced to that shown in Fig. 15.13. The Gwd and G/w transfers include the local feedback loops involving the fin controller and the autopilot, respectively: Gdw ¼ Gd =ð1 þ Cd Gd Gw Þ

ð15:9Þ

G/w ¼ G/ =ð1 þ Ca Ga G/ Þ

ð15:10Þ

Roll-stabilization ratio: The roll-stabilization ratio is a useful measure of roll-stabilization control system performance. It is defined as the magnitude of a particular sensitivity function as follows: SðjxÞ ¼ e/ ðjxÞ=wa ðjxÞ ¼ 1=ð1 þ Gdw G/w Gd/ Cad Þ

ð15:11Þ

15.3.2 NGMV Rudder Roll-Stabilization Controller The design of the NGMV controller for the rudder roll-stabilization problem requires an appropriate choice of the NGMV performance index. The disturbance to the autopilot loop must be minimized, and the desired shape of the roll sensitivity function must be achieved. The dynamic cost-function error weightings needed

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must have a maximum gain at the peak roll frequency and a lower gain at low and high frequencies. The NGMV control law can handle some nonlinearity naturally and by a proper choice of the nonlinear control weighting term can deal with rudder angle and rudder slew rate limiting. The control system objective is to reject the wave disturbances in the frequency range of interest, by shaping the sensitivity function using a feedback controller with roll output /ðtÞ and rudder input d/ ðtÞ. The ship rolling motion model is assumed to have a natural frequency of 0.8 rad/ s and a damping factor of 0.2: G/ ðsÞ ¼ 0:64=ðs2 þ 0:32s þ 0:64Þ The fin dynamics are not included and the fin controller is designed to give −12 dB reduction at the resonant frequency, using a roll rate measurement. The yaw dynamics are modelled as Gw ðsÞ ¼ 0:2=ðsð10s þ 1ÞÞ The NGMV feedback controller structure for the RRS problem is as shown in Fig. 15.14. A PI controller is assumed for the autopilot, which is designed for a bandwidth of 0.1 rad/s. Compared with the roll natural frequency of 0.8 rad/s, there is a factor of 8 difference in the frequency content of the roll and yaw motions. The rudder limits were set to 40° for the angle and 20 °/s for the rate. These are relatively high limits but attainable by current steering systems and necessary for effective rudder roll stabilization. The rudder-to-roll interaction is modelled as Gd/ ðsÞ ¼ 0:1ð1  4sÞ=ð1 þ 6sÞ The non-minimum-phase behaviour of this model imposes a fundamental limitation on the sensitivity function frequency response achievable. Roll reduction ratio: The roll reduction can be quantified by the Roll Reduction Ratio (RRR), defined as RRR ¼ 100 ð1  r2cl =r2ol Þ

ð15:12Þ

Wave disturbance d 0

+

G0 Pcd D2

1

+

ck

+

1

+ k

NGMV Controller

F0Yf 1

Rudder to Roll Ship Model Roll angle

Fig. 15.14 NGMV controller structure for the rudder roll-stabilization problem

15.3

Ship Rudder and Fin Roll Stabilization

717

This ratio represents the reduction in variance that the combined fin and rudder roll-stabilization scheme can provide (r2cl ), compared with the effect of the fin roll-stabilization system only (r2ol ). Design issues: The system model without the constraints is linear. The overall transfer-function from d/ ðtÞ to /ðtÞ involves the path shown in Fig. 15.13. This acts as the plant model for the total roll controller Cad . A polynomial form of the NGMV controller, described in Chaps. 4 and 5, will now be applied to this roll-stabilization problem. The dynamic weightings in the cost-function are chosen so that the control system rejects the wave disturbances in a specified frequency band (determined by the assumed sea state). The weightings should also ensure there is a relatively low controller gain at both low and high frequencies, to ensure the roll-stabilization system does not attempt to compensate for low or high-frequency roll motion errors, in the presence of the rudder angle and rate limits. It was noted in Chap. 5 that a starting point for the NGMV weighting selection is to set the magnitude of the control weighting to unity, and the error weighting equal to a controller known to stabilize the delay-free process. This approach has been applied in this example. The weighting choice is often based on a PID structure, but in this problem an LQG controller was used. That is, the continuous-time error and control weightings were chosen so that the ratio equalled an LQG controller design: Pc ðsÞ ¼

60:5ðs þ 0:2476Þðs2 þ 1:934s þ 0:9409Þ and F c ðsÞ ¼ 1 ðs þ 5:093Þðs2 þ 0:1034s þ 0:632Þ

The NGMV controller was assessed in terms of the wave-to-roll sensitivity function, as well as the simulated time responses with the nonlinear rate limited models. Due to the fundamental limitations on the achievable performance, it is generally not possible to provide a large roll reduction around the ship’s resonant frequency without amplifying the disturbances at other frequencies. The design, therefore, involves a trade-off situation in the weighting selection. NGMV frequency responses: The frequency responses of the nominal weighting functions are shown in Fig. 15.15, together with the open-loop transfer-function frequency response. Note that the peak in the error weighting that penalizes the roll motion around the ship’s natural frequency of 0.8 rad/s corresponds also to an envelope of expected wave spectra. The NGMV design philosophy assumes a linear disturbance model. Using a broader sea spectrum than for a particular sea state will make the controller more robust against varying operating conditions. The roll-stabilization ratio (sensitivity function relating the wave disturbance input to the roll output), for different values of the control weighting scalar gain F c ðsÞ ¼ qF , is shown in Fig. 15.16. The value of qF ¼ 0:7 was selected for further design analysis, and this corresponded to a roll reduction ratio of approximately RRR ¼ 90%. In reality, the rudder constraints limit the achievable performance, and the actual roll reduction obtained from a number of simulation runs was found to be about 84%.

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Fig. 15.15 NGMV weighting function magnitude frequency response and the open-loop frequency response

Fig. 15.16 Roll-stabilization ratio for a range of control weighting gains qF ¼ 0:5; 0:7; 1; 1:5

15.3

Ship Rudder and Fin Roll Stabilization

719

NGMV constraint handling: The use of the nonlinear control signal weighting function in the NGMV cost-index can ensure the nonlinearities are addressed. The constraints were handled in this problem by modifying the control weighting function, as discussed in Chap. 5.3: ^ck uÞðtÞ þ q=ð1  cz1 ÞðuðtÞ  g satðuðtÞÞÞ ðFck uÞðtÞ ¼ ðF

ð15:13Þ

where the nominal invertible part of the control weighting function is denoted as ^ck , and q, g are positive scalars. The second term in (15.13) does not affect the F overall weighted control signal in the “linear” region but becomes active under saturation conditions, i.e. when j u j  g. The binary parameter c can assume two values: 0 or 1. If the controller includes an integrator and the saturation limit g is set to the maximum rudder angle, then c ¼ 1 corresponds to a classical anti-windup scheme. However, since there is no integral action in the RRS controller, the parameter c was set to zero. Optimal control: The optimal NGMV control action can now be rewritten as h i 1 ^ck uð t Þ ¼ F F0 Yf1 ðWk uÞðtÞ  qu1 ðtÞ  G0 ðPcd D2 Þ1 eðtÞ

ð15:14Þ

where u1 ðtÞ ¼ ðuðtÞ  satðuðtÞÞÞ represents the deadzone operator. The corrective feedback action u1 ðtÞ is only active under saturation, and its magnitude is determined by the parameter q. For the nominal saturation limits, an equivalent of the anti-windup mechanism is obtained. On the other hand, if the saturation limits are smaller than dmax , then this arrangement corresponds to introducing soft constraints and is an implicit way of handling the constraints. Compared with the Model Predictive Control approach (where the constraints are taken into account explicitly), the NGMV controller is simpler to implement. Moreover, on-line optimization does not have to be performed, reducing the computational demands.

15.3.3 Rudder Roll Simulation The main requirement considered here is limiting the vessels roll angle but energy usage is becoming of much greater concern and the optimal approaches to design are therefore likely to be of more importance in future years. The weighting (15.13) was first used with the rudder saturation limit set to a ¼ 25° and rate limit set to b = 0 °/s, and optimized for q. It was possible to increase the roll reduction ratio from 68% for a well-tuned PID to 73%, which is lower than the “theoretical” 90% but is an improvement considering the presence of the system nonlinearities. Note that a penalty on both roll angle and the rate of change of angle can be introduced by replacing qu1 ðtÞ in (15.14) by

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Nonlinear Automotive, Aerospace, Marine and Robotics …

Fig. 15.17 Ships roll angle time response with both rudder angle and rate limits

Fig. 15.18 Ships rudder angle and rate of change of rudder angle time responses

15.3

Ship Rudder and Fin Roll Stabilization

721

qu1 ðtÞ ¼ q0 u1 ðtÞ þ q1 ð1  z1 Þu1 ðtÞ The output and input time responses using both fin roll and rudder stabilization action, simulated for 300 s, are shown in Figs. 15.17 and 15.18. The former also includes a comparison with only using fin roll stabilization. The rudder rate limit is also included for these results and is set to b = 15 °/s and the control costing parameters q0 ¼ 1 and q1 ¼ 0:1. The ships roll angle, with combined rudder and fin roll stabilization, or just with the fins active, is shown in Fig. 15.17. The roll reduction is evident but requires a rather higher rudder slew rate. The yaw angle (not shown) remains within 3° of the nominal heading. By varying the design parameters, it is possible to achieve different degrees of trade-off between the aggressiveness of rudder action and the roll reduction. The rudder roll rate limit is often reached but the rudder angle stays within the linear region. Sensitivity minimization: The H1 norm is particularly relevant to the ship roll-stabilization problem, since the roll reduction achieved depends upon the sensitivity function linking the wave disturbances to rolling motion. Roll amplification corresponds with peaks at low and high frequency in the sensitivity function that define the roll reduction ratio. It is interesting that limiting these peaks is a natural H1 control design problem. The H1 design produced by Hickey et al. [6] was tested successfully in sea trials on board the Brittany Ferries, M.V. Barfleur, during a number of crossings of the English Channel.

15.4

Dynamic Ship Positioning Systems

Dynamic ship positioning systems are used to control the position of a vessel in the horizontal plane (see [12–19]). A Dynamically Positioned (DP) vessel can maintain a position to a given accuracy that is needed in applications like drilling vessels. A significant advantage over the use of mooring lines and a cluster of anchors is the ability to establish a position and leave quickly. In some locations it may be very undesirable to deploy anchors because of underwater cables, pipelines and structures on the seabed. The use of dynamic positioning may also be the only solution available for deepwater exploration and production, due to the length of the mooring lines required. The position of a vessel is affected by the wave motion which can be represented by high frequency (first-order) oscillatory wave forces and the low-frequency (second-order) wave and current forces. In the special case of ship positioning, the control objective is normally to keep the vessel’s position approximately fixed, irrespective of the second-order wave forces, the wind and the current force disturbances. However, for some vessels like cable repair ships, the vessel must follow a given trajectory. A subtlety of the positioning problem is that a vessel must be controlled so that it is allowed to move with the first-order wave forces causing the oscillatory motions, but the low-frequency second-order wave and drift forces must be opposed. There is usually no need to oppose the first-order wave forces since these do not cause the vessel to move off position in the steady-state. Moreover, the ships thrusters will be

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Nonlinear Automotive, Aerospace, Marine and Robotics …

GPS and computer

Constant tension winch

T Stern thruster

D

Bow thruster

Hydrophones

d Beacon or transponder

't seconds

Taut wire system Sinker weight 2 tons

Well-head

Y Fig. 15.19 Dynamic ship positioning main components

worn unnecessarily and energy will be wasted trying to oppose the oscillatory components of wave motion. For most applications, like diving support, firefighting, supply and drilling vessels, it is not necessary to fix the position of the vessel absolutely. It is good enough to stop the vessel drifting off position in a steady-state sense, only allowing it to move with the first-order oscillatory wave forces. The main components in a Dynamic Positioning (DP) system are illustrated in Fig. 15.19 for an oil rig drilling vessel. A drilling vessel should maintain a position to within a small envelope so that the riser connecting the vessel to the well is approximately vertical. The vessel can be allowed to move with the waves but must not drift significantly off position. There are different combinations of actuators that may be used for positioning systems. These include the main engines, tunnel thrusters, azimuth thrusters (steerable thrusters) and cycloidal propellers. This also applies to measurement systems, where a number of measurements may be pooled including Global Positioning Systems (GPS), taut wire and acoustic position measuring systems.

