Design of Linear Multivariable Feedback Control Systems: The Wiener–Hopf Approach using Transforms and Spectral Factorization [1st ed.] 9783030443559, 9783030443566

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Joseph J. Bongiorno Jr. Kiheon Park

Design of Linear Multivariable Feedback Control Systems The Wiener–Hopf Approach using Transforms and Spectral Factorization

Design of Linear Multivariable Feedback Control Systems

Joseph J. Bongiorno Jr. Kiheon Park •

Design of Linear Multivariable Feedback Control Systems The Wiener–Hopf Approach using Transforms and Spectral Factorization

123

Joseph J. Bongiorno Jr. Department of Electrical and Computer Engineering, NYU Tandon School of Engineering Polytechnic Institute, New York University Brooklyn, NY, USA

Kiheon Park College of Information and Communication Engineering Sungkyunkwan University Suwon-si, Korea (Republic of)

ISBN 978-3-030-44355-9 ISBN 978-3-030-44356-6 https://doi.org/10.1007/978-3-030-44356-6

(eBook)

MATLAB is a registered trademark of The MathWorks, Inc. See https://www.mathworks.com/ trademarks for a list of additional trademarks. Mathematics Subject Classification (2010): 93-01, 93C05, 93C15, 93C35, 93C55, 93C57, 93C62, 93D09, 93D15, 93E20, 47A68 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to Dante C. Youla our mentor and colleague for his inspiration, encouragement, and collaboration.

Preface

This book is based on some of the research done by the authors and their colleague, Dante C. Youla, over a period that spanned almost 50 years beginning with Youla (1961) and continuing most recently with Park and Bongiorno (2009). Specifically, their work on the design of analog optimal time-invariant linear multivariable feedback control systems using transforms and the Wiener–Hopf methodology is presented within a unified framework for both the analog and the digital case. The authors are especially grateful to Dante C. Youla for helping us to resolve a number of mathematical issues connected with the material presented in this book. The focus is on feedback control systems that can be modeled as an interconnection of subsystems each of which is specified except for one that is chosen by the designer and called the controller. Attention is restricted to those systems in which all subsystems can be modeled with linear constant-coefficient differential equations (analog systems) or to those in which all subsystems can be modeled with linear constant-coefficient difference equations (digital systems). Exogenous inputs that are deterministic and persistent (steps, ramps, etc.) and/or ones that are stochastic are considered. In the book, a quadratic functional is used to measure system performance. The design task is to choose and synthesize the controller transfer matrix so that the desired overall system performance is achieved. The relationship of the works of others to the work presented here is described throughout the book in sections entitled “Historical Perspective and Discussion”. A key first step in the design process is the parameterization of the class of stabilizing controllers, which is the topic of Chap. 2. Larin, Naumenko, and Suntsev (1971, 1972, 1973) were the first to solve this problem. However, this achievement was not recognized at the time and solutions were also provided in Kučera (1974) and Youla, Jabr, and Bongiorno (1976b) with the parameterization in the latter attracting particular notoriety. In order to apply the Wiener–Hopf methodology when tracking of persistent deterministic reference inputs and rejection of persistent deterministic disturbance inputs is required, a refinement of the parameterization of stabilizing controllers is needed. In particular, with these inputs the subset of stabilizing controllers for which the error variables and the controller output variables are stable must be employed. The required parameterization is presented in Chap. 3 vii

viii

Preface

and used to obtain the optimal controller with respect to a quadratic performance functional for the standard configuration. In addition, a methodology is described that enables one to trade off optimality for improved stability margin and for reduced sensitivity to plant model uncertainty. Results are also provided for one-degree-of-freedom (1DOF) and three-degree-of-freedom (3DOF) systems. Chapter 3 contains material that first appeared in Park and Bongiorno (2009). Optimal design of decoupled systems is addressed in Chap. 4 which contains results not previously published. The numerical calculation of the optimal controller parameters and realization of the controller are treated in Chap. 5. With regard to applications, three examples are provided. Example 3.23 concerns the optimal H2 design of a 1DOF depth and pitch controller for a submerged submarine traveling at 30 knots speed using the linearized model for submarine motion given in Sect. 10.4 of Grimble (2006). Example 3.24 concerns the optimal H2 design of a 2DOF controller for a Rosenbrock process: i.e., a multivariable process whose transfer matrix exhibits a severe coupling phenomenon (Åström et al. 2002). Specifically, the transfer matrix has a nonminimum phase zero at s ¼ 1 and this yields fundamental limitations on control system performance. The design of an optimal decoupling 2DOF H2 controller is determined in Example 4.10 for the same Rosebrock process so that the price paid in performance for imposing the decoupling requirement could be assessed. This book is intended for readers familiar with linear systems and matrix theory. In particular, the reader should be familiar with the material in Appendix A and enough matrix theory to follow the presentation in Appendix B. Many examples with solutions are provided which contain important insights, extensions, and/or elaborations on the topics treated in the book. Examples of particular importance are identified with an asterisk. Brooklyn, USA Suwon-si, Korea (Republic of)

Joseph J. Bongiorno, Jr. Kiheon Park

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 System Models . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Stability, Signal Models, Performance Measures, and Saturation Constraints . . . . . . . . . . . . . . . . . 1.4 Notation and Terminology . . . . . . . . . . . . . . . . . 1.5 Historical Perspective and Commentary . . . . . . . 1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

............ ............ ............

1 1 2

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6 11 13 17

2 Stabilizing Controllers, Tracking, and Disturbance Rejection . 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Polynomial Matrix Descriptions . . . . . . . . . . . . . . . . . . . . 2.3 Stability Analysis of the Standard Configuration . . . . . . . . 2.4 Stability Analysis of the 2DOF Standard Configuration . . . 2.5 Stability Analysis of the 3DOF System . . . . . . . . . . . . . . 2.6 Tracking and Disturbance Rejection in the 3DOF System . 2.7 Parameterization and Realization of Proper Stabilizing Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Historical Perspective and Commentary . . . . . . . . . . . . . . 2.9 Additional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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23 23 24 26 42 48 55

..... ..... .....

75 84 85

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3 H2 Design of Multivariable Control Systems . . . . . . . . . . . . 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminary Considerations . . . . . . . . . . . . . . . . . . . . . . 3.3 Tracking and Disturbance Rejection with the Standard Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Tracking and Disturbance Rejection in the 3DOF System Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 H2 Design of Digital Multivariable Control Systems . . . 3.6 H2 Design of Analog Multivariable Control Systems . . .

. . . . . . 115 . . . . . . 115 . . . . . . 115 . . . . . . 117 . . . . . . 125 . . . . . . 133 . . . . . . 139

ix

x

Contents

H2 Design of 1DOF Systems . . . . . . . . . . . . . . . . . . . . . . H2 Design of 3DOF Systems . . . . . . . . . . . . . . . . . . . . . . Trade-off of Optimal Performance for Reduced Sensitivity to Plant Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2 Sensitivity to Small Plant Parameter Changes . . . . . 3.9.3 Stability Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.4 The Analytical Solution for Sensitvity and Stability Margin Combined . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Historical Perspective and Commentary . . . . . . . . . . . . . . . 3.11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7 3.8 3.9

4 H2 Design of Multivariable Control Systems with Decoupling 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Characterization of Realizable and Acceptable T . . . . . . . . 4.3 Characterization of Realizable Diagonal T and Acceptable Diagonal T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 H2 Design of Decoupled Systems for Acceptable Inputs . . 4.4.1 H2 Design for the Digital Case . . . . . . . . . . . . . . 4.4.2 H2 Design for the Analog Case . . . . . . . . . . . . . . 4.4.3 H2 Design of Decoupled Systems for Strictly Acceptable Inputs . . . . . . . . . . . . . . . . . . . . . . . . 4.5 H2 Design of 1DOF Decoupled Systems . . . . . . . . . . . . . 4.6 H2 Design of 3DOF Decoupled Systems . . . . . . . . . . . . . 4.7 Historical Perspective and Commentary . . . . . . . . . . . . . . 4.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 151 . . . . 166 . . . .

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185 185 186 189

. . . . 193 . . . . 197 . . . . 199

. . . . . 241 . . . . . 241 . . . . . 242 . . . .

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256 261 268 269

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272 276 286 302 304

5 Numerical Calculation of Wiener–Hopf Controllers . . . . . . . . . . 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Machinery for Analog Systems . . . . . . . . . . . . . . . . . 5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Basic Machinery for Digital Systems . . . . . . . . . . . . . . . . . . 5.5 State-Space Representation of the Wiener–Hopf Controllers: The Digital Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Historical Perspective and Commentary . . . . . . . . . . . . . . . . 5.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 319 . . . 319 . . . 319 . . . 324 . . . 337 . . . 341 . . . 353 . . . 354

Appendix A: Complex Functions, Transforms, Parseval’s Formula, and Stochastic Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Appendix B: Coprime Polynomial Matrix Fraction Descriptions for Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Contents

xi

Appendix C: Spectral Factorization of Rational Parahermitian-Positive Matrices . . . . . . . . . . . . . . . . . . . . . 435 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Chapter 1

Introduction

1.1

Overview

The design of analog and digital feedback control systems is addressed in this book. The focus is on control systems that can be modeled as an interconnection of subsystems each of which is specified except for one, called the controller, which is to be chosen by the designer. Associated with each subsystem is a set of signals that are the elements of a designated vector called the input and which generate responses that are the elements of a designated vector called the output. The input and/or output vectors can contain one or more elements; hence, the subsystems are multivariable ones that include single-input-output (single-variable) subsystems as a special case. Also associated with each subsystem is a transfer matrix, which connects the input vector to the contribution it makes in the output vector. The symbol used to designate the transfer matrix of a subsystem is shown within the block representing it in the system block diagram. The design task is to choose and synthesize the controller transfer matrix so that the desired overall system performance is achieved. In this book, quadratic measures of system performance are used exclusively. The essential elements of complex function theory, transforms, and stochastic processes needed are summarized in Appendix A. The essential elements of matrix theory needed are reviewed in Appendix B. When dealing with analog systems, attention is restricted to subsystems that can be modeled by a system of ordinary linear differential equations with real constant coefficients. As a consequence, all analog subsystem transfer matrices contain elements that are ratios of polynomials in the Laplace variable s with real coefficients or, equivalently, real rational functions of s. The traditional transform used by engineers for digital systems is the z-transform. It turns out for the problems of interest here that it is more convenient to use the k-transform, where k ¼ 1=z. When dealing with digital systems, attention is restricted to subsystems that can be modeled by a system of ordinary linear differential equations with real constant © Springer Nature Switzerland AG 2020 J. J. Bongiorno Jr. and K. Park, Design of Linear Multivariable Feedback Control Systems, https://doi.org/10.1007/978-3-030-44356-6_1

1

2

1

Introduction

coefficients. As a consequence, all digital subsystem transfer matrices contain elements that are ratios of polynomials in the variable k with real coefficients or, equivalently, real rational functions of k. So it should not be surprising that the methodologies used for analog and digital systems parallel one another. Sampled-data systems contain both analog and digital elements and represent a greater modeling challenge. Typically in such systems, all subsystems except for the controller are analog. The controller consists of a digital processor operating at a fixed rate followed by a hold circuit. The input to the digital processor is a sequence of sampled measurements and the digital processor output samples are the inputs to the hold circuit. Usually, the output of the hold circuit is held constant during the inter-sample interval at the value of the sample at the beginning of the interval. When attention is restricted to the behavior of the system at the sampling instants, the sampled-data system can be easily modeled as a digital system. When the system behavior in the interval between the samples needs to be taken into account, it is still possible with a more complex digital system model to account for the performance of the sampled-data system (Rosenwasser and Lampe 2006). Hence, all the results for digital systems presented in this book can be applied to sampled-data systems.

1.2

System Models

The block diagram of an analog control system that incorporates most of the features one might ever encounter in practice is shown in Fig. 1.1. All symbols designate Laplace transforms and the sizes of all column vectors representing the inputs and outputs in the system are consistent with the sizes of the subsystem transfer matrices. The subsystem that is to be controlled is referred to as the plant and the vector d(s) represents the environmental disturbances that impact the plant. The vector v(s) represents the variables that are available to control the plant. Whenever feasible and economical, a sensor capable of measuring some or all of the environmental disturbances impacting the plant is included in the feedforward path as shown. The matrix L(s) is the sensor transfer matrix and the vector nl ðsÞ accounts for sensor error. The feedforward path and the controller provide a mechanism for modifying the plant input v(s) so as to suppress the impact of the environmental disturbances d(s) on the plant output vector y(s). The same is true for the feedback path that contains a sensor that measures the plant output y(s). The matrix F(s) is the sensor transfer matrix and the vector nm ðsÞ accounts for sensor error. Often it is not physically possible or economical to measure the disturbance vector d(s). The task of suppressing the impact of disturbances on system performance in those cases must be accomplished only with the aid of the feedback path. Feedback also plays an essential role in reducing the sensitivity of system performance to changes in the plant transfer matrix P(s) and is essential for the control of unstable plants. The block with the transfer matrix Td ðsÞ does not represent a

1.2 System Models

3 nl (s )

+

Feedforward l( s ) L (s )

d (s )

+

Path

r (s ) +

+

n(s ) u (s)

z( s ) v (s )

C (s ) Controller

P (s)

+

+

y (s ) +

w( s ) n m (s )

Pd (s )

Plant

m ( s) +

+

Feedback

ε (s ) yd(s)

F (s ) Path Td ( s )

Desired Tracking

Fig. 1.1 Three-degree-of-freedom system (© 1990 Taylor and Francis Ltd. http://www. tandfonline.com. Reprinted, with permission, from Park and Bongiorno (1990))

component of the physical system. It is simply the mechanism used in the block diagram to define the system tracking error eðsÞ. Accordingly, the path in which it lies is indicated with a dashed line. Obviously, yd ðsÞ represents the desired tracking of the reference input r(s) that can be corrupted by noise n(s) when applied to the controller. There are three vector inputs to the controller and one can partition the controller transfer matrix accordingly so that vðsÞ ¼ Cw ðsÞwðsÞ  Cz ðsÞzðsÞ þ Cu ðsÞuðsÞ

ð1:1Þ

CðsÞ ¼ ½Cw ðsÞ  Cz ðsÞ Cu ðsÞ:

ð1:2Þ

and

The negative signs in (1.2) are introduced to reflect the fact that the feedforward path and feedback path are intended to cancel the undesired components present in the output y. Clearly, there are three submatrices associated with the controller that the designer can choose. For this reason, the system shown in Fig. 1.1 is referred to as a three-degree-of-freedom (3DOF) system. The second term on the right-hand side of (1.1) and the second submatrix on the right-hand side of (1.2) disappear when feedforward compensation is not included. The system is then called a twodegree-of-freedom (2DOF) system. In some control applications, there are

4

1

Introduction

restrictions on the implementation of the controller that correspond to Cw ¼ Cu , Cz ¼ 0. The controller is then of the form C ðsÞ ¼ Cu ðsÞ½ I

I ;

ð1:3Þ

where I denotes the identity matrix. Now there is only one matrix to choose in the design of the controller and the system is called a one-degree-of-freedom (1DOF) system. The block diagram of a 3DOF digital control system is identical to the one shown in Fig. 1.1 with the Laplace variable s replaced by the digital transform variable k. Whenever it is not essential to distinguish between analog and digital systems, the functional dependence on s or k is often omitted. All systems of interest here can be divided into two parts: one part is just the controller and the other part includes all the remaining subsystems. The block diagram shown in Fig. 1.2 is a generic way to show all such systems and it is called the standard configuration. The symbols G and C represent Laplace transform transfer matrices when the system is analog and k-transform transfer matrices when the system is digital. The block G is called the generalized plant and contains the physical plant and all subsystems other than the controller. The column vector e contains all the exogenous inputs to the system. Specifically, in the standard configuration representation of the 3DOF system shown in Fig. 1.1 the column vector e is an amalgamation of the column vectors r, n, nl , nm , and d. The measurements column vector ym is an amalgamation of the column vectors u, w, and z. For the column vector yc , one can choose an amalgamation of the controlled variables e and v since one desires small tracking errors with minimal control effort. One of the earliest papers in which the standard configuration is used to model linear multivariable feedback control systems is Cheng and Pearson (1981), but a similar configuration was also treated in Larin et al. (1971, 1972, 1973). The standard configuration is a useful representation for establishing fundamental properties of feedback control systems irrespective of the subsystems and their interconnection within the generalized plant. On the other hand, engineering insights are generally obtained from the analysis of specific feedback system implementations. A comprehensive approach covering both methodologies is employed in the sequel.

Fig. 1.2 Standard configuration (© 1989 IEEE. Reprinted with permission, from Park and Bongiorno (1989))

e

Generalized

yc

G v

Plant C Controller

ym

1.2 System Models

5

An insightful alternative to the standard configuration can be generated once the following observations are made. First, one can always view the exogenous input vector e as an amalgamation of a reference input vector r, the noise vector n corrupting r, and a disturbance vector d that accounts for all other exogenous system inputs. Second, one can view yc as an amalgamation of an error vector e ¼ yd  yt and a vector ys that contains all the system responses that need to be kept small. In particular, ys should include the signals which need to be kept small so that the linear system model remains valid and minimal control effort is required. The vector yt represents the system outputs, which need to track yd ¼ Td r. Third, r and its noise n involve only low power level signals that can only enter the system through the controller. An alternative to the standard configuration in this case is the one shown in Fig. 1.3. It is called the 2DOF standard configuration because the controller transfer matrix can be partitioned into two blocks so that v ¼ Cy y þ Cu u

ð1:4Þ

C ¼ ½ Cy

ð1:5Þ

and Cu :

A key element in the design of many feedback control systems is the requirement that the system tracks certain persistent reference input signals such as steps and ramps. In these cases, the 2DOF standard configuration is a natural representation for the system since it clearly exposes the tracking requirement and is especially useful when decoupling is needed. The system is decoupled when the transfer matrix connecting r to yt is diagonal. Two of the earliest papers in which the 2DOF standard configuration is used to model linear multivariable feedback control systems are Pernebo (1981a, b).

d

Generalized

ys y

v

G

t

ε

−

yd

+ Plant

Td y

C Controller

u n

Fig. 1.3 Two-degree-of-freedom standard configuration

+ +

r

6

1

1.3

Introduction

Stability, Signal Models, Performance Measures, and Saturation Constraints

As a consequence of superposition in linear systems, once the controller is selected, each of the output vectors yc ; ym ; and v in the standard configuration is given by the sum of two vectors: one completely determined by the exogenous input e and one completely determined by initial conditions in the system. So it is necessary for satisfactory performance that any contributions from the initial conditions or transients in response to exogenous inputs disappear in time. Then, the output vectors are determined essentially by e. The system is called stable in this case and system design can focus on the behavior of the system in response to e; this is the approach adopted here. First, the class of all stabilizing controllers is identified. Then the one within this class is sought for which performance requirements are met. Attention is restricted to exogenous input signals that are deterministic with real rational transforms, or are stochastic with real rational power spectral densities, or are combinations of these. Often an input signal is viewed as the product of a real random variable with a deterministic function. Such signals are called shape-deterministic signals and attention is restricted to those for which the deterministic part has a real rational transform. Since only physical inputs which have real rational transforms and since only physical subsystems with real rational transfer matrices are considered, it is assumed throughout whenever not explicitly stated otherwise that all polynomial matrices and all transfer matrices are real (i.e., all coefficients are real). It is useful in addition to the notion of system stability to define a stable transform. A Laplace transform whose finite poles all have negative real parts or a k-transform whose finite poles all have magnitudes greater than unity are called stable transforms. When the standard configuration is stable, when there are no stochastic inputs, when any persistent deterministic reference inputs are ones for which the controlled variables have zero steady-state value, and when all other deterministic inputs have stable transforms, then the output vector transform yc is stable. It follows under these conditions that in the digital case (see Appendix A) J¼

1 X k¼0

^y0c ðkÞ^yc ðkÞ ¼

1 ð 1 ð H  yc ðkÞyc ðkÞdk=k ¼ H  Tr½yc ðkÞyc ðkÞdk=k; 2pj 2pj jkj¼1

jkj¼1

ð1:6Þ where ^y0c ðkÞ is the transpose of the inverse k-transform of yc ðkÞ and yc ðkÞ is the transpose of yc ð1=kÞ. The notation Tr denotes taking the trace of a matrix which for a square matrix is the sum of its diagonal elements. The quantity J can be used as a measure of system performance. The validity of this lies in the fact that when any element of ^yc ðkÞ is nonzero a positive contribution is made to J and the larger the magnitude of this element, the larger the contribution. So the smaller J, the better the performance one can expect since ideally ^yc ðkÞ ¼ 0 is desired. Contrary to

1.3 Stability, Signal Models, Performance Measures, and Saturation Constraints

7

standard practice a circumflex over the inverse transform is used to distinguish it from its transform rather than the other way around. This is done to keep the notation simpler because transform quantities are primarily used here. Quadratic performance measures in the form of the last integral in (1.6) are the ones used in this work when designing a controller for digital systems that takes into account transient performance in response to deterministic inputs. Since this integral can also be used as a norm on the Hardy subspace of all real rational stable yc ðkÞ, these quadratic performance measures are also often called H2 performance measures in the literature. It is also typical in the analog case for yc to be strictly proper (i.e., yc is zero for s ¼ 1). It then follows from Parseval’s formula under the same conditions as cited above for the digital case that (see Appendix A) Z1 J¼ 0

^y0c ðtÞ^yc ðtÞdt

1 ¼ 2pj

Zj1 j1

1 yc ðsÞyc ðsÞds ¼ 2pj

Zj1 Tr½yc ðsÞyc ðsÞds; ð1:7Þ j1

where ^y0c ðtÞ is the transpose of the inverse Laplace transform of yc ðsÞ and yc ðsÞ is the transpose of yc ðsÞ. Again J can be used as a measure of system performance. The usefulness of (1.7) as a performance measure lies in the fact that when any element of ^yc ðtÞ is nonzero over any finite time interval a positive contribution is made to J and the larger the magnitude of this element, the larger the contribution. So the smaller J, the better the performance one can expect since ideally ^yc ðtÞ ¼ 0 is desired. Quadratic performance measures in the form of the last integral in (1.7) are the primary ones used here when designing the controller for analog systems to take into account transient performance in response to deterministic inputs. Since this integral can also be used as a norm on the Hardy subspace of all real rational stable strictly proper yc ðsÞ, these quadratic performance measures are also often called H2 performance measures in the literature. When the deterministic inputs are shape-deterministic instead, then \yc yc [ is used instead of yc yc in the integrands where \  [ denotes the ensemble average. When only stationary zero-mean stochastic signals are applied to a stable standard configuration, then for digital systems (see Appendix A) \^y0c ðkÞ^yc ðkÞ [ ¼

1 ð H  Tr[Uyc yc ðkÞdk=k 2pj

ð1:8Þ

jkj¼1

can be used as a measure of system performance since it represents the mean square value of the norm of the vector ^yc . The matrix Uyc yc ðkÞ is the two-sided k-transform of the covariance matrix \^yc ðk þ lÞy^0 c ðlÞ[ and must be free of poles on the unit circle for the integral to exist. The relationship corresponding to (1.8) for analog systems is (see Appendix A)

8

1

\^y0c ðtÞ^yc ðtÞ [

1 ¼ 2pj

Introduction

Zj1 Tr½Uyc yc ðsÞds:

ð1:9Þ

j1

The matrix Uyc yc ðsÞ is the two-sided Laplace transform of the covariance matrix \^yc ðt þ sÞ^y0c ðsÞ[ . For the integral to exist, s Uyc yc ðsÞ must be strictly proper and Uyc yc ðsÞ must be free of poles on the imaginary axis. It should be emphasized that (1.6) and (1.7) and their counterparts for shape-deterministic inputs are typically used to measure transient performance while (1.8) and (1.9) measure steady-state performance after all transients have disappeared. When a system is subjected to both shape-deterministic and stochastic inputs both measures can be combined into a single performance functional J¼

1 ð H  Tr½QðkÞUyc ðkÞdk=k; 2pj

ð1:10Þ

jkj¼1

where Uyc ðkÞ ¼ a1 \yc ðkÞyc ðkÞ [ þ a2 Uyc yc ðkÞ

ð1:11Þ

in the case of digital systems and 1 J¼ 2pj

Zj1 Tr½QðsÞUyc ðsÞds;

ð1:12Þ

j1

where Uyc ðsÞ ¼ a1 \yc ðsÞyc ðsÞ [ þ a2 Uyc yc ðsÞ

ð1:13Þ

in the case of analog systems. The matrix Q is included to allow for frequency weighting and is chosen so that Q ¼ Q and so that it is nonnegative definite on the unit circle in the case of digital systems and is nonnegative definite on the imaginary axis in the case of analog systems. The scalar constants a1 ; a2 are nonnegative and allow for weighting the relative importance of the contribution from deterministic and stochastic components to the controlled vector yc . Traditionally, Uyc yc is called a spectral density matrix. In the sequel, matrices that are of the form of the generalized spectral density matrix Uyc are also referred to as spectral density matrices and only a single subscript is used for simplicity to distinguish them from the traditional ones. The subset of all stabilizing digital controllers that yield finite values for (1.10) and the subset of all stabilizing analog controllers that yield finite values for (1.12) are parameterized in this book and the controllers that minimize these performance

1.3 Stability, Signal Models, Performance Measures, and Saturation Constraints

9

measures are identified. In addition, a methodology is presented that enables one to trade off optimality for improved stability margin and reduced sensitivity of the performance functional to plant model uncertainty. Moreover, specialized results for the systems shown in Figs. 1.1 and 1.3 are presented. The design of optimal 1DOF and 2DOF decoupled systems is also treated. A key element in the development of these results is the recognition that the deterministic components of the inputs to the physical process under control must be bounded. For example, the rudder angle of an aircraft or boat is limited. When such a limit is reached, saturation is said to have occurred. Clearly, a necessary condition for avoiding saturation is that the deterministic parts of the inputs to the physical plant have pseudo-stable transforms. That is, transforms which have no poles in Re s  0 for the analog case and no poles in jkj  1 for the digital case except possibly for simple poles on the stability boundaries (the finite s ¼ jx axis in the analog case or on jkj ¼ 1 in the digital case). Otherwise, the physical plant inputs would continue to grow until the saturation level is reached and the assumption of linearity would no longer hold. For the standard configuration shown in Fig. 1.2, one can write yc ¼ yc1 þ yc2 for the controlled variables, where yc1 represents the deterministic components and yc2 represents the stochastic ones. It is particularly convenient to set  yc1 ¼

yt1  ytd1 ys1  ysd1



 ¼

 et1 ; es1

ð1:14Þ

where ytd1 and ysd1 represent the transforms, respectively, of the desired responses of the deterministic tracking signals yt1 and the deterministic signals likely to cause saturation ys1 . When C is a stabilizing controller and et1 is stable, asymptotic tracking is assured. When v1 denotes the deterministic component of the output from C; when as is typical ys1 ¼ Gac v1 ; where Gac accounts for the actuators and any pre-compensation of the plant; and when Gac is stable except possibly for simple poles on the stability boundary; then ys1 is the pseudo-stable transform of a bounded temporal function for all stable v1 . Thus, attention is restricted to deterministic exogenous inputs e1 for which a stabilizing controller C exists such that et1 and v1 are stable. When ys1 is stable and ysd1 ¼ 0 or when ys1 is pseudo-stable and ysd1 is such that any poles of ys1 on the stability boundary are not poles of es1 , then es1 is also stable. In this case, the performance functional (1.6) with yc ¼ yc1 can be used for the digital case. The performance functional (1.7) can be used for the analog case provided yc1 is also strictly proper. An example in which ys1 is pseudo-stable and yt1 ¼ Gp ys1 ¼ Gt e1 is now considered. When ytd1 ¼ e1 ¼ md =s and md is a constant vector; when the output from ~ ac ðsÞv1 =s, where G ~ ac is stable a pre-compensator and actuator is ys1 ¼ Gac v1 ¼ G ~ ac ð0Þ 6¼ 0; when Gp ðsÞ is analytic at s ¼ 0 and Gpo ¼ Gp ð0Þ; and when a and G stabilizing controller is used so that the closed-loop transfer matrix Gt is stable and Gto ¼ Gt ð0Þ ¼ I; then the error

10

1

Introduction

et1 ¼ yt1  ytd1 ¼ ðGt  I Þmd =s

ð1:15Þ

lim ^et1 ðtÞ ¼ lim s et1 ðsÞ ¼ limðGt  I Þmd ¼ 0:

ð1:16Þ

is stable since t!1

s!0

s!0

Moreover, since ys1 is pseudo-stable and on the s ¼ jx axis possesses only a simple pole at s ¼ 0, one can find a constant vector m such that ys1 ¼ m=s þ ~ys1 ;

ð1:17Þ

where ~ys1 is stable. In addition, it follows from lim ^yt1 ðtÞ ¼ lim s yt1 ðsÞ ¼ lim sGp ys1 ¼ Gpo m ¼ Gto md ¼ md

t!1

s!0

s!0

ð1:18Þ

that the vector m must satisfy Gpo m ¼ md . In this case, ysd 1 ¼ m=s can be chosen and es1 ¼ ys1  ysd1 ¼ ~ys1

ð1:19Þ

is stable. A similar approach is used in Polyakov (2001) for single-variable sampled-data control systems. Typically, Gpo has row rank and there is at least one solution for m given by  1 m ¼ mo ¼ G0po Gpo G0po md :

ð1:20Þ

Clearly, all solutions are given by m ¼ mo þ mh ; where Gpo mh ¼ 0. Hence, m0h G0po ¼ 0 and one gets  1 m0h mo ¼ m0h G0po Gpo G0po md ¼ 0:

ð1:21Þ

Thus,  2   m  ¼ m0 m ¼ m0 þ m0 ðmo þ mh Þ ¼ kmo k2 þ kmh k2  kmo k2 : o h

ð1:22Þ

So the solution m ¼ mo is the minimum norm solution and is in keeping with the desire to avoid saturation. Moreover, mo as given by (1.20) is in terms of given data: Gpo and md .

1.4 Notation and Terminology

1.4

11

Notation and Terminology

The notation and terminology used throughout is collected here for easy reference. All results are obtained working exclusively with rational transforms. The Laplace transform in the complex variable s is used for analog systems and the k-transform is used for digital systems. The k-transform is related to the z-transform through the relationship k = 1/z. In the few instances that it is necessary to do so, a circumflex is used to distinguish an inverse transform from its transform as in (1.6) and (1.7) for example. Often function arguments are omitted for brevity when no confusion is possible or when the discussion is applicable to both analog and digital systems. Laplace transforms and k-transforms are distinguished from one another when necessary by showing explicitly the dependence on s or k. The complex variable p is used generically to represent either s or k when useful to do so. The conjugate of p is denoted by p. The real part of p is denoted by Re p and Re p  0 is used to denote the finite part of the closed right half p-plane. The notation j pj is used for the magnitude of p and j pj [ 1 represents the finite part of the open region of the complex p-plane outside the unit circle. A transform is called good in the analog case when it is free of poles on s = jx, x finite. It is called good in the digital case when it is free of poles on jkj ¼ 1. It is called stable in the analog case when all its finite poles lie in Re s\0, the open left-half s-plane. It is called stable in the digital case when all its finite poles lie in jkj [ 1, the region outside the unit circle. A transform that is not stable is called unstable. In general, the finite poles of a transform can be divided into those that lie in the stable region (Re s\0 or jkj [ 1) and those that do not. The former ones are called the stable poles and the latter ones are called the unstable poles. Transforms that have no poles in Re s  0 for the analog case and no poles in jkj  1 for the digital case, except possibly for simple poles on the stability boundaries (the finite s ¼ jx axis in the analog case or on jkj ¼ 1 in the digital case), are called pseudo-stable. A polynomial in s is called Hurwitz when all its zeros lie in Re s\0 and a polynomial in k is called Schur when all its zeros lie in jkj [ 1. A polynomial in p is called stable when its reciprocal is stable. Clearly, a polynomial is Hurwitz or Schur iff it is stable. A monic polynomial is one whose leading coefficient is unity. For any matrix G0 , G , det G, and Tr G are used for the transpose, conjugate transpose, determinant, and trace of G, respectively. A diagonal matrix G of order h is denoted by G = diag{g1, g2,…, gh}. The identity matrix is the special case of a diagonal matrix with ones on the diagonal and is denoted by I. A positive definite (nonnegative definite) matrix G is indicated by G > 0 (G > 0). A matrix of transforms is called good when all its elements are good. It is called stable when all its elements are stable. The matrix G ðsÞ is the conjugate transpose of GðsÞ. That is, G ðsÞ ¼ G ðsÞ which for a real rational matrix reduces to G ðsÞ ¼ G0 ðsÞ. The matrix G ðkÞ is the conjugate transpose of Gð1= kÞ. That is, G ðkÞ ¼ G ð1= kÞ which for a real rational matrix reduces to G ðkÞ ¼ G0 ð1=kÞ. It is important to note that G has a different interpretation for analog and digital systems, but it is always clear from the context which interpretation needs to be made. A real rational matrix

12

1

Introduction

G in s or k is called parahermitian when G ¼ G . Any parahermitian matrix GðsÞ or GðkÞ which is nonnegative definite on s ¼ jx or jkj ¼ 1, respectively, is called parahermitian-positive. A matrix GðpÞ is called proper (strictly proper)) when it is finite (zero) at p ¼ 1. Otherwise, it is called improper. A square proper matrix whose inverse is also proper is called biproper. The notation GðpÞ  0ðpm Þ means that no entry in GðpÞ grows faster than pm as p ! ∞. When GðpÞ is square and polynomial and the coefficient matrix of the highest power of p is nonsingular, it is called regular.. The rank of a matrix GðpÞ is its normal rank: i.e., the highest order of all non-identically zero minors of GðpÞ. The McMillan degree dðG ; po Þ of p ¼ po (finite or infinite) as a pole of the rational matrix GðpÞ is the largest multiplicity it possesses as a pole of any minor of GðpÞ. The McMillan degree d (G) of GðpÞ is the sum of the McMillan degrees of its distinct poles. In the partial fraction expansion of G(s), the contributions made by all its finite poles in Re s < 0, Re s > 0, and by its poles at s = ∞ are denoted by fGðsÞg þ ; fGðsÞg ; and fGðsÞg1 , respectively. Clearly, {G(s)}+ is analytic in Re s > 0, {G(s)}_ is analytic in Re s < 0, and both are strictly proper. The contribution {G(s)}∞ is polynomial. Similarly, in the partial fraction expansion of G(k), the contributions made by all its finite poles in jkj  1; jkj \1, and by its poles at k ¼ 1 are denoted by {G(k)}+, {G(k)}–, and {G(k)}∞, respectively. Clearly, {G (k)}+ is analytic in jkj\1; fGðkÞg is analytic in jkj  1, and both are strictly proper. The contribution {G(k)}∞ is polynomial. For any parahermitian-positive matrix GðsÞ, it is established in Youla (1961, 2015) that there are factors Xl ðs) and Xr ðsÞ analytic together with their respective left and right inverses in Re s [ 0 such that GðsÞ ¼ Xl ðsÞXl ðsÞ ¼ Xr ðsÞXr ðsÞ. Similarly, for any parahermitian matrix GðkÞ which is nonnegative definite on jkj ¼ 1, one can show as in Appendix C that there are factors Xl ðkÞ and Xr ðkÞ analytic together with their respective left and right inverses in jkj \1 such that GðkÞ ¼ Xl ðkÞXl ðkÞ ¼ Xr ðkÞXr ðkÞ. The matrix Xl ðXr Þ is called a left (right) Wiener–Hopf factor for G in both cases. Examples of parahermitian-positive matrices are power spectral density matrices. Such matrices are the transforms of ensemble averages and such averages are denoted by \  [ as in (1.8) and (1.9) for example. The Schur product G R of two same-size matrices G and R is the matrix whose i-row, j-column entry is gijrij. The Kronecker product G ⊗ R is the matrix whose ij-block is given by gijR. The vector vec G ¼ ½ g01 g02 . . . g0n 0 is formed by stacking all the columns of the matrix G. Useful references for the properties of Kronecker matrix products are Brewer (1978) and Horn and Johnson (1991). A useful reference for the properties of Schur matrix products is Horn and Johnson (1985) Additional notation and terminology needed for Appendices B and C is summarized in Sects. B.2 and C.2.

1.5 Historical Perspective and Commentary

1.5

13

Historical Perspective and Commentary

The methodologies employed to design linear time-invariant feedback control systems are generally either time-domain methodologies or frequency-domain methodologies. Time-domain methodologies are based on a state-variable representation for each given subsystem. Frequency-domain methodologies are based on a transfer matrix representation for each subsystem. The design methodologies are also distinguished by the performance measure employed. When tracking accuracy and control effort with the nominal plant model is key, then a quadratic or H2 performance measure is often used. When robustness or insensitivity to uncertainty or change in the plant model or exogenous inputs is key, then an H1 performance measure is often used. The H1 -norm of a real rational stable transfer matrix is the least upper bound for all the singular values of the matrix on the imaginary axis for analog systems or on the unit circle for digital systems. Examples of books focusing on the time-domain approach with an H2 performance measure are Kwakernaak and Sivan (1972), Anderson and Moore (1989), and Saberi et al. (1995). In Chen (2000), the time-domain approach with an H1 performance measure is considered and the results are used in practical applications: specifically, the control of the voice-coil-motor actuator of a hard disk drive, the control of a piezoelectric actuator, and the control of a gyro-stabilized mirror targeting system. Examples of books focusing on the time-domain approach and covering designs for both H2 and H1 performance measures are Chen and Francis (1995) and Zhou et al. (1996). Books focusing on the frequency-domain approach are Kučera (1979), Vidyasagar (1985), Grimble (2006), and Rosenwasser and Lampe (2006). In Kučera (1979), H2 performance measures are used exclusively; in Vidyasagar (1985), the emphasis is on H1 performance measures; and in Grimble (2006), both H2 and H1 performance measures are used. Grimble (2006) covers robustness in depth and contains applications of the theory that include designs of aircraft, ship, submarine, and industrial control systems. The H2 performance measure is used in Rosenwasser and Lampe (2006) and as in Chen and Francis (1995) the focus is on sampled-data systems. In Aliev and Larin (1998), H2 designs for both time-domain and frequency-domain methodologies are covered with an emphasis on computational algorithms. H2 designs for both time-domain and frequency-domain methodologies are also covered in Kučera (1991), with an emphasis on the relationships between the two methodologies. In addition to these analytically oriented design methodologies, a number of graphically oriented frequency-domain design methodologies for multivariable systems along classical lines have been proposed. The inverse Nyquist array of Rosenbrock, the characteristic locus method of MacFarlane, and the quantitative feedback theory of Horowitz are discussed in Maciejowski (1989) along with designs based on H2 and H1 performance measures. Another alternative to analytically oriented design methodologies is the use of convex programming along the lines taken in Boyd and Barratt (1991) or indirectly through linear matrix inequalities as described in Boyd et al. (1994) and in Skelton and Iwasaki (1995).

