New Trends in Observer-based Control: A Practical Guide to Process and Engineering Applications (Emerging Methodologies and Applications in Modelling, Identification and Control) [2] 0128170344, 9780128170342

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New Trends in Observer-Based Control: A Practical Guide to Process and Engineering Applications

NEW TRENDS IN OBSERVER-BASED CONTROL A Practical Guide to Process and Engineering Applications VOLUME 2 EDITED BY

Olfa Boubaker Quanmin Zhu Magdi S. Mahmoud José Ragot Hamid Reza Karimi Jorge Dávila

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-817034-2 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Mara Conner Acquisition Editor: Sonnini R. Yura Developmental Editor: Gabriela D. Capille Production Project Manager: Anitha Sivaraj Cover Designer: Miles Hitchen Typeset by SPi Global, India

Contributors Mohammed Abouheaf School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada Abdel Aitouche Automation, HEI Lille and CRIStAL Laboratory, Lille, France Ibrahim Alaridh CRIStAL Laboratory, University of Lille, Villeneuve-d’Ascq, France Marouane Alma University of Lorraine, Nancy, France Nezar Alyazidi Systems Engineering, College of Computer Sciences and Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Finn Ankersen European Space Agency, DTEC, Noordwijk, The Netherlands Olfa Boubaker University of Carthage, National Institute of Applied Sciences and Technology, Tunis, Tunisia Naceur Benhadj Braiek Advanced Systems Laboratory, Polytechnic School of Tunisia, La Marsa, Tunisia Mohamed Darouach University of Lorraine, Nancy, France Salwa Elloumi Advanced Systems Laboratory, Polytechnic School of Tunisia, La Marsa, Tunisia Afef Fekih Department of Electrical and Computer Engineering, University of Louisiana at Lafayette, Lafayette, LA, United States Eckhard Gauterin Department of Engineering I, Control Engineering, HTW Berlin, Berlin, Germany David Henry IMS Laboratory, University of Bordeaux, Talence, France Kshetrimayum Lochan Department of Electrical Engineering, National Institute of Technology Silchar, Silchar; Department of Mechatronics, Manipal Institute of Technology, Manipal, India Magdi S. Mahmoud Systems Engineering, College of Computer Sciences and Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Saleh Mobayen Advanced Control Systems Laboratory, Department of Electrical Engineering, University of Zanjan, Zanjan, Iran Florian Pöschke Department of Engineering I, Control Engineering, HTW Berlin, Berlin, Germany Binoy Krishna Roy Department of Electrical Engineering, National Institute of Technology Silchar, Silchar, India Horst Schulte Department of Engineering I, Control Engineering, HTW Berlin, Berlin, Germany

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CONTRIBUTORS

Adel Sharaf SHARAF Energy Systems, Inc., Fredericton, NB, Canada Jay Prakash Singh Department of Electrical Engineering, National Institute of Technology Silchar, Silchar, India Ahmet Sofyali Independent Researcher, Istanbul, Turkey Luigi Strippoli GMV Aerospace and Defence S.A.U., Madrid, Spain Bidyadhar Subudhi Department of Electrical Engineering, National Institute of Technology Rourkela, Rourkela, India Holger Voos Faculty of Science, Technology and Communication, University of Luxembourg, Luxembourg City, Luxembourg Ali Zemouche University of Lorraine, Nancy, France Quan Min Zhu Department of Engineering Design and Mathematics, University of the West of England, Bristol, United Kingdom Khadidja Chaib Draa Faculty of Science, Technology and Communication, University of Luxembourg, Luxembourg City, Luxembourg

Preface of Vol. 2, New Trends in Observer-Based Control A Practical Guide to Process and Engineering Applications In recent decades, it has been widely recognized that one of the challenging problems in modern control theory deals with observer design for dynamical systems where tremendous research activities have been developed covering different aspects. This edited book, New Trends in Observer-Based Control A Practical Guide to Process and Engineering Applications, aims to present a full picture of the state-of-the-art research and development of analysis techniques and design methodologies pertaining to observer-based control and its wide applications. The ultimate purpose is to provide a coherent dose of valuable information to researchers, professional engineers, graduate students, and interested readers. Particular attention is given to properly orient the various chapters so that the integrated book stands as a key reference in academia and international libraries. The book is triggered by ubiquitous applications of observer-based control (OBC for short), and the recurrent development of efficient algorithms for various types of dynamical systems. OBC plays an important role in modern process control infrastructures. The book contains two main volumes: 1. Vol. 1 covers Chapters 1 through 13, subdivided into three parts (Parts I–III). 2. Vol. 2 includes Chapters 1 through 10, subdivided into two parts (Parts I and II). This is the preface of Vol. 2, in which Part I deals with topics on Power, Renewable Energy, and Industrial Systems and Part II deals with Robotics, Flight Systems, and Vehicle Dynamics. We start with Part I where in Chapter 1, titled “Observer-Based Controller of Analytical Complex Systems: Application for a Large-Scale Power System,” the nonlinear control with a state observer of complex systems modeled in polynomial form is addressed. The basic idea of the chapter is to design a nonlinear control law with a state observer, guaranteeing the

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overall asymptotic stability of the augmented system formed by the nonlinear process, the control, and the observer. The design of the observer-based control is formulated using the Kronecker product and the power of matrices properties for the state-space description of polynomial systems. The stability of the observer-based polynomial system augmented is analyzed using the direct method of Lyapunov. Further extension was achieved for the case of several interconnected subsystems using a decentralized control law accompanied by a decentralized state observer. Simulation studies on three machine power systems demonstrate the potential performances of the design approach. Then in Chapter 2, titled “LMI Region-Based Nonlinear Disturbance Observer With Application to Robust Wind Turbine Control,” the authors investigate the Takagi-Sugeno (T-S) modeling approach for nonlinear control and observer design. A linear matrix inequality (LMI) constraint for T-S observers with unmeasurable premise variables based on the inputto-state property is established. By combining the presented LMIs for the application to a wind turbine, two different nonlinear control schemes based on the convex system description are derived, and the implications from introducing the observer for the wind turbine application are discussed. Next, Chapter 3, titled “Stochastic Control Approach for Distributed Generation Units Interacting on Graphs,” provides a novel online adaptive learning distributed control approach for a system of distributed generation units with disturbances in their dynamical environments. The interactions between the generation units are restricted by a graph topology to reflect intercoupling of the dynamics of the generation units. Distributed protocols are utilized to maintain synchronization among the generation units. In this case, the cost function is designed to take into account the neighborhood interactions and the graph topology. A distributed online reinforcement learning approach that employs a Kalman filter is employed to solve the optimal control problem of the multiagent system and is implemented in real time using a means of neural network approximations. The validity of the distributed control approach is tested using a system of distributed generation units working under disturbances. Chapter 4, titled “Control of Anaerobic Digestion Process,” deals with the LMI design of observer-based control strategies for the anaerobic digestion (AD) process. The AD process is represented by a mass balance nonlinear model where the key variables are not accessible for measurement. Therefore, the control scheme is composed by a nonlinear state observer and a feedback control. The objective is to track state trajectories reflecting desired biogas production. Stability analysis is performed by using the Lipschitz conditions and the Lyapunov function, leading to LMIbased stability conditions. Now moving to Part II, we begin with Chapter 5, titled “Finite-Time Disturbance Observer-Based Tracking Control Design for Nonholonomic

PREFACE OF VOL. 2

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Systems,” in which the authors examine a disturbance observer-based terminal sliding mode control (TSMC) approach for a class of nonholonomic systems subject to unknown perturbations. In this way, the approach integrates the robust stability properties of the sliding mode control with the compensation capacity of nonlinear disturbance observers (DOBs) and guarantees convergence behavior in finite time. The design performance is assessed using a wheeled mobile robot. In Chapter 6, titled “Design of a Composite Control in Two-Time Scale Using Nonlinear Disturbance Observer-Based SMC and Backstepping Control of a Two-Link Flexible Manipulator,” the authors investigate the problem of trajectory tracking control in the presence of unmatched disturbances and quick tip deflection suppression for a two-link flexible manipulator is considered. The solution approach is based on designing a composite (slow-fast) control technique in which the slow subsystem consists of the rigid body dynamics of the flexible manipulator and the fast subsystem consists of the flexible body dynamics of the flexible manipulator. A nonlinear disturbance observer-based SMC is designed on the slow subsystem under the presence of unmatched disturbances for the trajectory tracking control. A backstepping controller in used on the fast subsystem for the quick suppression of the links deflection. A composite controller is finally designed for the two-link flexible manipulator using the nonlinear disturbance observer-based SMC on the slow subsystem and the backstepping controller on the fast subsystem. Simulation results validate the effectiveness of the proposed composite. Next, in Chapter 7, titled “Design of Observer-Based Tracking Controller for Robotic Manipulators,” an observer-based tracking control design for an n-link robotic manipulator subject to external disturbances is described. Design of the tracking controller is based on the global sliding mode approach, which eliminates the need for the reaching phase by introducing an extra term in the manifold. The disturbance observer is designed to estimate system disturbances without requiring any knowledge about their upper bounds. The stability analysis of the proposed observer was carried out using Lyapunov stability theory. The performance of the proposed approach was assessed using a threedegrees-of-freedom rigid manipulator. Good tracking performance, robustness to disturbances, and eliminating the need for the reaching phase are among the positive features of the proposed approach. Chapter 8, titled “Disturbance Observer-Based Control of Spacecraft Attitude Dynamics Subject to Perturbations and Underactuation,” deals with the stabilization of nonlinear spacecraft attitude dynamics by using solely magnetic actuation. The considered control problem possesses instantaneous underactuation due to the magnetic torque production mechanism. The objective is to design an observer-based control system by integrating an already existing magnetic sliding mode attitude controller with a nonlinear disturbance observer where all major environmental

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disturbances and complete inertia matrix uncertainty are taken into account. It is proven that the obtained controller realizes sliding mode in the system, which implies robustness of the stabilization. Comparative simulation results indicate the superiority of the ultimate controller to the baseline controller in terms of control effort and chattering while the same state responses are output by both control systems. Following is Chapter 9, titled “A Class of Unknown Input Observers Under H∞ Performance for Fault Diagnosis: Application to the Mars Sample Return Mission,” in which the authors address the design and validation of a robust model-based fault diagnosis scheme to detect and isolate any thruster faults occurring in the thruster-based propulsion system of the chaser spacecraft during rendezvous on a circular orbit around Mars. The reference mission is the ESA Mars Sample Return mission. Key features of the proposed method are the use of a H∞ filter for robust fault detection and a bank of a new class of unknown input observers for fault isolation. A simulation campaign, based on a nonlinear high-fidelity simulator developed by GMV Space Industries, is conducted within the forced translation phase of the rendezvous phase of the MSR mission, under highly realistic conditions. Finally, in Chapter 10, titled “Actuator and Sensor Fault Detection Based on LPV Unknown Input Observer Applied to Lateral Vehicle Dynamics,” the estimation and the detection of actuator and sensor faults via the unknown input observer (UIO) applied to a lateral dynamics model of an automated steering vehicle are presented. The vehicle lateral dynamics have been described by an LPV model, taking into account the variations of the longitudinal velocity. The sensor faults are transformed into an augmented system with actuator faults and a first-order filter is needed to attenuate this noise. The work deals within the estimation of the actuator and sensor faults by converting sensor faults into actuator faults and designing an UIO to estimate both states and faults. The gains of the observer can be calculated by solving LMIs and the convergence of the observer is analyzed. In simulation, a vehicle lateral dynamics model with steering angle actuator fault and yaw velocity sensor fault has been tested and the ensuing results are presented to demonstrate the effectiveness of the proposed approach. Last but not least, the two volumes of the book would serve as an integrated and invaluable contemporary reference on observer-based control analysis and design. For this purpose, special thanks go to all of my colleagues (editors, authors, reviewers) for their dedicated work and remarkable effort in making this project a great success.

Magdi S. Mahmoud Saudi Arabia

C H A P T E R

1 Observer-Based Controller of Analytical Complex Systems: Application for a Large-Scale Power System Salwa Elloumi, Naceur Benhadj Braiek Advanced Systems Laboratory, Polytechnic School of Tunisia, La Marsa, Tunisia

1 INTRODUCTION In this chapter, we investigate observer-based controllers of interconnected large-scale systems. These complex systems constitute an important class of nonlinear systems that can describe the dynamical behavior of several physical processes, such as electrical power systems, industrial manipulators, and computer networks, to name a few. However, complex systems are often modeled by the interconnection of several subsystems, which makes it difficult to transfer information for process control. Therefore, possible control strategies are generally based on a decentralized solution because they use only local subsystem information, which reduces complexity and allow the control implementation to be more feasible [1]. Decentralized control of interconnected systems requires complete measurement of the state vector of each subsystem, which is not always valid. It is therefore necessary to use decentralized state observation techniques to reconstruct the nonmeasurable states. However, the synthesis of a separated decentralized control from that of a state observer that is itself decentralized is not obvious because the separation principle is not applicable in this situation. It is then necessary to simultaneously consider

New Trends in Observer-based Control https://doi.org/10.1016/B978-0-12-817034-2.00014-9

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© 2019 Elsevier Inc. All rights reserved.

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1. OBSERVER-BASED CONTROLLER OF ANALYTICAL COMPLEX SYSTEMS

the problem of decentralized control and state observation to ensure global asymptotic stabilization. The basic idea developed in this chapter deals mainly with the synthesis of nonlinear observer-based controller laws for complex systems (whether it is the general case of nonlinear systems or the particular case of interconnected systems), which are modeled in a polynomial form, guaranteeing the asymptotic stability of the augmented system composed of the nonlinear process, the control law, and the state observer. Our particular interest for polynomial modeling is justified by the fact that the polynomial systems allow the approximation of all the nonlinear analytical systems because the analytic functions can be approached by the Taylor series polynomial development. In addition, polynomial systems present the advantage of having the capacity of representing an important class of nonlinear systems and describing accurately the dynamic behavior of many physical processes such as electrical machines, power systems, manipulator robots, etc. The description of the polynomial system is based on the use of the Kronecker power of the state vector [2]. The study of polynomial systems has been developed in previous works [3–9], and shortly before in the work of Rotella [10] for the modeling, analysis, and synthesis of the feedback control as well as for the state observation. It is worth mentioning that according to our knowledge, there is still no work on the development of state observers for polynomial interconnected systems. Based on these considerations, this chapter aims to develop a new control approach with state observer nonlinear polynomial systems. This control law, where control and state observation gains are determined by a linear matrix inequality (LMI) resolution, must guarantee the stabilization of the overall augmented system (process + observer + control) by using the Lyapunov direct method for the study of stability, and this while verifying sufficient conditions of stability. The proposed approach is then extended to the case of large-scale interconnected systems. The elaborated developments are reported in this three-part chapter. The first part is interested in the theoretical basis to design a polynomial observer-based control for nonlinear systems and to determine sufficient LMI global stabilization conditions of the polynomial controlled system augmented by its observer. In the second part, we propose to enlarge the scope of our approach to cover analytical, interconnected, and largescale systems: we develop a decentralized control law with a decentralized state observer, which guarantees the asymptotic stability of the interconnected system together with its associated control and decentralized state observer. The last part is devoted to an interesting industrial case study of an interconnected large-scale system: an electrical power system composed of three interconnected machines. Our goal is to apply the approach of the second part in order to define a decentralized observer-based control

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

STATE OBSERVER-BASED CONTROLLER OF ANALYTICAL NONLINEAR SYSTEMS

5

that stabilizes the overall power system, and to prove the applicability and effectiveness of such control.

2 STATE OBSERVER-BASED CONTROLLER OF ANALYTICAL NONLINEAR SYSTEMS Observer-based control approaches have been widely studied in the literature. Nevertheless, without a common method to analyze or synthesize general nonlinear systems, only some special and limited classes of nonlinear systems have been considered up to now [11–17]. Obviously, many research activities are still in progress looking for new approaches to study and design valid and effective controls for nonlinear systems [18, 19]. This part focuses on the design of a new control approach with a state observer of analytical nonlinear systems modeled by a polynomial form. The basic idea of this study comes from the principle of the representation of an analytical nonlinear element by an infinite series that can be truncated, in practice, to an arbitrary finite order, considered sufficiently high. This principle, associated with the use of the tensor product of Kronecker, is used to formulate in a systematic way the analytical nonlinear vectorial processes and to design the feedback control and observation laws.

2.1 Problem Formulation of the Studied System The expression hereafter presents the state-space representation of a nonlinear continuous system we propose to study:  ˙ = F(X(t), U(t)) X(t) (1.1) Y(t) = CX(t) where ∀t ∈ R+ , the state X(t) ∈ Rn , the output Y(t) ∈ Rp , the input U(t) ∈ Rm , and F(.) is an analytic vector field from Rn to Rn where F(0) = 0. It is assumed that system (1.1) has a unique equilibrium for (X = 0). In the particular case, but sufficiently general, of nonlinear analytical systems, we will consider here their description in the state space using the Kronecker product. We expose the definition and properties of the Kronecker product in Appendix. Then, one can approximate the analytical function F(X(t)) with the following r-order polynomial form: F(X) =

r 

Ak X[k]

k=1

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(1.2)

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1. OBSERVER-BASED CONTROLLER OF ANALYTICAL COMPLEX SYSTEMS

where Ak , k = 1, . . . , r, are constant matrices of appropriate dimensions. The polynomial order r is considered odd (i.e., r = 2s − 1). Then, system (1.1) becomes ⎧ r  ⎪ ⎨ X(t) ˙ = Ak X[k] (t) + BU(t) (1.3) k=1 ⎪ ⎩ Y(t) = CX(t) We truncate the polynomial development to the third degree. This truncation is motivated by the fact that, at this order, we can move on our algebraic development and push it to deduce interesting research results without affecting its generality; in other terms, it will be easy to generalize the obtained results to any polynomial degree. Indeed, the approximation of any analytical system by a third-degree polynomial system is, generally, sufficient to obtain a polynomial model that is available in a sufficient domain around the operating point. The state representation (1.3) then becomes  ˙ = A1 X(t) + A2 X[2] (t) + A3 X[3] (t) + BU(t) X(t) (1.4) Y(t) = CX(t) i

where Ai ∈ Rn×n . Actually, we are not always sure to have the state measures; thus, we need to estimate this unmeasured state vector. Let us consider the following presentation for the state observer of nonlinear system (1.4):  ˙ˆ [2] ˆ [2] [3] ˆ [3] ˆ ˆ ˆ [2] ˆ [3] X = A1 X+A 2 X +A3 X +BU+L1 (Y− Y)+L2 (Y − Y )+L3 (Y − Y ) ˆ Yˆ = CX (1.5) i where Li ∈ Rn×p are the observer gain matrices. Based on Eq. (1.4), observer (1.5) can be rewritten as follows: ˙ˆ ˆ 3X ˆ 2X ˆ 1X ˆ +A ˆ [2] + A ˆ [3] + BU + L1 Y + L2 Y[2] + L3 Y[3] X =A

(1.6)

ˆ i = Ai − Li C[i] , i = 1, 2, 3. with A The state feedback control law is given by ˆ − K2 X ˆ [2] − K3 X ˆ [3] U = −K1 X

(1.7)

where K1 , K2 , and K3 are representing the control gain matrices of appropriate dimensions. ˆ Consider ε(t) = X(t) − X(t) the observation error (difference between the real and the observed states). Based on Eqs. (1.4), (1.6), one can express the observation error by ˆ [2] )+(A3 −L3 C[3] )(X[3] − X ˆ [3] ) (1.8) ε˙ = (A1 −L1 C) +(A2 −L2 C[2] )(X[2] − X

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Note that ˆ [2] = X ⊗ ε + ε ⊗ X − ε[2] X[2] − X

(1.9)

and ˆ [3] = X[2] ⊗ε +X⊗ε ⊗X−X⊗ε [2] +ε ⊗X[2] −ε ⊗X⊗ε −ε [2] ⊗X+ε [3] X[3] − X (1.10) Using Eqs. (1.4), (1.7), (1.8) gives the augmented system this new statespace representation:

¯ 2 Z2 + A ¯ 3 Z3 ¯ 1Z + A Z˙ = A



(1.11) T



X with Z = , Z2 = X[2] ε ⊗ X X ⊗ ε ε[2] , ε T  Z3 = X[3] ε ⊗ X[2] X ⊗ ε ⊗ X ε[2] ⊗ X X[2] ⊗ ε ε ⊗ X ⊗ ε X ⊗ ε[2] ε[3] , ¯1 = A ¯2 = A ¯3 = A





A1 + BK1 0

−BK1 A1 − L1 C

A2 + BK2 0

−BK2 A2 − L2 C[2]

−BK2 A2 − L2 C[2]

A3 + BK3 0

−BK3 A3 − L3 C[3]

−BK3 A3 − L3 C[3]

BK3 −(A3 − L3 C[3] )

,

BK3 −(A3 − L3 C[3] )

BK2 −(A2 − L2 C[2] )

BK3 −(A3 − L3 C[3] )

−BK3 [3] A3 − L3 C



−BK3 A3 − L3 C[3]

At this stage, we use the Kronecker product properties to simplify mathematically expression (1.11) and give it this compact form: ¯ 2 U−1 Z[2] + A ¯ 3 U−1 Z[3] ¯ 1Z + A Z˙ = A 2 3

(1.12)

where

⎛ ⎞ U2n×n2 0 0 0

⎜ 0 U Un×2n O2n2 0 0 ⎟ 2n×n2 ⎟ , U3 = (U2 ⊗ I2n ) · ⎜ U2 = ⎝ 0 0 U2n×n2 0 ⎠ O2n2 Un×2n 0 0 0 U2n×n2

Un×m is the permutation matrix defined in Appendix. It should be noted that the Kronecker jth power vectors, Z[j] , present redundancy within their elements. To bypass this problem and reduce complexity, we propose to rely on the nonredundant power vectors Z˜ [j] related to state vector Z = [Z1 , . . . , Zq ]T , and defined by ⎧ [1] [1] ˜ Z ⎪ ⎨ Z = Z j =j−1 j−1 j−2 j−2 j−2 [j] ˜ Z = [Z1 , Z1 Z2 , . . . , Z1 Zq , Z1 Z22 , Z1 Z2 Z3 , . . . , Z1 Z2q , (1.13) ⎪ ⎩ j−3 3 j−3 3 j T . . . , Z1 Z3 , . . . , Z1 Zq , . . . , Zq ] , ∀j ≥ 2

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1. OBSERVER-BASED CONTROLLER OF ANALYTICAL COMPLEX SYSTEMS

˜ [j] , The redundant and the nonredundant vectors, respectively, Z[j] and Z are related with this formula:

⎫ j q+j−1 ⎬ ∀j ∈ N, ∃!Tj ∈ Rq ×αj , αj = j (1.14) ⎭ ˜ [j] Z[j] = Tj Z where αj is the binomial coefficient. Expression (1.12) is then reformulated in terms of nonredundant power [j] ˜ Z as follows: ¯ 2 U−1 H2 T2 Z˜ [2] + A ¯ 3 U−1 H3 T3 Z˜ [3] ¯ 1Z + A Z˙ = A 2 3 with

⎛

⎞ ⎛ 0 0 In2 T2 0 ⎜ 0 ⎟ 0 ⎟ I n2 ⎝ 0 I n2 = , T H2 = ⎜ 2 ⎝ 0 Un×n 0 ⎠ 0 0 0 0 I n2 ⎛ ⎞ 0 0 0 In3 ⎜ 0 0 0 ⎟ I n3 ⎜ ⎟ ⎜ 0 In ⊗ Un×n 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ 0 I n3 ⎜ ⎟, H3 = ⎜ Un×n2 0 0 ⎟ ⎜ 0 ⎟ ⎜ 0 0 Un×n ⊗ In 0 ⎟ ⎜ ⎟ ⎝ 0 0 Un×n2 0 ⎠ 0 0 0 I n3 ⎛ ⎞ 0 0 0 T3 ⎜ 0 T2 ⊗ In 0 0 ⎟ ⎟ T3 = ⎜ ⎝ 0 0 In ⊗ T2 0 ⎠ 0 0 0 T3

(1.15)

⎞ 0 0 ⎠, T2

or, simply, as follows: ¯ 2 R2 Z˜ [2] + A ¯ 3 R3 Z˜ [3] ¯ 1Z + A Z˙ = A with R2 =

U2−1 H2 T2

and R3 =

(1.16)

U3−1 H3 T3 .

2.2 Stability Analysis of the Augmented Polynomial System This section is devoted to analyzing the stability of the nonlinear analytical system (1.16) augmented with the polynomial state observer-based control law. The goal is to determine stability-sufficient conditions for the overall system. Let us then consider the positive definite quadratic Lyapunov function V(X, ε) given by V(X, ε) = XT Pc X + εT Po ε

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

(1.17)

STATE OBSERVER-BASED CONTROLLER OF ANALYTICAL NONLINEAR SYSTEMS

9

This expression can also be reformulated as follows: V(Z) = ZT PZ with

P=

Pc 0

0 Po

(1.18)



where Pc and Po are symmetric positive definite matrices. If the derivative, with respect to time, of V(Z), given by ˙ V(Z) = Z˙ T PZ + ZT PZ˙

(1.19)

R2n ,

then the overall system is stable in the is negative definite for all Z ∈ sense of Lyapunov. Developing Eq. (1.19) along the trajectory of the system, one gets ˙ V(Z) =2

3 

ZT PAk Z[k]

(1.20)

k=1

¯ 1 , A2 = A ¯ 2 R2 , and A3 = A ¯ 3 R3 . Referring to Eq. (1.20), and with A1 = A  using property (A.9) of the “vec function presented in Appendix, it comes ˙ V(Z) =2

3  (vec PA)Z[k+1]

(1.21)

k=1

The use of expression (A.11) of the “mat function property, defined in Appendix, gives T T ˙ V(Z) = 2(ZT N1,1 Z + Z[2] N2,1 Z + ZT N1,2 Z[2] + Z[2] N2,2 Z[2] ) (1.22)

where Nk+1−j,j = mat(nk+1−j ,nj ) (vec(PAk ))

(1.23)

= Uni−1 ×n (P ⊗ Ini−1 ) · Mi−1,j (A) with

⎛ Mk+1−j,j

⎜ ⎜ =⎜ ⎜ ⎝

mat(nk−j ,nj ) (A1T k ) mat(nk−j ,nj ) (A2T k ) .. . mat(nk−j ,nj ) (AnT k )

(1.24)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

Aik is the ith line of matrix Ak . We insert Eq. (1.23) into Eq. (1.22), and we use the following property of Benhadj Braiek et al. [20]: ∀i, j ∈ N;

Uni ×nj X[i+j] = X[i+j]

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(1.25)

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1. OBSERVER-BASED CONTROLLER OF ANALYTICAL COMPLEX SYSTEMS

we obtain

 T ˙ V(Z) = 2 ZT PA1 Z + Z[2] (P ⊗ In )M2,1 Z T

+ ZT PM1,2 Z[2] + Z[2] (P ⊗ In )M2,2 Z[2]



(1.26)

which can be simply reformulated as follows: ˙ V(Z) = ZT (PM + MT P)Z with



Z=

Z Z[2]



,

P=

P 0

0 P ⊗ I2n



,

M=

(1.27) A1 M2,1 (A2 )

M1,2 (A2 ) M2,2 (A3 )



Using the nonredundant form (A.3), the vector Z is expressed by ˜ Z=T·Z where

T=

I2n 0

0 H2 T2

and

(1.28)

˜ = Z



Z˜ ˜ Z[2]



and therefore, Eq. (1.27) becomes ˜ T TT (PM + MT P)TZ˜ ˙ V(Z) =Z

(1.29)

The equilibrium (Z = 0) of system (1.16) is globally asymptotically stable ˙ if V(Z) is negative definite. We can also state the earlier sufficient condition for the stability of the studied system using the following inequalities:  P>0 (1.30) TT (PM + MT P)T < 0 The next section is devoted to solving this system of inequalities in order to obtain control and observation gains guaranteeing the stability of the augmented complex system.

2.3 A Linear Matrix Inequality Formulation of the Proposed Stability Conditions It is important to note that the second inequality of expression (1.30) contains several nonresolvable bilinear terms by the LMI tool. To bypass this problem, we propose to develop the matrix M expressed by Eq. (1.31), which is based on control and observation gain matrices:

M1,2 (A2 ) A1 (1.31) M= M2,1 (A2 ) M2,2 (A3 ) ¯ 1 , A2 = A ¯ 2 R2 , and A3 = A ¯ 3 R3 . We recall that A1 = A

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11

¯ 1, A ¯ 2 , and A ¯ 3 can be developed as According to Eq. (1.11), matrices A follows: ¯ i = Ai + BKi + Li Ci , i = 1, 2, 3 A (1.32) where



 A1 0 0 B , B= , K1 = −K1 K1 , L1 = , A1 = −L1 0 0 A1  C1 = 0 C

 A2 0 0 0 0 , K2 = −K2 K2 K2 −K2 , L2 = , A2 = L2 0 A2 A2 A2  C2 = 0 −C[2] −C[2] C[2]

A3 0 0 0 0 0 0 0 A3 = 0 A3 A3 A3 A3 A3 A3 A3  K3 = −K3 K3 K3 −K3 K3 −K3 −K3 K3

 0 , C3 = 0 −C[3] −C[3] C[3] −C[3] C[3] C[3] −C[3] L3 = L3 ¯ 3 = A3 R3 , K ¯ 2 = K2 R2 , K ¯ 3 = K3 R3 , C ¯ 2 = C2 R2 , and ¯ 2 = A2 R2 , A Let A ¯ = C R , the matrix M can be expressed as follows: C3 3 3

¯ 2 + BK ¯ 2 + L2 C ¯ 2) A1 + BK1 + L1 C1 M12 (A (1.33) M= ¯ 2 + BK ¯ 2 + L2 C ¯ 3 + BK ¯ 3 + L3 C ¯ 2 ) M22 (A ¯ 3) M21 (A Based on the following two lemmas [20], i+j Lemma 1. Let G ∈ Rn×m and H ∈ Rm×n , we have Mij (G.H) = (G ⊗ Ini )Mij (H)

(1.34)

Rn×m

and G.a(.) the resulting function of the product of Lemma 2. Let G ∈ the matrix G by the function a(.). Then, M(G.a) = G.M(a) with

⎛ ⎜ ⎜ G=⎜ ⎝

G

0

G ⊗ In ..

0

(1.35)

.

⎞ ⎟ ⎟ ⎟ ⎠

(1.36)

G ⊗ Ins−1

Which leads to the simplification of the matrix M into the following form: M = A + BK + LC

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(1.37)

12

1. OBSERVER-BASED CONTROLLER OF ANALYTICAL COMPLEX SYSTEMS

with

A=

¯ 2) M12 (A A1 ¯ ¯ 3) M21 (A2 ) M22 (A



;

B=

B 0

0 B ⊗ I2n

⎞ 0 C1 ¯ 2) ⎟ ⎜ 0 M12 (C K1 ⎟K = C=⎜ ¯ 2) ⎠ ⎝ M21 (C ¯ 2) 0 M21 (K ¯ 0 M22 (C3 )

L1 L2 0 0 L= 0 0 L2 ⊗ I2n L3 ⊗ I2n ⎛

;

¯ 2) M12 (K ¯ 3) M22 (K

The matrix inequality system (1.30) then becomes  P>0 TT (PA + PBK + PLC + AT P + KT BT P + CT LT P)T < 0

;

(1.38)

The unknown variables of problem (1.38) are P, K, and L. However, the second inequality is bilinear because it contains the coupled terms (P, K) and (P, L). To solve this problem, we try to relax the BMI (1.38) using the linearization techniques such as Schur lemma and the following separation lemma [21]: Lemma 3. For any vectors X ∈ Rn and Y ∈ Rn , and for any positive scalar δ, the following inequality is verified: XT Y + YT X ≤ δXT X + δ −1 YT Y

(1.39)

When applying the previous lemma, one has PBK + PLC + KT BT P + CT LT P ≤ δk PP + δk−1 (BK)T BK + δl PP + δl−1 (LC)T LC

(1.40)

with δk > 0 and δl > 0. Hence, the following inequality is sufficient to assure that the second inequality of system (1.38) is verified: TT (PA + AT P)T − TT P(−δk I)PT − TT P(−δl I)PT − TT (BK)T (−δk−1 I)BKT −TT (LC)T (−δl−1 I)LCT < 0 (1.41) Using the Generalized Schur complement, the inequality (1.41) can be given by this LMI form: ⎞ ⎛ T T (PA + AT P)T (PT)T (PT)T (BKT)T (LCT)T ⎟ ⎜ PT −δk−1 I 0 0 0 ⎟ ⎜ −1 ⎟ < 0 (1.42) ⎜ 0 0 PT 0 −δl I ⎟ ⎜ ⎠ ⎝ 0 BKT 0 0 −δk I LCT 0 0 0 −δl I

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when pre- and postmultiplying the inequality (1.42) by the bloc diagonal matrix Λ = diag(I, I, I, δk−1 , δl−1 ), we get ⎛ T ⎞ T (PA + AT P)T (PT)T (PT)T (BKδ T)T (Lδ CT)T ⎜ ⎟ PT −δk−1 I 0 0 0 ⎜ ⎟ ⎜ ⎟ −1 ⎜ ⎟ < 0 (1.43) I 0 0 PT 0 −δ ⎜ ⎟ l ⎜ ⎟ −1 0 0 −δk I 0 BKδ T ⎝ ⎠ −1 0 0 0 −δl I Lδ CT with Kδ = δk−1 K and Lδ = δl−1 L. So we have overcome the bilinearity problem between decision variables. The new obtained inequality is linear with respect to the sought variables. It then comes the following theorem [22] based on the resolution of an LMI problem: Theorem 1. The equilibrium (Z = 0) of augmented system (1.12) is globally asymptotically stabilizable if there exist • • • •

a symmetric positive definite matrix P, a control gain matrix K, an observation gain matrix L, and reals δk > 0 and δl > 0.

such that ⎧ P>0 ⎪ ⎪ ⎪ ⎛ T ⎪ T (PA + AT P)T ⎪ ⎪ ⎪ ⎜ ⎪ ⎨ ⎜ PT ⎜ ⎜ PT ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ BKδ T ⎪ ⎝ ⎪ ⎪ ⎩ Lδ CT

(PT)T

(PT)T

(BKδ T)T

(Lδ CT)T

−δk−1 I

0

0

0

−δl−1 I

0

0 0

0 0

0

−δk−1 I

0

0

0

−δl−1 I

⎞ ⎟ ⎟ ⎟ ⎟ are used. The Euclidean norm is denoted as x2 = x x. The real and imaginary part of a complex vector is described by Re(·) and Im(·), respectively. Congruence [10]: For a symmetric matrix X and a full column rank matrix Q, the following property holds X  0 ⇒ QT XQ  0. Completion of squares: For two matrices X and Y of appropriate dimensions and a symmetric positive definite matrix Q = QT  0, from (X − Q−1 Y)T Q(X − Q−1 Y) 0, the following inequality

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40

2. OBSERVER WITH APPLICATION TO ROBUST WIND TURBINE CONTROL

XT Y + YT X  XT QX + YT Q−1 Y 

can be derived. Schur complement [10]: For a symmetric matrix X = XT = the following conditions are equivalent  X22 ≺ 0 X≺0⇔ . T X11 − X12 X−1 X 22 12 ≺ 0

X11 XT12

 X12 , X22

3.1 Stability Conditions Consider a nonlinear closed-loop or autonomous TS system given by x˙ =

Nr 

hi (z) Gi x.

(2.3)

i=1

From the direct Lyapunov method, it is well known that a system is globally asymptotically stable if a function V(x) satisfying V(x) > 0 and

˙ V(x) 0 for which the following is verified ¯ x2 = −λx ¯ T x. xT (GTi P + PGi )x ≺ −λI 2

(2.9)

Employing a quadratic Lyapunov function approach, an upper bound of the corresponding derivative is denoted as [10] 2 2PFi λ¯ 2 ˙ δ22 , V(x) < − x2 + 2 λ¯

(2.10)

where PFi = maxPFi  can be calculated from the solution of the LMIi

based stability analysis of the system in Eq. (2.7). Thus, the derivative of the quadratic Lyapunov function in Eq. (2.10) is guaranteed to be negative as long as 2 λ¯ 2PFi δ22 < x22 ¯λ 2

(2.11)

holds. The ISS property of a system x˙ = f(x)+g(x)δ is presented in Ref. [14]. The concept proposes to assess the ISS property by finding a Lyapunov function candidate V(0) = 0, V(x) > 0 ∀x = 0 and monotonically increasing functions χ1 (·), χ2 (·), χ3 (·) : R≥0 → R≥0 , χ1 (0) = 0, χ2 (0) = ˙ ≤ −χ1 (x2 ) + χ2 (δ2 ) with x2 ≥ χ3 (δ2 ) 0, χ3 (0) = 0, such that V(x) for all x and δ. Then system (2.8) subject to a bounded input δ is ISS. From the definitions χ1 (x2 ) :=

2 λ¯ 2PFi x22 , χ2 (δ2 ) := δ22 , 2 λ¯

2

and χ3 (δ2 ) =

4PFi δ22 , λ¯ 2 (2.12)

it is apparent that Eqs. (2.10), (2.11) fulfill the ISS properties derived in Ref. [14]. Further, condition (2.11) allows for an estimate on the guaranteed resulting stability margin around the origin under impact of the bounded input, where V(x) is decreasing as long as Eq. (2.11) holds. In fact, for bounded inputs of the system, guaranteeing the autonomous closed-loop stability of the TS model (2.3) results in the ISS property because there is always ∃λ¯ in Eq. (2.9) [10]. However, as will be discussed for the observer design with unmeasurable premise variables, introducing

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2. OBSERVER WITH APPLICATION TO ROBUST WIND TURBINE CONTROL

a restriction in the form of Eq. (2.11) in the LMI design process can be used to impose a desired stability margin around the origin.

3.1.2 Decay Rate LMIs (2.7) constrain the solution of the eigenvalues at the vertices λi , that is, det(Gi − λi I) = 0, to lie anywhere within the left half of the complex plane. It is well known for linear systems that the transient response is governed by the location of the eigenvalues in the complex plane. Due to the nonlinear nature of the system, TS systems perform with variable transient responses depending on the current operating point. However, a good indicator of the transient properties of the overall nonlinear system is the location of the eigenvalues at the vertices of the polytopic description. Consequently, by influencing the location of the eigenvalues of the linear closed-loop submodels, the dynamic behavior of the nonlinear system can be designed to account for requirements that arise from, for example, the physical application. The introduction of an additional term to the LMIs (2.7) guarantees a decay rate α at which system (2.3) at least moves toward a stable equilibrium [9]. It results from restricting the Lyapunov function candidates ˙ derivative to an upper bound V(x) < −2αV(x) and is given in terms of LMIs as GTi P + PGi + 2αP ≺ 0

(2.13)

for i = 1, . . . , Nr , where the reformulation to GTi P + PGi ≺ −αP − αP directly indicates that all real parts of the eigenvalues of Gi attain at least Re(λi ) < −α.

3.1.3 LMI Region in the Complex Plane In many applications, there is a limit to the maximum attainable decay rate of the systems response, for example, from loading aspects or actuator saturation. Further, the transient response of the system is often restricted by certain natural frequencies of the process, which can be characterized as the imaginary part of the closed-loop eigenvalue Im(λi ), as discussed for the wind turbine in Section 4. Therefore, the excitation of unmodeled dynamics may be prevented by imposing a restriction of the resulting eigenvalues λi on the admissible subset in the complex plane, such that a desired transient behavior of the system occurs. In Ref. [15], an LMI representation for eigenvalues of a linear system to be located in region S(α, r, θ ), which is shown in Fig. 2.2, is presented. This formulation can be extended to TS models by the following LMIs for i = 1, . . . , Nr [16]

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LMI CONTROL AND OBSERVER SYNTHESIS

43



θ 

r

α

FIG. 2.2 Graphical representation of the subset S(α, r, θ ) in the complex plane defined by Eq. (2.14).



⎡ sin θ ⎣ cos θ



GT P + PGi

i  GTi P − PGi

GTi P + PGi + 2αP ≺ 0,   −rP PGi ≺ 0, GTi P −rP ⎤

cos θ PGi − GTi P

 ⎦ ≺ 0. sin θ GTi P + PGi

(2.14)

If the LMIs in Eq. (2.14) hold, the system is said to be quadratically D-stable and all poles of the vertices of the system Gi , i = 1, . . . , Nr lie inside of D, where S(α, r, θ ) is one possible representation of a subset D of the complex plane. It consists of an α-stability region (2.13), a disk with a center at the origin, and a cone given by S(0, 0, θ ) as the first, second, and third LMI in Eq. (2.14), respectively. 3.1.4 Disturbance Attenuation by H2 Approach Now consider a TS system subject to a disturbance d ∈ Rmd and given by x˙ =

Nr 

hi (z) (Gi x + Di d) ,

(2.15)

i=1

where Di is the nonlinear disturbance distribution matrix in TS formulation. The impact of the disturbance on the system can be characterized in terms of the L2 → L2 gain γ > 0 x2 ≤γ d2

⇐⇒

xT x − γ 2 dT d ≤ 0.

(2.16)

The aim of the control synthesis is to minimize gain γ , which implies an attenuation of the disturbance d on the system states x of γ . The

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44

2. OBSERVER WITH APPLICATION TO ROBUST WIND TURBINE CONTROL

characterization of the L2 → L2 gain in Eq. (2.16) is an LMI and can be introduced into the negativity condition of the Lyapunov function candidates derivative in Eq. (2.4) to form [10] ˙ V(x) + xT x − γ 2 dT d < 0.