15.4.1 Ship Motion Modelling Consider a ship given in the inertial Earth-based coordinate frame, as shown in Fig. 15.20. The objective here is to control the vector of the ship’s position and heading g ¼ ½x; y; wT via a thruster/propeller propulsion system so that the

15.4

Dynamic Ship Positioning Systems

Fig. 15.20 Dynamic ship positioning control problem

723

Ship-based frame

y

y (sway) s

xs (surge) (yaw)

x Earth-based frame

desired trajectory gref is followed. In the majority of applications, the reference position will be fixed for station keeping but there are applications such as minehunting where the desired path should be followed. The simplified linearized ship dynamics are described by the following differential equation: M v_ þ Dv ¼ s

ð15:15Þ

where M is the inertia matrix, D represents system damping, v ¼ ½ u t r T is a ship-based velocity vector, and s is a vector that includes the forces in xs and ys directions and the yaw torque s. This approximation is good for low speeds and for station keeping problems. The velocity vector is related to the Earth-based positions by the following kinematic equation: g_ ¼ RðwÞv

ð15:16Þ

where RðwÞ is the following 3-DOF rotation matrix: 2 3 cosw sinw 0 RðwÞ ¼ 4 sinw cosw 0 5 0 0 1

ð15:17Þ

Thrusters: A simple diagonal thruster configuration was assumed for this example. The following nonlinear static model for thrust forces and torque was used: Ti ¼ qd 4 KT ðni Þjni jni

i ¼ 1; 2; 3

ð15:18Þ

where q is the water density, d is the thruster diameter, n is the velocity in revolutions/second and KT is the additional nonlinear thruster coefficient. An example of a nonlinear thruster characteristic, as a function of n, is shown in Fig. 15.21. The models used for the simulation were obtained from the Marine Systems Simulator Toolbox for MATLAB, due to Thor Fossen and co-workers [11]. Assuming the thruster dynamics are fast relative to the ships dynamics, they can be neglected here but they are usually represented by a simple time-constant term.

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Nonlinear Automotive, Aerospace, Marine and Robotics …

Fig. 15.21 Nonlinear thruster characteristic

Based on the above system description, the open-loop Hammerstein model for the ship may be separated into nonlinear and linear components, as shown in Fig. 15.22, with the black-box model W 1k representing the thrusters. The nonlinear transformation matrix RðwÞ is not considered part of the subsystem W0 and will only be used to plot the ship’s position in the inertial frame. Wave model: The wave disturbance model includes a second-order resonant system, described by the transfer-function Ww ðsÞ ¼ k=ðs2 þ 2fxn s þ x2n Þ, where the natural frequency and damping factor were defined as xn ¼ 0:8 rad/s and f = 0.1, respectively. The scaling factor k was chosen so that realistic wave amplitudes were obtained in terms of an approximation of standard sea spectra [20]. The frequency responses of both the ship dynamics (without the thruster model), and the wave disturbance are shown in Fig. 15.23 (the cost-weightings are also shown in this figure). Some crude form of sea-state adaption is often included like switching the wave models in low, medium and high sea-state conditions. More elegant methods like extended Kalman filtering or self-tuning filtering [19] can also be used.

Position

R( ) Ship dynamics

n

W0 Thrusters

Fig. 15.22 Decomposition of the ship model

Position in earth coordinates

1 s

Position in ship coordinates

15.4

Dynamic Ship Positioning Systems

725

Fig. 15.23 Frequency responses of system models (ship axes motion model, wave disturbance and cost-weightings)

15.4.2 Dynamic Ship Positioning Control Design The dynamic ship-positioning control problem has been considered using several linear control design methods (for example, [13–17]). In the following, the NPGMV controller is assessed for this DP application but most of the previous work has been on what is often called Kalman iltering or Linear Quadratic Gaussian-(LQG) based systems [13, 14, 16–18]. In these linear control design approaches, nonlinearities were only addressed indirectly. An observer-based backstepping approach was proposed by Fossen and Grovlen [24] to deal with the nonlinearities in DP systems. For the NPGMV controller design, the linear ship dynamics were discretized using Tustin’s method, with a sample time of Ts = 0.1 s. A nominal one sample transport delay (k = 1) was assumed that might represent a combination of computing and physical plant transport delays. As mentioned above, the main objective of the ship positioning controller is to reduce the effect of the wind, current and steady wave motion forces, whilst ignoring the oscillatory (first-order) wave forces [21]. There are several types of ship positioning problem, and various control and estimation techniques have therefore been applied [22–24]. The controlled variables are assumed to be the ship’s position and its heading. The system includes integrators, but integral action in the controller is needed to compensate for disturbances that feed into the ship’s dynamics. Following the tuning guidance given in Sect. 5.2, the NPGMV controller design was first based on a PID-motivated weighting with the tuning gains selected as Kp = diag{12, 40, 10},

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Ki = diag{0.6, 2, 0.1} and Kd = diag{36, 20, 4}. Typical dynamic weighting functions are as shown in Fig. 15.23. Derivative filters with time constants of one second were included on the derivative terms of the PID controller. The continuous-time PID controller was discretized using the Tustin method to obtain CPID ðz1 Þ . The dynamic error and control weightings for the NPGMV control design were defined as Pc ðz1 Þ ¼ CPID ðz1 Þ and Fck ¼ VNT , respectively. The input weighting K2N was set to zero. It is interesting that, as noted in Theorem 4.1, for the case of N = 0, these weightings correspond to the limiting case of an NGMV controller but one that has a two degrees-of-freedom control structure. The NPGMV solution provides a predictive controller when the cost horizon N > 0. Two simulation scenarios were considered to illustrate the operation of the system: (i) Fixed point ship positioning subject to wave motion disturbances, which is the conventional DP problem. (ii) Elliptical trajectory following, which is a tracking problem. The simulation results for the limiting case of N = 0 are shown in Figs. 15.24 and 15.25. With the nominal weightings, the NPGMV controller performance (which is essentially NGMV control) gives only a small improvement over the classical PID control (not shown).

Fig. 15.24 Ship positioning using NPGMV control with N = 0

15.4

Dynamic Ship Positioning Systems

727

Fig. 15.25 Elliptic trajectory following using NPGMV control with N = 0

15.4.3 Dynamic Positioning Predictive Control Results The PID-based approach to weighting selection is a useful starting point for NPGMV control design. There is a rapid improvement with increasing N and the results that follow indicate that only a few steps are sufficient for a satisfactory design. The diagonal error weighting: Pc ðz1 Þ ¼ diag



46:32z2  91:43z þ 45:12 59:15z2  114:3z þ 55:15 13:81z2  26:67z þ 12:85 ; ; ðz2  1:905z þ 0:9048Þ ðz2  1:905z þ 0:9048Þ ðz2  1:905z þ 0:9048Þ



ð15:19Þ was modified by penalizing the frequency band corresponding to the wave motion spectrum. This involved an inverted band-pass (notch) wave filter: Hf ðz1 Þ ¼

z2  1:947z þ 0:9532 1:031z2  1:947z þ 0:922

ð15:20Þ

However, note that a lead term was also included in the control weighting function F ck ¼ Fck ¼ VNT to reduce the high-frequency noise amplification. The high-pass filter (appears approximately constant on the scale used in Fig. 15.23) was defined as

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Nonlinear Automotive, Aerospace, Marine and Robotics …

Hc ðz1 Þ ¼ ð1  0:1z1 Þ=0:9

ð15:21Þ

The aim of the notch filter on the error weighting was to reduce the controller gains near the dominant wave frequency so that the thrusters do not respond to the first-order wave forces (a phenomenon known as thruster modulation). The lead term on the control weighting is to ensure sufficient roll-off of the controller at high frequency. Simulation trials were performed for increasing values of the prediction horizon N, starting from the limiting case of N = 0 (Figs. 15.24 and 15.25). The responses for N = 4 are shown in Figs. 15.26 and 15.27, and the improved station keeping and tracking performance is evident. This is achieved at the expense of the more aggressive thruster action, which responds and compensates for the wave motion. Recall that the same prediction and control costing horizons have been used here although different horizons can be considered using the connection matrix defined in Chap. 11 (Sect. 11.4.1). In this case of common horizons, increasing the prediction horizon N often leads to smaller peak overshoots on sensitivity functions and step responses, which is due to better prediction but there is a secondary effect which makes the behaviour more uncertain. This is due to the relative importance of the error and control terms in the cost-index. If the control costing term is relatively more significant, an increase in N can tend to reduce controller gains countering the improvement in responses due to improved prediction.

Fig. 15.26 Ship positioning using NPGMV control with N = 4

15.4

Dynamic Ship Positioning Systems

729

Fig. 15.27 Elliptic trajectory following using NPGMV control with N = 4

15.4.4 Implementation Issues for NPGMV Ship Controller The algebraic-loop problem in implementing the controller can be solved in a similar method to that for the robot arm example in Sect. 15.2.3. However, in the DP control problem the black-box term is a static nonlinearity representing the thrusters. The NPGMV controller was described in Chap. 9 (Sect. 9.4.4) and the particular form for implementing the controller in the DP problem may now be explored. The nonlinear thruster model can be represented by the input nonlinearity W 1k , which is a static diagonal matrix. The structure of the model allows the algebraic-loop problem to be avoided. It is also assumed that the control weighting can be separated into a component affected by the current control (the direct through term) and a part dependent only on the past controls F ck ðsÞ ¼ F 0 ðsÞ þ F 1 ðsÞ, where F 1 ðsÞ contains at least a single-step delay. The NPGMV optimal control was given by (15.6), noting G0;N ¼ W 1k;N and G1;N ¼ 0, and follows as  1  ~ 0t þ k;N ðF 1;N  XN G1;N ÞUt;N þ C/ T0 BCI0 W 1k;N Ut;N Ut;N ¼ F 0;N  XN G0;N PCN R  1  ~ 0t þ k;N ðF 1;N  C/ T0 BCI0 W 1k;N ÞUt;N  PCN R ¼ F 0;N  XN W 1k;N

ð15:22Þ

730

15 Prediction

C N AN Ak

Rt

–

–

PCN

+

XN

CI 0

(t ) Torque demand

k N

1,N

+

C T0 BC I 0 xˆ(t t)

U t,N

1 1k , N

–

0

Rt

0, N

+

k ,N

Future reference

Nonlinear Automotive, Aerospace, Marine and Robotics …

1k , N

Position (t ) Estimation Kalman Filter

z

k

CI 0

Fig. 15.28 Detailed implementation of the NPGMV controller

where T0 ðk; z1 Þ contains a single-step transport delay. The right-hand side of (15.22) only involves the vector of controls computed in the previous step. The current control is therefore dependent on the past controls and inputs and no algebraic loop is present. To implement the control law (15.22), it is necessary to invert the static operator (F 0;N  XN W 1k;N ). In this situation, the NPGMV controller can be implemented without the algebraic loop (as in Chap. 5 for the NGMV control design). Based on (15.22) the structure of the controller to be implemented is as shown in Fig. 15.28, where the actuator saturation limits have also been included in the model and the Dt þ k;N term is null in this case. Thruster nonlinear compensation: One of the advantages of the NGMV and NPGMV approaches is that the control signal weighting can be nonlinear without additional complications to the solution. This feature may be exploited by using the 0 ðz1 Þ thruster nonlinearity model to redefine the control weighting as Fck ðuÞ ¼ Fck 0 1 HðuÞ. Here Fck ðz Þ is the nominal linear control weighting, and HðuÞ is the thruster characteristic shown in Fig. 15.21. The inverse of HðuÞ can be approximated by the pffiffiffiffiffiffiffi expression n ¼ 4:6078  signðTÞ  jTj. The varying nonlinear gain of the actuators can then compensate in some measure for the actuator nonlinearities.