14

1

Introduction

In this book attention is focused on an analytical frequency-domain design methodology using an H2 performance measure. The earliest efforts to design controllers for feedback systems by minimizing a quadratic or H2 performance measure are described in Newton et al. (1957). These efforts employ the calculus of variations to generate the integral equation, known as the Wiener–Hopf equation, which the impulse response function of the optimal controller satisfies. Equations of this kind had been treated in Wiener (1949). Transforms, partial fraction expansions, and spectral factorizations are then used to solve the Wiener–Hopf equation and an explicit formula for the controller transfer function is obtained for analog 1DOF single-input-output systems containing stable plants. Since the design methodology presented in this book also makes use of transforms, partial fraction expansions, and spectral factorizations, it is referred to as the Wiener–Hopf design methodology in recognition of the insightful work of these early researchers. In Weston and Bongiorno (1972), the Wiener–Hopf design methodology is extended to analog 1DOF multivariable systems with stable plants by utilizing the variational technique in Bongiorno (1969) and the matrix spectral factorization technique in Youla (1961). The Wiener–Hopf methodology is applied in Chang (1961) to analog 2DOF single-input-output systems containing stable plants. The restriction to stable plants is needed in the cases just cited because the methodologies involve first finding an optimal stable open-loop controller and then realizing it within a single-loop feedback structure. This methodology leads to pole-zero cancelations in Re s  0 when the plant is unstable; hence, the system is unstable. In Newton et al. (1957) and Chang (1961), it is suggested that unstable plants can be handled by first stabilizing the plant with minor-loop feedback. Although this approach allows one to make use of the methodology available for stable plants, it leaves open the question of whether or not the performance one can realize is different among the many possible choices one can make for the minor-loop transfer function. This is especially a problem in the multivariable case and/or when the plant cannot be stabilized with stable minor-loop feedback. That there exist unstable plants that cannot be stabilized with a proper stable minor-loop feedback transfer matrix is made clear in Youla et al. (1974). Also left open is whether the open-loop design can actually be realized with a stable closed-loop feedback system. Indeed, it is pointed out in Aliev and Larin (1998, pp. 66–67) that all stabilization by minor-loop feedback is not equivalent to all stabilizing controllers for the original plant. Obviously, an unequivocal approach is needed for unstable plants. The first Wiener–Hopf solution for unstable multivariable analog plants is contained in Larin et al. (1971, 1972, 1973). However, this notable achievement for its time went unrecognized by other researchers and similar results were published in Youla et al. (1976b) for analog systems and in Kučera (1979) for digital systems. In these works, all stabilizing controllers are effectively parameterized in terms of a free stable matrix and the optimal H2 choice for the free matrix parameter is determined. In Kučera (1979), it was shown that all stabilizing controllers can be generated from the parameterized solution set of a Bezout equation. (There is a difference of opinion on whether the Bezout equation should actually be called a

1.5 Historical Perspective and Commentary

15

Diophantine equation or an Aryabhatta equation as noted in the preface to Vidyasagar (1985).) Additional commentary on the history and attributes of stabilizing controller parameterizations is provided in Sect. 2.8. Also relevant are the comments in Aliev and Larin (2007) and the reply by Kučera (2007). The frequency-domain methodology in Kučera (1979) is applicable to 1DOF multivariable digital plants and is called the polynomial equation approach. In contrast with the Wiener–Hopf methodology in which a partial fraction expansion is needed, the polynomial equation approach involves finding solutions to a pair of linear matrix polynomial equations. Only open-loop optimal designs with zero initial conditions are considered in connection with the tracking of deterministic inputs. The stable closed-loop realization of these designs is not addressed for arbitrary initial conditions and with output feedback instead of state-variable feedback. Hence, the results for open-loop designs are of limited value for unstable plants. Optimal H2 closed-loop designs are given for the regulator problem with stochastic inputs; however, some important restrictions regarding signal and/or system poles on the unit circle are not adequately generalized although specific examples in which such poles can be handled successfully are provided. The next major step forward in connection with the Wiener–Hopf design method occurs in Youla and Bongiorno (1985) where analog 2DOF multivariable systems are treated. Not only is the optimal H2 design determined in this paper, but all designs which yield a finite value for the H2 performance measure are parameterized in terms of two free stable matrices. This important parameterization is extended to analog 3DOF systems in Park and Bongiorno (1990) and to the analog standard configuration in Park and Bongiorno (1989). Digital 2DOF and 3DOF systems are treated in Grimble (1988) using the polynomial equation approach. However, in contrast with the treatments cited above for the analog case, shape-deterministic inputs and plants with poles on the stability boundary are not treated directly. Instead, a limiting procedure is suggested. Moreover, only the optimal solution is given and not all solutions for which the H2 performance measure is finite. The standard configuration is also treated using a state-variable approach in Doyle et al. (1989) for analog systems and using the polynomial equation approach in Hunt et al. (1994) for digital systems. However, the first of these two papers does not deal directly with optimal designs for persistent inputs that have poles on the imaginary axis and the second paper does not deal fully with the issue of stability. Also, the second paper does not parameterize all designs for which the H2 performance measure is finite. It does provide a comprehensive chronological account of the publications related to the polynomial equation approach. The analog 2DOF standard configuration is treated in Corrêa and Da Silveira (1995), Xie et al. (2000), and Park et al. (2002). The first two papers allow persistent deterministic exogenous inputs with poles in Re s  0 where the plant has none and insists on asymptotic tracking with zero steady-state error. In this case, the plant input must have poles in Re s  0 and the H2 -norm of the plant input is not finite. Corrêa and Da Silveira (1995) circumvent this issue by only incorporating the stable part of the plant input into the system performance measure. Xie et al. (2000)

16

1

Introduction

circumvent this issue by only incorporating the H2 -norm of the transfer matrix between the reference input and the plant input into the system performance measure. As a consequence of these approaches, however, plant input saturation can become a problem. In Park et al. (2002) on the other hand, the only persistent deterministic exogenous inputs considered are ones with poles on the imaginary axis that the plant is capable of tracking with a stable plant input. In this case, the H2 norm of the plant input is finite and can be included in the system performance measure to avoid plant input saturation and, moreover, it is possible to make a connection between state-space and Wiener–Hopf design formulas. The parameterization of the subset of all stabilizing controllers that yield finite H2 performance in Park and Bongiorno (1990) led to the consideration of a trade-off between stability margin and nominal performance for 2DOF and 3DOF analog multivariable systems which is described in Bongiorno et al. (1997). Basically, an approximate measure of stability margin is embodied in an H2 performance measure which is a functional on the free stable matrix parameter. Then this stability margin performance measure is minimized subject to a constraint on the allowed increase in nominal performance that is given by the H2 -norm of the free stable matrix parameter. The extension of this methodology to the standard configuration of Fig. 1.2 is treated in Sect. 3.9. The trade-off problem turned out to be a special case of a more general one treated in Khargonekar and Rotea (1991) and involved obtaining a Wiener–Hopf solution in terms of Kronecker products. In the papers by Doyle et al. (1989), Khargonekar and Rotea (1991), and Hunt et al. (1994), the standard configuration exogenous input is assumed to be a white noise or simply an indeterminate to indicate an input channel. In this setting, the dynamics of the persistent inputs are absorbed into the generalized plant block. However, in these papers certain stabilizability assumptions are made which lead to a distinct disadvantage: the results are not applicable to 2DOF and 3DOF systems with unstable reference inputs. In Park and Bongiorno (1989), this problem is avoided by keeping the model of the persistent inputs separate from the generalized plant; however, persistent inputs with unstable poles only on the stability boundary can be accommodated. In Corrêa and Da Silveira (1995) and Xie et al. (2000), a generalized 2DOF configuration is considered and unstable persistent inputs are treated. The approach begins with a parameterization of the subset of stabilizing controllers for which the error transform is stable. However, the parameterization does not ensure that actual plant inputs remain bounded with time. That is, the real issue of plant saturation is not specifically respected.1 A methodology for overcoming these limitations is presented in Park and Bongiorno (2009), which begins with a parameterization of the subset of stabilizing controllers for which the error transform is stable and the actual plant inputs remain bounded with time. The optimal H2 solution is then obtained from within the subset of free parameters for which the cost functional is finite. A key element in this

© 2009 Taylor and Francis Ltd. http://www.tandfonline.com. This paragraph is reprinted, with permission, from Park and Bongiorno (2009). 1

1.5 Historical Perspective and Commentary

17

approach which distinguishes it from the earlier works cited above is the explicit recognition and imposition of the requirement that the controller output should be stable for the reasons described in the last few paragraphs of Sect. 1.3. This is the approach followed in Chap. 3 of this book. Additional remarks on Park and Bongiorno (2009) are contained in the reply by Bongiorno and Park (2010) to comments by Aliev and Larin (2010). The Wiener–Hopf design methodology has also been successful in connection with the design of optimal decoupled linear multivariable systems. This methodology is used in Lee and Bongiorno (1993a, b) for analog 2DOF and 3DOF systems and in both Youla and Bongiorno (2000) and Bongiorno and Youla (2001) for analog 1DOF systems. Optimal tracking under decoupling constraints has also been considered by Corrêa et al. (2001). In Park et al. (2002), the results contained in Lee and Bongiorno (1993a) are extended to the 2DOF standard configuration and computational algorithms based on state-variable representations are also given. An important step in the Wiener–Hopf solution of the optimal decoupling problem is the recognition in Lee and Bongiorno (1993a) that the Schur matrix product plays a key role. Another essential step in the solution of the optimization problem is the parameterization of all closed-loop transfer matrices that can be realized with stabilizing decoupling controllers. Relevant results in this regard for 1DOF systems are contained in Vardulakis (1987), Lin and Hsieh (1993), Linnemann and Wang (1993), and Gómez and Goodwin (2000) for unity-feedback systems. The extension to nonunity-feedback systems is given in Youla and Bongiorno (2000). Relevant results with regard to stabilizing decoupling controllers for 2DOF and 3DOF systems containing plants with square transfer matrices are given in Desoer and Gündes (1986) and the extension to plants with rectangular transfer matrices is given in Lee and Bongiorno (1993b). In this book, the comprehensive analog H2 Wiener–Hopf frequency-domain analytical design methodology developed by Youla, Bongiorno, and coworkers for linear time-invariant multivariable feedback control systems is presented in a unified way for analog and digital systems. Moreover, the tracking of persistent exogenous deterministic inputs is explicitly accounted for. The focus is on theoretical insights regarding engineering issues that are not easily obtained from a state-variable approach.

1.6

Examples

Example 1.1 The standard configuration in Fig. 1.2 represents the mathematical relationship 

yc ym



 ¼

G11 G21

G12 G22

    e e ¼G ; v v

18

1

Introduction

where the submatrices of G are compatibly sized with respect to the vectors yc , ym , e, and v. (a) Find Gyc e , the transfer matrix between e and yc for the standard configuration. (b) Find the expressions for the Gij that correspond to the 3DOF system shown in Fig. 1.1 when y0c ¼ ½ e0 v0 , y0m ¼ ½ w0 z0 u0 , and e0 ¼ ½ d 0 r 0 n0 n0m n0l . (c) Evaluate Gyc e for the 3DOF system. (d) Find explicit formulas which give e and v in response to d, r, n, nm , and nl for the 3DOF system using the results obtained in part (c). Solution (a) Clearly, v ¼ Cym and it follows that ym ¼ G21 e þ G22 v ¼ G21 e þ G22 Cym , ym ¼ ðI  G22 CÞ1 G21 e: Hence, from yc ¼ G11 e þ G12 v ¼ G11 e þ G12 Cym ¼ ½G11 þ G12 CðI  G22 CÞ1 G21 e one gets Gyc e ¼ G11 þ G12 CðI  G22 CÞ1 G21 : (b) It is evident from Fig. 1.1 that



 Pd

Td r  Pd d  Pv v

v 0 yc  w  FPd d  FPv  nm  FPd ym z L Ld  nl u 0 rn

Td 0 0 0 I

0 0 0 P 0 0 0 0 I 0 0 0 I I 0 0

Thus, for the 3DOF system

 Pd 0 G11 G12  FPd G21 G22 L 0

Td 0 0 0 I

0 0 0 P 0 0 0 I 0 I 0 FP 0 0 I 0 I 0 0 0

I

e FP v 0 0

1.6 Examples

19

(c) Using (1.2) and the Gij obtained in part (b) yields CðI  G22 CÞ1 ¼ ðI  CG22 Þ1 C ¼ ðI þ Cw FPÞ1 ½ Cw

Cz

Cu :

It then follows from part (a) that Gyc e ¼ G11 þ G12 CðI  G22 CÞ1 G21 ¼ G11 þ G12 ðI þ Cw FPÞ1 ½ Cw Cz Cu G21   P ¼ G11 þ ðI þ Cw FPÞ1 ½ ðCw FPd þ Cz LÞ Cu Cu Cw Cz : I

Hence, with  Gyc e ¼

Ged Gvd

Ger Gvr

Gen Gvn

Gem Gvm

Gel Gvl



 ¼

Gee Gve



one gets Ged ¼ Pd þ PðI þ Cw FPÞ1 ðCw FPd þ Cz LÞ; Ger ¼ Td  PðI þ Cw FPÞ1 Cu ; ½ Gen Gem Gel  ¼ PðI þ Cw FPÞ1 ½ Cu Cw

Cz ;

and Gve ¼ ðI þ Cw FPÞ1 ½ ðCw FPd þ Cz LÞ (d) From the fact that yc ¼ ½ e0 e ¼ ½I

0 yc ¼ ½ I

Cu

Cu

Cw

Cz :

v0 0 ¼ Gyc e e, it follows that 0 Gyc e e ¼ Gee e ¼ ed þ er þ en þ enm þ el ;

where ed ¼ Ged d ¼ ½PðI þ Cw FPÞ1 ðCw FPd þ Cz LÞ  Pd d; er ¼ Ger r ¼ ½Td  PðI þ Cw FPÞ1 Cu r; and en þ em þ el ¼ ½ Gen Gem Gel ½ n0 n0m n0l 0 ¼ PðI þ Cw FPÞ1 ðCu n  Cw nm  Cz nl Þ:

20

1

Introduction

Similarly, v ¼ ½ 0 I yc ¼ ½ 0 I Gyc e e ¼ Gve e ¼ ðI þ Cw FPÞ1 ½ ðCw FPd þ Cz LÞ

Cu

Cu

Cw

Cz e

1

¼ ðI þ Cw FPÞ ½Cw FPd þ Cz LÞd  Cu ðr þ nÞ þ Cw nm þ Cz nl ; Comment: Results similar to those obtained in part (d) are derived in a simpler and more direct way at the beginning of Sect. 2.6. That is, the determination of a particular result by treating a system as a special case of the standard configuration may not always be the best approach algebraically. Example 1.2 It is important in this example to replace G in Fig. 1.2 with ~¼ G¼G



~ 11 G ~ 21 G

~ 12 G ~ 22 G



so that it can be distinguished from the G in Fig. 1.3. With regard to Fig. 1.3, one can write 

    G11s G12s ¼ G11 d þ G12 v ¼ dþ v; yt G11t G12t     y G21 d þ G22 v ¼ ym ¼ u rþn yc ¼

ys



~ ij that corresponds to the 2DOF standard and e0 ¼ ½ d 0 r 0 n0 . (a) Find the G configuration. (b) Find Gyc e using the results obtained in Example 1.1 and evaluate it when C ¼ ½ Cy Cu . (c) Find explicit formulas which give e ¼ Td r  yt and ys in response to d, r, and n. Solution (a) For the 2DOF standard configuration, yc ¼ ½ G11

0

0 e þ G12 v

and 

G21 ym ¼ 0

0 I

   0 G22 eþ v: I 0

~ 11 ¼ ½ G11 G

0

~ 12 ¼ G12 ; 0 ; G

It is now evident that

1.6 Examples

21

 G21 ~ G21 ¼ 0

0 I

   0 ~ G22 ; G22 ¼ : I 0

(b) From the solution of Example 1.1(a), ~ 21 ¼ G ~ 11 þ G ~ 12 CðI  G ~ 22 CÞ1 G ~ 11 þ G ~ 12 ðI  C G ~ 22 Þ1 CG ~ 21 Gyc e ¼ G   1 G22 ~ 21 ¼ ½ G11 0 0  þ G12 I  ½ Cy Cu  ½ Cy Cu G 0 ¼ ½ G11

0

0  þ G12 ðI þ Cy G22 Þ1 ½ Cy G21

Cu

Cu :

(c) Clearly, 

 ys ¼ Gyc e e yt ¼ ½G11  G12 ðI þ Cy G22 Þ1 Cy G21 d þ G12 ðI þ Cy G22 Þ1 Cu ðr þ nÞ:

yc ¼

Now ys ¼ ½ I Since G11 ¼ ½ G011s one gets

0 yc ; e ¼ Td r  ½ 0

G011t 0 and G12 ¼ ½ G012s

I yc :

G012t 0 ,

ys ¼ ½G11s  G12s ðI þ Cy G22 Þ1 Cy G21 d þ G12s ðI þ Cy G22 Þ1 Cu ðr þ nÞ and yt ¼ ½G11t  G12t ðI þ Cy G22 Þ1 Cy G21 d þ G12t ðI þ Cy G22 Þ1 Cu ðr þ nÞ:

Chapter 2

Stabilizing Controllers, Tracking, and Disturbance Rejection

2.1

Overview

The family of controllers that stabilize the standard configuration is parameterized in terms of a free stable real rational matrix in this chapter. The family of controllers that stabilize the 2DOF standard configuration and the 3DOF system is then derived from this parameterization. Stability is only one of several objectives in feedback control system design, however. Often it is also required that the system steady-state error in response to certain persistent deterministic inputs be zero. In addition, as explained at the end of Sect. 1.3 with regard to saturation, it is important that not only is zero steady-state error realized, but also zero steady-state value at the controller output. Moreover, this is required for the finite value of the quadratic or H2 performance measures used when they include penalties for both nonzero error and nonzero control effort. There are also practical reasons for insisting that the controller output not contain any persistent deterministic components. Such components cause a persistent drain on the sources that supply power to the actuators providing the physical inputs to the plant. These steady-state requirements are equivalent to insisting that both the error transform and the controller output transform be stable. The family of persistent deterministic inputs for which this is the case in the 3DOF system is established in this chapter along with a parameterization of the subset of stabilizing controllers that achieve this desired objective. This is accomplished for both tracking of reference inputs and for rejection of disturbance inputs. The focus on the 3DOF system is justified by the insights it provides with regard to engineering and practical considerations: the generalizations possible for the 2DOF standard configuration are treated in Example 2.12 and for the standard configuration are presented in Sect. 3.3.

© Springer Nature Switzerland AG 2020 J. J. Bongiorno Jr. and K. Park, Design of Linear Multivariable Feedback Control Systems, https://doi.org/10.1007/978-3-030-44356-6_2

23

24

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Finally, the subset of all proper stabilizing controllers is parameterized and a tunable realization for all such controllers is described. These results are useful when the controller transfer matrix for which the performance measure is minimized turns out to be proper as is often the case. In this chapter, the stage is set for Chap. 3 in which the Wiener–Hopf design methodology is employed to make an optimal choice for the free parameters. The presentation of the material in this chapter relies heavily on Appendix B as a prerequisite.

2.2

Polynomial Matrix Descriptions

The polynomial matrix description for analog systems was introduced in Rosenbrock (1970) and also treated in Wolovich (1974), Kailath (1980), Callier and Desoer (1982), and Chen (1984). In Antsaklis and Michel (2006), polynomial matrix descriptions for both analog and digital systems are treated. Any system with such a description can be represented in terms of transformed quantities by equations of the form TðpÞnðpÞ ¼ UðpÞuðpÞ þ iT ðpÞ;

ð2:1Þ

yðpÞ ¼ VðpÞnðpÞ þ WðpÞuðpÞ þ iy ðpÞ;

ð2:2Þ

where p is the Laplace transform variable in the case of analog systems and the ktransform variable in the case of digital systems. The elements of the vector n are the transforms of the system variables that completely characterize system behavior. The elements of the vectors iT ðpÞ and iy ðpÞ are polynomials in p and account for the contribution from initial conditions in the analog case and from initial data strings in the digital case. The matrices T, U, V, and W are polynomial matrices in p with real coefficients and T is square; u is the transform of the system input vector; and y is the transform of the system output vector. Clearly, a unique solution for n exists iff (if, and only if) det T 6 0. In this case, the system is called nondegenerate. Attention is restricted to nondegenerate systems and this is to be implicitly understood throughout. The uniqueness of n, however, is not enough to establish the inverse transform ^ n: one must also invoke causality (see Appendix A). For analog systems, the causality requirement is met by specifying that the domain of existence of the Laplace transform is the half-plane to the right of all finite poles of n (s). For this case, one then gets by expanding n (s) into the sum of a vector with polynomial elements and a vector with strictly proper elements that the inverse Laplace transforms of the strictly proper elements are zero for t < 0. The inverse transform of the polynomial elements contains impulses, doublets, and/or impulses of a higher order.

2.2 Polynomial Matrix Descriptions

25

For digital systems and the k-transform, causality considerations require that one choose the circular contour for the inversion integral with a sufficiently small radius so that nðkÞ is analytic on and within this contour. This is possible for arbitrary iT ðkÞ only if det T ð0Þ 6¼ 0. Otherwise, T 1 ðkÞ has a pole at k ¼ 0: When uðkÞ is analytic at k ¼ 0, det T ð0Þ 6¼ 0 is also sufficient for the analyticity of nðkÞ at k ¼ 0. Hence, for digital systems det T ð0Þ 6¼ 0 is assumed which is more stringent than det T ðkÞ 6 0: The polynomial D ¼ ldet T; where l is a constant selected to make D monic, is called the characteristic polynomial of the system. For a nondegenerate system, it easily follows that n ¼ T 1 Uu þ T 1 iT

ð2:3Þ

y ¼ Gu þ VT 1 iT þ iy ;

ð2:4Þ

G ¼ VT 1 U þ W

ð2:5Þ

and

where

is the system transfer matrix whose elements are real rational functions of p. Definition 2.1 A system is called stable when with u  0, n is a stable transform for all iT . Lemma 2.1 A system is stable iff its characteristic polynomial is stable or, alternatively, D is Hurwitz in the analog case or Schur in the digital case. Proof With u  0, n ¼ T 1 iT is stable for any polynomial vector iT iff T 1 is stable. Since T is a polynomial matrix, T 1 is stable iff D ¼ ldet T is stable. It is important to note that when T 1 is stable, it follows from (2.5) that G is stable and VT 1 iT is stable. Hence, the transfer matrix of a stable system is stable and the output y is stable for all stable u since iy is polynomial. However, a stable transfer matrix does not imply a stable system: cancelation of unstable poles of T 1 by the polynomial matrices U and V is possible. As shown in Appendix B, Sect. B.11, the real rational transfer matrix GðpÞ can be expressed, respectively, in terms of left and right coprime polynomial matrix pairs Að pÞ, Bð pÞ and B1 ð pÞ, A1 ð pÞ that have real coefficients. Specifically, one can write G ¼ A1 B ¼ B1 A1 1

ð2:6Þ

26

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and, with g and g1 constants chosen to make wG monic, wG ¼ gdetA ¼ g1 detA1

ð2:7Þ

is the characteristic denominator for G. The representations of G in (2.6) are called coprime polynomial matrix fraction descriptions. For these descriptions, when po is a zero of wG of multiplicity mo , then po is a pole of G with McMillan degree mo (see Sect. B.6). That is, dðG; po Þ ¼ mo :

ð2:8Þ

It is clear from (2.5) since U, V, and W are polynomial matrices that po must be a zero of det T with at least the same multiplicity mo . Moreover, it is possible for det T to have zeros, where wG does not. Specifically, hG ¼ D=wG

ð2:9Þ

is a polynomial. The zeros of hG are called the hidden poles of the system.

2.3

Stability Analysis of the Standard Configuration

The standard configuration (Fig. 1.2) treated here consists of an interconnection of a generalized plant represented by the polynomial matrix description TG nG ¼ U1 e þ U2 v þ iTG

ð2:10Þ

yc ¼ V1 nG þ W11 e þ W12 v þ ic

ð2:11Þ

ym ¼ V2 nG þ W21 e þ W22 v þ im

ð2:12Þ

with a controller represented by the polynomial matrix description TC nC ¼ UC ym þ iTC

ð2:13Þ

v ¼ VC nC þ WC ym þ iv :

ð2:14Þ

Using (2.12) in (2.13) and (2.14) gives TC nC ¼ UC V2 nG þ UC W21 e þ UC W22 v þ UC im þ iTC

ð2:15Þ

v ¼ VC nC þ WC V2 nG þ WC W21 e þ WC W22 v þ WC im þ iv :

ð2:16Þ

2.3 Stability Analysis of the Standard Configuration

27

Equations (2.10) (2.15), and (2.16) are compactly described by the polynomial matrix description 2

TG 4 UC V2 WC V2

0 TC VC

3 2 3 32 3 2 i TG U2 nG U1 UC W22 54 nC 5 ¼ 4 UC W21 5e þ 4 UC im þ iTC 5 ðI  WC W22 Þ v WC W21 WC im þ iv : ð2:17Þ

Equation (2.17) together with (2.11) is a complete polynomial matrix description for the standard configuration. Hence, the system is stable iff det T is a stable polynomial, where 2 3 TG 0 U2 T ¼ 4 UC V2 TC UC W22 5 ð2:18Þ WC V2 VC ðI  WC W22 Þ : It is easy to confirm that the transfer matrix from input v to output ym for the generalized plant is G22 ¼ V2 TG1 U2 þ W22

ð2:19Þ

and that the transfer matrix from input ym to output v for the controller is C ¼ VC TC1 UC þ WC :

ð2:20Þ

Multiplying T on the right by the block triangular matrix 2

I

6 To ¼ 6 40 0 then gives

2

0 I

TG1 U2

7 TC1 UC G22 7 5

0

TG

6 T^ ¼ TTo ¼ 4 Uc V2 WC V2

3 ð2:21Þ

I

0

0

TC

0

VC

ðI  CG22 Þ

3 7 5

ð2:22Þ :

Since det T^ ¼ ðdet TÞðdet To Þ ¼ det T;

ð2:23Þ

28

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

it immediately follows that det T ¼ ðdet TG Þðdet TC Þ detðI  CG22 Þ:

ð2:24Þ

Hence, the standard configuration is stable iff det T is Hurwitz in the analog case and Schur in the digital case or, generically, iff det T is a stable polynomial. It is important to recognize that CG22 is the transfer matrix of the loop closed on the generalized plant by the controller. The transfer matrices C and G22 are real rational matrices since the polynomial matrices in the polynomial matrix description for the generalized plant and controller have real coefficients. In addition, to within multiplicative constants, det TG is the characteristic polynomial of the generalized plant and det TC is the characteristic polynomial of the controller. It is also insightful to introduce the left and right coprime polynomial matrix fraction descriptions C ¼ A1 C BC

ð2:25Þ

G22 ¼ B1 A1 1 :

ð2:26Þ

and

(The coprime polynomial matrix fraction description A1 B ¼ B1 A1 is used to 1 represent G, G22 , or P22 in different sections of this book. It is always made clear which is intended so that no confusion arises.) Since C and G22 are real rational matrices, the polynomial matrices in (2.25) and (2.26) have real coefficients. It now follows from (2.24) that  det T ¼

det TG det A1



 det TC det ðAC A1 þ BC B1 Þ: det AC

ð2:27Þ

Clearly, 

det TC det AC



 ¼ gC

DC wC

 ¼ gC hC ;

ð2:28Þ

where hC is a polynomial whose zeros account for any hidden poles in the controller and gC is a constant to account for the fact that DC and wC are monic polynomials. In order to make further progress, one needs to recognize that the transfer matrix of the generalized plant follows from (2.10) through (2.12) with iTG ; ic ; and im zero. Specifically,

2.3 Stability Analysis of the Standard Configuration



yc y¼ ym



29

  e ¼G ¼ Gu; v

ð2:29Þ

where  G¼

G11 G21

G12 G22



 ¼

 V1 1 T ½ U1 V2 G

 U2  þ

W11 W21

W12 W22

 ð2:30Þ :

Hence, by the same reasoning that led to (2.9),       det TG DG hG wG ¼ gG ¼ gG ; det A1 wG22 wG22

ð2:31Þ

where gG is an appropriate constant; the polynomial hG accounts for the hidden poles of the generalized plant; and wG and wG22 are the characteristic denominators of the generalized plant transfer matrix G and its submatrix G22 , respectively. Since G22 is a submatrix of G, every pole of G22 has a McMillan degree which is less or equal to its McMillan degree as a pole of G (see Sect. B.7). That is,  h22 ¼

wG wG22

 ð2:32Þ

is a polynomial and it follows from (2.27) and (2.28) that det T ¼ gC gG hC hG h22 detðAC A1 þ BC B1 Þ:

ð2:33Þ

Now the right-hand side of (2.33) is a product of polynomial factors and the choice of controller has no impact on hG or h22 . So if either one of these polynomials or, equivalently, their product is not stable, the standard configuration is unstable no matter what one chooses for the controller. This leads immediately to the notion of admissibility. Definition 2.2 The generalized plant in the standard configuration is said to be admissible  (i) when DG =wG and wG wG22 are stable polynomials or, equivalently,  (ii) their product DG wG22 is a stable polynomial. The two equivalent ways of defining admissibility in Definition 2.2 lead to different insights, which are otherwise not so easily derived from one another. These insights are particularly important when the stability and optimization of the 3DOF system is treated. Remarkably, it is possible to characterize all stabilizing controllers for an

30

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

admissible generalized plant and this is done next using polynomial matrices with real coefficients. It is first clear from (2.33) that the controller must be synthesized with no unstable hidden poles or, equivalently, hC must be a stable polynomial. Next one must choose a left coprime real polynomial matrix pair AC ; BC so that AC A1 þ BC B1 ¼ L;

ð2:34Þ

where L is a real polynomial matrix whose determinant is a stable polynomial. Since the pair B1 ; A1 is right coprime, there exists (see Sect. B.10) a real polynomial matrix pair X1 ; Y1 such that X1 A1 þ Y1 B1 ¼ I:

ð2:35Þ

LX1 A1 þ LY1 B1 ¼ L

ð2:36Þ

Hence,

and all polynomial matrix solution pairs to (2.34) are of the form AC ¼ LX1 þ ACo

ð2:37Þ

BC ¼ LY1 þ BCo ;

ð2:38Þ

and

where ACo and BCo are real polynomial matrices satisfying ACo A1 þ BCo B1 ¼ 0

ð2:39Þ

ACo ¼ BCo B1 A1 1 ¼ BCo G22 :

ð2:40Þ

or

One can now introduce the left coprime matrix fraction description G22 ¼ A1 B

ð2:41Þ

for which there exist (see Lemma B.5) real polynomial matrices X; Y such that AX þ BY ¼ I:

ð2:42Þ

2.3 Stability Analysis of the Standard Configuration

31

Then from (2.40) ACo ¼ BCo A1 B

ð2:43Þ

must be true. Premultiplying (2.42) by BCo A1 and using (2.43) gives BCo X  ACo Y ¼ BCo A1 :

ð2:44Þ

Since the left-hand side of (2.44) is a real polynomial matrix, M ¼ BCo A1

ð2:45Þ

must be a real polynomial matrix or BCo ¼ MA;

ð2:46Þ

where M is a real polynomial matrix. Using (2.46) in (2.43) immediately yields ACo ¼ MB

ð2:47Þ

which is also a real polynomial matrix as required. That is, all real polynomial matrix solution pairs (2.37) and (2.38) of (2.34) must be of the form AC ¼ LX1  MB

ð2:48Þ

BC ¼ LY1 þ MA:

ð2:49Þ

and

Conversely, it easily follows from BA1 ¼ AB1

ð2:50Þ

that the pair (2.48) (2.49) is a solution to (2.34) for any real polynomial matrix M. However, the solution pair needed is one which is left coprime. It is now established that all such pairs are obtained from (2.48) and (2.49) by choosing for L; M a left coprime polynomial matrix pair.

32

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

It follows from Appendix B, Sect. B.10, that AC ; BC is left coprime iff ½ AC BC  has row rank for all p. That is, AC ; BC is left coprime iff for every p the only solution to a½ AC

BC  ¼ a½ ðLX1  MBÞ

ðLY1 þ MAÞ ¼ 0

ð2:51Þ

is a ¼ 0. A useful identity which helps in the identification of the solution of (2.51) is obtained when (2.48) and (2.49) are substituted into (2.34). The result is ðLX1  MBÞA1 þ ðLY1 þ MAÞB1 ¼ L:

ð2:52Þ

Multiplying (2.52) on the left by a and using (2.51) then gives   A aL ¼ a½ ðLX1  MBÞ ðLY1 þ MAÞ 1 ¼ 0 B1

ð2:53Þ

Hence, (2.51) is equivalent to ½ aðLX1  MBÞ

aðLY1 þ MAÞ ¼ aM½ B

A¼0

ð2:54Þ

and it follows that  aM½ B

A

Y X

 ¼ aMðAX þ BYÞ ¼ aM ¼ 0:

ð2:55Þ

Combining (2.53) and (2.55) yields a½ L

M  ¼ 0:

ð2:56Þ

That is, any solution a of (2.51) must be a solution of (2.56). The converse is also true: if a satisfies (2.56), then aL ¼ 0 and aM ¼ 0 and it easily follows that a satisfies (2.51). So the pair AC ; BC given by (2.48) and (2.49) is left coprime iff L; M is left coprime. One additional constraint on the pair L; M remains. The representation (2.25) requires that det AC ¼ det ðLX1  MBÞ 6 0:

ð2:57Þ

It is now possible to state the key theorem concerning the stability of the standard configuration.

2.3 Stability Analysis of the Standard Configuration

33

Theorem 2.1 A standard configuration with given polynomial matrix description for the generalized plant and for the controller is stable iff the generalized plant is admissible, the controller is free of unstable hidden poles, and the controller transfer matrix can be written in the form C ¼ ðX1  KBÞ1 ðY1 þ KAÞ;

ð2:58Þ

where K is a stable real rational matrix such that det ðX1  KBÞ 6 0:

ð2:59Þ

Proof The necessity for admissibility and no unstable hidden poles in the controller has already been pointed out. From (2.48) and (2.49), it follows that the controller transfer matrix must have the form 1 1 C ¼ A1 C BC ¼ ðLX1  MBÞ ðLY1 þ MAÞ ¼ ðX1  KBÞ ðY1 þ KAÞ;

ð2:60Þ

where L is such that det L is a stable polynomial and K ¼ L1 M. Hence, K must be a stable real rational matrix. Moreover, det ðLX1  MBÞ ¼ ðdet LÞdet ðX1  KBÞ:

ð2:61Þ

Hence, Eq. (2.59) is equivalent to (2.57) since det L 6 0. The sufficiency of the stated conditions easily follows from the fact that any stable real rational matrix K satisfying (2.59) can be written as L1 M, where L; M is left coprime; det L is a stable polynomial; and both L and M have real coefficients. Then one can substitute K ¼ L1 M into (2.58) and (2.59) to confirm that (2.57) is satisfied and to make the identifications (2.48) and (2.49). It then follows from (2.33) that det T is Hurwitz in the analog case and Schur in the digital case or, equivalently, the system is stable. There are a number of important additional observations to make before closing this section. The first concerns an alternative form to (2.58) for the controller transfer matrix. The second concerns an algebraic result relating to the requirement  in connection with admissibility that wG wG22 be a stable polynomial. The third concerns showing that the stability results presented here hold irrespective of the polynomial matrix fraction description chosen for G22 . The presentation of these observations relies on an important identity, the generalized Bezout identity, assumed throughout this book and described in Lemma B.5 of Appendix B for the coprime polynomial matrix fraction descriptions of a rational matrix. Specifically, the real polynomial matrices in (2.26), (2.35), (2.41), and (2.42) can be chosen so that

34

2 Stabilizing Controllers, Tracking, and Disturbance Rejection



A Y1

B X1



X Y

B1 A1





X ¼ Y

B1 A1



A Y1

B X1





I 0 ¼ 0 I

 ð2:62Þ :

It is now possible to make the first observation and it is stated in the form of Lemma 2.2 The controller transfer matrix given by (2.58) satisfies the identity C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1 :

ð2:63Þ

Proof From the second matrix product in (2.62), one gets YA ¼ A1 Y1

ð2:64Þ

A1 X1 þ YB ¼ I:

ð2:65Þ

and

Solving (2.64) for Y1 , substituting the result into (2.58), and making use of (2.65) then leads to C ¼ ½I  ðY þ A1 KÞB1 ðY þ A1 KÞA:

ð2:66Þ

Invoking the easily verified matrix identity ðI  MNÞ1 M ¼ MðI  NMÞ1

ð2:67Þ

C ¼ ðY þ A1 KÞ½I  BðY þ A1 KÞ1 A:

ð2:68Þ

now gives

However, I  BðY þ A1 KÞ ¼ AX  BA1 K ¼ AX  AB1 K ¼ AðX  B1 KÞ

ð2:69Þ

and (2.68) reduces to the final expression for C in (2.63) which completes the proof of the Lemma. Attention is now directed to the second observation to be made. Specifically, Lemma 2.3 For the generalized plant transfer matrix in the standard configuration,  wG wG22 is a stable polynomial iff AG21 , G12 A1 , and G11 þ G12 A1 Y1 G21 are stable.