(2.17)

From integrating expression (2.17), assuming a resulting stable system V(x(∞)) = 0, and setting the initial condition to x(0) = 0  ∞ (γ 2 dT d − xT x)dt > 0 (2.18) 0

is obtained as the solution to the corresponding Lyapunov function. This is equivalent to the attenuation condition represented by Eq. (2.16). Therefore, using the system description subject to disturbance in Eq. (2.15), the derivative of the Lyapunov function candidate in Eq. (2.5), and expansion  r with respect to the property of the membership functions N i=1 hi (z) = 1, the stability condition (2.17) results in Nr 

 hi (z) xT GTi Px + xT PGi x + dT DTi Px + xT PDi d + xT x − γ 2 dT d < 0,

i=1

(2.19) or equivalently Nr  i=1

hi (z)



xT

  GT P + PGi + In i d DTi P T

PDi −γ 2 Imd

  x < 0. d

(2.20)

As a result, stability of the system and an attenuation γ > 0 from the disturbance to the states of the system are verified if   T PDi Gi P + PGi + In ≺0 (2.21) DTi P −γ 2 Imd with respect to a matrix P = PT  0 holds. The presented stability conditions form the basis for both the following observer and controller synthesis. In the remainder of the section, the proposed controller is described and the synthesis of the necessary feedback gains is formulated in terms of LMIs.

3.2 Control Synthesis The control design is based on the disturbed TS system in form of x˙ =

Nr 

hi (z) (Ai x + Bi u + Di d) .

i=1

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

(2.22)

LMI CONTROL AND OBSERVER SYNTHESIS

45

For control design, it is assumed that the Nr pairs (Ai , Bi ) are controllable. The proposed state feedback is the parallel-distributed compensation (PDC) control law [9, 10], which according to the TS structure consists of Nr feedback matrices Ki blended by the membership functions hi (z) or as will be derived for the wind turbine application on the estimate of the  r membership functions N z). Consequently, the PDC control law is j=1 hj (ˆ given by [9, 10] u=−

Nr 

hj (z)Kj x.

(2.23)

j=1

As a result, the closed-loop TS system under impact of the PDC control (2.23) is denoted as ⎛⎛ ⎞ ⎞ Nr Nr Nr    hi (z) ⎝⎝Ai − hj (z)Bi Kj ⎠ x + Di d⎠ = hi (z) (Gi x + Di d) , x˙ = i=1

j=1

i=1

(2.24) where from expansion with respect to the convex sum property  r N r hj (z) = 1 the closed-loop system matrix is given by Gi = N j=1 hj (z)  j=1  Ai − Bi Kj . 3.2.1 Control Synthesis: Decay Rate Assuming an undisturbed system, that is, d = 0, the stability condition (2.13) from a quadratic Lyapunov function approach for the closed-loop TS system in Eq. (2.24) reads as ATi P + PAi − KTj BTi P − PBi Kj + 2αP ≺ 0.

(2.25)

Because the objective of the LMI solver is to determine P and Kj simultaneously, the given constraint is bilinear. By employing the matrix X = XT = P−1 to derive a congruent inequality X(2.25)X, the control synthesis of the Nr feedback gains Ki with respect to a guaranteed decay rate α > 0 can be formulated as LMIs in the following way. Determine the matrix X = XT  0 and Nr matrices Mj with respect to a decay rate α constraint by [9] XATi + Ai X − MTj BTi − Bi Mj + 2αX ≺ 0

(2.26)

for every combination i, j = 1, . . . , Nr such that hi · hj = 0. The positive definite matrix fulfilling the Lyapunov function in Eq. (2.4) is given by P = X−1 and the Nr feedback matrices ensuring the stability of the closed-loop nonlinear TS system (2.24) are calculated by Kj = Mj X−1 .

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2. OBSERVER WITH APPLICATION TO ROBUST WIND TURBINE CONTROL

Remark 1. The combination of i and j in LMIs (2.26) of the control synthesis plays an important role with respect to both feasibility and performance of the resulting feedback gains. In general, stability of the TS system is verified if all possible combinations of i = 1, . . . , Nr and j = 1, . . . , Nr are respected in the synthesis. However, this results in a large number of LMIs and might impede the search for appropriate feedback gains Ki , which need to ensure the stability and performance imposed by the LMIs at every vertex of the polytopic description. The additional condition hi · hj = 0 accounts for the property of systems where not every membership function may be active, that is, hi = 0, at the same time (or operational point); see, for example, the discussion in Section 4.4 for the control syntheses of the wind turbine application. Thus, the conservativeness of the LMIs is reduced by accounting for hi · hj = 0. This consideration holds for all control and observer synthesis LMIs discussed in this contribution, where the double  Nr Nr N r  r z) occurs. sum N i=1 hi (z) j=1 hj (z) or i=1 hi (z) j=1 hj (ˆ Remark 2. For the derivation of the LMIs in the control synthesis    r by introducing Gi = N into the LMIs presented j=1 hj (z) Ai − Bi Kj in Section 3.1, all necessary terms can be expanded using the prop T Nr N r erty j=1 hj (z) = 1 to form, for example, j=1 hj (z) XAi + Ai X− MTj BTi − Bi Mj + 2αX ≺ 0 from Eq. (2.26), which holds if Eq. (2.26) holds for the necessary combinations of i and j as discussed in Remark 1. 3.2.2 Control Synthesis: LMI Region Constraint     Employing substitution P, PGi , GTi P ↔ X, Gi X, XGTi [15], the stability of the closed-loop system Gi in Eq. (2.24) constraint to the region S(α, r, θ ) can be expressed based on Eq. (2.14), such that the following synthesis results. Determine the matrix X = XT  0 and Nr matrices Mj for a desired α > 0, r > 0, and θ > 0 constraint by XATi + Ai X − MTj BTi − Bi Mj + 2αX ≺ 0,   −rX Ai X − Bi Mj ≺ 0, XATi − MTj BTi −rX      sin θ XATi + Ai X − MTj BTi − Bi Mj cos θ Ai X − Bi Mj − XATi + MTj BTi    ≺0  cos θ XATi − MTj BTi − Ai X + Bi Mj sin θ Ai X − Bi Mj + XATi − MTj BTi (2.27)

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LMI CONTROL AND OBSERVER SYNTHESIS

for all i, j = 1, . . . , Nr such that hi · hj = 0. If Eq. (2.27) is solved, the eigenvalues of the closed-loop system (2.24) are located in S(α, r, θ ), and the positive definite matrix fulfilling the Lyapunov function in Eq. (2.4) is given by P = X−1 . The Nr feedback matrices ensuring stability of the closed-loop nonlinear TS system are calculated by Kj = Mj X−1 . 3.2.3 Control Synthesis: H2 Disturbance Attenuation Inserting the closed-loop TS   system (2.24) into Eq. (2.21) and applying X 0 the congruence matrix yields 0 I   T Di XAi + Ai X − MTj BTi − Bi Mj + XX ≺0 (2.28) DTi −γ 2 Imd and from separation of the quadratic term gives     T Di XAi + Ai X − MTj BTi − Bi Mj X  X + T 2 0 D −γ Im i

 0 ≺ 0.

(2.29)

d

Applying the Schur complement to Eq. (2.29) results in the following set of LMIs for a control synthesis guaranteeing an attenuation of γ . Determine the matrix X = XT  0 and Nr matrices Mj with respect to a desired γ constraint by ⎤ ⎡ T XAi + Ai X − MTj BTi − Bi Mj Di X ⎥ ⎢ (2.30) −γ 2 Imd 0 ⎦≺0 DTi ⎣ X

0

−In

for i = 1, . . . , Nr , j = 1, . . . , Nr such that hi · hj = 0. Moreover, by a change of variable Γ = γ 2 , the LMI solver can be employed to minimize Γ . 3.2.4 Control Synthesis: Mixed H2 Disturbance Attenuation and LMI Region Constraint In some applications, such as those discussed for the wind turbine in Section 4, minimizing the disturbance attenuation within a particular LMI region might be favorable. For that reason, a mixed LMI constraint from the previously discussed LMIs can be formed similar to the approach for linear systems in Ref. [15]. Increasing the attenuation from the disturbance to the system states pushes the resulting eigenvalue of the closed-loop system further into the left half of the complex plane; see, for example, Table 2.1 in Section 4, and consequently has similar effects such as introducing a decay rate α in Eq. (2.27). Because stability of the closed-loop system is verified by Eq. (2.30), combining it with Eq. (2.27), where the first LMI respecting the guaranteed decay rate α is omitted, results in a mixed criterion ensuring

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2. OBSERVER WITH APPLICATION TO ROBUST WIND TURBINE CONTROL

an attenuation of γ with eigenvalues of the closed-loop system restricted to S(0, r, θ ). Formulated in terms of LMIs, this property is verified if for all i, j = 1, . . . , Nr such that hi · hj = 0, X = XT  0 and Nr matrices Mj the following LMI conditions hold ⎤ ⎡ T Di X XAi + Ai X − MTj BTi − Bi Mj ⎣ −γ 2 Imd 0 ⎦ ≺ 0, DTi X 0 −In   −rX Ai X − Bi Mj ≺ 0, XATi − MTj BTi −rX     T  T T T T T  sin θ XAi + Ai X − Mj Bi − Bi Mj   − MT BT − Ai X + Bi Mj cos θ XAT i j i

cos θ Ai X − Bi Mj − XAi + Mj Bi   sin θ Ai X − Bi Mj + XAT − MT BT i j i

≺ 0. (2.31)

3.3 Observer Synthesis An observer for the TS system represented by Eq. (2.22) with a linear output defined as y = Cx and estimated premise variables zˆ is denoted as [10] x˙ˆ =

Nr 

  hi (ˆz) Ai xˆ + Bi u + Li (y − y) ˆ ,

i=1

yˆ = Cˆx,



(2.32)



where in the following it is assumed that the Nr pairs Ai , C are observable. The structure of the observer described by Eq. (2.32) resembles the well-known Luenberger observer with a feedback term blended by the  r z)Li . By ensuring stability of the convex membership functions N i=1 hi (ˆ observation error e = x − xˆ , the convergence of the estimated states to the real states xˆ → x is verified. Therefore, consider the observer error dynamics e˙ of system (2.22) specified as [17] e˙ = x˙ − x˙ˆ +

Nr 

Nr      hi (ˆz) Ai x + Bi u + Di d − hi (ˆz) Ai x + Bi u + Di d (2.33)

i=1





i=1



=0

resulting in e˙ =

Nr  i=1

⎛ ⎞ Nr Nr       hi (ˆz) (Ai −Li C)e+Di d +⎝ hi (z) − hi (ˆz)⎠ Ai x + Bi u + Di d . 

i=1

i=1





:= (z,ˆz,x,u,d)

(2.34)

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LMI CONTROL AND OBSERVER SYNTHESIS

The premise vector z may be unmeasurable but estimated, which is  r accounted for by the notation N z). i=1 hi (ˆ 3.3.1 Observer Synthesis: Measurable Premise Variable In many applications, the premise variables may depend on an output or a measurable exogenous signal of the process. In those cases, where z is completely measurable, the error dynamic reduces to e˙ =

Nr 

  hi (z) (Ai − Li C)e + Di d .

(2.35)

i=1

The error dynamic given by Eq. (2.35) is compatible with the structure of the system represented by Eq. (2.24), and thus a quadratic Lyapunov candidate function V(e) = eT Pe can be applied accordingly. Therefore, the results from deriving the stability condition in Section 3.1 can be readily applied for the observer synthesis by introducing the closed-loop state matrices Gi = Ai − Li C. As a result, a “minimum decay rate” α in the convergence from the observer to the system states xˆ → x is guaranteed, if a positive definite matrix P = PT  0 and Nr matrices Ni for i = 1, . . . , Nr are found fulfilling ATi P + PAi − Ni C − CT NTi + 2αP ≺ 0,

(2.36)

P−1 Ni .

where the feedback gains are obtained from Li = In addition, D-stability implying that the closed-loop eigenvalues of the error dynamic at the vertices of the convex description are restricted to the subset S(α, r, θ ) for a given α > 0, r > 0, and θ > 0 is verified, if a positive definite matrix P = PT  0 and Nr matrices Ni for i = 1, . . . , Nr constrained by ATi P + PAi − Ni C − CT NTi + 2αP ≺ 0   −rP PAi − Ni C ≺0 ATi P − CT NTi −rP      cos θ PAi − Ni C − ATi P + CT NTi sin θ ATi P + PAi − Ni C − CT NTi   ≺0   cos θ ATi P − CT NTi − PAi + Ni C sin θ ATi P + PAi − Ni C − CT Ni (2.37) are determined. Accordingly, a disturbance attenuation to the error states e2 ≤ γ d2 of γ is verified, if a solution for P = PT  0 and Nr matrices Ni for i = 1, . . . , Nr constraint by  T  Ai P + PAi − CT NTi − Ni C + In PDi ≺0 (2.38) DTi P −γ 2 Imd is determined.

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3.3.2 Observer Synthesis: Unmeasurable Premise Variable For applications that depend on an unmeasurable state as premise variable, the estimated states of the observer can be used to calculate the membership functions of the TS feedback loop. However, in the transient region where the observer states xˆ converge to the real states x, this induces an additional disturbance term denoted as in Eq. (2.34), which stems  r from the error in the calculation of the membership function N i=1 hi (z) − N r h (ˆ z ). This error term converges → 0 as the states of the observer i=1 i move toward the real states of the system. To ensure stability with respect to unmeasurable premise variables that are states of the TS systems, in Ref. [18] a synthesis is proposed. However, as also stated in Ref. [19], the synthesis is conservative when accounting for the unmeasurable premise variable. For that reason in Ref. [19], an observer design is presented that employs an assumption on the maximum occurring mismatch from the calculation of the estimated membership functions hi (ˆx). The negativity of the Lyapunov derivative is ensured despite of (x, xˆ , u) influencing system (2.34). The impact on the Lyapunov function derivative is upper bounded by hi (x)x − hi (ˆx)ˆx2 ≤ Oi e2 and    hi (x) − hi (ˆx) u2 ≤ Ui e2 , where Oi and Ui are appropriate real matrices with all components being positive definite. Here, TS observers with states as unmeasurable premise variables are considered by a different approach. The error in the calculation will be accounted for by combining the necessary linear submodels in the observer synthesis to ensure convergence to a defined stability margin imposed by the ISS concept; see Section 3.1.1. The presented LMI condition is less restrictive with respect to the influence of the error term from the unknown premise variable, that is, to the bounded disturbance induced by the mismatch of the calculated membership functions. In the feedback synthesis, term (x, xˆ , u) is not entirely included in the presented approach. However, this property is derived at the cost of combining more linear submodels within the synthesis, imposing a different source of conservatism on this LMI-based approach. Therefore, consider rewriting the error dynamics with respect to convex r  r  r  r ˙ˆ and refor˙= N z) = N z)˙x − N ity N j=1 hj (ˆ i=1 hi (z) = 1 as e j=1 hj (ˆ i=1 hi (z)x mulating the state matrix of the observer as Aj = (Aj + Ai − Ai ). This yields e˙ =

Nr 

hi (z)

i=1

Nr 

  hj (ˆz) (Ai − Lj C)e + Ai,j xˆ + Bi,j u ,

(2.39)

j=1

with Ai,j = (Ai − Aj ) and Bi,j = (Bi − Bj ). By inserting xˆ = (x − e), the error dynamics are given by e˙ =

Nr  i=1

hi (z)

Nr 

  hj (ˆz) (Ai − Lj C − Ai,j )e + Ai,j x + Bi,j u .

j=1

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(2.40)

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51

As also argued in Ref. [19], if the membership functions are Lipschitz and the input u of the system is bounded, then the state x is also bounded. Thus the ISS property is applicable to the disturbance resulting from the mismatch in the estimation and the current state x and input u of the system. Therefore, only the closed-loop system matrix (Ai − Ai,j − Lj C) determines the ISS property. In case of an unmeasurable premise variable, here the disturbance in Eq. (2.34) is assumed to be d = 0. This is reasonable for the presented application of the wind turbine because the main task of the observer is to estimate the current wind speed, and thus the disturbance of the system is included as an augmented state in the system description. The following theorems provide the basis for the wind-speed observer design. Theorem 1. The observer with unmeasurable states as premise variables and a closed-loop error dynamic of Eq. (2.40) is ISS if a positive definite matrix P = PT  0 for V(e) = eT Pe and Nr matrices Nj for i, j = 1, . . . , Nr are found fulfilling (Ai − Ai,j )T P + P(Ai − Ai,j ) − Nj C − CT NTj ≺ 0,

(2.41)

where the feedback gains are obtained from Lj = P−1 Nj . Proof. The bounded inputs x and u can be omitted for applying ISS property, and thus the derivative of the quadratic Lyapunov function V(e) = eT Pe is denoted as ˙ V(e) =

Nr  i=1

hi (z)

Nr 

 hj (ˆz)eT (Ai − Ai,j )T P

j=1

 + P(Ai − Ai,j ) − CT NTj − Nj C e < 0,

(2.42)

which holds if Eq. (2.41) is verified. It is interesting to notice that by inserting Ai,j = (Ai − Aj ) in Eq. (2.41), condition (2.36) with α = 0 results because Ai − Ai,j = Aj . However, the given description with respect to a variation of i and j allows for the derivation of the following theorem that ensures the convergence to a defined stability margin with respect to a bounded input in case of an estimation error of the premise variable. Theorem 2. The observer with unmeasurable states as premise variables and a closed-loop error dynamic of Eq. (2.40) is ISS and the occurring error due to the bounded input x is confined to a stability margin defined by , if a positive definite matrix P = PT  0 for V(e) = eT Pe and Nr matrices Nj are determined for i, j = 1, . . . , Nr and a given matrix Q = QT  0 fulfilling

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(Ai − Ai,j )T P + P(Ai − Ai,j ) − Nj C − CT Nj +  2 Q

ATi,j P

 P Ai,j ≺ 0. −Q (2.43)

Proof. The resulting error from the bounded inputs is confined to a stability margin around the origin depending on x, u, and the resulting closed-loop dynamic of the observer in Eq. (2.40). However, a margin of stability can also be defined beforehand such that if the LMIs are solved, the desired margin around the origin is verified. So consider the case whenever x2 ≤ e2

(or

u2 ≤ e2 )

(2.44)

holds in operation, which by employing in the estimate of the Lyapunov function derivative implies that the observer error e decays to Eq. (2.44) as long as the condition holds; see also the margin given by Eq. (2.11). Because it can be addressed by the ISS property, neglecting the term introduced by u in Eq. (2.40) does not violate the stability conditions. The derivative of a quadratic Lyapunov function with respect to the bounded input x is given by  Nr Nr    ˙ hi (z) hj (ˆz) eT (Ai − Ai,j )T P + P(Ai − Ai,j ) V(e) = i=1

j=1

T

−C

NTj

  T T T − Nj C e + x Ai,j Pe + e P Ai,j x .

(2.45)

An upper bound of the Lyapunov function derivative outside the stability margin imposed by Eq. (2.44) and using the completion of squares property to achieve xT ATi,j Pe + eT P Ai,j x ≤ xT Qx + eT P Ai,j Q−1 ATi,j Pe ≤  2 eT Qe + eT P Ai,j Q−1 ATi,j Pe is denoted as ˙ V(e)
0 fulfilling Eq. (2.9). For the bounded input δ := v)β0,j − j=1 hj (ˆ N r i=1 hi (v)β0,i and PFi := maxPBi  the ISS property is verified. The i

¯ which is defined stability margin around the origin is governed by λ, by the maximum real parts of the Nr closed-loop eigenvalues λi by ¯ = max(λi ) < −α. −λ/2

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TS-I

The stability of the closed-loop system (2.54) under impact of control law (2.53) is verified and the resulting closed-loop poles in S(0, r, θ ),   are located if a matrix X = XT  0 and Nr matrices Mj = kj kI,j X can be found constrained by Eq. (2.31). Additionally, if Eq. (2.31) holds, an attenuation of γ from the wind disturbance to the rotational speed of the wind turbine is verified. The LMI solver is used to minimize Γ , which results from a change of variable Γ = γ 2 . The corresponding matrices in Eq. (2.54) are defined as       a 0 b b , Bi := i , and Di := d,i . Ai := i 0 0 1 0 Assuming that the LMI constraints are found feasible, there exists some  ¯λ > 0 fulfilling Eq. (2.9). For the bounded input δ := − Nr hi (v)β0,i and i=1 PFi := maxPBi  the ISS property is verified. The stability margin around i

¯ which is defined by the maximum real parts the origin is governed by λ, ¯ = max(λi ) < −α. of the Nr closed-loop eigenvalues λi by −λ/2 Remark 3. The term, which is handled by the ISS property, differs depending on whether the TS-v or the TS-I is designed. While for the observer-based TS-v approach, the bounded input decreases as the windspeed observer converges toward the real wind speed, the TS-I always shows an offset in the states, which will not vanish. This represents the integral state that needs to emerge to some value to reject the mismatch from the unknown operating point when employing the TS-I approach.

4.3 Wind-Speed Disturbance Observer As discussed previously, the wind field acting on the rotor of the wind turbine plays a major role from a control perspective. Therefore, a disturbance observer in TS formulation will be employed to derive an estimate of the current effective wind speed, such that the membership functions of the TS-v and the feedforward term βFF accounting for the current operating point in Eq. (2.50) can be calculated. The TS model of the wind turbine (2.49) is augmented by a state that represents the effective wind speed. For augmenting the system, the information from the disturbance distribution matrix bd,i is used. The wind dynamics are considered as a first-order transfer function v˙ = − τ1 v as suggested in Ref. [29], where the time constant is defined as τ = 4 s. The augmented TS model of the wind turbine is given by             Nr ai bd,i ω bd,i bi ω˙

β − v0,i , (2.55) hi (v) + x˙ = = 0 0 v v˙ 0 − τ1 i=1

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where the term −bd,i v0,i accounts for bd,i characterizing the disturbance distribution around the current operating point, that is, bd,i v = bd,i v − bd,i v0,i . The output is the deviation of the rotational speed from the desired value ω, such that C = [1 0] is obtained. As a result, by defining a TS observer for the augmented system (2.32) with the matrices Ai , Bi and vectors mi       ai bd,i bi bd,i and B m v0,i Ai := := = − i i 0 0 − τ1 0 and defining an observer similar to Eq. (2.32) x˙ˆ =

Nr 

  hj (ˆv = xˆ 2 ) Aj xˆ + Bj u + mj + Lj (Cx − Cˆx)

(2.56)

j=1

yields the dynamic of the observation error e = x − xˆ as e˙ =

Nr  j=1

hj (ˆv)

Nr 

  hi (v) (Ai − Lj C − Ai,j )e + Ai,j x + Bi,j u + mi,j , (2.57)

i=1

where mi,j = mi − mj is bounded and consequently can be addressed by the ISS property; see Section 3.1.1 with δ := mi,j . As discussed in the derivation of the TS observer with unmeasurable states as variables in Section 3.3, the convergence of the observer to a stability margin from determining  can be verified. Let us define the maximum state xmax = [ ω v]T = [1.5 rad/s 35 m/s]T . Condition (2.44) allows us to derive an error and therefore , for which the Lyapunov functions derivative is guaranteed to be negative definite if a feasible solution to the LMIs in Eq. (2.43) is found. Therefore, we define the maximum resulting error (only with regard to the offset stemming from the states) as emax = [0.5 rad/s 0.5 m/s]T . From these conditions, an  ≈ 42 results, which is used in the observer synthesis. For all observers discussed here, a design matrix of Q = 0.0001 · I is chosen in Eq. (2.43).

4.4 Simulation Studies The simulation studies were conducted using the FAST code [27] and employing the presented controllers in the full load region along the combination with other controllers for the adjacent operating ranges, for example, the quadratic control law Tg ∝ ω2 . For a detailed description of the entire control scheme of the wind turbine out of full load region, see Ref. [26]. However, in the full load region, where each of the presented results is located, the dynamics of the wind turbine are governed by the proposed TS approach. As suggested in Ref. [26], a single-pole filter on the generator speed measurement with a corner frequency of 0.25 Hz is used

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TABLE 2.1 Overview of the LMI Design Parameters Design notation

Type

LMIs

r

D

α

Attained γ

Slow

TS-I



0.8

0.7

0

0.1405

Nominal

TS-I

Eq. (2.31) w.r.t. Eq. (2.59) and Eq. (2.25) w.r.t. Eq. (2.60)

1

0.7

0

0.0907

Fast

TS-I



1.2

0.7

0

0.0651

Slow

TS-v



0.8

0.7

0

0.0505

Nominal

TS-v

Eq. (2.31) w.r.t. Eq. (2.59) and Eq. (2.25) w.r.t. Eq. (2.60)

1

0.7

0

0.0403

Fast

TS-v



1.2

0.7

0

0.0335

Slow

TS-obs



0.8

0.7

0.5

−

Nominal

TS-obs

Eq. (2.43) w.r.t. Eq. (2.58) and Eq. (2.37) w.r.t. i=j

1

0.7

0.7

−

Fast

TS-obs



1.2

0.7

0.8

−

Notes: TS-I: TS integral feedback control (2.53); TS-obs and TS-v: TS wind-speed disturbance observer and corresponding feedback (2.50). Same as for the corresponding nominal case.

before processing the measurement signal of the rotational speed to the controller. An overview of the simulation environment and the resulting signal flow for the two presented control schemes is depicted in Fig. 2.4. To illustrate and compare the two approaches (TS-I and TS-v&obs), a variation of the design parameters is conducted. The employed parameters for the syntheses are given in Table 2.1. For improved readability, the different parameters and dynamic feedback loops are denoted as slow, nominal, or fast depending on the design parameters forcing the closedloop poles further into the left half of the complex plane. 4.4.1 LMI Implementation With Respect to Wind Turbine Characteristics The variation of submodels indicated by a variation of i and j in the LMIs depends on the purpose of the design and an overview is given in Table 2.1. The TS-obs synthesis is conducted based on the LMI criterion emerging from the unmeasurable premise variable (2.43) with respect to (w.r.t.) a

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variation of i and j based on Eq. (2.58). It accounts for every combination of model and feedback gains as proposed in deriving the LMI to ensure stability despite a mismatch in the estimation of the wind speed. Additionally, LMIs (2.37) for i = 1, . . . , Nr , j = i in Table 2.1 constrain the solution to a desired LMI region at instances where the estimation error is close to 0. This accounts for a shaping of the dynamic in the proper operational point, that is, when the estimated wind speed is close to the real wind speed. The control design for both the TS-v and TS-I is based on the same synthesis, although as discussed, with different models and inputs to the system. The LMIs for the TS-v design given by Eq. (2.31) with respect to a variation of Eq. (2.59) account for two adjacent submodel combinations of the LMI criterion with regard to the LMI region and disturbance attenuation γ . LMI (2.25) with a decay rate α = 0 ensures stability of the six adjacent models before and after the ith submodel controller combination, and is introduced to ensure stability in case of an estimation error of the wind speed, regardless if it stems from the pitch angle measurement for the TS-I or from the wind-speed observer TS-obs. However, because the minimization of the disturbance attenuation gain yields a minimum decay rate, α is set to zero to verify stability, but we propose no further restrictions to the LMI solver. This approach verifies stability even if the information on the current operating point is inaccurate in a wider neighborhood of the real operating point. i = 1, . . . , Nr ,

j = 1, . . . , Nr ,

(2.58)

i = 1, . . . , Nr ,

j = i − 2, i − 1, i, i + 1, i + 2 if j ∈ [1, . . . , Nr ],

(2.59)

i = 1, . . . , Nr ,

j = i − 6, i − 5, . . . , i, . . . , i + 6 if j ∈ [1, . . . , Nr ]. (2.60)

The attainable disturbance attenuation gain γ as given in Table 2.1 decreases as the dynamic is getting faster and the LMI region is constrained further into the left half of the complex plane, indicated by an increasing r. The choice of the restricting radius r and the damping coefficient D is based on the description of the gain-scheduled PI controller for the employed wind turbine in Ref. [26]. The discussed gain-scheduled PI control scheme in Ref. [26] aims for a closed-loop natural frequency of ωn = 0.6 rad/ s with a damping of D = 0.7. This corresponds to a radius with $ ratio % length r = ωn2 + (ωn 1 − D2 )2 . In fact, from design with parameters in Table 2.1, the resulting closed-loop eigenvalues as shown in Fig. 2.8 are in the range of the proposed dynamic as suggested in Ref. [26], but because the damping is restricted by the LMI region in the complex plane by θ = cos−1 (D), the resulting damping ratios are well above the%desired damping ratio. Therefore, the achieved damped frequencies ωn 1 − D2

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TS-v

TS-I

0.5 0

0

− 0.5

− 0.5

−1

−1 −1

− 0.5

0

−1

− 0.5

0

−1

− 0.5

0 1

1

0.5

(l )

0.5

0

0

− 0.5

− 0.5

Optimized gains

( l )

0.5

Originial gains

1

1

−1

−1 −1

− 0.5 ( l )

0

−1

− 0.5 ( l )

0

−1

− 0.5 ( l )

0

FIG. 2.8 Eigenvalues of the derived closed-loop systems TS-I, TS-v, and TS-obs within the desired LMI region of the control synthesis in the complex plane. The eigenvalues are illustrated before and after the optimization procedure from Section 3.4 is conducted.

are below the considerable frequencies of the unmodeled wind turbine components given in Ref. [26]. 4.4.2 Application of the Gain Optimization Procedure The resulting eigenvalues of the nominal closed-loop TS models before and after the optimization are illustrated in Fig. 2.8 with respect to Eq. (2.59) for the TS-v and the TS-I controllers. The closed-loop poles of the disturbance observer TS-obs are illustrated with respect to a variation of i = 1, . . . , Nr , j = i because the LMI region was only imposed to this combination in the control design. As can be seen, the resulting closedloop eigenvalues are confined to the LMI region S(α, r, θ ). The subsequent optimization aim in this design step was changed from minimizing min(Γ ), with Γ = γ 2 to the minimization of the trace of the resulting feedback gains, that is, min(trace(MTj Mj )) for the control design and min(trace(NTj Nj )) observer. This is conducted to reduce the magnitude of the gains, such that with respect to the LMI region and the attained disturbance attenuation gains γ from the initial step, the feedback gains are as small as possible. This reduces the necessary actuator activity with respect to the dynamic constraints of the actuator, and the noise sensitivity in the implementation of the observer; see Ref. [20] for details. In Fig. 2.8, the optimization yields the biggest location change in the complex plane for the TS-obs, as the closed-loop eigenvalues are pushed toward the

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v

20

18 v vˆ

16 20

30

40

50

60

70

80 90 t in s

100

110

120

130

140

150

18

b

16 14 TS- I TS-v & obs

12 20

30

40

50

60

70

80 90 t in s

100

110

120

130

140

150

w/w d

1.05

1

0.95 20

TS- I TS-v & obs 30

40

50

60

70

80 90 t in s

100

110

120

130

140

150

FIG. 2.9 Step response of the nonlinear wind turbine simulated by FAST code, and governed by the presented control approaches TS-I and TS-v&obs in the full load region.

minimum decay rate α. For the control syntheses of TS-v and TS-I, the change from the optimization is rather small, however, resulting in a more even distribution of the eigenvalues along the range of attained values. 4.4.3 Step Response To assess the dynamical properties of the resulting closed-loop dynamic, a synthetic wind-speed signal is used. Due to the inertia of a natural incoming wind field, a wind signal such as that presented in the upper plot of Fig. 2.9 will not occur. Further, a single vector signal is used to represent the incoming wind field, such that no three-dimensional wind field is represented that acts distributed at the different rotor blade positions. However, this wind signal allows for a direct comparison with

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the estimated effective wind speed vˆ by the observer and the resulting feedback signals for the two evaluated control concepts TS-I and TS-v&obs. As can be seen in Fig. 2.9, the resulting wind-speed estimation from the observer based on the nominal design converges to the real wind speed after the step within ≈ 10 s. As a result, the offset in the pitch angle accounting for the operating point is adjusted through the feedforward term. In combination with the additional feedback by the nominal TS-v, the pitch signal in the middle plot of Fig. 2.9 is formed. Especially for steps to a higher wind speed, the pitch signal of the observer-based control approach decays faster toward its stationary value compared to the pitch signal from the nominal TS-I. This results in reduced over speeding of the rotor, as can be seen in the lower plot of Fig. 2.9. 4.4.4 Analysis of the Pitch Signal Components To directly assess the influence of the proposed observer, in Fig. 2.10 the components of the pitch angle signal in the upper plot of Fig. 2.10 are separated depending on their origin. As discussed, the signal is the sum of a component respecting the current operating point through the feedforward term and a direct feedback depending on the current deviation from the desired rotational speed ω. The feedback is depicted in the middle plot of Fig. 2.10, whereas the observer-based feedforward component βFF in Eq. (2.50) can be seen in the lower plot along with the resulting feedback from the integrator state in Eq. (2.53) for the TS-I. As can be seen, the TS-I signal resulting from the integral term in the lower plot of Fig. 2.10 declines before increasing the pitch angle to adjust the current operating point. This stems from a reduction of the feedback Nr v(β))kI,j due to an increase of the pitch angle from the gains j=1 hj (ˆ immediate feedback term in the middle plot of Fig. 2.10. This follows from N r v(β))kI,j decreasing for an increasing pitch angle, which is based j=1 hj (ˆ on the property of wind turbines to possess a higher sensitivity to changes in the pitch angle as the wind evolves toward higher speeds; see also the equilibria of the system in Fig. 2.3. Therefore, in the first seconds after the step change of the wind speed, the TS-I controller performs oppositely to the desired behavior until due to the integration of the error, the integrator state governs the increasing pitch angle. Contrarily, the observer-based approach TS-v&obs adjusts the pitch angle by estimating the current wind speed into the desired direction from the beginning. The feedback components shown in the middle plot of Fig. 2.10 are only active in the transient region, thus when a deviation from the desired rotational speed occurs. However, the term βFB produced by the TS-v remains below the counterpart in the TS-I, TS-v&obs due to the superior adjustment of the current operating from the deployment of the windspeed disturbance observer. This results in a faster convergence of the pitch

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FIG. 2.10 Detail view of the pitch angle signal β of the presented approaches TS-I and TSv&obs, which is separated into the steady state and transient component of the pitch signal to assess the effect of the derived disturbance observer.

angle to the desired value, as seen in the upper plot of Fig. 2.10 and to a smoother transition of the rotational speed into the desired value, as also illustrated in Fig. 2.9. 4.4.5 Wind Turbine Closed-Loop Dynamic Variation As mentioned in the introduction to the wind turbine application Section 4, the TS wind-speed disturbance observer allows us to decouple the calculation of the current operating point from the feedback in the estimated operating point. The TS-I approach combines the latter two, and thus the controller formulation incorporates both aspects. The design of the observer TS-obs allows us to define the dynamics of the transients

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from one operating point to the other independently from the closed-loop dynamic within the current operating point. From a wind turbine life prediction perspective, a lot of effort is devoted to the simulation of fatigue and extreme loads. Depending on the investigated scenario, the control in these analyses has different scopes, where in extreme events the maximum load from fast changes in the wind speed are of prior interest, in the fatigue case the number of cycles and corresponding magnitudes are considered from a load spectrum. Therefore, by shaping the dynamics of the two effects individually, a decoupling from the fatigue and extreme load might be attained to a certain degree. This can also be assessed by analyzing Fig. 2.11, where the slow and fast designed feedback loops are compared to the nominal design case. As can be seen in Table 2.1, the design parameters for the TS-I, TS-v, and TS-obs are equivalently adjusted to provide faster or slower dynamic responses. From Fig. 2.11, it is apparent that the small variation of the targeted dynamic in the design process is more directly adjustable for the TS-v&obs combination, as both the slow and fast variant significantly affect the resulting dynamic compared to the nominal case. The variation for the TS-I results in a faster decay compared to the nominal case, as can be seen in the upper plot of Fig. 2.11. However, imposing a slower dynamic within this small variation of desired LMI region results in

FIG. 2.11 Detail view of the step response for different LMI region constraints (slow, nominal, fast) of the presented approaches TS-I and TS-v&obs, where the resulting dynamic by variation of design parameters in the LMI synthesis is studied.

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a response of the closed-loop system with no significant difference from the nominal design. This indicates the additional complexity by the coupling of the adjustment of operating point and feedback loop through the integral state in a coupled design. As a result, the degree of freedom in the design is increased by employing a wind-speed disturbance observer compared to the integral-based approach TS-I.

5 CONCLUSION Within this contribution, the TS modeling technique is discussed as a framework for the control and observer design of nonlinear systems, where stability constraints and feedback gain syntheses are formulated as LMIs. By adding further constraints to the LMI stability problem, a desired performance can be imposed on the closed-loop dynamic. This facilitates the shaping of the resulting closed-loop system toward a desired dynamical response for nonlinear systems. Further, the formulation of a TS observer with unmeasurable premise variables enables the estimation of a wider class of systems compared to TS observers with the assumption of a measurable premise variable. The derived stability conditions of TS observers with unmeasurable premise variables differ from the existing approaches, such that the ISS property is used to account for the unmeasurable premise variables, and a stability margin is verified even in a mismatch from the estimation of the premise variable. This results in a satisfying performance for the developed windspeed disturbance observer of the wind turbine application. The estimation of the wind speed by the presented observer in TS form employs no additional sensor components in the presented wind turbine application compared to the state-of-the-art control concept. Therefore, compared to other approaches involving additional measurements, no hardware costs, or chances of hardware failure necessitating costly maintenance are introduced to the application. The approach can be interpreted as using the rotor of the wind turbine as the wind-speed sensor, and the model-based observer design is used to convert the information from the rotational dynamics into an estimated wind-speed signal. With regard to the different time scales governing the dynamics of the wind turbine, the information on the current wind speed is delayed by the inertia of the wind turbine, but by appropriate LMI-based observer design, a desired dynamic to the estimation error and the resulting feedforward term can be imposed. The resulting estimate of the wind speed represents an effective measure that yields the excitation of the rotational dynamics, and therefore the stochastic three-dimensional wind field cannot be recovered in detail. Anyhow, for the presented approach this is not necessary, as the aim is to calculate a collective signal for the pitch angle of all blades simultaneously.

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The obtained results from the simulation studies point toward the increased flexibility in the control design due the observer-based concept. By formulating the disturbance observer, the adjustment of the feedback terms to the current operating point is separated from the actual feedback signal. Stability and closed-loop dynamic constraints can be formulated in terms of LMIs in the design synthesis. As a result, a desired closed-loop dynamic of the combined TS-v&obs can be imposed, such that the effects of different time scales, that is, the rotational dynamics of the wind turbine and the wind-speed change can be shaped individually with respect to the limitations arising from the technical implementation such as conflicting natural frequencies of the application components.

References [1] J.G. Van Antwerp, R.D. Braatz, A tutorial on linear and bilinear matrix inequalities, J. Process Control 10 (4) (2000) 363–385. [2] G.A.M. van Kuik, J. Peinke, R. Nijssen, D. Lekou, J. Mann, J.N. Sørensen, C. Ferreira, J.W. van Wingerden, D. Schlipf, P. Gebraad, H. Polinder, A. Abrahamsen, G.J.W. van Bussel, J.D. Sørensen, P. Tavner, C.L. Bottasso, M. Muskulus, D. Matha, H.J. Lindeboom, S. Degraer, O. Kramer, S. Lehnhoff, M. Sonnenschein, P.E. Sørensen, R.W. Künneke, P.E. Morthorst, K. Skytte, Long-term research challenges in wind energy—a research agenda by the European Academy of Wind Energy, Wind Energy Sci. 1 (1) (2016) 1–39, https://doi.org/10.5194/wes-1-1-2016. [3] J.G. Njiri, D. Söfker, State-of-the-art in wind turbine control: trends and challenges, Renew. Sust. Energ. Rev. 60 (2016) 377–393, https://doi.org/10.1016/j.rser.2016.01.110. [4] K.Z. Ostergaard, P. Brath, J. Stoustrup, Estimation of effective wind speed, J. Phys. Conf. Ser. 75 (1) (2007) 012082. [5] M.N. Soltani, T. Knudsen, M. Svenstrup, R. Wisniewski, P. Brath, R. Ortega, K. Johnson, Estimation of rotor effective wind speed: a comparison, IEEE Trans. Control Syst. Technol. 21 (4) (2013) 1155–1167, https://doi.org/10.1109/TCST.2013.2260751. [6] D. Jena, S. Rajendran, A review of estimation of effective wind speed based control of wind turbines, Renew. Sust. Energ. Rev. 43 (2015) 1046–1062, https://doi.org/ 10.1016/j.rser.2014.11.088. [7] E. Gauterin, P. Kammerer, M. Kühn, H. Schulte, Effective wind speed estimation: comparison between Kalman filter and Takagi-Sugeno observer techniques, ISA Trans. 62 (2016) 60–72, https://doi.org/10.1016/j.isatra.2015.11.016. [8] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern. SMC-15 (1) (1985) 116–132, https://doi.org/10.1109/TSMC.1985.6313399. [9] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, John Wiley & Sons, Inc., New York, NY, 2001. [10] Z. Lendek, T.M. Guerra, R. Babuska, B.D. Schutter, Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models, Springer-Verlag, Berlin, Heidelberg, 2010. [11] F. Pöschke, J. Fortmann, H. Schulte, Nonlinear wind turbine controller for variable power generation in full load region, in: 2017 American Control Conference (ACC), 2017, pp. 1395–1400, https://doi.org/10.23919/ACC.2017.7963148. [12] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, 1994, https://doi.org/10.1137/1.9781611970777.