15.5

Diesel Engine Modelling and Control

There are many challenging control problems in automotive systems, in applications ranging from engine powertrains to autonomous vehicles. Regulators now demand low and less harmful emissions, and drivers seek improved fuel economy and good driveability. Driveability is hard to define but it is easy to appreciate in powertrains that are responsive, smooth and have intuitive behaviour [25].

15.5

Diesel Engine Modelling and Control

731

The complexity of automotive engines has increased over the last two decades as the legislation on reduced vehicle emissions has been introduced. This trend will continue and it will increase engine control system design time and development costs. There are also indirect costs in meeting the tighter emission standards such as the deterioration in fuel economy. This depends upon a number of design issues. For example, there is a correlation between engine NOx emissions and fuel consumption. Unfortunately higher engine efficiency and better fuel economy result in higher NOx. Exhaust after-treatment devices introduce an additional fuel economy penalty. It is unfortunate that emissions reduction and fuel economy are often contradictory goals. The focus in this section is on diesel engines (named after the German engineer Rudolf Diesel), that exhibit strongly interactive and nonlinear behaviour. The control of diesel engines is a challenging task and good performance is difficult to achieve using simple multi-loop linear controllers. A gain-scheduling scheme is normally required to take into account the nonlinearities in the engine and the differences in engine dynamics. The control challenges are mainly caused by the use of a turbocharger and the exhaust gas recirculation path. A Variable Geometry Turbocharger (VGT) can increase engine power, improve fuel economy and driveability. A turbocharger in a diesel engine consists of a compressor and a turbine on a common shaft. The exhaust gases drive the turbine, which in turn drives the compressor (Fig. 15.29). This compresses ambient air and directs it into the intake manifold. The increased volume of air allows a larger quantity of fuel to be burnt and a larger torque to be produced. It enables the fuel economy to be improved by improving operation at lean air-to-fuel ratios, and it increases the power density of the engine by forcing air into the cylinders. It allows additional fuel to be injected without reaching the smoke limit that is caused by rich combustion. The increased air-to-fuel ratio, air charge density and the temperature, that result from turbocharging, all tend to reduce the particulate emissions. However, the oxides of nitrogen (NOx) formation tend to increase. An intercooler is therefore

Intercooler

MAF

MAP Intake manifold

Compressor EGR cooler

EGR valve Exhaust manifold

turbine Catalyst

Fig. 15.29 Diesel engine layout

VGT position

EGR position

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used to reduce air charge temperature and partially offset the NOx increase caused by turbocharging. The intercooler increases the air charge density and thereby reduces the particulate emissions, but intercooling alone is not sufficient to reduce the NOx emissions to regulation levels. To reduce the emissions of the harmful oxides of nitrogen, a portion of the exhaust gas can be diverted back to the intake manifold [26, 27]. This dilutes the air supplied by the compressor, and is a process that is referred to as exhaust gas recirculation (EGR). Exhaust gas recirculation is controlled by an EGR valve and the path is as shown in Fig. 15.29 (based on [30]). The amount of exhaust gas that can be recirculated depends on the operating conditions because the exhaust gas reduces the amount of fresh air in the intake manifold, which is needed to boost engine torque and avoid smoke generation. The recirculated exhaust gas in the cylinders acts as an inert gas and increases the specific heat capacity of the air charge. It replaces oxygen in the inlet charge. This reduces the burn rate and the temperature profile of combustion, and the emissions of NOx. The actuators available in engines, such as electronic throttles, variable valve timing and EGR, assist in meeting the competing demands of emissions, performance and fuel economy. The management of the supply of air to the combustion chamber is an important aspect of the control of diesel engines. If emissions are to be reduced, the intake air properties must be controlled to suit the engine operating conditions. The nonlinearities are a complicating factor that arises from uncertainties in engine maps, and there are also unmodelled dynamics and disturbances due to both external and internal factors [28]. Several advanced control methods might be used but model-based predictive control has been researched extensively. A nonlinear model predictive control for a turbocharged diesel engine was described by Herceg et al. [29].

15.5.1 Current Practice in the Control of Diesel Engines The development of a new engine is very labour intensive. It often involves time-consuming engine calibration procedures, to obtain parameters that are stored in multidimensional tables for different operating conditions. It then involves lengthy procedures to tune controllers that have a classical feedback/feedforward structure. This traditional design approach does not exploit the sophisticated engine models and simulations that often exist, but it is easy to apply by engineers without much advanced controls background. The strength of model-based control methods is that they take into account plant dynamics, interactions, nonlinearities and saturation limits explicitly. The control of a diesel engine is challenging due to its multivariable nature, presence of severe nonlinearities, disturbances, uncertainties and interactions. A diesel engine has complex dynamic behaviour, due to the internal feedback loops introduced by the EGR and the turbocharger. These involve fast and slow dynamics, where the fast dynamics are due to pressure variations in the manifolds,

15.5

Diesel Engine Modelling and Control

733

and the slow dynamics are due to the turbocharger. These effects can result in non-minimum phase behaviour, varying undershoots and overshoots, and a possible DC gain (steady-state gain) sign reversal. Production controllers for diesel engines normally consist of single-input single-output PI (D) controllers. To cope with the nonlinearities, their gains are scheduled over the speed/load envelope, and features such as operating point dependent limits on the outputs are included. These controllers often have to be detuned to avoid instabilities due to cross-coupling effects. A lot of effort by calibration engineers is needed to map the engine characteristics and tune these controllers. This is a lengthy and costly process. Jung and Glover [31, 32] proposed the use of Linear Parameter Varying (LPV) model-based controls to reduce the calibration effort significantly. The use of LPV models was also explored by Mohammadpour et al. [33] using a decoupling control approach for diesel engines.

15.5.2 Model-Based Control Design Study In this design example, the diesel engine emissions and drivability requirements are represented by regulating two measurable variables: intake manifold pressure MAP and intake airflow MAF. The MAP and MAF setpoints are computed as a function of the engine conditions, i.e. the engine speed and the fuel flow rate, and are assumed given. These functions are normally determined by the optimization or calibration procedures. Both the engine speed and the fuel flow are treated as known external signals. The general block diagram of the diesel engine control system is shown in Fig. 15.30. The feedforward signals are denoted by the dashed lines, and are not included explicitly in the design that follows. The feedback controller includes a time-varying (state-dependent) Kalman filter to estimate the engine states, based on the model and the measurements of MAP, MAF and exhaust manifold pressure (EXMP). Experience suggests that this is one of the most reliable and effective parts Wf

+

SP

MAP Setpoints and Feedforward

SP

MAF

Feedback Controller

uvgt,FF uvgt

+ +

uegr

MAP

Engine MAF

+ uegr,FF EXMP

N Fig. 15.30 Schematic diagram of diesel engine control

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of the control solution, whether used with a state-space predictive control design, or some other control method. The signals may be identified as follows: • Outputs: The outputs are required to follow the MAP and MAF setpoints. • Control Inputs: The two control inputs are the Variable Geometry Turbine (VGT) vane lift uvgt (% closed) and EGR valve stem position uegr (% open). • Disturbances: The fuel flow rate Wf (g/s) variations are considered a disturbance representing the engine load changes. From the controller’s perspective, the engine speed N is also treated as a disturbance.

15.5.3 Engine Model The nonlinear engine model that is used here is a simplified first principles model, including the intake and exhaust manifold dynamics, external EGR and a simple mechanical model of a turbocharger (see Fig. 15.29). A common model approximation that is followed here is based on a three-state model, with the intake manifold pressure pi, exhaust manifold pressure px and compressor power Pc defined as the model states. This model has been used often and despite its relatively low order, it can replicate diesel engine phenomena, such as complex nonlinear behaviour, non-minimum phase dynamics and DC gain reversal [30]. In the following model description, it will be assumed that the air density is a constant parameter and that ambient pressure and temperature, and the intake and exhaust manifold temperatures are also approximately constant. Model equations: The summary of model equations are given below, with the constant parameter values collected in Table 15.1. The dynamics of intake and exhaust manifolds and the power transfer model are described as p_ i ¼ RTi ðWci þ Wxi  Wie Þ=Vi   p_ x ¼ RTx Wie  Wxi  Wxt þ Wf =Vx   P_ comp ¼ Pcomp þ gm Pt =sT The terms in the above model may be listed as follows: Compressor flow (MAF): Wci ¼

gc Pcomp =ððpi =pa Þl 1Þ cp T a

EGR flow (Wegr): Aegr ðxegr Þpx Wxi ¼ pffiffiffiffiffiffiffiffi RTx

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi 2pi pi 1 px px

ð15:23Þ

15.5

Diesel Engine Modelling and Control

735

Table 15.1 Constant parameters for the diesel engine model Parameter

Value

Unit

Parameter

Value

Unit

ηc ηt ηv ηm sT

0.569 0.76 0.877 0.98 0.121

– – – – s

Ti Tx pa Ta svgt pref Tref

313 509 98 298 0.01 101.3 298

K K kPa K s kPa K

Vd Vi Vx R Cp l = (c–1)/c a b c d segr r1 r2

2 6.26 0.2 287 1006.1 0.286 −0.1694 0.1989 0.1914 0.6 0.01 0.1243e−3 0.0946e−3

L L L J/kg K J/kg K – – – – – s – –

Cylinder flow (Wcyl): Wie ¼ gv pi NVd =ð120Ti RÞ Turbine flow (Wturb): rffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 

ffi px px Tref 2pa pa Wxt ¼ Avgt ðxvgt Þ c  1 þd 1 pa pref Tx px px Turbine power: Pt ¼ Wxt cp Tx gt ½1  ðpa =px Þl  Parameters that may be assumed constant may be listed as • Gas constant R, specific heat cp, specific heat ratio µ. • Volumes Vi, Vx, Vd, mechanical and volumetric efficiencies, turbine flow coefficients. Parameters that are assumed constant but may be measured follow as • Manifold temperatures Ti, Tx • Ambient conditions Ta, pa. The effective flow areas for the actuators are given as Avgt ðxvgt Þ ¼ axvgt þ b

and

Aegr ðxegr Þ ¼ r1 x2egr þ r2 xegr

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Fig. 15.31 Diesel engine static characteristics

The actuator dynamics are ignored in this example. Actuator constraints: Both the EGR valve position and VGT vane lift are physically constrained  between 0% and 100%. The   slew rate limits for these actuators are taken as x_ vgt   100 ð%=sÞ and x_ egr   100 ð%=sÞ. These limits are incorporated in the constrained Model Predictive Control solution. Static characteristics: The complex nonlinear nature of the model is illustrated by the engine static characteristics shown in Fig. 15.31. The steady-state values of MAP and MAF are plotted against increasing VGT, for four different EGR valve openings, and with constant speed and load. The nonlinearity is evident, and the MAF plots indicate a gain reversal with varying EGR. This behaviour can be very challenging for engine controllers.

15.5.4 LPV Modelling The state-dependent or qLPV model seems very suitable for engine control design whether diesel or SI engines. This type of model can be obtained in different ways: • The nonlinear physical engine model (15.23) can be rewritten in a qLPV model form that does not involve linearization or approximation. • Nonlinear system identification methods may be used to obtain an approximate model. • Jacobian linearization can be used at a general (non-equilibrium) operating point.