2.3 Stability Analysis of the Standard Configuration

35

This Lemma plays an important role in the next chapter where its significance is best appreciated. Before giving the proof of the Lemma, the basis for its conjecture is presented. From (2.29), (2.30), and the fact that v ¼ Cym in Fig. 1.2, one gets yc ¼ G11 e þ G12 Cym

ð2:70Þ

ym ¼ G21 e þ G22 Cym :

ð2:71Þ

and

~ where Solving (2.71) for ym and substituting the result into (2.70) gives yc ¼ Ge; ~ ¼ G11 þ G12 CðI  G22 CÞ1 G21 G

ð2:72Þ

is the transfer matrix for the standard configuration. Using the matrix identity (2.67) and the expression (2.58) for C then leads to ~ ¼ G11 þ G12 ½I þ ðX1  KBÞ1 ðY1 þ KAÞB1 A1 1 ðX1  KBÞ1 ðY1 þ KAÞG21 G 1 ð2:73Þ

or ~ ¼ G11 þ G12 A1 ½ðX1  KBÞA1 þ ðY1 þ KAÞB1 1 ðY1 þ KAÞG21 : G

ð2:74Þ

Using (2.35) and (2.50), one gets finally that ~ ¼ G11 þ G12 A1 Y1 G21 þ G12 A1 KAG21 ¼ G ~ 11 þ G12 A1 KAG21 : G

ð2:75Þ

~ are stable for every choice of stable K when the The standard configuration and G generalized plant is admissible and the controller has no unstable hidden poles. In ~ 11 ¼ G11 þ G12 A1 Y1 G21 must be this case then the choice K ¼ 0 indicates that G stable. It is also true when the standard configuration is stable that the transfer matrices from e to ym and from e to v must be stable. The former transfer matrix is given by ~ m ¼ ðX  B1 KÞAG21 G

ð2:76Þ

36

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and the latter by ~ v ¼ ðY þ A1 KÞAG21 : G

ð2:77Þ

~ m þ BG ~ v ¼ AG21 AG

ð2:78Þ

Hence,

 must be stable. Since the stability of the polynomial wG wG22 is equivalent to the . stability of the polynomial wG0 wG022 , it follows as shown in the proof below that G12 A1 must also be stable. Moreover, the only aspect of the admissibility of the  ~ is the stability of the polynomial wG wG , and generalized plant that impacts on G 22 it should now be clear why one might conjecture Lemma 2.3 for which a direct algebraic proof is now given. In the proof, extensive use is made of several properties (see Sect. B.7) of the McMillan degree.  Proof The polynomial wG wG22 is stable iff the unstable poles of G22 and G coincide and have identical McMillan degrees. Since G2 ¼ ½ G21

G22 

ð2:79Þ

is a submatrix of G, the McMillan degree of any pole of G2 is equal to or lower than its McMillan degree as a pole of G. Similarly, since G22 is a submatrix of G2 , the McMillan degree of any pole of G22 is equal to or lower than its McMillan degree as a pole of G2 . But the unstable poles of G22 and G coincide and have  identical McMillan degrees. Hence, the same is true for G2 and G22 . That is, wG2 wG22 must be a stable polynomial. Now one can always write G2 ¼ A1 2 B2 where A2 ; B2 ; is a left coprime polynomial matrix pair. Premultiplying (2.79) by A2 then gives that B2 ¼ ½ A2 G21

A2 G22  ¼ ½ A2 G21

A2 A1 B :

ð2:80Þ

Hence, A2 A1 B must be a stable matrix. But this is the case iff H2 ¼ A2 A1 is stable since A; B is left coprime and A2 is a polynomial matrix (see Appendix B, Lemma B.8). Moreover, ðdet H2 Þ1 ¼



det A det A2



¼ gwG22 w1 G2

ð2:81Þ

2.3 Stability Analysis of the Standard Configuration

37

is stable since g is simply a constant to account for the fact that det A and det A2 may not be monic. So H21 ¼

  adj H2 ¼ AA1 2 det H2

ð2:82Þ

must be stable. Premultiplying (2.79) by A then gives that AA1 2 B2 ¼ ½ AG21

AG22  ¼ ½ AG21

B

ð2:83Þ

must be a stable matrix. Thus, AG21 must be stable. Clearly, these same arguments can be applied to 

G011 G ¼ G012 0

G021 G022





G011 ¼ G012

G021 ðA01 Þ1 B01

 ð2:84Þ

to conclude that A01 G012 or, equivalently, G12 A1 must be stable. ~ 11 it is natural to post-multiply G by Now with regard to G  C¼

I A1 Y1 G21

0 I

 ð2:85Þ

to get 

 G12 ; G22

ð2:86Þ

~ 21 ¼ G21 þ G22 A1 Y1 G21 : G

ð2:87Þ

C21 ¼ A1 Y1 G21 ¼ A1 Y1 A1 AG21

ð2:88Þ

Go ¼ G C ¼

~ 11 G ~ 21 G

where

The matrix

is stable because AG21 must be stable and because it follows from the fact that A1 Y1 A1 B ¼ YB

ð2:89Þ

38

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

is stable that A1 Y1 A1 is stable (A; B is left coprime and A1 Y1 is a polynomial matrix). So C and 1

C



I ¼ A1 Y1 G21

0 I

 ð2:90Þ

are both stable and one gets from (2.86) that the unstable poles of G and Go coincide and have identical McMillan degrees (see Example B.3). Another important observation to make is that (2.87) is equivalent to ~ 21 ¼ ðI  B1 Y1 ÞG21 ¼ ðI  B1 Y1 ÞA1 AG21 : G

ð2:91Þ

Now ðI  B1 Y1 ÞA1 stable follows from 1 1 B1 X1 ¼ B1 ðI  Y1 B1 ÞA1 1 ¼ ðI  B1 Y1 ÞB1 A1 ¼ ðI  B1 Y1 ÞA B

ð2:92Þ

since B1 X1 is stable, ðI  B1 Y1 Þ is polynomial, and A; B is left coprime. This fact ~ 21 must be stable. So from and the fact that AG21 is stable yields that G 

0 Go ¼ ~ G21

  ~ 11 0 G þ 0 0

G12 G22

 ¼ Ga þ Gb

ð2:93Þ

it follows that the unstable poles of G; Go , and 

~ 11 G Gb ¼ 0

G12 G22

 ð2:94Þ

coincide and have identical McMillan degrees. Since the same is required of G and G22 , the unstable poles of Gb and G22 must coincide and have identical McMillan degrees. Now it is always possible to construct a submatrix of Gb given by  Sb ¼

g 0

 h0 ; S22

ð2:95Þ

~ 11 , h0 contains elements from G12 which are in where g is one of the elements of G the same row of Gb as g, and S22 is a square submatrix of G22 whose columns are the same as those from which the elements of h0 are taken. Then det Sb ¼ g det S22

ð2:96Þ

2.3 Stability Analysis of the Standard Configuration

39

is a minor of Gb . The matrix S22 can always be selected so that det S22 possesses any particular unstable pole of G22 with multiplicity equal to its McMillan degree as a pole of G22 . Since Sb is a submatrix of Gb and it is required that Gb and G22 possess this same pole with equal McMillan degree it follows from (2.96) that g must be ~ 11 must be stable. ~ 11 . Hence, G stable. Now g can be any one of the elements of G ~ This completes the proof for the necessity of AG21 , G12 A1 , and G11 stable in Lemma 2.3. With regard to sufficiency, it is first noted that when AG21 is stable one can again conclude as done immediately following (2.94) that the unstable poles of G and G22 coincide and have identical McMillan degrees when the same is true of Gb and G22 . ~ 11 is also stable, it then follows from When G  Gb ¼

~ 11 G 0

  0 0 þ 0 0

G12 G22

 ð2:97Þ

that all one needs to show is that the poles of  Gc ¼

G12 G22

 ð2:98Þ

and G22 coincide and have identical McMillan degrees. When G12 A1 is stable, it follows from (2.98) that 

G12 A1 Gd ¼ Gc A1 ¼ G22 A1





G12 A1 ¼ B1

 ð2:99Þ

is stable. Moreover, from Gc ¼ Gd A1 1

ð2:100Þ

one gets for every unstable pole po of Gc that dðGc ; po Þ  dðGd ; po Þ þ dðA1 1 ; po Þ:

ð2:101Þ

dðGd ; po Þ ¼ 0

ð2:102Þ

1 dðA1 1 ; po Þ ¼ dðB1 A1 ; po Þ ¼ dðG22 ; po Þ

ð2:103Þ

Now

since Gd is stable and

40

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

since B1 ; A1 are right coprime (see Lemma B.7 and Remark B.8). Thus, dðGc ; po Þ  dðG22 ; po Þ:

ð2:104Þ

On the other hand, since G22 is a submatrix of Gc , dðGc ; po Þ  dðG22 ; po Þ:

ð2:105Þ

dðGc ; po Þ ¼ dðG22 ; po Þ

ð2:106Þ

Hence,

for every unstable pole po of Gc and, therefore, the unstable poles of Gc and G22 coincide and have identical McMillan degrees. This completes the proof of Lemma 2.3. The final observation to be made concerns the fact that Theorem 2.1 holds irrespective of the coprime polynomial matrix fraction descriptions used for G22 . This is important because such descriptions are only unique to within the choice of certain arbitrary unimodular matrices. Indeed, as established in Lemma B.6 of Appendix B, G22 ¼ A1 B ¼ B1 A1 1

ð2:107Þ

~ 1 ~ 1 B ~¼B ~1A G22 ¼ A 1

ð2:108Þ

and

are both coprime polynomial matrix fraction descriptions of G22 iff ~ ¼ NA; B ~ ¼ NB A

ð2:109Þ

~ 1 ¼ A1 N1 ; B ~ 1 ¼ B1 N1 ; A

ð2:110Þ

and

where N and N1 are unimodular polynomial matrices. Moreover, there exist poly~1 ; Y~1 such that nomial matrices X ~ 1 þ Y~1 B ~ 1 ¼ I: ~1 A X Premultiplying (2.111) by N1 then leads to

ð2:111Þ

2.3 Stability Analysis of the Standard Configuration

~1 A1 þ N1 Y~1 B1 ¼ I: N1 X

41

ð2:112Þ

Subtracting (2.35) from (2.112) now yields ~1  X1 ÞA1 þ ðN1 Y~1  Y1 ÞB1 ¼ 0: ðN1 X

ð2:113Þ

Equation (2.113) is identical in form to (2.39) and so the same reasoning that led to (2.46) and (2.47) gives ~1  X1 Þ ¼ MB; ðN1 Y~1  Y1 Þ ¼ MA; ðN1 X

ð2:114Þ

where M is an arbitrary polynomial matrix. Now when one uses (2.108) and (2.111) in Theorem 2.1, the formula for the stabilizing controller transfer matrix becomes ~ ~ ¼ ðX ~1  K ~ BÞ ~ 1 ðY~1 þ K ~ AÞ: C

ð2:115Þ

~1 and Y~1 from (2.114) in (2.115) and setting Using (2.109) and the solutions for X ~ ¼ N11 ðK  MÞN 1 K

ð2:116Þ

~ ¼ ðX1  KBÞ1 ðY1 þ KAÞ C

ð2:117Þ

leads to

which is identical in form to (2.58). Moreover, since N and N1 are unimodular, for ~ in (2.116) and conversely for every stable K ~ in every stable K one gets a stable K (2.116) the solution for K is ~ K ¼ M þ N1 KN

ð2:118Þ

a stable matrix. So the set of all stabilizing controllers given by (2.58) as K ranges over all stable real rational matrices is identical to the set of all controllers given by ~ ranges over all stable real rational matrices. Because of the importance (2.115) as K of this observation, it is incorporated into Lemma 2.4 The set of all stabilizing controllers given by (2.58) as K ranges over all stable real rational matrices is invariant with respect to the choice of the real coprime polynomial matrix fraction descriptions used for G22 .

42

2.4

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Stability Analysis of the 2DOF Standard Configuration

The 2DOF standard configuration is shown in Fig. 1.3. It is important to recall that the block Td is not an actual component in the system and, therefore, plays no role with regard to stability of the interconnected physical components. Moreover, the physical system can be represented by the standard configuration in Fig. 1.2 when the identifications e ¼ ½ d0

r0

n0  0 ;

ð2:119Þ

yc ¼ ½ y0s

y0t 0 ;

ð2:120Þ

ym ¼ ½ y0

u0  0

ð2:121Þ

and

are made. In particular, when TG nG ¼ U1 d þ U2 v þ iTG ; ½ y0s

y0t 0 ¼ V1 nG þ W11 d þ W12 v þ ic ;

ð2:122Þ ð2:123Þ

and y ¼ V2 nG þ W21 d þ W22 v þ iy

ð2:124Þ

is a polynomial matrix description for the generalized plant in Fig. 1.3, it follows that TG nG ¼ ½ U1

0

yc ¼ V1 nG þ ½ W11

0 e þ U2 v þ iTG ;

ð2:125Þ

0 e þ W12 v þ ic ;

ð2:126Þ

     0 0 W22 i eþ vþ y I I 0 0

ð2:127Þ

0

and   V2 W21 ym ¼ n þ 0 0 G 

is the polynomial matrix description of the generalized plant in Fig. 1.2 for the 2DOF standard configuration. Hence, it is clear from a comparison of (2.10) through (2.12) with (2.125) through (2.127) that the stability results obtained for the standard configuration apply here when

2.4 Stability Analysis of the 2DOF Standard Configuration

~ 1 ¼ ½ U1 U

43

0 ;

ð2:128Þ

0 0 ;

ð2:129Þ

~2 ¼ ½ V20 00 0 ; V   ~ 21 ¼ W21 0 0 W 0 I I ;

ð2:130Þ

0

~ 11 ¼ ½ W11 W

ð2:131Þ

and 0 ~ 22 ¼ ½ W22 W

00 0

ð2:132Þ

are used instead of U1 ; W11 ; V2 ; W21 ; and W22 in (2.10) through (2.12). It then follows that in place of the Gij in (2.30) one gets ~ 11 ¼ V1 T 1 U ~1 þ W ~ 11 ¼ ½ G11 G G

0

0 ;

~ 12 ¼ V1 T 1 U2 þ W12 ¼ G12 ; G G 

~ 21 ¼ V ~1 þ W ~2 TG1 U ~ 21 ¼ G21 G 0

0 I

ð2:133Þ ð2:134Þ

0 I

 ð2:135Þ ;

and   ~ 22 ¼ V ~2 TG1 U2 þ W ~ 22 ¼ G22 ; G 0

ð2:136Þ

G11 ¼ V1 TG1 U1 þ W11 ;

ð2:137Þ

G21 ¼ V2 TG1 U1 þ W21 ;

ð2:138Þ

G22 ¼ V2 TG1 U2 þ W22 :

ð2:139Þ

where

and

Hopefully, there will be no confusion between the Gij introduced here and those used in Sect. 2.3. This risk is taken in order to avoid a profusion of symbols. In fact, the Gij introduced here are transfer matrices associated with (2.122) through (2.124), the polynomial matrix description of the generalized plant in the 2DOF standard configuration. The Gij introduced in Sect. 2.3 are transfer matrices associated with (2.10) through (2.12), the polynomial matrix description of the

44

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

~ ij is used here generalized plant in the standard configuration; hence, the symbol G to represent them (see Problem 1.2). It is especially important to recognize that G22 represents the transfer matrix between v and ym in the standard configuration of Sect. 2.3 and between v and y in the 2DOF standard configuration treated here. The two are connected by (2.136). ~ 22 ; po Þ ¼ dðG22 ; po Þ for any It is clear from (2.136) and Example B.2 that dðG pole po ; hence, wG~ 22 ¼ wG22 :

ð2:140Þ

It also follows from 

~ ~ ¼ G11 G ~ 21 G

~ 12 G ~ 22 G



2

G11 ¼ 4 G21 0

0 0 I

0 0 I

3 G12 G22 5 0

ð2:141Þ

that 2

G11 ~ C ¼ 4 G21 G 0

G12 G22 0

0 0 I

3  0 G 5 0 ¼ 0 I

0 I

 0 ; I

ð2:142Þ

where 2

I 0 60 0 C¼6 40 0 0 I

0 I 0 0

3 0 07 7 I5 0

ð2:143Þ

is nonsingular. Hence, when Example B.2 and Property B.5 are recalled, it is easy to see that wG~ ¼ wG~ C ¼ wG :

ð2:144Þ

1 Thus, wG~ 22 w1 ~ is stable iff wG22 wG is stable. Also, from (2.31), G

hG~ ¼

DG~ g det TG DG ¼ ¼ ¼ hG ; wG~ wG wG

ð2:145Þ

2.4 Stability Analysis of the 2DOF Standard Configuration

45

where g is a constant chosen so that g det TG is monic. In analogy with Definition 2.2, it is now natural to introduce Definition 2.3 The generalized plant in the 2DOF standard configuration is said to be admissible  (i) when DG =wG and wG wG22 are stable polynomials or, equivalently,  (ii) their product DG wG22 is a stable polynomial. Further progress requires determination of the connection between coprime ~ 22 and G22 . Specifically, one can polynomial matrix fraction descriptions for G always introduce G22 ¼ A1 B ¼ B1 A1 1

ð2:146Þ

AX þ BY ¼ I

ð2:147Þ

X1 A1 þ Y1 B1 ¼ I:

ð2:148Þ

along with the equations

and

It then follows from (2.136) that  ~ 22 ¼  A G 0

0 I

  1   B B ¼  1 A1 1 0 0

ð2:149Þ

and, moreover, 

A 0

0 I



X 0

   0 B þ ½Y I 0

 0 ¼

I 0

0 I

 ð2:150Þ

and 

X1 A1 þ ½ Y1

B 0 1 0

 ¼ I:

ð2:151Þ

Comparing (2.149) through (2.151) with (2.26), (2.35), (2.41), and (2.42) now indicates that A, B, X, Y, A1 , B1 , X1 , and Y1 in the latter group are replaced by  A ~ A¼ 0

0 I

 ;

  B ~ B¼ 0

ð2:152Þ

46

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

 X ~ X¼ 0

0 I



Y~ ¼ ½ Y

0

ð2:153Þ

;

~ 1 ¼ A1 ; B ~ 1 ¼ ½ B01 A

00 0

ð2:154Þ

~1 ¼ X1 ; Y~1 ¼ ½ Y1 X

0 :

ð2:155Þ

and

~ for K in (2.58) then gives Using these replacements and K C¼

  1  ~ B X1  K ½ Y1 0

~ 0þK



A 0

0 I

 ð2:156Þ :

It follows with ~ ¼ ½K K

H

ð2:157Þ

that C ¼ ðX1  KBÞ1 ½ ðY1 þ KAÞ

H :

ð2:158Þ

Comparing (1.5) and (2.158) immediately yields Cu ¼ ðX1  KBÞ1 H

ð2:159Þ

Cy ¼  ðX1  KBÞ1 ðY1 þ KAÞ:

ð2:160Þ

and

So in analogy with Theorem 2.1 one can state without any additional proof Theorem 2.2 A 2DOF standard configuration with given polynomial matrix description for the generalized plant and for the controller is stable iff the generalized plant is admissible, the controller is free of unstable hidden poles, and the controller transfer matrix can be written in the form C ¼ ½ Cy

Cu  ¼ ðX1  KBÞ1 ½ ðY1 þ KAÞ

H ;

ð2:161Þ

2.4 Stability Analysis of the 2DOF Standard Configuration

47

where K and H are stable real rational matrices and K is such that det ðX1  KBÞ 6 0:

ð2:162Þ

It is important to recognize that since the polynomial matrices in (2.146) through (2.148) are assumed chosen so that the generalized Bezout identity (2.62) is satisfied, one gets from (2.152) through (2.155) that 

~ A Y~1

~ B ~1 X



~ X Y~

~1 B ~1 A





~ X ¼ ~ Y

~1 B ~1 A



~ A Y~1

~ B ~1 X





I ¼ 0

0 I

 ð2:163Þ :

Indeed, that the first of the matrix products is an identity matrix easily follows once it is recognized that ~ X ~1 Y~ ¼ ½ Y1 Y~1 X

 0

X 0

 0  X1 ½ Y I

0  ¼ ½ ðY1 X  X1 YÞ 0  ¼ 0: ð2:164Þ

That the second matrix product is an identity matrix is an immediate consequence of the fact that MN ¼ I implies NM ¼ I for any two square matrices M; N. Hence, with the appropriate replacements, Lemmas 2.2, 2.3, and 2.4 are also applicable to the 2DOF standard configuration. The alternative formula for C that one can get from (2.63) is, however, not very useful in comparison with the one given by (2.161); therefore, it is not presented here. The replacements for the three key matrices in Lemma 2.3 are  ~G ~ 21 ¼ A A 0

0 I



G21 0

0 I

  0 AG21 ¼ I 0

0 I

0 I

 ð2:165Þ ;

~ 1 ¼ G12 A1 ; ~ 12 A G

ð2:166Þ

and ~ 1 Y~1 G ~ 21 ¼ ½ G11 ~ 11 þ G ~ 12 A G

 0

0  þ G12 A1 ½ Y1

¼ ½ ðG11 þ G12 A1 Y1 G21 Þ

0

G21 0

0 0 I I

 ð2:167Þ

0 0 :

Clearly, the left-hand sides of (2.165) through (2.167) are stable iff AG21 ; G12 A1 ; and ðG11 þ G12 A1 Y1 G21 Þ are stable. Thus, Lemma 2.5 For  the generalized plant transfer matrix in the 2DOF standard configuration, wG wG22 is a stable polynomial iff AG21 ; G12 A1 , and G11 þ G12 A1 Y1 G21 are stable.

48

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

It should also be clear that the appropriate replacement for Lemma 2.4 is Lemma 2.6 The set of all stabilizing controllers given by (2.161) as K and H range over all stable real rational matrices is invariant with respect to the choice of coprime polynomial matrix fraction descriptions used for G22 .

2.5

Stability Analysis of the 3DOF System

Most feedback control system implementations have the structure of the 3DOF system or some simplification of it (i.e., the 1DOF or the 2DOF system). So it is of interest and insightful to specialize the results for the 2DOF standard configuration to that of the 3DOF system. When the 3DOF system shown in Fig. 1.1 is viewed as a particular implementation of the 2DOF standard configuration shown in Fig. 1.3, the generalized plant takes the form shown in Fig. 2.1. The blocks Pa and Pp , with P ¼ Pp Pa ;

ð2:168Þ

are not shown explicitly in Fig. 1.1, but are included here so that the outputs from the actuator block Pa can be associated with the vector ys : these outputs are typically the internal variables of the physical plant that need to be kept small in order to minimize the control effort required and in order to assure that the linear model remains valid. The physical plant output y in Fig. 1.1 is designated by yt in Fig. 2.1 since it is in fact the output that is to be tracked. Moreover, this is best done to avoid confusion with the measurement vector y ¼ ½ w0

Fig. 2.1 Generalized plant for the 3DOF system

z0  0 :

ð2:169Þ

nl z

L ~ d

v

Pd

Pa

ys yt

Pp Physical Plant

nm

F

w

2.5 Stability Analysis of the 3DOF System

49

Similarly, the disturbance input d is denoted by d~ in Fig. 2.1 so as to avoid confusion with the exogenous input vector d ¼ ½ d~0

0

n0l

n0m 

ð2:170Þ

in Fig. 1.3. The objective here is to find the expressions for TG and G so that Definition 2.3, Theorem 2.2, and Lemma 2.5 can be specialized to the 3DOF system. Since the quantities sought do not in any way depend on the initial conditions or initial data strings, for simplicity these are taken to be zero. The first step is to introduce a polynomial matrix description for each subsystem in Fig. 2.1. Specifically, Ta na ¼ Ua v

ð2:171Þ

y s ¼ V a na þ W a v

ð2:172Þ

is used to represent the actuators and Tp np ¼ ½ Up1

0 d þ Up2 ys

0

yt ¼ Vp np þ ½ Wp1

0

0 d þ Wp2 ys

ð2:173Þ ð2:174Þ

is used to model the balance of the physical plant, where Pa ¼ Va Ta1 Ua þ Wa ;

ð2:175Þ

Pd ¼ Vp Tp1 Up1 þ Wp1 ;

ð2:176Þ

Pp ¼ Vp Tp1 Up2 þ Wp2 :

ð2:177Þ

and

Similarly, Tl nl ¼ ½ Ul z ¼ V l nl þ ½ W l

0 d

0 I

0 d

ð2:178Þ ð2:179Þ

with L ¼ Vl Tl1 Ul þ Wl

ð2:180Þ

50

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

is used for the feedforward transducer and Tf nf ¼ Uf yt

ð2:181Þ

w ¼ V f nf þ W f y t þ ½ 0

ð2:182Þ

0 Id

with F ¼ Vf Tf1 Uf þ Wf

ð2:183Þ

is used for the feedback transducer. Using (2.172) in (2.173) and (2.174) yields Tp np  Up2 Va na ¼ ½ Up1

0d þ Up2 Wa v

0

yt ¼ Vp np þ Wp2 Va na þ ½ Wp1

0

ð2:184Þ

0d þ Wp2 Wa v :

ð2:185Þ

0 d þ Uf Wp2 Wa v;

ð2:186Þ

Using (2.185) in (2.181) and (2.182) then gives Tf nf  Uf Vp np  Uf Wp2 Va na ¼ Uf ½ Wp1

0

w ¼ Vf nf þ Wf Vp np þ Wf Wp2 Va na þ ½Wf Wp1

Id þ Wf Wp2 Wa v :

0

ð2:187Þ

Hence, for the generalized plant in Fig. 2.1, Eq. (2.122) holds with nG ¼ ½ n0a

n0p

n0f

n0l 0

ð2:188Þ

and 2

Ta 6 Up2 Va TG ¼ 6 4 Uf Wp2 Va 0

0 Tp Uf Vp 0

0 0 Tf 0

3 0 07 7 05 Tl :

ð2:189Þ

The remaining coefficient matrices in (2.122) through (2.124) can also be identified from the above equations, but this is not necessary to complete the task at hand and is omitted for brevity. Clearly, for the 3DOF system, DG ¼ g det TG ¼ gðdet Ta Þðdet Tp Þðdet Tf Þðdet Tl Þ:

ð2:190Þ

2.5 Stability Analysis of the 3DOF System

51

The remaining item to be determined is the generalized plant transfer matrix. This can be done directly from Fig. 2.1. Clearly, 

ys yt



 ¼

0 Pd

   P 0 0 dþ a v 0 0 P

ð2:191Þ

and  y¼

  w FPd ¼ L z

0 I

   I FP dþ v 0 0

ð2:192Þ

So

P

and

wG22 ¼ wFP :

ð2:194Þ

Hence, in place of Definition 2.3 one has Definition 2.4 The 3DOF system is said to be admissible (i) when hG ¼

DG gðdet Ta Þðdet Tp Þðdet Tf Þðdet Tl Þ ¼ wG wG

ð2:195Þ

and wG =wFP are stable polynomials or, equivalently, (ii) their product gðdet Ta Þðdet Tp Þðdet Tf ÞðdetTl Þ DG ¼ wFP wFP

ð2:196Þ

is a stable polynomial. Now one can introduce coprime polynomial matrix fraction descriptions for FP satisfying the generalized Bezout identity (2.62) such that FP ¼ A1 B ¼ B1 A1 1 :

ð2:197Þ

52

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Then 

G22

  FP A ¼ ¼ 0 0 

A 0

0 I



X 0

0 I

1 

   B B1 1 ¼ A1 ; 0 0

   0 B þ ½ Y I 0



I 0 ¼ 0

0 I

ð2:198Þ

 ð2:199Þ ;

and 

X1 A1 þ ½ Y1

B1 0 0

 ¼ I:

ð2:200Þ

It is clear from a comparison of (2.198) through (2.200) with (2.146) through (2.148) that the results sought for the 3DOF system can be obtained by replacing A, B, X, Y, A1 , B1 , X1 , and Y1 in the results previously obtained for the 2DOF standard configuration with     ~¼ A 0 ;B ~ ¼ B ð2:201Þ A 0 I 0  ~¼ X X 0

 0 ~ ; Y ¼ ½ Y I

0

ð2:202Þ

  ~ 1 ¼ A1 ; B ~ 1 ¼ B1 A 0

ð2:203Þ

and ~1 ¼ X1 ; Y~1 ¼ ½ Y1 X

0 :

ð2:204Þ

In particular, when these replacements are made in (2.161), when K is partitioned compatibly as K ¼  ½ K1

H2 ;

ð2:205Þ

and when H is replaced with H1 , one gets  C¼



X1 þ ½ K1

B H2  0

1   ½ Y1

0   ½ K1

 A H2  0

0 I

 

 H1 ð2:206Þ

2.5 Stability Analysis of the 3DOF System

53

which reduces to C ¼ ðX1  K1 BÞ1 ½ ðY1 þ K1 AÞ

H2

H1  :

ð2:207Þ

So the equivalent to Theorem 2.2 for the 3DOF system is Theorem 2.3 An admissible 3DOF system is stable iff the controller is free of unstable hidden poles and the controller transfer matrix can be written in the form C ¼ ½ Cw

Cz

Cu  ¼ ðX1  K1 BÞ1 ½ ðY1 þ K1 AÞ H2

H1 ;

ð2:208Þ

where K1 , H1 , and H2 are stable real rational matrices and K1 is such that det ðX1  K1 BÞ 6 0:

ð2:209Þ

Attention is turned next to Lemma 2.5. For the 3DOF system, the three key matrices become  ~ 21 ¼ A AG 0

0 I



FPd L

  0 I AFPd ¼ I 0 L

0 I

A 0

 ð2:210Þ ;

    Pa Pa A1 ~ A ¼ G12 A1 ¼ ; P 1 PA1 and

ð2:211Þ

   0 0 0 Pa  A ½ Y FP 0 Y  P 0 0 P 1 1 d  1  d 0 Pa A1 Y1 Pa A1 Y1 FPd ¼ ðPd  PA1 Y1 FPd Þ 0 PA1 Y1 :

~ 1 Y~1 G21 ¼ G11 þ G12 A



ð2:212Þ

So since A and Y1 are polynomial matrices one gets in place of Lemma 2.5 Lemma 2.7 For the 3DOF system wG =wFP is a stable polynomial iff AFPd ;L, Pa A1 , PA1 , Pa A1 Y1 FPd , and ðI  PA1 Y1 FÞPd are all stable. Another fundamental result with regard to admissibility can be obtained from (2.196) using the polynomial matrix description of the physical plant contained in (2.171) and (2.184), and the physical plant transfer matrix which is easily determined from Fig. 2.1. Specifically, 

Ta Up2 Va

0 Tp



na np





0 ¼ Up1

Ua Up2 Wa

  d~ v

ð2:213Þ

54

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and 

ys yt



 ¼

0 Pd

Pa P

  d~ v :

ð2:214Þ

Hence, the characteristic polynomial for the physical plant is 

Ta Dp ¼ gp det Up2 Va

0 Tp

 ¼ gp ðdet Ta Þðdet Tp Þ;

ð2:215Þ

where gp is the constant needed for Dp to be monic, and the transfer matrix for the physical plant is  Po ¼

0 Pd

Pa P

 ð2:216Þ :

It follows that Dp ¼ ho wPo ;

ð2:217Þ

where wPo is the characteristic denominator of Po and ho is the monic polynomial accounting for the hidden poles of the physical plant. Moreover, since P is a submatrix of Po , wPo ¼ h1 wP ;

ð2:218Þ

where h1 is a monic polynomial. It should also be clear that Df ¼ gf det Tf ¼ hf wF

ð2:219Þ

Dl ¼ gl det Tl ¼ hl wL ;

ð2:220Þ

and

where hf and hl are the monic polynomials which account for the hidden poles of the feedback and feedforward transducers, respectively, and wF and wL are the characteristic denominators of F and L, respectively. Thus, (2.196) becomes Dp Df Dl hwP wF wL DG ¼ ¼ ; wFP wFP wFP

ð2:221Þ

2.5 Stability Analysis of the 3DOF System

55

where h ¼ ho h1 hf hl :

ð2:222Þ

Now when po is a finite pole of FP, it follows that dðFP; po Þ  dðF; po Þ þ dðP; po Þ:

ð2:223Þ

Hence, wFP divides into wF wP without remainder and (2.221) is a stable polynomial iff h, wL , and wF wP =wFP are all stable polynomials. Clearly, h is a stable polynomial iff ho , h1 , hf , and hl are all stable polynomials. That is, iff the physical plant, the feedback transducer, and the feedforward transducer, are all free of unstable hidden poles and, in addition, h1 is a stable polynomial. The polynomial h1 can be determined from (2.218):  h1 ¼ wPo wP :

ð2:224Þ

Finally, it is noted that wL is a stable polynomial iff L is stable. It now follows from (ii) of Definition 2.4 without the need for any additional proof that Lemma 2.8 The 3DOF system is admissible iff (i) the feedback transducer, the feedforward transducer, and the physical plant are free of unstable hidden poles and (ii) L is stable and both wPo wP and wF wP =wFP are stable polynomials. Lemmas 2.7 and 2.8 establish important properties that hold for any admissible 3DOF system. These properties play a key role in the formulation of the Wiener– Hopf design methodology for such systems and are essential in connection with Theorem 2.3. It should also be emphasized that as in earlier cases the results for 3DOF systems do not depend on the choice of coprime polynomial matrix fraction descriptions used for FP. Specifically, in place of Lemma 2.6 one has Lemma 2.9 The set of all stabilizing controllers given by (2.208) as K1 , H1 , and H2 range over all stable real rational matrices is invariant with respect to the choice of coprime polynomial matrix fraction descriptions used for FP.

2.6

Tracking and Disturbance Rejection in the 3DOF System

The tracking and disturbance rejection capabilities of the 3DOF system shown in Fig. 1.1 are investigated in this section. It is assumed that the system is admissible and the controller is a stabilizing one as described in Theorem 2.3 with transfer matrix given by (2.208). The reference noise vector n and the transducer noise vectors nl and nm are typically stochastic in nature and, therefore, are devoid of any

56

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

deterministic or shape-deterministic components. The elements of the disturbance input vector d (d~ in Fig. 2.1) and the reference input vector r, however, often contain deterministic or shape-deterministic components that are transforms of such signals as steps, ramps, and exponentially growing functions of time. In this case, d and r possess unstable poles. The objective here is to identify the class of inputs d and r of this type for which the tracking error e and the controller output v have stable transforms and to parameterize the subset of stabilizing controllers that achieve this objective. Definition 2.5 The reference input r is called trackable if, when it is the only exogenous input, there exists a stabilizing controller for which both the error transform and the controller output transform are stable. Definition 2.6 The disturbance input d is called rejectable if, when it is the only exogenous input, there exists a stabilizing controller for which both the error transform and the controller output transform are stable. The reason for insisting on the stability of v is a consequence of the following considerations. The use of a quadratic performance measure that includes a penalty on the ys shown in Fig. 2.1 requires that Pa v be stable. So any unstable poles of v would have to be canceled by zeros of Pa . Since for admissibility any actuator poles and/or zeros in the unstable region (Re s  0 for analog systems, jkj  1 for digital systems) must appear in P ¼ Pp Pa as well, and since such poles and zeros typically degrade the performance achievable, it is customary to pick actuators for which Pa is stable and maintains full column rank in the unstable region. Indeed, Pa ¼ I is often used to model the actuators. In these cases, stability of ys is assured iff v is stable. There are also mathematical benefits from insisting on v stable: with this restriction, the task of parameterizing the subset of all stabilizing controllers for which e is stable is made more transparent than in Cheng and Pearson (1981), Pernebo (1981), and Saeks and Murray (1981). Since a stabilizing controller is assumed, the contributions to e and v from initial conditions in the analog case or from initial data strings in the digital case are stable. So there is no loss in generality in ignoring the contribution from these sources. That is, one can work exclusively with exogenous inputs and transfer matrices. From Fig. 1.1 and (1.1), it follows that v ¼ Cw ðm þ nm Þ  Cz ðLd þ nl Þ þ Cu ðr þ nÞ;

ð2:225Þ

m ¼ FðPv þ Pd dÞ:

ð2:226Þ

where

2.6 Tracking and Disturbance Rejection in the 3DOF System

57

Eliminating m yields v ¼  ðRw FPd þ Rz LÞd þ Ru r þ vn ;

ð2:227Þ

where ½ Rw

Rz

Ru  ¼ ðI þ Cw FPÞ1 ½ Cw

Cz

Cu 

ð2:228Þ

and vn ¼ Rw nm  Rz nl þ Ru n:

ð2:229Þ

It now immediately follows from e ¼ Td r  Pv  Pd d

ð2:230Þ

in Fig. 1.1 that the error transform is given by e ¼ ðTd  PRu Þr  ðPd  PRw FPd  PRz LÞd  Pvn :

ð2:231Þ

When the parameterizations of Cu , Cw , and Cz given in (2.208) for a stabilizing controller are used and when (2.62) and (2.197) are recalled, the closed-loop transfer matrices Ru , Rw , and Rz simplify significantly. Specifically, one gets 1 ðI þ Cw FPÞ1 ¼ ½I þ ðX1  K1 BÞ1 ðY1 þ K1 AÞB1 A1 ¼ A1 ðX1  K1 BÞ 1 

ð2:232Þ and it easily follows from (2.228) that ½ Rw

Rz

Ru  ¼ ½ A1 ðY1 þ K1 AÞ A1 H2

A1 H1 :

ð2:233Þ

Since cancelations between unstable terms in (2.227) or in (2.231) cannot be counted on in practice, the contributions to v and e from d and r must be stable separately. In addition, the contributions from r depend only on Ru , which is determined by the free stable real rational matrix H1 and the contributions from d depend only on Rw and Rz which are determined, respectively, by the free stable real rational matrices K1 and H2 . An important consequence of this separation of the design parameters is that one can address the issues of tracking error in response to reference inputs independently from the issue of disturbance rejection. Initially, attention is focused on the contributions from r given by

58

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

vr ¼ Ru r ¼ A1 H1 r

ð2:234Þ

er ¼ ðTd  PRu Þr ¼ ðTd  PA1 H1 Þr:

ð2:235Þ

and

The first observation to make is that admissibility together with Lemma 2.7 requires that PA1 be stable. In addition, stability requires that H1 be stable. So PA1 H1 is stable and there is no good reason to choose Td to be anything but a stable matrix. In this case, Td  PA1 H1 and A1 H1 are both stable. With r ¼ A1 r br

ð2:236Þ

a left coprime real polynomial matrix fraction description for r, it then follows that vr and er are stable iff the real rational matrices W ¼ A1 H1 A1 r

ð2:237Þ

1 U ¼ ðTd  PA1 H1 ÞA1 r ¼ Td Ar  PW

ð2:238Þ

and

are both stable. So the question of interest is whether or not one can choose a stable H1 so that U and W are stable? Clearly, the answer is yes iff there exist stable solution pairs U; W satisfying U þ PW ¼ Td A1 r

ð2:239Þ

and among such solution pairs there is at least one for which the solution for H1 in (2.237) is a stable matrix. Actually, under reasonable assumptions, it is established in the sequel that one can exhibit all stable solution pairs U; W and for every such pair the solution for H1 in (2.237) is a stable matrix. The first step is the introduction of the coprime polynomial matrix fraction descriptions 1 P ¼ A1 p Bp ¼ Bp1 Ap1 ;

ð2:240Þ

where 

Ap Yp1

Bp Xp1



Xp Yp

Bp1 Ap1



 ¼

Xp Yp

Bp1 Ap1



Ap Yp1

Bp Xp1



 ¼

I 0 0 I

 ð2:241Þ :

2.6 Tracking and Disturbance Rejection in the 3DOF System

59

Then (2.239) yields Ap U þ Bp W ¼ Ap Td A1 r :

ð2:242Þ

So if U and W are both stable, it is necessary that T~d ¼ Ap Td A1 r

ð2:243Þ

be a stable matrix. Given this is the case, the family of all stable solution pairs U; W can be parameterized with the aid of Ap Xp þ Bp Yp ¼ I

ð2:244Þ

which is a consequence of (2.241). Indeed, it should be evident that all stable solution pairs U; W are of the form U ¼ Xp T~d þ Uo

ð2:245Þ

W ¼ Yp T~d þ Wo ;

ð2:246Þ

and

where Uo and Wo are stable matrices satisfying 1 Uo ¼ A1 p Bp Wo ¼ Bp1 Ap1 Wo :

ð2:247Þ

Since Bp1 ; Ap1 is right coprime and Wo must be stable, Uo is stable iff Kr ¼ A1 p1 Wo

ð2:248Þ

U ¼ Xp T~d  Bp1 Kr

ð2:249Þ

W ¼ Yp T~d þ Ap1 Kr

ð2:250Þ

is stable. That is,

and

parameterizes all stable real rational solution pairs U; W in terms of the arbitrary stable real rational matrix Kr .

60

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Thus, one can choose a stable H1 for which both W and U are stable iff one can choose a stable Kr for which 1 1 H1 ¼ A1 1 WAr ¼ A1 Yp Ap Td þ A1 Ap1 Kr Ar

ð2:251Þ

is stable. Now, from (2.241) Yp Ap ¼ Ap1 Yp1 ;

ð2:252Þ

H1 ¼ A1 1 Ap1 ðYp1 Td þ Kr Ar Þ:

ð2:253Þ

hence,

Clearly, when A1 1 Ap1 is stable, H1 is stable for all stable Kr since Td is stable and since Yp1 and Ar are polynomial matrices. In this regard the following Lemma plays an important role. Lemma 2.10 When wFP ðwF wP Þ1 is stable, A1 1 Ap1 is stable iff F is stable. Proof When A1 1 Ap1 is stable, det ðA1 1 Ap1 Þ ¼

  det Ap1 w w ¼ g F P w1 F det A1 wFP

ð2:254Þ

must be stable where g is the appropriate constant needed because the w’s are monic polynomials. Hence, 1 1 1 w1 F ¼ g wFP ðwF wP Þ det ðA1 Ap1 Þ

ð2:255Þ

must be stable or, equivalently, F must be stable. Conversely, when F is stable A1 1 Ap1 ¼ ðX1 þ Y1 FPÞAp1 ¼ X1 Ap1 þ Y1 FBp1

ð2:256Þ

is stable. This completes the proof. Since in almost all practical cases the feedback transducer transfer matrix F is stable, there is ample justification to make this hypothesis. In this circumstance, it follows from Lemmas 2.8 and 2.10 that (2.253) parameterizes all stable H1 for which er and vr are stable and one has with no additional proof needed Theorem 2.4 When (1) r ¼ A1 r br is a left coprime real polynomial matrix fraction description for the reference input and (2) Td and F are stable, the reference input is trackable with a stabilizable 3DOF system iff Ap Td A1 r is stable. Moreover, all stabilizing controllers for which r is trackable are given by ( 2.208) with

2.6 Tracking and Disturbance Rejection in the 3DOF System

H1 ¼ A1 1 Ap1 ðYp1 Td þ Kr Ar Þ;

61

ð2:257Þ

where Kr is a stable real rational matrix. 1 Remark 2.1 Theorem 2.4 is primarily of interest when A1 r is unstable. When Ar is stable, r is stable and it follows from (2.234) and (2.235) that vr and er are stable for all stable H1 . That is, the desired tracking capability is realized in this case without the need to impose any additional restriction on H1 beyond that required by system stability.