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[13] J.F. Sturm, Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones, Optim. Methods Softw. 11 (1–4) (1999) 625–653, https://doi.org/ 10.1080/10556789908805766. [14] E.D. Sontag, On the input-to-state stability property, Eur. J. Control 1 (1) (1995) 24–36, https://doi.org/10.1016/S0947-3580(95)70005-X. [15] M. Chilali, P. Gahinet, Hinf design with pole placement constraints: an LMI approach, IEEE Trans. Autom. Control 41 (3) (1996) 358–367, https://doi.org/10.1109/9.486637. [16] D. Rotondo, V. Puig, F. Nejjari, M. Witczak, Automated generation and comparison of Takagi–Sugeno and polytopic quasi-LPV models, Fuzzy Set. Syst. 277 (2015) 44–64, https://doi.org/10.1016/j.fss.2015.02.002. [17] H. Schulte, On nonlinear observers in Takagi-Sugeno form based on fixed and variable structure, in: 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2016, pp. 2364–2371, https://doi.org/10.1109/FUZZ-IEEE.2016.7737989. [18] P. Bergsten, Observers and Controllers for Takagi-Sugeno Fuzzy Systems (Ph.D. thesis), Orebro University, Schweden, 2001. [19] D. Ichalal, B. Marx, J. Ragot, D. Maquin, Brief paper: state estimation of Takagi-Sugeno systems with unmeasurable premise variables, IET Control Theory Appl. 4 (5) (2010) 897–908, https://doi.org/10.1049/iet-cta.2009.0054. [20] F. Pöschke, H. Schulte, Optimization of Takagi-Sugeno observers with application to fault estimation, in: Proceedings of the 3rd IFAC Conference on Embedded Systems, Computational Intelligence and Telematics in Control (CESCIT 2018), 2018. [21] T. Burton, N. Jenkins, D. Sharpe, E. Bossanyi, The controller, in: Wind Energy Handbook, John Wiley & Sons, Ltd, New York, NY, 2011, pp. 475–523, https://doi.org/ 10.1002/9781119992714.ch8. [22] T. Burton, N. Jenkins, D. Sharpe, E. Bossanyi, Design loads for horizontal axis wind turbines, in: Wind Energy Handbook, John Wiley & Sons, Ltd, New York, NY, 2011, pp. 193–323, https://doi.org/10.1002/9781119992714.ch5. [23] P. Towers, B.L. Jones, Real-time wind field reconstruction from LiDAR measurements using a dynamic wind model and state estimation, Wind Energy 19 (1) (2016) 133–150, https://doi.org/10.1002/we.1824. [24] T. Pedersen, G. Demurtas, F. Zahle, Calibration of a spinner anemometer for yaw misalignment measurements, Wind Energy 18 (11) (2015) 1933–1952, https://doi.org/ 10.1002/we.1798. [25] F. Dunne, L.Y. Pao, Optimal blade pitch control with realistic preview wind measurements, Wind Energy 19 (12) (2016) 2153–2169, https://doi.org/10.1002/we.1973. [26] J. Jonkman, S. Butterfield, W. Musial, G. Scott, Definition of a 5-MW reference wind turbine for offshore system development, Tech. Rep., National Renewable Energy Laboratory, 2009. [27] B. Jonkman, J. Jonkman, FAST v8.16.00a-bjj, National Renewable Energy Laboratory, 2016. [28] J.M. Jonkman, M.L. Buhl Jr., FAST user’s guide, Tech. Rep., National Renewable Energy Laboratory, 2005. [29] T. Ekelund, Speed control of wind turbines in the stall region, in: 1994 Proceedings of IEEE International Conference on Control and Applications, vol. 1, 1994, pp. 227–232, https://doi.org/10.1109/CCA.1994.381194.

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C H A P T E R

3 Stochastic Control Approach for Distributed Generation Units Interacting on Graphs Nezar Alyazidi*, Mohammed Abouheaf † , Magdi S. Mahmoud*, Adel Sharaf ‡ *Systems Engineering, College of Computer Sciences and Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia † School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada ‡ SHARAF Energy Systems, Inc., Fredericton, NB, Canada

1 INTRODUCTION Smartgrids are shown to be effective and reliable sources of energy compared to the traditional power plants operated using fossil fuels [1, 2]. The smartgrids are combinations of active loads and distributed generation units such as fuel cells, wind schemes, tidal energy, and photovoltaic cells. The active and reactive power support for the power system networks can be achieved using the storage units. The renewable generation units work in two main operation modes. In the stand-alone mode, the generation unit is responsible for providing the local area’s frequency and voltage support while in the grid-connected mode, the control tasks have local and global goals [2]. In this chapter, an online distributed Kalman filter is introduced to estimate the dynamics of the coupled generation units, where the interactions are restricted by a graph topology. This is done using the optimal control theory framework and the pinning control ideas. Applying the multiagent control approaches to networks of distributed generation units is a challenging task, due to the multidisciplinary nature of these problems [3]. The multiagent control approaches are explored for large networks in Ref. [4]. The practical side of the multiagent control

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schemes for power system networks is shown in Ref. [1]. A cooperative scheme with a secondary-level control structure has been used for a network of isolated smartgrids in Ref. [1]. Distributed control approaches for systems with sparse communication topologies are introduced in Ref. [4]. Multiagent control techniques, taking into consideration the information flow between the agents, are successfully deployed in robotic systems, unmanned air vehicles, and sensor networks [5–14]. The cooperative control problems are classified, in terms of the objectives, into leader-tracking or team-consensus problems [5–14]. In the consensus control problem, the team reaches one common goal while in the tracking problem, the team follows the leader [9, 15]. The dynamic programming framework is broadly applied to solve the optimal control problems. Unfortunately, it cannot be used to solve the complex and nonlinear problems due to the associated required massive computations in the action and state spaces [16]. A suboptimal solution for the control problem is found using the principles of dynamic programming [17]. Approximate dynamic programming (ADP) approaches are introduced to handle the dynamic programming’s main complexities. They use neurodynamic programming structures to solve Bellman equations of the underlying control problems [17, 18]. The approximate dynamic programming’s classes optimize the performance index or the utility function in order to find the optimal policies [19–22]. These are classified based on the structures of the solving value functions and the associated policies [23]. Reinforcement learning (RL) approaches are temporal difference techniques that are used to implement the ADP solutions [6, 24–27]. These approaches employ two-step policy or value iteration processes [28, 29]. The value iteration process does not start with the initial admissible policy [18, 30] while in the policy iteration (PI), the initial admissible policies are needed in order to guarantee stability [31, 32]. The optimal strategies are found by optimizing online forms of the temporal difference equations in Ref. [25]. RL has been applied to solve the multiagent control problems in Refs. [6–8]. RL uses dynamic learning environments where the optimal actions are obtained through consecutive learning for the optimal strategies [8, 31]. An approach based on RL is developed to solve the game with finite-state systems in Refs. [11, 25]. The differential games are solved using online multiagent RL techniques in Refs. [3, 12, 33]. The adaptive critic techniques work as reliable implementation platforms for many RL approaches. The critic network approximates and evaluates the different solving value function structures while the actor network approximates the optimal strategy that will minimize the cost function [23, 34]. Differential graphical games are used to develop online control schemes for the distributed systems in Ref. [35]. The dynamic games combine game theory with the optimal control theory [35, 36]. A solution of the cost-togo objective function of a multiagent system is equivalent to solving the

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underlying Hamiltonian-Jacobi-Bellman (HJB) equations [36]. The solution for the underlying Hamilton-Jacobi equations yields a Nash equilibrium outcome for the game [23, 24, 28, 33, 36]. Differential graphical games are solved using online adaptive learning techniques in Ref. [6]. The HamiltonJacobi-Bellman equations are solved simultaneously using online adaptive integral RL techniques in Refs. [37, 38]. Two communicating constrained linear systems are controlled using a game theoretic approach with a distributed model predictive control approach in Ref. [39]. Several schemes have been proposed to overcome the disturbance problem in a network of the multiagent systems. The H∞ -control is solved using a model-free ADP scheme in disturbed environments [40]. The robust process of the parameters and dynamics estimations have made Kalman filtering a useful tool in the uncertain dynamical environments. A Kalman filter is used to estimate the states of a system in a noisy communication network in Ref. [41]. The nonlinear state-estimation problems are solved using the extended Kalman filter in Ref. [42]. A combined Kalman filter and model predictive control scheme are employed to address the issues of faulttolerant control in distributed systems [43]. The contributions in this chapter are fourfold: ◦ A real-time RL approach that employs Kalman filtering is introduced to tackle the challenging control problem of distributed generation units interacting on graphs under disturbances. This approach solves the optimal control problem in a distributed fashion using the local neighborhood information and it does not need the drift dynamics of the generation units. ◦ Novel temporal difference equations are developed that are based on a set of coupled Bellman optimality equations that are introduced herein. ◦ A novel distributed PI process in the presence of disturbances is employed to achieve coordination objectives among a set of generation units. The convergence analysis of the PI process is introduced. ◦ The distributed Kalman filter approach is implemented using one layer of neural network approximations for each generation unit. This RL solution does not consider inverse-based calculations. The chapter is formed as follows. Section 2 shows the mathematical setup of the synchronization problem. Section 3 introduces the cooperative control problem associated with the Kalman filtering structure. Section 4 proposes an online adaptive learning approach to solve the distributed control problem using a Kalman filter. The proof of convergence of the online adaptive technique is provided. Section 5 shows the critic neural network implementation of the graphical game’s solution (the distributed control problem). Section 6 explains the dynamic model of the distributed generation unit. Section 7 highlights the simulation outcomes.

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2 DYNAMIC GRAPHICAL GAMES In the sequel, the objectives of the multiagent problem are defined where the generation systems are exchanging information through a communication graph topology [6]. The units select their optimal control policies in order to follow the leader’s dynamics.

2.1 Graph Notations ˆ prescribes the information flow between the units in A directed graph G two directions. The connectivity matrix is Eˆ = [eij ], where eij , ∀i, j represent the connection weights between any two units i, j. The connectivity weight eij has a positive value if there is a link between the units i,j [13].

2.2 Problem Formulation The dynamical equation for each unit i in terms of the local stochastic disturbance is given by xi(k+1) = Ai xik + Bi uik + Bdi wik , yik = Ci xik + vik ,

(3.1) (3.2)

n

p

describes the states of unit i, yik ∈ is a vector where the vector xik ∈ of the output signals of unit i, the vector uik ∈ m describes the control inputs of unit i, and wik ∈ q , vik ∈ p are the input and measurement disturbances, respectively, and they are independent zero-mean secondorder random vectors [41, 44]. The leader follows the following dynamics xt(k+1) = Axtk ,

(3.3)

n

is a vector of the states of the leader. where xtk ∈ The local tracking error protocol is given by [45]  eij (xjk − xik ) + gi (xtk − xik ), εik =

(3.4)

j∈Ni

where Ni describes the systems in the neighborhood of system i. The following expression describes the tracking error dynamics for each unit i [45]   eij Bj ujk + eij Bdj wjk εi(k+1) = Aεik − (dj + gi )Bi uik + j∈Ni

j∈Ni

− (di + gi )Bdi wik .

(3.5)

The goal is to select the control strategy such that the error systems (3.5) are asymptotically stable. I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

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The performance for each unit i is measured by ⎡ ⎛ ⎞⎤ N−1   1 ⎝εT Qεik + uT Ruik + uTjk Rij ujk ⎠⎦ , Ji (εik , uik , u−ik ) = E ⎣ ik ik 2 K=0

(3.6)

j∈Ni

where Rii > 0, Rij > 0, Qii ≥ 0 are the weighting matrices. The utility function for each unit i is given by ⎛ ⎞  1⎝ T uTjk Rij ujk ⎠ . εik Qεik + uTik Ruik + Ui (εik , uik , u−ik ) = 2 j∈Ni

The objective is to find the conditions that will minimize Eq. (3.6) in order to obtain the optimal value Ji∗ , ∀i so that  N−1 

∗ ∗ Ji (εik , uik , u−ik ) = min E (3.7) Ui εik , uik , u−ik , uik

K=0

where the neighbors’ optimal strategies are denoted by u∗−ik for each unit −i ∈ Ni .

3 KALMAN FILTER FOR DYNAMIC GRAPHICAL GAMES In the sequel, an online optimal least-square estimation strategy using a Kalman filter is found to solve the synchronization problem formulated using the framework of the graphical games. The expected value of the dynamics of each unit i is given by xˆ ik = E[xik ],

(3.8)

and the state-estimation error for each unit i is represented by ξik = xik − xˆ ik . Then, the covariance of the error for each unit i is   Ψik = E ξik ξikT   T  . = E xik − xˆ ik xik − xˆ ik

(3.9)

(3.10)

The error’s covariance expression follows     E wTik wik = Λiw ; E vTik vik = Λiv , which represents the estimated states so that xˆ i(k+1|k) = Ai xˆ (k|k) + Bi uik .

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(3.11)

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Therefore, using Eq. (3.9), the state-estimation error dynamics are given by ξi(k+1|k) = xi(k+1) − xˆ i(k+1) , = Ai ξik + Bdi wik .

(3.12)

The error’s covariance expression follows Ψi(k+1|k) = [Ai ξik + Bdi wik ][Ai ξik + Bdi wik ]T , = Ai Ψik Ai + Λiw .

(3.13)

The Kalman observer equation is given by xˆ i(k+1|k) = Ai xˆ ik + Bi uik + Kik (yik − Ci xˆ ik ),

(3.14)

n×m

where Kik ∈ is the Kalman gain matrix. The estimated dynamics for each unit i (3.11) are weighted by matrix Γi [41]. Thus, Eq. (3.14) yields xˆ i(k+1|k+1) = Γi(k+1) xˆ i(k+1|k) + Ki(k+1) Ci xi(k+1|k) + Ki(k+1) vi(k+1|k) ,

(3.15)

where the weighted matrix Γi(k+1) is given by [41] Γi(k+1) = I − Ki(k+1) Ci ,

(3.16)

xˆ i(k+1|k) = Γi(k+1) xˆ i(k+1|k) + Ki(k+1) Ci xˆ i(k+1|k) − Ki(k+1) vi(k+1) .

(3.17)

where Γik ∈ n×n , then

The update equation of the covariance is given by T T Ψi(k+1|k+1) = Γi(k+1) Ψi(k+1|k) Γi(k+1) + Ki(k+1) Λiv Ki(k+1) .

(3.18)

Minimizing the covariance (3.18) yields   L = min trace Ψi(k+1|k+1) Ki(k+1)

  T T + Ki(k+1) Λiv Ki(k+1) = min trace Γi(k+1) Ψi(k+1|k) Γi(k+1) . Ki(k+1)

(3.19)

Consequently, ∂L = −2Γi(k+1) Ψi(k+1|k) CTi + 2Ki(k+1) Λiv = 0. ∂Ki(k+1) The Kalman gain is given by  −1 Ki(k+1) = Ψi(k+1|k) CTi Ci Ψi(k+1|k) CTi + Λiv .

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(3.20)

KALMAN FILTER FOR DYNAMIC GRAPHICAL GAMES

83

The estimation of the leader’s dynamics (3.3) can be done in a similar fashion. The estimation of synchronization error for each unit i is calculated by  eij (ˆxjk − xˆ ik ) + gi (ˆx0k − xˆ ik ). (3.21) εˆ ik = j∈Ni

Thus, the estimation dynamics are given by εˆ i(k+1) = Aˆεik − (dj + gi )Bi uik + +





eij Bj ujk

j∈Ni

ˆ jk − (gi + di )Bdi w ˆ ik . eij Bdj w

(3.22)

j∈Ni

Remark 1. This dynamical equation reflects the coupled nature of the problem, and shows how the dynamics of the existing disturbances are affected due to the graph topology.

3.1 Bellman Equation Formulation The solving value structure for unit i can be expressed such that Vi (ˆεik ) =

∞ 

Ui (ˆεik , πik , π−ik ),

(3.23)

k=0

where π−ik and πik represent stabilizing strategies for the units in the neighborhood of unit i and those of unit i, respectively. A coupled form of Bellman equation for each system i [6] is given by ⎛ ⎞  1 T Qii εˆ ik + πikT Rii πik + πikT Rij πik ⎠ Viπ (ˆεik ) = ⎝εˆ ik 2 j∈Ni

+ Viπ (ˆεi(k+1) ),

(3.24)

where Viπ (0) = 0. The optimal policies are obtained by applying the Bellman optimality principles such that  ∞ 

o (3.25) Ui εˆ il , uil , π−il . Vi (ˆεik ) = minVi (ˆεik ) = min ui

ui

l=k

Thus, the optimal control policy is given by uoik = Mi Vi (ˆεi(k+1) ),

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(3.26)

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o ¯ T where Mi = R−1 εik ) = ii ([· · · (gi + di ) · · · − eji · · · ] ⊗ Bi )Vi (εˆ i(k+1) ) and Vi (ˆ

∂Vi (ˆεik ) . The coupled Bellman optimality equation for each unit i is given by ∂ εˆ ik

⎛ ⎞  1 T o o⎠ Qii εˆ ik + uoT uoT Vio (ˆεik ) = ⎝εˆ ik ik Rii uik + jk Rij ujk 2 j∈Ni

+ Vio (ˆεi(k+1) ).

(3.27)

4 ONLINE ADAPTIVE RL SOLUTION An online value iteration-based control approach is developed for the autonomous smartgrids in Refs. [46, 47]. Herein, an RL process that employs PI is developed for a system of interacting distributed generation units in order to achieve cooperative control objectives. This algorithm does not need to know the full dynamics of the distributed generation units. The PI algorithm solves the coupled Bellman optimality (3.27). The proposed PI algorithm is structured as follows

Algorithm 1 ALGORITHM

ADAPTIVE

LEARNING

CONTROL

˜ 0 (ˆεik ), ∀i. Step 1: Use stabilizing strategies u0ik , ∀i, and initialize V i Step 2: Kalman filter estimation (ˆεik , uik , Ψik ). Step 2.1: Dynamics estimation xˆ i(k+1|k) = Ai xˆ (k|k) + Bi uik . Calculate the error ei(k+1|k) = Ai ξik + Bdi wik . Calculate the covariance Ψi(k+1|k) = Ai Ψik Ai + Λiw . Step 2.2: Correct the Kalman filter gain  −1 T Ki(k+1) = Ψi(k+1|k) CT . i Ci Ψi(k+1|k) Ci + Sv Perform Kalman filter observer calculations as follows

xˆ i(k+1) = Ai xˆ ik + Bi uik + Kik yik − Ci xˆ ik , T T + Ki(k+1) Sv Ki(k+1) . Ψi(k+1|k+1) = Γi(k+1) Ψi(k+1|k) Γi(k+1)

˜ l (. . . ), ∀i Step 3: Solve for V i

 l l    u ,u ˜ l (ˆεik ) = Ui εˆ ik , ul , ul ˜ l εˆ ik −ik , V + V i ik −ik i i(k+1)

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(3.28)

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ONLINE ADAPTIVE RL SOLUTION

where l is the outer process calculation steps. Step 4: Strategy update l ˜0 = R−1 ([· · · (gi + di ) · · · − eji · · · ] ⊗ BT ul+1 i )Vi (ˆεi(k+1) ) , ik ii    ˜ l+1 ˜ l  , ∀i, End. − V Step 5: On convergence of V i i

∀i.

(3.29)

The PI process does not require knowing and using the drift dynamics of the distributed generation units; it only uses the control input matrices. Remark 2. PI Algorithm 1 introduces an online distributed Kalman filter that is employed to solve the coordination problem of the distributed generation units. PI is known with its fast policy convergence characteristics; however, its implementation requires structured algorithms based on leastsquare approaches. In the next section, an easy approach based on neural network approximations is introduced to tackle this difficulty. The following development shows the conditions at which Algorithm 1 converges, when it is performed simultaneously by all the generation units. Theorem 1. Assume that all the initial values of the control signals u0ik , ∀i are stabilizing, and the highest singular values of (R−1 jj Rij ), ∀i, j are selected small enough. Then, 1. The optimal control values uoik , ∀i are stabilizing strategies. 2. The PI algorithm results in a set of value function sequences so that ˜ l+1 ≤ V ˜ l , ∀i and these sequences converge to the ˜∗ ≤ ··· ≤ V 0 ≤ ··· ≤ V ik ik ik ∗ ˜ optimal value functions Vik , ∀i or the solutions of Eq. (3.27). Proof. 1. Bellman Eq. (3.27) yields  l l   l l ˜ l εˆ uik ,uik − V ˜ l εˆ uik ,uik < 0, V i i i(k+1) ik

∀i, l.

(3.30)

˜ l , ∀i, l are shown to be Lyapunov Therefore, the functions V i functions. Let the strategies u0i , ∀i be admissible, then the value functions l ˜ Vi , ∀i, l satisfy  l l    l+1 l  ∞   uik ,u−ik u ,u l l l l ˜ ˜ Ui εˆ i(m) , ui(m) , u−i(m) = Vi εˆ ikik −ik = Vi εˆ ik m=k

  ˜ k ul , ul+1 . + U i(m) i(m)

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(3.31)

86

3. STOCHASTIC CONTROL APPROACH FOR DISTRIBUTED GENERATION

where

∞      T   1 l l+1 l ˜ Uk ui(m) , ui(m) = Rii uli(m) − ul+1 ui(m) − ul+1 i(m) i(m) 2 m=k   l+1 l + u(l+1)T R − u u . (3.32) ii i(m) i(m) i(m)

The policies ul+1 ik , ∀i, l are given by Eq. (3.29). Therefore,  l+1 l ˜ Uk u , u and Eq. (3.31) yield i(m)

i(m)

 l l   l+1 l  uik ,u−ik u ,u l l ˜ ˜ > Vi εˆ ikik −ik . Vi εˆ ik

(3.33)

Similarly,  l+1 l    l+1 l+1  ∞   uik ,u−ik l+1 l l ˜ ˜ l εˆ uik ,u−ik = U , u , u ε ˆ = V V i im i(m) −i(m) i i εˆ ik ik m=k

  l+1 ˜ k ul + U , u −i(m) −i(m) .   ˜ k ul , ul+1 > 0 ensures The condition U −i −i  l+1 l   l+1 l+1  uik ,u−ik u ,u l l ˜ ˜ > Vi εˆ ikik −ik . Vi εˆ ik

(3.34)

(3.35)

  ˜ k ul , ul+1 > 0 is a sufficient condition to guarantee the Thus, U −i −i stability so that  T   1  (l+1)T l+1 l Rij − ujk Rij uljk − ul+1 ujk − ujk > 0. (3.36) jk 2 j∈Ni

Applying the norm properties, Eq. (3.36) yields     l l    1  ujk ,u−jk   l  −1 l  ˜ ¯ Ψ (Rij )  ujk  > (gj +dj )Ψ (Rjj , Rij ) (gj +dj )j Vj εˆ j(k+1)   2 j∈Ni j∈Ni ⎞   l l      ujk ,u−jk  l  ⎠ Bjk  , (3.37) ejo  + o Vj εˆ j(k+1)  o∈Nj

where Ψ (. . . ) and Ψ¯ (. . . ) are the  lowest and  highest singular values of

some matrix (. . . ) and uljk = uljk − ul+1 . jk Under this assumption, inequalities (3.33), (3.35) yield  l l   l+1 l   l+1 l+1  ˜ l εˆ uik ,u−ik > V ˜ l εˆ uik ,u−ik . ˜ l εˆ uik ,u−ik > V V i i i ik ik ik

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

(3.38)

87

ONLINE ADAPTIVE RL SOLUTION

The careful choice of the weighting matrices guarantees this result. Therefore, ul+1 ik , ∀i, l are admissible. The inequalities (3.30), (3.38) yield  l+1 l+1   l+1 l+1  uik ,u−ik u ,u l l ˜ ˜ − Vi εˆ ikik −ik < 0. (3.39) Vi εˆ i(k+1) 2. Using Eq. (3.28), yields  l+1 l+1   l+1 l+1    uik ,u−ik u ,u l+1 l+1 ˜ ˜ k εˆ ik , ul+1 , ul+1 = 0. (3.40) ˜ εˆ i(k+1) − Vi εˆ ikik −ik + U Vi ik −ik Inequalities (3.28), (3.39), (3.40) yield  l+1 l+1   l+1 l+1   l+1 l+1   l+1 l+1  uik ,u−ik uik ,u−ik uik ,u−ik u ,u l+1 l+1 l l ˜ ˜ ˜ ˜ − Vi εˆ ik ≤ Vi εˆ i(k+1) − Vi εˆ ikik −ik . Vi εˆ i(k+1) (3.41) Evaluating an infinite summation of Eq. (3.41) yields  l+1 l+1   l+1 l+1  ∞   uik ,u−ik u ,u l l ˜ ˜ Vi εˆ i(k+1) − Vi εˆ ikik −ik k=K˜


0 is the gain from ω to x˜ , and ν > 0 is to be determined. Usually, we use Lyapunov functions to get checkable conditions guaranteeing (Eq. 4.25). In the LMI framework, we take a quadratic Lyapunov function V(˜x) V(˜x) = x˜ T P˜x

(4.26)

such that dV (4.27) (˜x) + ˜x2Ln − μω2Ln ≤ 0 2 2 dt By analogy to the results presented in Refs. [7, 60], we obtain the following proposition, which provides sufficient LMI conditions under which the inequality (4.27) holds. Theorem 1. If there exist symmetric positive definite matrices P, Zij , i = 1, . . . , m, j = 1, . . . , ni , and matrices Yj , j = 0, . . . , s, of appropriate dimension, such that the following convex optimization problem holds: ϑ(t) 

min(μ) subject to Eq. (4.29) ⎡

 Σ1

⎢Θ ⎢ ⎢ ⎢ ⎣

⎤ 

Σm ⎥ ⎥ ⎥ ≤ 0, ⎥ ⎦

∀ ∈ {ρmin , ρmax }

(4.29)

−ΛZ

() with

Σ

 ...

(4.28)

⎡ Θ11

Θ=⎣

()

 ⎤ P 0 ⎦, −In

 Θ11 =

A (P, Y, ) ()

A (P, Y, ) = PAT0 +A0 P−Y0 B −BY 0 +

s 

In −μIn

 (4.30)

j PATj + A0 P − Yj B − BY j

j=1

(4.31) I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

!

110

4. CONTROL OF ANAEROBIC DIGESTION PROCESS

" # n  Σi = Ni1 (P, Y, Zi1 ) . . . Ni i P, Yi , Zini ⎡ T⎤ ⎡ ⎤ Hij PHi   j Ni P, Y, Zij = ⎣ 0 ⎦ + ⎣ 0 ⎦ Zij 0 0

(4.32) (4.33)

Λ = block-diag (Λ1 , . . . , Λm ) $ % 2 2 In , . . . , In , Λi = block-diag ϕ¯i1 i ϕ¯ini i

(4.34)

Z = block-diag (Z1 , . . . , Zm )   Zi = block-diag Zi1 , . . . , Zini

(4.36)

(4.35)

(4.37)

then the H∞ criterion (4.25) is satisfied with the tracking controller gains −1 Kj = Y j P ,

j = 0, . . . , s

The disturbance attenuation level μ is the minimum value returned by Eq. (4.28), and ν = λmax (P). Proof. The Lyapunov function candidate is given by V(˜x) = x˜ P−1 x˜ . now, we can deduce easily that a sufficient condition ensuring inequality (4.27) is ⎞ ⎛ 

XTij

AK (ρ ud ) + In ()

   Y ⎟ ⎜  m,n i ⎟ ⎜ −1   i 

 −1 P ⎟ ⎜ P Hij T Hi 0 +Yi Xij ⎟ ≤ 0 (4.38) ϕij (t) ⎜ + −μIn 0 ⎟ ⎜ i,j=1 ⎠ ⎝

where

  T  AK (ρ ud )  A(ρ ud ) − BK(ρ ud ) P−1 + P−1 A(ρ ud ) − BK(ρ ud )

(4.39)   P 0 , By pre- and postmultiplying the left-hand side of (4.38) by 0 In applying the Schur lemma and using Young’s inequality in the following manner:

T

1 Yi + Zij Xij Yi + Zij Xij Z−1 XTij Yi + YTi Xij ≤ ij 2 we obtain Theorem 1. This ends the proof. Remark 4. It is worth noticing that when ϕ ij < 0, we have to use

ϕ¯ ij = ϕ¯ij − ϕ ij instead of ϕ¯ij and rearrange the matrix Θ to obtain the exact corresponding terms.

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OBSERVER-BASED REFERENCE TRAJECTORY TRACKING

4 OBSERVER-BASED REFERENCE TRAJECTORY TRACKING 4.1 Formulation of the Problem In the reference trajectory tracking problem (this is the case with most control design problems), the state of the system, x(t), is generally not available for feedback. That is why often a state observer is necessary. This section is devoted to this issue. We will provide two different LMI-based tracking design techniques. The first one gives some separation results, where the design of the observer and the controller gains are computed separately by solving two quasiindependent LMI conditions [66]. The second method consists of designing the observer-based controller gains simultaneously by solving a single LMI condition [67]. As a state observer, we consider the same nonlinear state observer designed in Ref. [7] due to its fast convergence and systematic implementation. Therefore, the structure of the observer-based trajectory tracking model is defined as the following: x˙ˆ = A(ρ u )ˆx +

m 

  Gi γi (ϑˆ i ) + Bu + L(ρ u ) y − Cˆx

(4.40a)

i=1

  u = −K(ρ ud ) xˆ − xd + ud

(4.40b)

ϑˆ i = Hi xˆ + Ki (ρ )(y − Cˆx)

(4.40c)

u

where xˆ represents the estimate of the state x, the control gain given as previously (Eq. 4.21), and the observer gains given by u

L(ρ ) = L0 +

s 

ρju Lj ,

j=1

u

Ki (ρ ) = Ki0 +

s 

ρju Kij

(4.40d)

j=1

The dynamics of the estimation error, e = x − xˆ , are given by ⎞ ⎛ m,n i   φij (t)Hij Hi − Ki (ρ u )C ⎠ e(t) e˙(t) = ⎝AL (ρ u ) +

(4.41)

i,j=1

where AL (ρ u ) = A(ρ u ) − L(ρ u )C. Hence, the dynamics of the reference tracking error (4.22) become ⎞ ⎛ m,n i   ϕij (t)Hij Hi⎠ x˜ + BK(ρ ud )e + A(ρ u ) − A(ρ ud ) xd x˙˜ = ⎝A(ρ u ) − BK(ρ ud ) +    i,j=1

ω(t)

(4.42) In the next two sections, we will provide two different LMI techniques to handle the problem of trajectory tracking based on the state observer.

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4. CONTROL OF ANAEROBIC DIGESTION PROCESS

4.2 First LMI Technique: Parallel Design This section is devoted to an LMI technique that ensures the exponential convergence of the state estimation error to zero and guarantees the H∞ asymptotic stability of the tracking error. We will present a kind of separation principle for nonlinear systems. Note that in the linear case, the separation principle means that we can investigate separately the convergence of the estimation error and the stability of the tracking error by using the concept of eigenvalues. However, in the nonlinear case, the separation results that we will provide are based on the Lyapunov analysis and on the use of the well-known Barbalat’s lemma. Because the dynamics (4.41) do not depend on the reference tracking error x˜ (t) and the functions φij (t) are bounded, then we can study the convergence of the estimation error e(t) separately and will use it in the dynamics of the tracking error as a bounded disturbance exponentially converging toward zero. The following theorem provides the synthesis conditions expressed in term of LMIs. Theorem 2. The closed-loop system (4.42) is H∞ asymptotically stabilizable by the observer-based feedback (4.40), if there exist symmetric positive definite matrices P, Q, Zij , Sij , i, j = 1, . . . , n, and matrices Yi , Xi , Xij of appropriate dimensions such that for given two positive scalar β, the LMI conditions (4.43) are fulfilled and the convex optimization problem (4.48) is solvable. 1. LMIs for the observer gains: ⎡ ⎢A (Q, X, ) + βQ ⎢ ⎣

 Π1

⎤ 

Πm ⎥ ⎥ ≤ 0, ⎦

Π

 ...

∀ ∈ {ρmin , ρmax }

−ΛS

()

(4.43) with s !  j ATj Q + QAj − CT Xj − XTj C A (Q, X, ) = AT0 Q+QA0 −CT X0 −XT0 C+ j=1

(4.44) and

" #  j n  Πi = M1i (Q, Si1 ) . . . Mi i Q, Sini , Mi Q, Sij = QHij + HiT Sij − C Xij (4.45) S = block-diag (S1 , . . . , Sm )   Si = block-diag Si1 , . . . , Sini

(4.46) (4.47)

The observer gains Lj and Kij are computed as Lj = Q−1 XTj ,

T Kij = S−1 ij Xij

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OBSERVER-BASED REFERENCE TRAJECTORY TRACKING

113

2. Optimization problem for the controller gains: min(μ) subject to Eq. (4.49) ⎡

 Σ1

⎢Θ ⎢ ⎢ ⎢ ⎣

Θ=

⎤ 

Σm ⎥ ⎥ ⎥ ≤ 0, ⎥ ⎦

∀ ∈ {ρmin , ρmax }

(4.49)

−ΛZ

() with

Σ

 ...

(4.48)

⎡ ⎣Θ11 ()

 ⎤ P 0 ⎦, −In



Θ11

A (P, Y, ) = ()

A (P, Y, ) = PAT0 + A0 P−Y0 B −BY 0 +

s 

In −μIn

 (4.50)

j PATj +A0 P−Yj B −BY j

!

j=1

(4.51) The matrix blocks Σi and Z are defined in Theorem 1. Thus, the H∞ criterion (4.25) is satisfied with the tracking controller gains −1 Kj = Y j P ,

j = 1, . . . , s

The disturbance attenuation level μ is the minimum value returned by Eq. (4.48), and ν = λmax (P). Proof. The proof is easy and standard. It is based on the use of the Barbalat’s lemma because the dynamics of the augmented system with  x˜ have a triangular structure. For more details, we refer the the state e reader to Refs. [61, 68]. For the observer convergence, we use the Lyapunov function V1 (e) and for the tracking error, we use V2 (˜x) and the H∞ criterion (4.25), where V1 (e) = e Qe,

V2 (˜x) = x˜ P−1 x˜

(4.52)

4.3 Second Approach: Simultaneous Design This section is dedicated to a second observer-based trajectory tracking method. Contrary to the first method, the second one provides a unified LMI synthesis condition ensuring the convergence of the global augmented system containing the tracking error vector and the estimation

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4. CONTROL OF ANAEROBIC DIGESTION PROCESS

error. For simplicity, we assume without loss of generality that the observer does not contain output feedback in the nonlinear terms, that is, Ki (ρt ) ≡ 0. x˜ Using the augmented vector ζ  , the dynamics of the tracking error e and the estimation error given by Eqs. (4.42), (4.41), respectively, can be rewritten under the following unified form: ⎞ ⎤ ⎛⎡ AK (ρ ud )  m,n i ϕij (t)Hij Hi    0 ⎠ζ ζ˙ = ⎝⎣A(ρ u ) − BK(ρ ud ) BK(ρ ud )⎦ + (t)H 0 φ ij ij Hi u i,j=1 0 AL (ρ ) ⎡ ⎤ In   + ⎣ ⎦ A(ρ u ) − A(ρ ud ) xd (4.53)   0  ω(t)

The aim consists of finding the gain matrices Lj and Kj so that the augmented error ζ satisfies the following H∞ criterion: ( ζ L2n ≤ μω2Ln + νζ0 2 2n (4.54) 2

2

L2

where μ > 0 is the gain from xd to ζ , and ν > 0 is to be determined. In order to satisfy Eq. (4.54), we use a quadratic Lyapunov function V(ζ ), such that dV (4.55) (ζ ) + ζ 2 − μω2 ≤ 0 ϑ(t)  dt By analogy to the previous results, we obtain the following proposition, which provides sufficient LMI conditions under which the inequality (4.55) is satisfied. Theorem 3. Assume that there exist symmetric positive definite matrices P, Q, Sij , S¯ ij , i, j = 1, . . . , n, and matrices Xi , Yi , i = 0, . . . , s, of appropriate dimensions, such that for a given positive scalar , the following convex optimization problem holds: min(μ) subject to Eq. (4.57) ⎡ ⎢Θ ⎢ ⎢ ⎢ ⎣ ()

 Σ1

Σ

 ...

⎤ 

Σm ⎥ ⎥ ⎥ ≤ 0, ⎥ ⎦

∀(, ) ¯ ∈ Vρ × V¯

−ΛM

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

(4.56)

(4.57)

OBSERVER-BASED REFERENCE TRAJECTORY TRACKING

where

⎡

⎤⎤ ¯ BY () ⎦⎥ ⎣ In ⎥ ⎥ 0 ⎥ ⎦ 0 −2P

⎡ ⎤ P ⎣0⎦ 0 −In ()

115

⎡

⎢Θ11 ⎢ Θ=⎢ ⎢ ⎣ () () ⎡ B (P, Y, ) ¯ () Θ11 = ⎣ In

0 A (Q, X, ) 0

A (Q, X, ) = AT0 Q+QA0 −CT X0 − XT0 C +

s 

(4.58)

⎤ In 0 ⎦ −μIn

(4.59) !

j ATj Q+QAj −CT Xj −XTj C

j=1

B (P, Y, ) ¯ = PAT0 + A0 P−Y0 B −BY 0 +

s 

(4.60) ! ¯ j PATj + A0 P−Yj B −BY j

j=1

(4.61) ¯ = Y0 + Y ()

s 

¯ j Yj

(4.62)

j=1

"   # n  Σi = Ni1 Si1 , S¯ i1 . . . Ni i Sini , S¯ ini ⎡⎡ T ⎤ ⎡ ⎤ Hij PHi ⎢ 0 ⎥ ⎢ 0 ⎥ ! ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ j ⎥ ⎢ ⎥ ⎢ Ni Sij , S¯ ij = ⎢ ⎢⎢ 0 ⎥ + ⎢ 0 ⎥ Sij ⎣⎣ 0 ⎦ ⎣ 0 ⎦ 0

0

⎤ ⎡ ⎤ ⎤ 0 0 ⎢QHij ⎥ ⎢HT ⎥ ⎥ ⎥ ⎢ i⎥ ⎥ ⎢ ⎢ 0 ⎥ + ⎢ 0 ⎥ S¯ ij ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎦

(4.63)

⎡

0

(4.64)

0

Λ = block-diag (Λ1 , . . . , Λm ) ! n Λi = block-diag Λ1i , . . . , Λi i ) * 2 j I2n Λi = block-diag bij i M = block-diag (M1 , . . . , Mm ) ! n Mi = block-diag M1i , . . . , Mi i $ % j ¯ Mi = block-diag Sij , Sij

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

(4.65) (4.66) (4.67) (4.68) (4.69) (4.70)

116

4. CONTROL OF ANAEROBIC DIGESTION PROCESS

Then, the H∞ criterion (4.54) is satisfied with the observer-based tracking controller gains Lj = Q−1 XTj ,

−1 Kj = Y j P ,

j = 1, . . . , s

The disturbance  μ is the minimum value returned by Eq. (4.56),  attenuation level and ν = max λmax P−1 , λmax Q . Proof. In order to satisfy criterion (4.54), we use the following quadratic Lyapunov function  −1  0 T P ζ V(ζ ) = ζ 0 Q where Q = QT > 0 and P = PT > 0 By calculating the derivative of V(ζ ) along the trajectories of Eq. (4.53), we obtain ⎡

⎛

−1 u ˙ ) = x˜ T ⎢ V(ζ ⎣P ⎝AK (ρ d ) +

m,n i

⎞

⎛

ϕij Hij Hi ⎠ + ⎝AK (ρ ud ) +

i,j=1

m,n i

⎤

⎞T

⎥ ϕij Hij Hi ⎠ P−1 ⎦ x˜

i,j=1

⎡ ⎛ ⎞ ⎛ ⎞T ⎤ m,n m,n i i   ⎢ ⎥ φij Hij Hi ⎠ + ⎝AL (ρ u ) + φij Hij Hi ⎠ Q⎦ e + eT ⎣Q ⎝AL (ρ u ) + i,j=1

i,j=1

+ 2˜xT P−1 BK(ρ ud )e + 2˜xT P−1 ω(t)

Hence, ϑ(t) ≤ 0 (Eq. 4.55) if the following inequality is fulfilled: ⎡



ud u ⎣D (AK (ρ ), AL (ρ )) + I2n

() ⎛

XTij

P−1 0

  ⎤ 0 In Q 0 ⎦ −μIn ⎞

⎟ ⎜⎡  ⎤ Yi ⎟ ⎜ −1 m,n P H    i ⎟ ⎜ ij 

⎟ ⎜⎣ T ϕij (t) ⎜ + 0 ⎦ Hi 0 0 +Yi Xij ⎟ ⎟ ⎜ 0 ⎟ ⎜ i,j=1 ⎠ ⎝

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

(4.71)

117

OBSERVER-BASED REFERENCE TRAJECTORY TRACKING

⎛

⎞

¯T X ij

  ⎤ ⎜⎡ ⎜ 0 ⎜  ⎣QHij ⎦ 0 + φij (t) ⎜ ⎜ ⎜ 0 i,j=1 ⎝

¯ Y

i Hi

m,n i

⎡ ⎤ 0 + ⎣In ⎦ (BK)T P−1 0 where

⎟ ⎡ ⎤ ⎟ P−1 BK 

⎟ ¯ TX ¯ ⎟ ⎣ 0 ⎦ 0 0 +Y i ij ⎟ + ⎟ 0 ⎠

In

0 0 ≤0

(4.72)

T  −1  AK (ρ ud ) 0 P 0 D AK (ρ ), AL (ρ )  0 AL (ρ u ) 0 Q  −1   u 0 P 0 AK (ρ d ) + 0 AL (ρ u ) 0 Q 

ud

u



0





(4.73)

By pre- and postmultiplying the right-hand side of Eq. (4.72) by ⎤ ⎡ P 0 0 ⎣ 0 In 0 ⎦ 0 0 In we get the following equivalent inequality: ⎛ ⎡ T(1.1) + PP ⎣ 0 In

0

T(2.2) + In 0 ⎛

⎞

¯T X ij

  ⎤ ⎜⎡ ⎜ m,n 0 i ⎜  ⎣QHij ⎦ 0 φij (t) ⎜ + ⎜ ⎜ 0 i,j=1 ⎝ ⎡ ⎤ 0 + ⎣In ⎦ (BK)T 0

0

XTij

  ⎤ ⎜⎡ ⎤ Y ⎜ Hij  m,n In i i  ⎜ ⎣ ⎦ Hi P 0 0 ⎦+ ϕij (t) ⎜ ⎜ 0 ⎜ 0 −μIn i,j=1 ⎝

¯i Y

 Hi

⎞ ⎟ ⎟ 

⎟ 0 +YTi Xij ⎟ ⎟ ⎟ ⎠

VT

  ⎤ ⎟ ⎡ U ⎟ BK   



⎟ T¯ ⎟ ¯ ⎣ ⎦ 0 +Yi Xij ⎟ + 0 0 In 0 ⎟ 0 ⎠

0 ≤0

where ud ud T(1.1) = PA K (ρ ) + AK (ρ )P u u T(2.2) = A L (ρ )Q + QAL (ρ )

I. POWER, RENEWABLE ENERGY, AND INDUSTRIAL SYSTEMS

(4.74)

118

4. CONTROL OF ANAEROBIC DIGESTION PROCESS

Finally, the use of the Schur lemma and Young’s inequality in the following manner:

T

1 Yi + Sij Xij Yi + Sij Xij S−1 XTij Yi + YTi Xij ≤ ij 2



1 ¯ ¯ ij + S¯ ij Y ¯T ¯ ¯ ¯ i T S¯ −1 X ¯i ¯ TY Xij + S¯ ij Y X ij i + Yi Xij ≤ ij 2 1 VT U + UT V ≤ [U + ZV]T (Z)−1 [U + ZV] 2 leads to Theorem 3.