15.5

Diesel Engine Modelling and Control

737

The latter approach is used here and the linearization was performed analytically, giving an LPV model for deviations around the operating point, of the form: 2

3 2 3 dpi ðtÞ dp_ i ðtÞ

 dNðtÞ dAv ðtÞ 6 7 6 7 þ Gðpt Þ 4 dp_ x ðtÞ 5 ¼ f ðxt ; ut Þ þ Aðpt Þ4 dpx ðtÞ 5 þ Bðpt Þ dWf ðtÞ dAr ðtÞ dP_ c ðtÞ dPc ðtÞ 2 3 dpi ðtÞ
 dMAP 6 7 ¼ Cðpt Þ4 dpx ðtÞ 5 dMAF dPc ðtÞ ð15:24Þ where the system parameter-varying matrices A, B, G and C are obtained by differentiating the model (15.23). The parameter vector p contains the variables (pi, px, N, Wf, pa, Ta, Ti, Tx), which are assumed to be measured or estimated at time t. For the design of the predictive controller, the state-dependent or qLPV model (15.24) is discretized with a sample time of Ts = 5 ms. The NGMV controller is simple to construct since in this case the discretized model of the engine will all be placed within the black-box input subsystem W 1k . These models are used in a more fundamental way to compute the outputs over the prediction horizon in the predictive controllers.

15.5.5 Baseline Controller A simple decentralized PID control solution was used for comparison with the more advanced control designs. In this control scheme, the VGT vanes were used to control the intake manifold pressure, whilst the EGR valve controlled the mass airflow. This choice ensured the monotonic relationship between the respective inputs and outputs. Following the approach detailed in van Nieuwstadt [34], the PI controller parameters were gain-scheduled using the inverse of the plant DC gain matrix (determined for a range of engine speeds and fuel flows). The PI gains can be computed as Kp;vgt ¼ Kp11 Kg11 ðN; Wf Þ;

Ki;vgt ¼ Ki11 Kg11 ðN; Wf Þ

Kp;egr ¼ Kp22 Kg22 ðN; Wf Þ;

Ki;egr ¼ Ki22 Kg22 ðN; Wf Þ

ð15:25Þ

1 where the matrix Kg is defined as Kg ¼ KDC . This leaves only four parameters to select, simplifying the tuning procedure.

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15.5.6 Design Approach and Weighting Options Predictive control seems to have a bright future in diesel engine control. The use of predictive control for actuator allocation in diesel engine control was considered by Vermillion et al. [25, 35]. In fact, model-based multivariable control methods are needed in today’s automotive diesel engines because of the range of actuators and measurements now available [36, 37]. This also applies to applications in marine diesel engines [38] and engines for power generation. Good results for the NGMV controller were obtained relatively easily with little design effort. These results may be compared with the Nonlinear Predictive Generalized Minimum Variance (NPGMV) controller. This employs a simplified state-dependent/qLPV model. For small cost horizons, the predictive control results should relate to the NGMV control solution since the algorithms become equivalent to two degrees-of-freedom NGMV controls. The predictive control solution includes a dynamic cost-function weighting on the prediction error and on the input signal (incorporated via an extended state-equation model). It is more flexible since it involves more options, such as the choice of cost horizons and an option to apply constraint handling. Alas, it is also more intensive computationally. Cost-function weighting choices: There are several possible ways to choose the cost-function weightings, described in Chap. 5. The two main categories are 1. PID-inspired dynamic cost-function weightings structure. 2. Free choice of dynamic weighting function structure. The first approach uses the idea that if a PID controller or classical controller stabilizes a system, then it may be used as a starting point for the error weighting term choice. The control signal costing, in this case, can be normalized to the identity for both the NGMV and NPGMV related designs. For the free choice of weightings to achieve good reference tracking and disturbance rejection, the error weighting should be set to have a large gain at low frequencies. The control weighting should include a lead term to penalize high-frequency actuator movements and limit measurement noise amplification. The two approaches to weighting selection provide cost-weightings that are similar in the frequency characteristics employed. The Kalman filter required for state estimation includes the plant and the disturbance model states, as well as the states of the dynamic cost-function weightings. This is a time-varying estimator and is implemented as a recursive algorithm. Scheduled weightings: In some applications, when the desired performance cannot be achieved with linear time-invariant weightings, the weightings need to be time-varying or scheduled with an operating point. For example, the cost-weightings can be represented in a state-dependent form, providing an approximation to a nonlinear weighting term. There is also the more unusual option of using nonlinear or gain-scheduled control weighting terms (possibly in just F ck ). This type of scheduled weightings can improve performance in both the NGMV and the NPGMV control design solutions.

15.5

Diesel Engine Modelling and Control

739

Choice of cost-function horizons: The cost-function horizon in the predictive control problems considered here is usually chosen to be the same for the error and control cost-function terms, but there is the option to use different horizon lengths. In addition, the future controls can be frozen for a pattern of control steps into the future. This technique is referred to as move blocking and reduces the control signal computations (see Sect. 11.4.1). Hard constraints: The quadratic programming (QP) solution can be used to address the system constraints explicitly in the controller solution. However, the soft constraints approach is often preferred since it avoids the increased computational burden associated with the QP solver.

15.5.7 Simulation Results The diesel engine is a difficult system to control, especially at higher loads (which implies higher MAP and MAF), where the controller gains need to be reduced to avoid stability issues. This is common practice with the gain-scheduled PID control. The results for a scenario of reference filtered step changes is shown in Fig. 15.32. MAP [kPa]

145

GS-PID NGMV

140

MAF [g/s]

40

38

135 36

130 125

34 120 115

32 2

4

6

8

10

12

VGT lift [%]

100

2

6

8

10

12

EGR valve position [%]

30

90

4

25

80 20 70 15

60 50

10 2

4

6

8

10

12

2

4

6

8

10

12

Fig. 15.32 Diesel engine control system responses for gain-scheduled PID and NGMV controls. The reference signals are denoted by the dotted lines

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Fig. 15.33 NGMV control weighting frequency responses

MAP [kPa]

145

GS-PID SD-NPGMV

140

MAF [g/s]

40

38

135 130

36

125 34 120 115

32 2

4

6

8

10

12

VGT lift [%]

100

2

50

80

40

70

30

60

20

50

6

8

10

12

EGR valve position [%]

60

90

4

10 2

4

6

8

10

12

2

4

6

8

10

Fig. 15.34 Comparison of gain-scheduled PID and state-dependent NPGMV responses

12

15.5

Diesel Engine Modelling and Control

741

NGMV control: The NGMV control design approach is the simplest, and the frequency responses of the weightings (singular values in the MIMO case) are shown in Fig. 15.33. These have the typical characteristics discussed earlier, of integral type action for the error weighting and a lead term for the control weighting, and were obtained by modifying the PID-based weighting designs. The NGMV control time responses are also shown in Fig. 15.32 and can be compared against the scheduled PID controller. The two are quite similar although the NGMV control action is generally smoother. This is for the nominal PID-inspired weightings but before retuning to increase gain and thereby performance. Predictive control: The ability of predictive control to “look” further into the future can be particularly beneficial for the difficult diesel engine dynamics that involve non-minimum-phase behaviour. Compared with the NGMV design, the penalty on the tracking errors, relative to the control actions, may now be increased, resulting in improved performance. The time-response results are illustrated in Fig. 15.34 for the unconstrained case. The prediction horizon of N = 15 steps was used. This was sufficient to capture the non-minimum-phase behaviour of the intake manifold pressure and to provide tighter tracking. Constraints: A major advantage of the predictive controller for the diesel engine control design is the ability to handle constraints. For example, physical actuator MAP [kPa]

145

MAF [g/s]

40 Unconstrained Constrained

140

38 135 130

36

125 34 120 32

115 2

4

6

8

10

VGT lift [%]

100

2

12

4

6

8

10

12

EGR valve position [%]

60 50

90

40 80 30 70

20 10

60 2

4

6

8

10

12

2

4

6

Fig. 15.35 NPGMV-constrained case outputs and controls and setpoints

8

10

12

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Nonlinear Automotive, Aerospace, Marine and Robotics …

rate limits can be incorporated into the design. The limits may also represent the design values introduced to reduce wear and tear on the actuators. Such design limits have been included on both the EGR and VGT inputs, and the results are as shown in Fig. 15.35. It can be seen that the control signals rate of change is indeed limited, whilst setpoint tracking is only slightly less tight. A less computationally demanding approach would be to include “soft constraints” in the design, using cost penalties on the actuator variations.

15.6

Sightline Stabilization of Aircraft Electro-optical Servos

The defence of civilian and military aircraft against missile attacks requires the rapid identification of hostile threats before countermeasures may be taken. Effective and reliable stabilized airborne electro-optic (EO) systems are needed to mount the instrumentation. The pointing, stabilization, tracking and image processing of electro-optic systems is a subject that is known as sightline control. Turrets for airborne electro-optical sensor systems are supplied by a number of companies, for fixed wing, rotary wing and unmanned aircraft platforms. Multi-sensor turret systems fall into two broad classes: • Intelligent Surveillance and Reconnaissance (ISR) systems. • Defensive Aid Suite (DAS) systems. These systems combine high-performance sensors and a turret for airborne observation, surveillance and reconnaissance [42–44]. These systems use similar optical technology, but the operational differences demand different solutions. The ISR system should have a long-range imaging capability. These require high-resolution imagers and accurate sightline stabilization that can minimize jitter. A DAS system needs a very agile sightline to ensure that incoming threats to the host platform are properly tracked and interrogated or defeated. Two important features of EO countermeasures devices may benefit from the application of advanced controls, namely the tracking of an agile moving target and ship motion sightline stabilization.

15.6.1 Nadir Problem When a ship or aircraft comes under attack by a surface-to-air missile, a fast and high accuracy sightline control loop is needed. The aim is then to jam the missile seeker. The tracking system must be able to operate over a full hyper-hemispherical Field-of-Regard (FoR) (as shown in Fig. 15.36). A 2-axis gimbal device with one gimbal rotating over the azimuth axis and the other over the elevation axis can

15.6

Sightline Stabilization of Aircraft Electro-optical Servos

743

provide a sufficient FoR. Unfortunately, as described in [44, 45], there is a problem when the target moves so that the Line-of-Sight (LoS) vector approaches the azimuth axis. This occurs at around −90° in elevation, where the system loses a degree of freedom. This compromises the tracking accuracy and the loss of a degree of freedom is referred to as the “nadir cone.” The tracking in the neighbourhood of the nadir needs sufficient agility from the outer azimuth gimbal axis to the limit of the LoS vector tracking. Unfortunately, the singularity when the LoS and azimuth axis are collinear results in infinite theoretical acceleration and rate demands. The acceleration demands in this region saturate the servos, leading to significant tracking errors. The approach described below employs a Nonlinear Generalized Minimum Variance (NGMV) sightline controller that includes a model of the nonlinearity to mitigate the effect of the nadir singularity on the tracking error. NGMV control: Operation in the neighbourhood of the nadir involves two major nonlinearities. The first is the singularity described above, and the second is a kinematic nonlinearity. This is caused by the difference between the cross-elevation axis (measured relative to the imager frame) and the outer gimbal azimuth axis. This suggests that a true nonlinear control design technique should be used and some that have been evaluated include predictive control, sliding mode control, adaptive backstepping, LMI optimization, nonlinear H∞ control, feedback linearization and also Lyapunov methods. However, many of these nonlinear control design techniques are difficult to understand and they can also be difficult to tune. The NGMV control law provides a simple design framework that accommodates the nonlinearities. Once the nonlinearities in the system are captured in the modelling process, the NGMV controller can be tuned via parameterized cost-function weightings. The controller yields a nonlinear control solution that is simple to design and commission.