Remark 2.2 The equations that characterize e and v in the 2DOF system obtained when the feedforward path in Fig. 1.1 is removed and the controller submatrix Cz is deleted correspond to setting Rz ¼ 0 in (2.227) and (2.231). Clearly, this has no impact on vr and er so Theorem 2.4 is also applicable to the 2DOF system. Remark 2.3 Since Ar ; br is left coprime, Ap Td A1 r is stable iff Ap Td A1 r br ¼ Ap Td r

ð2:258Þ

is stable or, when Td is square and possesses an inverse, iff r ¼ Td1 A1 p ro ;

ð2:259Þ

where ro is a stable real rational vector. In this case, (2.259) parameterizes the family of all real rational reference inputs that can be tracked with zero steady-state error and zero steady-state controller output under Theorem 2.4. Typically, Td1 is also stable; then the unstable poles of r must be a subset of the unstable poles of A1 p which are the unstable poles of the plant transfer matrix P. Application of Theorem 2.4 requires finding coprime polynomial matrix fraction descriptions for both FP and P. By imposing a somewhat stronger assumption on F and Td , however, one can avoid the need of finding a coprime polynomial matrix fraction description for P and obtain a simpler formula in place of (2.257). Indeed, it is next established that Theorem 2.5 When (1) r ¼ A1 r br is a left coprime real polynomial matrix fraction description for the reference input and (2) Td , Td1 , and ðTd F  IÞP are stable, the reference input is trackable with a stabilizable 3DOF system iff A A1 r is stable. Moreover, all stabilizing controllers for which r is trackable are given by ( 2.208) with H1 ¼ Y1 þ Kr Ar ; where Kr is a stable real rational matrix.

ð2:260Þ

62

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Proof Equation (2.238) can be rewritten as U ¼ Td A1 r  Td FPW þ ðTd F  IÞPW:

ð2:261Þ

The last term on the right-hand side of (2.261) is stable when W is stable. Hence, 1 U is stable in this case iff Td ðA1 r  FPWÞ is stable or, since Td and Td are stable, 1 ðAr  FPWÞ is stable. That is, there exists a stable solution pair U; W to (2.261) iff ~ W to there exists a stable solution pair U; ~ ¼ A1  FPW ¼ A1  A1 BW U r r

ð2:262Þ

~ þ BW ¼ AA1 : AU r

ð2:263Þ

or, equivalently,

A necessary and sufficient condition for this to be the case is that AA1 r be stable. One can now apply the same reasoning as in connection with Theorem 2.4 to get ~ W are given by that all stable real rational solution pairs U; W ¼ YAA1 r þ A1 Kr

ð2:264Þ

~ ¼ XAA1  B1 Kr : U r

ð2:265Þ

and

It then follows from (2.237) that 1 H1 ¼ A1 1 WAr ¼ A1 YA þ Kr Ar ¼ Y1 þ Kr Ar

ð2:266Þ

and the proof is complete. Remark 2.4 Obviously, Remarks 2.1 and 2.2 apply here as well. Remark 2.5 The assumption that Td , Td1 , and C ¼ ðTd F  IÞP ¼ Td FP  P

ð2:267Þ

are stable is a stronger assumption than just assuming that Td and F are stable. This can be seen from the following reasoning. When Td , Td1 , and C are stable, one gets from (2.267) for any unstable pole located at po that

2.6 Tracking and Disturbance Rejection in the 3DOF System

dðFP; po Þ ¼ dðTd FP; po Þ ¼ dðC þ P; po Þ ¼ dðP; po Þ:

63

ð2:268Þ

That is, wP w1 FP is stable. Now when the system is also admissible, one has from Lemma 2.8 that wFP ðwF wP Þ1 is stable. So 1 1 w1 F ¼ ½wP wFP ½wFP ðwF wP Þ 

ð2:269Þ

is stable or, equivalently, F must be stable. However, admissibility and a stable F and Td are not enough to guarantee a stable C and Td1 . Remark 2.6 It has already been pointed out that one benefit of the stronger Assumption (2) in Theorem 2.5 is that a coprime polynomial matrix fraction description for P is not needed and a simpler parameterization of H1 is obtained. There are other benefits as well. It easily follows from (2.267) that CA1 ¼ Td B1  PA1 :

ð2:270Þ

Hence, PA1 is stable and this is consistent with one of the requirements needed for admissibility in Lemma 2.7. Assumption (2) in Theorem 2.5 also turns out to be significant with regard to tracking when 1DOF systems are treated (see Examples 2.13 and 3.12). Remark 2.7 Since Ar ; br is left coprime, A A1 r is stable iff A A1 r br ¼ Ar

ð2:271Þ

r ¼ A1 ro ;

ð2:272Þ

is stable. This is the case iff

where ro is a stable real rational vector. Equation (2.272) parameterizes the family of all real rational reference inputs that can be tracked with zero steady-state error and zero steady-state controller output under Theorem 2.5. Since (2.268) holds, it follows from (2.197) that the unstable poles of A1 and P are the same. It is then clear from (2.272) that the unstable poles of r must be a subset of the unstable poles of the plant transfer matrix P. Attention is now turned to the contribution to v and the contribution to e on account of d in (2.227) and (2.231), respectively. These contributions are given by vd ¼ A1 Hd

ð2:273Þ

64

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and ed ¼  ðPd  PA1 HÞd;

ð2:274Þ

where, when (2.233) is recalled, H ¼ ðY1 þ K1 AÞFPd þ H2 L:

ð2:275Þ

A1 H ¼ ðY þ A1 K1 ÞAFPd þ A1 H2 L

ð2:276Þ

Using (2.64) one gets

and Pd  PA1 H ¼ ðI  PA1 Y1 FÞPd  PA1 ðK1 AFPd þ H2 LÞ

ð2:277Þ

which are clearly stable because of admissibility and Lemma 2.7. With d ¼ A1 d bd

ð2:278Þ

a left coprime real polynomial matrix fraction description for d, it then follows that vd and ed are stable iff the real rational matrices W ¼ A1 HA1 d

ð2:279Þ

1 U ¼ ðPd  PA1 HÞA1 d ¼ Pd Ad  PW

ð2:280Þ

and

are both stable. These two equations are identical in form to (2.237) and (2.238) with Pd replacing Td and H replacing H1 ; however, here H need not be stable, only K1 and H2 in H must be stable. It still follows in similar fashion, however, that ~ d ¼ Ap Pd A1 P d

ð2:281Þ

must be a stable matrix and all stable real rational solution pairs U; W of (2.280) are given by ~ d  Bp1 Km U ¼ Xp P

ð2:282Þ

2.6 Tracking and Disturbance Rejection in the 3DOF System

65

and ~ d þ Ap1 Km ; W ¼ Yp P

ð2:283Þ

where Km is an arbitrary stable real rational matrix. Hence, in an admissible 3DOF system it is possible for both vd and ed to be stable iff one can choose stable K1 , H2 , and Km so that det ðX1  K1 BÞ 6 0 and so that ~ W ¼ A1 HA1 d ¼ Yp Pd þ Ap1 Km

ð2:284Þ

or, equivalently, 1 ~ A1 ðK1 Qd þ H2 L A1 d Þ  Ap1 Km ¼ Yp Pd  YAFPd Ad ;

ð2:285Þ

where 1 ~ Qd ¼ AFPd A1 d ¼ AFAp Pd :

ð2:286Þ

Now it is again consistent with practice to assume F is stable in which case it 1 follows from AFP ¼ AFA1 p Bp ¼ B, a stable matrix, that AFAp and, therefore, Qd are stable matrices. Equation (2.285) is equivalent to ~ H2 L A1 d  Km ¼ Cd  K1 Qd ;

ð2:287Þ

where 1 1 1 ~ ~ Cd ¼ A1 1 ðYp Pd  YAFPd Ad Þ ¼ ðA1 Yp  Y1 FAp ÞPd

ð2:288Þ

and ~ m ¼ A1 K 1 Ap1 Km :

ð2:289Þ

1 Using A1 1 ¼ X1 þ Y1 FP and Ap ¼ Xp þ PYp in (2.288) yields

~d : Cd ¼ ðX1 Yp  Y1 FXp ÞP

ð2:290Þ

Clearly, when F is stable, Cd is stable. Moreover, one also has from Lemma 2.10 that A1 1 Ap1 is stable. In addition, admissibility together with Lemma 2.7 requires 1 ~ that PA1 ¼ Bp1 A1 p1 A1 be stable or, equivalently, Ap1 A1 be stable. Hence, Km is stable iff Km is stable and the problem of finding stable real rational matrices K1 , H2 ,

66

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and Km that satisfy (2.285) has been reduced to one of finding stable real rational ~ m that satisfy (2.287). matrices K1 , H2 , and K The real coprime polynomial matrix fraction description 1 L A1 d ¼ Ag Bg

ð2:291Þ

Ag Xg þ Bg Yg ¼ I

ð2:292Þ

satisfying

is now introduced. Then (2.287) yields ~ H2 A1 g Bg ¼ Cd  K1 Qd þ Km ;

ð2:293Þ

hence, the left-hand side must be a stable matrix. Since a stable H2 is required and 1 the pair Ag ; Bg is left coprime, H2 A1 g Bg is stable iff H2 Ag is stable or, equivalently, H2 ¼ Kg Ag ;

ð2:294Þ

where Kg is a stable matrix. Then (2.293) is satisfied for any choice of stable K1 when ~ m ¼ Kg Bg þ K1 Qd  Cd : K

ð2:295Þ

That is, vd and ed are stable under the assumptions here iff ½ K1

H2  ¼ ½ K1

Kg Ag 

ð2:296Þ

with K1 and Kg arbitrary stable real rational matrices. It is of course essential to establish that there is at least one choice of K1 for which det ðX1  K1 BÞ 6 0. Obviously, for the choice K1 ¼ 0 this is the case since one can always construct X1 so that det X1 6 0 (Appendix B, Remark B.6); so there is at least one stabilizing 3DOF controller described in Theorem 2.3 for which d is rejectable. It is now possible to state without additional proof that Theorem 2.6 When (1) d ¼ A1 d bd is a left coprime real polynomial matrix fraction description for the disturbance input, (2) the real polynomial matrix pair Ag ; Bg satisfies (2.291) and (2.292), and (3) the feedback sensor transfer matrix F is a stable matrix, then the disturbance input d is rejectable with a stabilized 3DOF system iff Ap Pd A1 d is stable. Moreover, all stabilizing controllers for which d is rejectable are given by ( 2.208) with ½ K1

H2  ¼ ½ K1

Kg Ag ;

ð2:297Þ

2.6 Tracking and Disturbance Rejection in the 3DOF System

67

where K1 and Kg are stable real rational matrices belonging to the nonempty set of such matrices for which det ðX1  K1 BÞ 6 0. Remark 2.8 Equation (2.297) parameterizes in terms of arbitrary stable matrices K1 and Kg all stable K1 ; H2 for which all pairs vd ; ed given by (2.273), (2.274) are stable. Although this parameterization may include cases in which a particular choice of stable K1 yields det ðX1  K1 BÞ  0, it can be demonstrated that in most cases of practical interest this possibility does not occur. In analog systems, for example, it is typical that K1 is chosen so that the sensitivity matrix satisfies S ¼ ðI þ Cw FPÞ1 ! I;

s ! 1:

ð2:298Þ

It immediately follows from (2.232) since det A1 6 0 that det ðX1  K1 BÞ ¼ det S=det A1 6 0:

ð2:299Þ

In digital systems it is usually the case that det A1 ðkÞ 6¼ 0, B1 ðkÞ ¼ 0, and BðkÞ ¼ 0 at k ¼ 0. Also, since a stable K1 ðkÞ is analytic in jkj  1, K1 ðkÞ is finite at k ¼ 0. In this case then, X1 ð0Þ  K1 ð0ÞBð0Þ ¼ X1 ð0Þ

ð2:300Þ

X1 ð0ÞA1 ð0Þ þ Y1 ð0ÞB1 ð0Þ ¼ X1 ð0ÞA1 ð0Þ ¼ I:

ð2:301Þ

and

It follows that at k ¼ 0 det ðX1  K1 BÞ ¼ det X1 ¼ 1=det A1 6¼ 0;

ð2:302Þ

therefore, det ðX1  K1 BÞ 6 0 for any stable K1 . Remark 2.9 Theorem 2.6 is primarily of interest when A1 d is unstable. Otherwise, d is stable and it easily follows from (2.276) and (2.277) being stable that vd and ed are stable. That is, the desired disturbance rejection capability is realized in this case without the need to impose any additional restrictions on K1 and H2 beyond that required for system stability. Remark 2.10 The equation that characterizes vd and the one that characterizes ed in the 2DOF system are obtained when the feedforward path in Fig. 1.1 is removed and the controller submatrix Cz is deleted; these equations are the same as those obtained by setting H2 ¼ 0 in (2.275) and (2.276). Equation (2.287) is then replaced with

68

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

~ m ¼ K1 Qd  Cd K

ð2:303Þ

~ m for any stable K1 . It now easily follows that Theorem 2.6 which gives a stable K holds for the 2DOF system except that all stabilizing controllers for which d is rejectable are now given by C ¼ ½ Cw

Cu  ¼ ðX1  K1 BÞ1 ½ ðY1 þ K1 AÞ

H1 ;

ð2:304Þ

where K1 and H1 are stable real rational matrices and K1 is such that det ðX1  K1 BÞ 6 0:

ð2:305Þ

That is, a rejectable d is rejectable in a 2DOF system with any stabilizing controller; no additional constraints on K1 or H1 are needed. ~ d Ad and ~ d ¼ Ap Pd A1 is stable, then Ap Pd ¼ P Remark 2.11 Clearly, when P d ~ d bd are stable. Conversely, when Ap Pd and Ap Pd d ¼ Ap Pd A1 Ap Pd d ¼ P d bd are 1 stable, it follows from Ad ; bd left coprime that Ap Pd Ad is stable. That is, Ap Pd A1 d is stable iff Ap Pd and Ap Pd d are stable. Hence, Pd d ¼ A1 d ; where d is a stable o o p vector and Pd d cannot have any unstable poles that are not poles of the plant. Moreover, when Ap Pd d is stable and Ap Pd has a stable left inverse given by ðAp Pd Þ# , then ðAp Pd Þ# Ap Pd d ¼ d must be stable and Remark 2.9 applies. It is again of interest to determine conditions under which the need to find a coprime polynomial matrix fraction description for P can be avoided. These conditions are embodied in Theorem 2.7 When (1) d ¼ A1 d bd is a left coprime real polynomial matrix fraction description for the disturbance input, (2) the real polynomial matrix pair Ag ; Bg satisfies (2.291) and (2.292), and (3) Td , Td1 , ATd A1 , and ðTd F  IÞP are stable matrices, the disturbance input d is rejectable with a stabilized 3DOF system iff APd A1 d is stable. Moreover, all stabilizing controllers for which d is rejectable are given by (2.208) with ½ K1

H2  ¼ ½ K1

Kg Ag ;

ð2:306Þ

where K1 and Kg are stable real rational matrices belonging to the nonempty set of such matrices for which det ðX1  K1 BÞ 6 0: Proof One again obtains (2.280) or

ð2:307Þ

2.6 Tracking and Disturbance Rejection in the 3DOF System

U þ PW ¼ U þ ðTd F þ I  Td FÞPW ¼ Pd A1 d :

69

ð2:308Þ

Multiplying (2.308) on the left by Td1 and introducing ~ ¼ T 1 ½U  ðTd F  IÞPW ¼ T 1 Pd A1  FPW U d d d

ð2:309Þ

~ þ FPW ¼ T 1 Pd A1 : U d d

ð2:310Þ

gives

~ is Under (3) in the assumptions, it is clear from (2.309) when W is stable that U stable iff U is stable. So a stable solution pair U; W exists for (2.308) iff a stable ~ W exists for (2.310). solution pair U; The coprime polynomial matrix fraction descriptions (2.197) that satisfy the generalized Bezout identity (2.62) can now be introduced to get from (2.310) ~ þ BW ¼ P ~ do ; AU

ð2:311Þ

1 1 1 ~ do ¼ ATd1 Pd A1 P d ¼ ðATd A ÞðAPd Ad Þ:

ð2:312Þ

where

~ do must be stable. Since Td1 and Sd ¼ ATd A1 are It is evident from (2.311) that P 1 assumed stable, it follows that ATd1 A1 ¼ S1 d ¼ det ðTd Þ adj (Sd Þ is stable. ~ do is stable iff APd A1 Hence, P d is stable and the necessity of this condition is clear. Invoking similar arguments to the ones used to get (2.284) now yields ~ W ¼ A1 HA1 d ¼ Y Pdo þ A1 Km

ð2:313Þ

or in view of (2.275), (2.286), and (2.291) H2 A1 g Bg ¼ Cdo  K1 Qd þ Km ;

ð2:314Þ

where 1 1 ~ do ¼ Y1 Td1 ðTd F  IÞA1 Cdo ¼ Y1 ðFTd  IÞA1 P p ðAp A ÞAPd Ad :

ð2:315Þ

1 It is next established that Cdo is stable by showing that ðTd F  IÞA1 p and ðAp A Þ are stable. In accordance with Remark 2.5, the assumption that ðTd F  IÞP is stable

70

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

forces F to be stable. From ðTd F  IÞP ¼ ðTd F  IÞA1 p Bp it then follows that 1 ðTd F  IÞAp must be stable. Also, AðTd F  IÞP ¼ ATd FP  AP ¼ ATd A1 B  AP

ð2:316Þ

must be stable. Since ATd A1 is stable, it follows that AP ¼ AA1 p Bp must be stable must be stable. Moreover, or, equivalently, AA1 p .  1 1 det ðAA1 p Þ ¼ det Ap det A ¼ gðwF wP =wFP ÞwF

ð2:317Þ

must be stable because F is stable and wF wP =wFP is polynomial. Hence, Ap A1 ¼ 1 ðAA1 must be stable and Cdo is indeed stable. Thus, the problem has been p Þ reduced to finding stable real rational matrices K1 , H2 , and Km that satisfy (2.314). Clearly, H2 A1 g Bg must be stable and this is the case iff

H2 ¼ Kg Ag ;

ð2:318Þ

where Kg is a stable matrix. Then (2.314) is satisfied for any choice of stable K1 when Km ¼ Kg Bg þ K1 Qd  Cdo :

ð2:319Þ

That is, vd and ed are stable under the assumptions here iff ½ K1

H2  ¼ ½ K1

Kg Ag 

ð2:320Þ

with K1 and Kg arbitrary stable real rational matrices. It is again essential to establish that there is at least one choice for which det ðX1  K1 BÞ 6 0. Clearly, for K1 ¼ 0, det ðX1  K1 BÞ ¼ det X1 6 0

ð2:321Þ

since one can always construct X1 so that det X1 6 0 (Appendix B, Remark B.6); this shows that there is at least one stabilizing 3DOF controller described in Theorem 2.3 for which d is rejectable. This completes the proof of Theorem 2.7. Remark 2.12 It should be apparent that with only minor adjustments Remarks 2.8 through 2.11 are applicable here as well.

2.6 Tracking and Disturbance Rejection in the 3DOF System

71

Remark 2.13 When the contributions to v and e from inputs r ¼ A1 r br , d ¼ 0 or 1 from r ¼ 0, d ¼ Ad bd are stable, it follows from superposition that the contri1 butions to v and e from inputs r ¼ A1 r br , d ¼ Ad bd are stable. So Theorems 2.4 through 2.7 are equally useful when both reference inputs and disturbance inputs are present simultaneously as in the practical case. It should also be clear that the controller parameterizations given in Theorems 2.4 through 2.7 do not depend explicitly on br or bd . Hence, v and e are stable no matter what the polynomial vectors br and bd are; so in this regard the controllers are robust. More specifically, it should be clear that vr and er are stable for all possible polynomial vectors br , whether coprime to Ar or not, iff the matrices W and U given by (2.237) and (2.238) are stable. Again when Td and F are stable this is the case iff the conditions of Theorem 2.4 are satisfied; that is iff Ap Td A1 r and Kr are stable and H1 is given by (2.257). Similarly, vd and ed are stable for all possible polynomial vectors bd , whether coprime to Ad or not, iff the matrices W and U given by (2.279) and (2.280) are stable. Again when F is stable this is the case iff the conditions of Theorem 2.6 and Kd are stable and ½ K1 H2  is given by are satisfied; that is iff Ap Pd A1 d (2.297). Clearly, analogous observations also apply in connection with Theorems 2.5 and 2.7. Remark 2.14 Practical requirements often demand that vr and er both be stable when any one of the elements of r is not zero while all of the other elements are. That is, one might desire instead of vr ¼ A1 H1 r, er ¼ ðTd  PA1 H1 Þr stable that vr ¼ A1 H1 Dr , er ¼ ðTd  PA1 H1 ÞDr be stable, where Dr ¼ diag fr1 ; r2 ; . . .; rN g and r ¼ ½ r1 r2 . . .rN 0 . In this case, a coprime polynomial matrix fraction description Dr ¼ A1 r Br can be introduced and one can easily replicate the appropriately modified and nearly identical replacements for Theorems 2.4 and 2.5. Similarly, in connection with disturbance inputs d ¼ ½ d1 d2 . . .dM 0 one might require instead of vd ¼ A1 Hd, ed ¼ ðPd  PA1 HÞd stable that vd ¼ A1 HDd , ed ¼ ðPd  PA1 HÞDd be stable, where Dd ¼ diag fd1 ; d2 ; . . .; dM g. In this case, a coprime polynomial matrix fraction description Dd ¼ A1 d Bd can be introduced and one can easily replicate the appropriately modified and nearly identical replacements for Theorems 2.6 and 2.7. Example 2.1 This example offers additional insights and illustrates the application of the theory. An analog plant with transfer matrix Pp ¼

  1 ðs  1Þ ðs  1Þ 1 2 s2 ðs  2Þ

ð2:322Þ

is considered in a 3DOF system with L ¼ F ¼ Pa ¼ Td ¼ I

ð2:323Þ

and Pd ¼ Pp . It is assumed that the feedback transducer, the feedforward transducer, and the physical plant are free of unstable hidden poles. So this 3DOF

72

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

system is admissible provided (ii) of Lemma 2.8 is satisfied. Obviously, L ¼ I is stable. Since F ¼ I, it follows that wF ¼ 1 and wFP ¼ wP . Hence, ðwF wP =wFP Þ ¼ 1 is a stable polynomial. Moreover,  Po ¼

0 Pd

Pa P



 ¼

0 Pd

Pa Pp Pa



 ¼

I 0 0 Pp



0 I

I I

 ð2:324Þ 

Now, the second matrix in the product at the far right-hand side of (2.324) is nonsingular and the characteristic denominator of the first is wPp ¼ wP . It therefore  follows that wPo ¼ wP and ðwPo wP Þ ¼ 1 is a stable polynomial; hence, the system is indeed admissible. Clearly, 1 P ¼ Pa Pp ¼ Pp ¼ A1 p Bp ¼ Bp1 Ap1

ð2:325Þ

Ap ¼ Ap1 ¼ s2 ðs  2ÞI

ð2:326Þ

when

and  Bp ¼ Bp1 ¼

ðs  1Þ ðs  1Þ 1 2

 ð2:327Þ 

Moreover, since det Bp ð0Þ 6¼ 0 and det Bp ð2Þ 6¼ 0, rank ½ Ap Ap ; Bp is left coprime. Similarly, Bp1 ; Ap1 is right coprime. Consistent with Remark 2.3, the reference input r ¼ A1 r br ¼

  1 ðs  1Þ 1 s2 ðs  2Þ

Bp  ¼ 2 for all s; so

ð2:328Þ

should be trackable. Now one cannot choose Ar ¼ s2 ðs  2ÞI because in this case no matter what choice for br is made rank ½ Ar br  ¼ 2 for all s is impossible. However, with the alternative choice  Ar ¼ one gets

1 0

ð1  sÞ s2 ðs  2Þ

 br ¼ ;

  0 1

ð2:329Þ

2.6 Tracking and Disturbance Rejection in the 3DOF System

 rank ½ Ar

br  ¼ rank

1 ð1  sÞ 0 s2 ðs  2Þ

73

 0 ¼2 1

ð2:330Þ

for all s or, equivalently, the pair in (2.329) is left coprime. So according to Theorem 2.4, the reference input (2.328) is indeed trackable when the Ar of (2.329) is used in (2.257) since in this case Ap Td A1 r is stable. In order to complete the determination of the H1 given by (2.257), the matrices A1 and Yp1 have to be found. Since FP ¼ P ¼ Pp in this example, one can take A ¼ A1 ¼ Ap ¼ Ap1 ; B ¼ B1 ¼ Bp ¼ Bp1 :

ð2:331Þ

Hence, A1 1 Ap1 ¼ I. The determination of Yp1 requires also the determination of Xp1 , Xp , and Yp so that (2.241) is satisfied. Actually, it is established next that one can use any solution pair Xp1 ; Yp1 to Xp1 Ap1 þ Yp1 Bp1 ¼ I:

ð2:332Þ

~p ; Y~p such that Since Ap ; Bp is left coprime, there exist matrices X ~p þ Bp Y~p ¼ I: Ap X

ð2:333Þ

Then ~p  Bp1 M; Xp ¼ X

Yp ¼ Y~p þ Ap1 M

ð2:334Þ

gives AXp þ BYp ¼ I

ð2:335Þ

for any polynomial matrix M which can always be chosen so that ~p  Xp1 Y~p  M ¼ 0 Yp1 Xp  Xp1 Yp ¼ Yp1 X

ð2:336Þ

assuring (2.241) is satisfied. So it only remains to find a solution pair Xp1 ; Yp1 for (2.332). This can be accomplished by assuming Xp1 is a constant matrix, setting Yp1 ¼ s2 Yp12 þ sYp11 þ Yp10 , and finding the unknown constant matrices Xp1 , Yp12 , Yp11 , and Yp10 so that (2.332) is satisfied. The results obtained yield  Xp1 ¼

2 2 1 1

 ð2:337Þ

74

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and  Yp 1 ¼

2ðs2  s  1Þ 1 ðs2  s  1Þ 1

 ð2:338Þ 

Hence, from (2.257) 

2ðs2  s  1Þ H1 ¼ ðs2  s  1Þ

1 1



 þ Kr 

1 0

ð1  sÞ s2 ðs  2Þ

 ð2:339Þ :

It is important to note that at s ¼ 0 or at s ¼ 2 no restriction is placed on the first column of the product Kr Ar when Ar is given by (2.329). On the other hand, if instead of tracking (2.328) it was required that r¼

  1 b1 ¼ 2 s ðs  2Þ b2

A1 r br

ð2:340Þ

be tracked for all b1 ; b2 then in accordance with Remark 2.13 one would have to use Ar ¼ s2 ðs  2ÞI instead. In this case, Kr Ar ¼ 0 at s ¼ 0 or at s ¼ 2 for any stable Kr . So by requiring more inputs of the form (2.340) to be tracked instead of the single one given by (2.328), design freedom is sacrificed. It is important for this reason to limit the class of inputs to be tracked to include only those truly expected. For example, when r ¼ A1 r br ¼ br =hðsÞ and hðsÞ is a strict Hurwitz polynomial, then the choice Ar ¼ hðsÞI places no restriction on the product Kr Ar because Kr ¼  ~ r any stable real rational matrix. ~ r h can be chosen with K K Attention is now turned to the issue of disturbance rejection. In accordance with Theorem 2.6 and Remark 2.11, since Ap Pd ¼ Ap Pp ¼ Bp is stable, any input d such that Ap Pd d ¼ Bp d ¼ do

ð2:341Þ

is a stable vector is rejectable. In particular, 1 d ¼ B1 p do ¼ Bp ½ a

b0 ¼

 1 2 ðs  1Þ 1

ð1  sÞ ðs  1Þ

  a ; b

ð2:342Þ

where a; b are polynomials is rejectable. Moreover, when að1Þ 6¼ 0 is assumed, it follows that  rank ½ Bp

do  ¼ rank

ðs  1Þ ðs  1Þ 1 2

 a ¼2 b

ð2:343Þ

2.6 Tracking and Disturbance Rejection in the 3DOF System

75

for all s. Hence, with Ad ¼ Bp and bd ¼ ½a b0 , d ¼ A1 d bd is a left coprime polynomial matrix fraction description for d. Then 1 LA1 d ¼ Bp

ð2:344Þ

and one can choose in accordance with (2.291) and (2.292) Ag ¼ Bp ; Bg ¼ I

ð2:345Þ

Xg ¼ 0; Yg ¼ I:

ð2:346Þ

and

It is of interest to note from (2.290) and (2.336) that Cd ¼ ðXp1 Yp  Yp1 Xp Þ ¼ 0

ð2:347Þ

since on account of (2.331) one can take X1 ¼ Xp1 , Y1 ¼ Yp1 . So Cd is stable as expected. Finally, (2.297) yields ½ K1

H2  ¼ ½ K1

Kg Ag  ¼ ½ K1

Kg Bp 

ð2:348Þ

and, since det X1 ¼ det Xp1  0 for the Xp1 given by (2.337), it remains to show that at least one stable K1 can be chosen so that det ðX1  K1 BÞ ¼ det ðXp1  K1 Bp Þ 6 0:

ð2:349Þ

Indeed, the choice K1 ¼ I yields  det ðX1  K1 BÞ ¼ det ðXp1 þ Bp Þ ¼ det

ðs  3Þ 2

 ðs  3Þ ¼ ðs  3Þ 6 0: 3 ð2:350Þ

2.7

Parameterization and Realization of Proper Stabilizing Controllers

Two approaches have been taken in the literature with regard to the parameterization of stabilizing controllers. The one adopted in the earlier sections of this chapter relied on the use of coprime polynomial matrix fraction descriptions as in Youla et al. (1976b). The other approach introduced in Desoer et al. (1980) makes use of proper stable coprime rational matrix fraction descriptions instead, motivated

76

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

by practical considerations which often dictate that the controller transfer matrix be proper. However, to insist on proper stabilizing controllers at the outset for analog systems excludes the possibility of using proportional–integral–derivative (PID) controllers which is common in practice when the exogenous input signal spectra permit. (In the February 2006 issue, vol. 26, no. 1, of the IEEE Control System Magazine the extensive use of PID controllers in industrial applications is reported on.) Moreover, when H2 designs are being considered, one should allow the design methodology to dictate whether or not a proper controller is needed or is best. Also, the use of proper stable coprime rational matrix fractions involves selecting an arbitrary set of stable poles for the proper rational matrices used and these poles must all cancel out in the end anyway. In addition, the use of k transforms for digital systems removes the need to impose properness on the digital controller transfer matrix: the issue of causality is addressed instead with the requirement that det (X1  K1 BÞ 6¼ 0 at k ¼ 0. For these reasons, the temptation to follow the preference of others (Vidyasagar 1985; Gündes and Desoer 1990; and Zhou et al. 1996) to use proper stable coprime real rational matrix fractions was rejected here. Moreover, in the analog case even when it may be desirable to parameterize all proper stabilizing controllers, this can be done in terms of coprime polynomial matrix fraction descriptions of the plant as shown in the sequel. Attention is focused on the standard configuration of Fig. 1.2 for generality and on the case in which G22 is strictly proper for simplicity. It is of course assumed that the generalized plant is admissible and that the controller is free of unstable hidden poles. The objective is the parameterization of the stable matrix K in (2.63) so that the controller transfer matrix C belongs to the subset of all stabilizing proper controllers. In this regard, it is easier to work with R ¼ CðI  G22 CÞ1 ¼ ðI  CG22 Þ1 C

ð2:351Þ

instead of directly with C. Clearly, R is proper when C is proper since R ! C when G22 C ! 0. Conversely, from C ¼ ðI þ RG22 Þ1 R ¼ RðI þ G22 RÞ1 ;

ð2:352Þ

C is proper when R is proper since C ! R when G22 R ! 0. That is, when G22 is strictly proper C is proper iff R is proper; hence, parameterizing all proper C is equivalent to parameterizing all proper R. It is noted before proceeding that the requirement that det ðX1  KBÞ 6 0 assures that the inverses in (2.352) exist. This fact follows from I þ RG22 ¼ I þ ðI  CG22 Þ1 CG22 ¼ ðI  CG22 Þ1 ¼ A1 ðX1  KBÞ

ð2:353Þ

2.7 Parameterization and Realization of Proper Stabilizing Controllers

77

which yields det ðI þ G22 RÞ ¼ det ðI þ RG22 Þ ¼ ðdet A1 Þdet ðX1  BKÞ 6 0:

ð2:354Þ

For the C given by (2.63) one gets R ¼ ðY þ A1 KÞA ¼ A1 ðY1 þ KAÞ:

ð2:355Þ

It is convenient at this point to introduce two stable real rational matrices K and X which have the property that K1 and X1 are stable matrices and, moreover, A1 K1 and X1 A together with their inverses are proper matrices. [The structure of the matrix R in (45) of Park and Bongiorno (1989) motivated the introduction of the matrices K and X.] There are a number of ways in which to construct such matrices. When the polynomial matrix A is row reduced and the polynomial matrix A1 is column reduced (see Kailath 1980), diagonal matrices K and X can be chosen with stable polynomials for the diagonal elements. Then, the matrix A1 K1 and its inverse are proper matrices when the degree of the diagonal element in the j-column of K is chosen equal to the column degree of the j-column of A1 for each possible j and the matrix X1 A and its inverse are proper matrices when the degree of the diagonal element in the i-row of X is chosen equal to the row degree of the i-row of A for each possible i. It should also be noted for analog systems that in Antsaklis ~ are acceptable choices for K1 and X1 , respectively. (1989) the matrices P and P Another way to construct the matrices K and X for analog systems which works even if A is not row reduced and A1 is not column reduced is given in Park et al. (1999). A description of this methodology is presented later. Introducing the transformation K ¼ K1 QX1

ð2:356Þ

into the first expression for R in (2.355) yields 1 R ¼ A1 K1 ðKA1 1 YX þ QÞX A:

ð2:357Þ

Since K, X, and each of their inverses are stable matrices, it follows from (2.356) that K is stable iff Q is stable. Another important observation to make is that one can always use the polynomial division algorithm if needed to write 1 1 KA1 1 YX ¼ fKA1 YXg1 þ fKA1 YXgsp ;

ð2:358Þ

where the first term on the right-hand side is a polynomial matrix and the second is a strictly proper rational matrix. One can then make the substitution

78

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Q ¼ Z  fKA1 1 YXg1 ;

ð2:359Þ

clearly, Q is stable iff Z is stable. Hence, for any stabilizing controller C 1 R ¼ A1 K1 ðfKA1 1 YXgsp þ ZÞX A;

ð2:360Þ

where Z is a stable real rational matrix. Moreover, since A1 K1 and X1 A are biproper matrices (i.e., they are proper together with their inverses), R is proper iff Z is proper. Hence, the controllers described by (2.63) are proper stabilizing controllers iff 1 K ¼ K1 ðZ  fKA1 1 YXg1 ÞX ;

ð2:361Þ

where Z is a proper stable real rational matrix such that v ¼ det ðX1  KBÞ 6 0:

ð2:362Þ

It is established next that (2.362) is automatically satisfied and imposes no additional constraints on Z. Substituting (2.361) into (2.362) and using (2.358) one easily gets after (2.65) is recalled that v ¼ vo =det A1 ; where 1 vo ¼ det ½I  A1 K1 ðfKA1 1 YXgsp þ ZÞX B:

ð2:363Þ

Hence, v 6 0 iff vo 6 0. Now from X1 B ¼  X1 AG22 it follows that X1 B is strictly proper since X1 A is proper and G22 is strictly proper. Since vo ! 1 when 1 A1 K1 ðfKA1 1 YXgsp þ ZÞX B ! 0;

ð2:364Þ

it readily follows that (2.362) is automatically satisfied for any proper Z. That is Theorem 2.8 When the generalized plant in the standard configuration is admissible, G22 is strictly proper, and the controller is free of unstable hidden poles, then all proper stabilizing controller transfer matrices are given by C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1

ð2:365Þ

with 1 K ¼ K1 ðZ  fKA1 1 YXg1 ÞX ;

ð2:366Þ

2.7 Parameterization and Realization of Proper Stabilizing Controllers

79

where Z is a proper stable real rational matrix and where the real rational matrices K and X are stable together with their inverses and selected so that A1 K1 and X1 A are biproper. It is always possible to construct the matrices K and X. Remark 2.15 Substituting (2.366) into (2.365) leads to ~ ¼ ðY~ þ A ~ 1 ZÞðX ~1  Z BÞ ~ 1 ðY~1 þ Z AÞ ~ B ~ 1 ZÞ1 C ¼ ðX

ð2:367Þ

where ~ ¼ X1 A; B ~ 1 ¼ A1 K1 ; B ~ ¼ X1 B; A ~ 1 ¼ B1 K1 ; A

ð2:368Þ

1 1 1 ~ ¼ XX þ B1 K1 fKA1 X 1 YXg1 ¼ A X  B1 K fKA1 YXgsp ;

ð2:369Þ

1 1 Y~ ¼ YX  A1 K1 fKA1 1 YXg1 ¼ A1 K fKA1 YXgsp ;

ð2:370Þ

1 1 1 1 ~1 ¼ KX1 þ fKA1 X 1 YXg1 X B ¼ KA1  fKA1 YXgsp X B;

ð2:371Þ

1 1 1 Y~1 ¼ KY1  fKA1 1 YXg1 X A ¼ fKA1 YXgsp X A

ð2:372Þ

and

are all stable proper real rational matrices that satisfy (see Example 2.15) 

~ A ~ Y1

~ B ~1 X



~ X Y~

~1 B ~1 A



 ¼

~ X Y~

~1 B ~1 A



~ A ~ Y1

~ B ~1 X



 ¼

I 0

0 I

 ð2:373Þ 

~ and Y~1 are strictly proper and X ~ and X ~1 are biproper. This ~ B ~ 1 ; Y; Actually, B; corresponds to the parameterization of stabilizing controllers derived using proper stable coprime real rational matrix fractions in the literature cited previously in this section. Remark 2.16 It is easy to see from (2.351) and (2.352) when G22 is proper that R is strictly proper iff C is strictly proper. Hence, Theorem 2.8 holds for proper G22 except that now all strictly proper stabilizing controller transfer matrices are given by (2.365) and (2.366), where Z is a strictly proper stable real rational matrix. Remark 2.17 It should be clear from (2.366) that K is stable iff Z is stable. Hence, (2.367) is an alternative parameterization of all stabilizing controller transfer matrices in terms of a stable real rational matrix Z. When G22 is proper, but not ~ and X ~1 are strictly proper, it still follows from (2.368) through (2.372) that X ~ ~ biproper; now, however, B and B1 are proper, but not strictly proper. So it is no ~ and ~1  Z BÞ longer automatically true for the subset of all proper stable Z that ðX ~ ~ ðX  B1 ZÞ together with their inverses are proper. That is, it is no longer assured without additional restrictions that C is proper when Z is proper or that it is even