5 SIMULATION RESULTS In this section, we illustrate by numerical simulation the proposed control strategies to track a constant reference trajectory planned by the plant operator. We will investigate the case when the full state vector is available for measurement and the case when only its partial measurement is available, which is the most realistic case.

5.1 State-Feedback Trajectory Tracking In order to simulate the control strategy proposed in Section 3, we use the model parameters given in Table 4.1, and we take ρmin = 0.1 day−1 , ρmax = 0.8 day−1 . After solving the optimization problem (4.28) given by Theorem 1, by using the LMI toolbox of Matlab, we have obtained the following controller gain  K0 =

0.0007 −0.0008

0.0020 −0.0021

0.0002 −0.0008

0.0722 −0.0209

−0.0000 0.0000

−0.0001 0.0000



K1 = 0 and μ = 0.5. In this example, the system is initialized at x(0) = [1.8, 0.4, 12, 0.7, 109.15, 55]T and we want to track the desired reference given by xd = [1.9572, 0.6058, 5.4, 1.3893, 242.8, 240.3413]T and ud = [0.4966, 0.0436]T , which corresponds to an enhanced quality of biogas at the steady state, containing only 20.5% of CO2 gas. The simulation results are reported in Figs. 4.1–4.9. In the first seven former figures, we compare the state trajectories and the biogas quality when applying the control strategy and when not applying the control. As can be noticed from those figures, despite the initial gap between the initialization and the desired reference, the controlled system is tracking the desired reference and the biogas quality is enhanced from 30% to 20.5% of the countenance of CO2 gas. Moreover, the required

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119

SIMULATION RESULTS

Organic substrate concentration x1 (g/L)

3

x1 x1d

2.8

x1c 2.6 2.4 2.2 2 1.8 1.6 0

5

10

15

20

25

30

35

40

45

50

Time(days)

FIG. 4.1 Organic substrate concentration x1 (g/L). 0.75

Acidogenic bacteria concentration (g/L)

0.7 0.65 0.6 0.55 0.5

x2 x2d

0.45

x2c 0.4 0.35 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.2 Acidogenic bacteria concentration x2 (g/L). 18

x3

Acetate concentration (mmol/L)

16

x3d x3c

14

12

10

8

6

4 0

5

10

15

20

25

30

35

40

Time (days)

FIG. 4.3 Acetate concentration x3 (mmol/L).

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45

50

120

4. CONTROL OF ANAEROBIC DIGESTION PROCESS

Mathenogenic bacteria concentration (g/L)

1.8

1.6

1.4

1.2

x4 1

x4d x4c

0.8 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.4 Mathenogenic bacteria concentration x4 (g/L). Inorganic carbon concentration (mmol/L)

280 260 240 220 200 180 160 140 120

x5

100

x5d x5c

80 60 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.5 Inorganic carbon concentration x5 (mmol/L). 280

Alkalinity concentration (mmol/L)

260 240 220 200 180 160 140

x6

120

x6d

100

x6c

80 60 0

5

10

15

20

25

30

35

40

Time (days)

FIG. 4.6 Alkalinity concentration x6 (mmol/L).

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0.32

Biogas quality ( CO2 %)

0.3 0.28

Quality 0.26

Qualityd

0.24

Quality

c

0.22 0.2 0.18 0

5

10

15

20

25

30

35

40

45

50

30

35

40

45

50

30

35

40

45

50

Time (days)

FIG. 4.7 Biogas quality (CO2 %). 0.525

Control input u1 (1/day)

0.52

0.515

0.51

0.505

0.5

0.495 0

5

10

15

20

25

Time (days)

FIG. 4.8 Control input u1 (1/day). 0.044

Control input u2 (1/day)

0.043 0.042 0.041 0.04 0.039 0.038 0.037 0.036 0.035 0.034 0

5

10

15

20

25

Time (days)

FIG. 4.9 Control input u2 (1/day).

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control inputs u1 and u2 , which are depicted in Figs. 4.8–4.9, respectively, show satisfactory behavior. Indeed, the dilution rate of the system changes smoothly and remains in its acceptable domain of variation.

5.2 Observer-Based Reference Trajectory Tracking Often, in real applications, not all the state variables of the system are measurable, thus only the partial measurement of the state vector is available. Moreover, in the AD process, it is known that measurement of the different bacteria concentrations is difficult, time consuming, and costly to process. Therefore, in this section, we will suppose that only measurements of the substrate concentrations, x1 , x3 , and x6 are available with the CO2 gas flow rate (qc (x)). Thus, we will investigate the second control strategy, observer-based reference trajectory tracking (Section 4). In order to show the efficiency of the proposed two design methodologies, parallel and simultaneous design, for the observer-based control to track the desired reference, we will run the same numeric simulation as previously (using the same model parameters as well as operating and initial conditions as in Section 5.1). Then, we will compare the obtained results. 5.2.1 Parallel Design After solving the LMI conditions (4.43) that have been found to be feasible for β = 0.06 by using the LMI toolbox of Matlab, and the optimization problem (4.48) given by Theorem 2, we have found the following controller gain 

K0 =

0.0007 −0.0008

0.0020 −0.0021

0.0002 −0.0008

0.0722 −0.0209

−0.0000 0.0000

−0.0001 0.0000

K1 = 0, μ = 0.43, and the following observer parameters ⎡

77.0743 ⎢ −1.9248 ⎢ ⎢ −36.2480 L0 = ⎢ ⎢ −0.7587 ⎢ ⎣−318.3944 −0.0000

−29.9151 0.7410 429.9256 −1.4631 −408.7394 −0.0000

0.0000 −0.0000 0.0000 −0.0000 −0.0000 0.9954

⎡ 120.7145 ⎢ −3.0766 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , L1 = ⎢ 133.0371 ⎢ −2.0359 ⎥ ⎢ ⎥ ⎣ −751.4685 ⎦ −0.0000 ⎤

26.3467 −0.6662 90.6119 −0.7101 −243.3368 −0.0000



⎤ 0.0000 −0.0000 ⎥ ⎥ −0.0000 ⎥ ⎥ −0.0000 ⎥ ⎥ 0.0000 ⎦ −0.8690

    0.8492 0.1310 0.0000 −0.0090 0.0129 −0.0000 , K11 = K10 = −0.0240 −0.0000   −0.7609 0.6798 −0.0000   0.0140 −0.0283 0.7706 0.0000 −0.0008 −0.0060 −0.0000 K20 = , K21 = −0.1384 −1.1604 −0.0000 0.0008 0.0091 −0.0000

Moreover, to run the simulation, we have initialized the system as in the previous section and the observer by xˆ 0 = [1.8, 0.6, 12, 0.3, 45, 55]T . We depict the simulation results in Figs. 4.10–4.17, which show that although the large initial estimation error the observer is converging to the simulated

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SIMULATION RESULTS

Organic substrate concentration x1 (g/L)

3.2

x1 x1d x1c x ˆ1

3 2.8 2.6 2.4 2.2 2 1.8 1.6 0

5

10

15

20

25

30

35

40

45

50

Time (days)

Acidogenic bacteria concentration (g/L)

FIG. 4.10 Organic substrate concentration x1 (g/L). 0.75 0.7 0.65 0.6 0.55 0.5

x2 x2d x2c x ˆ2

0.45 0.4 0.35 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.11 Acidogenic bacteria concentration x2 (g/L). 20

x3 x3d x3c x ˆ3

Acetate concentration (mmol/L)

18 16 14 12 10 8 6 4 0

5

10

15

20

25

30

35

40

Time (days)

FIG. 4.12 Acetate concentration x3 (mmol/L).

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Mathenogenic bacteria concentration (g/L)

1.8 1.6 1.4 1.2 1

x4

0.8

x4d 0.6

x4c x ˆ4

0.4 0.2 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.13 Mathenogenic bacteria concentration x4 (g/L). Inorganic carbon concentration (mmol/L)

260 240 220 200 180 160 140 120

x5 x5d x5c x ˆ5

100 80 60 40 0

5

10

15

20

25

30

35

40

45

50

Time (days)

Alkalinity concentration (mmol/L)

FIG. 4.14 Inorganic carbon concentration x5 (mmol/L).

250

200

150

x6 x6d x6c

100

x ˆ6 50 0

5

10

15

20

25

30

35

40

Time (days)

FIG. 4.15 Alkalinity concentration x6 (mmol/L).

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SIMULATION RESULTS 0.56

Control input u1 (1/day)

0.55

0.54

0.53

0.52

0.51

0.5

0.49 0

5

10

15

20

25

30

35

40

45

50

30

35

40

45

50

Time (days)

FIG. 4.16 Control input u1 (1/day).

0.044

Control input u2 (1/day)

0.042 0.04 0.038 0.036 0.034 0.032 0.03 0.028 0.026 0

5

10

15

20

25

Time (days)

FIG. 4.17 Control input u2 (1/day).

state of the system and the closed-loop system tracks the desired reference trajectory. Moreover, the behavior of the controller remains smooth and very acceptable. 5.2.2 Simultaneous Design In this section, we simulate the simultaneous design approach where we compute the controller and observer parameters by solving unified LMI conditions. Hence, we solve the optimization problem (4.56) given by Theorem 3. Using the LMI toolbox of Matlab as well as the same parameter values and operating conditions as previously (Sections 5.1 and 5.2.1), we have

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found the LMI conditions (4.57) feasible for  = 10. Hence, we have obtained the following observer and control parameters ⎡

L0

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ ⎡

L1

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ 

K0

=

−7101.3 169.51 77,611 −216.27 −65,770 −0.076595

1941 −46.037 −7212.1 6.8699 29.072 0.015661 −4522.5 108.25 81,081 −255.86 −82,473 −0.17246

0.0019 −0.0005

−1174.2 27.94 12,913 −36.065 −10,981 −0.045071

0.0061 −0.0017

16.193 −0.42981 −194.51 0.5592 172.44 1.1282 −10.376 0.2354 64.577 −0.13418 −33.41 −1.0134

0.0007 −0.0002

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0.1989 −0.0571

0.0000 −0.0000

−0.0004 0.0004



K1 = 0 and μ = 0.48. According to the simulation results, which are presented in Figs. 4.18–4.26, we can see that the tracking error is converging to zero. Hence, the system is tracking the desired reference and this is despite the large initial estimation error and the initial gap between the simulated system and the desired reference. Indeed, both the observer and the control show positive results. Besides, the control values (u1 and u2 ) remain acceptable to avoid washout of bacteria and emptying the digester. Nevertheless, if we compare the results obtained by the parallel and the simultaneous design approaches, we may notice that at the initial time and for the same initial conditions, more control efforts are required by the simultaneous design, Figs. 4.27 and 4.28, and higher picks are obtained for the state reconstruction, Figs. 4.29 and 4.30. This may be due to the larger values of both the observer and the controller gains ensuring the feasibility of the LMI conditions (4.57). However, the results are not conclusive; more simulation studies and deep analysis need to be performed. Moreover, the impact of the higher control amplitude obtained by the simultaneous design approach at the transient time is reflective of the transient response of the state variables. For example, we compare the time response of the states x1 and x3 obtained by the two approaches (xiparc and xisimc correspond to the controlled state xi obtained by parallel and simultaneous design approaches, respectively) in Figs. 4.29 and 4.30, respectively. As can be seen, the transient response of the two state variables is more aggressive when using the simultaneous design.

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SIMULATION RESULTS

Organic substrate concentration x1 (g/L)

4

x1 x ˆ1

3.5

x1d x1c

3

2.5

2

1.5 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.18 Organic substrate concentration x1 (g/L).

Acidogenic bacteria concentration (g/L)

0.7 0.65 0.6 0.55 0.5

x2 0.45

x ˆ2

0.4

x2c

x2d

0.35

0

5

10

15

20

25

30

35

40

45

50

Time (days)

Acetate concentration (mmol/L)

FIG. 4.19 Acidogenic bacteria concentration x2 (g/L).

x3

25

x ˆ3 x3d

20

x3c 15

10

5

0 0

5

10

15

20

25

30

35

40

Time (days)

FIG. 4.20 Acetate concentration x3 (mmol/L).

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Mathenogenic bacteria concentration (g/L)

1.8 1.6 1.4 1.2 1 0.8

x4

0.6

x ˆ4 x 4d

0.4

x 4c

0.2 0

5

10

15

20

25

30

35

40

45

50

Time (days)

Inorganic carbon concentration (mmol/L)

FIG. 4.21 Mathenogenic bacteria concentration x4 (g/L).

250

200

150

100

50

80

x5

60

x ˆ5

40

x5d 0

0.1

0.2

0.3

0.4

0.5

x5c

0 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.22 Inorganic carbon concentration x5 (mmol/L).

Alkalinity concentration (mmol/L)

280 260 240 220 200 180 160 140

x6

120

x ˆ6

100

x6d

80

x6c

60 0

5

10

15

20

25

30

35

40

Time (days)

FIG. 4.23 Alkalinity concentration x6 (mmol/L).

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SIMULATION RESULTS 0.35

Biogas quality (CO2 %)

0.3 0.25 0.2 0.15

Quality Qualitye

0.1

Qualityd 0.05

Qualityc

0 0

5

10

15

20

25

30

35

40

45

50

30

35

40

45

50

30

35

40

45

50

Time (days)

FIG. 4.24 Biogas quality (CO2 %).

0.64

Control input u1 (1/day)

0.62 0.6 0.58 0.56 0.54 0.52 0.5 0.48 0

5

10

15

20

25

Time (days)

FIG. 4.25 Control input u1 (1/day).

0.06

Control input u2 (1/day)

0.055

0.05

0.045

0.04

0.035

0.03 0

5

10

15

20

25

Time (days)

FIG. 4.26 Control input u2 (1/day).

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0.7

u1par Control input u1 (1/day)

0.65

u1sim

0.6

0.55

0.5

0.45 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.27 Control input u1 (1/day) (par: parallel, sim: simultaneous). 0.06

u2par

Control input u2 (1/day)

0.055

u2sim 0.05

0.045

0.04

0.035

0.03

0.025 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.28 Control input u2 (1/day) (par: parallel, sim: simultaneous).

Organic substrate concentration x1 (g/L)

3.6

x1d

3.4

x1parc

3.2

x1simc

3 2.8 2.6 2.4 2.2 2 1.8 1.6 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.29 Organic substrate concentration x1 (g/L) (parc: parallel control, simc: simultaneous control).

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CONCLUSION

Acetate concentration (mmol/L)

25

x3d x3parc x3simc

20

15

10

5 0

5

10

15

20

25

30

35

40

45

50

Time (days)

FIG. 4.30 Acetate concentration x3 (mmol/L) (parc: parallel control, simc: simultaneous control).

Finally, we conclude that both design methodologies for the observerbased feedback control allow finding the appropriate observer and controller parameters that enable the system to track the desired reference. According to the simulation results, the control amplitude remains in a suitable interval of variation. However, more investigations are required to deal with the saturation of the control inputs and the stability of the closed system under such saturation.

6 CONCLUSION In this chapter, we have proposed a state-feedback control to track a reference trajectory. In order to prove the stability of the closed-loop system, we have provided nonrestrictive LMI conditions numerically tractable by convex optimization algorithm. Moreover, due to the lack of sensors to measure all the state variables of the system in AD applications, we have considered the state estimation through the inclusion of an exponential observer in the control design. In the sequel, we have provided two methodologies to find the observer and controller parameters that ensure the stability of the closed-loop system (system composed from the system, controller, and the observer). We point out that the design has been done for a full state-feedback control. However, it can be easily applied for partial feedback control or linear combination of the state variables by taking, for example, the tracking error x˜ = T(x − xd ), where T is a linear matrix of appropriate dimension. Moreover, it is not difficult to investigate the case of disturbed dynamics and measurements of the system. However, further research is required to account for saturation constraints in the control inputs.

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[43] A. Isidori, Nonlinear Control Systems, Springer Science & Business Media, New York, NY, 2013 . [44] H.O. Méndez-Acosta, R. Femat, V. González-Alvarez, J.P. Steyer, Substrate regulation in an anaerobic digester based on geometric control, in: Proceedings of the IFAC—Congreso Latinoamericano de Control Automatico, 2002. [45] H.O. Méndez-Acosta, R. Femat, D.U. Campos-Delgado, Improving the performance on the chemical oxygen demand regulation in anaerobic digestion, Ind. Eng. Chem. Res. 43 (1) (2004) 95–104. [46] B. Palacios-Ruiz, H.O. Méndez-Acosta, V. Alcaraz-Gonzalez, V. Gonzalez-Alvarez, C. Pelayo-Ortiz, Regulation of volatile fatty acids and total alkalinity in anaerobic digesters, IFAC Proc. Vol. 41 (2) (2008) 13611–13616. [47] H.O. Méndez-Acosta, D.U. Campos-Delgado, R. Femat, V. González-Alvarez, A robust feedforward/feedback control for an anaerobic digester, Comput. Chem. Eng. 29 (7) (2005) 1613–1623. [48] H.O. Méndez-Acosta, B. Palacios-Ruiz, V. Alcaraz-González, J.P. Steyer, V. González-Álvarez, E. Latrille, Robust control of volatile fatty acids in anaerobic digestion processes, Ind. Eng. Chem. Res. 47 (20) (2008) 7715–7720. [49] H.O. Méndez-Acosta, B. Palacios-Ruiz, V. Alcaraz-González, V. González-Álvarez, J.P. García-Sandoval, A robust control scheme to improve the stability of anaerobic digestion processes, J. Process Control 20 (4) (2010) 375–383. [50] A. Teel, L. Praly, Tools for semiglobal stabilization by partial state and output feedback, SIAM J. Control Optim. 33 (5) (1995) 1443–1488. [51] R. Flores-Estrella, G. Quiroz, H. Mendez-Acosta, R. Femat, H∞ control of anaerobic digester for winery industry wastewater treatment, Ind. Eng. Chem. Res. 52 (7) (2013) 2625–2632. [52] J. Liu, G. Olsson, B. Mattiasson, Monitoring and control of an anaerobic upflow fixed-bed reactor for high-loading-rate operation and rejection of disturbances, Biotechnol. Bioeng. 87 (1) (2004) 43–53. [53] J. Liu, G. Olsson, B. Mattiasson, Control of an anaerobic reactor towards maximum biogas production, Water Sci. Technol. 50 (11) (2004) 189–198. [54] K. Boe, J.P. Steyer, I. Angelidaki, Monitoring and control of the biogas process based on propionate concentration using online VFA measurement, Water Sci. Technol. 57 (5) (2008) 661–666. [55] A.J. Guwy, F.R. Hawkes, S.J. Wilcox, D.L. Hawkes, Neural network and on-off control of bicarbonate alkalinity in a fluidised-bed anaerobic digester, Water Res. 31 (8) (1997) 2019–2025. [56] P. Holubar, L. Zani, M. Hager, W. Fröschl, Z. Radak, R. Braun, Advanced controlling of anaerobic digestion by means of hierarchical neural networks, Water Res. 36 (10) (2002) 2582–2588. [57] D. Gaida, Dynamic Real-Time Substrate Feed Optimization of Anaerobic Co-Digestion Plants, Leiden Institute of Advanced Computer Science (LIACS), Faculty of Science, Leiden University, 2014. [58] J. Jimenez, E. Latrille, J. Harmand, A. Robles, J. Ferrer, D. Gaida, C. Wolf, F. Mairet, O. Bernard, V. Alcaraz-González, Instrumentation and control of anaerobic digestion processes: a review and some research challenges, Rev. Environ. Sci. Bio/Technol. 14 (4) (2015) 615–648. [59] E. Mauky, S. Weinrich, H.J. Nägele, H.F. Jacobi, J. Liebetrau, M. Nelles, Model predictive control for demand-driven biogas production in full scale, Chem. Eng. Technol. 39 (2016) 652–664. [60] K. Chaib-Draa, H. Voos, M. Alma, A. Zemouche, M. Darouach, LMI-based H∞ nonlinear state observer design for anaerobic digestion model, in: 25th Mediterranean Conference on Control and Automation (MED), 2017, pp. 678–683.

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C H A P T E R

5 Finite-Time Disturbance Observer-Based Tracking Control Design for Nonholonomic Systems Saleh Mobayen*, Afef Fekih † , Olfa Boubaker ‡ , Quan Min Zhu § *Advanced Control Systems Laboratory, Department of Electrical Engineering, University of Zanjan, Zanjan, Iran † Department of Electrical and Computer Engineering, University of Louisiana at Lafayette, Lafayette, LA, United States ‡ University of Carthage, National Institute of Applied Sciences and Technology, Tunis, Tunisia § Department of Engineering Design and Mathematics, University of the West of England, Bristol, United Kingdom

1 INTRODUCTION Nonholonomic systems represent a wide class of mechanical systems such as rigid spacecraft, unmanned aerial vehicles, underactuated satellites, cars towing several trailers, car-like vehicles, vertical rolling wheels, and wheeled mobile robots [1, 2]. Mathematically, nonholonomic systems are modeled using Lagrange equations with linear nonintegrable constraints [3]. The underlying geometry of nonholonomic systems, however, makes designing stabilizing controllers for them a challenging task [4]. The presence of disturbances and parameter variations further complicates the control design problem [5]. A number of representative approaches were proposed in the literature for finite-time control and stabilization of nonholonomic systems [6–9]. Fortunately, under certain circumstances, these systems can be transformed into chain-shaped systems, allowing some

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control approaches such as continuous feedback control, discontinuous feedback control, and hybrid control methods to stabilize such dynamics [10]. A summary of the latest developments in the control of high-order chained-form nonholonomic systems can be found in Huang et al. [11]. Motivated by the widespread application of mobile robots and the inherent nonlinearities in their dynamics, the stabilization issue of nonholonomic wheeled mobile robots has attracted much attention [12]. It is considered that wheeled mobile robotic systems have nonholonomic constraints because they have restricted mobility in that the wheels roll without slipping. Hence, linearized models of mobile robots are considered to have deficiencies in controllability, thus hindering the application of linear control techniques and generating interest toward nonlinear control techniques. Owing to Brockett’s theorem, it was shown that nonholonomic systems with restricted mobility cannot be stabilized by differentiable or continuous state-feedback controllers [4]. Using the direct Lyapunov approach, a stable tracker was proposed in Kanayama et al. [13] for autonomous mobile robots. Subsequently, various robust control techniques were proposed in the literature for nonholonomic systems [14– 16]. Even though the aforementioned techniques are quite effective at controlling and stabilizing nonholonomic systems, they lack the inherent robustness to uncertainties and disturbances often induced by unmodeled dynamics and random disturbances. Sliding mode control (SMC) has been recognized as an effective tool in designing control approaches for nonlinear systems operating under uncertainties and unmeasurable external disturbances [17, 18]. It owes its popularity to its ability to render the closed-loop response entirely insensitive to a specific class of perturbations, parameter variations, and unmodeled dynamics. However, its discontinuous nature makes conventional SMC suffer from chattering, singularity, and sensitivity to mismatched uncertainties and disturbances, which limit its practical implementation [19] because a large class of practical systems is affected by unmatched uncertainties. SMC has several stimulating properties that cannot be effortlessly attained by the other methods. Further, while conventional SMC guarantees asymptotic stability, there is no guarantee that this will happen in finite time, especially under unmatched disturbances. When a system is in its sliding mode, it follows a determined reduced-order system and it becomes insensitive to parametric variations and disturbances. To overcome some of these limitations, a variety of control approaches have been proposed to handle mismatched uncertainties and disturbances and alleviate chattering [20–25]. An SMS-based tracking controller was proposed in Yang and Kim [26] to asymptotically stabilize a nonholonomic wheeled mobile robot. Other approaches worth mentioning are (1) SMC tracking control of nonholonomic wheeled mobile robots in polar coordinates [27];

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(2) control of nonholonomic systems with parametric uncertainties via a second-order SMC approach [28]; (3) an SMC formation controller for electrically driven nonholonomic mobile robotic systems in the existence of parametric uncertainties and external disturbances [29]; (4) dealing with nonlinearities, nonholonomic constraints, external disturbances, and dynamic coupling between the wheeled mobile robot and its mounted robotic manipulator, an adaptive backstepping-based SMC approach [30]; (5) an adaptive SMC dynamic scheme with an integrator in the loop is planned for nonholonomic wheeled mobile robots [31]; and (6) a leader-follower-based formation control approach is presented for nonholonomic mobile robotic systems with mismatched uncertainties via integral SMC (ISMC) [32]. Attributes such as chattering reduction, finite-time convergence, highprecision performance, and ease of implementation have led to the adoption of terminal sliding mode control (TSMC) for the control of nonholonomic systems [33–37]. TSMC replaces the linear sliding surfaces in standard SMC by nonlinear switching hyperplanes to improve the system’s transient performance and reduce the ubiquitous chattering effects. A TSMC-based approach was proposed recently for a class of nonholonomic systems with parametric uncertainties [38]. In most of the aforementioned approaches, however, the bounds of the disturbances were directly employed in the design process of the TSMC control law. This problem can be overcome via the augmentation of the control scheme with an observer that can provide proper estimation of parametric uncertainties and external disturbances [39]. The remarkable features of disturbance observers, such as their ability to effectively estimate external disturbances, and simple design make their integration with control approaches an effective way to improve a system’s tracking performances [40–43]. A TSMC tracker was established in Chen et al. [44] via a disturbance-observer procedure for a class of uncertain SISO nonlinear systems in the presence of a control singularity and unknown nonsymmetric input saturation. However, some considerable chattering can be observed in the control signal for a Duffing forced oscillation system. Nonlinear disturbance observer (DOB)-based SMC approaches were also suggested in Yang et al. [45] for the stabilization of uncertain nonlinear systems. Nevertheless, highfrequency oscillations are the significant drawbacks in the results of Yang et al. [45]. In Kim et al. [43], Kim [46], and Chen [40], DOBs were applied for robust control and approximation of exterior disturbances, although the finite-time convergence of the state trajectories is not observed in any of these references. For autonomous mobile robots, TSMC with a disturbance observer was suggested in Kun and Mou [47]. However, the singularity problem has not been considered in a derivative of the sliding surface

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of Kun and Mou [47]. To the best of the authors’ information, very few attempts have been investigated for the disturbance observer-based finitetime control tracking control approach of nonholonomic systems. In this chapter, we propose an observer-based TSMC approach for a class of nonholonomic systems subject to disturbances. The disturbance observer is designed to make the disturbance approximation error converge to zero in finite time. A reaching control tracker is then derived to ensure the existence of the sliding mode around the innovative recursive TSMC surface in finite time and guarantee system stability. The main contributions of this chapter are as follows: • A disturbance observer-based TSMC approach that ensures the finite-time-tracking performance of nonholonomic systems while guaranteeing robustness and disturbance rejection properties. • A design that does not require prior knowledge about the disturbance bounds but rather relies on a disturbance observer that guarantees the finite-time convergence of the disturbance estimation errors to the origin. • A design that ensures the finite-time stability for nonholonomic systems despite the external disturbances. This chapter is organized as follows. Section 2 presents some preliminaries and states the problem description. Section 3 elaborates the DOB-based TSMC approach. Section 4 provides some simulation results pertaining to the implementation of the proposed approach to a wheeled mobile robot. Finally, some concluding remarks are drawn in Section 5.

2 PROBLEM STATEMENT AND PRELIMINARIES Consider the following uncertain nonholonomic system: x˙ 1 = u1 , x˙ 2 = u2 , x˙ 3 = x2 u1 + f (x) + d,

(5.1)

where x = [x1 , x2 , x3 ]T denotes the state vector of the system (5.1), u1 , u2 ∈ R are the control inputs, f (x) ∈ R is a nonlinear function assumed to be known, and d ∈ R is an unknown disturbance. Fig. 5.1 shows the diagram of a chained-form nonholonomic system. Let the desired trajectory vector xd = [x1d , x2d , x3d ]T be designed as follows [48]: x˙ 1d = u1d , x˙ 2d = u2d , x˙ 3d = x2d u1d + f (xd ),

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(5.2)

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x• 1

u1

x• 2

u2

1/ s

x1

1/ s

x2

x• 3

f (x )

1/ s

x3

d FIG. 5.1 Chained-form nonholonomic system.

where u1d and u2d are the reference control signals and f (xd ) ∈ R denotes a nonlinear function assumed to be known. Define the tracking error vector as xe = x − xd . Using Eqs. (5.1), (5.2), we can express the error dynamics by x˙ 1e = x˙ 1 − x˙ 1d = u1 − u1d , x˙ 2e = x˙ 2 − x˙ 2d = u2 − u2d , x˙ 3e = x˙ 3 − x˙ 3d = x2 u1 − x2d u1d + f (x) + d − f (xd ) = (x2e + x2d ) u1 − x2d u1d + f (x) − f (xd ) + d = (x2e + x2d ) u1 − (x2e + x2d ) u1d + (x2e + x2d ) u1d − x2d u1d + f (x) − f (xd ) + d = (x2e + x2d ) (u1 − u1d ) + u2e u1d + f (x) − f (xd ) + d. Let for system (5.1): u˙ 1 = v1 , u˙ 2 = v2 ,

(5.3)

where v1 and v2 denote new control inputs to be designed. Using Eqs. (5.3), (5.3), the following subsystem is obtained: x˙ 1e = u1 − u1d , u˙ 1 = v1 .

(5.4)

Choosing the following control law for v1 [34]: v1 = −5(u1 − u1d + 5x1e 3/5 )3/5 − 3x1e −2/5 (u1 − u1d ) + u˙ 1d ,

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(5.5)

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guarantees that x1e and u1 − u1d approach zero in finite time and remain in its vicinity thereafter. Using Eqs. (5.3), (5.3) yields the following third-order nonholonomic system, x˙ 2e = u2 − u2d , x˙ 3e = d + f (x) − f (xd ) + x2e u1d , (5.6) u˙ 2 = v2 . The main objective of this study is to design a disturbance observerbased SMC approach that ensures the finite-time-tracking performance of system (5.6) while guaranteeing robustness and disturbance rejection properties. Lemma 1. Assuming a positive-definite continuous functional V(t) fulfills the subsequent inequality [49]: ˙ V(t) ≤ −μV η (t),

∀t ≥ t0 , V(t0 ) ≥ 0,

(5.7)

where t0 is the initial time, μ represents a positive constant, and η denotes a ratio of odd positive integers (1 > η > 0) . Then, for any specified t0 , the function V(t) fulfills the following inequality: V 1−η (t) ≤ V 1−η (t0 ) − μ(1 − η)(t − t0 ),

t0 ≤ t ≤ ts .

(5.8)

Thus, the Lyapunov functional V(t) converges to the origin in a finite time ts , given by V 1−η (t0 ) . (5.9) ts = t 0 + μ(1 − η)

3 DISTURBANCE OBSERVER-BASED TSMC APPROACH Designing the disturbance observed-based TSMC approach requires two steps: 1. Designing a finite-time DOB to estimate the external disturbances. 2. Deriving a TSMC tracking controller to ensure the finite-time stability of the controlled system.

3.1 Disturbance Observer Finite-Time Convergence Theorem 1. The estimation error vector between the nonholonomic system (5.1) and the TSMC-based DOB defined by x˙ 1 = u1 , x˙ 2 = u2 ,

(5.10) p0 /q0

z˙ = x2 u1 − βsgn(s) − ks − |f (x)|sgn(s) − εs

,

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145

converges to the origin in finite time for the estimation disturbance: dˆ = −ks − βsgn(s) − εsp0 /q0 − |f (x)|sgn(s) − f (x),

(5.11)

where s is a switching surface defined by s = −x3 + z,

(5.12)

and p0 and q0 are odd positive integers such that p0 < q0 and k, β, and ε are positive constant parameters to be designed with β ≥ |d| (the absolute value of d). Proof. Using the third equations of Eqs. (5.1), (5.10), and differentiating Eq. (5.12), yields s˙ = −˙x3 + z˙ = −d − βsgn(s) − ks − |f (x)|sgn(s) − εsp0 /q0 − f (x).

(5.13)

Constructing the Lyapunov function as V(s) =

1 2 s , 2

(5.14)

gives the following time derivative along the trajectory:   ˙ V(S) = s −|f (x)|sgn(s) − f (x) − ks − d − βsgn(s) − εsp0 /q0 ≤ −εs(p0 +q0 )/q0 − ks2 − β|s| + |s| |d| − sf (x) − |f (x)| |s| ≤ −εs(p0 +q0 )/q0 − ks2

(5.15)

≤ −2(p0 +q0 )/2q0 εV(s)(p0 +q0 )/2q0 − 2kV(s). Based on Lemma 1 and Eq. (5.15), the switching surface s is guaranteed to converge to the origin in the finite time. Using Eqs. (5.1), (5.11), (5.13), we can derive the approximation error d˜ as follows: d˜ = dˆ − d = −βsgn(s) − d − εsp0 /q0 − |f (x)|sgn(s) − f (x) − ks = −βsgn(s) − εsp0 /q0 − |f (x)|sgn(s) − f (x) + f (x) + x2 u − ks − x˙ 3 (5.16) = x˙ 3 + x2 u − βsgn(s) − εsp0 /q0 − ks − |f (x)|sgn(s) = −˙x3 + z˙ = s˙. Because the switching surface s approaches zero in finite time and so does its derivative, we can confirm the convergence of the approximation error d˜ to zero in the finite time.

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Remark 1. The existence and boundedness of s˙ and s¨ are derived from Eq. (5.16), and Barbalat’s lemma [50] allows us to assume the convergence of s˙ to zero.

3.2 TSMC Tracking Control Approach Consider the following recursive sliding surface structure: σ1 = x2e + x3e , σ2 = σ˙ 1 + ρ1 σ1 p1 +q1 , σ3 = s + σ˙ 2 + ρ2 σ2

(5.17)

p2 +q2

,

where ρi (i = 1, 2) are positive constant coefficients; qi and pi (i = 1, 2) are two positive odd integers with pi < qi . For the sliding surfaces σ2 and σ3 , one can d  p1 /q1  σ , dt 1 d  p2 /q2  σ + s˙, σ˙ 3 = σ¨ 2 + ρ2 dt 2 where the jth-order time derivatives of σ2 and σ3 are achieved as σ˙ 2 = σ¨ 1 + ρ1

(j)

d(j) dt(j) d(j) (j+1) = σ2 + ρ2 (j) dt (j+1)

σ2 = σ1 (j) σ3

+ ρ1

 

p /q1

σ1 1

p /q σ2 2 2

(5.18)

 , 

(5.19) +s . (j)

The following result can be derived using Eqs. (5.3), (5.17): ... σ 1 = d¨ + x¨ 2e u1d + v˙ 2 − u¨ 2d + x2e u¨ 1d + f¨ (x) − f¨ (xd ) + 2˙x2e u˙ 1d .

(5.20)

Based on Eqs. (5.17)–(5.20), one obtains σ˙ 3 =

2  j=1

ρj

d(3−j)  pj /qj  ... σj + s˙ + σ 1 (3−j) dt

= x¨ 2e u1d + 2˙x2e u˙ 1d + v˙ 2 − u¨ 2d + f¨ (x) + s˙ + x2e u¨ 1d − f¨ (xd ) + d¨ +

2  j=1

ρj

d(3−j)  pj /qj  σ . dt(3−j) j

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(5.21)

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147

Theorem 2. For any δ > 0, μ > 0, and ρi (i = 1, 2) > 0, the nonholonomic system (5.1) under the DOB-based TSMC law designed as v˙ 2 = −2˙x2e u˙ 1d + u¨ 2d − x¨ 2e u1d − x2e u¨ 1d 2 (3−j)  p /q  ... p /q ¨ˆ  d j j ¨ + f (xd ) − d − ρj (3−j) σj + s − f¨ (x) − s˙ − δσ3 − μσ3 3 3 dt j=1

(5.22) tracks any desired trajectories (5.3) in a finite time where the switching surfaces σi (i = 1, . . . , 3) are described by the relations (5.17). Proof. Replacing Eq. (5.22) into Eq. (5.21), one has p /q3

σ˙ 3 = −μσ3 3

− δσ3 .

(5.23)

Considering the subsequent Lyapunov function: V(σi ) = 0.5σi2 ,

(5.24)

computing its time derivative along the trajectory of switching surface σ3 yields ˙ 3 ) = σ3 σ˙ 3 V(σ

  p /q = −σ3 δσ3 + μσ3 3 3

(5.25)

≤ −2δV(σ3 ) − 2η3 μV(σ3 )η3 , p3 + q3 . 2q3 Thus, the switching surface σ3 = 0 is reached in the finite time. From Eq. (5.17), we can derive the following result:

where η3 =

p /q2

σ˙ 2 = −ρ2 σ2 2

− s.

(5.26)

The finite-time convergence property of TSMC along with Theorem 1 allows us to confirm the reachability of the sliding surface σ2 = 0. Thus, we can derive from Eq. (5.17): p /q1

σ˙ 1 = −ρ1 σ1 1

.

(5.27)

Deriving Eq. (5.24) along the trajectories of the switching surfaces σi (i = 1, 2) incomes q1

˙ 1 = σ1 σ˙ 1 ≤ −ρ1 σ1  p1 +1 , V p2

˙ 2 = σ2 σ˙ 2 ≤ −ρ2 σ2  q2 +1 . V

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(5.28)

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Using Lemma 1, we conclude that the switching surfaces converge to the origin in the finite time and the zero tracking error is reached in the finite time.

4 SIMULATION RESULTS To assess the performances of the proposed DOB-based TSMC methodology, we implemented it to the wheeled mobile robotic benchmark described by x˙ 1 = u1 , x˙ 2 = u2 ,   √  (5.29)  2 x˙ 3 = 2 1 − x1 x2 − x1 + x2 u1 + cos(0.3π t) − 2 cos 0.2 2t + 3 , where the known nonlinear function is given by   f (x) = 2 1 − x21 x2 − x1 , and the unknown disturbance is assumed to be modeled by   √ d = cos(0.3π t) − 2 cos 0.2 2t + 3 .

(5.30)

Considering the following transformations for the states and control inputs [9]: x1 x2 x3 u˙ 1 u˙ 2

= θ, = y cos θ − x sin θ, = y sin θ + x cos θ , = w2 ,

(5.31)

= w1 − (x cos θ + y sin θ )w2 ,

where (x, y) represent the coordinates of the center of mass, θ is the heading angle measured from the x-axis, and w1 and w2 are the forward velocity of center mass and the angular velocity, respectively. Adopting the reference model described by Eq. (5.2) with:   (5.32) f (xd ) = 2 1 − x21d x2d − x1d . Defining the dynamical model of the tracking errors xie = xi − xid (i = 1, 2, 3) by Eq. (5.3). Choosing the design parameters as k = 10, β = 2, q0 = 5, ε = 3, p0 = 3, q1 = 9, p1 = 5, q2 = 7, p2 = 5, ρ1 = 3, ρ2 = 2, δ = 5, and μ = 1.5, and arbitrarily selecting the initial conditions as u1 (0) = 2, u2 (0) = −1, xe (0) = [1, 0, −1], and z(0) = 0.5 and selecting the reference inputs as u1d = 1 − e−t and u2d = 1 − e−3t , yields the results illustrated in Figs. 5.2–5.4. The time histories of the tracking errors are depicted in Fig. 5.2. II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

149

SIMULATION RESULTS

2

X1e

1 0 –1 0

0.5

1

1.5

2

2.5

3

3.5

4

2.5

3

3.5

4

2.5

3

3.5

4

t (s)

(A)

X2e

0.5

0

–0.5 0

0.5

1

1.5

2 t (s)

(B) 1

X3e

0 –1 –2 0

(C)

0.5

1

1.5

2 t (s)

FIG. 5.2 Dynamics of the tracking errors x1e (A), x2e (B), and x3e (C).

Note the quick convergence of the tracking errors to the origin along with the adequate transient response of the system. The time histories of the switching surfaces are depicted in Fig. 5.3. Note the finite-time convergence of the proposed TSMC sliding surfaces to the origin. The time histories of the control signals are reported in Fig. 5.4. Based on the previous simulation results, we can confirm the effectiveness of the proposed tracking approach in controlling nonholonomic systems with time-varying external disturbances and guaranteeing robust stability and chattering-free dynamics.

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5. FINITE-TIME DISTURBANCE OBSERVER-BASED TRACKING CONTROL DESIGN

1

2

0

1 s1

s

150

–1

0

–2

–1 0

2

4

t (s)

(A)

0

2 t (s)

(B) 200

5

100 s3

10

s2

4

0

0

–100

–5 0

2 t (s)

(C)

0

4

2

4

t (s)

(D)

FIG. 5.3 Time histories of the switching surfaces s (A), σ1 (B), σ2 (C), and σ3 (D). 2

u1

0 u1 –2 u1d –4 0

0.5

1

1.5

2

2.5

3

3.5

4

t (s)

(A)

u2

2

0

u2 u2d

–2 0

(B)

0.5

1

1.5

2

2.5

3

3.5

t (s)

FIG. 5.4 Time histories of the control inputs u1 (A) and u2 (B). II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

4

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5 CONCLUSION This chapter has proposed a DOB-based TSMC approach for nonholonomic systems subject to unknown perturbations. Its main objective is to ensure zero-tracking error in finite time. The proof of finite-time convergence of the proposed approach is established using the Lyapunov theory. Implementation of the proposed approach to a wheeled mobile robotic benchmark proved its robustness, tracking capabilities, and stability. Apart from counteracting the effects of uncertainties and ensuring overall good tracking performance, the proposed approach substantially alleviated the chattering problem. This work can further be extended in the future by implementing an adaptive super twisting global sliding control approach for TSMC to remove the reaching phase and improve the robustness of nonholonomic systems.