Fig. 15.36 Hemispherical coverage of the electro-optical tracking device mounted on a platform (Based on the thesis and by courtesy of Savvidis [45])

Sightline 180o Base Tilt

0o

Electro-Optical Device

Waterline

FoR

Turre t Bas

e

Blind Region

360o

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15.6.2 Experimental Configuration The state-space form of Nonlinear Generalized Minimum Variance (NGMV) control algorithm, described in Chap. 8, was applied to the stabilization of an electro-optical (EO) gyroscopic turret used in surveillance applications. The algorithm was deployed successfully into a National Instruments PXI real-time target to provide stabilization for a 2-axis gyroscopic servomechanism (gimbal) in a target tracking application. The algorithm was described in Savvidis et al. [44] and the problem and results are summarized below. The algorithm involved an internal approximation of the gimbal that was executed at each iteration to produce the ksteps ahead predictions of its response. As a result, the time on target increased significantly, particularly when tracking near the nadir cone area. The real-time control experimental work was conducted at the University of Glasgow with the support of SELEX (now Leonardo, Edinburgh). This involved hardware-in-the-loop functional testing. A TigerEye 2-axis visible band electro-optic turret was used (TigerEye is a trademark and brand of AeroMech Engineering, Inc.). The turret was mounted on a tripod with 1.50 m height and 2.50 m distance from a projector screen. The projector was used to show a video of different operational scenarios. These were chosen to push the tracking loop so that it reached the performance limits in both the creation of a suitable track/ association and in kinematic prediction. In the first set of experiments, only the tracking-loop dynamic response was of interest and consequently the projected target used was a white ball against a blue background. This moved along either a circular or a vertical trajectory (specified in the world coordinate system). For a circular trajectory, this consisted of the y- and zaxis of the ball, defined as sinusoidal functions, where the parameters defined the circle radius and the revolution time. The target motion frequency used was 0.1 Hz with a sample rate for the projected image of 100 Hz. The TigerEye can operate in rate mode. The commands sent to the turret were interpreted as rate-loop demands for each axis. A schematic diagram of the system is shown in Fig. 15.37. The measurement for the elevation displacement was taken as the vertical pixel displacement (the camera was fixed to the inner gimbal). This measurement was scaled from pixel error to angle error, and is then converted into a rate demand. The azimuth measurement displacement is also taken in terms of

Fig. 15.37 Tracking system physical configuration (By courtesy of Savvidis et al. [44] and Savvidis [45])

15.6

Sightline Stabilization of Aircraft Electro-optical Servos

745

horizontal pixel displacement (measurement obtained from the camera on the inner gimbal). The second error measurement is the azimuth with respect to the inner gimbal, referred to as the cross-elevation. Simulation model: The overall tracking system is shown in Fig. 15.38, with the various transformations required to convert the target vector from the world to azimuth/elevation coordinate frames. The transformation matrices Rw/b, Rb/og and Rog/ig are, respectively, for the world to base, base to outer gimbal and outer gimbal to inner gimbal. The controller area network (CAN) bus delays and the individual rate-loop transfer-functions were obtained by using system identification methods. Each channel comprises an external position and an internal rate loop. For the duration of the experiments, the base declination angle remained constant, although in practice base motion will occur (when mounted on an airborne platform). Turret kinematics: The way a change in target position relates to the gimbal angles demanded for accurate tracking will now be considered. Assume that the line-of-sight rests on the waterline (zero azimuth and elevation angles) before tracking a displacement in the target axes. There are three principal axes of rotation: • Base tilt h, which transforms from world axes to base axes, • Azimuth angle η (outer gimbal) rotation about the TigerEye z-axis and • Elevation angle e (inner gimbal) rotation about the y-axis. Since the camera is carried with the inner gimbal, the image and inner gimbal axes may be assumed coincident, neglecting alignment tolerances. The expression for the target position on the image axes may be obtained as xlos

im

  ¼ Te Tg Th xlosgeo

ð15:26Þ

Target trajectory

Rw/b

θ

Base Tilt

ηo

deg/rad

Sightline position

1

s

o

-

R b/og

e Frame rate

+

Rate limits

Tilt Rate loop

Rate limits

Pan Rate loop

pixel deg/s Frame rate

.

ε

.

ε demand

εo R og/ig

CAN bus delay

ε demand

Cartesian to Angle

MIMO controller

.

η demand

η demand e Frame rate

+ -

deg/rad

CAN bus delay

pixel deg/s Frame rate

.

η

1 Azimuth position

s

Fig. 15.38 Tracking system simulation configuration (Based on the thesis and by courtesy of Savvidis [45])

746

15

where

Nonlinear Automotive, Aerospace, Marine and Robotics …

2

3 2 cosðeÞ 0 sinðeÞ cosðgÞ Te ðeÞ ¼ 4 0 1 0 5; Tg ðgÞ ¼ 4 sinðgÞ sin 0 2 ðeÞ 0 cosðeÞ 3 cosðhÞ 0 sinðhÞ 5 1 0 Te ð h Þ ¼ 4 0 sinðhÞ 0 cosðhÞ

sinðgÞ cosðgÞ 0

3 0 0 5; 1

The matrices Te ; Tg ; Th are the direction-cosine transformation matrices, for the elevation, azimuth and base tilt angles, respectively. Thence we obtain xlos 2 ¼4

cosðeÞcosðgÞcosðhÞ  sinðeÞsinðhÞ cosðhÞsinðgÞ cosðeÞsinðgÞ þ cosðgÞcosðhÞsinðeÞ

im

cosðeÞsinðgÞ cosðgÞ sinðeÞsinðgÞ

3 cosðhÞsinðeÞ  cosðeÞcosðgÞsinðhÞ 5xlosgeo sinðgÞsinðhÞ cosðeÞcosðhÞ  cosðgÞsinðeÞsinðhÞ

To recover the off-boresight angle errors needed to drive each tracking-loop controller, the actual target position is projected onto the surface of a unit sphere centred at the image plane. A simple conversion from Cartesian to polar coordinates recovers the error angles as measured by the TigerEye imager. It will be assumed that the LoS is expressed in the geographical frame, when aligned with the initial target position, as follows: xlosgeo ðe; g; hÞ0 ! xtarget ðx; y; zÞ0

ð15:27Þ

If the target moves to a different position xtarget ðx; y; zÞ1 , then, given the angles x, y, z, h, the elevation e and azimuth h angles can be computed that are required by the gimbal to realign its LoS with the new position. The main kinematic problem that arises is when the turret is tracking a target, which moves near or past the −90° elevation angle. The sightline axis then becomes aligned with the outer gimbal rotation axis and the system loses a degree of freedom. A 180° azimuth turn is needed in minimum time to maintain accurate tracking (as shown in Fig. 15.39). Nadir nonlinearity: The operation of the line-of-sight system near this condition, inside the nadir cone, introduces two problems. A mechanical singularity arises from the loss of a degree of freedom at the nadir. This results in the inability of the outer gimbal (azimuth) to perform an instantaneous 180° rotation, due to the drive/motor saturation, so that tracking is lost (Fig. 15.39). The second nadir nonlinearity arises from the cross-elevation error which is the measurement driving the azimuth motor. The transformation that refers the cross-elevation back to the azimuth angle (shown in (15.31)) involves the cross-elevation error du multiplied by the cosine of the elevation gimbal angle. Let 2

dgerr

cosðeÞ 0 1 ¼ Grad=pix 4 0 sinðeÞ 0

32 3 2 3 sinðeÞ 0 0 5 0 54 0 5 ¼ 4 0 cosðeÞ du cosðeÞdu

ð15:28Þ

15.6

Sightline Stabilization of Aircraft Electro-optical Servos

747

miss

track 90

90

90

el

B

90

Back

-90

F Front

gimbal reset

-90 az -90

el

az -90

Fig. 15.39 Tracking loss for missile through the nadir (By courtesy of Savvidis et al. [44] and Savvidis [45])

For an increase in the angle e, there is effectively a reduction in the azimuth tracking-loop gain equivalent to cos(e). The cross-elevation and azimuth axes are orthogonal at the nadir, leading to no effective control in the azimuth trackingloop. The NGMV controller accommodates this nadir problem by handling the nonlinearities directly.

15.6.3 Model Equations Assume that initially the target rests 2.5 m away from the gimbal in coordinates xtarget ðx; y; zÞ ! ð2:5; 1; 1Þ and that the base is resting at the waterline (0° base declination). Also, assume that the target is moving along a circular trajectory on a two-dimensional plane perpendicular to the LoS of the gimbal. If the target is moving at an angular velocity of x ¼ 2p  0:1 rad/s, then the individual y and z trajectories are determined by the following equations of motion: ytarget ¼ sinðxtÞ;

ztarget ¼ cosðxtÞ

ð15:29Þ

The simplicity of this configuration allows us to compute the analytical solution to the Inverse Kinematics (IK) problem and to obtain the elevation and azimuth angle requirements to track the target. Since the two degrees of freedom are clearly decoupled, we can conveniently invert the forward kinematics and therefore calculate the angle demands from the following relationships:

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Nonlinear Automotive, Aerospace, Marine and Robotics …

  e ¼ tan1 ytarget =ztarget

ð15:30Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  g ¼ tan1 xtarget = y2target þ z2target

ð15:31Þ

The kinematic responses of the two axes given the previous parameters were simulated and validated for different base declinations. Equations for gimbal axes: The turret system consists of two servo motors, one for each gimbal. These rate loops can be considered decoupled in this particular application, having no interacting reaction torque effects from one to the other. Consequently, the two paths are linear and can be described by Newton’s second law for rotation as JI x ^I ðtÞ ^ I ðtÞ  JI x ^ I ðt ÞÞ ¼ L ^_ I ðtÞ þ ðx

ð15:32Þ

^I ðtÞ ¼ ½TIC ; TIro   ½TIU   ½TIf xðtÞ  and with L   JO x ^ I ðt Þ ¼ L ^O ðtÞ ^ O ðtÞ  JO x ^ O ðt ÞÞ þ L ^_ O ðtÞ þ ðx O

ð15:33Þ

^O ðtÞ ¼ ½TOC ; TOro   ½TOU   ½TOf xðtÞ . In these expressions J is the inertial with L matrix, L is the sum of the kinematic torques around the equivalent gimbal. These include the following: • • • •

Control torques TC Friction and cable restraining torques Tf xðtÞ Reaction torques Tro (acting on the outer gimbal by the inner) Mass imbalance torques TU about each gimbal.

Cosecant correction term: One method to accommodate the nadir tracking involves directly adjusting the cross-elevation error signal with the use of a secant function to cancel out the cosine in the denominator as shown Fig. 15.40. To implement the cosecant correction, the inner gimbal angle is obtained from the

Fig. 15.40 Signal flow for the cosecant correction controller (By courtesy of Savvidis et al. [44] and Savvidis [45])

15.6

Sightline Stabilization of Aircraft Electro-optical Servos

749

message the TigerEye transmits over the CAN bus. In the results there are some slight synchronization errors between CAN bus reporting and the imager frame rate, but not sufficient to disrupt the control action. Note the cosecant correction controller should have a roll-off close to the actual nadir; otherwise, infinite gain would be injected into the azimuth axis tracking loop. Model development: Prior to commissioning the controller in the NI LabVIEW environment, the models of the system were developed and validated and the NGMV controller was then designed. An accurate model of the TigerEye was required, and a frequency-domain identification strategy was used to obtain the transfer-function (TF) for each axis. Each TF relates the rate command to the actual gimbal axis rate (reported by the stabilization loop gyros over the CAN bus). The CAN bus trajectory reporting frequency was 10 Hz and the imager frame rate was 30 Hz. Sinusoids of frequencies ranging from 0.05 to 4 Hz were used to excite each axis and the system frequency response functions were computed for logarithmically spaced input frequencies. This data was used to obtain the gain and phase plots. The characteristics for the tilt and pan rate loops were compared against the models identified, as shown in Fig. 15.41. Validation: The accuracy of the model in the region of the nadir was assessed using a frequency response analysis. The moving target, gimbal kinematics and turret model were implemented in MATLAB. The case where the turret tracks a ball moving along a circular trajectory in 0.1 Hz and under various base tilt angles from 0 (waterline) to 90° (nadir) was then simulated. The physical hardware for the turret system was then tested under these scenarios, by comparing the rate commands and spectral error with those from the simulation (see Figs. 15.42, 15.43, 15.44). The saturation is apparent in the azimuth gimbal motor but the camera captures the target again in the case where the target moves relatively slowly (Fig. 15.44). There is mismatch due to phase differences and potential jitter on the CAN bus (varying sample rate), but the identified model data suggests that the model is sufficiently accurate for the control design stage.