80

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

necessary that Z be proper for proper C. However, when one imposes well-posedness of the feedback loop as the proponents of proper stable coprime rational matrix fraction descriptions do (e.g., see Zhou, Doyle, and Glover, Sect. 5.2, 1996) then this ambiguity disappears. Well-posedness of the feedback loop requires that not only is C a proper stabilizing controller transfer matrix, but one for which all closed-loop transfer matrices are proper including ðI  G22 CÞ1 . ~1 C  One then gets from the first expression for C in (2.367) that Z ¼ ðX 1 1 ~ 1 ~ ~ ~ ~ ~ Y1 ÞðA þ BCÞ ¼ ðX1 C  Y1 ÞðI  G22 CÞ A must be proper. Moreover, from ~ 1 þ ðY~1 þ Z AÞ ~ B ~ 1 þ CB ~ A ~ 1 ¼ I, it follows that ðX ~1  Z BÞ ~ 1 ¼ A ~ 1 must be ~1  Z BÞ ðX ~1 is biproper and proper when C is proper. So, from the fact that X 1 ~ ~ ~ ~ ~ det ðX1  Z BÞ ¼ ðdet X1 Þdet ðI  Z BX1 Þ, the first expression for C in (2.367) is ~ ~11 ð1Þ 6¼ 0. That is, when G22 is proper, the first proper iff det ½I  Zð1ÞBð1Þ X expression for C in (2.367) parameterizes all proper stabilizing controller transfer matrices for which the feedback loop is well-posed in terms of a proper stable real ~ ~11 ð1Þ 6¼ 0. In similar fashion, rational matrix Z for which det ½I  Zð1ÞBð1Þ X one can establish when G22 is proper that the second expression for C in (2.367) parameterizes all proper stabilizing controllers for which the feedback loop is well-posed in terms of a proper stable real rational matrix Z for which ~ 1 ð1ÞB ~ 1 ð1ÞZð1Þ 6¼ 0. det ½I  X The procedure given in Park et al. (1999) for constructing the matrices K and X for analog systems is especially relevant in connection with the Wiener–Hopf methodology; hence, the essential ideas are presented here. Basically, K and X are chosen so that A1 A1 þ B1 B1 ¼ K K

ð2:374Þ

AA þ BB ¼ XX

ð2:375Þ

and

where K is a right Wiener–Hopf factor and X is a left Wiener–Hopf factor (Appendix C). The matrices K and X are polynomial matrices because the left-hand sides of the equations are polynomial matrices (see Corollary 2 of Theorem 2 in Youla 1961). Hence, K and X are stable. Moreover, K1 and X1 are analytic in Re s [ 0. In addition, since A1 ; B1 is right coprime and A; B is left coprime, ½ A01 B01 0 has full column rank and ½ A B has full row rank at any point on the finite s ¼ jx axis. It then follows that on the finite imaginary axis 

a ½ A1



 A1 B1  a ¼ a K Ka ¼ kKak2 6¼ 0 B1

ð2:376Þ

2.7 Parameterization and Realization of Proper Stabilizing Controllers

81

and b ½ A

 B

 A b ¼ b XX b ¼ kX bk2 6¼ 0 B

ð2:377Þ

for all nonzero constant vectors a; b. That is, det K 6¼ 0 and det X 6¼ 0 on s ¼ jx, x finite; therefore, K1 and X1 are stable matrices. One also has from (2.374) that 1 I þ G22 G22 ¼ A1 1 K KA1

ð2:378Þ

1 1 þ K1 ¼ I: K1  A1 A1 K  B1 B1 K

ð2:379Þ

and

Recalling that G22 is strictly proper, setting s ¼ jx, and taking the trace of each of these equations leads to the fact that each element of KðjxÞA1 1 ðjxÞ and each 1 element of A1 ðjxÞK ðjxÞ must have a finite magnitude as x ! 1. Hence, KA1 1 and A1 K1 are proper. A similar development beginning with (2.346) would confirm that X1 A and A1 X are proper. An interesting question concerning the controller transfer matrix given by (2.367) is whether or not it can be realized as an interconnection with one separate block for Z. Then all other blocks in the realization would be fixed by the given data and any later change in the choice of the design parameter Z would require only a change in the one block realizing it. Specifically, a tunable realization of the controller in the form shown in Fig. 2.2 is desired. Straightforward analysis yields v ¼ Cym ; where C ¼ Gvy þ Gvz ZðI  Gzz ZÞ1 Gzy :

Fig. 2.2 Tunable controller realization

ð2:380Þ

v

ym zo

GC

G vy G zy

Z

Gvz Gzz

zi

82

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

~ 1 is square, it follows from (2.367) that Z has the same number of rows as C; Since A hence, Gvz is square. Assuming it is also invertible and using the identity ZðI  Gzz ZÞ1 ¼ ðI  ZGzz Þ1 Z then gives 1 1 C ¼ Gvy þ ðG1 vz  ZGzz Gvz Þ ZGzy

ð2:381Þ

or 1 1 1‘ 1 C ¼ ðG1 vz  ZGzz Gvz Þ ½Gvz Gvy þ ZðGzy  Gzz Gvz Gvy Þ:

ð2:382Þ

Equation (2.382) has the same form as the first expression for C in (2.367); it is now clear that one can choose ~11 ; Gvz ¼ X

~X ~11 ; Gzz ¼ B

~11 Y~1 ; Gvy ¼ X

ð2:383Þ

and ~þB ~X ~11 Y~1 : Gzy ¼ A

ð2:384Þ

~1 is biproper. It is The assumption that Gvz is invertible is indeed sustained since X also important to recognize that the controller subsystem transfer matrix  GC ¼

Gvy Gzy

Gvz Gzz



 ¼

I ~ B

0 I



~11 X 0

0 I



Y~1 ~ A

I 0

 ð2:385Þ

is proper and, therefore has a state-variable realization. Moreover, for any unstable ~11 Þ. On the other hand, pole po , it follows from (2.385) that dðpo ; GC Þ  dðpo ; X 1 ~1 is a submatrix of GC , one also has dðpo ; GC Þ  dðpo ; X ~11 Þ. Thus, since Gvz ¼ X ~11 Þ. From (2.371), dðpo ; GC Þ ¼ dðpo ; X ~1 ¼ K L1 X o ðLo X1  Mo BÞ;

ð2:386Þ

where Lo ; Mo is a left coprime polynomial matrix pair satisfying 1 1 1 Ko ¼ L1 o Mo ¼ K fK A1 YXg1 X :

ð2:387Þ

Now Ko is a stable matrix and Lo is a polynomial matrix; hence, Lo and L1 o are stable. Since the same is true of K, it follows that

2.7 Parameterization and Realization of Proper Stabilizing Controllers

83

dðpo ; GC Þ ¼ dðpo ; ðLo X1  Mo BÞ1 Lo K1 Þ ¼ dðpo ; ðLo X1  Mo BÞ1 Þ ¼ mo ; ð2:388Þ where mo is the multiplicity of po as a zero of wo ¼ go det ðLo X1  Mo BÞ;

ð2:389Þ

the characteristic denominator of ðLo X1  Mo BÞ1 . These facts are used in the sequel to establish the conditions under which the tunable controller realization will have no unstable hidden poles. In order to address the issue of unstable hidden poles, the characteristic polynomial for the tunable controller realization is needed. Clearly, the controller realization shown in Fig. 2.2 has the same form as the standard configuration shown in Fig. 1.2. It then follows by analogy with (2.24) that the characteristic polynomial for the controller is given by DC ¼ DG DZ det ðI  ZGzz Þ ¼

~1  ZX1 BÞ DG DZ det ðX ~1 det X

ð2:390Þ

or DC ¼

~1  ZX1 BÞ go DG DZ ðdet Lo Þdet ðX ; wo det K

ð2:391Þ

where DG and DZ are the characteristic polynomials for the subsystems GC and Z, respectively. Recalling (2.361) yields ~1  ZX1 B ¼ K ðX1  KBÞ ¼ K L1 ðLX1  MBÞ X

ð2:392Þ

when K ¼ L1 M is a left coprime polynomial matrix fraction description for K. It also follows from (2.365) that C ¼ ðLX1  MBÞ1 ðLY1 þ MAÞ:

ð2:393Þ

It has already been established that (2.393) is a left coprime polynomial matrix fraction description for C: see (2.51) through (2.57). Hence, wC ¼ gC det ðLX1  MBÞ

ð2:394Þ

84

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

is the characteristic denominator for the tunable controller realization. Using (2.392) and (2.394) in (2.391) gives  DC ¼

go gC

   det Lo DG DZ wC : det L wo

ð2:395Þ

Now go and gC are simply the scaling constants that make wo and wC monic polynomials. The polynomials det Lo and det L are stable. So there are no unstable hidden poles iff there are no zeros of DC =wC in the unstable region, Re s  0 for analog systems or jkj  1 for digital systems. Clearly, this is the case iff DG DZ hG hZ wG wZ ¼ wo wo

ð2:396Þ

has no zeros in the unstable region. The polynomials hG and hZ account for the hidden poles and wG and wZ are the characteristic denominators of GC and Z, respectively. On account of (2.388) and (2.389), it follows that every zero of wG in the unstable region is canceled exactly, multiplicity included, by a corresponding zero of wo . Moreover, wZ is stable because Z is stable. Hence, there are no unstable hidden poles iff hG and hZ are stable polynomials. That is, there are no unstable hidden poles iff the block GC and the block Z are realized without unstable hidden poles. This is of course assured when minimal state-variable realizations of GC and Z are employed.

2.8

Historical Perspective and Commentary

The use of polynomial matrix descriptions and of coprime polynomial matrix fractions in modeling linear systems was popularized by Rosenbrock (1970). Early advances in the characterization and parameterization of stabilizing multivariable controllers were made in Larin et al. (1971, 1972), Kučera (1974), and Youla et al. (1976b) with the parameterization of all stabilizing controllers in the latter attracting particular notoriety. Specific results for 2DOF and 3DOF systems are given in Youla and Bongiorno (1985) and in Park and Bongiorno (1990), respectively; a parameterization of all stabilizing controllers for the standard configuration is employed in Park and Bongiorno (1989). A comprehensive review and comparison of these parameterizations of all stabilizing controllers is given in Aliev and Larin (2008) including ones cited next which involve real rational matrix fractions. The alternative parameterization of stabilizing controllers in terms of stable coprime real rational matrix fractions was initiated for 1DOF systems in Desoer et al. (1980) and applied to 2DOF systems in Desoer and Gustafson (1984) and in

2.8 Historical Perspective and Commentary

85

Vidyasagar (1985). This approach was followed by Nett (1986) to treat stability for the most general of configurations. In all of these works, stability is considered from an input–output viewpoint. That is, instead of defining stability in terms of the response to arbitrary initial conditions as done here, stability is defined in terms of all possible closed-loop transfer matrices with auxiliary inputs injected. Such an approach was not followed in this chapter since more fundamentally stability is connected to system response to initially stored energy or, equivalently, initial conditions. It then follows for stabilized systems that all closed-loop transfer matrices of interest are stable—the definition of stability in these papers. That is, stability with respect to initial conditions assures stable closed-loop transfer matrices, but the converse is not automatically true on account of the possibility of unstable hidden poles. It is appropriate to point out however, that Lemma 2.3 corresponds to Lemma 31 of Nett (1986) where the complete set of conditions of this kind first appeared. Of the early papers on tracking and disturbance rejection (or regulation), the one by Saeks and Murray (1981) is closest conceptually with the material presented in Sect. 2.6. Here, however, stable controller output transforms in addition to stable error transforms are required on the one hand while on the other hand no restriction that the controller be proper is imposed. In this way, the parameterization of the subset of stabilizing controllers for which tracking and disturbance rejection are achieved is made more pertinent and more transparent. The Wiener–Hopf methodology in which an H2 performance measure that includes penalties for large plant inputs and system errors then determines whether or not an analog controller should be proper. The question of causality in digital systems is not linked to proper controllers when k-transforms are used and so there is no need to impose such a condition in this case. The structure of the tunable controller realization for proper stabilizing controllers first appeared in Doyle et al. (1989).

2.9

Additional Examples

Example 2.2 The generalized plant in the standard configuration of Fig. 1.2 can typically be viewed as an interconnection of nondegenerate dynamic subsystems as depicted in Fig. 2.3. Each subsystem Gq , q ¼ 1 ! N, has the polynomial matrix description Tq ðpÞnq ðpÞ ¼ Uq ðpÞuq ðpÞ þ iTq ðpÞ yq ðpÞ ¼ Vq ðpÞnq ðpÞ þ Wq ðpÞuq ðpÞ þ iyq ðpÞ

86

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

e

Nondynamic Connection Network

v

Fig. 2.3 Generalized plant configuration for Example 2.2 (© 1989 IEEE. Reprinted with permission, from Park and Bongiorno (1989))

and the associated transfer matrix Gq ðpÞ ¼ Vq ðpÞTq1 ðpÞUq ðpÞ þ Wq ðpÞ: The dynamical subsystems can be compactly described in terms of the block diagonal matrix G ¼ diag fG1 ; G2 ; . . .; GN g and the vectors n ¼ ½n01 n02    n0N 0 ; u ¼ ½u01 u02    u0N 0 ; and y ¼ ½y01 y02    y0N 0 : The connection network is characterized by a constant matrix M according to 2

3 2 M11 u 4 yc 5 ¼ 4 M21 ym M31

M12 M22 M32

32 3 2 3 M13 e e M23 5 4 v 5 ¼ M 4 v 5: y y M33

It is also convenient to introduce T ¼ diag fT1 ; T2 ; . . .; TN g; U ¼ diag fU1 ; U2 ; . . .; UN g; V ¼ diag fV1 ; V2 ; . . .; VN g; W ¼ diag fW1 ; W2 ; . . .; WN g; iT ¼ ½i0T1 i0T2    i0TN 0 and iy ¼ ½i0Y1 i0Y2    i0YN 0 :

2.9 Additional Examples

87

(a) Establish that the characteristic polynomial for the generalized plant is given by DG ¼ g ðdet TÞ det ðI  GM13 Þ; where g is a constant chosen so that DG is monic. ~ denote the transfer matrix from ½e0 v0 0 to ½y0 y0 0 in Fig. 2.3. That is, G ~ is (b) Let G c m the transfer matrix of the generalized plant. Show that the appropriate expression to use for G22 in (2.24) is ~ 22 ¼ M32 þ M33 ðI  GM13 Þ1 GM12 : G (c) It is important to recognize that the formula for the characteristic polynomial DG applies to any set of interconnected dynamical systems and the matrix M13 is determined by the feedback from the vector of subsystem outputs y to the vector of subsystem inputs u through the connection network. For example, the standard configuration in Fig. 1.2 is a special case of an interconnection with ~ G2 ¼ C, u1 ¼ ½e0 v0 0 , y1 ¼ ½y0 y0 0 , u2 ¼ ½0 Iy1 ¼ ym , y2 ¼ v. N ¼ 2, G1 ¼ G, c m Then from

show that one gets (2.24). Solution (a) Clearly, Tn ¼ Uu þ iT y ¼ V þ Wu þ iy : Using u ¼ ½ M11

  e M12  þ M13 y v

then gives   e Tn ¼ U½ M11 M12  þ UM13 y þ iT v   e y ¼ Vn þ W½ M11 M12  þ WM13 y þ iy v

88

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

or 

UM13 ðI  WM13 Þ

T V

    n U ¼ ½ M11 y W

    i e M12  þ T : iy v

In addition one has 

yc ym



 ¼

    M21 M23 n ½0 I  þ M33 M31 y

M22 M32

  e : v

This pair of equations is of the form TG 

  n

yc ym

y 

¼ UG

¼ VG

  e

v   n y

 þ

þ WG

 iT iy   e v

and represents a polynomial matrix description for the generalized plant. Hence,    T UM13 I DG ¼ g det TG ¼ g det ¼ g det V ðI  WM13 Þ VT 1   T UM13 ¼ g ðdet TÞ det (I  GM13 Þ ¼ g det 0 ðI  GM13 Þ

0



I

T

UM13

V

ðI  WM13 Þ

Note that the generalized plant is nondegenerate iff det (I  GM13 Þ 6 0 (b) One has with iT ¼ 0 and iy ¼ 0 that y ¼ Gu ¼ G ½ M11

M12 

  e þ GM13 y: v

It is assumed that the generalized plant is nondegenerate so that   e 1 : y ¼ ðI  GM13 Þ G ½ M11 M12  v From 

yc ym



      e M23 n M21 M22 ¼ ½0 I  þ M M M y 33 31 32       v M23 M21 M22 e ~ e ; ¼ yþ ¼G M33 M31 M32 v v 

 :

2.9 Additional Examples

89

it then follows that  ~ ¼ M21 G M31 Thus, ~ 22 ¼ ½ 0 G

M22 M32



 þ

 M23 ðI  GM13 Þ1 G ½ M11 M33

M12 :

  0 ~ I G ¼ M32 þ M33 ðI  GM13 Þ1 GM12 : I

Now in (2.24), G22 is the matrix in the second block row, second block column ~ 22 . of the generalized plant transfer matrix which in this example is given by G (c) For the standard configuration, it is evident that

Hence, 2

I

0

0

I C

6 ðdet TÞ det (I  GM13 Þ ¼ ðdet TG~ Þ ðdet TC Þ det 4 0 " ¼ ðdet TG~ Þ ðdet TC Þ det " ¼ ðdet TG~ Þ ðdet TC Þ det

~ 22 G

I C I 0

3 ~ 12 G ~ 22 7 5 G I #

I

~ 22 G ~ 22 Þ ðI  CG

# ~ 22 Þ; ¼ ðdet TG~ Þ ðdet TC Þ det ðI  C G

which corresponds to (2.24). AX þ BY ¼ I, and Example 2.3 Suppose G22 ¼ A1 B ¼ B1 A1 1 , ~ ~ X1 A1 þ Y1 B1 ¼ I. In accordance with Theorem 2.1 for the standard configuration, ~ 1 ðY~1 þ KAÞ; ~ ~1  KBÞ any stabilizing controller transfer matrix is given by C ¼ ðX ~ where K is a stable real rational matrix. Suppose also with M 6 0 that 

A Y~1

B ~1 X



X Y

B1 A1





I ¼ M

0 I



90

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and so a relationship in the form of the generalized Bezout identity (2.62) does not hold. (a) Show nevertheless that as in Lemma 2.2 it is still true that C ¼ ðY þ A1 KÞðX  B1 KÞ1 ; where K is a stable real rational matrix. (b) Show that the ~1 ; Y~1 is not unique and that there always exists another choice for which a pair X relationship in the form of (2.62) does hold. Solution (a) Setting ~1  KBÞ ~ 1 ðY~1 þ KAÞ ~ ¼ ðY þ A1 KÞðX  B1 KÞ1 C ¼ ðX leads to ~ ~1  KBÞðY ~  B1 KÞ ¼ ðX þ A1 KÞ ðY~1 þ KAÞðX ~ ~ ~ ~ ~ ~1 A1 K  KBY ~ ~ 1K Y1 X  Y1 B1 K þ KAX  KAB1 K ¼ X1 Y þ X  KBA ~  M: ~ X ~1 Y þ Y~1 X ¼ K K¼K ~ is a So K is a stable real rational matrix since M is a real polynomial matrix and K stable real rational matrix. (b) Clearly, ~ ~ ~ ~1  MBÞA ðX 1 þ ðY1 þ MAÞB1 ¼ I: ~1 ; Y~1 can be replaced with ðX ~1  MBÞ; ~ ~ ~ is any Hence, X ðY~1 þ MAÞ; where M polynomial matrix. Then as in the generalized Bezout identity (2.62), one gets 

A ~ ðY~1 þ MAÞ

B ~1  MBÞ ~ ðX



X Y

B1 A1



 ¼

I ~  MÞ ðM

  0 I ¼ I 0

0 I



~ ¼ M is chosen. when M Example 2.4 For the two coprime polynomial matrix fraction descriptions of G22 given by (2.107) and (2.108) it follows from (2.114) that ~1 ¼ N11 ðX1  MBÞ, Y~1 ¼ N11 ðY1 þ MAÞ; where M is an arbitrary real polynomial X matrix. (a) In similar fashion, establish when AX þ BY ¼ I that ~ ¼ ðX  B1 M1 ÞN 1 , Y~ ¼ ðY þ A1 M1 ÞN 1 ; where M1 is an arbitrary real polyX nomial matrix. In addition, show when the generalized Bezout identity (2.62) is satisfied and M1 ¼ M is chosen that 

~ A Y~1

~ B ~1 X



~ X Y~

~1 B ~1 A



 ¼

~ X Y~

~1 B ~1 A



~ A Y~1

~ B ~1 X



 ¼

I 0

 0 : I

2.9 Additional Examples

91

(b) It follows from (a) and from Lemma 2.2 by analogy that ~ ¼ ðY~ þ A ~ 1 KÞð ~ ¼ ðX ~ X ~ B ~ 1 KÞ ~ 1 ~1  K ~ BÞ ~ 1 ðY~1 þ K ~ AÞ C ~ as K ~ ranges over all stable real when M1 ¼ M. Confirm that the set of all C rational matrices is identical to the set of all C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1 as K varies over all stable real rational matrices. Solution ~X ~ þB ~ Y~ ¼ NðAX ~ þ BYÞ ~ ¼ I ¼ NN 1 gives AXN ~ þ BYN ~ ¼ I ¼ AX þ BY or (a) A ~  YÞ ¼ ~ ~ ~ AðXN  XÞ þ BðYN  YÞ ¼ 0. Thus, ðXN  XÞ ¼ A1 BðYN 1 ~ B1 A1 ðYN  YÞ and so 1 ~ ~ A1 1 ðYN  YÞ ¼ ðX1 A1 þ Y1 B1 ÞA1 ðYN  YÞ ~  YÞ þ Y1 B1 A1 ~ ¼ X1 ðYN 1 ðYN  YÞ

~  YÞ  Y1 ðXN ~  XÞ ¼ M1 ¼ X1 ðYN

~  Y ¼ A1 M1 , XN ~  X ¼ B1 M1 or a polynomial matrix. That is, YN 1 1 ~ ¼ ðX  B1 M1 ÞN . It is now easy to confirm that Y~ ¼ ðY þ A1 M1 ÞN , X ~X ~ þB ~ Y~ ¼ I. Hence, when the generalized Bezout identity (2.62) is satisfied A and M1 ¼ M is chosen, ~ X ~1 Y~ ¼ N11 ðY1 þ MAÞðX  B1 M1 ÞN 1  N11 ðX1  MBÞðY þ A1 M1 ÞN 1 Y~1 X ¼ N11 ðY1 X  Y1 B1 M1 þ MAX  MAB1 M1  X1 Y  X1 A1 M1 þ MBY þ MBA1 M1 ÞN 1 ¼ N11 ðY1 X  X1 Y  M1 þ MÞN 1 ¼ 0

and it follows that   ~ ~ ~ X A B ~1 Y~ Y~1 X

~1 B ~1 A





~ X ¼ ~ Y

~1 B ~1 A



~ A Y~1

~ B ~1 X





I ¼ 0

 0 : I

(b) One only needs to show the following since (2.115) and (2.117) are already established. By straightforward substitution ~ 1 KÞð ~ ¼ ðY~ þ A ~ X ~ B ~ 1 KÞ ~ 1 ¼ ðY þ A1 M1 þ A1 N1 KNÞðX ~ ~ 1 C  B1 M1  B1 N1 KNÞ ~ ¼ M þ N1 KN ~ ¼ ðY þ A1 KÞðX  B1 KÞ1 where K ¼ M1 þ N1 KN as in (2.118). Since M1 ¼ M is polynomial and N and N1 are unimodular, there ~ So is a one-to-one correspondence between every stable K and every stable K. ~ and the set of all C are identical. the set of all C

92

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Example 2.5 For the 2DOF standard configuration shown in Fig. 1.3, establish Definition 2.3 and Theorem 2.2 directly from the polynomial matrix description of the system instead of using the results for the standard configuration as was done in 1 BCu ; Sect. 2.4. Hint: One can always write C ¼ ½ C1 C2  ¼ A1 C BC ¼ AC ½ BCy where AC ; BC is a left coprime polynomial matrix pair. Then first establish that C ¼ ðX1  KBÞ1 ½ ðY1 þ KAÞ H is necessary, where K and H are stable real rational matrices. Solution Equations (2.122) through (2.124) describe the generalized plant and are repeated here for convenience: TG nG ¼ U1 d þ U2 v þ iTG ½ y0s y0t 0 ¼ V1 nG þ W11 d þ W12 v þ ic y ¼ V2 nG þ W21 d þ W22 v þ iy : Next a polynomial matrix description for the controller is introduced: TC nC ¼ UC1 y þ UC2 u þ iTC v ¼ VC nC þ WC1 y þ WC2 u þ iv : Thus, TC nC ¼ UC1 V2 nG þ UC1 W21 d þ UC1 W22 v þ UC1 iy þ UC2 u þ iTC v ¼ VC nC þ WC1 V2 nG þ WC1 W21 d þ WC1 W22 v þ WC1 iy þ WC2 u þ iv and one gets 2

32 3 0 U2 nG TG 4 UC1 V2 TC UC1 W22 5 4 nC 5 W2C1 V2 VC ðI 3 WC1 W22 v 2Þ 3   i TG 0 U1 d þ 4 UC1 iy þ iTC 5: ¼ 4 UC1 W21 UC2 5 u WC1 iy þ iv WC1 W21 WC2 Hence, the system is stable iff T 1 is stable, where 2

TG T ¼ 4 UC1 V2 WC1 V2

0 TC VC

3 U2 UC1 W22 5: ðI  WC1 W22 Þ

2.9 Additional Examples

93

Now 2

TG 6 det T ¼ det 4 UC1 V2 WC1 V2 2 TG 6 ¼ det 4 UC1 V2 WC1 V2 2 TG 6 ¼ det 4 UC1 V2 WC1 V2

0 TC VC 0 TC VC 0 TC VC

0

3

7 UC1 ðV2 TG1 U2 þ W22 Þ 5 1 ðI  WC1 ðV2 TG U2 þ W22 ÞÞ 3 0 7 UC1 G22 5 ðI  WC1 G22 Þ 3 0 7 0 5 ¼ ðdet TG Þðdet TC Þ det ðI  C1 G22 Þ 1 ðI  ðVC TC UC1 þ WC1 Þ G22 Þ

since in accordance with (1.4) and (1.5) C ¼ ½ C1

C2  ¼ ½ ðVC TC1 UC1 þ WC1 Þ

ðVC TC1 UC2 þ WC2 Þ ¼ ½ Cy

Cu :

Writing 1 C ¼ A1 C BC ¼ AC ½ BCy

1 BCu ; G22 ¼ B1 A1 1 ¼ A B;

where AC ; BC and A; B are left coprime pairs and B1 ; A1 is a right coprime pair, one then gets    det TG det TC det T ¼ det (AC A1 þ BCy B1 Þ det A1 det AC or    DG DC D ¼ gdet T ¼ g det (AC A1 þ BCy B1 Þ w wC   G22 wG ¼ ghG hC det ðAC A1 þ BCy B1 Þ: wG22  Since G22 is a submatrix of G, wG wG22 is a polynomial. Hence, a necessary condition for the existence of a stabilizing controller is that hG ¼ DG =wG and  wG wG22 be stable polynomials or, according to Definition 2.3, that the generalized plant be admissible. Given an admissible plant, one must then pick C so that hC is stable (i.e., the controller has no unstable hidden poles) and AC A1 þ BCy B1 ¼ L, where L is a real polynomial matrix whose determinant is stable. Employing the same methodology that yielded (2.48) and (2.49) one then gets that AC and BCy are given by AC ¼ LX1  MB, BCy ¼ LY1 þ MA. It then follows that

94

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

C ¼ ðLX1  MBÞ1 ½ ðLY1 þ MAÞ BCu  ¼ ðX1  KBÞ1 ½ ðY1 þ KAÞ L1 BCu  ¼ ðX1  KBÞ1 ½ ðY1 þ KAÞ

H ;

where K ¼ L1 M and H ¼ L1 BCu are stable matrices. It remains to show that any such C is a stabilizing controller. First, given a stable K one can always write K ¼ L1 M; where L; M is a left coprime polynomial matrix pair and det L is stable. Then C ¼ ðX1  L1 MBÞ1 ½ ðY1 þ L1 MAÞ ¼ ½ C1 C2 :

1 H ¼ ðLX1  MBÞ ½ ðLY1 þ MAÞ LH

Using the same argument as with (2.51) through (2.56) yields LX1  MB, LY1 þ MA is a left coprime polynomial pair. Hence, the characteristic denominator for C1 is wC1 ¼ gC1 det ðLX1  MBÞ: It now follows from det T ¼ ðdet TG Þðdet TC Þ det ðI  C1 G22 Þ that 

  DG DC D ¼ g det T ¼ g det ½ðLX1  MBÞA1 þ ðLY1 þ MAÞB1  wG22 wC1   wG wC ¼ ghG hC det L: wG22 w C1  When hG , hC , and wG wG22 are stable polynomials, it follows that D is a stable  polynomial provided wC wC1 is a stable polynomial. This is indeed the case for the following reasons. When po is an unstable finite pole of C, one has that dðC; po Þ  dððLX1  MBÞ1 ; po Þ because ½ ðLY1 þ MAÞ LH  is a stable matrix. On the other hand, dðC; po Þ  dððLX1  MBÞ1 ; po Þ since C1 ¼ ðLX1  MBÞ1 ðLY1 þ MAÞ is a submatrix of C. Thus, dðC; po Þ ¼ dððLX1  MBÞ1 ; po Þ ¼ mo , where mo is the multiplicity of po as a zero of det ðLX1  MBÞ or, equivalently, wC1 . But mo is also the multiplicity of po as a zero of wC . So every such  unstable zero of wC is canceled exactly by the corresponding zero of wC1 and wC wC1 is a stable polynomial: that wC1 divides wC without remainder is a consequence of the fact that wC1 is the characteristic denominator of a submatrix of C.

2.9 Additional Examples

95

Example 2.6 Suppose AFPd , L, Pa , and PA1 are stable. Suppose in addition that F is square and F and F 1 are stable. Use Lemma 2.7 for the 3DOF system to establish that wG =wFP is a stable polynomial. Solution Obviously, all one has to show is that the conditions of Lemma 2.7 are satisfied. Clearly, this is the case if Pa A1 , Pa A1 Y1 FPd , and ðI  PA1 Y1 FÞ Pd are stable. That Pa A1 is stable follows immediately from Pa stable and A1 polynomial. It is also apparent that Pa A1 Y1 FPd ¼ Pa YAFPd is stable. Finally, on account of the assumed properties of F, one has that ðI  PA1 Y1 FÞ Pd is stable because FðI  PA1 Y1 FÞ Pd ¼ ðI  FPA1 Y1 ÞF Pd ¼ ðI  B1 Y1 ÞFPd ¼ XAFPd is stable. Example 2.7 Suppose for (2.216) in the case of the 3DOF system one has  Po ¼

0 Pd

Pa P



 ¼

0 Pp Pb

 Pa ; Pp Pa

where Pa and Pb are square stable matrices with stable inverses. Show that dðP o ; po Þ ¼ dðP ; po Þ for any finite unstable pole po of Po and, therefore, that wPo wP is a stable polynomial. Hence, this condition in Lemma 2.8 is always satisfied in these cases. Solution From  Po ¼

0 0

  0 Pa þ P 0 p Pb

0 Pp Pa



and the stability of Pa and Pb one gets that dðPo ; po Þ ¼ dðPp ½ Pb Pa  ; po Þ  d ðPp ; po Þ. Since Pa and P1 a are stable, dðP ; po Þ ¼ dðPp Pa ; po Þ ¼ dðPp ; po Þ. Hence, dðP o ; po Þ  dðP ; po Þ. But P is a submatrix of Po . Thus, dðPo ; po Þ  dðP ;po Þ and wPo wP is a polynomial. It then follows that dðPo ; po Þ ¼ dðP ; po Þ and wPo wP is a stable polynomial. Example 2.8 From Lemma 2.8, a necessary condition for the 3DOF system to be admissible is that wF wP =wFP must be a stable polynomial. Show that this condition in turn requires that PA1 and AF be stable where A1 B and B1 A1 1 are left coprime and right coprime polynomial matrix fraction descriptions for FP, respectively. Hint: Introduce the coprime matrix fraction descriptions F ¼ A1 f Bf and 1 P ¼ Bp1 Ap1 . Let the polynomial matrix M be the greatest common right divisor of Bf Bp1 and Ap1 . Let the polynomial matrix N be the greatest common left divisor of Af and Bf Bp1 . Then establish first that det M and det N must be stable polynomials. Next show that FPA1 ¼ B1 and AFP ¼ B lead to PA1 and AF stable, respectively.

96

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Solution With M and N as defined, Bf Bp1 ¼ Bm1 M, Ap1 ¼ Am1 M; where Bm1 ; Am1 is a right coprime polynomial matrix pair and Af ¼ NAn , Bf Bp1 ¼ NBn , where An ; Bn is a left coprime polynomial matrix pair. That Am1 ; Bm1 is right coprime is easily established. For if M1 is a common right divisor of Bm1 ; Am1 then M1 M is also a common right divisor of Bf Bp1 ; Ap1 and so must divide M, the greatest common right divisor, on the right. That is, MðM1 MÞ1 ¼ M11 must be a polynomial matrix. Hence, det ðM11 Þ ¼ 1=det M1 must be a polynomial or det M1 must be a constant. Thus, M1 must be unimodular. That An ; Bn is left coprime is also easily established. For if N1 is a common left divisor of An ; Bn a dual argument yields that N1 must be unimodular. Clearly, 1 1 1 1 1 1 FP ¼ A1 B ¼ B1 A1 1 ¼ Af Bf Bp1 Ap1 ¼ Af Bm1 Am1 ¼ An Bn Ap1

and it follows that at any finite point po 1 dðFP; po Þ ¼ dðA1 ; po Þ  dðA1 f ; po Þ þ dðAm1 ; po Þ

and 1 dðFP; po Þ ¼ dðA1 ; po Þ  dðA1 n ; po Þ þ dðAp1 ; po Þ:

Hence, det A is a zero-remainder divisor of both ðdet Af Þ ðdet Am1 Þ and ðdet An Þ ðdet Ap1 Þ. So from ðdet Af Þ ðdet Ap1 Þ wF wP ðdet Af Þ ðdet Am1 Þ ðdet MÞ ¼g ¼g det A det A wFP ðdet NÞ ðdet An Þ ðdet Ap1 Þ ¼g det A one gets that wF wP =wFP is a stable polynomial only if det M and det N are stable polynomials or, equivalently, M 1 and N 1 are stable matrices. One also has from 1 1 1 FPA1 ¼ A1 f Bf Bp1 Ap1 A1 ¼ Af Bm1 Am1 A1 ¼ B1

that Bm1 A1 m1 A1 ¼ Af B1 is stable. Since Bm1 ; Am1 is right coprime, it follows that 1 1 A is stable. Hence, M 1 A1 A1 m1 1 m1 A1 ¼ Ap1 A1 and, therefore, PA1 ¼ Bp1 Ap1 A1 must be stable. Similarly, from 1 1 1 AFP ¼ AA1 f Bf Bp1 Ap1 ¼ AAn Bn Ap1 ¼ B

2.9 Additional Examples

97

one gets that AA1 n Bn ¼ BAp1 is stable. Since An ; Bn is a left coprime pair, it follows 1 1 that AAn must be stable. Hence, AA1 ¼ AA1 and, therefore, AF ¼ AA1 n N f f Bf must be stable. 1 Example 2.9 Let Gg ¼ A1 g Bg ¼ Bg1 Ag1 ; where the real polynomial matrix pairs Ag ; Bg and Bg1 ; Ag1 satisfy         Bg Bg Xg Bg1 Xg Bg1 Ag Ag I 0 ¼ ¼ : Yg1 Xg1 Yg Ag1 Yg Ag1 Yg1 Xg1 0 I

Establish that all stable real rational matrices Ka ; Kb that satisfy Ka Gg þ Kb ¼ C ~ ¼ CAg is a stable matrix are given by when C 1 ~ g þ Kc Ag ; Kb ¼ CX ~ g  Kc Bg ; Ka ¼ CY 1 1 where Kc is a stable real rational matrix. ~ is a stable matrix, it then Solution It is evident that Xg1 Ag1 þ Yg1 Bg1 ¼ I. Since C follows that a particular solution pair for the equation Ka Gg þ Kb ¼ C or, equiva~ is Ka ¼ CY ~ g , Kb ¼ CX ~ g . Hence, all potential solution lently, Ka Bg1 þ Kb Ag1 ¼ C 1 1 ~ ~ g þ Kb , where Ka and Kb are pairs can be written as Ka ¼ CYg1 þ Ka , Kb ¼ CX 1 stable real rational matrices satisfying Ka Bg1 þ Kb Ag1 ¼ 0 or Kb ¼ Ka Bg1 A1 g1 ¼ 1 Ka Ag Bg . Now Ag ; Bg is a left coprime polynomial matrix pair. Thus, with Ka stable, Kb is stable iff Ka A1 g ¼ Kc or Ka ¼ Kc Ag ; where Kc is a stable real rational matrix. It immediately follows that Kb ¼ Ka A1 g Bg ¼ Kc Bg . That is, all solution pairs are given by ~ g þ Kc Ag Ka ¼ CY 1 ~ g  Kc Bg ; Kb ¼ CX 1

where Kc is a stable real rational matrix. Example 2.10 Under the assumptions in Theorem 2.6, (2.297) is the parameterization, in terms of stable matrices K1 and Kg , of all the allowable K1 and H2 in (2.275) for which vd and ed are stable. (a) Show by direct substitution of (2.297) into (2.275) that the parameterization of H directly in terms of K1 and Kg is given by H ¼ ðY1 þ K1 AÞFPd þ Kg Bg Ad . (b) Show that this parameterization of H directly in terms of K1 and Kg agrees with the parameterization one obtains starting instead from the solution of (2.284) for H; specifically, from 1 ~ ~ H ¼ A1 1 ðYp Pd þ Ap1 Km ÞAd ¼ A1 Yp Ap Pd þ Km Ad

98

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

~ m given by (2.295). (c) Confirm that vd and ed are indeed stable when H is with K parameterized as in part (a). Solution (a) Using (2.275), (2.291), and (2.297), one gets H ¼ ðY1 þ K1 AÞFPd þ H2 L ¼ ðY1 þ K1 AÞFPd þ Kg Ag A1 g Bg Ad ¼ ðY1 þ K1 AÞFPd þ Kg Bg Ad : (b) Using (2.292) and (2.295), one gets 1 ~ H ¼ A1 1 Yp Ap Pd þ Km Ad ¼ A1 Yp Ap Pd þ ðKg Bg þ K1 Qd  Cd ÞAd

¼ A1 1 Yp Ap Pd  Cd Ad þ Kg Bg Ad þ K1 AFPd : One also has from (2.288) that 1 ~ Cd Ad ¼ A1 1 ðYp Pd Ad  YAFPd Þ ¼ A1 ðYp Ap Pd  A1 Y1 FPd Þ 1 ¼ A1 Yp Ap Pd  Y1 FPd :