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[36] M.P. Aghababa, Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique, Nonlinear Dyn. 69 (1–2) (2012) 247–261. [37] S. Mobayen, Finite-time tracking control of chained-form nonholonomic systems with external disturbances based on recursive terminal sliding mode method, Nonlinear Dyn. 80 (1–2) (2015) 669–683. [38] J. Guo, Y. Luo, K. Li, Dynamic coordinated control for over-actuated autonomous electric vehicles with nonholonomic constraints via nonsingular terminal sliding mode technique, Nonlinear Dyn. 85 (1) (2016) 583–597. [39] S. Li, J. Yang, W.-H. Chen, et al., Generalized extended state observer based control for systems with mismatched uncertainties, IEEE Trans. Ind. Electron. 59 (12) (2012) 4792–4802. [40] W.H. Chen, Disturbance observer based control for nonlinear systems, IEEE/ASME Trans. Mechatron. 9 (4) (2004) 706–710. [41] L. Guo, W.H. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, Int. J. Robust Nonlinear Control 15 (3) (2005) 109–125. [42] Y.-S. Lu, Sliding-mode disturbance observer with switching-gain adaptation and its application to optical disk drives, IEEE Trans. Ind. Electron. 56 (9) (2009) 3743–3750. [43] K.-S. Kim, K.-H. Rew, S. Kim, Disturbance observer for estimating higher order disturbances in time series expansion, IEEE Trans. Autom. Control 55 (8) (2010) 1905–1911. [44] M. Chen, Q.X. Wu, R.X. Cui, Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems, ISA Trans. 52 (2) (2013) 198–206. [45] J. Yang, S. Li, X. Yu, Sliding-mode control for systems with mismatched uncertainties via a disturbance observer, IEEE Trans. Ind. Electron. 60 (1) (2013) 160–169. [46] E. Kim, A fuzzy disturbance observer and its application to control, IEEE Trans. Fuzzy Syst. 10 (1) (2002) 77–84. [47] X. Kun, C. Mou, Terminal sliding mode control with disturbance observer for autonomous mobile robots, in: Control Conference (CCC), 2015 34th Chinese, IEEE, 2015, pp. 765–770. [48] Y.P. Tian, K.C. Cao, Time-varying linear controllers for exponential tracking of non-holonomic systems in chained form, Int. J. Robust Nonlinear Control 17 (7) (2007) 631–647. [49] E. Moulay, W. Perruquetti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appl. 323 (2) (2006) 1430–1443. [50] J.-J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ, 1991.

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C H A P T E R

6 Design of a Composite Control in Two-Time Scale Using Nonlinear Disturbance Observer-Based SMC and Backstepping Control of a Two-Link Flexible Manipulator Kshetrimayum Lochan*, † , Jay Prakash Singh*, Binoy Krishna Roy*, Bidyadhar Subudhi ‡ *Department of Electrical Engineering, National Institute of Technology Silchar, Silchar, India † Department of Mechatronics, Manipal Institute of Technology, Manipal, India ‡ Department of Electrical Engineering, National Institute of Technology Rourkela, Rourkela, India

1 INTRODUCTION Flexible manipulators (FMs) are being used for many applications in industry [1], aerospace [2], medical science [3], home use [4], education [1, 5], chaos theory [6], etc. The above-stated applications of FMs are increasing gradually because of their many inherent advantages [1]. Various types of control problems are considered for an FM in the literature. The commonly used problem is the trajectory tracking for the joint angle and tip position [1]. The precise operation of FMs depends on the modeling method used for the design of the controller. The commonly used modeling methods of an FM are assumed mode method (AMM) [7], lumped parameter method (LPM) [8, 9], and finite element method (FEM) [1, 10]. Among these, the

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most widely used method is AMM [1]. Another interesting modeling method used in co-ordination with the AMM is the singular perturbation (SP) technique [1]. In this technique, the two-time scale separation principle is used where the dynamics are divided into two parts: slow and fast subsystems [7, 11]. The slow system consists of the rigid dynamics and the fast subsystem consists of the flexible dynamics [11]. Thus, using this method, it is easy to design the separate control inputs to achieve the desired performances. The suppression of link deflection of a flexible robot manipulator is also considered an interesting control problem. Various control techniques along with modeling methods are reported in the literature for the quick suppression of links’ deflection of an FM [12–14]. The SP is also used for the suppression of the links’ deflection. The SP modeling method is more appropriate for suppression of link deflections. This is because, in the SP, a separate control can be designed using the fast subsystem for the suppression of links’ deflection. Many types of control techniques are reported in the literature on the control of a TLFM. Some classical control techniques are state feedback control [15] and observer-based control [16] while some robust control techniques such as backstepping control [17], extended state observer [18], sliding mode control (SMC) [19–22], adaptive control [23], adaptive SMC [19, 24, 25], second-order SMC [26–28], hybrid control technique [29], etc. Some soft computing techniques (intelligent control techniques) are also used to control FMs such as fuzzy logic control [30], an artificial neural network [30], genetic algorithm [30], etc. Most of the reported control techniques on the control of a TLFM use a particular modeling method such as AMM, LPM, or FEM. But, the use of the SP modeling approach along with the AMM for designing the controller of a two-link FM is rare. The available literature on the use of the SP modeling approach along with the AMM for a TLFM is classified in Table 6.1. It is apparent from Table 6.1 that the use of the SP for designing the controller for a TLFM is still less explored. It is also observed from the literature that designing of a composite controller for TLFM with unmatched disturbances is not available. Motivated by the previous discussions, this chapter attempts to design a composite control using the SP technique for the trajectory tracking of a TLFM. In this chapter, a composite controller is designed for the trajectory tracking control of a two-link FM. The composite controller consists of a dynamic surface control for the slow subsystem and a backstepping control for the fast subsystem. The nonlinear disturbance observer-based SMC is designed for the exponentially varying trajectory tracking and the backstepping controller is designed for the quick suppression of the links’ deflection.

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INTRODUCTION

TABLE 6.1 Categorization of Composite Controllers Applied on the SP Model of a TLFM References of paper

Slow subsystem

Fast subsystem

Types of controller

Types of controller

PD

PD

[31]

Feedback linearization

Linear quadratic regulator (LQR)

[32]

Computed torque control

LQR

[7]

PD

State feedback control

[33]

VSC

Virtual force control

[34]

PID + ANN

H∞

[35]

Adaptive normal SMC with H∞

LQR

[36]

PID feedback control

PID

[37]

Fuzzy TSMC

Observer-based LQR

[38]

PD

Lyapunov based

[39]

SMC

H∞

[40]

VSC

Lyapunov based

[41]

NN

LQR

[42]

Nonsingular TMC

Observer-based LQR

[43]

LMI-SMC

LMI-based state feedback control

[44]

Nonlinear disturbance observerbased SMC

Backstepping

This work

The following are the contributions of the chapter: A composite control is proposed in this chapter for the trajectory tracking control and quick links’ deflection suppression for a TLFM in the presence of bounded and unmatched disturbances. (ii) A nonlinear disturbance observer is designed to estimate the unmatched disturbances. (iii) The composite controller is designed with the nonlinear disturbance observer-based SMC for the slow subsystem and a backstepping control for the fast subsystem. (iv) The nonlinear disturbance observer-based SMC is designed on the rigid body dynamics of the slow subsystem for the exponentially varying such as signal trajectory tracking and the backstepping control is designed on the flexible body dynamics of the fast subsystem for the quick link deflection suppression. (i)

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The organization of the chapter is as follows. Section 2 discusses the modeling of a two-link FM. Modeling using the SP is shown in Section 3. Section 4 presents the designing of a composite control for the trajectory tracking and quick links’ deflection suppression of a two-link FM. Results and discussion are given in Section 5. This chapter is concluded in Section 6.

2 MODELING OF THE TWO-LINK FM A pictorial view of a two-link FM is shown in Fig. 6.1. The variables used in the schematic of the two-link FM are given in Table 6.2.

FIG. 6.1 Pictorial representation of a two-link FM.

TABLE 6.2 Variables and Their Description for the Two-Link FM Shown in Fig. 6.1 Symbol/variable

Description

τi

Actuator torque applied at ith link

θi

Joint angle of ith link

ui (li , t)

Link deflection of ith link

Mp

Payload mass attached with tip of link 2

(Xi , Yi )

Rigid body coordinate frame with ith link

ˆ ˆ i , Y) (X i

Flexible coordinate frame

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The rigid body dynamics and the flexible body motion of the two-link FM are presented by using θi and ui (li , t), respectively. Here, the dynamic model of the two-link FM is obtained by using the Euler-Bernoulli beam theory [7, 23]. The dynamic model of the manipulator is obtained in the form of a partial differential equation (PDE) along with the boundary conditions describing the links’ motion. The system energy of the dynamics of the two-link FM is obtained using the Lagrangian formation approach with the help of the assumed mode modeling method [7, 23]. The Lagrangian dynamics of the flexible body motion of a two-link FM are described in Eq. (6.1).   ∂((Eki )i − (Epe )i ) d ∂((Eki )i − (Epe )i ) = τi (6.1) − dt ∂qi ∂qi where (Eke )i and (Epi )i are the total kinetic energy and potential energy, respectively, of the ith link and qi is the generalized coordinates consisting of joint angles, joint velocities, and modal coordinates [7]. The total kinetic energy is obtained as (Ek )i = (Kinetic energy due to ith joint) + (Kinetic energy due to ith link)+(Kinetic energy due to Mp ) in the absence of gravity. The PDE for the link deflections is given as (FI)i

∂ 4 ui (li , t) ∂l4i

+ ρi

∂ 2 ui (li , t) ∂l2i

=0

(6.2)

where i: ith link; (EI)i : flexural rigidity; li : length of the link; ρi : density of the link; t: time; and ui (li , t): deflection of the link. A solution of Eq. (6.2) can be obtained by applying proper boundary conditions as discussed in Refs. [45–47]. Assuming that the mass of the links is negligible compared with the mass of the payload, we can write as [45–47]:   ⎧ 4 i (li ,t) ⎨(FI)i ∂ ui (l2 i ,t) = −Jmi d2 ∂u∂l dt i ∂li (6.3) ⎩(FI)i ∂ 3 ui (li ,t) = Mm d2 (ui (li , t)) 3 i dt2 ∂li

where Jmi and Mmi are the mass inertia and moment of inertia at the end of the ith. The finite dimensional equation of the links flexibility ui (li , t) can be expressed using AMM [45] as ui (li , t) =

n

Øik (li )∂ik (t)

j=1

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(6.4)

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where Øik : kth mode shapes (special coordinates) of the ith link; ∂ik : kth modal coordinates (time coordinates) of the ith link; and n: number of assumed modes. The solution of Eq. (6.1) can be obtained using Eq. (6.4) in the form of time-harmonic function and space eigenfunction as in Eq. (6.5) [45].

∂ij (t) = ejϕik t ϕij = G1,i sin(αi , li ) + G2,i cos(αi , li ) + G3,i sin h(αi , li ) + G4,i cosh(αi , li ) (6.5) where ϕi is the natural frequency and αi4 = ϕi4 ρi /(FI)i . Now using the claimed free boundary condition for AMM [7, 23], the constants in Eq. (6.5) are obtained as ⎧ ⎪ ⎨G3,i = −G 1,i , G4,i = −G2,i (6.6) G1,i ⎪ =0 ⎩[f (αi , li )] G2,i The values of αi can be obtained by solving Eqs. (6.5), (6.6). Finally using the Lagrangian expression (6.1), the dynamic model of motion equation of a TLFM obtained using AMM is described as [7, 45]:           θ¨i 0 H1 (θi , δi , θ˙i , δ˙i ) θ˙ τ (6.7) +K + D ˙i = i M(θi , δi ) ¨ + 0 δi H2 (θi , δi , θ˙i , δ˙i ) δi δi where τi : actuated torques; δi , δ˙i : modal displacements and velocities; θi , θ˙i : joint angle and velocity; B: positive definite mass inertia matrix; H1 , H2 : vectors of centrifugal and Coriolis forces; K: positive definite stiffness matrix; and D: positive definite damping matrix.

3 SINGULAR PERTURBATION MODELING OF A TLFM In this section, the SP technique is used to divide the dynamics of a twolink FM into the slow and fast subsystems. The rigid body dynamics of the manipulator help in the design of the slow subsystem and the flexible mode dynamics of the manipulator help in the design of the fast subsystem. Now, the dynamics obtained in these two subsystems are used to design two separate controllers to obtain the desired performances.

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161

The dynamics model obtained in Eq. (6.7) of a two-link FM can be rewritten as [7, 45]:       Hr + Dr θ˙ θ¨ τ (6.8) M ¨ + = i 0 Hf + Df δ˙ + Kδ δ or in a simplified form as [7, 45]: θ¨ = −M11 (Hr + Dr θ˙ ) − M12 (Hf + Df δ˙ + Kδ) + M11 τi δ¨ = −M21 (Hr + Dr θ˙ ) − M22 (Hf + Df δ˙ + Kδ) + M21 τi

(6.9) (6.10)

where θ = [θ1 , θ2 ]T ∈ R2 : vector of joint angle; δ = [δi1 , δi2 ]T ∈ R4 : vector of flexible modes; Dr ∈ R2×2 , Df ∈ R4×4 : damping matrices; K ∈ R4×4 : stiffness matrix; Hr ∈ R2 , Hf ∈ R4 : matrices contacting gravitational, Coriolis, and centripetal forces; and M ∈ R6×6 : inertia matrix. The inertial matrix M can be represented as    Br M11 M12 = M= M21 M22 (Brf )T

Brf Bf

−1 (6.11)

where M11 ∈ R2×2 , M12 ∈ R2×4 , M21 ∈ R4×2 , M22 ∈ R4×4 and Br = [M11 − M12 (M22 )−1 M21 ]−1

(6.12)

Consider new state coordinates δ = εq and variable Ks = εK, where ε is a new parameter and is defined as ε = K1m , and Km is the value of the smallest stiffness. Using the new state coordinates and state variable, the singularly perturbed model of the two-link flexible robot manipulator dynamics (6.8) can be presented [7, 45] as θ¨ = −M11 (Hr + Dr θ˙ ) − M12 (Hf + Df ε˙q + Ks q) + M11 τi ¨ = −M21 (Hr + Dr θ˙ ) − M22 (Hf + Df ε˙q + Kδ) + M21 τi εq

(6.13) (6.14)

A composite control τi is described as τi = τs + τf

(6.15)

where τs and τf are the slow and fast control inputs, respectively.

3.1 Dynamic Model of the Slow Subsystem Choosing ε = 0 in Eq. (6.14), we can obtain the slow subsystem dynamics of the two-link FM given in Eq. (6.7), and solving for q we get [7, 45] II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

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¯ 22 )−1 (M ¯ 21 D ¯ 22 H ¯ 21 τs ) ¯ r θ˙¯ + M ¯r+M ¯f −M ¯ 21 H q¯ = Ks−1 (M

(6.16)

where overbar in the terms represent the variables with ε = 0. Using Eq. (6.16) in Eq. (6.13), we get ¯ 12 (M ¯ 22 )−1 M ¯r−D ¯ r θ¯˙ + τs ) ¯ 21 )(−H ¯ 11 − M θ¨¯ = (M

(6.17)

The expression given in Eq. (6.17) represents the rigid body dynamics of the two-link flexible robot manipulator. Now using Eq. (6.12), the dynamics of the slow subsystem are written in Eq. (6.18). ¯r−D ¯ r θ˙¯ + τs ) θ¨¯ = (B¯ r )−1 (−H

(6.18)

3.2 Dynamic Model of the Fast Subsystem Now, in order to get the dynamics of the fast subsystem, √ the two-time ε and boundary scale method is used. Consider a fast time scale t = τ √ correction terms z1 = q − q¯ and z2 = ε q˙¯ . Thus, using Eq. (6.14), the boundary layer system is written [7, 45] as  dz 1 dτ = z2 (6.19) dz2 = −B21 (Hr + Dr θ˙ ) − B22 (Hf + Df ε˙q + Kδ) + B21 τi dτ

Using the SP method, the slow dynamics variables can be treated as √ dq˙¯ negligible [11], thus dτ = εq˙¯ = 0. Using Eq. (6.16) into Eq. (6.19) with ε = 0, we can write as dz2 ¯ 22 Ks y1 + M21 τf = −M dτ

(6.20)

or the dynamics of the fast subsystem are written in Eq. (6.21). y˙ = Af z + Bf τf where z = [z1 , z2 ]T ∈ R8 and Refs. [7, 45]   0 1 Af = ¯ 22 Ks 0 , −M

(6.21) 

Bf =

0 ¯ 21 M

 (6.22)

The state-space dynamics of the fast subsystem given in Eq. (6.21) are like a linear system with system and input matrices given in Eq. (6.22).

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4 DESIGN OF A COMPOSITE CONTROL USING NDO-BASED SMC AND BACKSTEPPING CONTROL This section discusses the design of a composite control input τi = τs + τf for the two-link FM dynamics discussed in Eq. (6.11). The composite controller is designed with the help of two separate controllers τs and τf for the slow and fast subsystems, respectively.

4.1 Nonlinear Disturbance Observer-Based Control of the Slow Subsystem The dynamics surface control is designed to track the desired trajectory of the TLFM dynamics in Eq. (6.7). The controller is designed using the slow subsystem dynamics of TLFM (6.8). The dynamics of the slow subsystem of the two-link FM are reproduced from Eq. (6.18) as ¯r−D ¯ r θ˙¯ + τs ) θ¨¯ = (B¯ r )−1 (−H

(6.23)

Considering new state variables as x1 = θ¯ , x2 = θ˙¯ , the dynamics of the slow subsystem given in Eq. (6.23) can be written as  x˙ 1 = x2 = θ˙¯ (6.24) ¯r−D ¯ r x2 + τs ) x˙ 2 = (B¯ r )−1 (−H Now, the slow subsystem dynamics defined in Eq. (6.24) are used to design nonlinear disturbance observer-based SMC for the regulation of the joint angle. Let y = x1 , x = [x1 , x2 ]T , then Eq. (6.24) can be written as x˙ = f (x) + f1 (x)u + f2 (d)d

(6.25)

¯ r −D ¯ r x2 +τs )]T , f1 (x) = [0, (B¯ r )−1 ]T , let (B¯ r )−1 = where f (x) = [x2 , (B¯ r )−1 (−H T g(x), f2 (d) = [1, 0] . A nonlinear DOB (NDOB), which can estimate the disturbances in Eq. (6.25), is introduced and is given in Eq. (6.26) [48].

˙ = −nf2 R − n[f2 nx + f (x) + f1 u] R (6.26) dˆ = R + nx ˆ R, and n are the estimation of disturbance, the internal state of where d, the nonlinear disturbance observer, and the observer gain to be designed, respectively. Let xd be the desired trajectory for the slow subsystem dynamics of the manipulator defined in Eq. (6.24), then the tracking error is defined as e = x − xd

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(6.27)

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Now, using the tracking error e = [e1 , e2 ]T , a sliding surface s is defined in Eq. (6.28) for system (6.25) using unmatched disturbances estimation of Eq. (6.26). s = e2 + ke1 + dˆ

(6.28)

The designed disturbance observer-based SMC law is defined as ¯ r g(x) − x¨ d + ke2 + kdˆ + ρ tanh(s)] τs = −g(x)−1 [−g(x)Dr x2 − H

(6.29)

Note 1. In the control input (6.29), the tan hyperbolic function is used instead of the signum function, which is preferred to avoid chattering. where g(x) = (B¯ r )−1 . The block diagram for the implementation of the nonlinear disturbance observer-based SMC designed is given in Fig. 6.2. 4.1.1 Stability Analysis Assumption 1. The derivative of the disturbances (d) is bounded and ˙ = 0. satisfies limt→∞ d(t) Lemma 1 (Yang et al. [48]). Suppose that Assumption 1 is satisfied for system (6.24) and if the observer gain n is chosen such that the nf2 > 0 holds, that is, e˙dt (t) + nf2 (x)edt (t) = 0

FIG. 6.2 Block diagram of the nonlinear disturbance observer-based SMC.

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(6.30)

DESIGN OF A COMPOSITE CONTROL

165

then the estimation of disturbance dˆ of NDO (6.26) can track the disturbance d of system (6.24). Assumption 2. Suppose the estimation error (in Eq. 6.30) of the disturbance is bounded, and is given by e∗dt = max |edt (t)| t>0

(6.31)

Theorem 1. Suppose that the system in Eq. (6.24) is controlled by Eq. (6.30) with the assumption (6.1) and if the switching gain (ρ) in Eq. (6.29) is designed such that ρ > (k + nf2 )e∗dt and nf2 > 0 holds, then the joint angle of the manipulator follows the desired trajectory. Proof. Consider a Lyapunov function candidate as 1 T ss 2 Taking the derivative of Eq. (6.32) we can write as V1 (s) =

˙ 1 (s) = s˙sT V

(6.32)

(6.33)

Now the derivative of the sliding surface form (6.28) along with Eq. (6.24) is written as d2 x1 ˙ − x¨ d + ke2 + kd + dˆ dt2 Now using Eq. (6.24) in Eq. (6.34) we can write as s˙ =

¯r−D ¯ r x2 + τs ) − x¨ d + ke2 + kd + d˙ˆ s˙ = g(x)(−H

(6.34)

(6.35)

Now using the control torque of the slow system (6.29) in Eq. (6.35), it yields ˆ + d˙ˆ s˙ = −ρ tanh(s) + k[d − d]

(6.36)

From Eq. (6.26) we can write as ˙ dˆ = −nf2 (dˆ − d)

(6.37)

Now using Eq. (6.37) in Eq. (6.35), we can write as ˆ − nf2 (dˆ − d) s˙ = −ρ tanh(s) + k[d − d]

(6.38)

In terms of the error of the disturbance and arranging some terms, we can write Eq. (6.38) as s˙ = −ρ tanh(s) + [k + nf2 ]edt

(6.39)

Substituting Eq. (6.39) in Eq. (6.33), the derivative of the Lyapunov function is given as ˙ 1 (s) = sT {−ρk tanh(s) + [k + nf2 ]e_dt } V

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(6.40)

166

6. DESIGN OF A COMPOSITE CONTROL IN TWO-TIME SCALE

Now, using the boundedness of the error considered in Assumption 2, Eq. (6.40) can be written as ˙ 1 (s) ≤ −{ρ − [k + nf2 ]e∗ }|s| V d

(6.41)

Finally, Eq. (6.41) can be written in terms of the Lyapunov function as √ ˙ 1 (s) ≤ − 2{ρ − [k + nf2 ]e∗ }V 1/2 (6.42) V dt 1 It is seen from Eq. (6.42) that the derivative of the Lyapunov function is negative definite when ρ > [k + nf2 ]e∗dt . Thus, we can say that the error dynamics of the manipulator reach the defined sliding surface s = 0 (6.28) in the finite time. The error dynamics operate on the sliding surface when it satisfies s = 0, s˙ = 0. Using s˙ = 0, the equivalent sliding mode dynamics are described as ˙ˆ e˙2 = −(ke2 + d)

(6.43)

Now, to establish the sliding mode dynamics using Lyapunov theory, let us consider a candidate positive definite Lyapunov function as 1 2 1 ˜2 e + d 2 2 2 Taking the derivative of Eq. (6.44) and using Eq. (6.43), we write     ˙ ˙ v˙ s1 = e2 −ke2 − dˆ + d˜ −dˆ vs1 =

(6.44)

(6.45)

Separating the variables in Eq. (6.45) and using Eq. (6.37), it is written as ˜ 2 + d) ˜ v˙ s1 = −ke22 − nf2 d(e ˜ 2 − d˜ 2 nf2 e2 v˙ s1 = −ke2 − nf2 de 2

Writing Eq. (6.47) in quadratic form as   −k v˙ s1 = e2 d˜ nf − 22

nf

− 22 −nf2

  e2 d˜

(6.46) (6.47)

(6.48)

It is seen from Eq. (6.48) that −˙vs1 is positive definite if the matrix is n2 f 2

positive definite if the condition satisfied (i) k > 0 and (ii) (nf2 )k > − 4 2 . Therefore, according to the Lyapunov stability theory, the sliding motion on the sliding surface s = 0 is stable and ensures the convergence of error dynamics. Remark 1. To ensure the complete stability of the sliding surface s, the gain of the sliding surface of the nonlinear disturbance observer-based SMC switching law should be selected such that it satisfied ρ > [k + nf2 ]e∗dt . Now, it is ensured that the disturbance is estimated effectively using the ˆ is disturbance observer and the magnitude of the disturbance error |d − d| expected to converge to zero.

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DESIGN OF A COMPOSITE CONTROL

167

Remark 2. From Eq. (6.37), it is seen that in the absence of disturbances, ˙ ˙ ˆ which concludes that dˆ = 0 if the initial value we can write as dˆ = −nf2 d(t), ˆ is selected as d(t) ˆ = 0. of d(t)

4.2 Backstepping Control for the Fast Subsystem In the previous section, a nonlinear disturbance observer-based SMC was designed for the tracking control of the slow system of the FM. In this section, a backstepping control technique is designed on fast subsystem dynamics for quick suppression of link deflection of the FM. Considering zf = y, the dynamics of the fast subsystem of the FM from Eqs. (6.21), (6.22) are rewritten as

z˙ 1f = z2f (6.49) z˙ 2f = −Af 3 z1f + Bf 2 τf where Af 3 = B¯ 22 Ks and Bf 2 = B¯ 21 . The links deflection errors are obtained from Eq. (6.49) in Eqs. (6.50), (6.51). e1ft = zdt − z1f

(6.50)

e2ft = vdt − z2f

(6.51)

where zdt is a twice-differentiable desired link deflection for the FM and vdt is a virtual control term. Now, using Eqs. (6.50), (6.51), the error dynamics are obtained in Eqs. (6.52), (6.53). e˙1ft = z˙ dt − z2f

(6.52)

e˙2ft = v˙ dt + Af 3 z1f − Bf 2 τf

(6.53)

The control law designed for quick suppression of the tip deflection of the FM using the backstepping control on the fast subsystem is obtained using Theorem 2. Theorem 2. Suppose the fast subsystem of manipulator dynamics (6.49) is controlled using the backstepping control law defined in Eq. (6.54) along with the error variables (6.52), (6.53). Then, the fast subsystem of the two-link FM follows the desired deflection zdt . τf = (Bf 2 )−1 (˙vd + Af 3 z1f + k2b e2ft )

(6.54)

Proof. The designing of a backstepping controller for the fast subsystem of TLFM (Eq. 6.49) is achieved using the following steps: Step 1: Consider a Lyapunov function candidate as v1f =

1 2 e 2 1ft

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(6.55)

168

6. DESIGN OF A COMPOSITE CONTROL IN TWO-TIME SCALE

The time derivative of Eq. (6.55) is written in Eq. (6.56) using Eqs. (6.52), (6.53) as v˙ 1ft = e1ft (˙zd + e2ft − vd ) = e1ft z˙ d − e1ft vd + e1ft e2ft

(6.56)

The derivative of the Lyapunov function in Eq. (6.56) is negative definiteness if the virtual control variable vd is selected as vd = z˙ dt + R1b e1ft + e2ft

(6.57)

where R1b > 0 is a positive definite matrix. Now using Eq. (6.57) the derivative of the Lyapunov function (6.56) can be written as v˙ 1f = −R1b e21ft

(6.58)

It is seen that in Eq. (6.58), v˙ 1f is a negative definite function. Thus, the first state variable z1f of the fast subsystem in Eq. (6.49) is now stabilized. The next step in designing the backstepping control is to show the stability of the second state variable and to obtain the control input τf for the fast subsystem. Step 2: The stability of the second state variable z1f is shown by considering another Lyapunov function candidate as 1 (6.59) v2f = v1f + e22ft 2 Now, using the second error variable from Eq. (6.56) and the derivative of the Lyapunov function (6.58), the time derivative of Eq. (6.59) is written in Eq. (6.60). v˙ 2f = −R1b e21ft + e2ft (˙vd + Af 3 z1f − Bf 2 τf )

(6.60)

It is seen from Eq. (6.60) that it is negative definite if the torque input for the fast subsystem τf is selected as in Eq. (6.61). τf = (Bf 2 )−1 (˙vd + Af 3 z1f + R2b e2ft )

(6.61)

Using the torque input of the fast subsystem τf defined in Eq. (6.61), the derivative of the Lyapunov function v˙ 2f in Eq. (6.60) is written Eq. (6.62). v˙ 2f = −(R1b e21ft + R2b e22ft )

(6.62)

Now, using the Lyapunov stability theory we can say that Eq. (6.62) is a negative definite function with requiring the condition that R1b , R2b are positive constant matrices. Therefore, we can say that the error variables e1ft and e2ft are asymptotically stable. The quick and fast stabilization of error variables e1ft and e2ft depends on the selection of the parameter R1b , R2b . Thus, the

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RESULTS AND DISCUSSION FOR THE COMPOSITE CONTROL

169

FIG. 6.3 Block diagram of the designed composite control technique.

deflections of the links of the TLFM are suppressed to their desired values, that is, at zero by designing the backstepping control on its fast subsystem.

Structure of Composite Controller for the Two-Link FM (6.11)

It is seen in Sections 4.1 and 4.2 that two different types of controllers, namely the nonlinear disturbance observer-based SMC and a backstepping controller, are designed on the slow and fast subsystem dynamics, respectively, of the two-link FM (6.11). Using the controller in the slow and fast subsystems, a composite controller is designed. The structure of the composite controller designed for the trajectory tracking and the quick suppression of the links deflections of the two-link FM (6.11) is presented in Fig. 6.3.

5 RESULTS AND DISCUSSION FOR THE COMPOSITE CONTROL This section discusses the results and the trajectory tracking control of the TLFM. The parameters of the TLFM used for system (6.7) are given in Table 6.3. The initial condition considered for simulating the TLFM dynamics in Eq. (6.11) and the disturbance observer (6.37) are (θ (0), δ(0)) = ˆ ˙ = (0, 0, 0, 0, 0, 0)T , d(0) = (0, 0)T . The (0.1, 0.1, 0.0, 0.0, 0.0, 0.0)T , (θ˙ (0), δ(0))

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6. DESIGN OF A COMPOSITE CONTROL IN TWO-TIME SCALE

TABLE 6.3 Parameters of a Physical Two-Link FM [49] Mass of link 1, m1 = 0.15268 kg

Coefficients of viscous damping, Beq1 = 4 Nms/rad, Beq2 = 1.5 Nms/rad

Mass of link 2, m2 = 0.0535 kg

Efficiency of gear boxes, ηg1 = 0.85, ηg2 = 0.9

Length of link 1, L1 = 0.201 m

Efficiency of motors, ηm1 = 0.85, ηm2 = 0.85

Length of link 2, L2 = 0.201 m

Constants of back EMF, Km1 = 0.119 V/rad, Km2 = 0.0234 V/rad

Resistance of armatures, Rm1 = 11.5 , Rm2 = 2.32 

Gear ratio, Kg1 = 100, Kg2 = 50

Equivalent MI at load, Jeq1 = 0.17043 kg m2

Motor torque constants Kt1 = 0.119 Nm/A, Kt2 = 0.0234 Nm/A

Equivalent MI at load, Jeq2 = 0.0064387 kg m2

Stiffness of the links, Ks1 = 22 Nm/rad, Ks2 = 2.5 Nm/rad

Link 1 MI, Jarm1 = 0.002035 kg m2

Link 2 MI, Jarm2 = 0.0007204 kg m2

value of the gains used for the nonlinear disturbance observer-based SMC and the backstepping controller is ⎡ ⎤ 10 0 0 0     ⎢ 0 10 0 0 ⎥ 10 0 10 0 ⎥ k= , n= , ρ = [10, 10]T , R1b = ⎢ ⎣ 0 0 10 0 ⎦ , 0 10 0 10 0 0 0 10 ⎡ ⎤ 10 0 0 0 ⎢ 0 10 0 ⎥ 0 ⎥ R2b = ⎢ ⎣0 0 10 0 ⎦ 0 0 0 10 These gains are considered in a manner to achieve the better tracking performances but using lesser control efforts. The desired trajectory considered for the trajectory tracking is given in Eq. (6.63).   3   ⎧ 7 ⎨yd1 = π4 − 76 e− 2 t + 19 e−2t     (6.63) ⎩y = π − 7 e− 32 t + 7 e−2t d2 6 5 11 External disturbances are considered as in Eq. (6.64).  d = [0.1 sin(t), 0.1 sin(t)]T

II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

(6.64)

171

RESULTS AND DISCUSSION FOR THE COMPOSITE CONTROL

5.1 Simulation Results With the Nominal Payload (0.145 kg) Trajectory tracking and link deflection suppression with the nominal payload (0.145 kg) and in the presence of unmatched and bounded disturbances in Eq. (6.64) for the two-link FM (6.11) using nonlinear disturbance observer-based SMC and backstepping control in the fast subsystem are discussed here. Here, the exponentially varying desired trajectory is used for the slow system given in Eq. (6.63). Fig. 6.4 shows the behavior of the trajectory tracking for both the links of the TLFM. It is observed from Fig. 6.4 that the trajectory tracking for an exponentially varying desired signal (6.63) is achieved within 0.1 s. The modes of both the links with the nominal payload of 0.145 kg using the composite controller are shown in Figs. 6.5 and 6.6. It is noted from Figs. 6.5 and 6.6 that the flexible modes are suppressed quickly and are of low value. The nature of the tip deflections for both the links of the two-link FM are given in Fig. 6.7. It is observed from Fig. 6.7 that the tip deflections of both the links are suppressed within 10−2 and 10−3 ranges. The required control torque inputs in the slow subsystem using the nonlinear disturbance observer-based SMC are shown in Fig. 6.8. The control inputs required in the fast subsystem backstepping control are shown in Fig. 6.9. The nature of the composite control inputs consisting

q1, q1d (rad)

1 0.5

q1 q1d

0 –0.5

(A)

0

1

2

3

4

5

t (s)

q2, q2d (rad)

0.6 0.4 q2 0.2 0 –0.2

(B)

q2d

0

1

2

3

4

5

t (s)

FIG. 6.4 Trajectory tracking for both the links of the two-link FM (6.11) using nonlinear disturbance observer-based SMC with a 0.145 kg payload. (A) Link-1 and (B) link-2.

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6. DESIGN OF A COMPOSITE CONTROL IN TWO-TIME SCALE

(rad)

0.05

11

0

–0.05 4

2

0

(A)

6 t (s)

8

10

12

10–4

(rad)

5

12

0

–5 2

0

4

6

8

10

12

t (s)

(B)

FIG. 6.5 Characteristics of the modes of link 1 with a 0.145 kg payload in the fast subsystem of the two-link FM (6.11) using backstepping control. (A) Link-1 and (B) link-2.

10–3

d 21 (rad)

10

5

0

2

0

(A)

4

6

8

10

12

8

10

12

t (s) 10–5

d 22 (rad)

5

0

–5 0

(B)

2

4

6

t (s)

FIG. 6.6 Characteristics of the modes of link 2 with a 0.145 kg payload in the fast subsystem of the two-link FM (6.11) using backstepping control. (A) Link-1 and (B) link-2.

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RESULTS AND DISCUSSION FOR THE COMPOSITE CONTROL

173

u1 (rad)

0.1

0

–0.1 0

2

4

2

4

(A)

6 t (s)

8

10

12

6

8

10

12

u1 (rad)

0.01

0

–0.01 0

(B)

t (s)

FIG. 6.7 Characteristics of tip deflections of both the links for the two-link FM with a 0.145 kg payload. (A) Link-1 and (B) link-2.

t s1 (Nm)

10

0

–10 0

1

2

(A)

3

4

5

3

4

5

t (s)

t s2 (Nm)

40 20 0 –20 0

(B)

1

2 t (s)

FIG. 6.8 Control inputs for the trajectory tracking of the slow subsystem of the two-link FM (6.11) using nonlinear disturbance observer-based SMC with payload 0.145 kg. (A) Link-1 and (B) link-2.

of nonlinear disturbance observer-based SMC and backstepping control is given in Fig. 6.10. It is noted from Fig. 6.10 that the required control inputs using the composite control inputs for both the links are initially high but after some time, they decayed within small values. It is seen from Figs. 6.4, 6.7, and 6.10 that the trajectory tracking and deflection suppression for the

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174

6. DESIGN OF A COMPOSITE CONTROL IN TWO-TIME SCALE

t f1 (Nm)

50 0 –50 –100 –150

0

0.5

1

1.5

2

0

0.5

1

1.5

2

(A)

2.5 t (s)

3

3.5

4

4.5

5

2.5

3

3.5

4

4.5

5

60 t f2 (Nm)

40 20 0 –20

(B)

t (s)

FIG. 6.9 Control inputs for the suppression of links deflection of the fast subsystem of the two-link FM (6.11) using backstepping control with payload 0.145 kg. (A) Link-1 and (B) link-2.

t 1 (Nm)

100 0 –100 –200

0

1

2

(A)

3

4

5

3

4

5

t (s)

t 2 (Nm)

100 50 0 –50 0

(B)

1

2 t (s)

FIG. 6.10 Composite control inputs for the trajectory tracking and link deflection of the two-link FM (6.11) using nonlinear disturbance observer-based SMC and backstepping control with payload 0.145 kg. (A) Link-1 and (B) link-2.

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RESULTS AND DISCUSSION FOR THE COMPOSITE CONTROL

two-link FM (6.7) are achieved properly with the effective control torque inputs, even in the presence of unmatched disturbances.

5.2 Trajectory Tracking and Link Deflection Suppression With a 0.3 kg Payload This section describes the robustness of the composite controllers designed for the trajectory tracking and link deflection suppression of the two-link FM in Eq. (6.7) with a payload of 0.3 kg. The trajectory tracking of both the links with a 0.3 kg payload is shown in Fig. 6.11. The behaviors of modes of both the links with a 0.3 kg payload are shown in Figs. 6.12 and 6.13. The tip deflections of both the links are shown in Fig. 6.14. It is noted from Fig. 6.14 that with an increase in a payload, there is a small increase in the magnitude of deflection for both the links. The behavior of the control inputs used for the slow and fast subsystem with the increase in the 0.3 kg payload using nonlinear disturbance observer-based MSC and backstepping control in the slow and fast subsystems, respectively, is shown in Figs. 6.15 and 6.16. The behavior of the composite control inputs with a 0.3 kg payload is shown in Fig. 6.17. It is noted from Fig. 6.17 that with an increase in the payload the required control inputs are more in comparison with the nominal payload. Thus, it is apparent from Figs. 6.11, 6.14, and 6.17 that the trajectory tracking and

q 1,q 1d (rad)

1 0.5

q1 0 –0.5

q 1d 0

1

2

(A)

3

4

5

t (s)

q 2,q 2d (rad)

1 0.5

q2 0 –0.5

(B)

q 2d 0

1

2

3

4

5

t (s)

FIG. 6.11 Trajectory tracking for both the links of the two-link FM (6.11) using nonlinear disturbance observer-based SMC with a 0.3 kg payload. (A) Link-1 and (B) link-2.

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6. DESIGN OF A COMPOSITE CONTROL IN TWO-TIME SCALE

d11 (rad)

0.1

0

–0.1 2

0

4

(A)

6

8

10

12

8

10

12

t (s) 10

-4

d12 (rad)

5

0

–5 0

2

4

(B)

6

t (s)

FIG. 6.12 Characteristics of the modes of link 1 with a 0.3 kg payload in the fast subsystem of the two-link FM (6.11) using backstepping control. (A) First and (B) second modes.

10–3

d 21 (rad)

10 5 0 –5

0

2

4

2

4

(A)

6 t (s)

8

10

12

8

10

12

10–5

d 22 (rad)

5

0

–5 0

(B)

6 t (s)

FIG. 6.13 Characteristics of the modes of link 2 with a 0.3 kg payload in the fast subsystem of the two-link FM (6.11) using backstepping control. (A) First and (B) second modes.

deflection suppression for the two-link FM (6.7) are achieved properly in the presence of an increase in payload, and with unmatched and bounded disturbances. Therefore, the composite controller designed in this chapter works effectively.

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177

u1 (rad)

0.1

0

–0.1

0

2

4

0

2

4

(A)

6 t (s)

8

10

12

6

8

10

12

u2 (rad)

0.01

0

–0.01

(B)

t (s)

FIG. 6.14 Characteristics of tip deflections of both the links for the two-link FM with a 0.3 kg payload. (A) Link-1 and (b) link-2.

t s1 (Nm)

10

0

–10

0

1

2

(A)

3

4

5

3

4

5

t (s)

t s2 (Nm)

40 20 0 0

(B)

1

2

t (s)

FIG. 6.15 Control inputs for the trajectory tracking of the slow subsystem of the two-link FM (6.11) using nonlinear disturbance observer-based SMC with payload 0.3 kg. (A) Link-1 and (b) link-2.

5.3 Comparison With the Results in Lochan et al. [50] This section presents the comparison of the performance of the controller designed in this chapter to the controller available in Lochan et al. [50]. The comparison of the controller is presented in Table 6.4.