Fig. 15.41 Turret frequency fit: elevation axis (left) and azimuth axis (right)

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Fig. 15.42 Turret-identified model validation for 25° base declination

Fig. 15.43 Turret-identified model validation for 45° base declination

15.6

Sightline Stabilization of Aircraft Electro-optical Servos

751

Fig. 15.44 Turret-identified model validation for 90° base declination

15.6.4 NGMV Control Design and Results The rate loops are linear but the total system includes the inverse cosine nonlinearity, which appears as a singularity on the reference signal. This makes a classical control design difficult. The NGMV controller has the advantage that it accommodates the various nonlinearities naturally in the controller design. To quantify the performance of the NGMV controller, the tracking error for target motion in the zaxis was considered. This tracking error was found for a PI controller, a PI + cosecant correction and for the NGMV controller. The tracking performance for a close nadir pass scenario is shown in Fig. 15.45. This is close to the nadir singularity where the azimuth demand changes quickly. There is a large error prior to the nadir, due to the rate limit on the azimuth gimbal. In all three responses, the underlying PI controller is identical. The differences are due to the additional nonlinearities in the cosecant correction and the NGMV controllers. All three controllers are well damped, with minimal overshoot, and little ringing. The PI controller is the slowest due to the presence of the cosine gain term in the measurement path (the large error causes saturation in the rate loop). The key requirement of a directed infrared countermeasures (DIRCM) tracker is to maximize energy on target. The NGMV controller settles down to acceptable tracking after the first nadir pass at 10 s. The PI + cosecant control settles down after 11 s and the PI control after 12 s. The NGMV controller would, therefore, provide more time-on-target of the DIRCM defeat laser. This would increase the probability that the missile would be confused and the countermeasures would be successful.

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Fig. 15.45 Kinematic and achieved azimuth angles for a 1 cm nadir pass scenario

The tracking performance close to the nadir is important since the controller performance must be maintained throughout the FoR. It was found that the radial tracking error representing the angle between boresight and target vectors, for four nadir pass conditions: 1, 5, 20 and 40 cm was also reasonable. The same trend was observed in each case. The cosecant correction outperformed the PI and the NGMV controller provided the best tracking performance [43]. Comments on the sightline stabilization design: The models of the gimbal axes and the kinematics were validated against experimental results and the performance of the NGMV tracking-loop controller designed was good near the nadir range of operation. The main benefit was the ability to include the system nonlinearities, including the azimuth axis measurement singularity, directly in the control design. The NGMV controller was the best performing controller with minimal additional tuning, and seems to offer a practical design technique for sightline controllers.

15.7

15.7

Design of Flight Controls

753

Design of Flight Controls

The aerodynamic characteristics and operational requirements of high-performance aircraft have led to a greater focus on the severe nonlinearities within the system and the effects on control loops. However, the design of flight control systems has traditionally used linear systems analysis and design tools. Mathematical models of aircraft behaviour can be constructed based upon physical laws of nature, and parameters can be found using aerodynamic data obtained from wind tunnel testing. The control laws must, of course, be tuned to cover the entire flight envelope to obtain the best handling characteristics for the aircraft. A grid of operating points is normally defined to cover the entire operating envelope so that local linear controllers may be computed for particular regions. These then have to be integrated together to cover the envelope of operation, using controller parameter scheduling. For example, an inverse schedule can be applied to compensate for dynamic pressure changes and thereby maintain a constant loop gain. The variables for scheduling depend upon the external pressure and airflow sensor measurements. Unfortunately, a set of linearized models can provide a poor approximation to the true nonlinear model in certain operating regions. Model uncertainties are therefore introduced due to the plant model mismatch in addition to the extraneous disturbances, such as wind gusts, that affect the motion of the aircraft. The aircraft modes to be controlled can be listed as • Modes concerned with rotational degrees of freedom, including short-period, roll, and Dutch-roll modes. • The phugoid mode that involves the translational degrees of freedom. • The spiral mode that depends on the aerodynamic moments. The responsiveness of an aircraft to manoeuvring commands is determined by the speed of the rotational modes. The frequencies of these modes are so high that a pilot would find it difficult to control the aircraft if the modes were lightly damped or unstable. An augmentation system is needed to control these modes, and to provide the pilot with a type of response to the control inputs. The control augmentation system is usually split into two control systems that can handle the longitudinal and lateral dynamics. They are implemented by feedback controllers using accelerometers and rate gyros as the sensors and elevators, ailerons, or rudder, as the control surfaces [56]. For a fighter aircraft, the gain scheduling depends upon the payload being carried or even the needs of reconfiguration in fault conditions [46]. The aim is to optimize the handling and robustness of operation, whilst for military aircraft flying at supersonic speeds the prime requirement is for good mission effectiveness. There is an obvious benefit of LPV models in such cases [47]. Fighter aircraft are often designed to be inherently unstable allowing for a high degree of manoeuvrability but emphasizing the need for robust feedback controls. This suggests the use of Quantitative Feedback Theory [48].

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15.7.1 Model Predictive Flight Control The ability of Model Predictive Control to handle multivariable processes makes it an ideal candidate for aircraft and gas-turbine engine control, as it is capable of optimizing a range of control inputs to manage multiple reference outputs. The availability of good aircraft models derived from the physics of the problem and from first principles also lends itself to MPC applications. However, some shortcomings of MPC have limited its applications in aerospace, particularly the problems of validation when hard constraints are included and the level of computational effort that is required. Youssef et al. [56] considered the design of a control system for a precision tracking task involving a so-called pitch-rate control augmentation system for an F-16 fighter aircraft. A small signal model for states, outputs and controls was used where, respectively: xðtÞ ¼ xðtÞ þ zðtÞ, yðtÞ ¼ yðtÞ þ wðtÞ and uðtÞ ¼  uðtÞ þ vðtÞ. The linearized plant model and a cost-function based on the small signal deviations can then be used to compute a GPC controller to obtain DVt; N1 and thereby the  t;N1 þ Vt;N1 . This provides an approximate vector of future controls Ut;N1 ¼ U NGPC control strategy. The receding horizon control algorithm optimized a tracking quadratic cost-function for the nonlinear continuous-time control system with input constraints. The optimisation procedure involved a dynamic linearization around a predicted future control trajectory. The proposed control algorithm [56] may be summarized as follows: • Measure or estimate the current state vector xðtÞ of the plant.  t1; N and • Take the previously computed vector of predicted future controls U   compute the predicted states Xt þ 1; N and outputs Yt þ 1; N and the next predicted t; N . state vector X t;N1 calculate the future plant system and input matrices for  • Using Xt; N ; X t ¼ 0; . . .; N  1, linearized around the predicted trajectory. • From the GPC solution, calculate the incremental control DVt; N1 and the  t; N1 þ Vt; N1 . Take the first element uðtÞ of resulting Vt; N1 and Ut; N1 ¼ U Ut; N1 for the control to be applied by again invoking the receding horizon principal. The flight controller was found to be reasonably robust and the desired performance objectives were achieved over the flight envelope. This early work on predictive flight control illustrated the great potential of the predictive algorithms although more recent algorithms probably provide more developed solutions. Nevertheless, the approach raises some questions that are still relevant. For example, is linearization about a trajectory a reliable approach? It clearly introduces modelling errors but in some applications, good results are still obtained even if there is a lack of comforting stability results. Another question explored was the robustness issue which is obviously important for flight controls.

15.7

Design of Flight Controls

755

It is a sobering thought that for this last application discussed in the text, attention is drawn to the rather practical problem of ensuring the control is truly robust. There are many ways to influence the different types of robustness of nonlinear controls which have been discussed. However, there is not one method or tuning procedure that will guarantee robustness of this type of NGPC controller. A second problem that particularly applies to flight controls is the certification process, which involves a rigorous verification and validation process. Such tests can account for more than half the flight control system development costs. This is likely to be problematic for a linearization about a trajectory type of algorithm. The qLPV model-based predictive controls discussed in Chap. 11 may have some advantages in this respect.

15.8

Concluding Remarks

The simple NGMV nonlinear controller was used in the rudder roll-stabilization problem, the diesel engine control and the sightline stabilization system control. It is the simplest nonlinear optimal control method to apply, and the black-box plant input subsystem model provides a unique capability. When companies require a comparison of techniques to be provided, it is often the first on the list of possible options. Predictive control, where an extended prediction horizon is used, has the advantage that the control action can begin well before the changes in the reference signal occur. There is, of course, the assumption that future setpoint knowledge is available. This is often the case in trajectory tracking applications. The use of predictive control is natural for some robotic applications like the two-link robot arm where the future reference trajectories may be defined precisely. For this type of problem, multi-pass process control solutions [49] may provide some learning capabilities. It was demonstrated that predictive control may be valuable for ship manoeuvring systems, including dynamic ship positioning and tracking problems. Because of the changing sea-state conditions, marine systems must be robust to stochastic disturbances. There is, therefore, a possible role for the use of H∞ control design [50–52]. In ship positioning systems, the nonlinear thruster characteristics can be included in the black-box plant model used in NGMV or NPGMV controllers. The nonlinear control weighting that can be used in these controllers provides one method to compensate for nonlinearities. The use of the dynamic weighting functions to penalize specific frequency ranges is also valuable. The control law tuning procedures were facilitated by the use of existing PID control solutions. These methods do of course have some peculiarities; namely, the nonlinear part of the black-box input model is assumed stable, and the predictions of disturbances are based on the output or qLPV subsystem. It was found that the NPGMV controller has some advantages relative to the basic NGMV control design; exploiting the predictive control law properties, at the expense of some additional complexity.

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The control of the diesel engine revealed that both the NGMV and the predictive control methods provided possible solutions to this difficult control problem. It was observed that similar techniques could be applied to the control of Spark Ignition (SI) engines [39–41]. However, from the control law design viewpoint, the diesel engine seems to include more challenges than SI engines [53–55]. However, in both cases the future seems to lie with model-based control design approaches, which the automotive industry and other industries are exploring enthusiastically. A range of nonlinear control solutions has been presented with different merits but the best solution is usually the simplest that will do the job. The message here is that it is best to choose a spanner that fits the nut rather than the newest and brightest.