Hence, A1 1 Yp Ap Pd  Cd Ad ¼ Y1 FPd and it follows that H ¼ A1 1 Yp Ap Pd  Cd Ad þ Kg Bg Ad þ K1 AFPd ¼ Y1 FPd þ Kg Bg Ad þ K1 AFPd ¼ ðY1 þ K1 AÞFPd þ Kg Bg Ad as in part (a). (c) From (2.273), vd ¼ A1 Hd ¼ A1 ½ðY1 þ K1 AÞFPd þ Kg Bg Ad A1 d bd ¼ ½ðY þ A1 K1 ÞAFPd þ A1 Kg Bg Ad A1 d bd ¼ ½ðY þ A1 K1 ÞQd þ A1 Kg Bg bd is stable because K1 , Kg , and Qd are stable. From (2.274), ed ¼ ðPd  PA1 HÞd ¼ Pd d  Pvd ¼ ½Pd A1 d  PðY þ A1 K1 ÞQd  PA1 Kg Bg bd ¼ ½Pd A1 d  PYQd  PA1 ðK1 Qd þ Kg Bg Þbd : Now 1 1 1 ~ 1 ~ Pd A1 d  PYQd ¼ Pd Ad  PYAFPd Ad ¼ Ap Pd  PYAFAp Pd 1 ~ ~ ¼ ½A1 p  PYAFðXp þ PYp ÞPd ¼ ½Ap  PYBYp  PA1 Y1 FXp Pd

~ ¼ ½A1 p  PðI  A1 X1 ÞYp  PA1 Y1 FXp Pd ~d : ¼ ½Xp þ PA1 ðX1 Yp  Y1 FXp ÞP

2.9 Additional Examples

99

Hence, ~ d bd  PA1 ðK1 Qd þ Kg Bg Þbd ed ¼ ½Xp þ PA1 ðX1 Yp  Y1 FXp ÞP ~ d , and F are stable. is stable because K1 , Kg , Qd , PA1 , P Example 2.11 Some of the comments made in Remark 2.13 are confirmed in this problem. (a) Suppose Ar is given and r ¼ A1 r br ; where the elements of Ar and br are real polynomials. Establish for an admissible 3DOF system that r is trackable for all br when Td and F are stable iff Ap Td A1 r is stable. Moreover, all stabilizing controllers for which r is trackable for all br are given by (2.208) with H1 ¼ A1 1 Ap1 ðYp1 Td þ Kr Ar Þ; where Kr is a stable real rational matrix. (b) Suppose Ad is given and d ¼ A1 d bd , where the elements of Ad and bd are real polynomials. Establish for an admissible 3DOF system that d is rejectable for all bd when F is stable and the real polynomial matrix pair Ag ; Bg satisfies (2.291) and (2.292) iff Ap Pd A1 d is stable. Moreover, all stabilizing controllers for which d is rejectable for all bd are given by (2.208) with ½ K1 H2  ¼ ½ K1 Kg Ag ; where K1 and Kg are stable real rational matrices belonging to the nonempty set of such matrices for which det ðX1  K1 BÞ 6 0. Solution (a) From (2.234) through (2.238), vr ¼ A1 H1 r ¼ A1 H1 A1 r br ¼ Wbr er ¼ ðTd  PA1 H1 Þr ¼ ðTd  PA1 H1 ÞA1 r br ¼ Ubr must be stable for all br with real polynomial elements. Hence, they must be stable when all elements of the column vector br are zero except for unity in the j-row. Thus, the j-column of W and the j-column of U must be stable. Clearly, this holds no matter what the choice of j and so the real rational matrices W and U must be stable. Conversely, if this is the case, vr and er are stable for all br . That is, vr and er are stable iff W and U in (2.237) and (2.238) are stable. One again gets (2.242) which has stable solutions U and W iff T~d ¼ Ap Td A1 r is stable and these solutions are given by (2.249) and (2.250). Any such solution pair is realized iff the H1 given by (2.253) is chosen. But a stabilizing controller is required and so H1 must be stable. The fact that Kr , F, and Td are stable guarantees this is the case when Lemma 2.10 is invoked. (b) From (2.273), (2.274), (2.279), and (2.280) vd ¼ A1 Hd ¼ A1 HA1 d bd ¼ Wbd ed ¼ ðPd  PA1 HÞd ¼ ðPd  PA1 HÞA1 d bd ¼ Ubd :

100

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Applying the same reasoning as in part (a), one concludes that vd and ed are stable for all bd iff the real rational matrices W and U are stable. From (2.280), one gets ~ Ap U þ Bp W ¼ Ap Pd A1 d ¼ Pd ~ d is stable and these solutions are given by (2.282) which has stable solutions iff P and (2.283). So proceeding exactly as was done in the steps leading to Theorem 2.6 in which the stability of F is invoked one finds that all stabilizing controllers for which d is rejectable for all bd are the ones described in Theorem 2.6. Example* 2.12 The tracking and disturbance rejection capabilities of the 2DOF standard configuration shown in Fig. 1.3 are investigated in this problem. It is assumed that the system is admissible and the controller is a stabilizing one as described in Theorem 2.2 with transfer matrix given by (2.161). The reference noise vector n is typically stochastic in nature and, therefore, devoid of any deterministic or shape-deterministic components. The elements of the disturbance input vector d and the reference input vector r, however, often contain deterministic or shape-deterministic components that are transforms of such signals as steps, ramps, and exponentially growing functions of time. In this case, d and r possess unstable poles. The objective here is to identify the class of inputs d and r of this type for which the tracking error e and the controller output v have stable transforms and to parameterize the subset of stabilizing controllers that achieve this objective. Since a stabilizing controller is assumed, the contributions to e and v that result from initial conditions in the analog case and initial data strings in the digital case are automatically stable and can therefore be ignored. That is, one can work exclusively with transfer matrices. Specifically, for Fig. 1.3, ½ y0s y0t y0 0 ¼ G½ d 0 v0 0 ; where

and the Gij for i; j ¼ 1; 2 are given by (2.134) and (2.137) through (2.139). (a) Establish that v ¼ A1 Hr þ A1 ðY1 þ KAÞG21 d þ A1 Hn e ¼ ðTd  Gtv A1 HÞr  ½Gtd þ Gtv A1 ðY1 þ KAÞG21 d  Gtv A1 Hn: (b) Introduce the coprime polynomial matrix fraction descriptions Gtv ¼ A1 p Bp ¼ 1 1 Bp1 Ap1 that satisfy (2.241). Then establish when (1) r ¼ Ar br is a left coprime real polynomial matrix fraction description and (2) Td and A1 1 Ap1 are stable

2.9 Additional Examples

101

1 that r is trackable iff Ap Td A1 r is stable or, equivalently, Ap Td r ¼ Ap Td Ar br is stable. Moreover, all H for which r is trackable are given by H ¼ A1 1 Ap1 ðYp1 Td þ Kr Ar Þ; where Kr is a stable real rational matrix.

(c) Let d ¼ A1 d bd be a left coprime real polynomial matrix fraction description. Introduce the coprime polynomial matrix fraction descriptions 1 1 1 Gtv ¼ A1 p Bp ¼ Bp1 Ap1 ; AG21 Ad ¼ Ag Bg

satisfying (2.241) and (2.292), respectively. Then establish that d is rejectable ~ td þ Y1 G21 A1 Þ ~ td ¼ Ap Gtd A1 is stable when A1 Ap1 and Cd ¼  ðA1 Yp G iff G d 1 1 d are stable (Note that the comments in Remark 2.11 apply here as well when Pd is replaced by Gtd ). Moreover, any K for which d is rejectable is given by K ¼ Kg Ag ; where Kg is a stable real rational matrix. (d) Show that the matrix Cd in (c) is stable when ½ G21 G22  ¼ M½ Gtd Gtv  and M is a stable matrix. This is the case for the system shown in Fig. 1.1 when L  0 (a 2DOF system), the feedback sensor transfer matrix F is stable, and one is concerned only about the contribution to e and v caused by the disturbance d (e) This part concerns the case when Cd is not stable. Now one needs to introduce 1 the coprime polynomial matrix fraction descriptions AG21 A1 d ¼ Ag Bg ¼ Bg1 A1 g1 satisfying         Bg Bg Xg Bg1 Xg Bg1 Ag Ag I 0 ¼ ¼ Yg1 Xg1 Yg Ag1 Yg Ag1 Yg1 Xg1 0 I : ~ ~ Establish that d is rejectable when A1 1 Ap1 is stable iff Gtd and Cd ¼ Cd Ag1 are stable and there exists a real rational matrix Kd for which ~ d Yg þ Kd Ag ÞB 6 0. det ½X1  ðC 1 Solution (a) From y ¼ G21 d þ G22 v; one gets v ¼ Cy y þ Cu u ¼ Cy G21 d  Cy G22 v þ Cu r þ Cu n or v ¼ ðI þ Cy G22 Þ1 ½Cy G21 d þ Cu r þ Cu n: Now from (2.161) 1 1 Ry ¼ ðI  Cy G22 Þ1 Cy ¼ ½I þ ðX1  KBÞ1 ðY1 þ KAÞB1 A1 1  ðX1  KBÞ ðY1 þ KAÞ

¼ A1 ½ðX1  KBÞA1 þ ðY1 þ KAÞB1 1 ðY1 þ KAÞ ¼ A1 ðY1 þ KAÞ

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2 Stabilizing Controllers, Tracking, and Disturbance Rejection

and 1 1 Ru ¼ ðI þ Cy G22 Þ1 Cu ¼ ½I þ ðX1  KBÞ1 ðY1 þ KAÞB1 A1 1  ðX1  KBÞ H

¼ A1 ½ðX1  KBÞA1 þ ðY1 þ KAÞB1 1 H ¼ A1 H: Hence, v ¼ Ru r þ Ry G21 d þ Ru n ¼ A1 Hr þ A1 ðY1 þ KAÞG21 d þ A1 Hn: Then e ¼ Td r  yt ¼ Td r  ðGtd d þ Gtv vÞ ¼ Td r  Gtd d  Gtv ½A1 Hr þ A1 ðY1 þ KAÞG21 d þ A1 Hn ¼ ðTd  Gtv A1 HÞr  ½Gtd þ Gtv A1 ðY1 þ KAÞG21 d  Gtv A1 Hn: (b) The reference input r is trackable iff vr ¼ A1 Hr ¼ A1 HA1 r br er ¼ ðTd  Gtv A1 HÞr ¼ ðTd  Gtv A1 HÞA1 r br are stable. Since an admissible 2DOF standard configuration is assumed, it follows from Lemma 2.5 that 

   Gsv Gsv A1 G12 A1 ¼ A ¼ Gtv 1 Gtv A1 is stable. Hence, Gtv A1 is stable. Also, since a stabilizing controller is assumed, H is stable. Thus, with Td stable one has that A1 H and ðTd  Gtv A1 HÞ are stable. It then follows from the fact that Ar ; br is a left coprime polynomial matrix pair that r is trackable iff W ¼ A1 HA1 r and 1 U ¼ ðTd  Gtv A1 HÞA1 r ¼ Td Ar  Gtv W

are stable. Introducing Gtv ¼ A1 p Bp yields ~ Ap U þ Bp W ¼ Ap Td A1 r ¼ Td :

2.9 Additional Examples

103

Hence, r is trackable iff there exist stable real rational matrices U; W satisfying this equation for which H ¼ A1 1 WAr is a stable real rational matrix. Now by direct analogy with (2.249) and (2.250) in the 3DOF case, stable solution pairs U; W exist iff T~d is stable and all such pairs are given by U ¼ Xp T~d  Bp1 Kr ; W ¼ Yp T~d þ Ap1 Kr ; where Kr is a stable real rational matrix. Moreover, when A1 1 Ap1 is stable so is the real rational matrix 1 1 ~ H ¼ A1 1 WAr ¼ A1 ðYp Td þ Ap1 Kr ÞAr ¼ A1 Ap1 ðYp1 Td þ Kr Ar Þ

which gives all H for which r is trackable. (c) The disturbance input d is rejectable iff vd ¼ A1 ðY1 þ KAÞG21 A1 d bd ed ¼  ½Gtd þ Gtv A1 ðY1 þ KAÞG21  A1 d bd are stable. Since an admissible 2DOF standard configuration is assumed, it follows from Lemma 2.5 that AG21 is stable. Hence, A1 ðY1 þ KAÞG21 ¼ ðY þ A1 KÞAG21 is stable since K is stable for a stabilizing controller. Also from Lemma 2.5, 

G11 þ G12 A1 Y1 G21

     Gsd Gsd þ Gsv A1 Y1 G21 Gsv ¼ þ A Y G ¼ Gtd Gtv 1 1 21 Gtd þ Gtv A1 Y1 G21

is stable. Hence, Gtd þ Gtv A1 ðY1 þ KAÞG21 ¼ Gtd þ Gtv A1 Y1 G21 þ Gtv A1 KA G21 is stable. It then follows from the fact that Ad ; bd is a left coprime polynomial matrix pair that d is rejectable iff W ¼ A1 Hw A1 d and 1 U ¼ ðGtd þ Gtv A1 Hw ÞA1 d ¼ Gtd Ad þ Gtv W

are stable, where Hw ¼ ðY1 þ KAÞG21 . Introducing Gtv ¼ A1 p Bp in the expression for U leads to ~ Ap U  Bp W ¼ Ap Gtd A1 d ¼ Gtd :

104

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Hence, d is rejectable iff there exist stable real rational matrices U; W satisfying this equation for which Hw ¼ ðY1 þ KAÞG21 ¼ A1 1 WAd has a stable solution for K satisfying det ðX1  KBÞ 6 0. Now by the same arguments that yielded (2.282) and (2.283) in the 3DOF case, one can conclude that stable solution pairs U; W exist ~ td is stable and all such pairs are given by iff G ~ td þ Bp1 Km ; W ¼ Yp G ~ td þ Ap1 Km ; U ¼ Xp G where Km is a stable real rational matrix. Using these results in the expression for Hw then gives ~ td þ Ap1 Km ÞAd Hw ¼ ðY1 þ KAÞG21 ¼ A1 ðYp G 1

or 1 1 1 ~ KAG21 A1 d  A1 Ap1 Km ¼  ðA1 Yp Gtd þ Y1 G21 Ad Þ ¼ Cd :

Thus, d is rejectable iff there exist stable real rational matrices K; Km satisfying this equation for which det ðX1  KBÞ 6 0. Now A1 1 Ap1 stable is given and 1 1 admissibility requires Gtv A1 ¼ Bp1 A1 A stable. Hence, A1 is p1 1 p1 A1 ¼ ðA1 Ap1 Þ 1 ~ m ¼ A1 Ap1 Km , which is stable iff Km is stable, also stable. So one can introduce K

1 and AG21 A1 d ¼ Ag Bg to get that d is rejectable iff there exist stable real rational ~ m satisfying matrices K; K

~ KA1 g Bg ¼ Cd þ Km for which det ðX1  KBÞ 6 0. The first observation to make is that the left-hand side ~ m , is stable. For of this equation must be stable because the right-hand side, Cd þ K ¼ K stable K it immediately follows that this is the case iff KA1 g , a stable matrix. g Hence, all solution pairs are given by ~ m ¼ Cd þ Kg Bg ; K ¼ Kg Ag ; K where Kg is a stable real rational matrix. The second observation to make is that ~ m ¼ Cd for which det ðX1  KBÞ ¼ for Kg ¼ 0 the solution pair is K ¼ 0; K det X1 6 0 since one can always choose X1 so that this is the case. That is, d is rejectable and all stabilizing controllers for which d is rejectable are given by (2.161) with K ¼ Kg Ag ; where Kg is a stable real rational matrix. 1 1 1 1 ~ ~ (d) Cd ¼ A1 1 Yp Gtd  Y1 G21 Ad ¼ A1 Yp Gtd  Y1 MAp Ap Gtd Ad

~ td ¼  ½ðX1  Y1 G22 ÞYp þ Y1 MðXp þ Gtv Yp ÞG ~ td ¼  ðX1 Yp þ Y1 MXp Þ G ~ td ¼  ðX1 Yp  Y1 MGtv Yp þ Y1 MXp þ Y1 MGtv Yp Þ G

2.9 Additional Examples

105

~ td and M are stable. (e) The initial steps in part (c) up to the which is stable when G 1 ~ m apply here also. However, since Cd is not stable now, equation KAg Bg ¼ Cd þ K 1 one cannot proceed as before. Instead, the substitution A1 g Bg ¼ Bg1 Ag1 is made to ~ d . Clearly, C ~ d stable for stable solutions K; K ~ m Ag1 ¼ Cd Ag1 ¼ C ~ m is get KBg1  K ~ ~ ~ necessary. It is also sufficient since in this case K ¼ Cd Yg1 , Km ¼ Cd Xg1 are particular solutions. Using the results of Example 2.9, it is clear that all solutions are given by ~ d Xg þ Kd Bg ; ~ d Yg þ Kd Ag ; K ~ m ¼ C K¼C 1 1 where Kd is a stable real rational matrix. Hence, d is rejectable under the assumption ~ ~ A1 1 Ap1 is stable iff Gtd and Cd are stable and there exists a stable real rational matrix Kd for which ~ d Yg þ Kd Ag ÞB 6 0: det ðX1  KBÞ ¼ det ½X1  ðC 1 Example* 2.13 Consider an admissible 3DOF system with stabilizing controller given by (2.208). This controller reduces to a 1DOF controller when H1 ¼ H2 ¼ Y1 þ K1 A. For in this case C ¼ ðX1  K1 BÞ1 ðY1 þ K1 AÞ½ I

I

I  ¼ C1 ½ I

I

I

and the 1DOF design freedom available resides in the choice of the stable real rational matrix K1 . The objective in this example is to investigate the tracking capabilities of this 1DOF system with respect to the reference input r. In this case, r is the only exogenous input which is not identically zero and it follows from Fig. 1.1 and Eq. (1.1) that v ¼ C1 q; where q ¼ w  z þ u ¼ Fy þ r since n ¼ 0, z ¼ 0, and nm ¼ 0. That is, the issue of r being trackable is equivalent to r being trackable for the system shown in Fig. 2.4. It is assumed throughout this example that F and Td are stable.

Fig. 2.4 1DOF tracking model for reference input

r

qr

wr

yr

vr C

r

P

F

Td

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2 Stabilizing Controllers, Tracking, and Disturbance Rejection

(a) Let r ¼ A1 r br be a left coprime real polynomial matrix fraction description for the reference input r. Introduce the coprime polynomial matrix fraction descriptions 1 1 AA1 r ¼ Bh1 Ah1 ¼ Ah Bh

that satisfy   Ah Bh Xh Yh1 Xh1 Yh

Bh1 Ah1





Xh ¼ Yh

Bh1 Ah1



Ah Yh1

Bh Xh1





I ¼ 0

0 I



1 1 ~ ~ and let Ch ¼ A1 1 Yp Td  Y1 Ar , where Td ¼ Ap Td Ar . Then establish that r is ~ h ¼ Ch Ah and T~d are stable and there exists a stable matrix Kh for trackable iff C 1 which det ðX1  K1 BÞ 6 0 when the choice K1 ¼ Ch Ah1 Yh1 þ Kh Ah is made. ~ h is stable iff ðFTd  IÞA1 Ah is stable. That (b) When T~d is stable show that C 1 r ~ is, r is trackable iff Td and ðFTd  IÞA1 A are stable and there exists a stable h 1 r matrix Kh for which det ðX1  K1 BÞ 6 0 when the choice K1 ¼ ~ h is stable iff Ch Ah1 Yh1 þ Kh Ah is made. Hint: Establish and use the fact that C ~ B1 Ch is stable. (c) From Fig. 2.4, qr ¼ r  Fyr ¼ r  FðTd r  er Þ ¼ ðI  FTd Þr þ Fer :

With a trackable r, Fer is stable. Hence, qr is stable iff ðI  FTd Þr or, equivais stable. This is not assured by the stability of ðFTd  lently, ðFTd  IÞA1 r IÞA1 A and it is possible for r to be trackable with qr unstable. Show that this h 1 r is indeed the case for an analog 1DOF system when F ¼ 1, P ¼ 1=s, Td ¼ 1  s, and r ¼ 1=sð1  sÞ. Also, demonstrate for 1 1 sþ2 þ ¼ s þ Kh ¼ s þ 2 s þ 1 ðs þ 1Þ ðs þ 1Þ2 that qr , vr , and er are strictly proper and C1 is proper. (d) Suppose that T~d and ðFTd  I ÞA1 are stable. Establish that AA1 is stable. Also confirm that r is r r trackable and qr is stable with any stabilizing controller. (e) Show that Ch is stable when Td1 , ðTd F  I ÞP, and T~d are stable. Then establish in this case that r is trackable and K1 ¼ Ks Ah ; where Ks is a stable real rational matrix. (f) Under the assumptions for (e), confirm directly that vr and er are stable for K1 ¼ Ks Ah ; where Ks is a stable real rational matrix. Also prove that qr is stable iff XA1 h is stable. Solution (a) For the 1DOF system one gets from (2.234) and (2.235) that vr ¼ A1 ðY1 þ K1 AÞA1 r br er ¼ ½Td  PA1 ðY1 þ K1 AÞA1 r br :

2.9 Additional Examples

107

Now K1 is stable for a stabilizing controller, PA1 is stable for an admissible system from Lemma 2.7, and Td is stable by assumption. Since Ar ; br is a left coprime pair, it follows that vr and er are stable iff W ¼ A1 ðY1 þ K1 AÞA1 r 1 U ¼ ½Td  PA1 ðY1 þ K1 AÞA1 r ¼ Td Ar  PW

are stable matrices. So r is trackable iff there exist stable matrices U  1 real rational  1 for which K ¼ A WA  Y is stable A and W satisfying U þ PW ¼ Td A1 1 r 1 r 1 and detðX1  K1 BÞ 6 0. Introduce the coprime polynomial matrix fraction 1 descriptions P ¼ A1 p Bp ¼ Bp1 Ap1 satisfying (2.241). Then U and W are solutions 1 of Ap U þ Bp W ¼ Ap Td Ar ¼ T~d . It then follows from the same reasoning that led to (2.249) and (2.250) that these solutions are stable iff T~d is stable. In this case, the solutions are given by U ¼ Xp T~d  Bp1 K2 ; W ¼ Yp T~d þ Ap1 K2 ; where K2 is a stable real rational matrix. Substituting the solution for W into the  

1 ~ or expression for K1 yields K1 ¼ A1 1 Yp Td þ Ap1 K2 Ar  Y1 A 1 1 1 ~ K1 AA1 r  A1 Ap1 K2 ¼ A1 Yp Td  Y1 Ar ¼ Ch : 1 ~ 2 ¼ A1 Now K 1 Ap1 K2 is stable iff K2 is stable since A1 Ap1 is stable by Lemma 1 2.10 and admissibility requires PA1 ¼ Bp1 Ap1 A1 stable or, equivalently, A1 p1 A1 stable. Thus, the tracking problem has been reduced to finding stable matrices K1 ~ 2 satisfying K1 AA1 ~ and K r  K2 ¼ Ch or

~h ~ 2 Ah1 ¼ C K1 Bh1  K for which detðX1  K1 BÞ 6 0. It follows from the same reasoning as used in ~ h is stable and all such solutions ~ 2 exist iff C Example 2.8 that stable solutions K1 ; K are given by ~ h Yh þ Kh Ah ; K ~ h Xh þ Kh Bh : ~ 2 ¼ C K1 ¼ C 1 1 ~ h are Hence, r is trackable under the assumptions for this problem iff T~d and C stable and there exists a stable real rational matrix Kh for which detðX1  K1 BÞ 6 0.

108

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

(b) It follows from ~ h ¼ ðA1 Yp T~d  Y1 A1 ÞAh ¼ A1 ðYp T~d Ah  A1 Y1 A1 Ah Þ C 1 1 1 1 r 1 r 1 1 ~ ~ ðY A  YAA A Þ ¼ A ðY A  YB Þ ¼ A1 T T p d h1 h1 p d h1 h1 1 r 1   ~ h is stable because Yp T~d Ah  YBh is stable. Clearly, ~ h is stable iff B1 C that C 1 1 ~ h ¼ B1 A1 ðYp T~d Ah  YBh Þ ¼ FPðYp Ap Td A1 Ah  YAA1 Ah Þ B1 C 1 1 1 1 1 r r h  i   1 1 ¼ FPYp Ap Td  FPA1 Y1 Ar Ah1 ¼ F Ap  Xp Ap Td  B1 Y1 A1 r Ah1  1   1 ¼ FTd  FXp Ap Td  I þ XA Ar Ah1 ¼ ðFTd  I ÞAr Ah1  FXp Ap Td  XA A1 r Ah1 1 ¼ ðFTd  I ÞAr Ah1  FXp T~d Ah1 þ XBh1

~ is stable iff ðFTd  I ÞA1 r Ah is stable because FXp Td Ah1 and XBh1 are stable. (c) From the given data: FP ¼ P ¼

1 ) s

A ¼ A1 ¼ Ap ¼ Ap1 ¼ s; B ¼ B1 ¼ Bp ¼ Bp1 ¼ 1; X ¼ X1 ¼ Xp ¼ Xp1 ; Y ¼ Y1 ¼ Yp ¼ Yp1 :

AX þ BY ¼ 1 ) sX þ Y ¼ 1 ) X ¼ 1; Y ¼ 1  s 1 ) Ar ¼ sðs  1Þ; br ¼ 1 r¼ sð1  sÞ 1 ) Ah ¼ Ah1 ¼ s  1; Bh ¼ Bh1 ¼ 1; Xh ¼ Xh1 ; Yh ¼ Yh1 : AA1 r ¼ s1 Ah Xh þ Bh Yh ¼ 1 ) ðs  1ÞXh þ Yh ¼ 1 ) Xh ¼ 1; Yh ¼ s: 1 1 Clearly, ðFTd  IÞA1 ¼ A1 r Ah1 ¼ 1 is stable, but ðFTd  IÞAr h1 ¼ ðs  1Þ is unstable. In addition, T~d ¼ Ap Td A1 r ¼ 1 and 1 ~ Ch ¼ A1 p Yp Td  Y1 Ar ¼

s1 1 þ ¼ 1: s s

Thus, K1 ¼ Ch Ah1 Yh1 þ Kh Ah ¼ sðs  1Þ þ Kh ðs  1Þ ¼ ðs þ Kh Þðs  1Þ and X1  K1 B ¼ 1  K1 ¼ ðs2  s  1Þ  Kh ðs  1Þ 6 0

2.9 Additional Examples

109

for any real rational stable Kh . So r is trackable and qr is unstable. In particular, C1 ¼ ðX1  K1 BÞ1 ðY1 þ K1 AÞ ¼ 

ðs  1Þðs2 þ Kh s  1Þ s2 þ ðKh  1Þs  ðKh þ 1Þ

and it easily follows from qr ¼ r  FPC1 qr that qr ¼ ðI þ FPC1 Þ1 r ¼ s 

1 þ Kh s1

which is unstable for any stable Kh . When Kh ¼ s þ

1 1 sþ2 þ ¼ s þ 2 s þ 1 ðs þ 1Þ ðs þ 1Þ2

a little algebra yields C1 ¼ 

s1 ; sþ3

qr ¼ 

sþ3 ðs  1Þðs þ 1Þ

2

;

vr ¼ C1 qr ¼

1 ðs þ 1Þ2

;

and er ¼ Td r  Pvr ¼

1 1 sþ2  ¼ : s sðs þ 1Þ2 ðs þ 1Þ2

Clearly, C1 is proper and qr , vr , and er are strictly proper. Moreover, vr and er are stable; hence, the analog performance measure in (1.7) with yc ¼ ½ e0r v0r 0 is finite in this case. 1 (d) It is given that M1 ¼ T~d ¼ Ap Td A1 r and M2 ¼ ðFTd  IÞAr are stable. Hence, 1 1 A1 r ¼ FTd Ar  M2 ¼ FAp M1  M2

and 1 AA1 r ¼ AFAp M1  AM2 : 1 Since AF is stable, AFA1 p is stable iff AFAp Bp ¼ AFP ¼ B is stable. This is 1 1 indeed the case and it follows that AAr ¼ Ah Bh is stable. Therefore, A1 h is stable. It is also obvious that M2 Ah1 ¼ ðFTd  IÞA1 A is stable. Moreover, for any h1 r

110

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

stabilizing controller, K1 is stable and det ðX1  K1 BÞ 6 0. One then gets from ~ h Yh þ Kh Ah for Kh is stable. Clearly, (b) that r is trackable if the solution of K1 ¼ C 1 1 ~ ~ h , and A1 are stable. Hence, Kh ¼ ðK1  Ch Yh1 ÞAh is indeed stable since K1 , C h from (c), qr is stable because r is trackable and M2 ¼ ðFTd  IÞA1 is stable. r ~ (e) Since Td is stable, 1 1 1 1 ~ ~ Ch ¼ A1 1 Yp Td  Y1 Ar ¼ ðA1 Yp  Y1 Td Ap ÞTd

is stable if 1 1 1 ðA1 1 Yp  Y1 Td Ap Þ ¼ ðX1 þ Y1 FPÞYp  Y1 Td ðXp þ PYp Þ

¼ X1 Yp  Y1 Td1 Xp þ Y1 Td1 ðTd F  IÞPYp is stable. This is indeed the case when Td1 and ðTd F  IÞP are stable. Hence, ~ h ¼ Ch Ah is stable. So from part (a) r is trackable if one can choose a stable Kh so C 1 ~ h Yh þ Kh Ah ¼ Ch Ah Yh þ Kh Ah one gets det ðX1  K1 BÞ 6 0. that with K1 ¼ C 1 1 1 Clearly, Kh ¼ Ch Yh is stable and gives det ðX1  K1 BÞ ¼ det X1 6 0 since K1 ¼ 0 in this case and X1 can always be selected so that det X1 6 0. Moreover, for any stable Kh , K1 ¼ Ch Yh Ah þ Kh Ah ¼ Ks Ah ; where Ks ¼ Ch Yh þ Kh is stable since Ch is stable with the assumptions for this part of the example. (f) The response vr is stable iff W ¼ A1 ðY1 þ K1 AÞA1 r is stable. With K1 ¼ Ks Ah , 1 W ¼ A1 ðY1 þ Ks Ah AÞA1 r ¼ ðY þ A1 Ks Ah ÞAAr 1 ~ ¼ YAA1 r þ A1 Ks Bh ¼ A1 Y1 Ar þ A1 Ks Bh ¼ Yp Td  A1 Ch þ A1 Ks Bh

is indeed stable since T~d , Ch , and Ks are stable. The response er is stable iff U ¼ ½Td  PA1 ðY1 þ K1 AÞA1 r is stable. With K1 ¼ Ks Ah , 1 1 1 ~ U ¼ Td A1 r  PA1 Y1 Ar  PA1 Ks Ah AAr ¼ Td Ar  PðYp Td  A1 Ch Þ  PA1 Ks Bh 1 ¼ Ap ðI  Bp Yp ÞT~d þ PA1 ðCh  Ks Bh Þ ¼ Xp T~d þ PA1 ðCh  Ks Bh Þ

is indeed stable since T~d , PA1 , Ch , and Ks are stable. Clearly, qr ¼ r  Fyr ¼ r  FPvr ¼ r  FPA1 ðY1 þ K1 AÞr ¼ ½I  B1 ðY1 þ K1 AÞr 1 ¼ ðI  B1 Y1  B1 K1 AÞA1 r br ¼ ðX  B1 K1 ÞAAr br

2.9 Additional Examples

111

is stable iff 1 Qr ¼ ðX  B1 K1 ÞAA1 r ¼ ðX  B1 Ks Ah ÞAh Bh

or, equivalently, ~ r ¼ ðX  B1 Ks Ah ÞA1 ¼ XA1  B1 Ks Q h h is stable because Ar ; br and Ah ; Bh are coprime pairs. Thus, qr is stable iff XA1 h is stable. Example 2.14 (a) Show that Theorem 2.4 is unaffected by the choice of coprime polynomial matrix fraction descriptions used for r, P, and FP. (b) Show that Theorem 2.6 is unaffected by the choice of coprime polynomial matrix fraction descriptions used for d, P, FP, and LA1 d . ~ 1 , and FP ¼ A ~ 1 be ~ 1 B ~ 1 B ~ 1 ~br , P ¼ A ~p ¼ B ~¼B ~1A ~ p1 A Solution (a) Let A r p p1 1 ~ ~ alternative coprime descriptions for Theorem 2.4. Then Ar ¼ Nr Ar , Ap ¼ Np Ap , ~ 1 ¼ A1 N1 , and Np1 Y~p1  Yp1 ¼ Mp Ap ; where Nr , Np , Np1 , and N1 are ~ p1 ¼ Ap1 Np1 , A A unimodular polynomial matrices and Mp is an arbitrary real polynomial matrix (see ~ p1 ¼ ~ 1 A the material leading to Lemma 2.4 and Example 2.4 and its solution). So A 1 1 1 1 1 1 1 ~ ~ N1 A1 Ap1 Np1 and Ap Td Ar ¼ Np Ap Td Ar Nr are stable iff A1 Ap1 and Ap Td A1 r are stable, respectively. Hence, in light of (2.234) and 2.235), it only remains to show that the set of all ~1H ~ r Þ ¼ Ap1 Np1 ðY~p1 Td þ K ~ p1 ðY~p1 Td þ K ~1 ¼ A ~rA ~ r Nr Ar Þ A ~ r ranges over all stable real rational matrices is the same as the set of all A1 H1 as K given by (2.257) as Kr ranges over all stable real rational matrices. Now, ~1H ~ 1 ¼ Ap1 ðNp1 Y~p1 Td þ Np1 K ~ r Nr Ar Þ A ~ r Nr Ar Þ ¼ Ap1 ðYp1 Td þ Kr Ar Þ ¼ A1 H1 ¼ Ap1 ðYp1 Td þ Mp Ap Td þ Np1 K when ~ ~ ~ Kr ¼ Mp Ap Td A1 r þ Np1 Kr Nr ¼ Mp Td þ Np1 Kr Nr : ~ r is stable. So Since, Np1 and Nr are unimodular and T~d is stable, Kr is stable iff K ~1H ~ 1. there is a one-to-one correspondence between every realizable A1 H1 and A 1 ~ ~ ~ (b) Let Ad bd be an alternative coprime description for d. Then Ad ¼ Nd Ad , Nd a ~ 1 ¼ Np Ap Pd A1 N 1 is stable iff ~ p Pd A unimodular polynomial matrix. Hence, A d d d

112

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

Ap Pd A1 d is stable. So, in light of (2.273), (2.274), and (2.276), it only remains to show that the set of all ~1H ~gL ~1K ~ d þA ~1H ~1K ~ d þA ~1K ~ 1 ¼ ðY~ þ A ~ 1 ÞAFP ~ 2 L ¼ ðY~ þ A ~ 1 ÞAFP ~gA A ~ 1 and K ~ g range over all stable real rational matrices is the same as the set of all as K A1 H1 ¼ ðY þ A1 K1 ÞAFPd þ A1 H2 L ¼ ðY þ A1 K1 ÞAFPd þ A1 Kg Ag L as K1 and Kg range over all stable real rational matrices. Now one has ~ 1 ¼ LA1 N 1 ¼ A1 Bg N 1 ¼ A ~ 1 B ~g: LA d d d g d g Moreover, 

½ Ag

Bg Nd1

 ¼ ½ Ag

I 0 Bg  0 Nd1



has full row rank everywhere since Ag ; Bg is a left coprime pair and Nd is unimodular. Hence, Ag ; Bg Nd1 is another left coprime polynomial matrix pair in ~ 1 . ~g; B ~ g which can be used in the matrix fraction description for LA addition to A d 1 ~ ~ g ¼ Ng Bg Nd ; where Ng is unimodular. So one gets That is, Ag ¼ Ng Ag ; B ~1H ~ 1 ¼ ½ðY þ A1 MÞN 1 þ A1 N1 K ~ 1 NAFPd þ A1 N1 K ~ g Ng Ag L A ~ 1 NÞAFPd þ A1 ðN1 K ~ g Ng ÞAg L; ¼ ½Y þ A1 ðM þ N1 K ~ 1 N and where M is an arbitrary real polynomial matrix. Clearly, K1 ¼ M þ N1 K ~ g Ng are stable iff K ~ 1 and K ~ g are stable and there is a one-to-one correKg ¼ N1 K ~1H ~ 1. spondence between every realizable A1 H1 and A Example* 2.15 Confirm the validity of (2.373). ~B ~ 1 . Then it suf~ 1 ¼ X1 AB1 K1 ¼ X1 BA1 K1 ¼ B ~A Solution First note that A ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ fices to show that AX þ BY ¼ I, X1 A1 þ Y1 B1 ¼ I, and Y1 X ¼ X1 Y. Clearly,

2.9 Additional Examples

113

~X ~ A ~ 1  B ~ 1 YXg Þ þ B ~ 1 fA ~ 1 YXg ¼ I ~ þB ~ Y~ ¼ Að ~ 1 fA ~A A sp sp 1 1 1 1 1 ~ ~ ~B ~ ~ ~ ~ ~ ~ ~1 ¼ I ~ X1 A1 þ Y1 B1 ¼ ðA1  fA1 YXgsp BÞA1 þ fA1 YXgsp A ~ 1  fA ~ 1 YXg BÞ ~ 1 ~ ~ ~ 1 ~ ~ ~ 1 ~1 Y~ ¼ ðA X sp A1 fA1 YXgsp ¼ ðI  fA1 YXgsp BA1 ÞfA1 YXgsp 1 1 ~ 1 YXg Að ~ A ~ 1  B ~ 1 YXg Þ ¼ fA ~ 1 YXg ðI  A ~B ~ 1 YXg Þ ~ ¼ fA ~ 1 fA ~ 1 fA Y~1 X 1

sp

1

sp

1

sp

1

sp

~ 1 YXg B ~ ~ ~ 1 ~ ~ ¼ ðI  fA sp A1 ÞfA1 YXgsp ¼ X1 Y: 1 Example 2.16 The purpose of this problem is to specialize Theorem 2.8 to the case of the 3DOF system shown in Fig. 1.1 when FP ¼ A1 B ¼ B1 A1 1 and the generalized Bezout identity (2.62) is satisfied. In particular, when the 3DOF system is admissible, FP is strictly proper, and the controller is free of unstable hidden poles, establish that all proper stabilizing controller transfer matrices are given by (2.208) with 1 1 1 K1 ¼ K1 ðZw  fKA1 1 YXg1 ÞX ; H1 ¼ K Zu ; H2 ¼ K Zz ;

where Zu , Zw , and Zz are proper stable real rational matrices and the real rational matrices K and X are stable together with their inverses and selected so that A1 K1 and X1 A are biproper. Solution The approach that is taken here is to specialize the G22 for the standard configuration to that of the 3DOF system. This is easily accomplished in a two-step process from the results already available by first specializing to the 2DOF standard configuration and then specializing from the 2DOF standard configuration to the 3DOF system. In particular, from (2.141) it is clear that G22 ! ½ G022 0 0 on the first step. On the second and final step, it follows from (2.193) that ½ G022 0 0 ! ½ ðFPÞ0 0 0 0 . So G22 in the standard configuration in the particular case of the 3DOF system is given by the strictly proper matrix 3 3 2 1 3 2 FP B1 A1 A B 1 ¼ 4 0 5 ¼ 4 0 5 ¼ 4 0 5; 0 0 0 2

G22

where the first block row has the same number of rows as w, the second block row has the same number of rows as z, and the third block row has the same number of rows as r. Hence, 2

G22

A ¼ 4 0 0

0 I 0

3 31 2 2 3 B B1 0 0 5 4 0 5 ¼  4 0 5A1 1 ; 0 I 0

114

2 Stabilizing Controllers, Tracking, and Disturbance Rejection

2

3 B1 0 0 4 0 5 ¼ I; X1 ½ Y 0

X1 A1 þ ½ Y1

2

0  ¼ ½ Y1

0

X 0 4 0 0

0

0 I 0

3 0 0 5; I

and 2

A 40 0

32 0 0 X I 0 54 0 0 0 I

0 I 0

3 2 3 0 B 0 5 þ 4 0 5½ Y I 0

2

0

I 0 ¼ 40 0

0 I 0

3 0 0 5: I

Notice that the matrices X1 and A1 are unaffected. Also, in conformity with (2.157) and (2.205), the matrix K is replaced with ½ K1 H2 H1 . In addition, it should be clear from the equations above that for the X-matrix in Theorem 2.8 one can choose the block diagonal matrix diag fX; I; Ig. Then in place of (2.365) one gets 2

311 B 6 7C H1 4 0 5A

0 B C ¼ @X1  ½ K1

H2

0

0 B @½ Y1

0

0  þ ½ K1

H2

2

A 0 6 H1 4 0 I 0 0

¼ ðX1  K1 BÞ1 ½ ðY1 þ K1 AÞ H2

31 0 7C 0 5A I

H1  ¼ ½ Cw

Cz

Cu ;

which is in agreement with (2.208). In place of (2.366), one gets 0 ½ K1

H2

B H1  ¼ K1 @½ Zw 2

X1 6 4 0 0 ¼ K1 ð½ Zw

Zz

Zu  

8 > < > :

2

X 6 0 0 4 0 0

KA1 1 ½ Y

3 0 0 7 I 05 0

I

Zz

2

Zu  þ f½ KA1 1 YX

1 ¼ ½ K1 ðZw  fKA1 1 YXg1 ÞX

0

K1 Zz

39 1 0 0 = > 7 C I 05 A > ; 0 I 1

X1 6 0 g1 Þ4 0 0 K1 Zu :

3 0 0 7 I 05 0 I

Chapter 3

H2 Design of Multivariable Control Systems

3.1

Overview

The design of analog and digital linear multivariable control systems for which performance is measured by a quadratic or H2 cost functional of the kind described in Chap. 1 and Appendix A is treated in this chapter. First, the parameterization of the tracking and disturbance rejection controllers for the 3DOF system presented in Chap. 2 is extended to the standard configuration of Fig. 1.2. Next, for this subset of stabilizing controllers, the ones for which the cost functional is finite are parameterized and the one for which the cost functional is minimum is established. Specialized results for 1DOF and 3DOF systems are then presented.