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6. DESIGN OF A COMPOSITE CONTROL IN TWO-TIME SCALE

t f1 (Nm)

100 0 –100 –200

0

0.5

1

1.5

2

0

0.5

1

1.5

2

(A)

2.5 t (s)

3

3.5

4

4.5

5

2.5

3

3.5

4

4.5

5

t f2 (Nm)

100 50 0 –50

(B)

t (s)

FIG. 6.16 Control inputs for the suppression of link deflection of the fast subsystem of the two-link FM (6.11) using backstepping control with payload 0.3 kg. (A) Link-1 and (b) link-2.

t 1 (Nm)

10 5 0 –5 0

5

0

5

10 t (s)

15

20

10

15

20

t 2 (Nm)

20

0

–20

t (s)

FIG. 6.17 Composite control inputs for the trajectory tracking and link deflection of the two-link FM (6.11) using nonlinear disturbance observer-based SMC and backstepping control with payload 0.3 kg. (A) Link-1 and (B) link-2.

It is seen from Table 6.4 that the results obtained using the controller designed in this chapter have better performance than the results obtained using the controller designed in Lochan et al. [50], in terms of smaller tracking time and lesser control efforts.

II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

Details of controller

Controller in Lochan et al. [50] with a 0.145 kg payload

Controller in this chapter with a 0.145 kg payload

Structure of composite controller

Slow subsystem

Fast subsystem

Slow subsystem

Fast subsystem

Adaptive SMC

Backstepping

Nonlinear disturbance observer-based SMC

Backstepping

Trajectory tracking time

Link 1: 6 s, Link 2: 6 s

Link 1: 0.3 s, Link 2: 0.4 s

Maximum amount of tip deflection (in mm)

Link 1: 0.02, Link 2: 0.002

Link 1: 0.08, Link 2: 0.009

Maximum required control torque (in Nm)

Link 1: 200, Link 2: 40

Link 1: 40, Link 2: 20

RESULTS AND DISCUSSION FOR THE COMPOSITE CONTROL

II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

TABLE 6.4 Comparison of the Results With the Controller in Lochan et al. [50] and the Controller Designed in This Chapter

179

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6 CONCLUSION In this chapter, a composite control technique is designed for trajectory tracking control and link deflection suppression of a two-link FM is reported. We know that the dynamics of a two-link FM are nonlinear and complex; hence the trajectory tracking and link deflection suppression in the presence of unmatched disturbances for such a system is a challenging task. Therefore, the dynamics of a two-link FM are first obtained using the AMM. Then, the complete dynamics are divided into two subsystems consisting of the slow and fast subsystems using the two-time scale separation principle with the help of the SP technique. The slow subsystem deals with the rigid body dynamics of the manipulator and the fast subsystem deals with the flexible mode dynamics of the manipulator. Using these two subsystems, a composite controller using the controller of each subsystem is designed. The composite controller consists of a nonlinear disturbance observer-based SMC control in the slow subsystem for trajectory tracking and a backstepping-based controller in the fast subsystem of the two-link FM for the suppression of the link deflection. The disturbance observer is used to estimate the unmatched disturbances and the SMC is designed for the faster and precise tracking of the desired trajectory. It is seen from the Matlab simulation results that the desired trajectory tracking in the presence of unmatched disturbances and quick link deflection suppression is achieved effectively for a two-link FM.

References [1] K. Lochan, B.K. Roy, B. Subudhi, A review on two-link flexible manipulators, Annu. Rev. Control 42 (2016) 346–367, https://doi.org/10.1016/j.arcontrol.2016.09.019. [2] M. Sabatini, P. Gasbarri, R. Monti, G.B. Palmerini, Vibration control of a flexible space manipulator during on orbit operations, Acta Astronaut. 73 (2012) 109–121, https://doi.org/10.1016/j.actaastro.2011.11.012. [3] A. Arora, Y. Ambe, T.H. Kim, R. Ariizumi, F. Matsuno, Development of a maneuverable flexible manipulator for minimally invasive surgery with varied stiffness, Artif. Life Robot. 19 (2014) 340–346, https://doi.org/10.1007/s10015-014-0184-7. [4] T. Nakamura, N. Saga, M. Nakazawa, T. Kawamura, Development of a soft manipulator using a smart flexible joint for safe contact with humans, in: IEEE/ASME Int. Conf. Adv. Intell. Mechatronics, AIM, 2003, pp. 441–446, https://doi.org/10.1109/AIM.2003.1225136. [5] C.T. Kiang, A. Spowage, C.K. Yoong, Review of control and sensor system of flexible manipulator, J. Intell. Robot. Syst. Theory Appl. 77 (2014) 187–213, https://doi.org/10.1007/s10846-014-0071-4. [6] J.P. Singh, K. Lochan, N.V. Kuznetsov, B.K. Roy, Coexistence of single- and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria, Nonlinear Dyn. 90 (2017) 1277–1299, https://doi.org/10.1007/s11071-017-3726-4. [7] B. Subudhi, A.S. Morris, Dynamic modelling, simulation and control of a manipulator with flexible links and joints, Rob. Auton. Syst. 41 (2002) 257–270, https://doi.org/10. 1016/S0921-8890(02)00295-6.

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[27] K. Lochan, B.K. Roy, Second-order SMC for tip trajectory tracking and tip deflection suppression of an AMM modelled nonlinear TLFM, Int. J. Dyn. Control (2017) 1–25, https://doi.org/10.1007/s40435-017-0371-1. [28] K.L.J.P. Singh, B.K.R.B. Subudhi, Adaptive time-varying super-twisting global SMC for projective synchronisation of flexible manipulator, Nonlinear Dyn. 93 (2018) 2071–2088, https://doi.org/10.1007/s11071-018-4308-9. [29] S.K. Pradhan, B. Subudhi, Real-time adaptive control of a flexible manipulator using reinforcement learning, IEEE Trans. Autom. Sci. Eng. 9 (2012) 237–249, https://doi.org/10.1109/TASE.2012.2189004. [30] B. Subudhi, A.S. Morris, Soft computing methods applied to the control of a flexible robot manipulator, Appl. Soft Comput. 9 (2009) 149–158, https://doi.org/10.1016/j. asoc.2008.02.004. [31] F. Khorrami, S. Jain, A. Tzes, Experiments on rigid body-based controllers with input preshaping for a two-link flexible manipulator, IEEE Trans. Robot. Autom. 10 (1994) 55–65. [32] F. Khorrami, S. Jain, Non-linear control with end-point acceleration feedback for a two-link flexible manipulator: experimental results, J. Robot. Syst. 10 (1993) 505–530, https://doi.org/10.1002/rob.4620100407. [33] X.U. Bo, Y. Bakakawa, Control two-link flexible manipulators using controlled Lagrangian method, in: SICE Annu. Conf. Sapporo, 2004, pp. 289–294. [34] S.H. Lee, C.W. Lee, Hybrid control scheme for robust tracking of two-link flexible manipulator, J. Intell. Robot. Syst. 34 (2002) 431–452, https://doi.org/10.1023/A: 1014286327294. [35] Y. Li, G. Liu, T. Hong, K. Liu, Robust control of a two-link flexible manipulator with quasi-static deflection compensation using neural networks, J. Intell. Robot. Syst. 44 (2005) 263–276, https://doi.org/10.1007/s10846-005-9019-z. [36] Y. Zhang, Y. Mi, M. Zhu, F. Lu, Adaptive sliding mode control for two-link flexible manipulator with H infinity tracking, in: Proc. Fourth Int. Conf. Mach. Learn. Cybern., Guangzhou, 2005, pp. 702–707. [37] F. Matsuno, K. Yamamoto, Dynamic hybrid force/position control of two degree of freedom flexible manipulator, J. Robot. Syst. 11 (1994) 355–366. [38] Y. Wang, Y. Feng, X. Yu, Fuzzy terminal sliding mode control of two-link flexible manipulators, in: 34th IEEE Annu. Conf. Ind. Electron., 2008, pp. 1620–1625, https://doi.org/10.1109/IECON.2008.4758196. [39] A. Ashayeri, M. Eghtesad, M. Farid, Trajectory tracking for two-link flexible arm via two-time scale and boundary control methods, in: Proc. IMECE2008, 2008 ASME Int. Mech. Eng. Congr. Expo., 2008, pp. 1–9. [40] Y.C. Li, B.J. Tang, Z.X. Shi, Y.F. Lu, Experimental study for trajectory tracking of a two-link flexible manipulator, Int. J. Syst. Sci. 31 (2010) 3–9, https://doi.org/10.1080/002077200291398. [41] E. Mirzaee, M. Eghtesad, S.A. Fazelzadeh, Maneuver control and active vibration suppression of a two-link flexible arm using a hybrid variable structure/Lyapunov control design, Acta Astronaut. 67 (2010) 1218–1232, https://doi.org/10.1016/j.actaastro.2010. 06.054. [42] D. Yue-Jiao, C. Xi, Z. Ming, R. Jun, Anti-windup for two-link flexible arms with actuator saturation using neural network, in: 2010 Int. Conf. E-Product E-Service E-Entertainment (ICEEE), 2010, pp. 6–9. [43] Y. Wang, F. Han, Y. Feng, H. Xia, Hybrid continuous nonsingular terminal sliding mode control of uncertain flexible manipulators, in: 40th IEEE Annu. Conf. Ind. Electron. Soc., 2014, pp. 190–196, https://doi.org/10.1109/IECON.2009.5415316. [44] K. Lochan, R. Dey, B.K. Roy, B. Subudhi, Tracking control with vibration suppression of a two-link flexible manipulator using singular perturbation with composite control design, in: V. Balas, L. Jain, M. Balas (Eds.), Soft Comput. Appl. SOFA 2016. Adv. Intell. Syst. Comput., Springer, Berlin, Heidelberg, 2018, pp. 365–377.

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[45] A. De Luca, B. Siciliano, Closed-form dynamic model of planar multilink lightweight robots, IEEE Trans. Syst. Man Cybern. 21 (1991) 826–839, https://doi.org/10.1109/21.108300. [46] B. Subudhi, S. Ranasingh, A.K. Swain, Evolutionary computation approaches to tip position controller design for a two-link flexible manipulator, Arch. Control Sci. 21 (2011) 269–285, https://doi.org/10.2478/v10170-010-0043-2. [47] B. Subudhi, S.K. Pradhan, A flexible robotic control experiment for teaching nonlinear adaptive control, Int. J. Electr. Eng. Educ. (2016), https://doi.org/10.1177/ 0020720916631159. [48] J. Yang, S. Li, S. Member, X. Yu, Sliding-mode control for systems with mismatched uncertainties via a disturbance observer, IEEE Trans. Ind. Electron. 60 (2013) 160–169. [49] QUANSER, Equation for the First (Second) Stage of the 2DOF Serial Flexible Link Robot, QUANSER, 2006, https://www.quanser.com/products/2-dof-serial-flexible-link/. [50] K. Lochan, J.P. Singh, B.K. Roy, Hidden chaotic path planning and control of a flexible robot manipulator, in: V.-T. Pham, S. Vaidyanathan, C. Volos, T. Kapitaniak (Eds.), Nonlinear Dyn. Syst. With Self-Excited Hidden Attractors, Springer, India, 2018, pp. 433–463.

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C H A P T E R

7 Design of Observer-Based Tracking Controller for Robotic Manipulators Saleh Mobayen*, Olfa Boubaker † , Afef Fekih ‡ *Advanced

Control Systems Laboratory, Department of Electrical Engineering, University of Zanjan, Zanjan, Iran † University of Carthage, National Institute of Applied Sciences and Technology, Tunis, Tunisia ‡ Department of Electrical and Computer Engineering, University of Louisiana at Lafayette, Lafayette, LA, United States

1 INTRODUCTION The field of robotics has attracted a great deal of research in recent decades. Robots with widespread applications in various fields such as space exploration, military applications, underwater vehicles, assembly lines, painting, and medical equipment have motivated this interest [1, 2]. As a result, various categories of robotic manipulators have been established and studied, such as serial manipulators, parallel manipulators, and cable-driven manipulators [3]. Robotic manipulators have been widely employed in applications involving tasks that require high precision and high-speed trajectory tracking [4]. For instance, they can perform some action functions of arms, handle objects, and control devices. However, their highly coupled, nonlinear, and time-varying dynamics [5] make their stabilization/tracking control a challenging task [6–14]. Further, their dynamics suffer from parametric uncertainties and external disturbances such as nonlinear friction and payload variations, sensor noises, and unmodeled dynamics [15–19]. This makes designing controllers for the robotic manipulators a challenging task [11, 20–24]. Various tracking control approaches, such as adaptive control [10], finite-time H∞ control [25],

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PID control [26], neural-network switching control [13], suboptimal control [27], and backstepping control [28], have been proposed in the literature for robotic manipulators. The recent trend toward increased automation and the progress in robotic technology have triggered the need to improve robotic performance so as to attain higher precision and higher speed while reducing the cost [29]. Attaining this objective, however, requires overcoming the difficulties caused by perturbations. In the robotic manipulators, perturbations are divided into two main parts: that is, internal perturbations and external perturbations (disturbances). The internal perturbations represent the unmodeled dynamics, which stem from the flexibilities of the links and joints and parameter variations [30, 31]. The external disturbances represent the exterior forces on the end effector, joint friction, and actuators’ torque ripple. Various control techniques have been proposed in the past decades to either decrease or remove the effects of the perturbations [32]. Sliding mode control (SMC) is a robust technique well suited for systems with uncertainties and disturbances [33–35]. It has attractive features such as robustness against parameter variations, remarkable transient performance, fast dynamic response, reasonable computational simplicity, guaranteed stability, and insensitivity to matched uncertainties and disturbances [36–39]. In sliding mode-based control techniques, the closedloop system is not robust to perturbations during the reaching phase, and even matched uncertainties/disturbances can destabilize the system [40–43]. The idea of a global SMC method has recently been proposed as a general framework to remove the reaching phase and overcome the unwanted chattering phenomenon [44, 45]. By providing an extra term in the manifold, the global SMC removes the reaching phase and enables the states to move on the manifold right from the beginning [46, 47]. The estimation/observation problem is a very important problem in the control of robotic systems. Various types of observers can be found in the literature [48–51]. An improved hybrid position/force control technique using a sliding observer was proposed in Farooq et al. [52] for a flexible robotic manipulator. However, a model-reference adaptive control algorithm is employed in Farooq et al. [52] for the estimate of friction forces at the contact point between the end effector and the environment. A robust control approach for a module manipulator was presented in Shi et al. [53] based on the sliding mode controller and extended state observer. Nevertheless, some high-frequency oscillations can be observed in the control signals of Shi et al. [53]. In Van et al. [54], using the neural second-order sliding observer, an algorithm for the fault diagnosis of robotic manipulators is investigated. In Van et al. [55], an output-feedback tracking control technique was proposed for the robotic manipulators with parametric uncertainties via a fuzzy compensator and a higher-order sliding observer. Although, in both these works, the

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singularity problem was not considered in the adaptation law. A decentralized fault-tolerant control approach for reconfigurable manipulators was studied in Zhao et al. [56] using the sliding mode observer, where the proposed technique is meaningfully simplified in terms of controller structure owing to its dual closed-loop architecture. In He et al. [57], an adaptive fuzzy sliding control scheme based on the nonlinear observer is designed for a redundant robotic manipulator, where the precise trajectory tracking of the robotic manipulator in the existence of the perturbations is achieved. But, the chattering phenomenon is an important drawback in the results of He et al. [57]. An adaptive integral backstepping controller based on the sliding disturbance observer is suggested in Zhang and Yan [58] for the piezoelectric nanomanipulators. However, a suitable tracking performance is not observed in Zhang and Yan [58]. The problem of sliding mode switching control based on the disturbance observer for uncertain robotic manipulators is addressed in Yu et al. [59]. Though in this reference, the tracking performance has not been satisfied in the finite time. In Zhao et al. [60], the trajectory tracking controller for a one-degree-of-freedom (1DOF) manipulator using the switched sliding control scheme and an extended state observer is investigated. Nevertheless, the position error, estimation, and control input signals of Zhao et al. [60] have high-frequency oscillations. In Zhu et al. [61], a composite control technique based on the adaptive sliding disturbance observer is proposed for the space manipulators with uncertainties and disturbances. However, there are slight errors in the trajectory tracking performance. The problem of position control of the Stewart manipulator is addressed in Navvabi and Markazi [62] via the extended adaptive fuzzy sliding control technique and a disturbance observer. But, the tracking performance is satisfied in infinite time. In Kallu et al. [63], a sensor-less reaction force estimation approach of the end effector of the dual-arm robotic manipulator via the sliding control methodology with a perturbation observer is investigated. However, the trajectory tracking purpose is obtained slowly and after a noticeable time. This chapter proposes an observer-based global SMC approach for an n-link robotic manipulator subject to perturbations. The disturbance observer is designed to overcome the perturbations without requiring any information about their bounds. In most of the existing works, the theory of the disturbance observer has been established only for constant disturbance while in the current chapter, by defining the external disturbances as the state observers, it is possible to eliminate the effects of the time-varying disturbances. In this chapter, a disturbance observer-based tracking control approach is developed for n-link robotic manipulators with external disturbances. The tracking controller is based on the adaptive global sliding mode approach. The disturbance observer is designed to estimate system

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disturbances without requiring any knowledge about their upper bounds. The main contributions of this chapter are twofold: (a) A robust global sliding mode design that mitigates exterior disturbances without requiring any knowledge about their upper bounds. (b) An approach that guarantees the convergence of the state trajectories of the n-link robotic manipulator to the switching manifold in the finite time. This chapter is organized as follows: The mathematical model of the nlink robotic manipulator is provided in Section 2. The stability analysis and design procedure of the proposed control scheme are presented in Section 3. Computer simulations illustrating the performance of the proposed approach on a three-degrees-of-freedom (3DOF) rigid robotic manipulator are illustrated in Section 4. Conclusions are finally drawn in Section 5.

2 MATHEMATICAL MODEL OF ROBOTIC MANIPULATORS The Euler-Lagrange equation of an n-link rigid robotic manipulator is given as [64, 65] B(q)¨q + C(q, q˙ )˙q + g(q) = u(t),

(7.1)

q˙ ∈ and q¨ ∈ represent the vectors of joint where q ∈ position, velocity, and acceleration, correspondingly; u(t) illustrates the control input, which is the applied torque on the joints; B(q) is the bounded positive-definite inertia matrix; C(q, q˙ ) is the centripetal Coriolis matrix; and g(q) indicates the gravity vector. The dynamical model of an n-link rigid robotic manipulator can be represented by [66] Rn ,

Rn ,

Rn

B(q)¨q + C(q, q˙ )˙q + Fd q˙ + Fs (˙q) + τd (q, q˙ ) + g(q) = u(t),

(7.2)

where τd (q, q˙ ) is the external disturbance, Fs (˙q) ∈ is the static friction vector, and Fd ∈ Rn×n is the dynamic friction coefficient matrix. Eq. (7.2) is rewritten as Rn

B(q)¨q + n(q, q˙ ) + Fd q˙ + Fs (˙q) + τd (q, q˙ ) = u(t),

(7.3)

where n(q, q˙ ) = C(q, q˙ )˙q + g(q). From Eq. (7.3), one can find q¨ = −B0 (q)−1 {n0 (q, q˙ ) + Fd q˙ + Fs (˙q) + τd (q, q˙ ) + B(q)¨q + n(q, q˙ ) − u(t)}, (7.4) where B(q) = B0 (q) + B(q) with B0 (q) and B(q) being the known and unknown parts of B(q), respectively. n(q, q˙ ) = n0 (q, q˙ ) + n(q, q˙ ) with N0 (q, q˙ ) and n(q, q˙ ) represent the known and unknown parts of n(q, q˙ ), respectively. II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

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The simplified model of an n-link rigid robotic manipulator is obtained from Eq. (7.4) as q¨ = f0 (q, q˙ ) + B0 (q)−1 u(t) + D(t),

(7.5)

where D(t) = −B0 (q)−1 {Fd q˙ + Fs (˙q) + τd (q, q˙ ) + B(q)¨q + n(q, q˙ )} is the lumped perturbations term and f0 (q, q˙ ) = −B0 (q)−1 n0 (q, q˙ ) is the bounded known nonlinear function.

3 OBSERVER-BASED TRACKING CONTROLLER In this section, an observer-based tracking control approach is developed for n-link rigid robotic manipulators. Its main objective is to guarantee the convergence of the tracking errors to the sliding manifold in finite time and ensure robustness against nonlinearities and exterior disturbances. Consider the following sliding manifold: s(t) = G(r(t) − r(0) exp(−βt)),

(7.6)

where r(t) = e˙(t)+λe(t), G is a constant row vector, and λ and β are positive constants. e(t) = q − qd and e˙(t) = q˙ − q˙ d are, respectively, the tracking and velocity errors, with qd representing the desired trajectory and q˙ d the desired velocity. Differentiating Eq. (7.6) with respect to time and using Eq. (7.5) yields s˙(t) = G(f0 (q, q˙ ) − q¨ d + λ˙e(t) + βr(0) exp(−βt) + B0 (q)−1 u(t) + D(t)). (7.7) Equating s˙(t) to zero yields the equivalent controller defined by ueq (t) = −B0 (q)(f0 (q, q˙ ) − q¨ d + λ˙e(t) + βr(0) exp(−βt)).

(7.8)

In practice, the upper bound of perturbations is unknown and D(t) is difficult to determine. Assume that the perturbation is unknown and bounded with Γ > D(t), where Γ is the unknown positive constant. The disturbance-observer adaptation law is defined as Γ˙ˆ = κs(t)GT ,

(7.9)

where Γˆ is an estimate of Γ and κ is a positive constant. The auxiliary controller is designed as uaux (t) = −B0 (q)(ks(t)GT + Γˆ sgn(s(t)GT )),

(7.10)

where k is a positive coefficient. The total controller is obtained from Eqs. (7.8), (7.10) as u(t) = −B0 (q)(f0 (q, q˙ )− q¨ d +λ˙e(t)+βr(0) exp(−βt)+ks(t)GT + Γˆ sgn(s(t)GT )). (7.11)

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Theorem 1. Consider the sliding manifold of Eq. (7.6), the disturbanceobserver law defined by Eq. (7.9), and system perturbations defined as an unknown and bounded term with Γ > D(t). If the adaptive controller (7.11) is applied, then the states of the robotic manipulator (7.2) converge to the manifold s = 0 in finite time. Proof. Substituting Eq. (7.11) into Eq. (7.7), we have s˙(t) = G(−ks(t)GT − Γˆ sgn(s(t)GT ) + D(t)).

(7.12)

Construct the Lyapunov functional as V(t) =

1 1 s(t)2 + γ (Γˆ − Γ )2 . 2 2

(7.13)

˙ Taking V(t) and using Eqs. (7.9), (7.12) yields ˙ V(t) = s(t)G(D(t) − ks(t)GT − Γˆ sgn(s(t)GT )) + γ (Γˆ − Γ )Γ˙ˆ ≤ s(t)G D(t)− Γˆ s(t)G+γ κs(t)G(Γˆ −Γ )+s(t)GΓ − s(t)GΓ ≤ −s(t)G(Γ − D(t)) − (1 − γ κ)s(t)G(Γˆ − Γ ). (7.14) Because Γ > D(t) and γ κ < 1, one can find from Eq. (7.14) that  √ 2 |s(t)| Γˆ − Γ ˙ s(t)G (1 − γ κ)  V(t) ≤ 2G(Γ − D(t)) √ − 2 γ 2 γ ⎛ ⎞ (7.15) ˆ −Γ |s(t)| Γ 1/2 ⎠ ≤ −ΩV(t) , ≤ −Ω ⎝ √ +  2 2 γ

 2G(Γ − D(t)), γ2 (1 − γ κ)s(t)T G . Hence, using where Ω = min the adaptive disturbance observer-based control law (7.11) guarantees the convergence of the sliding manifold (7.6) in finite time. √

4 SIMULATION RESULTS In this section, we implement the proposed control approach to the 3DOF rigid robotic manipulator illustrated in Fig. 7.1. Here l1 , l2 , and l3 represent the distances of the center of mass of three links from their joint axis; Il1 , Il2 , and Il3 are the moments of inertia of the links; mm1 , mm2 , and mm3 are the masses of the rotors; m1 , ml2 , and ml3 are the masses of the three links; and Im1 , Im2 , and Im3 are the moments of inertia of the rotors.

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y

u3

u2 u1

q3

q2

q1

x

FIG. 7.1 Three-degrees-of-freedom robotic manipulator.

Consider the dynamical model of the 3DOF rigid robotic manipulator as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ b11 b12 b13 h11 h12 h13 fd q˙ 1 fs sgn(˙q1 ) τ1 ⎣b21 b22 b23 ⎦ q¨ + ⎣h21 h22 h23 ⎦ q˙ + ⎣fd q˙ 2 ⎦ + ⎣fs sgn(˙q2 )⎦ + ⎣τ1 ⎦ b31 b32 b33 h31 h32 h33 fd q˙ 3 τ1 fs sgn(˙q3 ) ⎡ ⎤ g1 (q) + ⎣g2 (q)⎦ = u(t), (7.16) g3 (q) where the parameters are defined as b11 = Il1 + ml1 l21 +kr21 Im1 +Il2 +mm2 a21 +Im2 +ml2 (a21 +l22 +2a1 l2 c2 )+Il3 +Im3 + mm3 (a21 +a22 +2a1 a2 c1 )+ml3 (a21 +a22 +l33 +2a1 a2 c2 + 2a1 l3 c23 +2a2 l3 c3 ), (7.17) 2 b22 = Il2 + Il3 + kr2 Im2 + Im3 + mm3 a22 + ml2 l22 + ml3 (a22 + l23 + 2a2 l3 c3 ), (7.18)

b33 = Il3 + kr23 Im3 + ml3 l23 ,

(7.19)

b12 = b21 = Il2 + Il3 + kr2 Im2 + Im3 + mm3 (a22 + a1 a2 c2 ) + ml2 (l22 + a1 l2 c2 ) b13 = b31 =

+ ml3 (a22 + l23 + a1 a2 c2 + a1 l3 c23 + 2a2 l3 c3 ),

(7.20)

Il3 + kr3 Im3 + ml3 (l23 Il3 + kr3 Im3 + ml3 (l23

(7.21)

+ a1 l3 c23 + a2 l3 c3 ),

+ a2 l3 c3 ), b23 = b32 = h11 = −mm3 a1 a2 s1 q˙ 1 − (ml2 + a1 l2 s2 + ml3 a1 a2 s2 + ml3 a1 l3 s23 )˙q2 − (ml3 a1 l3 s23 + ml3 a2 l3 s3 )˙q3 , h22 = −(ml3 a2 l3 s3 )˙q3 , h33 = −(ml3 a1 a2 s2 )˙q2 − (ml3 a2 l3 s3 )˙q3 ,

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(7.22) (7.23) (7.24) (7.25)

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7. DESIGN OF OBSERVER-BASED TRACKING CONTROLLER

h12 = −(ml2 a1 l2 s2 + ml3 a1 a2 s2 + ml3 a1 l3 s23 )˙q1 − (ml3 a1 a2 s2 + ml3 a1 l3 s23 + mm3 a1 a2 s2 + ml2 a1 l2 s2 )˙q2 h13

− (ml3 a1 l3 s23 + ml3 a1 l3 s23 )˙q3 , = −(ml3 a1 l3 s23 + ml3 a2 l3 s3 )˙q1 − (ml3 a1 s23 + ml3 a2 l3 s3 )˙q2

(7.26)

− (ml3 a2 l3 s2 + ml3 a1 l3 s23 )˙q3 , = (ml2 a1 l2 s2 + ml3 a1 a2 s2 + ml3 a1 l3 s23 )˙q1 − (ml3 a2 l3 )˙q3 , = −(ml3 a2 l3 s3 )˙q1 − (ml3 a2 l3 s3 )˙q2 − (ml3 a2 l3 s3 )˙q3 ,

(7.27) (7.28) (7.29)

h21 h23 h31 = −(ml3 a1 l3 s23 + ml3 a2 l3 s3 )˙q1 + (ml3 a2 l3 s3 )˙q2 , h32 = −(ml3 a2 l3 s3 )˙q1 + (ml3 a2 l3 s3 )˙q2 − (ml3 a2 l3 s3 )˙q3 ,

(7.30) (7.31)

g1 (q) = (ml1 I1 + ml2 a1 + mm2 a1 + ml3 a1 + mm3 a1 )gc1 + (ml2 I2 + ml3 a2 + mm3 a2 )gc12 + ml3 l3 gc123 , g2 (q) = (ml2 l2 + ml3 a2 + mm3 a2 )gc12 + ml3 l3 gc123 , g3 (q) = ml3 l3 gc123 ,

(7.32) (7.33) (7.34)

with c1 = cos q1 , s1 = sin q1 , c2 = cos q2 , s2 = sin q2 , c3 = cos q3 , c12 = cos(q1 + q2 ), c123 = cos(q1 + q2 + q3 ), s12 = sin(q1 + q2 ), and s123 = sin(q1 + q2 + q3 ). The constant parameters of the system are fd = 5, fs = 6, l1 = 0.6, l2 = 0.5, l3 = 0.4 m, ml1 = 12 kg, ml2 = 10 kg, ml3 = 8 kg, Il1 = Il2 = Il3 = 1 kg m2 , mm1 = 1.3  mm2 =1.1 kg, mm3 = 0.9 kg, τ1 =  kg, 10 1 + sin(2t) + sin(1.5t) + cos π2 t + sin π2 t , a1 = a2 = 1, Im1 = Im2 =  T Im3 = 0.01 kg m2 , κ = 3, λ = 3, β = 1.5, k = 2.5, q(0) = −0.1 −0.3 0.2 ,  T   qd = 0.4 cos(2t), 0.35 sin(2t), 0.3 sin(2t) , and G = 1.5 0.9 1.2 . The uncertainties terms are given as ⎡ ⎤ 0.3 cos(q1 ) 0.1 sin(q2 ) −.3 sin(q2 ) B(q) = ⎣0.4 sin(q2 ) 0.3 sin(q1 ) 0.45 cos(q1 )⎦ and 0.1 sin(q2 ) 0.2 cos(q1 ) 0.4 sin(q2 ) ⎡ ⎤ 0.3˙q1 q˙ 2 cos(q2 ) n(q, q˙ ) = ⎣0.3 sin(q1 − q2 ) + cos(q1 + q2 )⎦ . q˙ 1 q˙ 2 sin(q1 + q2 ) The control law illustrated in Eq. (7.11) was implemented to the previous system. The time histories of the position of the joints are shown in Fig. 7.2. Note the suitable tracking performance of the position trajectories of the joints. The time histories of the control signals are illustrated in Fig. 7.3. The time response of the proposed sliding manifold is illustrated in Fig. 7.4. Note that the fast convergence of the manifold to the origin is obtained. The dynamics of the disturbance-observer gain are illustrated in Fig. 7.5. The final value of the adaptation gain is Γˆ = 1.878. Note that the overall

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CONCLUSION

q1(t)

0.5

0

–0.5 0

2

4

6

8

10

12

8

10

12

10

12

t

q2(t)

0.5

0

–0.5 0

2

4

6 t

q3(t)

0.5

0

–0.5 0

2

4

6

8

t

Planned method Reference signal

FIG. 7.2 Time responses of the positions of joints.

robustness and good tracking performance of the proposed approached is satisfied. Consequently, as can be observed from the simulation results, the tracking performance of the positions of all three joints is admissible, the tracking errors are converged to the origin in finite time, and the robust performance of the disturbance-observer controller against perturbations is guaranteed.

5 CONCLUSION This chapter proposed an observer-based tracking control design for an n-link robotic manipulator. The tracking controller was designed based on the global sliding mode approach so as to eliminate the reaching phases and ensure the convergence of the tracking errors to the sliding manifold in the finite time. The disturbance observer was designed to estimate system disturbances without requiring any knowledge about their upper bounds. Implementation of the proposed approach to a 3DOF rigid manipulator

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u1(t)

1000 500 0 –500 0

2

4

6 t

8

10

12

0

2

4

6 t

8

10

12

0

2

4

6 t

8

10

12

10

12

u2(t)

400 200 0 –200

100

u3(t)

50 0 –50

FIG. 7.3 Time histories of the control inputs.

2

s(t)

0 –2 –4 0

2

4

6

8

t

FIG. 7.4 Trajectory of the sliding manifold.

showed good tracking performance and robustness to disturbances. The proposed approach eliminated the reaching phase and ensured robustness against nonlinearities and external disturbances, ubiquitous problems in robotic manipulators.

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REFERENCES

2

GL

1.5 1 0.5 0 0

2

4

6

8

10

12

t

FIG. 7.5 Dynamics of the disturbance-observer gain.

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[56] B. Zhao, C. Li, D. Liu, et al., Decentralized sliding mode observer based dual closed-loop fault tolerant control for reconfigurable manipulator against actuator failure, PLoS ONE 10 (7) (2015) e0129315. [57] J. He, M. Luo, Q. Zhang, et al., Adaptive fuzzy sliding mode controller with nonlinear observer for redundant manipulators handling varying external force, J. Bionic Eng. 13 (4) (2016) 600–611. [58] Y. Zhang, P. Yan, Sliding mode disturbance observer-based adaptive integral backstepping control of a piezoelectric nano-manipulator, Smart Mater. Struct. 25 (12) (2016) 125011. [59] L. Yu, J. Huang, S. Fei, Sliding mode switching control of manipulators based on disturbance observer, Circuits Syst. Signal Process. 36 (6) (2017) 2574–2585. [60] L. Zhao, Q. Li, B. Liu, et al., Trajectory tracking control of a one degree of freedom manipulator based on a switched sliding mode controller with a novel extended state observer framework, IEEE Trans. Syst. Man Cybernet. Syst. 49 (6) (2019) 1110–1118. [61] Y. Zhu, J. Qiao, L. Guo, Adaptive sliding mode disturbance observer-based composite control with prescribed performance of space manipulators for target capturing, IEEE Trans. Ind. Electron. 66 (3) (2019) 1973–1983. [62] H. Navvabi, A. Markazi, Position control of Stewart manipulator using a new extended adaptive fuzzy sliding mode controller and observer (E-AFSMCO), J. Frankl. Inst. 355 (5) (2018) 2583–2609. [63] K.D. Kallu, W. Jie, M.C. Lee, Sensorless reaction force estimation of the end effector of a dual-arm robot manipulator using sliding mode control with a sliding perturbation observer, Int. J. Control Autom. Syst. 16 (3) (2018) 1367–1378. [64] M. Liaquat, M.B. Malik, Sampled data output regulation of n-link robotic manipulator using a realizable reconstruction filter, Robotica 34 (4) (2016) 900–912. [65] A. Fenili, J.M. Balthazar, The rigid-flexible nonlinear robotic manipulator: modeling and control, Commun. Nonlinear Sci. Numer. Simul. 16 (5) (2011) 2332–2341. [66] M.W. Spong, M. Vidyasagar, Robot Dynamics and Control, John Wiley & Sons, Hoboken, NJ, 2008.

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C H A P T E R

8 Disturbance Observer-Based Control of Spacecraft Attitude Dynamics Subject to Perturbations and Underactuation Ahmet Sofyali Independent Researcher, Istanbul, Turkey

1 INTRODUCTION Disturbance observer (DOB)-based control has been regarded as a promising disturbance attenuation technique [1]. If the control system designer has a baseline controller at hand that provides the considered control problem with satisfactory stability and performance characteristics under nominal conditions, proper integration of the nominal controller with a disturbance observer may result in an ultimate control system that performs as well as the baseline system under realistic conditions. On the other hand, if the baseline control system is designed by taking perturbations into account, which means that it already possesses stability and performance robustness, the same state responses may be obtained by reduced control effort when the estimated disturbance is logically fed back to the system. In the case of this work, the baseline controller provides the purely magnetic spacecraft attitude control problem with a robustly and globally stable solution. Such a solution had been developed by making use of sliding mode control theory to enhance the capability of the solely magnetic actuation. The considered system is underactuated due to the lack of control authority along the instantaneous orientation of the local

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geomagnetic field vector. However, by making state trajectories reach at a novel sliding manifold, which adjusts itself according to the local field direction, in finite time, robust stabilization was achieved. The baseline control law guarantees that, when used to drive three magnetic torquers that are orthogonal to each other, the spacecraft’s attitude is carried from an arbitrary state to the vicinity of the reference and is kept there under the effects of environmental disturbances and inertia matrix uncertainty. In this chapter, the aforementioned baseline control system with proven robust and global stability is going to be aided by a disturbance observer to attain the same characteristics with significantly reduced control power consumption. Because the baseline controller is a sliding mode controller, the switching control signal causes high-frequency oscillations in state responses. A significant reduction in the amplitude of control signals will lead to highly smoothed system outputs; this function justifies why the disturbance observer-based control approach has been employed with sliding mode controllers for chattering suppression in the last decade.

2 PROBLEM FORMULATION The problem that is going to be solved via disturbance observer-based control is inertial pointing of rigid spacecraft under solely magnetic actuation. The aimed qualitative property of the developed control system is global and robust stability, which is achievable with a significantly reduced control effort thanks to the inclusion of a nonlinear disturbance observer in the structure. Because the baseline control system to be enhanced by the disturbance observer operates relying on a sliding mode control law, the reduction in the discontinuous actuation signal is expected to lead suppression in the amplitude of high-frequency oscillations in system outputs, which is called chattering in the literature.

2.1 State-Space Representation of Baseline Control System In this section, the state-space expression of spacecraft attitude dynamics subject to environmental disturbances and inertia matrix uncertainty is going to be presented. 2.1.1 Dynamic Equations This work deals with the rotational motion of a rigid spacecraft about its center of mass (CoM). The spacecraft’s CoM translates along a closed orbit around the Earth. The angular motion, namely the attitude of the spacecraft’s (satellite’s) body reference frame (B in Fig. 8.1) with respect to the Earth-centered inertial reference frame (N in Fig. 8.1), is aimed to be stabilized so that those two reference frames coincide with each other.

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PROBLEM FORMULATION n3

n2

Earth N

b1 a3

n1

a1

B

b3

orbit

satellite A

a2

b2

FIG. 8.1 Reference frames.

The rotational dynamics of the satellite can be written in B as follows: d + T  mc . Jω ˙ + ω  × Jω  =T

(8.1)

Here, ω  is the angular velocity vector of size 3 × 1, J is the inertia matrix,  mc is the  d is the resultant environmental disturbance torque vector, and T T magnetic control torque vector. The highly probable uncertainty in the knowledge of the six distinct elements of the full inertia matrix J is taken into account as J  Jn + J. Jn , which is defined as

⎡

J1 Jn  ⎣ 0 0

0 J2 0

(8.2)

⎤ 0 0 ⎦ J3

(8.3)

is the nominal part of J, and corresponds to the principal inertia matrix consisting of the spacecraft’s three principal moments of inertia. This means that, ideally, the body axis system B is assumed to coincide with the principal axis system. In reality, such a coincidence is nearly impossible, thus an inertia uncertainty matrix is defined as the sum of one diagonal matrix and one full parametric uncertainty matrix: J  J1 + J2 , ⎡ δ 1 J1 0 δ2 J2 J1  ⎣ 0 0 0 

J2  1 + δ¯J1



(8.4a)

⎤

0 0 ⎦, δ3 J 3

(8.4b) ⎡

δ11 max (J1 , J2 , J3 ) ⎣ δ12 δ13

δ12 δ22 δ23

⎤ δ13 δ23 ⎦ . δ33

(8.4c)

J1 models the uncertainty in the knowledge of principal moments of inertia, and J2 represents the uncertainty in the angular orientation of

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B with respect to the principal reference frame. The expected variation intervals for δi (i = 1, 2, 3) and δjk (j, k = 1, 2, 3) are symbolized by  (8.5a) δi ∈ −δ¯J1 ; +δ¯J1 ,  (8.5b) δjk ∈ −δ¯J2 ; +δ¯J2 . For the accepted values of the interval limits and the related justifications, the reader is kindly expected to refer to Sofyalı and Jafarov [2]. There are four major environmental effects on the attitude motion: the gravity gradient (gg) along the distributed mass of the spacecraft, the aerodynamic drag (aero) due to the residual atmosphere, the solar radiation pressure (solar) originating from energetic solar particles impacting spacecraft surfaces, and the residual magnetic moment (mag) induced by the hardware onboard. The formulas of the corresponding disturbance torques are presented in Sofyalı and Jafarov [2]. They together constitute the resultant disturbance torque:  gg + T  aero + T  solar + T  mag . d = T T

(8.6)

Because the control torque is solely obtained by the interaction of the resul c , which is produced by the three orthogonally tant magnetic moment M  placed magnetic actuators onboard, with the local geomagnetic field B,  the magnetic control torque Tmc is restricted to lie in the plane that is instantaneously orthogonal to the local field according to the following interaction law:  c × B.   mc = M T

(8.7)

As a result, in a purely magnetic attitude control system, there is no control  which leads to underactuation. If the authority in the direction along B, spacecraft’s orbital plane, which is defined by the third (a3 ) and first (a1 ) unit vectors of the orbital axis system A (see Fig. 8.1), does not coincide with the equatorial plane of the geomagnetic field, the components of  will continuously vary. Because all the defined vectors are written in B B, it can be concluded that the underactuated direction will constantly rotate with respect to B while the spacecraft moves along the orbit. This property specific to the magnetic control problem implies instantaneous underactuation [3]. Because that orthogonal plane of continuous control authority will eventually cover the three-dimensional space of attitude dynamics as long as the actuation goes on, the controllability of such a system can be deduced [4]. 2.1.2 Kinematic Equations The angular orientation of B with respect to N is expressed in nondimensional Euler parameters (quaternions). This representation is preferred to the one by three Euler angles to obtain a singularity-free and faster attitude

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PROBLEM FORMULATION

propagation, but adds an additional dimension to the state vector due to its redundant nature. Quaternions are related to the angular velocity vector through the following kinematic equations:  1  + q × ω  , (8.8a) q4 ω q˙ = 2  1  . (8.8b) q˙ 4 = − q · ω 2 Here, q is the quaternion vector of size 3 × 1 whereas q4 is the scalar part of the quaternions. The mentioned redundancy relies on the following interrelation among quaternions: q21 + q22 + q23 + q24 = 1.

(8.9)

As a result, the spacecraft’s attitude state can be represented by the following state vector: x =



qT

q4

ω T

T

.

(8.10)

Thus, the inertial pointing state is written as xN 



01×3

1

01×3

T

.