15.9

The Final Word

As mentioned at the start of this text, many of the problems in advanced control have been solved or partially solved. The general area of control and systems engineering is of course changing rapidly as new application problems emerge, but practical solutions are often available, even if they are not the “best” or “optimal” in some sense. However, the behaviour of control systems containing significant nonlinearities is very unpredictable and made more so by the large range of possible nonlinear effects. There is therefore a need for simple and effective control laws to tackle the real difficulties that nonlinearities cause. This text has hopefully provided the motivation to focus on practical design procedures for nonlinear systems. Although not all of the techniques described are as simple as we would wish, the basic theoretical concepts required are straightforward. We therefore hope there are some practitioners in the industry that will benefit from our many years of enjoyable and interesting study. Mike J. Grimble and Pawel Majecki

References 1. Hrovat D, Di Cairano S, Tseng HE, Kolmanovsky IV (2012) The development of model predictive control in automotive industry: a survey. In: IEEE international conference on control applications, Dubrovnik, Croatia 2. Deori L, Garatti S, Prandini M (2015) A model predictive control approach to aircraft motion control. In: American control conference, Palmer House Hilton, Chicago 3. Slotine JJE, Li W (1991) Applied nonlinear control. Prentice-Hall, Englewood Cliff, New Jersey 4. Asada H, Slotine JJE (1986) Robot analysis and control. Wiley, New York 5. Grimble MJ, Ordys AW (2001) Non-linear predictive control for manufacturing and robotic applications. In: IEEE conference on methods and models in automation and robotics, Poland 6. Hickey NA, Grimble MJ, Johnson MA, Katebi MR, Wood D (1995) Fin roll control system design. In: IFAC conference, control applications in marine systems, Trondheim, Norway

References

757

7. Hickey NA, Johnson MA, Katebi MR, Grimble M J (1999) PID controller optimization for fin roll stabilization. In: IEEE International Conference on Control Applications, Kohala Coast Island of Hawaii, pp 1785–1790 8. Katebi MR, Hickey NA, Grimble MJ (2000) Evaluation of fin roll stabiliser controller design. IFAC Manoeuvring and Control of Marine Craft, Aalborg, Denmark 9. Hickey NA, Grimble MJ, Johnson MA, Katebi MR, Melville R (1997) Robust fin roll stabilisation of surface ships. In: 36th IEEE conference on decision and control, San Diego, California, pp 4225–4230 10. Grimble MJ, Katebi MR, Zhang Y (1993) H∞ based ship fin rudder-roll stabilisation design. In: 10th Ship control symposium, Ottawa, Canada 11. Fossen TI (1994) guidance and control of ocean vehicles. Wiley, Chichester 12. Morgan MJ (1978) Dynamic positioning of offshore vessels. Petroleum Publishing Company 13. Balchen JG, Jenssen NA, Saelid S (1976) Dynamic positioning using Kalman filtering and optimal control theory. Automation in offshore oil field operation, pp 183–188 14. Balchen JG, Jenssen NA, Mathisen E, Sælid S (1980) A dynamic positioning system based on Kalman filtering and optimal control. Model Ident Control 1:135–163 15. Sælid S, Jenssen NA, Balchen JG (1983) Design and analysis of a dynamic positioning system based on Kalman filtering and optimal control. IEEE Trans Autom Control 28:331– 339 16. Grimble MJ, Patton RJ, Wise DA (1979) The design of dynamic ship positioning control systems using extended Kalman filtering technique. IEEE Oceans 79 conference, San Diego, California, pp 488–497 17. Grimble MJ, Patton RJ, Wise DA (1980) Use of Kalman filtering techniques in dynamic ship positioning systems. IEEE Proc 127(3):93–102 18. Grimble MJ, Patton RJ (1980) The design of dynamic ship positioning control systems using stochastic optimal control theory. Optimal Control Appl Methods 1:167–202 19. Fung PT, Grimble MJ (1983) Dynamic ship positioning using a self-tuning Kalman filter. IEEE Trans Autom Control 28:339–350 20. Pierson WJ, Moskowitz L (1963) A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodsku. US Naval Oceanographic Office 21. Grimble MJ, Johnson MA (1988) Optimal control and stochastic estimation, vols I and II. Wiley, Chichester 22. Yamamoto I, Daigo K, Matsuura M, Yamaguchi Y, Tanabe A, Shimizaki K (1998) A dynamic position system for FPSOs with fuzzy control logic based on a nonlinear programming algorithm. In: 17th ASME international conference on offshore mechanics and arctic engineering 23. Johannesen E, Egeland O (1993) Dynamic positioning with wave filtering using a passive controller structure. In: IEEE 32nd conference on decision and control, San Antonio, Texas, pp 1903–1908 24. Fossen T, Grovlen A (1998) Nonlinear output feedback control of dynamically positioned ships using vectorial observer backstepping. IEEE Trans Control Syst Technol 6(1):121–128 25. Vermillion C, Butts K, Reidy K (2010) Model predictive engine torque control with real-time driver-in-the-loop simulation results. In: American control conference, Baltimore, pp 1459– 1464 26. Stefanopoulou A, Kolmanovsky I, Freudenberg J (2000) Control of variable geometry turbocharged diesel engines for reduced emissions. IEEE Trans Control Syst Technol:733– 745 27. Jankovic M, Jankovic M, Kolmanovsky I (1998) Robust nonlinear controller for turbocharged diesel engines. In: American control conference, Philadelphia, pp 1389–1394 28. Kolmanovsky V, Moraal PE, van Nieuwstadt MJ, Stefanopoulou AG (1997) Issues in modelling and control of intake flow in variable geometry turbocharged engines. In: 18th IFIP conference on system modelling and optimization, Detroit

758

15

Nonlinear Automotive, Aerospace, Marine and Robotics …

29. Herceg M, Raff T, Allgöwer F (2006) Nonlinear model predictive control of a turbocharged diesel engine. In: IEEE international conference on control applications, Munich, Germany, pp 2766–2771 30. Jung M, Glover K (2003) Control-oriented linear parameter-varying modelling of a turbocharged diesel engine. IEEE conference on control applications, Istanbul, Turkey, pp 155–160 31. Jung M, Glover K (2003) Calibratable linear parameter-varying modelling of a turbocharged diesel engine. In: IEEE international conference on control applications, Istanbul, pp 155–160 32. Jung M, Glover K (2006) Calibratable linear parameter-varying modelling of a turbocharged diesel engine. IEEE Trans Control Syst Technol 14(1):45–62 33. Mohammadpour J, Grigoriadis K, Franchek M, Wang Y, Haskara I (2010) LPV decoupling and input shaping for control of diesel engines. In: American control conference, Baltimore, pp 1477–1482 34. Van Nieuwstadt MJ, Kolmanovsky IV, Moraal PE, Stefanopoulou A, Jankovicegr M (2000) VGT control schemes: experimental comparison for a high-speed diesel engine. IEEE Trans Control Syst 20:63–67 35. Vermillion C, Sun J, Butts K (2007) Model predictive control allocation for over-actuated systems: stability and performance. In: IEEE conference on decision and control, New Orleans, pp 1251–1256 36. Jankovic M, Jankovic M, Kolmanovsky I (2000) Constructive lyapunov control design for turbocharged diesel engines. IEEE Trans Control Syst Technol 8(2):288–299 37. Wahlstrom J, Eriksson L, Nielsen L (2010) EGR-VGT control and tuning for pumping work minimization and emission control. IEEE Trans Control Syst Technol 18(4):993–1003 38. Banning R, Johnson MA, Grimble MJ (1997) Advanced control design for marine diesel engine propulsion systems. J Dyn Syst Meas Contr 119(2):167–174 39. Majecki P, Javaherian H, Grimble MJ (2009) Supervisory multiple-model approach to multivariable lambda and torque control of SI engines. In: IFAC workshop on engine and powertrain control, simulation and modeling IFP, Rueil-Malmaison, France, vol 42, No. 26, pp 24–31 40. Majecki P, van der Molen GM, Grimble MJ, Haskara I, Hu Y, Chang CF (2015) Real-time predictive control for si engines using linear parameter-varying models. In: 5th IFAC conference on nonlinear model predictive control, Seville, and IFAC-papers on-line, vol 48, No. 23, pp 94–101 41. Majecki P, Grimble MJ, Haskara I, Hu Y, Chang CF (2017) Total engine optimization and control for SI engines using linear parameter-varying models. American Control Conference, Seattle 42. Martin P, Brignall N, MacDonald M, Grimble MJ (2008) Constrained predictive control of a servo-driven tracking turret. UKACC international conference on control, University of Manchester 43. Anderson D, McGookin M, Brignall N (2009) Fast model predictive control of the nadir singularity in electro-optic systems. J Guid Control Dyn 32(2):626–632 44. Savvidis P, Anderson D, Grimble MJ (2010) Application of nonlinear generalised minimum variance to the nadir problem in 2-axis gimbal pointing and stabilization. In: SPIE conference on defence, security, and sensing, Orlando, Florida 45. Savvidis P (2017) Nonlinear control: an LPV nonlinear predictive generalised minimum variance perspective, Ph.D. Thesis, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow 46. Huzmezan M, Maciejowski J (1998) Reconfiguration and scheduling in flight using quasi-LPV high-fidelity models and MBPC control. In: American control conference, Philadelphia, pp 3649–3653 47. Marcos, Balas GJ (2004) Development of linear-parameter varying models for aircraft. J Guid Control Dyn 27(2):218–228 48. Wu S-F, Grimble MJ, Breslin SG (1998) Introduction to quantitative feedback theory for lateral robust flight control systems design. Control Eng Pract 6(7):805–828

References

759

49. Owens DH (2016) Iterative learning control, an optimization paradigm. AIC Series Monograph, Springer 50. Grimble MJ, Zhang Y, Katebi MR (1993) H∞ based ship autopilot design. In: 10th ship symposium, Canada 51. Desanj DS, Grimble MJ, Katebi MR (1997) State-space adaptive H∞ controller with application to a ship control system. Institute of Meas. and Control, vol 19, No. 3 52. Dohna DC, Desanj DS, Katebi MR, Grimble MJ (1998) H∞ adaptive controllers for autopilot applications. Int J Adaptive Control Signal Process 132:623–648 53. Brahma A, Upadhyay D, Serrani A, Rizzoni G (2004) Modelling, identification and state estimation of diesel engine torque and NOx dynamics in response to fuel quantity and timing excitations. In: American control conference, Boston, pp 2166–2171 54. Ponti F, Ravaglioli V, Moro D, Serra G (2013) MFB50 on-board estimation methodology for combustion control. Control Eng Pract 1821–1829 55. Maruyama T, Shimura T, Ejiri A, Ikai Y, Shimotani K (2011) Model predictive control applied to a diesel engine air-path system with dead time. In: SICE annual conference, Waseda University, Tokyo, Japan 56. Youssef A, Grimble MJ, Ordys A, Dutka A, Anderson D (2003) Robust nonlinear flight predictive control. In: European control conference, Cambridge, pp 2487–2492

Index

A Absolute stability, 17 Aerospace, 24, 426, 700, 754 Artificial intelligence, 238, 239 Asymptotic stability, 14, 15, 20, 30, 313, 327 Automotive engine, 18, 40, 43, 52, 216, 239, 377, 410, 425, 462, 699, 731 Autonomous vehicles, 239, 420, 730 Auto-tuning, 50, 56, 58 B Backlash, 8, 10 Backlash in gear trains, 10 Backstepping, 24, 201, 725 Benchmarking, 108, 109, 113, 147, 174, 189, 191, 420, 566, 584, 623, 624, 645 BIBO stable, 15–17 Bifurcation, 13 Black-box model, 131, 136, 141, 160, 238, 366, 455, 463, 511, 555, 558, 568, 573 C Circle criterion, 17 Constrained optimization, 296, 332–334, 393, 700 Constraints, 46, 120, 217, 220, 297, 330–334, 544, 739 Cost-function, 49, 80, 90, 100, 111, 119, 138, 253, 272, 277, 314, 358, 389, 396, 403, 419, 489, 500, 527, 562, 617 Cost-function weightings, 82, 85, 91, 147, 186, 206, 208, 210, 213, 216, 274, 282, 315, 330, 357, 409, 419, 710, 738 Coulomb friction, 11, 12 Criterion, 65, 81, 100, 138, 314, 398, 527, 562