3.2

Preliminary Considerations

It is important to keep separate the modeling of an interconnected physical system from the modeling of the performance measures and the input signals. Otherwise, important issues with regard to system stability can be obscured and/or confused. That is, it is best to employ the standard configuration of Fig. 1.2 to represent all possible interconnections of the physical components in any feedback configuration, but it is best to keep separate the modeling of the exogenous inputs e and the e and C represent the physical compodesired responses yd as in Fig. 3.1 where G e e nents while G e and G d account for the mathematical models of the exogenous signals e and the desired system responses yd, respectively. In particular, it is e d do not represent physical components and e e and G important to recognize that G have no impact on the stability of the system as determined by the response to e and C. The modeling of e and initial conditions or data strings within the blocks G 0 0 0 yd is accomplished by setting q ¼ ½ r l  where r is a vector whose elements are © Springer Nature Switzerland AG 2020 J. J. Bongiorno Jr. and K. Park, Design of Linear Multivariable Feedback Control Systems, https://doi.org/10.1007/978-3-030-44356-6_3

115

3 H2 Design of Multivariable Control Systems

116 Fig. 3.1 Persistent input standard configuration (© 2009 Taylor & Francis Ltd, http://www.tandfonline. com. Reprinted, with permission, from Park and Bongiorno (2009))

real random variables with zero mean and covariance matrix \rr0 [ ¼ I and l is a stochastic vector whose power spectral density matrix is given by Ull ¼ I. It is assumed that r and l are independent processes. Clearly,  e eq ¼ G e e1 e¼G

e e2 G

   r e e1 r þ G e e2 l ¼ e1 þ e2 ¼G l

ð3:1Þ

 e dq ¼ G e d1 yd ¼ G

e d2 G

   r e d1 r þ G e d2 l ¼ yd1 þ yd2 ¼G l

ð3:2Þ

and

where e1 and yd1 account for the shape-deterministic components of e and yd, respectively, and e2 and yd2 account for the stochastic components of e and yd, respectively. With persistent inputs such as steps, ramps, and growing exponentials e e1 is unstable. However, there is no loss in generality in assuming that the matrix G e e2 is stable when the power spectral density matrix for e2 is good and nonsingular G on the finite jx-axis in the analog case or is good and nonsingular on jkj ¼ 1 in the digital case since, in accordance with Corollaries C.3 and C.4 of Appendix C, one e e2 a stable left Wiener–Hopf factor of the power spectral density can choose for G e d2 and so it also is assumed to be stable. matrix for e2. Similar reasoning applies to G The elements of yc are the physical plant outputs (e.g., yt and ys in Fig. 2.1) that are required to track reference inputs as well as other signals likely to cause saturation whose deviations from a required response must be kept small. That is, a fundamental objective is that yc track yd. As a first step toward satisfying this requirement, C is chosen from among the stabilizing controllers so that yd1 , the shape-deterministic component of yd , is stable. Then the inverse transform of each element of yd1 approaches zero asymptotically. Of course, no progress can be made e is admissible. Otherwise, no stabilizing controllers without the assumption that G even exist. An additional consideration in the choice of the controller is the need in response to persistent exogenous inputs to prevent unbounded inputs to the actual plant (e.g., the inputs to Pp in Fig. 2.1) in order to avoid saturation. Hence, the shape-deterministic component of the actual plant input must be stable except

3.2 Preliminary Considerations

117

possibly for simple poles on the stability boundary. Typically, the controller output v drives any precompensation of the actual plant and for robustness any needed simple poles of the actual plant input on the stability boundary are provided by the precompensation. (For example, in Fig. 2.1 any precompensation is included in Pa.) Then all permissible actual plant inputs can be generated with stable v along the lines described in the last few paragraphs of Sect. 1.3. So attention is restricted here to optimizing system performance over the subset of controllers for which both v1, the shape-deterministic component of v, and yd1 are stable. The first step in the process involves establishing the conditions under which such a subset of controllers exists and to parameterize this subset. Remark 3.1 It is possible to include the elements of v among the elements of yc e d to contain only zeros so that the elements and to pick the corresponding rows of G of v are also included among the elements of yd . Then the stability of v1 is assured by that of yd1 . However, since v is an explicit system variable, it is simpler and more insightful to take into account directly the requirement that v1 be stable whether or not v is included in yc.

3.3

Tracking and Disturbance Rejection with the Standard Configuration1

Clearly, ee  G e w q þ G12 v e 11 G e d Þq þ G e 12 v ¼ G11 G yd ¼ ð G

ð3:3Þ

e eq þ G e w q þ G22 v e 21 G e 22 v ¼ G21 G ym ¼ G

ð3:4Þ

and

where  e 11 G11 ¼ G

 e 12 ; G21 ¼ ½ G e 21 I ; G12 ¼ G

e 22 ; 0 ; G22 ¼ G

ð3:5Þ

and ew ¼ G

1



ee G e Gd



 ¼

e e1 G e G d1

e e2 G e G d2



 e w1 ¼ G

 e w2 : G

ð3:6Þ

This section contains, with permission, some material equivalent to that in Park and Bongiorno (2009), © 2009 Taylor & Francis Ltd, http://www.tandfonline.com.

3 H2 Design of Multivariable Control Systems

118

Using (3.4) in v ¼ Cym , solving for v, and substituting the result into (3.3) then yields e wq v ¼ ðI  CG22 Þ1 CG21 G

ð3:7Þ

e w q: yd ¼ ½G11 þ G12 ðI  CG22 Þ1 CG21  G

ð3:8Þ

and

Now in accordance with Lemma B.5, one can introduce coprime polynomial matrix fraction descriptions e 22 ¼ A1 B ¼ B1 A1 G22 ¼ G 1

ð3:9Þ

satisfying 

A Y1

B X1



X Y

B1 A1





X ¼ Y

B1 A1



A Y1

B X1





I ¼ 0

 0 : I

ð3:10Þ

(The coprime polynomial matrix fraction description A1 B ¼ B1 A1 1 is used to represent G, G22 , or P22 in different sections of this monograph. It is always made clear which is intended so that no confusion arises.) It then follows from Theorem 2.1 and Lemma 2.2 that all stabilizing controller transfer matrices are given by C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1

ð3:11Þ

where K is a stable rational matrix such that det ðX1  KBÞ 6 0. Substituting the first expression for C in (3.11) into (3.7) and (3.8) gives after straightforward calculations e w q ¼ Gv G e w1 r þ Gv G e w2 l ¼ v1 þ v2 v ¼ Gv G

ð3:12Þ

e w q ¼ Gd G e w1 r þ Gd G e w2 l ¼ yd1 þ yd2 yd ¼ Gd G

ð3:13Þ

Gv ¼ A1 ðY1 þ KAÞG21 ¼ ðY þ A1 KÞAG21

ð3:14Þ

and

where

3.3 Tracking and Disturbance Rejection with the Standard …

119

and Gd ¼ G11 þ G12 A1 ðY1 þ KAÞG21 ¼ G11 þ G12 Gv :

ð3:15Þ

The following lemma plays an important role in the sequel and provides some e useful results associated with the admissibility of the generalized plant G. Lemma 3.1 The matrices G12 A1 , AG21 , and G11 þ G12 A1 Y1 G21 are all stable e is admissible. Moreover, when G e is admissible and K is stable, then Gv when G and Gd are stable. e is admissible it follows that G e 12 A1 , Proof On account of Lemma 2.3, when G e 21 are stable. Hence, e 11 þ G e 12 A1 Y1 G e 21 , and G AG  e 12 A1 ; AG21 ¼ A G e 21 G12 A1 ¼ G

  e 21 0 ¼ AG

 0 ;

ð3:16Þ

and   e 12 A1 Y1 ½ G e 11 I þ G e 21 0  G11 þ G12 A1 Y1 G21 ¼  G  e 21 Þ I e 11 þ G e 12 A1 Y1 G ¼ ðG

ð3:17Þ

are stable. It now readily follows from (3.14) and (3.15) that Gv and Gd are stable since K is stable.   e w and e 11 I G Remark 3.2 One can instead of as in (3.5) choose G11 ¼ G   e w so that the system in Fig. 3.1 is absorbed into a standard e 21 0 G G21 ¼ G configuration with 

yd ym



 ¼

G11 G21

G12 G22

  q v

ð3:18Þ

and impose on this structure the admissibility constraints AG21 , G12 A1 , G11 þ G12 A1 Y1 G21 stable. But it is unwise to do so. For it then follows from ym ¼ G21 q þ G22 v ¼ G21 q þ G22 Cym or, equivalently, ym ¼ ðI  G22 CÞ1 G21 q that the shape-deterministic component of ym , ym1

    r I ¼ ½I þ A BðY þ A1 KÞðX  B1 KÞ  G21 ¼ ðX  B1 KÞAG21 r; 0 0 1

1 1

ð3:19Þ must be stable. Now for the 3DOF system shown in Fig. 1.1 and the 2DOF standard configuration shown in Fig. 1.3, the elements of the controller input u ¼ r þ n are included in ym. Since the noise n is typically purely stochastic, the shape-deterministic components of u are those of r which then have to be stable.

3 H2 Design of Multivariable Control Systems

120

That is, systems of the type shown in Figs. 1.1 and 1.3 with persistent shape-deterministic reference inputs are excluded from consideration. In contrast, the choices made in (3.5) lead to no such restriction. Specifically, (3.3) and (3.4) correspond to the generalized persistent input standard configuration shown in Fig. 3.2 for which G, on account of Lemma 3.1, does satisfy G12 A1 , AG21 , G11 þ G12 A1 Y1 G21 stable without imposing the undesirable restrictions on 2DOF and 3DOF systems cited above. It is also important to point out for Fig. 3.2 that the transfer matrices from w to v and from w to yd are, respectively, Gv and Gd. In addition, Fig. 3.2 is a more general representation in that it includes Fig. 3.1 when e might the choices (3.5) are made, but can represent other configurations in which G be embedded (e.g., see Remark 3.6 in the sequel). Attention is of course restricted e is admissible, C belongs to the set of all stabilizing to those cases in which G e and G22 ¼ G e 22 . In this regard, it is useful to introduce controllers for G, Definition 3.1 The subsystem G in Fig. 3.2 is said to be compatible when the e within it is admissible; G22 ¼ G e 22 ; and G12 A1 , AG21 , and physical system G G11 þ G12 A1 Y1 G21 are stable. It is assumed that the set of all possible outcomes for r is sufficiently broad so e w1 r and yd1 ¼ Gd G e w1 r are both stable for all outcomes r iff Gv G e w1 that v1 ¼ Gv G e and Gd G w1 are both stable. For example, this is the case when the set of all possible outcomes includes all the possibilities in which only one element of r is nonzero.   e w1 and e w2 ¼ G e0 e 0 0 are stable, it follows that Gv G Since Gv , Gd , and G G e2 d2 e e e Gd G w1 are both stable iff Gv G w and Gd G w are both stable. In the optimization problem treated in the sequel one has to identify the subset of all stabilizing controllers for which Gv Gwo and Gd Gwo are both stable where with a1 a positive  pffiffiffiffiffi ffie  e w1 pffiffiffiffi constant and a2 a nonnegative constant, Gwo ¼ a2 G a1 G w2 . Clearly, e w1 and e w2 is stable, the subset of all stabilizing controllers for which Gv G when G e Gd G w1 are both stable and the subset of all stabilizing controllers for which Gv Gwo and Gd Gwo are both stable is the same. This observation is used to advantage in Sects. 3.7 and 3.8 and provides an alternative approach to the one taken in Sects. 3.5 and 3.6 (see Example 3.9). Clearly, conditions on an arbitrary matrix Gw under which one can choose a stable K so that GvGw and GdGw are both stable are of interest. These conditions are now derived and a parameterization of all such K in

Fig. 3.2 Generalized persistent input standard configuration (© 2009 Taylor & Francis Ltd, http://www. tandfonline.com. Reprinted, with permission, from Park and Bongiorno (2009))

w

q

v

3.3 Tracking and Disturbance Rejection with the Standard …

121

terms of an arbitrary stable rational matrix is given. For this purpose, it is useful to introduce Definition 3.2 A matrix Gw is said to be acceptable for a compatible G when there exists a stable matrix K such that GvGw and GdGw are both stable when Gv and Gd are given by (3.14) and (3.15), respectively. It is sometimes appropriate and possible with persistent inputs to choose a stable e w1 r is also stable, where K so that in addition to v1 and yd1 being stable, ym1 ¼ Gm G Gm is the transfer matrix between w and ym in Fig. 3.2: ym ¼ Gm w and Gm ¼ ðG21 þ G22 Gv Þ. With regard to this case, it is convenient to introduce Definition 3.3 A matrix Gw is said to be strictly acceptable for a compatible G when there exists a stable matrix K such that GvGw, GdGw and GmGw are stable. Treatment of this concept is relegated to Examples 3.21 and 3.22 for the standard configuration in Fig. 3.2 and justified in connection with 1DOF systems in Sect. 3.7. The first step in establishing the conditions under which Gw is acceptable for a compatible G is to introduce the left-coprime polynomial matrix description Gw ¼ A1 w Bw . Since Gv and Gd are stable, it follows from Lemma B.8 that GvGw and GdGw are stable iff 1 W ¼ Gv A1 w ¼ ðY þ A1 KÞAG21 Aw

ð3:20Þ

1 1 U ¼ Gd A1 w ¼ ðG11 þ G12 Gv ÞAw ¼ G11 Aw þ G12 W

ð3:21Þ

and

are stable. Of course this is the case when A1 is stable. When persistent w e shape-deterministic inputs are present, however, G w1 and, therefore, A1 are w unstable. In this case, the choice of stable K in the controller must be further constrained so that both U and W are stable. Clearly, there is no choice of stable K for which both U and W are stable unless (3.21) or, equivalently, U  G12 W ¼ G11 A1 w

ð3:22Þ

admits stable solution pairs U, W: For the purpose of addressing this issue, it is convenient to introduce the coprime polynomial matrix fraction descriptions 1 G12 ¼ A1 o Bo ¼ Bo1 Ao1

ð3:23Þ

3 H2 Design of Multivariable Control Systems

122

where 

Ao Yo1

Bo Xo1



Xo Yo

Bo1 Ao1



 ¼

Xo Yo

Bo1 Ao1



Ao Yo1

Bo Xo1



 ¼

 I 0 : ð3:24Þ 0 I

Then (3.22) is equivalent to D

Ao U þ Bo W ¼ Ao G11 A1 w ¼ Cw :

ð3:25Þ

Since U and W must be stable, Cw must be a stable matrix; this is the first necessary condition on the data. Moreover, U and W must be stable solutions of ed ¼ (3.25). Equation (3.25) has the same form as (2.242) with Ap, Bp, and T 1 Ap Td Ar replaced by Ao, Bo, and Cw , respectively. Hence, by the same reasoning that yielded (2.249) and (2.250) one gets when Cw is stable that all possible stable solution pairs U; W of (3.25) are given by U ¼ Xo Cw  Bo1 Kw ; W ¼ Yo Cw þ Ao1 Kw

ð3:26Þ

where Kw is a stable rational matrix. These solution pairs assure that (3.21) is satisfied. So all that remains is the satisfaction of (3.20) when W is given by (3.26). In particular, one has to identify the stable K, Kw pairs for which WAw ¼ ðYo Cw þ Ao1 Kw ÞAw ¼ ðY þ A1 KÞAG21

ð3:27Þ

ðY þ A1 KÞAG21 A1 w ¼ ðYo Cw þ Ao1 Kw Þ:

ð3:28Þ

or, equivalently,

In order to complete this final step, one more coprime polynomial matrix fraction description is needed. Specifically, 1 1 AG21 A1 w ¼ Ag Bg ¼ Bg1 Ag1

ð3:29Þ

where 

Ag Yg1

Bg Xg1



Xg Yg

Bg1 Ag1





Xg ¼ Yg

Bg1 Ag1



Ag Yg1

Bg Xg1





 I 0 ¼ : ð3:30Þ 0 I

It readily follows from (3.28) and (3.29) that ðY þ A1 KÞBg1 ¼ ðYo Cw þ Ao1 Kw ÞAg1

ð3:31Þ

3.3 Tracking and Disturbance Rejection with the Standard …

123

or D

Ko Bg1  Kw Ag1 ¼ A1 o1 ðYo Cw Ag1  YBg1 Þ ¼ Co

ð3:32Þ

Ko ¼ A1 o1 A1 K:

ð3:33Þ

where

When G is compatible, G12 A1 ¼ Bo1 A1 o1 A1 is stable; this is the case according 1 to Lemma B.8 iff Ao1 A1 is stable. So Ko must be stable and it follows from (3.32) that Co must be stable. This is the second necessary condition on the data. It also follows from the solution of Example 2.9 that when Co is stable all stable solution pairs KoKw of (3.32) are given by Ko ¼ Co Yg1 þ Km Ag ; Kw ¼ Co Xg1 þ Km Bg

ð3:34Þ

where Km is a stable matrix. So from (3.33), a stable solution pair K,Kw exists iff 1 K ¼ A1 1 Ao1 Ko ¼ A1 Ao1 ðCo Yg1 þ Km Ag Þ

ð3:35Þ

is stable for some stable Km. When A1 1 Ao1 is stable this is the case for every stable Km. Moreover, Kg ¼ A1 A K is stable iff Km is stable because as previously o1 m 1 1 pointed out Ao1 A1 stable follows from G12 A1 stable. So when A1 1 Ao1 is stable, (3.35) is equivalent to K ¼ A1 A C Y þ K A where K is a stable matrix. Thus, o1 o g1 g g g 1 without the need of further proof one has Theorem 3.1 A matrix Gw ¼ A1 w Bw is acceptable for a compatible G iff Cw ¼ 1 Ao G11 A1 and C ¼ A ðY C A  YBg1 Þ are both stable and one can choose a o o w g1 w o1 stable matrix Km in (3.35) for which K is stable. Moreover, when A1 1 Ao1 is stable one gets for every stable matrix Km a stable matrix K in (3.35) and all such K are given by 1 K ¼ A1 1 Ao1 ðCo Yg1 þ Km Ag Þ ¼ A1 Ao1 Co Yg1 þ Kg Ag

ð3:36Þ

where Kg is a stable matrix. Remark 3.3 There is no guarantee under Theorem 3.1 that det ðX1  KBÞ 6 0 for any of the K given by (3.36) as required by (3.11). However, practical design objectives for analog systems usually demand that S ¼ ðI  CG22 Þ1 ¼ A1 ðX1  KBÞ ! I;

s ! 1:

ð3:37Þ

3 H2 Design of Multivariable Control Systems

124

Hence, for any K meeting this requirement one gets det S ¼ ðdet A1 Þdet ðX1  KBÞ ! 1;

s!1

ð3:38Þ

and det ðX1  KBÞ 6 0. For digital systems it is usually the case in practice that det A1 ðkÞ 6¼ 0, B1 ðkÞ ¼ 0, and BðkÞ ¼ 0 at k ¼ 0. Then, since a stable KðkÞ is analytic in jkj  1, X1 ð0Þ  Kð0ÞBð0Þ ¼ X1 ð0Þ:

ð3:39Þ

X1 ð0ÞA1 ð0Þ þ Y1 ð0ÞB1 ð0Þ ¼ X1 ð0ÞA1 ð0Þ ¼ I;

ð3:40Þ

In addition,

hence, det X1 ð0Þ ¼ 1=detA1 ð0Þ 6¼ 0. That is, det ½X1 ð0Þ  Kð0ÞBð0Þ ¼ det X1 ð0Þ ¼ 1=det A1 ð0Þ 6¼ 0

ð3:41Þ

and so det ðX1  KBÞ 6 0 for any stable K. For additional insights regarding the realization of C when (3.36) applies see Example 3.19. Also, see Example 3.3 for alternative expressions one can use to check whether or not Cw and Co are stable. Remark 3.4 When, irrespective of whether or not A1 1 Ao1 is stable, there exists a stable matrix Kmo for which Ks ¼ A1 1 Ao1 ðCo Yg1 þ Kmo Ag Þ

ð3:42Þ

Km ¼ Kmo þ A1 o1 A1 Kg ;

ð3:43Þ

is stable and one chooses

then Km is stable for all stable Kg and (3.35) yields K ¼ Ks þ Kg Ag

ð3:44Þ

which is stable for any stable matrix Kg. That is, when there is one stable Km for which the right-hand side of (3.35) is stable, then there is a family of such matrices given by (3.43) as Kg ranges over all stable matrices. However, this family may not be the complete set of stable matrices Km for which the right-hand side of (3.35) is stable when A1 1 Ao1 is unstable: in this case there may be unstable Kg for which 1 both Ao1 A1 Kg and KgAg are stable.

3.3 Tracking and Disturbance Rejection with the Standard …

125

1 Remark 3.5 When AG21 A1 w is stable, it follows from (3.29) that Ag is stable. Hence, in this case, (3.35) has a stable solution for Km given by Km ¼ ðA1 o1 A1 K  1 Co Yg1 ÞAg for every stable matrix K. That is, when G is compatible and when AG21 A1 w , Cw , and Co are stable then GvGw and GdGw are stable for any stable K (See Example 3.2).

Remark 3.6 It is sometimes the case, as with the 3DOF system tracking and e e1 e d1 ¼ T ed G disturbance rejection problem addressed in Sect. 3.4, that l ¼ 0 and G e where T d is a stable matrix. Then the final forms of (3.3) and (3.4) still hold with e e1 in place of G e w , r in place of q, and G e 11  T e 12 ; G21 ¼ G e 21 ; G22 ¼ G e 22 ed ; G12 ¼ G G11 ¼ G

ð3:45Þ

in place of (3.5). Now e 21  T e 11 þ G e 12 A1 Y1 G ed G11 þ G12 A1 Y1 G21 ¼ G

ð3:46Þ

e is admissible since T e is admissible and T ed is stable. So when G ed is is stable when G stable it again readily follows from (3.45) and (3.46) that G is compatible because e 22 . Hence, when G12 A1 , AG21 , and G11 þ G12 A1 Y1 G21 are all stable and G22 ¼ G e e Gw ¼ G e1 , the Gij are given by (3.45), and T d is stable, Theorem 3.1 gives the e e1 to be acceptable (See Example 3.7). necessary and sufficient conditions for G

3.4

Tracking and Disturbance Rejection in the 3DOF System Revisited

It is instructive and informative to reconsider the tracking and disturbance rejection problem treated in Sect. 2.6 for the 3DOF system using Theorem 3.1 and Remark 3.6 instead. Specifically, the system shown in Fig. 1.1 is considered with r and d solely deterministic and the only nonzero inputs. That is,    1    r d Ad Bd rd 1 e ¼ e1 ¼ ¼ ¼ Ae Be d r rr A1 B r r r r

ð3:47Þ

where 

Ad Ae ¼ 0

0 Ar





Bd Be ¼ 0

 0 : Br

ð3:48Þ

3 H2 Design of Multivariable Control Systems

126

When Ad, Bd and Ar, Br are both left-coprime polynomial matrix pairs, it follows that x0 ½ A e

 Be  ¼ ½ x01

x02 

Ad 0

0 Ar

Bd 0

0 Br

 ¼0

ð3:49Þ

iff x01 ½ Ad

Bd  ¼ 0; x02 ½ Ar

Br  ¼ 0

ð3:50Þ

or x01 ¼ 0 and x02 ¼ 0 since ½ Ad Bd  and ½ Ar Br  have full row rank everywhere (Lemma B.4). Hence, x ¼ 0 is the only solution possible in (3.49) and the pair e e1 ¼ A1 Bw with Aw ¼ Ae , Bw ¼ Be . Ae ; Be is left-coprime. Then one can set G w The controlled output yc is taken to be y and obviously ym ¼ ½ w0 z0 u0 0 . The noise inputs n, nl, and nm are usually purely stochastic and, therefore, do not contribute to the shape-deterministic components of yc and ym. This justifies the restriction of concern here to the case in which only r and d are nonzero inputs. Clearly, yc ¼ y ¼ Pv þ Pd d ¼ ½ Pd

0 e þ Pv

ð3:51Þ

3 2 3 0 FP 0 5e þ 4 0 5v: I 0

ð3:52Þ

and 3 2 3 2 FPd w FPv þ FPd d 5¼4 L Ld ym ¼ 4 z 5 ¼ 4 r 0 u 2

It readily follows that e 11 ¼ ½ Pd G

e 12 ¼ P; 0 ; G12 ¼ G

ð3:53Þ

and 2

e 21 G21 ¼ G

FPd ¼4 L 0

3 2 3 0 FP e 22 ¼ 4 0 5 0 5; G22 ¼ G I 0

ð3:54Þ

when (3.45) is invoked. Moreover, yd is connected to the error e by yd ¼ yc  yd ¼ y  Td r ¼ e

ð3:55Þ

3.4 Tracking and Disturbance Rejection in the 3DOF System Revisited

127

and ed e: yd ¼ Td r ¼ ½ 0 Td e ¼ T

ð3:56Þ

Thus, e 11  T ed ¼ ½ Pd G11 ¼ G

Td :

ð3:57Þ

Attention is restricted to the case in which Td is stable. Clearly (see Remark 3.6), e is admissible and Theorem 3.1 can be invoked. G is then compatible when G In order to proceed, the coprime polynomial matrix fraction descriptions defined in (3.9), (3.23), and (3.29) have to be identified. For this purpose, the following coprime polynomial matrix fraction descriptions are introduced: FP ¼ A1 B ¼ B1 A1 1

ð3:58Þ

with 

A Y1

B X1



X Y

B1 A1



 ¼

X Y

B1 A1



A Y1

B X1



 ¼

I 0

0 I

 ð3:59Þ

and 1 P ¼ A1 p Bp ¼ Bp1 Ap1

ð3:60Þ

with 

Ap Yp1

Bp Xp1



Xp Yp

Bp1 Ap1





Xp ¼ Yp

Bp1 Ap1



Ap Yp1

Bp Xp1





 I 0 ¼ : ð3:61Þ 0 I

Clearly, 2

G22

3 FP e 1 e 1 B e ¼ B e1A ¼ 4 0 5 ¼ A 1 0

ð3:62Þ

where 2

A e ¼40 A 0

0 I 0

3 2 3 0 B e¼4 0 5 0 5; B I 0

ð3:63Þ

3 H2 Design of Multivariable Control Systems

128

and 2

3 B1 e 1 ¼ A1 ; B e 1 ¼ 4 0 5: A 0

ð3:64Þ

Moreover, one can introduce 2

X e ¼40 X 0

3 0 0 e ¼ ½ Y I 0 5; Y 0 I

0

0

ð3:65Þ

and e 1 ¼ X1 ; Y e1 ¼ ½ Y1 X

0 0 :

ð3:66Þ

e B, e 1, B e 1, X e, Y e, X e 1 , and Y e1 are the e A That is, for the 3DOF system, A, replacements for A, B, A1, B1, X, Y, X1, and Y1, respectively, in (3.9) and (3.10). It should also be obvious from (3.23) and (3.53) that Ao ¼ Ap ; Bo ¼ Bp ; Ao1 ¼ Ap1 ; Bo1 ¼ Bp1

ð3:67Þ

Xo ¼ Xp ; Yo ¼ Yp ; Xo1 ¼ Xp1 ; Yo1 ¼ Yp1 :

ð3:68Þ

and

The next step involves the determination of the coprime polynomial matrix fraction descriptions in (3.29). Clearly, they are determined by 2

e 21 A1 AG w

A ¼40 0

0 I 0

32 0 FPd 0 54 L 0 I

3 0  A 05 d 0 I

0 Ar

1

2

AFPd A1 d 4 ¼ LA1 d 0

3 0 0 5 ð3:69Þ A1 r

for the 3DOF system. Further progress is possible with the introduction of the coprime polynomial matrix fraction descriptions 

AFPd A1 d LA1 d



1 ¼ A1 q Bq ¼ Bq1 Aq1

ð3:70Þ

with 

Aq Yq1

Bq Xq1



Xq Yq

Bq1 Aq1





Xq ¼ Yq

Bq1 Aq1



Aq Yq1

Bq Xq1





 I 0 : ð3:71Þ ¼ 0 I

3.4 Tracking and Disturbance Rejection in the 3DOF System Revisited

Then (3.69) yields  Aq 1 e AG21 Aw ¼ 0

0 Ar

1 

Bq 0

  0 Bq1 ¼ I 0

0 I

It is now easy to confirm that      Aq 0 Bq 0 Aq1 Ag ¼ ; Ag1 ¼ ; Bg ¼ 0 Ar 0 I 0



Aq1 0

129

0 Ar

1 :

ð3:72Þ

  0 Bq1 ; Bg1 ¼ Ar 0

0 I



ð3:73Þ and  Xg ¼

Xq 0

  0 Yq ; Yg ¼ 0 0

  0 Xq1 ; Xg1 ¼ I 0

  0 Yq1 ; Yg1 ¼ 0 0

0 I

 ð3:74Þ

satisfy (3.30). Hence, coprime polynomial matrix fraction descriptions that can be associated with (3.29) are provided by (3.73). All the information needed for the evaluation of Cw and Co in Theorem 3.1 is now available. One easily gets  1   Ad 0 : Cw ¼ Ao G11 A1 ¼ A ½ P T  ¼ Ap Pd A1 Ap Td A1 p d d w d r 0 Ar ð3:75Þ Also,

⎡A ~ Γo = Ao−11 (Yo Γw Ag1 − Y Bg1 ) = − Ap−11Y p Γw ⎢ q1 ⎣ 0

0⎤ − Ap−11[− Y Ar ⎥⎦

⎡B 0 0]⎢ q1 ⎣ 0

0⎤ I ⎥⎦ ð3:76Þ

or Co ¼ ½ Co1

Co2  where

1 1 Co1 ¼ A1 p1 Yp Ap Pd Ad Aq1 þ Ap1 ½ Y

1 0 Bq1 ¼ Yp1 Pd A1 d Aq1 þ Ap1 ½ Y

0 Bq1 ð3:77Þ

and Co2 ¼ A1 p1 Yp Ap Td ¼ Yp1 Td :

ð3:78Þ

Clearly, Co2 is stable when Td is stable. When F and Cw are also stable, it can be shown as follows that Co1 is stable. Solving (3.70) for Bq1 and using the result in (3.77) yields 1 1 1 1 1 Co1 ¼ Yp1 Pd A1 d Aq1 þ Ap1 YAFPd Ad Aq1 ¼ Ap1 ðYp þ YAFAp ÞAp Pd Ad Aq1 :

ð3:79Þ

3 H2 Design of Multivariable Control Systems

130

Now Yp þ YAFA1 p ¼ YAFðXp þ PYp Þ  Yp ¼ YAFXp þ ðYB  IÞYp ¼ A1 ðY1 FXp  X1 Yp Þ

ð3:80Þ

and one gets from (3.79) 1 Co1 ¼ A1 p1 A1 ðY1 FXp  X1 Yp ÞAp Pd Ad Aq1 :

ð3:81Þ

Since G is compatible, one has that e1 ¼ G e 1 ¼ PA1 ¼ Bp1 A1 A1 e 12 A G12 A p1

ð3:82Þ

is stable; hence, A1 p1 A1 is stable. It is now easy to see when F and Td are stable that 1 Cw and Co are both stable iff Ap Pd A1 d and Ap Td Ar are both stable. Necessity is obvious from (3.75) and sufficiency follows immediately from (3.75), (3.78), and (3.81). That is, when F and Td are stable and the 3DOF system is admissible then 1 1 Gw ¼ Ge1 ¼ A1 e Be is acceptable iff Ap Pd Ad and Ap Td Ar are both stable. Clearly, this result is consistent with Theorems 2.4 and 2.6. Moreover, e 1 Ao1 ¼ A1 Ap1 ¼ ðX1 þ Y1 FPÞAp1 ¼ X1 Ap1 þ Y1 FBp1 A 1 1

ð3:83Þ

is stable and (3.36) applies. It is also of interest to examine the formula (3.36) which for the 3DOF system becomes K ¼ A1 1 Ap1 Co Yg1 þ Kg Ag . After the appropriate substitutions one gets  K ¼ Cd Aq1 Yq1

 A1 1 Ap1 Yp1 Td þ ½ Kg1

 Kg2 

Aq 0

0 Ar

 ¼ ½ ½ K1

H2  H1  ð3:84Þ

where Cd ¼ ðY1 FXp  X1 Yp ÞAp Pd A1 d :

ð3:85Þ

The final expression in (3.84) evolves from (2.157) and (2.205). One immediately gets H1 ¼ A1 1 Ap1 Yp1 Td þ Kg2 Ar

ð3:86Þ

3.4 Tracking and Disturbance Rejection in the 3DOF System Revisited

131

1 where Kg2 is an arbitrary stable matrix. Since A1 p1 A1 and A1 Ap1 are both stable, there is no loss in generality to set Kg2 ¼ A1 1 Ap1 Kr in (3.86) to get in agreement with (2.257)

H1 ¼ A1 1 Ap1 ðYp1 Td þ Kr Ar Þ

ð3:87Þ

where Kr is an arbitrary stable matrix. Equation (3.84) also gives, since Aq1 Yq1 ¼ Yq Aq , ½ K1

H2  ¼ ðCd Yq þ Kg1 ÞAq ¼ Kgo Aq

ð3:88Þ

where Kgo is an arbitrary stable matrix: one can always choose Kg1 ¼ Kgo  Cd Yq because Cd is stable. The final expression in (3.88) is not yet in agreement with (2.297). Further progress requires exploiting the fact that the left-hand side of (3.70) can be expressed as 

M N



 ¼

AFPd A1 d LA1 d



 ¼

A1 m Bm A1 n Bn



 ¼

Am 0

0 An

1 

Bm Bn

 ð3:89Þ

where Am ; Bm and An ; Bn are left-coprime polynomial matrix pairs and recognizing that 1 1 M ¼ AFPd A1 d ¼ ðAFAp ÞðAp Pd Ad Þ

ð3:90Þ

1 is stable when Ap Pd A1 d and AFAp are stable. The former follows from (3.75) and the latter is the case because

AFA1 p Bp ¼ AFP ¼ B

ð3:91Þ

1 is stable and this is true when F is stable iff AFA1 p is stable. So Am is a stable matrix. When, moreover,

 Aq ¼

Am 0

   Bm 0 ; Bq ¼ An Bn

ð3:92Þ

constitutes a left-coprime polynomial matrix pair, then it follows from (3.70) and  in (3.88) one gets (3.89) that with Kgo ¼  Km A1 K n m ½ K1

 H2  ¼ Km A1 m

Kn

  Am 0

0 An

 ¼ ½ Km

Kn An 

ð3:93Þ

where Km and Kn are arbitrary stable matrices. Equation (3.93) is in agreement with (2.297) when notational differences are taken into account.

3 H2 Design of Multivariable Control Systems

132

The fact that Am ; Bm and An ; Bn are left-coprime polynomial matrix pairs, however, is not sufficient to assure that the pair Aq ; Bq given by (3.92) is left-coprime although this is true when det Am and det An have no common zeros which is typically the case (see Example 3.6). Nevertheless, one can still arrive at (3.93) by directly specializing (3.28) to the 3DOF system beginning with e 1 KÞ AG e 21 A1 ¼ ðYo Cw þ Ao1 Kw Þ: eþA ðY w

ð3:94Þ

From (3.69) and (3.89) 2

e 21 A1 AG w

M ¼4N 0

3 0 0 5: A1 r

ð3:95Þ

One then gets from (3.94) that 2

ð½ Y

0

0  þ A1 ½ K1

H2

3 M H1 Þ4 N 5 ¼ Yp Ap Pd A1 d þ Ap1 Kw1 ð3:96Þ 0

where Kw1 is the appropriate first block column of Kw . Hence, ðY þ A1 K1 ÞM þ A1 H2 N ¼ Yp Ap Pd A1 d  Ap1 Kw1

ð3:97Þ

1 1 H2 N ¼ A1 1 ðYp Ap Pd Ad  YMÞ  ðA1 Ap1 Kw1 þ K1 MÞ:

ð3:98Þ

or

1 1 Now ðA1 1 Ap1 Kw1 þ K1 MÞ is stable and using A1 ¼ X1 þ Y1 FP and Ap ¼ Xp þ PYp one gets 1 1 1 1 1 A1 1 ðYp Ap Pd Ad  YMÞ ¼ A1 Yp Ap Pd Ad  A1 YAFPd Ad 1 1 1 ¼ ðA1 1 Yp  Y1 FAp ÞAp Pd Ad ¼ ðX1 Yp  Y1 FXp ÞAp Pd Ad

ð3:99Þ 1 which is also stable when F and Ap Pd A1 d are stable. So H2 N ¼ H2 An Bn is stable and this is true for stable H2 iff H2 ¼ Ks An where Ks is an arbitrary stable matrix. In particular, for any stable K1 and stable Ks one can always pick a stable Kw1 so that (3.98) and, therefore, (3.96) is satisfied. Hence, consistent with (2.297) when notational differences are taken into account,

½ K1

H2  ¼ ½ K1

where K1 and Ks are arbitrary stable matrices.