(8.11)

2.1.3 State Equation The state equation is as follows:        + d x, t . x˙ (t) = fn x + bn x, t u

(8.12)

Here, 

 ⎤  q ω  + q × ω  4     fn x = ⎣ ⎦ − 12 q · ω  −1  −Jn (ω  × Jn ω) ⎡

1 2

(8.13)

is the nominal system vector of size (7 × 1), bn



⎡

⎤ 03×3 ⎦ x, t = ⎣ 01×3   Jn−1 CB x, t 

is the nominal control matrix of size (7 × 3),       d x, t  dn x, t + dunc x, t

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(8.14)

(8.15)

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8. DISTURBANCE OBSERVER-BASED CONTROL OF SPACECRAFT ATTITUDE

is the total disturbance vector, ⎤ ⎡ 03×1   ⎥ ⎢ 0 dn x, t = ⎣ ⎦           aero x, t + T  solar x, t + T  mag x, t  gg x + T Jn−1 T ⎤ ⎡ 03×1 ⎦ 0 =⎣    d x, t Jn−1 T (8.16) is the nominal disturbance vector, dunc

⎡

⎤ 03×1 ⎦ 0 x, t = ⎣   −1  unc x, t Jn T





(8.17)

is the disturbance vector solely due to inertia matrix uncertainty, and       d x, t  mc + T  unc x, t = −ω  × Jω  − JJ−1 −ω (8.18)  × Jω  +T T is the disturbance torque vector solely due to inertia matrix uncertainty. For the derivation of both the presented form of the state equation and  unc , the reader is kindly expected to refer to Sofyalı and Jafarov [2]. T  , which is The matrix CB in Eq. (8.14) projects the control vector u computed according to the control law as if there is no underactuation in the system, onto the plane instantaneously orthogonal to the local geomagnetic field vector. The output of the projection is the magnetic control torque vector:      mc x, t = CB x, t u . (8.19) T Because the magnetic moment vector in the baseline control system is evaluated according to the projection-based law [5] as    x, t × u   B   c x, t =  , M     2 B x, t 

(8.20)

2

CB is defined according to Eq. (8.7) as follows: T   B˜ x,t B˜ x,t CB x, t  ( ) (2 )  (x,t) B ⎡

=

1 B21 +B22 +B23

B22 + B23 ⎣ −B2 B1 −B3 B1

−B1 B2 B23 + B21 −B3 B2

⎤ −B1 B3 −B2 B3 ⎦ . B21 + B22

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(8.21)

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PROBLEM FORMULATION

2.1.4 Baseline Controller The baseline controller used in this work is the magnetic sliding mode attitude controller. It was introduced to the literature in Sofyalı et al. [6]. It has been designed based on the equivalent control method, which leads to a control law consisting of two parts:  reach . u  eq + u u

(8.22)

Here,     1  eq x = − Jn Kq q4 ω (8.23)  + q × ω  +ω  × Jn ω  − nJn Kq Kintq q u 2 is its continuous part, namely the equivalent control vector whereas         reach x, t = −Kss sgn s x, t − Kss x, t (8.24) u is the reaching control vector. The magnetic sliding mode controller provided the purely magnetic attitude stabilization problem with a robust and global solution thanks to the utilized novel sliding manifold structure that was specifically proposed to the problem, which is  t  t       dτ . s x, t = ω qdτ + Jn−1 DB x, τ u (8.25)  + Kq q + nKq Kintq t0

t0

 onto the local The matrix DB in the sliding surface vector s projects u geomagnetic field vector. The output of the projection is the component  that cannot be exerted on the spacecraft body. Thus of u     (8.26) DB x, t = I3×3 − CB x, t holds, and this complementary matrix is defined as    x,t B  x,t T B DB x, t  ( ) ( 2)  (x,t) B ⎡ =

1 B21 +B22 +B23

B21 ⎣ B2 B1 B3 B1

B1 B2 B22 B3 B2

⎤ B1 B3 B2 B3 ⎦ . B23

(8.27)

Kq and Kintq are sliding surface design matrices while Kss and Ks are reaching law design matrices. All are diagonal, constant, and positive definite. For the sake of design simplicity, their diagonal elements are accepted to be the same and equal to the design parameters kq , kintq , kss , and ks , respectively. For the sliding mode control method, Refs. [7–10] may be consulted. It was theoretically shown under the ideal sliding assumption, which means if d = 07×1 , that the proposed sliding manifold guarantees the asymptotical convergence of the attitude state to the reference.

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8. DISTURBANCE OBSERVER-BASED CONTROL OF SPACECRAFT ATTITUDE

Provided that the derived reaching (existence) conditions are satisfied, the magnetic sliding mode controller given in Eq. (8.22) carries the state trajectories to the sliding manifold in finite time, which indicates the existence of a sliding mode in the purely magnetic attitude control problem [6]. In the next and main section of this work, a nonlinear disturbance observer is going to be merged to the magnetic sliding mode controller in such a manner that the final control system will retain the global property of robust stability.

3 DISTURBANCE OBSERVER-BASED MAGNETIC SLIDING MODE ATTITUDE CONTROLLER  mc , and thus on u  unc is dependent on T  , which is a switching signal. If T  d and T  unc is called the lumped disturbance torque vector, it can the sum of T be concluded that the estimation of the complete lumped disturbance is  unc is divided into two parts as problematic. Therefore, T

where

∗ ∗∗  unc  unc  unc  T +T , T

(8.28)

  ∗ d  unc  eq + T = −ω  × Jω  − JJ−1 −ω  × Jω  + CB u T

(8.29)

is its continuous part while ∗∗  unc  reach = −JJ−1 CB u T

(8.30)

is the discontinuous one. As a result, the continuous component of the lumped disturbance vector can be symbolized by   ∗  T d − ω d .  eq + T T  × Jω  − JJ−1 −ω  × Jω  + CB u (8.31) d Although the condition of slow time-variation, which can be mathematically expressed by ˙ ∗ ≈ 03×1 T d  ∗ , is not satisfied, the following condition is satisfied: for T d    ˙ ∗  Td  < ∞. 2

(8.32)

(8.33)

3.1 Low-Pass Filtering of Input Signals of Observer The dynamics of the disturbance observer should be smooth. In  and ω the baseline control system, the signals belonging to u  are not smooth. Thus, the high-frequency oscillations in those signals have to be

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DOB-BASED MAGNETIC SLIDING MODE ATTITUDE CONTROLLER

207

filtered out. The same first-order low-pass filter with linearly decreasing τ  and ω: is used to smooth both u   filt = u , ˙ filt + u τu τω ˙ filt + ω  filt = ω. 

(8.34a) (8.34b)

Then, the following smooth state equation, which is suitable to be used in the observer equation, can be written:  ∗.  filt + d∗  fn + bn u  filt + gT x˙ = fn + bn u d

(8.35)

3.2 Disturbance Observer The utilized nonlinear disturbance observer is as follows:     ∂ p x fn + bn u ˆ ∗ ,  filt + gT z˙ = − d ∂ x   ˆT ∗   z + p x , d    T  −1 T T  04×3 −1 = 03×4 Jn = . g  03×4 Jn Jn−1

(8.36a) (8.36b) (8.36c)

Remark 1 (Li et al. [1]). The observer in Eq. (8.36) can track some fast time-varying disturbances with bounded error as long as the derivative of disturbances is bounded. Lemma 1 (Li et al. [1]). Suppose that the condition in Eq. (8.33) is satisfied. The disturbance estimation error system   ˙e + ∂ p x ge = T ˙ ∗ , (8.37a) d ∂ x ∗ − T ˆ ∗ . e  T (8.37b) d

d

is locally input-to-state stable (ISS) if the observer gain is chosen such that   ˙e + ∂ p x ge = 0 (8.38) ∂ x is asymptotically stable. If   ∂ p x (8.39a) g ≡ Ke , ∂ x ⎡ ⎤ ke 0 0 (8.39b) Ke  ⎣ 0 ke 0 ⎦ , 0 0 ke

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8. DISTURBANCE OBSERVER-BASED CONTROL OF SPACECRAFT ATTITUDE

ke > 0,

(8.39c)

∂ p  ≡ 03×4 Ke Jn ∂ x can be written, which leads to   ∂ p  filt . x ≡ Ke Jn ω p x = ∂ x The resulting nonlinear disturbance observer is      04×3  ˙z = − 03×4 Ke Jn   filt , z + Ke Jn ω  filt + fn + bn u Jn−1 ˆ ∗ = z + Ke Jn ω  filt . T d

(8.40)

(8.41)

(8.42a) (8.42b)

3.3 Disturbance Observer-Based Controller In this section, the disturbance observer-based control law is going to be derived in two steps. 3.3.1 Equivalent Control Part In the designed disturbance observer-based sliding mode control system, the lumped disturbance component that cannot be estimated is defined as ⎡ ⎤ 03×1 ⎦. 0 (8.43) d∗∗ = ⎣ ∗∗  unc Jn−1 T Then, under ideal sliding assumption, that is, for s = s˙ = 03×1 , ≡u  eq , u

(8.44b)

e = 03×1 ,

(8.44c)

∗∗

d

= 07×1 .

(8.44a)

(8.44d)

The equivalent control vector can be solved for from −1  eq s˙ = Gx˙+ nKq Kintq q + J n DB u ˆ  eq + d∗ + nKq Kintq q + Jn−1 DB u  eq = G fn + bn u

ˆ  eq = Gfn + Gd∗ + nKq Kintq q + Jn−1 (CB + DB ) u ˆ ∗ −1 −1    eq = Gfn + Jn Td + nKq Kintq q + Jn u = 0

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(8.45)

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DOB-BASED MAGNETIC SLIDING MODE ATTITUDE CONTROLLER

as   1 ˆ ∗ .  eq = − Jn Kq q4 ω  + q × ω  +ω  × Jn ω  − nJn Kq Kintq q − T (8.46) u d 2 The utilized approach of negatively feeding the estimated lumped disturbance component back to the system by the equivalent control law is present in the literature [11].

3.3.2 Reaching Control Part In the reaching mode, the following inequality holds:   ˆ ∗ + nKq Kintq q + Jn−1 u  = 03×1 . s˙ = G fn + ge + d∗∗ + Jn−1 T d

(8.47)

Theorem 1. Assume that the spacecraft’s orbital plane does not coincide with the equator plane of the geomagnetic field. Then, the attitude motion of the rigid spacecraft on such an orbit (8.10)–(8.21) controlled by the disturbance observerbased magnetic sliding mode controller (8.22), (8.24)–(8.27), (8.46) reaches the sliding mode in finite time for any value of ks if √ (8.48a) L1 > 3L2 ,   L1 e kss > (8.48b) √ ∞ L1 − 3L2 are satisfied. L1 and L2 in Eq. (8.48a) were defined in Sofyalı and Jafarov [2] as     L1  1 − δ¯J1 min (J1 , J2 , J3 )  1 − δ¯J1 Jmin , (8.49a)      L2  δ¯J1 + δ¯J2 1 + δ¯J1 max (J1 , J2 , J3 )  δ¯J1 + δ¯J2 + δ¯J1 δ¯J2 Jmax . (8.49b) Proof. Derive the Lyapunov function candidate 1 (8.50) V = sT Jns > 0 2 with respect to time to arrive at    ˆ ∗ + nKq Kintq q + Jn−1 u ˙ = sT Jns˙ = sT Jn G fn + ge + d∗∗ + Jn−1 T  V d     1  q J K ω  + q × ω  − ω  × J ω  · · · n q n 4 2 = sT ∗∗ + u ˆ unc ˆ ∗ + nJn Kq Kintq q + e + T  eq + u  reach ··· + T d   ⎡ 1 ⎤  + q × ω  −ω  × Jn ω  ··· 2 Jn Kq q4 ω ⎢ ⎥ ∗∗ + u  unc ˆ ∗ + nJn Kq Kintq q + e + T  reach · · · = sT ⎣ · · · + T ⎦ d   ˆ ∗  + q × ω  +ω  ×J ω  − nJ K K q − T · · · − 1J K q ω 2 n q

4

  ∗∗  unc  reach = sT e + T +u

n

n q intq

d

(8.51)

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8. DISTURBANCE OBSERVER-BASED CONTROL OF SPACECRAFT ATTITUDE

after successive substitutions of the equalities in Eq. (8.47) and then in  reach is substituted into Eq. (8.51) and the result is Eq. (8.46). If then u mathematically manipulated,     ∗∗ + u  unc ˙ s = sT e + T  reach V     reach = sT e + I − JJ−1 CB u      = −sT I − JJ−1 CB Kss sgn s + Kss − e     = −sT I − JJ−1 CB Kss sgn s − e   −sT I − JJ−1 CB Kss   = −sT Kss sgn s − e  +sT JJ−1 CB Kss sgn s −sT Kss + sT JJ−1 CB Kss 3     |si | ≤ − kss − e∞  −1  i=1      + Ji2 J i2 CB i2 Kss sgn s 2 s2    2 −sT Kss + Ji2 J−1 i2 CB i2 Ks i2 s2 3     |si | ≤ − kss − e∞ i=1   2  2     + LL21 Kss i2 sgn s 2 s2 − ks s2 + LL21 ks s2    3    √   2 |si | + LL2 kss 3 s2 − 1 − LL2 ks s2 = − kss − e∞ 1 1 i=1

 3 3    √  |si | + 3 LL2 kss |si | − 1 − ≤ − kss − e∞ 1 

=−



i=1

i=1

  3    √ |si | − 1 − 1 − 3 LL21 kss − e∞ i=1

L2 L1



L2 L1



 2 ks s2

 2 ks s2

(8.52) is obtained. The last row indicates the same two reaching conditions given in Eq (8.48).

4 COMPARISON THROUGH SIMULATIONS For the simulations, a nearly circular orbit is used. Because it is nearly polar, the spacecraft’s orbital motion is not coplanar with the geomagnetic field equator. The altitude is in the low-Earth-orbit range, where purely magnetic actuation is feasible in terms of producible magnetic torque magnitude. The orbital angular velocity (i.e., the mean motion n) is equal to 6.07 × 10−2 deg/s, which corresponds to an orbital period T of 5939 s. A model in the class of microsatellites, which limits the mass by 100 kg,

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TABLE 8.1 Input Data for Simulations Input

Value

Initial conditions

Value

δ¯J1 ; δ¯J2 [–] √ 3L2 /L1 [–]

0.1; 0.06

[q1 q2 q3 q4 ]0 [–]

[0.123 0.707 0 0.696]

0.859 < 1

[ω1 ω2 ω3 ]0 [×10−2 deg/s]

[5 7 6]

kq [1/s]

3.5 × 10−3

[1.18 2.99 1.047]

kintq [–]

7.5 × 10−1

[s1 s2 s3 ]0 [×10−3 1/s]   ω  0 2 [n]

ks [Nms]

1 × 10−1

ke [1/s]

2.7

τ (t0 ) [T]

1 × 10−3

1.728

is chosen: J1 , J2 , J3 = 2.904, 3.428, 1.275 kg m2 . The class of the selected satellite model is operable by solely magnetic actuation in the selected orbital altitude. Note 1. The reaching condition in Eq. (8.48a) is satisfied by the simulation input data as seen in Table 8.1. Remark 2. The reaching conditions satisfied by the baseline controller are [6] √ (8.53a) L1 > 3L2 , 















 

√        L√ 1 L + L  3 T 1   gg  + T aero  + T 1 2 solar  + Tmag  L 1 L1 − 3L2 ∞ ∞ ∞ ∞       kq L2  2 +J   ¯    q L ω  J ω  + 2+ δ +nJ k k + L√1 max 2 max q intq 1 J1 max 2 2 2 L − 3L2 L1 √ 1                  3(L√ +L ) 1 2  gg  + T  aero  + T  solar  + T  mag  = T L1 − 3L2 ∞ ∞ ∞ ∞       kq L√ 2 2 ¯   L1 + 2 + δJ1 Jmax ω  2 + nJmax kq kintq q2 .  2 + Jmax 2 ω + L1 − 3L2

kss >

(8.53b)

In the simulation of the baseline controller, the state-dependent switching gain varies according to the        √          1 +L2 )    + + kss x (t) = 1.1 × 3(L√ T T Tgg  + T      aero mag  solar L1 − 3L2 ∞ ∞ ∞ ∞    kq L√ 2 2 ¯ +  (t)2 + Jmax 2 ω L1 + 2 + δJ1 Jmax ω  (t)2 L1 − 3L2    +nJmax kq kintq q (t)2 , (8.54) which satisfies the condition in Eq. (8.53b). The state responses are presented in Figs. 8.2 and 8.3.

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FIG. 8.2 Quaternions by baseline controller.

FIG. 8.3 Angular velocities by baseline controller.

213

COMPARISON THROUGH SIMULATIONS

FIG. 8.4 Sliding surface vector components by baseline controller.

Because the component of the lumped disturbance vector along the local geomagnetic field vector cannot be counteracted due to the lack of control action along that direction, steady-state errors exist in space responses. The attitude motion is kept in sliding mode once that mode is entered at the end of a very short reaching mode (see Fig. 8.4). Fig. 8.5 shows how the switching gain is being adjusted according to the decision rule in Eq. (8.54). As seen from Fig. 8.6, magnetic torquers are required to produce magnetic moments of nearly 50 A m2 . In Figs. 8.7–8.10, the environmental disturbances are depicted as limited by their infinity norm values, which were evaluated in Sofyalı and Jafarov [2].  unc is plotted in Fig. 8.11 against its infinity norm envelope, which is T computed by its derived formula     Tunc 

∞

⎫ ⎧   √   k ⎪ ⎪ ¯ ⎬  2 + 3kss + ks s2 · · ·  22 + Jmax 2q ω L2 ⎨ L1 + 2 + δJ1 Jmax ω          = .   √         ⎪ L1 ⎪  gg  + T  aero  + T  solar  + T  mag  ⎭ ⎩ · · · + nJmax kq kintq q + 3 T 2 ∞

∞

∞

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∞

(8.55)

FIG. 8.5 Switching gain by baseline controller.

FIG. 8.6 Magnetic moments by baseline controller.

FIG. 8.7 Gravity-gradient torque.

FIG. 8.8 Aerodynamic drag torque.

FIG. 8.9 Solar pressure torque.

FIG. 8.10 Residual magnetic torque.

COMPARISON THROUGH SIMULATIONS

217

FIG. 8.11 Inertia matrix uncertainty torque.

The nonlinear disturbance observer-based magnetic sliding mode attitude controller is taken into operation after half an orbit. As seen in  ∗ can be estimated by the observer. Figs. 8.12–8.14, the components of T d The disturbance observer-based control system outputs the same system responses (see Fig. 8.15) with the baseline controller. In addition, if the angular velocity curves are zoomed in as in Fig. 8.16, the alleviation of chattering can be observed. To satisfy the reaching condition in Eq. (8.48b), the switching gain of the observer-based controller is calculated by kss = 1.1 ×

 L1 e , √ ∞ L1 − 3L2

(8.56)

where the infinity norm of the estimation error in the interval beginning at 0.5T is decided to be taken as 1 × 10−9 Nm after simulation trials. The drop in kss occurring at the moment when it is switched to the observerbased controller is shown in Fig. 8.17. This change results in a highly decreased control effort (see Fig. 8.18) while the robustness property of the purely magnetic attitude control system’s global stability is preserved; Fig. 8.19 shows that the attitude motion is kept in sliding mode, both at the switching moment and thereafter.

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FIG. 8.12 Disturbance estimation: First component.

FIG. 8.13 Disturbance estimation: Second component.

FIG. 8.14 Disturbance estimation: Third component.

FIG. 8.15 Quaternions by disturbance observer-based controller.

FIG. 8.16 Angular velocities by disturbance observer-based controller.

FIG. 8.17 Switching gain.

FIG. 8.18 Reaching control signals.

FIG. 8.19 Sliding surface vector components.

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8. DISTURBANCE OBSERVER-BASED CONTROL OF SPACECRAFT ATTITUDE

5 CONCLUSION The considered purely magnetic control problem is formulated for nonlinear spacecraft attitude dynamics. Thus, a nonlinear disturbance observer had to be employed. There are four major environmental disturbance components in spacecraft attitude control, all of which were taken into account in the design of the baseline controller with their realistic mathematical models. In addition, the complete effect of the inertia matrix uncertainty on attitude dynamics was expressed in the form of a fifth external disturbance torque vector, which enables its summation with environmental torque vectors to construct the so-called lumped disturbance torque vector. The employed observer requires that the two-norm of the lumped disturbance’s time derivative is bounded. Because the inertia matrix uncertainty torque is explicitly dependent on the control torque vector, which is discontinuous, the observer dynamics have to be defined so that only the continuous component of the lumped disturbance torque vector is being estimated. The results showed that the disturbance estimation was achieved. The integration of the observer with the baseline controller was accomplished by negatively feeding the output of the observer, which is the estimated continuous lumped disturbance component, to the system via the continuous part of the sliding mode control law, namely the equivalent control law. The angular velocities and the control vector are inputs to the disturbance observer according to its design. Angular velocities are not smooth as a result of the chattering phenomenon, thus high-frequency oscillations in their signals had to be attenuated by a low-pass filter before they entered the observer. Because the equivalent control vector is evaluated by the baseline controller, it is always known for the control system. However, inputting the equivalent control signal instead of the whole sliding mode control signal would degrade the robustness property of the ultimate system. Therefore, the same filtering was applied on the discontinuous control signals at the input of the disturbance observer. The resulting scheme performed well, and the amplitude of the discontinuous reaching control signals became incomparably lower than the one in the baseline sliding mode attitude control system. The disturbance observer-based sliding mode attitude control system retains the stability robustness of the baseline system.

References [1] S. Li, J. Yang, W.H. Chen, X. Chen, Disturbance Observer-Based Control: Methods and Applications, CRC Press, Boca Raton, FL, 2014. [2] A. Sofyalı, E.M. Jafarov, New sliding mode attitude controller design based on lumped disturbance bound equation, J. Aerosp. Eng. 31 (1) (2018) 1–15.

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[3] J.R. Forbes, C.J. Damaren, Geometric approach to spacecraft attitude control using magnetic and mechanical actuation, J. Guid. Control Dyn. 33 (2) (2010) 590–595. [4] S.P. Bhat, Controllability of nonlinear time-varying systems: applications to spacecraft attitude control using magnetic actuation, IEEE Trans. Autom. Control 50 (11) (2005) 1725–1735. [5] F. Martel, P.K. Pal, M. Psiaki, Active magnetic control system for gravity gradient stabilized spacecraft, in: Proceedings of the Second Annual IAA Conference on Small Satellites, Utah State University, Utah, 1988. [6] A. Sofyalı, E.M. Jafarov, R. Wisniewski, Robust and global attitude stabilization of magnetically actuated spacecraft through sliding mode, Aerosp. Sci. Technol. 76 (2018) 91–104. [7] V.I. Utkin, Sliding Modes in Control and Optimization, Springer-Verlag, London, 1992. [8] C. Edwards, S.K. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor & Francis, London, 1998. [9] E.M. Jafarov, Variable Structure Control and Time-Delay Systems, WSEAS Press, Athens, 2009. [10] Y. Shtessel, C. Edwards, L. Fridman, A. Levant, Sliding Mode Control and Observation, Springer, New York, NY, 2014. [11] M. Chen, W.H. Chen, Sliding mode control for a class of uncertain nonlinear system based on disturbance observer, Int. J. Adapt. Control Signal Process. 24 (1) (2010) 51–64.

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C H A P T E R

9 A Class of Unknown Input Observers Under H∞ Performance for Fault Diagnosis: Application to the Mars Sample Return Mission David Henry*, Finn Ankersen † , Luigi Strippoli ‡ Laboratory, University of Bordeaux, Talence, France † European Space Agency, DTEC, Noordwijk, The Netherlands ‡ GMV Aerospace and Defence S.A.U., Madrid, Spain

*IMS

1 INTRODUCTION The introduction is organized such that we will provide a list of real space missions that have exhibited severe faults in flight in order to set the context for the chapter. This is then followed by a discussion of the fundamental types of FDI and estimation filters used and concludes with a short summary of the notations used throughout. Many types of faults may occur in spacecraft during a space mission. Dealing with actuator faults, one can mention the following on-orbit failure accidents of previous space missions (to name a few): • The Far Ultraviolet Spectroscopic Explorer (FUSE) spacecraft lost two of its four reaction wheels [1], causing a mission interruption. • NASA’s Earth-orbiting Lewis Spacecraft failed due to excessive thruster firing. This led to the shutdown of all thrusters and left the spacecraft in an uncontrolled attitude, draining most of its battery charge [2].

New Trends in Observer-based Control https://doi.org/10.1016/B978-0-12-817034-2.00022-8

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© 2019 Elsevier Inc. All rights reserved.

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• The GPS BII-07 spacecraft of the NAVSTAR GPS constellation suffered from a reaction wheel failure that led to a three-axis stabilization failure and a total loss of the spacecraft [3]. • The Iridium 27 spacecraft failed due to a thruster anomaly that depleted operational fuel, and the mission of Iridium 42 was interrupted due to a wheel tachometer failure [3]. • The Nozomi spacecraft consumed more fuel than expected during Earth swing by due to a thruster valve being stuck open. • The GOES-9 total loss was caused by a momentum wheel onboard that was drawing high current, resulting in hot operating conditions [4]. • The primary pitch momentum wheel of the Radarsat-1 spacecraft failed due to excessive friction and temperature, causing a mission degradation [5]. • On the Galaxy 8i spacecraft, three of four xenon ion thrusters failed in 2000 [3]. • The EchoStar V mission was interrupted due to a failure of one of the three momentum wheels [4]. • The spacecraft EchoStar VI was hit by micrometeorites, causing a propellant leak in one of the thrusters. • The TOPEX spacecraft was not able to perform attitude maneuvers due to a failure of the pitch reaction wheel [6]. • The spacecraft JCSat-1B experienced attitude loss due to a thruster anomaly [7]. • Wheels 2 and 3 of the GPS BI-05 spacecraft stopped completely [4]. • The Myriade satellites experienced depointing due to the occurrence of bubbles in fuel lines, causing an intermittent lack of thrust [8]. This quick overview clearly demonstrates that model-based fault detection isolation (FDI) solutions can be used on-board to engage autonomous recovery actions and thus significantly increase the autonomy of a spacecraft. This has begun to be admitted by space agencies and industries, see for instance the interesting talks and papers from the space agencies and industries [9–23]. The work presented in this chapter should be understood in this context. It deals with the development of a model-based FDI solution for thruster faults in the propulsion system of the chaser spacecraft during rendezvous on a circular orbit around Mars. The reference mission is the European Space Agency (ESA) Mars Sample Return (MSR) mission. This is of prime importance for such deep space missions because it is obvious that failures cannot be diagnosed and recovered through telemetry actions. This is due to the significant communication delays between the spacecraft and the Earth’s ground stations and the potential lack of communication.

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Numerous model-based FDI techniques have been studied in the past decades by the academic community. The proposed techniques can be brought down to two basic concepts: • Estimation methods where the goal is to estimate a set of parameters or input signals. It follows that any estimation technique can be used on an adequately formulated problem, one of the most famous being the Kalman-based techniques. In this context, one can refer to the linear Kalman filter [24]; the extended Kalman filter (EKF) technique and its improved versions such as the unscented Kalman filter proposed in Refs. [25, 26] and further investigated in Ref. [27]; the cubature Kalman filter (CKF) [28] (in contrast to the Kalman-based filters, the CKF avoids linearization issues), the divided-difference filter (DDF) initially proposed by Norgaard et al. [29] (the DDF uses divided-difference approximations of derivatives based on Stirling’s interpolation formula), and the particle filtering approaches [30–32]. • Residual generation is different from fault estimation because it only requires the disturbances and model perturbations attenuation. The residual has to remain sensitive to faults while guaranteeing robustness against unknown inputs. This means that a fault-estimation problem is fundamentally a minimization problem, whereas residual generation is a minimization/maximization problem. One usually distinguishes three main categories: • the parity space approach [33–39]; • the observer-based approach through the so-called unknown input observers (UIO) approaches, the eigenstructure assignment technique [34, 40–46], the iterative learning observer (ILO) technique [47], and the sliding mode observer (SMO) approach; see for instance Refs. [48–50] and the books [51, 52] for background material about sliding mode theories and the very recent interval approaches [53–57]; and • the so-called norm-based approaches, sometimes referred to as the approximate decoupling approach. These approaches can be further classified according to the previous discussion as fault-estimation approaches [58–65] and residual-generation approaches [65–75]. With regard to the problem of spacecraft thruster fault diagnosis, one can mention the work presented in Ref. [76] that proposes an ILO (jointly with a sliding mode technique in Ref. [77]) to achieve estimation of thruster faults. An SMO-based approach is considered too in Ref. [50] for the Mars Express spacecraft during the sun acquisition mode. The work reported in Ref. [78] also addresses the Mars Express experiment and is based on both state estimation of a linear model for the spacecraft system and unknown input decoupling. In Ref. [79], an EKF is used to estimate the

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bias of the torque. FDI is done by matching it with the torque directions of each thruster. In Ref. [66], the proposed method is based on a H(0) filter with robust pole assignment for fault detection and a cross-correlation test for fault isolation in thrusters. The H∞ /H− approach is considered for micro-Newton colloidal thruster faults during the experimental phase for the LISA Pathfinder experiment in Refs. [71, 80]; H∞ /H− filter-based strategies are also proposed to diagnose thruster faults for the Microscope spacecraft. The same technique is also proposed in Ref. [75] for thruster faults in the chaser spacecraft of the MSR mission. Thruster faults during station-keeping maneuvers are considered for telecom satellites in Ref. [15]. Finally, a pure H∞ FDI strategy is proposed in Refs. [81, 82] for control surface faults in the Hopper reentry vehicle. Note that this overview of existing works covers only studies that deal with real space missions. There exist plenty of papers that address the design of model-based FDI solutions for spacecraft, but they are thought to be more or less an academic exercise because they are not representative of the fully problematic (in view of the authors of this chapter). Often, they consider the spacecraft as a rigid body and/or do not consider the coupling between the rotational and translational motion, the effect of large solar arrays with more than four flexible modes, or the effect of propellant sloshing on, for example, the variation of the center of mass of the spacecraft. The work presented in this chapter overcomes these drawbacks. It addresses the design of a complete model-based FDI unit able to detect and isolate a single thruster fault occurring in the propulsion unit of the chaser spacecraft during the forced translation rendezvous phase of the ESA MSR mission, under worst case situations (maximum magnitude of disturbances, no membranes in tanks for annihilating propellant sloshing, worst case of flexible solar arrays, etc.). The investigated faults have been defined in accordance with ESA, GMV Space, and Thales Alenia Space industries. They follow their experience on real spacecraft, that is, the fault profiles are concerned by (i) stuck-open cases that correspond to a thruster opening at 100%: this case provides maximum force regardless of the demand and is very propellant-consuming; and (ii) a thruster closing itself: in this case, the faulty thruster does not generate any thrust. The proposed method belongs to the residual-based approach. More precisely, a fault detector based on the H∞ /H− technique is designed for robust fault detection with guaranteed fault sensitivity performance (in the H− -sense). The fault detector is also designed to ensure robustness against a large range of uncertainties such as uncertainties in the flexible modes of the solar arrays and propellant-sloshing modes. For fault isolation, a bank of a novel class of UIOs satisfying a H∞ performance is proposed. Intensive simulations from a high-fidelity simulator of the rendezvous phase of the MSR mission demonstrate that the proposed FDI scheme

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is able to cope with the considered thruster faults under highly realistic conditions. This chapter is organized as follows: Section 2 presents the rendezvous mission and describes the spacecraft characteristics. Section 3 is dedicated to modeling issues. Section 4 is devoted to the design of the actuator FDI unit and Section 5 presents the results obtained from an intensive simulation campaign from a high-fidelity simulator. It should be outlined that developments within the H∞ /H− setting [68, 69, 71, 74] are obviously presented in the following due to the use of the H∞ /H− fault detector. However, the central theoretical contribution of this chapter is focused on the novel class of the UIOs satisfying the H∞ performance presented in Section 4.1.

1.1 Notations Throughout the chapter, the following notations are used: R and C denote the real and complex sets, respectively. R+ := [0, ∞) ⊂ R is the set of nonnegative real numbers. Rn (Cn ) is the n-dimensional real (complex) space. Rn×m (Cn×m ) is the set of real (complex) n × m matrices. If A ∈ Rn×m (A ∈ Cn×m ), then AT ∈ Rm×n is the transpose of A (AH ∈ Cm×n is the transpose conjugate). I and O denote, respectively, the identity and the null matrices of the appropriate dimensions. When there exists no ambiguity, these dimensions are not given. Otherwise, they are specified as indices, that is, In denotes the identity matrix of dimension n and On×m denotes the null matrix of dimension n × m. A > 0 means the definite positiveness of A. A ≥ (>)B means A − B is positive semidefinite (positive definite). σ (A)/σ (A) denote the maximum/minimum singular values of the matrix A. Let w(t): R+ → Rn .1 The L2 -norm of w (often called simply the 2-norm  ∞ 1/2 . The shorthand w ∈ L2 of w(t)) is defined as w2 := 0 w(t)T w(t)dt where L2 refers to the space of signals mapping R+ to Rn with finite L2 norm is equivalent to w  2 < ∞. A linear state-space model (A, B, C, D) is  A B . Its associated Laplace transform P(s) (or simply denoted P = C D P) is defined according to P(s) = C(sI − A)−1 B + D. P(s) is assumed to be in RH∞ , real rational function with P∞ = sup σ (P(j )) < ∞. P∞ is also called the H∞ -gain of P. In a similar way, P− is used to denote the smallest gain of a transfer matrix P. However, this is not a norm. Despite the fact that a clear definition of the H∞ norm exists, there exist different definitions for the H− gain. For example, the H− gain for LTI systems is defined on a finite frequency range in Refs. [60, 68, 83] whereas it is defined on an infinite frequency horizon in Refs. [84, 85]. If the smallest gain of P(s) is defined according to inf  σ P(j ) , then for strictly proper 1

The case w(t): R+ → Cn is not considered in this manuscript.

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  P, inf  σ P(j ) = 0 because the frequency range of interest is infinite. Toward this end, the H− index of P is  defined  here as in Refs. [68, 69, 86], P(j ) to a finite frequency domain σ that is, it is the restriction of inf    Ω: P− = inf  ∈Ω σ P(j ) < ∞. Thus, P− = 0 is always true for any P; see Refs. [68, 86] if necessary. The H− criterion is sometimes considered for  → 0 and is thus defined by P(0) = lim →0 P(0)− . This criterion is called the H(0) gain; see Ref. [66]. The linear fractional transformation (LFT) paradigm is used For appropriately dimensioned   in the following. M11 M12 , the lower LFT is defined according matrices N and M = M21 M22 to Fl (M, N) = M11 + M12 N(I − M22 N)−1 M21 and the upper LFT according to Fu (M, N) = M22 +M21 N(I −M11 N)−1 M12 , under the assumption that the involved matrix inverses exist. This assumption is discussed in the chapter when it is judged necessary. Otherwise, it is assumed to be satisfied.

2 THE RENDEZVOUS SCENARIO The MSR mission consists of two vehicles directly injected toward Mars by launchers. The first module enters the Martian atmosphere, lands on the surface, fetches Martian samples, inserts them into a spherical container called the orbiting sample (OS) (we will call it the target spacecraft in the following), and is then launched from the surface through a Martian ascent vehicle (MAV), to reach a low Mars orbit where the OS is released by the MAV. Fig. 9.1 illustrates the rendezvous orbit. Its main characteristics are a semimajor axis of a = 3919.6 km, an eccentricity e = 0, and an inclination of i = 40 degrees. Meanwhile, the chaser spacecraft (also the orbiter) is inserted on a similar orbit as the target at a relative distance of some hundred meters behind. It performs the rendezvous operations with the target to reach a relative distance of 100 m and places it into a dedicated reentry capsule (the earth reentry capsule [ERC]), and finally returns to Earth, where it ejects the ERC into the Earth’s atmosphere. Fig. 9.2 illustrates the geometry of the two spacecrafts. In this work, we will focus on the final translation in forced motion of the rendezvous with the canister containing the Martian samples. The rendezvous conditions are driven by the capture or docking mechanism. The docking mechanism is basket-like with a funnel followed by a cylindrical aperture, which is part of the sample handling system. It is oriented along the chaser x-axis (see Fig. 9.2) and located with a lateral offset on the +x face, hence not on the axis symmetry line of the chaser. The rendezvous

i, Y i, Z i 2 and the orbit around Mars in the inertial frame Fi = OM ; X i, Y i, Z i The origin of this frame coincides with the center of the Mars planet and the axes X are oriented orthogonally, as shown in Fig. 9.1. 2

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zi Chaser Target

3 yi 2

1

Approach corridor Chaser trajectory

z (m)

Ascent vehicle

0 xi Orbit of the redezvous

–1

–2

The rendezvous trajectory Capture of the canister (target)

–3 –3

–2 6

´10

2 0

–1

0

1

2

–2 3

THE RENDEZVOUS SCENARIO

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Initial position of the chaser and target (relative position = 100 m)

6

´10

´106

y (m)

x (m)

FIG. 9.1 The rendezvous phase of the ESA’s Mars Sample Return mission. The inclination of the orbit is with the plane (X i , Y i ).

231

232

9. NEW TRENDS ON FAULT DIAGNOSIS METHODS Flaps HGA SHS

ZOM

OM

X

SA1

ERC

Y

tanks

SA2 Solar cell SC base

10 6

X

12

4RF antennas

2

SC lid

8

4

9 5

I/F for end-effector with central screw Separation bolt

11

1 3

7

Optical connector

FIG. 9.2 MSR Orbiter (chaser spacecraft), position of its thrusters, and the canister containing the Martian samples (target spacecraft).

chaser relative rendezvous trajectory

in the local vertical local horizontal l, Y l, Z l 3 are illustrated in Fig. 9.1 for a better (LVLH) frame Fl = Ol ; X understanding of the mission.4 The performance objective of the control is to dock with an accuracy better than 5 cm laterally and a speed less than 10 cm/s. During the rendezvous, the control of the relative attitude and position of the chaser is continuous and applied by the thrusters. Nominally 12, with 12 redundant thrusters, are on the chaser; see Fig 9.2 where the position of the 12 thrusters can be visualized. The relative attitude is controlled in order to keep the target within the camera field of view and to align the docking mechanism to fulfill its docking requirements. At the sensor level, the chaser uses two inertial measurement units (IMU) in hot redundancy,5 a Star-Tracker in cold redundancy, and two cameras in hot redundancy. The set of sensors and actuators is minimized l is pointing to OM , The origin of the LVLH frame is at the target center of mass, the axis Z l is aligned with the negative orbital momentum vector, and X l = −Z l × Y l. the axis Y

3

4

The presented figure comes from the high-fidelity nonlinear simulator.

5

Hot and cold redundancy is terminology used by space industries to outline that all redundant sensors are switched on (hot), the others being switched off and waiting for a possible use (cold).

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TABLE 9.1 Hierarchical Fault Detection Levels Levels

Interest

Level 1 sensor checks

Monitoring of the outputs of all sensors. This level covers most of the sensor faults such as sudden sensor death and lock-in-place fault types

Level 2 IMU/IMU-IMU/STR

Interest is limited to the detection of failures not seen by Level 1, such as slow drifts

Level 3 thruster/IMU

Interest is faults in thrusters. The IMU hot redundancy enables discarding IMU failures, leading model-based techniques based on the IMUs to be viable candidates

Level 3 wheel/tachometer

Covers wheels faults. The isolation is immediate because a tachometer is available on each wheel

Level 4 approach corridors

Monitor the position/velocity of the chaser versus the approach corridors

Level 4 collision risks

Detect if a collision may occur between the spacecraft

Level 4 mode success

Detect the divergence of the controller outputs

Level 5 power alarm

Protection against ground operation errors and electrical subsystem failures

to reduce the risk of fault occurrence and to reduce the power consumption and mass. Two large solar arrays (SA1 and SA2) are attached to the chaser body, which embeds two large tanks. Both tanks are half-filled during the rendezvous and this induces the propellant-sloshing phenomena. The propellant mass represents about 30% of the total mass and its impact on the overall center of mass location can reach 15 cm.

3 MODELING ISSUES Common fault diagnosis and recovery architectures used by space industries rely on a hierarchical implementation of failure detection and management. Recent studies for the case of the MSR mission tend to demonstrate that the failure management system that covers completely all system and subsystem failures consists of several functions, all being hierarchically organized into five levels [87, 88]. These levels, with the main functions they are dealing with, are summarized in Table 9.1. As can be seen in this table, the model-based FDI unit that is presented in this chapter is intended to be implemented at Level 3. Because the IMU hot redundancy enables discarding IMU failures, that is, failures in the IMU are diagnosed and accommodated at Level 2 (by switching to the faultfree IMU), it seems clear that the proposed model-based FDI unit has to be developed based only on the IMU measurements so that it is a viable solution from a global point of view. Toward this end, the design FDI will II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

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be based on the dynamics that links the thruster commands and the IMUs measurements, that is, the attitude dynamics.