Cross product cost terms, 66, 85, 90, 95, 96, 122, 146 D Dead-time, 667, 668 Dead-zone, 57, 231, 367, 369 Describing function methods, 18 Design issues, 24, 36, 122, 244, 506, 586, 717, 731 Deterministic disturbances, 412, 414 Diesel engines, 296, 410, 699, 730–736, 738, 739, 741, 755, 756 Diophantine equation, 69, 79, 93, 96, 105, 139, 143, 188, 229, 254, 271, 276, 303, 564, 583 Distributed Control System (DCS), 293, 294 Disturbance rejection, 88, 129, 143, 185, 371, 441, 663, 666, 673 Dynamic inversion, 24 Dynamic weightings, 80, 90, 138, 150, 204, 205, 356, 685, 738 E Edmunds algorithm, 111 Engine control, 40, 43, 216, 410, 411, 413, 415, 425, 463, 569, 625, 731, 736, 754 Equilibrium points, 13, 14, 26, 28, 29, 32–34, 40, 682–684, 691 Equivalence between LQG and MV, 100 Escape time, 13 Exogenous input, 78, 147, 172, 261 Explicit predictive control, 529 Extended Kalman filter, 25, 51, 52, 352, 353, 366, 453, 511, 593, 597, 600–603, 631

© Springer-Verlag London Ltd., part of Springer Nature 2020 M. J. Grimble and P. Majecki, Nonlinear Industrial Control Systems, https://doi.org/10.1007/978-1-4471-7457-8

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762 F Feedback linearization, 22–24, 161, 348, 700 Feedforward control, 129–134, 138, 150, 181–183, 185, 190, 650 Finite escape time, 13 Finite-gain stability, 506 Friction nonlinearities, 8, 199 Fuzzy set theory, 239 G Gain scheduling, 19, 24, 28–30, 34, 40, 115, 470, 650, 677, 686, 753 Generalized linear quadratic gaussian, 65, 67, 82, 90, 91, 100 Generalized linear quadratic gaussian control, 65 Generalized minimum variance, 65–67, 76, 81, 82, 87, 108, 159, 347, 426, 455 Generalized minimum variance control, 65, 75, 76, 82, 87, 108, 138, 159, 202, 248, 252, 270, 271, 288, 347, 425, 426, 455 Generalized plant, 80, 97, 171, 173, 174, 191, 230, 237, 271, 328, 357, 364, 374, 457 Generalized Predictive Control (GPC), 77, 120, 170, 293, 308–317, 377, 388–397, 488 Generalized spectral factor, 104, 164 Gyro stabilized platform, 712, 749 H Hard constraints, 196, 219, 291, 293, 308, 331, 332, 336, 488, 505, 544, 547, 739, 754 H∞ control, 241, 242, 248, 268, 272, 278, 288, 743 H∞ control design, 65, 100, 101, 210, 240, 245, 272, 721, 755 Hierarchical control, 42 High-gain control, 23 Hybrid systems, 3, 27, 42–48, 121 Hysteresis, 7, 8, 10, 11 I Implicit predictive control, 291, 297 Implied equation, 86, 95, 105, 256, 258, 259, 261 Incremental form, 525 Inferred output, 67, 78, 138, 141, 146, 187, 252, 447, 448 Initialisation, 39 Integral-action, 177, 204, 656, 681 Integral wind-up, 196, 283, 402 Internal model control, 18, 202, 231, 234, 359, 365 Inverse simulation, 458 I-O pairing, 102, 114

Index J Jacobian, 31, 32, 34, 37–41, 115, 600, 602, 603, 691, 736 K Kalman filtering, 362, 445, 453, 505, 553, 555, 562, 567, 579, 585, 593, 598, 609, 613–615, 724, 725 Kronecker product, 111 L Lambda control, 412, 463 Limit cycles, 13 Linearization, 26, 28, 32, 34, 38, 470, 602, 691 Linear Matrix Inequalities, 30, 121, 122, 241, 242 Linear Parameter Varying systems, 3, 466 Linear predictive control, 330, 377, 509, 510 Linear Quadratic Gaussian control, 76, 353 Linear systems, 6, 7, 22, 26–28, 30–41, 69, 115, 311, 430 Liquid level control, 175 Looper control hot mills, 661, 662 LPV control, 29–31 LPV modelling, 30, 41, 58, 661, 691, 699, 736 LQG control, 82, 85, 87, 88, 98, 101, 123, 248, 255, 267, 288, 353, 419 Lyapunov design, 22 Lyapunov methods, 16, 743 M Marcinkiewicz space, 162, 250, 435, 609 Measurable disturbance, 132, 134, 136, 137, 142, 146, 151, 152, 183–186, 192, 193 Minimum variance control, 67–69, 71–73, 87, 98, 101, 122, 123, 164, 171, 317, 553 Model Predictive Control (MPC), 48, 50, 68, 77, 194, 239, 291–299, 308, 309, 313, 326, 330–332, 341, 377, 388, 405, 411, 419–422, 431, 470, 509 Modified Jacobian linearization, 34 Multi-parametric quadratic programming, 297, 298 Multiple model control, 29 Multiple models, 102, 114–117, 119, 121, 123 Multivalued nonlinearities, 9, 11 Multivariable control, 35, 53, 89, 90, 135, 291, 338, 347, 366, 462, 645, 660, 661, 663, 667, 687, 701, 709, 738 N Neural networks, 131, 141, 156, 176, 238, 239, 467, 555, 568, 587, 593, 631

Index Nonlinear control, 3, 4, 7, 9, 15, 18, 22, 26, 28, 31, 35, 52, 58, 65, 75, 97, 101, 115, 122, 129, 132, 139, 148, 160–162, 175, 196, 200, 206, 214, 218, 222, 241, 244, 268, 464, 499, 503 Nonlinear filtering, 25, 553–555, 557, 574, 578, 593, 597, 606, 607, 638 Nonlinear generalized minimum variance control, 27, 138, 164, 199, 202, 248, 252, 270, 271, 348, 425, 426, 460, 606, 675 Nonlinear Generalized Predictive Control (NGPC), 58, 341, 469–471, 475, 488, 490, 493–496, 498, 499, 501, 503 Nonlinear identification methods, 52 Nonlinear Model Predictive Control (NMPC), 294–296, 299 Nonlinear predictive control, 58, 102, 296, 309, 317, 334, 340, 341, 421, 470, 493, 506, 509, 691, 701 Nonlinear predictive generalized minimum variance, 314, 341, 378, 397, 475, 486, 488, 495, 499, 500, 503–505, 513, 514, 516, 519–525, 545–547, 738 Non-minimum phase, 24, 72, 73, 75, 77, 80, 81, 87–90, 122, 130, 159, 160, 199, 203, 266, 275, 284, 287, 314, 335, 336 Non-smooth nonlinearities, 8 O Open-loop, 129–133, 138–141, 664, 671, 679, 702, 714 Optimal control, 6, 48–50, 65, 70, 76, 78, 82, 85, 87, 90, 91, 95, 138, 147, 164, 186, 202, 253, 268, 272, 295, 309, 312, 355, 388, 398, 419, 447, 488–490, 499, 541 Optimisation, 47, 167, 183, 242, 333, 334, 393, 555, 754 Oscillations, 10, 11, 13, 14, 18, 19, 56, 57, 658, 666 P Parametric optimisation, 102, 105, 111 Parametric uncertainty, 236 Phase-plane, 20 PID control, 4, 18, 58, 180, 201, 207–210, 212–216, 369, 409, 458, 529, 673–677, 751 Piecewise-affine systems, 26, 27 Pointing systems, 742 Polynomial system description, 160, 593 Popov stability, 16 Predictive Functional Control, 294 Preview control, 416

763 Process control, 175, 645, 646, 670, 677, 687, 691, 692, 695, 755 Q Quadratic Program (QP), 295, 297, 332–334, 488, 544, 547, 739 Quantitative Feedback Theory (QFT), 239–241, 546, 753 Quasi-LPV systems, 35, 425, 452 R Reference governor, 333, 334 Reference signal, 74, 162, 349, 416–418, 432–434, 459 Relative gain array, 114 Relay auto-tuning, 56, 57 Relay characteristics, 8, 57 Restricted structure control, 101, 243, 288, 469, 526, 527, 529, 540, 542, 546, 594 Restricted structure predictive control, 541, 546 Robotics, 22, 293, 420, 516, 554, 699, 701, 705, 755 Robot manipulator, 699, 701, 703 Robustness, 101, 236, 239, 241, 247, 267–270, 274, 278, 279, 287, 509–511, 527, 546, 586, 753 S Saturating inputs, 120 Saturation, 8–11, 14, 26, 27, 81, 119, 120, 123, 130, 184, 199, 217–221, 244, 283, 331, 333, 334, 361, 505, 506, 670–672, 709, 719 Saturation functions, 8, 9, 26, 27, 220 Schur complement, 242 Sector-bounded nonlinearity, 23 Sensitivity functions, 85, 101, 204, 237, 248, 268, 278, 282, 283, 286, 313, 330, 354, 407, 584, 715–717, 721, 728 Servo-system, 7, 11, 58, 131, 160, 200, 295, 699, 700, 714 Ship control, 713 Ship positioning, 25, 132, 133, 186, 200, 699, 721–723, 725, 726, 728, 755 Ship roll-stabilization, 334, 699, 713, 721 Singular perturbations, 494 Sliding mode control, 19–21, 743 Small gain theorem, 16, 237 Smith predictor, 18, 202, 207, 231–235, 237, 238, 244, 329, 341, 347, 364–366, 374, 407, 410, 426, 460–462, 513 Smooth nonlinearities, 8 Soft constraints, 195, 291, 331, 333, 334, 544, 547, 700, 719, 739, 742

764 Soft sensing, 138, 139 Spark-ignition engines, 411, 412, 462, 463 Spectral-factorization, 78, 105, 178, 254, 257, 264, 554, 561, 583, 613–615 Stability, 13–17, 72, 86, 88, 95, 150, 191, 192, 206, 237, 250, 261, 262, 287, 326–329, 407, 428, 456, 462, 506, 508 State-dependent Riccati equation method, 426 State estimation, 25, 267, 351, 352, 377, 388, 453, 487, 489, 500, 505, 738 Static nonlinearities, 8, 9, 160, 184, 282, 366, 625, 729 Steel industry, 165, 231 Stiction, 10, 231 Stochastic disturbances, 66, 69, 143, 153, 162, 193, 203, 235, 247, 248, 278, 301, 302, 322, 348, 359, 379, 380, 430, 432 Successive approximation algorithm, 107, 108, 111, 115, 117 Supervisory control, 292, 648, 661 Supervisory Control and Data Acquisition (SCADA), 174, 293, 294 Switching systems, 20, 46, 121 System identification, 3, 25, 50, 52–55, 115, 176, 268, 295, 299, 425, 682, 736, 745 T Tank-level control, 172 Tanks, 175 Tensor product, 111 Time-delay compensation, 214, 235 Torque control, 462

Index Tracking, 179–183, 186, 384, 388, 432, 726, 728, 741–749 Transport delays, 132, 135, 171, 202, 211, 231, 244, 249, 263, 269, 283, 308, 348, 350, 361–365, 367, 381, 422, 431, 432, 462 Two and a half degrees of freedom, 182 U Uncertainty, 114, 116, 236–238, 240, 241, 267, 269, 270, 510, 559, 586, 606, 624, 628, 636 Unmeasurable disturbance, 130, 132–135, 137, 151, 181, 183, 185, 193 Unmodelled dynamics, 21, 23, 236, 267, 732 Unstructured uncertainty, 236 V Variable structure systems, 19, 20, 47 Velocity based LPV, 37–40 Velocity-dependent LPV models, 38 W Weighting selection, 117, 150, 152, 172, 203, 204, 208, 210, 212, 244, 278, 366, 369, 374, 407, 457, 466, 717, 727, 738 Wind farm control, 647 Wind turbine control, 420, 646, 655, 658 Z Ziegler–Nichols PID tuning, 4, 57