3.1 Modeling the Chaser Rotational Motion The rotational motion of the chaser caused by an applied moment (sum of all torques acting on it) can be derived from Euler’s second law in the b, Z b, Y b } (the origin of this frame is fixed to the body frame Fb = {Ob ; X b, Z b, Y b are center of mass of the chaser and the directions of the vectors X the directions of X, Y, ZOM , illustrated in Fig. 9.2):  Tk − J−1 ω × Jω (9.1) ω˙ = J−1 k

]T

ω = [ωx , ωy , ωz is the angular velocity vector and J ∈ R3×3 is the inertia dyadic about the chaser’s CoM. In Eq. (9.1), k Tk ∈ R3 describes the sum of torques. • The external torques, which are the torque Tp ∈ R3 due to the thruster-based propulsion unit and the orbit disturbances torque Td ∈ R3 due to the solar pressure, the Mars gravity gradient, and the atmospheric drag. • The internal torques, which are mainly the torques caused by the solar arrays Tsa ∈ R3 and the propellant-sloshing phenomena Ts ∈ R3 . Remark 1. In this work, Td is considered as an exogenous vector and it is not required to have a mathematical model. Toward this end, its derivation is not explained in the following. The interested reader can refer to Ref. [89]. Modeling the flexible appendages: Tsa comes from the flexible modes of the two solar arrays. Flexible modes can be modeled using a second-order vector-based equation, that is, ˙ q¨ + 2ξ ω0 q˙ + ω02 q = LT ω,

q ∈ Rns ·np

(9.2)

ns denotes the number of flexible modes and np is the number of solar arrays. Here, np = 2 and for the considered solar arrays, it is assumed ns = 4, meaning that the solar arrays have four flexible modes. Thus, it follows that q ∈ R8 . ξ and ω0 are matrices of adequate dimensions that correspond to the damping ratios and the frequencies of each flexible mode. Then Tsa = −L¨q −

np 

JSAi ω˙

i=1

(9.3)

 The matrices L = . . . Li . . . , i = 1, . . . , np : L ∈ R3×(ns ·np ) and Jsai , i = 1, . . . , np : JSAi ∈ R3×3 characterize the solar arrays: Jsai is the total inertia of the ith solar array. It corresponds to the sum of the nominal inertia

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MODELING ISSUES

ZSA

SA1 Z OM YSA

Z z Tank2

X

z Tank1

Y X

y

y

Y

XSA

FIG. 9.3 The frames FSA (left) and Fti , i = 1, 2 (right).

J0i ∈ R3×3 and the so-called transport inertia. The matrices Li are given according to: Li = RSA BRi + S(d)RSA BTi ,

i = 1, . . . , np

(9.4)

The matrices BRi , BTi , i = 1, . . . , np are the flexible appendages participation SA , Y SA , Z SA }; see factor matrices, all given in the frame FSA = {OSA ; X Fig. 9.3 for the definition (center and direction of vectors) of FSA . The role of the rotation matrix RSA is to transform BRi , BTi , i = 1, . . . , np , given in FSA , into the body frame Fb ; see Fig. 9.3 for an illustration of the two frames. S(d) denotes the skew symmetric matrix of the vector d ∈ R3 , d being the distance vector between the centers of the frames FSA and Fb ; see Fig. 9.3. In this work, we address only the rendezvous part final translation, so it is assumed that the two solar arrays do not rotate so that RSA and L are constant matrices. Modeling the propellant sloshing: Propellant sloshing in the two tanks is modeled in this work as a three-dimensional spring-mass model, given in their associated frames Fti , i = 1, 2; see Fig. 9.3 that gives an illustration of Fti , i = 1, 2. Because the chaser is equipped by two tanks, it follows that Ts = 2i=1 Tsi , the torque contribution due to each tank Tsi being deduced from the following mass-spring damper vector-based equation, given in the body frame Fb (note that the orientation of Fti , i = 1, 2 coincides with the orientation of the body frame Fb ) li ki x˙ s + xs = γti , mi i mi i Tsi = ri × (li x˙ si + ki xsi )

x¨ si +

xsi ∈ R3 , γti ∈ R3 ,

i = 1, 2

(9.5) (9.6)

In Eq. (9.6), xsi is the vector from the center of the ith tank to the center of the liquid, and li and ki , i = 1, 2 are matrices of adequate dimension, defining the damping and stiffness coefficients. mi is the sloshing mass associated II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

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9. NEW TRENDS ON FAULT DIAGNOSIS METHODS

with the ith tank and ri ∈ R3 is the distance vector between the spacecraft CoM and the center of mass of the ith tank, in the body frame. γti ∈ R3 is the acceleration on the considered fuel tank, which is defined according to  γti = − γ + 3k=1 γki where the sign “–” is due to the used orientation of the frames, and 3k=1 γki describes the sum of the following accelerations about the ith tank: • The Coriolis acceleration: γ1i = 2ω × x˙ si ,

i = 1, 2

• The centrifugal acceleration:   γ2i = ω × ω × (ri + xsi ) ,

i = 1, 2

(9.7)

(9.8)

• The Euler acceleration: γ3i = ω˙ × (ri + xsi ), γ ∈ R3 is the acceleration vector mγ =



i = 1, 2

Fk

(9.9)

(9.10)

k

where m is the mass of the chaser. The term k Fk describes the sum of forces acting on the chaser, that is, the forces due to the propulsion system Fp ∈ R3 , the orbit disturbance forces Fd ∈ R3 , and the forces generated by the two solar arrays Fsa ∈ R3 . The determination of Fsa follows the same reasoning used for Tsa . Then, it can be verified that: Fsa = −L¨q −

np 

mSAi γ

(9.11)

i=1

L = [. . . Li . . .],

Li = RSA BTi ,

i = 1, . . . , np

(9.12)

where mSAi denotes the mass of the ith solar array. Remark 2. Similarly to Remark 1, Fd is considered an exogenous vector and does not require a mathematical model. Toward this end, it is not explained in this work. Remark 3. As can be noted, the term k Fk does not account for sloshing forces. This is because sloshing forces have been revealed to have no impact on the FDI performance. The interested reader can refer to, for example, Ref. [89] for modeling issues.

3.2 Thrusters With Fault Model Consideration The thrusters are physically organized as illustrated in Fig. 9.2. They are in charge of producing a force Fp ∈ R3 and a moment Tp ∈ R3 . Let Sall = {1, 2, . . . , 12} denote the set of all the thruster indices. The numbering II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

237

MODELING ISSUES

is shown in Fig. 9.2. All thrusters have fixed directions and each one is able to produce a maximum thrust of 10N. The electronics driving the thrusters initiate the opening of each thruster valve for the commanded duration 0 ≤ uk ≤ 1, ∀k ∈ Sall , which are in fact scaled ON times. The scaling is done versus the sampling period Ts of the control unit and is defined according to ui (tk ) = Toni (tk )/Ts , where Toni (tk ) is the actual firing duration (ON time) of the ith thruster at control cycle tk = kTs . The propulsion system is obviously a source of uncertainty in the system. Here, this uncertainty is modeled as a time delay τ that aims at modeling the effect of the unknown delay induced by the electronics and the uncertainties on the thruster rise times. The delay τ is assumed to be unknown, but is upper bounded by a known constant τ , that is, τ ≤ τ . Let uk (t−τ ), ∀k ∈ Sall be the commanded opening duration vector of the kth thruster delayed by τ . Then the net moments Tp and forces Fp generated by thrusters (in fault-free case) are given in the chaser body fixed frame by Tp = [MT1 . . . MT12 ]u = MT u(t − τ ), Fp = [MF1 . . . MF12 ]u = MF u(t − τ )   MT , M ∈ R6×12 M= MF

u ∈ R12

(9.13) (9.14) (9.15)

The columns of MT and MF are the influence coefficients defining how each thruster affects each component of Tp and Fp , respectively. By analyzing the matrices MT and MF in terms of directional properties, the following is revealed: Sall can be classified into the following five different subsets of thrusters S1 = {1, 2, 3, 4},

S2 = {5, 12},

S3 = {6, 11},

S4 = {7, 10},

S5 = {8, 9} (9.16) Each of these subsets plays a particular role, especially the directions spanned by the column vectors of MT and MF , which reveal that MT1 = −MT4 , MT2 = −MT3 , MT5 ≈ MT12 , MT6 ≈ MT11 , MT7 ≈ MT10 , MT8 ≈ MT9 MF1 = MF2 = MF3 = MF4 , MF5 = −MF12 , MF6 = −MF11 , MF7 = −MF10 , MF8 = −MF9 (9.17)

This means that the thruster pairs of the sets Sk , k = 2, . . . , 5 produce exactly opposite forces and close moment directions. The subset S1 produces exact collinear forces while it can be noticed that the pairs of the subsets S14 = {1, 4} and S23 = {2, 3} produce exactly opposite moments. These directional properties are illustrated in Fig. 9.4. Thus, the complete set Sall can be classified into the following seven different subsets of thrusters S1 = {1, 2, 3, 4}, S2 = {5, 12}, S3 = {6, 11}, S5 = {8, 9}, S14 = {1, 4}, S23 = {2, 3}

S4 = {7, 10},

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(9.18)

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9. NEW TRENDS ON FAULT DIAGNOSIS METHODS Force directions (normalized)

Moment directions (normalized)

12 2

1

7

10

11

4 1 12 5

0 8

-0.5

6 11

9

0.2 0

y

-0.5

0

-0.2 -1

-0.4

1

7 10

0

1

9 0.4

0.5

3

-0.5

3 -1 1

2 4

8

0.5

1

z

z

0.5

0.5

6 -1 0.4

5 0.2

x

0

y

-0.2

0 -0.4

-0.5

x

FIG. 9.4 The directional properties of MT (left) and MF (right).

With regard to the thruster faults considered in this work, the focus is on the stuck-open (fully open) and stuck-closed (closed thruster) faults. The following mathematical model can be used to describe these faults:  max{uk (t), 1} if stuck open (9.19) φk (t) = 0 if stuck closed where the index k refers to the kth thruster. Assuming no simultaneous faults, the considered thruster faults can be modeled in a multiplicative way according to (the index f outlines the faulty case) uf (t) = (I12 − Ψ (t)) u(t)

(9.20)

with Ψ (t) = diag (ψ1 (t), . . . , ψ12 (t)), where 0 ≤ ψk (t) ≤ 1, ∀k ∈ Sall are unknown. The status of the kth thruster is modeled by ψk as follows:  0 if healthy (9.21) ψk (t) = 1 − φk (t)/uk (t) if faulty where φk enables considering the two different fault scenarios.

3.3 Toward an LFT Model

T

Let x be the vector defined according to x = ωT qT q˙ T xTs x˙ Ts where T

xs = xTs1 xTs2 . With Eqs. (9.1)–(9.3), (9.6), it can be verified that the angular acceleration ω˙ is given by   −1 −1 Tp + Td (9.22) f (x) + Jeq ω˙ = Jeq where the nonlinear function f (x) is defined according to 2    f (x) = L 2ξ ω0 q˙ + ω02 q − ω × Jω + ri × (li x˙ si + ki xsi ) i=1

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(9.23)

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n p and where Jeq = J + LLT + i=1 JSAi plays the role of an equivalent inertia. Then with Eq. (9.2), it follows that   −1 Tp + Td (9.24) q¨ = g(x) + LT Jeq where the nonlinear function g(x) is defined according to   −1 f (x) − 2ξ ω0 q˙ + ω02 q g(x) = LT Jeq

(9.25)

Now consider Eqs. (9.10), (9.11), (9.24), it can be verified that the acceleration γ satisfies the following relation   1  −1 (Tp + Td ) (9.26) Fp + Fd − L g(x) + LT Jeq γ = meq np where meq = m+ i=1 mSAi . Then, with Eqs. (9.5), (9.7)–(9.9), it can verified that the following expression holds for x¨ si , i = 1, 2  1  T −1 x¨ si = hi (x) + LL Jeq (Tp + Td ) − (Fp + Fd ) , i = 1, 2 (9.27) meq In this equation, the function hi (x) is defined according to hi (x) =

  1 Lg(x) − 2ω × x˙ si − ω × ω × (ri + xsi ) meq li ki − x˙ si − xs , i = 1, 2 − ω˙ × (ri + xsi ) − mi mi i

(9.28)

Combining Eqs. (9.22), (9.24), (9.27) into a state-space form leads to the following state-space representation that is nonlinear into the state x, but linear into the torques and forces Tp , Fp due to the propulsion unit and the torques and forces Td , Fd due to the spatial disturbances, the measurement vector y being provided by the IMUs ⎧ ⎤ ⎤ ⎡ −1 ⎤ ⎡ ⎡ −1 Jeq O ⎪ Jeq f (x) ⎪ ⎪ ⎥ ⎢ O ⎥ ⎪ O ⎪ ⎢ q˙ ⎥ ⎢ ⎥ ⎥ ⎢ ⎪ ⎥ ⎢ ⎪ ⎢ T −1 ⎥ ⎥ ⎪ ⎢ ⎪ ⎢ g(x) ⎥ ⎢ L Jeq ⎥   ⎢ ⎢ O ⎥ ⎨ ⎥ ⎢ + + T + T x˙ = ⎢ ⎥ ⎥ (Fp + Fd ) ⎢ ⎢ p d ⎥ O ⎥ ⎥ ⎢ O ⎢ x˙ s ⎥ ⎢ 1 ⎥ ⎥ ⎢ ⎪ ⎢ 1 T −1 ⎪ ⎣ h1 (x) ⎦ ⎣ m LL Jeq ⎦ ⎪ ⎣− meq I3 ⎦ ⎪ eq ⎪ ⎪ 1 T −1 ⎪ h2 (x) − m1eq I3 ⎪ meq LL Jeq ⎪ ⎩ y = ω + nω = [I3 O] x + nω (9.29) Here nω denotes the IMU noise. Now, considering the model of the faulty thrusters given by Eqs. (9.13), (9.20), it follows that the rotational dynamic of the chaser (with possible

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faults in its propulsion unit) obeys the following nonlinear state-space model, ⎧ ⎤ −1 ⎤ ⎡ ⎡ −1 O Jeq ⎪ f (x(t)) J ⎪ eq ⎪ ⎪ O O ⎥ ⎪ ⎢ q˙ (t) ⎥ ⎢ ⎪ ⎥ ⎪ ⎥ ⎢ ⎢ −1 ⎪ ⎢ LT Jeq O ⎥ ⎪ ⎥ ⎢ ⎪ ⎥ ⎢ g(x(t)) ⎪ ⎥ ⎪ M (I12 − Ψ (t)) u(t − τ ) x˙ (t) = ⎢ ⎪ ⎢ x˙ s (t) ⎥ + ⎢ O O ⎥ ⎪ ⎥ ⎢ ⎪ ⎥ ⎢ 1 ⎢ ⎪ −1 − 1 I ⎥ ⎪ ⎣ h1 (x(t)) ⎦ ⎣ m LLT Jeq ⎪ meq 3 ⎦ ⎪ eq ⎪ ⎪ 1 1 T −1 ⎪ (x(t)) h 2 ⎪ ⎨ meq LL Jeq meq I3 ⎤ ⎡ −1 O Jeq ⎪ ⎪ ⎢ ⎪ O O ⎥ ⎪ ⎥ ⎢ ⎪  ⎪ T −1 ⎢ ⎪ 0 ⎥ L Jeq ⎪ ⎥ Td (t) ⎢ ⎪ ⎪ + ⎥ ⎢ ⎪ O O ⎥ Fd (t) ⎪ ⎢ ⎪ ⎪ ⎢ 1 LLT J−1 − 1 I ⎥ ⎪ ⎪ ⎣ meq ⎪ eq meq 3 ⎦ ⎪ ⎪ ⎪ 1 1 T J −1 ⎪ LL ⎪ eq meq meq I3 ⎪ ⎩ y(t) = ω(t) + nω (t) = [I3 O] x(t) + nω (t) (9.30) that is rewritten into the general form ⎧   Td (t) ⎨ x˙ (t) = ϕ(x(t)) + BM (I12 − Ψ (t)) u(t − τ ) + B Fd (t) (9.31) ⎩ y(t) = ω(t) + nω (t) = [I3 O] x(t) + nω (t) Considering an approximation of this nonlinear state space around small angular and linear accelerations ω∗ , γ ∗ (which is the case from a practical point of view), it follows that the rotational motion of the chaser can be modeled according to: ⎧    Td (t) ⎨ ∂ϕ(x)  x˙ (t) = Ax(t) + BM (I12 − Ψ (t)) u(t − τ ) + B Fd (t) with A = ⎩ ∂x ω∗ ,γ ∗ y(t) = ω(t) + n (t) = [I O] x(t) + n (t) ω

3

ω

(9.32) The delay τ is next approximated using a Padè approximation. To proceed, let ud (t) = u(t − τ ), ud ∈ R1 , u ∈ R1 and consider its associated transfer function ud (s) (9.33) = e−τ s , ud ∈ R1 , u ∈ R1 u(s) of the time delay τ being an irrational transfer. The Padè approximation substitutes e−τ s with an approximation in the form of the following rational transfer function e−τ s =

1 − α1 s + α2 s2 − · · · ± αp sp 1 + α1 s + α2 s2 + · · · + αp sp

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where p is the order of the approximation and the coefficients αi are functions of p. In this work, a first-order approximation is considered when the Padè coefficients become α1 = τ2 and αi = 0, i = 2, . . . , p. If this approximation is considered for the MIMO case, then the first-order Padè approximation is equivalent to the following state-space representation  x˙ d = Ad (τ )xd + Bd u (9.35) ud = Cd (τ )xd + Dd u where xd is the delayed state and the matrices Ad (τ ), Bd , Cd (τ ), and Dd are given as follows: 2 4 (9.36) Ad (τ ) = − I, Bd = I, Cd (τ ) = I, Dd = −I τ τ where the identity matrix is of dimension u (12 in our case). Then, using an additive approximation [90] of the fault multiplicative fault model stated by Eqs. (9.19)–(9.21), it can be verified that the model of the chaser can be written according to ⎧   Td ⎨ + Kf x˙ = Ax + Bu + E Fd (9.37) ⎩ y = Cx + nω = ω + nω In this equation, x is the augmented state vector x = [xT xTd ]T and the matrices A, B, E, and C are given by       BMDd B A BMCd (τ ) B= (9.38) C = [I3 O] E = A= Bd O O Ad (τ ) The ith column of the matrix K, namely Ki , is the ith fault signature associated with the ith fault mode fi . The indices i = 1, . . . , 12 also coincide with the numbering of the thrusters. The model (9.37) depends obviously on some uncertain parameters. The list of these parameters is given in Table 9.2. To tackle these uncertainties in an efficient manner, the uncertain state-space model (9.37) is formulated as an LFT. To proceed, let ρ = [. . . ρi . . .]T , ∀i be the vector of the uncertain parameters and ρi = ρi0 + wi δi . The notation ".0 " refers to the median value of the associated parameter. The terms wi δi refer to the variations of the parameters around their median value, wi being weights introduced  O C . to normalize δi so that |δi | ≤ 1. Let Σ(δ) = [B(δ) E(δ) K(δ)] A(δ) The derivation of the LFT can be stated as follows: given Σ(δ), find an interconnection matrix M and a block structure (r1 , . . . , rp ) such that   Σ(δ) = Fu (M, ), with  = diag δ1 Ir1 , . . . , δp Irp . The order of this p representation is r = i=1 ri . Of all representations for Σ(δ), we are particularly interested in minimal ones where r is the smallest possible. Algebraically, this problem can be

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TABLE 9.2 Uncertainties in the Model (9.37) Subsystem

Uncertain parameters

Flexible appendage

Participation matrix BR for mode 2 along x Participation matrix BR for mode 1 along y Participation matrix BR for mode 3 along y Participation matrix BR for mode 4 along z Participation matrix BT for mode 1 along x Participation matrix BT for mode 3 along x Participation matrix BT for mode 4 along x Participation matrix BT for mode 2 along y Damping coefficients ξ along x, y, z Frequencies ω0 along x, y, z

Propellant-sloshing model

Sloshing masses ms1 , ms2 Damping coefficients ls Stiffness coefficients ks

Chaser spacecraft

Position of center of mass CoM w.r.t. the geometrical center of the spacecraft Main characteristics of the inertia Jxx , Jyy , Jzz Appendage angle (angle between the z axes of the frames FSA and Fb , see Fig. 9.3) Position of the IMU along x, y, z axes

Thrusters

Matrix M (due to position misalignment of thrusters) Delay τ

regarded as a multivariable multidimensional (n-D) minimal realization problem that can be solved efficiently by the method proposed in Ref. [91]. The approach to the minimal representation problem exploits the structure of the uncertainty to directly obtain a low-order, possibly minimal, LFT. The method consists of breaking down the n-D polynomial matrices entering in Σ(δ) into sums and products of simple factors for which minimal LFT realizations can be obtained. The minimal LFTs are then combined together according to the obtained decomposition sequence. For that purpose, let us consider the following three definitions: Definition 1 (Direct-Sum Decomposition). Let (α1 , α2 , . . . , αk ) be a par  tition of a finite set α ⊂ N, that is, kj=1 αj = α and αi αj = ∅∀i = j, 1 ≤ i, j ≤ k. Denote δα = {δi : i ∈ α} and R[δ]m×n as the set of m × n matrices whose elements are polynomial functions in δi .

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Consider a matrix P ∈ R[δα ]m×n with α = (1, 2, . . . , p) with its expansion P(δα ) = P0 + i∈β mi (δ)Pi , β = (1, 2, . . . , l). m(δ) is a monomial, that is, a k

k

kp

product of the form cδ11 δ22 . . . δp with ki , i = 1, p nonnegative integers and c ∈ R (when c = 1, m(δ) is called monic). P(δα ) has a direct-sum decomposition if there exists a nontrivial partition (α1 , . . . , αk ) of the parameter index set α such that S0 = P(0) = S0 + S(δα1 ) + S(δα2 ) + · · · + S(δαk ) where S0 = P(0),

S(δαi ) =



mi (δαi )Pi ,

1 ≤ j ≤ k,

(9.39)

βj = {i ∈ β: mi ∈ R[δαi ]}

i∈βj

(9.40) Definition 2 (Affine Factorization). With the same notation as the one used in Definition 1, consider polynomial matrices P ∈ R[δ]m×n that exhibit rows of columns with common factors. Assume, without loss of generality, that P(0) = 0 (otherwise set P := P − P(0)). Suppose that δj is a common factor of the ith row (respectively, column) of P. Then P can be written ˜ I˜ j P(δ) = Ij R(δ) = R(δ)

(9.41)

where Ij and I˜ j are identity matrices with their ith entries replaced by δj . The matrices Ij and I˜ j are elementary left and right affine factors of P(δ). Product factorizations, such as Eq. (9.41), capture the directional structure of the uncertain parameter qj and can be exploited to reduce the order ˜ of R(δ) (R(δ), respectively). The affine product decomposition (9.41) that maximizes the order reduction is called affine factorization. Definition 3 (Weighted-Sum Decomposition). With the same notation as the one used in Definition 1, consider a polynomial matrix P ∈ R[δ]m×n . For each δi ∈ δ, factorize P(δ) into two additive factors, one that contains all the terms in δi and one that does not contains any term in δi so that P(δ) = Vi (δ) + Wi (δ|δi ), Vi (δ) = k∈βi mk (δ)Pk , (9.42) Wi (δ|δi ) = k∈β|βi mk (δ)Pk , βi = {k ∈ β: degδi mk (δ) > 0} Finding the parameters qi , i = 1, . . . , p for which the affine factorization of Vi leads to maximal order reduction (see Definition 2), is called weightedsum decomposition. Let DIRECT-SUM, AFFINE-FACTOR, and WEIGHTED-SUM be procedures that implement the direct-sum decomposition, the affine factorization and the weighted-sum decomposition, respectively. Each of these procedures takes as input argument a polynomial matrix. The procedure DIRECT-SUM returns a list of polynomial matrices that form a directsum decomposition. AFFINE-FACTOR and WEIGHTED-SUM return the factors that maximize order region. II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

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A high-level description of the decomposition algorithm is given by Algorithm 1.

Algorithm 1 STREED(P) P ← DIRECT-SUM(π(P)) for S ∈ P if S not simple if S has affine factor [M, N] ← AFFINE-FACTOR(π(S)) else [M, N] ← WEIGHTED-SUM(π(S)) STREED(M) STREED(N) else return

This recursive algorithm generates an ordered tree where the internal nodes represent sum or product decompositions and the leaves are simple polynomial matrices. To find a representation for the original polynomial matrix, it is sufficient to obtain minimal representations for the leaves of its structure tree and traverse the tree performing the specified operations. In Algorithm 1, π(P) is an operator that takes a polynomial matrix P(δ) and removes its constant rows and columns. This operator is introduced to simplify the factorization process. Associated with each projection π(P), there are a series of sets that include a row index set containing the indices of the rows of P(δ) with nonzero entries, a column index set containing the indices of the columns of P(δ) with nonzeros entries, and a constant matrix E = P(0) so that it is possible to recover P from its projection π(P). Breaking down the n-D polynomial matrices entering in Σ(δ) into sums and products of simple polynomial matrices such as affine polynomials and 1-D polynomials is done by calling Algorithm 1. Then, as explained, minimal LFT representations for the leaves are established and the tree is traversed performing the specified operations to obtain the LFT for Eq. (9.37), that is, ⎤ = Td ⎢ Fd ⎥ ⎥ y = Fu (P, ) ⎢ ⎣ f ⎦ + nω , u ⎡

block-diag(δ1 I2 , δ2 I2 , δ3 I2 , δ4 I2 , δ5 I4 , δ6 I4 , δ7 I4 , δ8 I4 , δ9 I4 , δ10 I4 , δ11 I4 , δ12 I2 , δ13 I2 , δ14 I2 , δ15 I8 , δ16 I2 , δ17 I2 , δ18 I2 , δ19 I2 , δ20 I2 , δ21 I2 , δ22 I2 , δ23 I2 , δ24 I2 , δ25 I2 , δ26 I4 , δ27 I4 , δ28 I4 , δ29 I4 , δ30 I3 , δ31 I3 , δ32 I3 , δ33 I3 , δ34 I3 , δ35 I3 , δ36 I3 ),  ∈ R107×107 , ∞ ≤ 1 (9.43)

3.4 Issues on Fault Isolation Coming back to the definition of the sets Sk , k = 2, . . . , 5 and S14 , S23 given by Eqs. (9.18)–(9.17) and as has been already noticed, the thruster II. ROBOTICS, FLIGHT SYSTEMS, AND VEHICLE DYNAMICS

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pairs of the sets Sk , k = 2, . . . , 5 produce similar moment directions and the pairs of the subsets S14 = {1, 4} and S23 = {2, 3} produce exactly opposite moments; see Fig. 9.4. These properties suggest that a fault that occurs on a thruster in a given set cannot be distinguished from a fault occurring on the other thruster from the same set. In other words, the fault isolation property is not satisfied and it is thus impossible to distinguish a faulty thruster from another one, in a given set Sk , k = 2, . . . , 5, S14 and S23 , using a model-based solution based on the attitude model. To demonstrate formally this lack of fault isolation, the following property can be used Property 1. Consider the model given by Eq. (9.37). Two faults fi and fj are isolable if and only if     C O C O = Im (9.44) Im A − sI Kj A − sI Ki Because it can be verified that |K1 | = |K4 |, |K2 | = |K3 |, K5 ≈ K12 , K6 ≈ K11 , K7 ≈ K10 , and K8 ≈ K9 where |Ki | states for the matrix of the absolute value of its components, that is, |Ki | = ⎡ ⎤ |Ki1 |     C O C O ⎢ .. ⎥ ⎣ . ⎦, it follows that Im A − sI K = Im A − sI K for (i, j) = j i |Kin | (1, 4); (2, 3); (5, 12); (6, 11); (7, 10); (8, 9). This gives the formal proof of the fault isolation problem. Thus, in the following section, a model-based FDI unit will be proposed so that it circumscribes thruster faults to S14 , S23 , Sk , k = 2, . . . , 5.

4 DESIGN OF THE THRUSTER FDI UNIT Based on the LFT model (9.43), we are now ready to address the design of the robust model-based thruster FDI unit. To proceed, consider the diagram given by Fig. 9.5. On this figure, Katt (s) stands for the attitude control law (in the Laplace domain), which consists of a PID controller. Θ ∈ R3 refers to the vector of attitude angles, that is, using the individual rotation matrices from the Euler (3, 2, 1) rotation, the

+

+ +

+ + +

+

FIG. 9.5 The setup of the fault detector design problem.

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relationship between the rotational velocities ω and the rate of the Euler angles Θ = [θx θy θz ]T is given by ⎡ ⎤ cos(θy ) sin(θx ) sin(θy ) cos(θx ) sin(θy ) 1 ⎣ 0 cos(θx ) cos(θy ) − sin(θx ) cos(θy )⎦ ω Θ˙ = (9.45) cos(θy ) cos(θ ) 0 sin(θ ) x

x

Then, including the linear approximation of Eq. (9.45) around Θ ∗ = 0 into the LFT Fu (P, ) (see Eq. 9.43) leads to the LFT model Fu (P, ), as illustrated in Fig. 9.5. M stands for the torque allocation unit that is modeled in this work, as the Moore-Penrose inverse of MT ; see Eq. (9.13). The navigation unit is omitted for simplicity. As can be seen, the proposed FDI strategy consists of two main functions: (i) A robust fault detector composed of a H∞ /H− dynamical filter F (s) associated with a generalized likelihood ratio (GLR) test for decision making. (ii) A bank of seven UIOs that is in charge to circumscribe the fault to Sk , k = 1, . . . , 5, S14 , S23 ; see Eq. (9.18). As already mentioned, this part is the theoretical contribution of this chapter because it is based on a new class of UIOs under H∞ performance. The GLR test is built on the Neyman-Pearson lemma [92]. It is now a well-mastered technique. Toward this end, theoretical developments are not presented here. The interested reader can refer to, for example, Refs. [93, 94]. Thus, the following sections are dedicated to the design of the H∞ /H− dynamical filter F (s) and the isolation strategy based on a bank of UIOs. Remark 4. The reasons for using a H∞ /H− filter for fault detection in spite of the UIOs are twofold: • First, as will be seen later (see Section 4.1), the UIO technique will not be able to consider the uncertainties  (see Eq. 9.43) without approximating them into additive unknown inputs. • Second, even if we admit the possibility of approximating the uncertainties in terms of additive unknown inputs, it seems obvious that there will not exist a solution to the design problem of the UIOs because with three measurements, it is well known from the decoupling principle that it is not possible to decouple more than three unknown inputs. Here, we are speaking about 36 uncertainties, so possibly 36 unknown inputs and even if we use an adequate method to reduce the distribution matrix of the unknown inputs, it is quite reasonable to think that the dimension of the problem is too huge for using the UIO theory.

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Coming back to the diagram depicted in Fig. 9.5, it can be seen that the H∞ /H− filter F (s) generates a signal r (that we constrain to be of dimension 1) so that r(s) = F (s)ey (s)

ey ∈ R21

r ∈ R,

(9.46)

where ey (s) is the output estimation error of the seven UIOs (in the Laplace domain), the output estimation error eyk , k = 1, . . . , 7 of each UIO being defined as eyk = y − yˆ

eyk ∈ R3 ,

k = 1, . . . , 7

(9.47)

so that each output estimation error eyk , k = 1, . . . , 7 possesses different directional properties. In this equation, y is the output of the model (9.43), that is, the angular velocities ωm measured by the IMUs. Toward this end, it is thought more judicious to first explain the design of the UIOs and then address the H∞ /H− filter design.

4.1 The Isolation Strategy: The UIOs As it is outlined in Section 3.2, the thruster pairs of the sets Sk , k = 2, . . . , 5 produce exactly opposite forces while they produce nearly collinear moments. The subset S1 produces exact collinear forces while in terms of moment directions, the pairs of the subsets S14 = {1, 4} and S23 = {2, 3} produce exactly opposite moments; see Fig. 9.4 that gives an illustration. These directional properties are used to enforce the following directional properties of the output estimation error eyk , k = 1, . . . , 7: • The first UIO is designed so that the estimation error ey1 (t) behaves in the orthogonal space induced by the moment directions of the thruster set S1 . • The second UIO is designed so that the estimation error ey2 (t) behaves in the orthogonal space induced by the moment directions of the thruster set S2 . • The three other UIOs follow the same decoupling principle for the thruster sets Sk , k = 3, 4, 5. • Finally, the two last UIOs are designed similarly for the thruster sets S14 and S23 . Then, the UIO with the minimum estimation error in the sense of the 2norm, that is, mini eyk 2 , k = 1, . . . , 7 reveals that a fault occurs in the associated set of thrusters. To proceed, let us consider the design of the first UIO. The index “1” of ey1 is omitted from now on, for better clarity. The design of the UIO is based on the LFT model (9.43) under the assumption that all uncertainties except those related to CoM and the thruster configuration M are neglected, that is, the adequate components δi of  are fixed to zero to annihilate the

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uncertainty and thus fix the associated parameters to their nominal value ρi0 . Furthermore, the influence of the exogenous disturbances is neglected, that is, Td = 0, Fd = 0, and nω = 0. In other words, no robustness constraints are considered against the flexible appendage uncertainties, the propellant-sloshing model uncertainties, the chaser’s inertia, the position of the IMUs, the delay τ , and the exogenous disturbances so that all degrees of freedom of the UIO are used for fault isolation; see the discussion in Remark 4. However, it should be noted that these uncertainties will be considered for the H∞ /H− fault detector. Then, the following model can be derived from Eq. (9.37) ⎡ ⎤ f1  12

 ⎢ .. ⎥ x˙ = Ax + Bu + i=1 Ki (ρ)fi f = ⎣ . ⎦ ∈ R12 , K(ρ) = K1 (ρ), . . . , K12 (ρ) y = Cx = ω f12 (9.48) where ρ and f are the parameter vectors associated with the nonneglected uncertainties and the fault vector, respectively. f is then split into two subsets according to the definition of S1 and Sk , k = 2, . . . , 5, namely f = [f1 f2 f3 f4 ]T , which will play the role of the unknown input vector to be decoupled and f = [f5 . . . f12 ]T , which are the remaining components of f . The partition of K(ρ) follows so that Eq. (9.48) can be rewritten according to  x˙ = Ax + Bu + K(ρ)f + K(ρ)f (9.49) y = Cx = ω Let us now rewrite this equation as follows 

!

 x˙ = Ax + Bu + . . . E(ρi ) . . . d + . . . E(ρj ) . . . w + K(ρ)f y = Cx = ω

,

i = 1, . . . , N1 , j = 1, . . . , N2

(9.50) 



so that . . . E(ρi ) . . . d + . . . E(ρj ) . . . w = K(ρ)f . Such a decomposition of K(ρ) can be obtained from linear algebra using, for example, a singular value decomposition to retain some a priori chosen dominant components of K(ρ) or simply from a grid of K(ρ). Then, the goal turns out to be the design of the following UIO  z˙ = Nz + Hu + L1 y ey = y − yˆ = C(x − xˆ ) (9.51) xˆ = z − L2 y

in such a way that ey is decoupled from the unknown inputs d and robust to w, in the H∞ -norm sense. The solution to this problem  the following theorem

is given by ) . . . . . . E(ρ is of full column rank and that Theorem 1. Assume that i # " # " rank C . . . E(ρi ) . . . = rank . . . E(ρi ) . . . . The UIO exists if and only if

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there exists a Lyapunov matrix X = XT > 0, matrices Y, K of adequate dimension and a scalar β so that ⎡

 T ⎢X(I + UC)A + YVCA − KC + !X(I + UC)A + YVCA!− KC ⎢ T ⎢ X(I + UC) . . . E(ρj ) . . . + YVC . . . E(ρj ) . . . ⎣ I

! ! X(I + UC) . . . E(ρj ) . . . + YVC . . . E(ρj ) . . . −βI O

⎤ I ⎥ ⎥ 0, matrices Y, K of adequate dimension and a scalar α so that  T X(I + UC)A + YVCA − KC + X(I + UC)A + YVCA − KC < 0 (9.60) Y − αI > 0 (9.61) where U and V are defined as in Theorem 1. Then N, H, L1 , L2 , Y, and K are deduced from Y and K as in Theorem 1.

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Proof. The proof is similar to the one of Theorem 1. By a suitable choice of α, it is possible to enhance the fault sensitivity performance of the UIO (9.51). Of course, because the H∞ robustness constraint against w has been removed in the linear matrix inequality (LMI) (9.60), there is no robustness guarantee against w. That is especially why the UIOs are not used for fault detection and why a robust H∞ /H− fault detector is preferred for that.

4.2 Design of the H∞ /H− Filter F (s) The goal is now to derive the dynamical filter F so that the residual signal r(t) defined by Eq. (9.46) achieves some required robustness and fault sensitivity objectives. Here, robustness is required against nω and all uncertainties listed in Table 9.2. This requirement is formulated as the minimization of the H∞ -norm of the transfer function from nω to r, that is, min γ1 F

∀ : ∞ ≤ 1 s.t.Tnω r ()∞ < γ1

(9.62)

The robust fault sensitivity requirement is expressed using the H− -index according to max γ2 ∀ ∈ Ω, ∀ : ∞ ≤ 1 (9.63) F s.t.Tfr ()− > γ2 where Ω denotes the frequency range where it is required to enforce fault sensitivity. Following the method proposed in Ref. [68], the requirements (9.62), (9.63) are expressed in terms of desired gain responses for the appropriate closed-loop transfers. To proceed, let Wn and Wf denote the (dynamical) shaping filters associated with Eqs. (9.62), (9.63), respectively. Based on the knowledge of the characteristics of the IMUs used to measure the angular velocities ω, Wn is fixed to a low-pass filter with crossover frequency n and static gain γn , that is, Wn (s) = γn

1 + τs I3 , 1 + τn s

τn =

1 , n

τ=

1 

(9.64)

This means that it is desired to reject the measurement noise nω at frequencies higher than n . The frequency  is introduced to make Wn invertible; see Theorem 3 and   n . For the fault sensitivity objectives and due to the definition of f , it is natural to define the shaping filter Wf as   (9.65) Wf (s) = diag Wf1 (s), . . . , Wf12 (s) Because it is required to enforce fault sensitivity in low frequencies, Wfi are chosen as low-pass filters with cutting frequencies fi and static gain γfi ,

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DESIGN OF THE THRUSTER FDI UNIT

251

i = 1, . . . , 12, that is, Wfi (s) = γfi

1 , 1 + τfi s

τn =

1 , fi

i = 1, . . . , 12

(9.66)

The positive constants γn , γfi , i = 1, . . . , 12 and n , fi , i = 1, . . . , 12 are introduced to manage the gain and the frequency behavior of Wn and Wfi , i = 1, . . . , 12. By this parameterization, proper weights Wn and Wf for the MIMO objective described by Eqs. (9.62), (9.63) can be adequately tuned in order to obtain the best achievable robustness and fault sensitivity performance. The following theorem solves the problem: Theorem 3 (Henry and Zolghadri [68]). Consider the shaping filters Wn and Wf defined by Eqs. (9.64)–(9.66). We assume that Wn and Wf have been scaled such that Wn ∞ ≤ γ1 and Wf − ≥ γ2 . Then, Eq. (9.62) yields if and only if Tnω r ()Wn−1 ∞ < 1

∀ : ∞ ≤ 1

(9.67)

where Tnω r denotes the transfer between r and nω . Now, introduce WF , a right invertible transfer matrix so that γ2 (9.68) Wf − = WF − λ WF − > λ, λ = 1 + γ2 (9.69) Define the signal r˜(s) such that r˜(s) = r(s) − WF (s)f (s)

(9.70)

Then a sufficient condition for the fault sensitivity specification (9.63) to hold is Tf r˜ ()∞ < 1

∀ : ∞ ≤ 1

(9.71)

where Tf r˜ () denotes the transfer between r˜ and f . Proof. The proof of this theorem can be found in Ref. [68]. Using Theorem 3, the H∞ /H− residual generator design problem can be formulated in a fictitious H∞ framework by combining Eqs. (9.67), (9.71) into a single H∞ constraint. Fig. 9.6 also gives the block diagram associated with this problem. On this figure, P corresponds to the closed loop illustrated in Fig. 9.5, which is deduced from the attitude controller Katt , the allocation matrix M, and the LFT Fu (P, ) using LFT algebra. Furthermore, the seven UIOs given by Eq. (9.51) are also included into P, hence everything to the left of ey . Then, a sufficient condition for Eqs. (9.67), (9.71) to hold is $  $ $ $ ˜ (9.72) F (s) $ < 1 $Fl P(s), ∞

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9. NEW TRENDS ON FAULT DIAGNOSIS METHODS

−1 The model P˜ is deduced  as illustrated in Fig. 9.6 by including Wn and WF 

into the LFT Fu P,  using LFT algebra so that:        n(s) ˜ r(s) ˜ = Fl Fu P(s),  , F (s) f (s) r˜(s)

(9.73)

where ˜ n(s) = Wn (s)nω (s)

(9.74)

The interested reader can refer to Refs. [68, 69] to compute the explicit state˜ space model of P. 4.2.1 The LMI Solution The H∞ /H− fault detector design problem being put into a standard H∞ setup (see Eq. 9.72 and Fig. 9.6) can be solved using the LMI technique described, for example, in Ref. [96]. However, there exist some structural properties in the problem definition because the filter F operates in open loop versus Fu (P, ). Thus, F does neither affect the state of P nor its output ey . To this end, the same philosophy of the LMI procedure proposed in Ref. [68] should be preferred to those presented in Ref. [96] because it enables taking into account this structural property. This is done through the following theorem. Theorem P˜ admits the state-space model ⎤ ⎡ 4. Assume that ˜ ˜ ˜ A B1 B2 ⎢ ˜ ˜ B˜ 1 , B˜ 2 , C˜ 1 , C˜ 2 , D ˜ 11 , D ˜ 12 , D ˜ 21 , and ˜ 11 D ˜ 12 ⎥ P˜ = ⎦ where A, ⎣ C1 D ˜2 D ˜ 21 D ˜ 22 C ˜ D22 are  matrices whose  dimensions are determined such that ˜ 2 sI − A ˜ B˜ 2 + D ˜ 22 r(s) : ey ∈ R21 , r ∈ R. It can be verified that, by ey (s) = C ˜ 22 = 0, showing that F operates in open loop versus construction B˜ 2 = 0 and D ˜ 21 ). Then Fu (P, ). Let W denote an orthonormal base of the null space of (C˜ 2 D

+

FIG. 9.6 The equivalent forms of the design problem of F (s).

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DESIGN OF THE THRUSTER FDI UNIT

Eq. (9.72) yields if and only if there exist γ < 1 and two matrices R = RT > 0 and S = ST > 0 satisfying the following system of LMIs ⎞ ⎛ ˜ + RA ˜ T RC ˆ T B˜ 1 AR 1 ⎜ ˆ 1R ˆ 11 ⎟ (9.75) C −γ I D ⎠