Ks An 

ð3:100Þ

3.5 H2 Design of Digital Multivariable Control Systems

3.5

133

H2 Design of Digital Multivariable Control Systems

The case in which the generalized persistent input standard configuration of Fig. 3.2 represents a digital system is considered in this section. In this instance the H2 performance measure used for transient responses is Jtr ¼

¼

1 2pj

ð H 

Tr ½hyd1 ðkÞyd1 ðkÞi

jkj¼1

dk k

h i 1 ð e w1 hrr0 i G e w1 Gd dk H  Tr Gd G 2pj k jkj¼1

¼

h i 1 ð e w1 G e w1 Gd dk H  Tr Gd G 2pj k

ð3:101Þ

jkj¼1

and Jss ¼

  dk 1 ð H  Tr Uyd2 yd2 ðkÞ 2pj k jkj¼1

¼

h i 1 ð e w2 Ull G e w2 Gd dk H  Tr Gd G 2pj k jkj¼1

¼

h i 1 ð e w2 G e w2 Gd dk H  Tr Gd G 2pj k

ð3:102Þ

jkj¼1

is used to measure steady-state stochastic performance. These two measures are incorporated into the single one J¼

1 ð dk H  Tr ½QGd Gw Gw Gd  2pj k

ð3:103Þ

jkj¼1

where flexibility for weighting the relative importance of the rows of Gd Gw or, equivalently, the elements of yd is accomplished with the addition of a parahermitian-positive weighting matrix Q that is good. In addition, Gw ¼

 pffiffiffiffiffi e w1 a1 G

pffiffiffiffiffi e  a2 G w2

ð3:104Þ

3 H2 Design of Multivariable Control Systems

134

where a1 is a positive constant and a2 is a nonnegative constant which allow for weighting the relative importance of the transient and steady-state contributions to system performance. Invoking Theorem C.2, one can always write Q ¼ W W where W is a stable matrix because Q is good. Then J¼

1 ð dk : H  Tr ½ðWGd Gw ÞðWGd Gw Þ  2pj k

ð3:105Þ

jkj¼1

The design objective is to find the stabilizing controller C or, equivalently, find the stable matrix K for which GvGw and GdGw are stable and the cost functional J is minimum when G is compatible and Gw is acceptable for G. Attention is restricted to the case in which A1 1 Ao1 is stable so that in accordance with Theorem 3.1 all permissible K for which GvGw and GdGw are stable are given by either one of the expressions in (3.36) with Km or Kg an arbitrary stable matrix. It is preferable to work with Km as the arbitrary stable matrix parameter when W is a polynomial matrix: in this case, the matrices that must be factored to obtain the solution of the optimization problem being treated are polynomial. This is the approach taken here. For all such Km, WGdGw is stable and the cost function J has practical significance as pointed out in Chap. 1; hence, it does make sense to seek that Km for which J is a minimum. It easily follows from (3.21), (3.26), and (3.34) that WGd Gw ¼ WUBw ¼ WðXo Cw þ Bo1 Co Xg1  Bo1 Km Bg ÞBw ¼ ra  rb Km rc ð3:106Þ where ra ¼ WðXo Cw þ Bo1 Co Xg1 ÞBw ; rb ¼ WBo1 ; rc ¼ Bg Bw

ð3:107Þ

are all stable and determined by the given data. So the task remaining is to find the stable Km for which J is minimized. In this regard it is convenient to introduce q ¼ Tr ½ðra  rb Km rc Þðra  rb Km rc Þ  ¼ Tr ½ðra  rb Km rc Þðra  rc Km rb Þ

ð3:108Þ

which is 2pjk times the integrand of J. Using Tr ðM þ NÞ ¼ Tr ðMÞ þ Tr ðNÞ and Tr ðMNÞ ¼ Tr ðNMÞ one easily gets q ¼ Tr ðra ra Þ þ Tr ðKm rc ra rb  rb ra rc Km þ Km rc rc Km rb rb Þ: ð3:109Þ

3.5 H2 Design of Digital Multivariable Control Systems

135

It is important to recognize that the two terms on the right-hand side of (3.109) cannot in general be combined into one because neither ∇a nor ra need have the same number of rows as Km. It follows from (3.109) that q ¼ qa þ Tr ðKm C  CKm þ Km XX Km K KÞ

ð3:110Þ

qa ¼ Tr ðra ra Þ; C ¼ rb ra rc ;

ð3:111Þ

where

and X and K are the Wiener–Hopf spectral factors satisfying Gb ¼ rb rb ¼ K K; Gc ¼ rc rc ¼ XX :

ð3:112Þ

Clearly, Gb and Gc are square and parhermitian-positive. Since ∇b and ∇c are good, so are Gb and Gc. Also, it is assumed here that both of the square matrices Gb and Gc are nonsingular on the unit circle. Then, in accordance with Theorem C.2 and Remark C.3, the matrices X and K can be chosen so that in addition to being square they are together with their inverses analytic in jkj  1. There is no need to distinguish between left and right inverses since X1 is both a left and right inverse of X and similarly for K1 when Gb and Gc have full rank on jkj ¼ 1 as assumed. It easily follows from (3.110) utilizing the properties of the trace and then completing the square that e e q ¼ qo þ Tr ½ðKKm X  CÞðKK m X  CÞ 

ð3:113Þ

eC e Þ e ¼ K1 CX1 ; qo ¼ qa  Tr ð C C  

ð3:114Þ

where

are both good because ∇a, ∇b, ∇c, K1 , and X1 are good. Clearly, consistent with the notation for partial fraction expansions introduced in Sect. 1.4, e ¼ Z  f Cg e KKm X  C 

ð3:115Þ

e e Z ¼ KKm X  f Cg þ  f Cg1

ð3:116Þ

where

3 H2 Design of Multivariable Control Systems

136

e e is stable for every stable Km: the matrix f Cg þ is stable because the matrix C is good. Conversely, for every stable Z 1 e e Km ¼ K1 ðZ þ f Cg þ þ f Cg1 ÞX

ð3:117Þ

is stable. So minimizing J over all stable Km is equivalent to minimizing J¼

o 1 ð dk 1 ð n e ÞðZ  f Cg e Þ  þ qo dk ¼ H  qðkÞ H  Tr ½ðZ  f Cg    2pj k 2pj k jkj¼1

jkj¼1

ð3:118Þ over all stable Z. Now for any real rational matrix MðkÞ one gets    Zp 1 ð dk 1 ð 1 dk 1 ¼ ¼ H  Tr ½M ðkÞ H  Tr M Tr ½Mðejh Þdh ð3:119Þ 2pj k 2pj k k 2p jkj¼1

jkj¼1

p

and Zp Zp 1 ð dk 1 1 jh ¼ H  Tr ½MðkÞ Tr ½Mðe Þdh ¼ Tr ½Mðejh Þdh: ð3:120Þ 2pj k 2p 2p jkj¼1

p

p

Thus, both integrals are equal to one another. Hence, 1 ð dk 1 ð e e Z g dk ¼ 0 ¼ H  Tr ½Zf Cg H  Tr ½f Cg    2pj k 2pj k jkj¼1

ð3:121Þ

jkj¼1

e where the last equality holds for the following reasons. First, Z and f Cg  are e analytic in jkj  1. Moreover, f Cg is strictly proper and, therefore, every nonzero e has a zero at k ¼ 1. Thus, every element of f Cg e element of f Cg   has a zero at k ¼ 0 and the integrand of the first integral in (3.121) is analytic in jkj  1; hence, the integral is zero as a consequence of the Cauchy Integral Theorem (Appendix A). It now readily follows from (3.118) that J¼

1 ð dk e f Cg e : H  fTr ½ZZ  þ Tr ½f Cg    þ qo g 2pj k jkj¼1

ð3:122Þ

3.5 H2 Design of Digital Multivariable Control Systems

137

When Z is an nr  nc matrix, 1 2pj

ð H 

Tr ½ZZ 

jkj¼1

1 ¼ 2p

Zp

dk 1 ð dk ¼ H  Tr ½Z Z k 2pj k jkj¼1

0

Tr ½Z ðe ÞZðe jh

jh

p

1 Þdh ¼ 2p

Zp X nc nr X zrc ðejh Þ 2 dh 0 p

ð3:123Þ

r¼1 c¼1

and the performance functional J is minimized when Z  0 is chosen. In this case one gets from (3.117) and (3.122) that Km and J are given by, respectively, 1 e e Kmo ¼ K1 ðf Cg þ þ f Cg1 ÞX

ð3:124Þ

and Jo ¼

1 ð dk e f Cg e : H  fTr ½f Cg    þ qo g 2pj k

ð3:125Þ

jkj¼1

Using the same reasoning as in connection with (3.123) it should be clear from (3.105) that J 0 for every choice of stable Km including Kmo. Thus, Jo 0. It is now convenient and possible without the need of further proof to state the key results derived above for digital systems as a theorem. Theorem 3.2 Under the assumptions (1) ∇a, ∇b, and ∇c stable; (2) det ðrb rb Þ 6¼ 0, jkj ¼ 1; and (3) det ðrc rc Þ 6¼ 0, jkj ¼ 1; the performance functional J¼

1 ð dk 1 ð dk ¼ ð3:126Þ H  q H  Tr ½ðra  rb Km rc Þðra  rb Km rc Þ  2pj k 2pj k jkj¼1

jkj¼1

is finite for all stable Km. Moreover, one can construct Wiener–Hopf spectral factors K and X that are square and together with their inverses are analytic in jkj  1 such that rb rb ¼ K K; rc rc ¼ XX

ð3:127Þ

3 H2 Design of Multivariable Control Systems

138

is satisfied. In addition, any stable Km can be written as 1 e e Km ¼ K1 ðZ þ f Cg þ þ f Cg1 ÞX

ð3:128Þ

e ¼ K1 CX1 ¼ K1 rb ra rc X1 . In this where Z is a stable matrix and where C     case, J¼

1 ð dk eC e  Þ þ Tr ½f Cg e f Cg e ; H  fTr ½ZZ  þ Tr ðra ra Þ  Tr ð C   g 2pj k jkj¼1

ð3:129Þ which is equivalent to (see the solution for Example 3.25) J¼

1 ð dk e e e e ;  fTr ½ZZ  þ Tr ½ra ra   Tr ½ðf Cg þ þ f Cg1 Þðf Cg þ þ f Cg1 Þ g 2pj k H

jkj¼1

ð3:130Þ and the choice Z  0 results in the minimum value for J being realized. Remark 3.7 Theorem 3.2 still holds when Assumption (1) is replaced with ∇a, ∇b, and ∇c good. However, in this case ra  rb Km rc need not be stable for stable Km and the relevance of J as a measure of system performance no longer pertains: the inverse transform of ra  rb Km rc may contain elements whose amplitude can increase without bound. (See Example 3.10 for additional insights.) Remark 3.8 The implications of Assumptions (2) and (3) of Theorem 3.2 on the identifications made in (3.107) is clearly of interest. The first observation to make is that Assumption (2) holds iff rb ¼ WBo1 has full column rank on jkj ¼ 1. Similarly, Assumption (3) holds when rc ¼ Bg Bw iff BgBw has full row rank on jkj ¼ 1. At this point there is no guarantee that using (3.128) for Km in (3.36) yields a K such that det ðX1  KBÞ 6 0

ð3:131Þ

when Z is admissible. However, this is typically the case on account of Remark 3.3. For easy reference and completeness, the results for digital systems with persistent inputs obtained in this section are now summarized without any additional proof needed in Theorem 3.3 When G is compatible and Gw is acceptable for G, A1 1 Ao1 is stable, W is stable and on the unit circle WBo1 has full column rank and BgBm has full row rank, then the controller transfer matrix C given by (3.11) for which GvGw and GdGw are stable and the performance functional

3.5 H2 Design of Digital Multivariable Control Systems

J¼

1 ð dk H  Tr ½ðWGd Gw ÞðWGd Gw Þ  2pj k

139

ð3:132Þ

jkj¼1

is finite is the one for which K ¼ A1 1 Ao1 ðCo Yg1 þ Km Ag Þ

ð3:133Þ

1 e e Km ¼ K1 ðZ þ f Cg þ þ f Cg1 ÞX ;

ð3:134Þ

is chosen with

provided det ðX1  KBÞ 6 0 which is typically the case on account of Remark 3.3. In the formula for Km, Z is an arbitrary stable matrix, e ¼ K1 CX1 ¼ K1 rb ra rc X1 C    

ð3:135Þ

where ra ¼ WðXo Cw þ Bo1 Co Xg1 ÞBw ; rb ¼ WBo1 ; rc ¼ Bg Bw ;

ð3:136Þ

and K and X are square Wiener–Hopf spectral factors satisfying rb rb ¼ K K; rc rc ¼ XX :

ð3:137Þ

All other matrices needed are as defined in Sect. 3.3. Finally, the minimum value for J is obtained when Z  0 is chosen and is given by J¼

1 ð dk eC e  Þ þ Tr ½f Cg e f Cg e ; H  fTr ðra ra Þ  Tr ð C   g 2pj k

ð3:138Þ

jkj¼1

which is equivalent to J¼

1 ð dk e e e e : H  fTr ½ra ra   Tr ½ðf Cg þ þ f Cg1 Þðf Cg þ þ f Cg1 Þ g 2pj k jkj¼1

ð3:139Þ

3.6

H2 Design of Analog Multivariable Control Systems

The case in which the generalized persistent input standard configuration of Fig. 3.2 represents an analog system is considered in this section. In this instance the H2 performance measure used for transient responses is

3 H2 Design of Multivariable Control Systems

140

1 Jtr ¼ 2pj 1 ¼ 2pj 1 ¼ 2pj

Zj1 Tr ½hyd1 ðsÞyd1 ðsÞids j1

Zj1

e w1 hrr0 i G e w1 Gd ds Tr ½Gd G

j1

Zj1

e w1 G e w1 Gd ds Tr ½Gd G

ð3:140Þ

j1

and 1 Jss ¼ 2pj 1 ¼ 2pj 1 ¼ 2pj

Zj1 Tr ½Uyd2 yd2 ðsÞds j1

Zj1

e w2 Ull G e w2 Gd ds Tr ½Gd G

j1

Zj1

e w2 G e w2 Gd ds Tr ½Gd G

ð3:141Þ

j1

is used to measure steady-state stochastic performance. These two measures are incorporated into the single one 1 J¼ 2pj

Zj1 Tr ½QGd Gw Gw Gd ds

ð3:142Þ

j1

where flexibility for weighting the relative importance of the rows of GdGw or, equivalently, the elements of yd is accomplished with the addition of a parahermitian-positive weighting matrix Q that is good. In addition, Gw ¼

 pffiffiffiffiffi e w1 a1 G

pffiffiffiffiffi e  a2 G w2

ð3:143Þ

where the positive constant a1 and the nonnegative constant a2 allow for weighting the relative importance of the transient and steady-state contributions to system performance. Invoking Theorem C.1, one can always write Q ¼ W W where W is a stable matrix because Q is good. Then

3.6 H2 Design of Analog Multivariable Control Systems

1 J¼ 2pj

141

Zj1 Tr ½ðWGd Gw ÞðWGd Gw Þ ds:

ð3:144Þ

j1

The design objective is to find the stabilizing controller C or, equivalently, find the stable matrix K for which GvGw and GdGw are stable and the cost functional J is minimum when G is compatible and Gw is acceptable for G. Attention is restricted to the case in which A1 1 Ao1 is stable so that in accordance with Theorem 3.1 all permissible K for which GvGw and GdGw are stable are given by either one of the expressions in (3.36) with Km or Kg an arbitrary stable matrix. It is preferable to work with Km as the arbitrary stable matrix parameter when W is a polynomial matrix: in this case, the matrices that must be factored to obtain the solution of the optimization problem being treated are polynomial. This is the approach taken here and the alternative approach with Kg an arbitrary stable matrix is treated in Example 3.20. For all stable Km, WGdGw is stable. In contrast with the treatment for digital systems, however, the cost functional J is finite in the case of analog systems only if WGdGw is strictly proper as well. As a consequence, a modification of the approach taken for digital systems is needed in order to find that stable Km, if one exists, for which J is a minimum. From (3.14) and (3.15), Gd ¼ G11 þ G12 ðY þ A1 KÞAG21 :

ð3:145Þ

Then using the first expression for K in (3.36) one gets r ¼ WGd Gw ¼ WG11 Gw þ WG12 ðY þ Ao1 Co Yg1 þ Ao1 Km Ag ÞAG21 Gw :

ð3:146Þ

When (3.23) and (3.29) are recalled, it easily follows that e o þ Km ÞBg Bw r ¼ WG11 Gw  WBo1 ð C

ð3:147Þ

e o ¼ A1 ðY þ Ao1 Co Yg1 ÞA1 : C o1 g

ð3:148Þ

where

The stability of ∇ is automatically guaranteed for all stable Km by Theorem 3.1. The next step is the parameterization of the subset of such Km for which ∇ is strictly proper as well. It is again convenient to introduce the Wiener–Hopf spectral factors K and X satisfying Gb ¼ rb rb ¼ K K; Gc ¼ rc rc ¼ XX

ð3:149Þ

3 H2 Design of Multivariable Control Systems

142

where rb ¼ WBo1 ; rc ¼ Bg Bw :

ð3:150Þ

Since ∇b and ∇c are stable, Gb and Gc are good. Also, attention is restricted here to the case in which both of the square matrices Gb and Gc are nonsingular on the finite jx-axis. Then, in accordance with Theorem C.1 and Remark 3.2, the matrices X and K can be chosen so that in addition to being square they are together with their inverses analytic in Re s 0. There is no need to distinguish between left and right inverses since X1 is both a left and right inverse of X and similarly for K1 when Gb and Gc have full rank on the finite jx-axis as assumed. It is also important to recognize from 1 K1 ¼ I; X1 rc rc X1  ¼I  rb rb K

ð3:151Þ

Ub ¼ rb K1 ¼ WBo1 K1

ð3:152Þ

Uc ¼ X1 rc ¼ X1 Bg Bw

ð3:153Þ

Ub Ub ¼ I; Uc Uc ¼ I:

ð3:154Þ

that

and

are paraunitary matrices:

Since attention is restricted to real rational matrices, it is easily inferred from Lemma C.1 that Ub, Ub , Uc, and Uc are proper matrices. Using (3.152) and (3.153) in (3.147) yields e o X þ KKm XÞUc : r ¼ WG11 Gw  Ub ðK C

ð3:155Þ

A reasonable assumption consistent with practical applications is that WG11 Gw  0ðs1 Þ:

ð3:156Þ

Clearly, in this case, ∇ is strictly proper iff e ¼ Ub ðK C e o X þ KKm XÞUc r

ð3:157Þ

3.6 H2 Design of Analog Multivariable Control Systems

143

is strictly proper. Since e o X þ KKm XÞUc Uc ¼ ðK C e o X þ KKm XÞ; e c ¼ Ub Ub ðK C Ub rU

ð3:158Þ

e is strictly proper iff ðK C e o X þ KKm XÞ is strictly proper. For the choice r e o Xg X1 ; e m  K1 fK C Km ¼ K 1

ð3:159Þ

e m X is strictly proper. Moreover, Km is stable e m is such that K K this is the case iff K e m is stable. Any K e m satisfying these restrictions is called admissible. Clearly, iff K (3.159) parameterizes the subset of all stable Km for which ∇ is strictly proper and e m which includes stable when (3.156) holds in terms of the set of admissible K e m ¼ 0. K Using (3.159) in (3.155) and setting e o Xg þ fK C e o Xg ÞUc ra ¼ WG11 Gw  Ub ðfK C þ 

ð3:160Þ

one gets e m rc : e m XUc ¼ ra  rb K r ¼ WGd Gw ¼ ra  Ub K K

ð3:161Þ

e m including K e m ¼ 0, it is Since ∇ is strictly proper and stable for all admissible K evident that ∇a is strictly proper and stable (also see Example 3.8). In addition, it is clear from (3.150) that ∇b and ∇c are stable. That is, (3.161) is identical in form with (3.106) except that now ∇a has the additional property of being strictly proper as required for the analog case. Following the same steps that lead to (3.113) again yields e e  e m X  CÞðK e m X  CÞ q ¼ Tr ðrr Þ ¼ qo þ Tr ½ðK K K 

ð3:162Þ

where e ¼ K1 CX1 ¼ K1 rb ra rc X1 ¼ Ub ra Uc C    

ð3:163Þ

eC e  Þ: qo ¼ Tr ðra ra Þ  Tr ð C

ð3:164Þ

and

3 H2 Design of Multivariable Control Systems

144

e is strictly proper and good because ∇a is. Thus, f Cg e e Clearly, C 1 ¼ 0 and f Cg þ 2 e m can be is stable. In addition, qo  0ðs Þ. Moreover, every stable matrix K expressed as e ÞX1 e m ¼ K1 ðZ þ f Cg K þ

ð3:165Þ

e mX e m is stable iff Z is stable and K K where Z is a stable matrix. Indeed, the matrix K is strictly proper iff Z is strictly proper. That is, under the assumptions made here any stable Km for which ∇ is stable and strictly proper is given by 1 e o Xg X1 ¼ K1 ðZ þ f Cg e e e m  K1 fK C Km ¼ K 1 þ  fK C o Xg1 ÞX

ð3:166Þ

where Z is admissible (i.e., is an arbitrary stable and strictly proper rational matrix). It now easily follows for any such Km that the performance functional (3.144) is given by 1 J¼ 2pj ¼

1 2pj

Zj1 j1 Zj1

1 qds ¼ 2pj

Zj1 Tr ðrr Þds j1

e ÞðZ  f Cg e Þ gds fqo þ Tr ½ðZ  f Cg   

ð3:167Þ

j1

and is finite for all admissible Z. Hence, the problem of minimizing J over all stable Km is equivalent to minimizing J over all stable and strictly proper Z. The solution to the optimization problem easily follows after some key observations are made. First (Appendix A), for any real rational matrix MðsÞ  0ðs2 Þ one has that Zj1

Zj1 Tr ½MðsÞds ¼

j1

Z Tr ½M ðsÞds ¼

j1

Tr ½MðsÞds

ð3:168Þ

C

where the contour C includes the jx-axis and is closed by a semicircular arc into either the left or right-half planes. Hence, Zj1 j1

e Tr ½Zf Cg  ds ¼

Zj1 j1

e Z ds ¼ Tr ½f Cg 

Z C

e Z ds ¼0 Tr ½f Cg 

ð3:169Þ

3.6 H2 Design of Analog Multivariable Control Systems

145

e Z is analytic in Re s  0, f Cg e Z  0ðs2 Þ, and the contour C can because f Cg   be closed into the left-half plane. It follows from (3.167) that 1 J¼ 2pj

Zj1

e f Cg e fqo þ Tr ½ZZ þ f Cg   gds:

ð3:170Þ

j1

When Z is an nr  nc matrix, 1 2pj

Zj1 j1

1 Tr ½ZZ ds ¼ 2p

Z1 1

1 Tr ½ZðjxÞZ ðjxÞdx ¼ 2p 0

Z1 X nc nr X 1

jzrc ðjxÞj2 dx 0

r¼1 c¼1

ð3:171Þ and the performance functional J is minimized when Z = 0. It is now convenient and possible without the need of further proof to state the key results derived above for analog systems as a theorem. Theorem 3.4 Under the assumptions (1) (2) (3) (4)

∇a, ∇b, and ∇c stable; det ðrb rb Þ 6¼ 0 on the finite jx-axis; det ðrc rc Þ 6¼ 0 on the finite jx-axis; and ∇a strictly proper;

one can construct Wiener–Hopf spectral factors K and X that are square and together with their inverses are analytic in Re s 0 such that rb rb ¼ K K; rc rc ¼ XX

ð3:172Þ

e m can be written as is satisfied. In addition, any stable K e ÞX1 e m ¼ K1 ðZ þ f Cg K þ

ð3:173Þ

e ¼ K1 CX1 ¼ K1 rb ra rc X1 is where Z is a stable matrix and where C     strictly proper and good. In this case, the performance functional 1 J¼ 2pj

Zj1 j1

1 qds ¼ 2pj

Zj1 j1

e m rc Þðra  rb K e m rc Þ ds ð3:174Þ Tr ½ðra  rb K 

3 H2 Design of Multivariable Control Systems

146

is finite iff Z is strictly proper and is given by 1 J¼ 2pj

Zj1

eC e  Þ þ Tr ½f Cg e f Cg e fTr ½ZZ  þ Tr ðra ra Þ  Tr ð C   gds;

j1

ð3:175Þ which is equivalent to (see the solution for Example 3.25) 1 J¼ 2pj

Zj1

e f Cg e fTr ½ZZ  þ Tr ½ra ra   Tr ½f Cg þ þ  gds;

ð3:176Þ

j1

and the choice Z  0 results in the minimum value for J being realized. Remark 3.9 Theorem 3.4 still holds when Assumption (1) is replaced with ∇a, ∇b, e m rc need not be stable for stable K em and ∇c good. However, in this case ra  rb K and the relevance of J as a measure of system performance no longer pertains: the e m rc may contain elements whose amplitude can inverse transform of ra  rb K increase without bound. (See Example 3.10 for additional insights.) Remark 3.10 The implications of Assumptions (2) and (3) of Theorem 3.4 on the identifications made in (3.150) is clearly of interest. The first observation to make is that Assumption (2) holds iff rb ¼ WBo1 has full column rank on the finite jx-axis. Similarly, Assumption (3) holds when rc ¼ Bg Bw iff BgBw has full row rank on the finite jx-axis. At this point there is no guarantee that using (3.166) for Km in (3.36) yields a K such that det ðX1  KBÞ 6 0

ð3:177Þ

when Z is admissible. However, this can be assured as described in the sequel with the introduction of order restrictions on the data. In this regard it is useful to recognize from (3.166) that eo Km ¼ K1 Ca X1  C

ð3:178Þ

e e e Ca ¼ Z þ f Cg þ þ fK C o Xg þ þ fK C o Xg

ð3:179Þ

where

3.6 H2 Design of Analog Multivariable Control Systems

147

is strictly proper when Z is admissible. Using (3.178) in (3.36) and recalling (3.148) then yields 1 1 K ¼ A1 1 ðAo1 K Ca X Ag  YÞ

ð3:180Þ

and it follows that A1 ðX1  KBÞ ¼ A1 X1 þ YB  Ao1 K1 Ca X1 Ag B ¼ I  Ao1 K1 Ca X1 Ag B: ð3:181Þ Clearly, A1 ðX1  KBÞ ! I, s ! 1 when the second term in the final expression of (3.181) is strictly proper and in this case det ðX1  KBÞ 6 0. Conditions that assure this behavior are easily obtained with the help of the following lemma. Lemma 3.2 A necessary and sufficient condition for M M  0ðs2k Þ or MM  0ðs2k Þ is M  0ðsk Þ. Proof For any rational matrix M(s) one can always write NðsÞ 1 X i ¼ Ni s dðsÞ dðsÞ i¼1 q

MðsÞ ¼

ð3:182Þ

where d(s) is the lcd of M(s) and N(s) is a polynomial matrix with Nq 6¼ 0. Clearly, Nq Nq 6¼ 0 because with Nq 6¼ 0 there is at least one diagonal entry of Nq Nq which is not zero: each diagonal element of Nq Nq is the norm of a different column of Nq and these cannot all be zero. Hence, q P q P

M M ¼

i¼1 j¼1

ð1Þi Ni Nj si þ j dd

 0ðs2k Þ

ð3:183Þ

iff q  dðdÞ  k or M  0ðsk Þ. The modification of this proof to accommodate the case MM is obvious. It follows from (3.149) and (3.150) that ðrb rb Þ1 ¼ ðBo1 W WBo1 Þ1 ¼ K1 K1 

ð3:184Þ

or 1 1 ðAo1 K1 ÞðAo1 K1 Þ ¼ ðA1 ¼ ðG12 W WG12 Þ1 : o1 Bo1 W WBo1 Ao1 Þ

ð3:185Þ

3 H2 Design of Multivariable Control Systems

148

Hence, by Lemma 3.2, ðG12 W WG12 Þ1  0ðs2k1 Þ , Ao1 K1  0ðsk1 Þ:

ð3:186Þ

In addition, 1 ðrc rc Þ1 ¼ ðBg Bw Bw Bg Þ1 ¼ ½ðAg AG21 Gw ÞðAg AG21 Gw Þ 1 ¼ X1  X

ð3:187Þ or ðG21 Gw Gw G21 Þ1 ¼ ðX1 Ag AÞ ðX1 Ag AÞ:

ð3:188Þ

Hence, by Lemma 3.2, ðG21 Gw Gw G21 Þ1  0ðs2k2 Þ , X1 Ag A  0ðsk2 Þ:

ð3:189Þ

X1 Ag AG22 ¼ X1 Ag B

ð3:190Þ

X1 Ag B  0ðsk2 þ k3 Þ

ð3:191Þ

G22  0ðsk3 Þ:

ð3:192Þ

Moreover, from

one gets

when

Finally, when the order relationships (3.186), (3.189), and (3.192) hold and Z is admissible so that Ca is strictly proper, then Ao1 K1 Ca X1 Ag B  0ðsk1 þ k2 þ k3 1 Þ:

ð3:193Þ

Hence, from (3.181), det ðX1  KBÞ 6 0 when k1 þ k2 þ k3  0 and Z is admissible. It is also of interest to investigate the implication of these order conditions on the controller obtained with the K given by (3.180). With the aid of (3.181) it is clear that 1 1 1 C ¼ ðX1  KBÞ1 A1 1 A1 ðY1 þ KAÞ ¼ ðI  Ao1 K Ca X Ag BÞ ðA1 Y1 þ A1 KAÞ:

ð3:194Þ

3.6 H2 Design of Analog Multivariable Control Systems

149

Since A1 KA ¼ Ao1 K1 Ca X1 Ag A  YA

ð3:195Þ

and YA ¼ A1 Y1 , one easily gets C ¼ ðI  Ao1 K1 Ca X1 Ag BÞ1 Ao1 K1 Ca X1 Ag A:

ð3:196Þ

When k1 þ k2 þ k3  0 and Z is admissible, it immediately follows that C  0ðsk1 þ k2 1 Þ

ð3:197Þ

because Ao1 K1 Ca X1 Ag B is strictly proper in this case. It is very typical in practical applications that (3.186) and (3.189) are satisfied with k1 ¼ k2 ¼ 0; it then follows from (3.197) that C(s) is strictly proper. e o and Ca are possible. Substituting Some simplification in the expressions for C into (3.148) the expression for Co defined in (3.32) yields e o ¼ A1 ðY þ Yo Cw Ag1 Yg1  YBg1 Yg1 ÞA1 C o1 g 1 1 ¼ A1 o1 ½Y þ Yo Cw Yg Ag  YðI  Xg Ag ÞAg ¼ Ao1 Ya

ð3:198Þ

where Ya ¼ Yo Cw Yg þ YXg :

ð3:199Þ

Using (3.160) in (3.163) and recalling (3.154) gives e o Xg  fK C e o Xg : e ¼ Ub WG11 Gw Uc  fK C C þ 

ð3:200Þ

It then follows from (3.179) that e o Xg Ca ¼ Z þ fUb WG11 Gw Uc g þ þ fK C 

ð3:201Þ

or 1 1 Ca ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKAo1 Ya Xg :

ð3:202Þ

For easy reference and completeness, the results for analog systems obtained in this section are now summarized without the need for any additional proof in Theorem 3.5 When G is compatible and Gw is acceptable for G; when A1 1 Ao1 and W are stable; when WBo1 has full column rank and BgBw has full row rank on the finite jx-axis; and when the order relationships

3 H2 Design of Multivariable Control Systems

150

WG11 Gw  0ðs1 Þ; ðG12 W WG12 Þ1  0ðs2k1 Þ; ðG21 Gw Gw G21 Þ1  0ðs2k2 Þ; G22  0ðsk3 Þ

ð3:203Þ

are satisfied with k1 þ k2 þ k3  0; then all controller transfer matrices C given by (3.11) for which GvGw and GdGw are stable and the performance functional 1 J¼ 2pj

Zj1 Tr ½ðWGd Gw ÞðWGd Gw Þ ds

ð3:204Þ

j1

is finite are the ones for which 1 1 K ¼ A1 1 ðAo1 K Ca X Ag  YÞ

ð3:205Þ

is chosen where 1 1 Ca ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKAo1 Ya Xg

ð3:206Þ

with Z a strictly proper stable matrix and Ya ¼ Yo Cw Yg þ YXg :

ð3:207Þ

rb ¼ WBo1 ; rc ¼ Bg Bw

ð3:208Þ

In the formulas for K and Ca ,

and K and X are the square stable Wiener–Hopf spectral factors with stable inverses satisfying rb rb ¼ K K; rc rc ¼ XX :

ð3:209Þ

In this case, the controller transfer matrix is given by C ¼ ðI  Ao1 K1 Ca X1 Ag BÞ1 Ao1 K1 Ca X1 Ag A

ð3:210Þ

C  0ðsk1 þ k2 1 Þ:

ð3:211Þ

and satisfies

Finally, the minimum value for J is obtained when Z  0 is chosen and is given by

3.6 H2 Design of Analog Multivariable Control Systems

1 J¼ 2pj

Zj1

151

eC e  Þ þ Tr ½f Cg e f Cg e fTr ðra ra Þ  Tr ð C   gds

ð3:212Þ

j1

which is equivalent to 1 J¼ 2pj

Zj1

e f Cg e fTr ½ra ra   Tr ½f Cg þ þ  gds

ð3:213Þ

j1

where 1 1 ra ¼ WG11 Gw  rb K1 ðfKA1 o1 Ya Xg þ þ fKAo1 Ya Xg ÞX rc

ð3:214Þ

easily follows from (3.152), (3.153), (3.160), and (3.198) and e ¼ K1 rb WG11 Gw rc X1  fKA1 Ya Xg  fKA1 Ya Xg C þ    o1 o1

ð3:215Þ

easily follows from (3.200).

3.7

H2 Design of 1DOF Systems

It is instructive and insightful and a useful illustration of the theory to consider the 1DOF system shown in Fig. 3.3. Without loss in generality, n ¼ nm þ nl  nr incorporates any stochastic noises nr, nl, and nm contaminating respectively r, l, and m. Also, L must be stable because the feedforward transducer is a physical block

d l

Pd

L

_ r

+

u

v

C

_

P

+

+

y _ +

+ +

n

m

F

Td

Fig. 3.3 1DOF system (© 1976 IEEE. Reprinted with permission, from Youla et al. (1976b))

3 H2 Design of Multivariable Control Systems

152

outside the feedback loop. The first step is to establish the conditions for and the implications of admissibility of the system. In this regard it is convenient to set Po ¼ FPd þ L

ð3:216aÞ

9 ym ¼ u = : yc ¼ ½ y0 v0  0 ; e ¼ ½ d 0 r 0 n0 0

ð3:216bÞ

and to make the identifications

It then follows from

Pd d + Pv ⎡ y⎤ ⎡ ⎤ ⎡ Pd ⎢v ⎥ = ⎢ ⎥=⎢ 0 v ⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎣ u ⎥⎦ ⎢⎣ r − n − Po d − FPv ⎥⎦ ⎢⎣ − Po that



yc ym



0

0

0 0 I −I

   e e e ¼ G 11 ¼G e 21 v G

e 12 G e 22 G

⎡d ⎤ P ⎤⎢ ⎥ r I ⎥⎢ ⎥ ⎥⎢n ⎥ − FP ⎥⎦ ⎢ ⎥ ⎣v ⎦   e v

ð3:217Þ

ð3:218Þ

where  e 11 ¼ Pd G 0

0 0

 0 ; 0

  e 12 ¼ P ; G I e 21 ¼ ½ Po G

I

ð3:219Þ ð3:220Þ

I ;

ð3:221Þ

and e 22 ¼ FP: G

ð3:222Þ

⎡ y − Td r ⎤ ~ ~ ~ ⎡ yδ ⎤ ⎢ ⎥ = ⎡(G11 − Td ) G12 ⎤ ⎡ w⎤ = v ⎢ ~ ⎢y ⎥ ⎢ ~ ⎥⎢ ⎥ G22 ⎦ ⎣ v ⎦ ⎣ m ⎦ ⎢ u ⎥⎥ ⎣ G21 ⎦ ⎣

ð3:223Þ

Moreover, with w = e, the choice

3.7 H2 Design of 1DOF Systems

153

where  e d ¼ 0 Td T 0 0

0 0

 ð3:224Þ

is obviously of interest with regard to the final objective of keeping ^vðtÞ and be ðtÞ small. Hence, it follows that the 1DOF system is a particular case of the one shown in Fig. 3.2 with e 12 ; G21 ¼ G e 21 ; G22 ¼ G e 22 ; G12 ¼ G

ð3:225Þ

and 

G11

Pd ¼ 0

Td 0

 0 e 11  T ed : ¼G 0

ð3:226Þ

The usual coprime polynomial matrix fraction descriptions for FP and P given by (3.58) through (3.61) are again introduced. Then from G12 ¼ 

Ap 0

0 I



     Bp1 1 P Ap ¼ Ap1 ¼ Ap1 0 I

Xp Yp

   0 Bp ½ Yp þ I I

0 I

1  

0 ¼

I 0

 Bp ; I

ð3:227Þ

 0 ; I

ð3:228Þ

and Xp1 Ap1 þ ½ Yp1

  Bp1 0 ¼I Ap1

ð3:229Þ

it is clear that (3.23) and (3.24) are satisfied with 

Ap Ao ¼ 0

  Xp 0 ; Xo ¼ Yp I

0 I

 ð3:230Þ

Ao1 ¼ Ap1 ; Xo1 ¼ Xp1

ð3:231Þ

    B B Bo ¼  p ; Bo1 ¼  p1 Ap1 I

ð3:232Þ

Yo ¼ ½ Yp

0 ; Yo1 ¼ ½ Yp1

0 :

ð3:233Þ

3 H2 Design of Multivariable Control Systems

154

On account of Definition 2.2 and Lemma 2.3, the 1DOF system is admissible only if   e 12 A1 ¼ G12 A1 ¼ PA1 ; G A1 e 21 ¼ AG21 ¼ ½ APo AG

ð3:234Þ

A ;

A

ð3:235Þ

and e 11 þ G e 21 ¼ G11 þ G12 A1 Y1 G21 þ T e 12 A1 Y1 G ed G

ð3:236Þ

are all stable. Hence, PA1 and AFPd ¼ APo  AL must be stable. It then follows from  ðPd  PA1 Y1 Po Þ PA1 Y1 e e e G 11 þ G 12 A1 Y1 G 21 ¼ YAPo A1 Y1

PA1 Y A1 Y1

 ð3:237Þ

that Pd  PA1 Y1 Po ¼ ðI  PA1 Y1 FÞPd  PA1 Y1 L

ð3:238Þ

must be stable. Hence, ðI  PA1 Y1 FÞPd must be stable. In fact, it is not difficult to confirm that e 21 are stable when e 12 A1 , A G e 21 , and G e 11 þ G e 12 A1 Y1 G Lemma 3.3 The matrices G L is stable iff PA1, AFPd, and ðI  PA1 Y1 FÞPd are stable. Since G11 þ G12 A1 Y1 G21 stable is also needed for Theorem 3.1, it follows from ed stable or, equivalently, Td stable should be assumed as well. (3.236) that T One also has from (ii) of Definition 2.2 that the 1DOF system is admissible iff . e 22 ¼ FP that De we is a stable polynomial. It readily follows from G G G 22 we ¼ wFP . The determination of De can be accomplished using the procedure of G 22

G

Example 2.1. In particular, with the identifications G1 ¼ ½ Pd P ; G2 ¼ F; G3 ¼ L; e ¼ ½ d 0 r 0 n0 0 u1 ¼ ½ d 0 v0 0 ; u2 ¼ y ¼ y1 ; u3 ¼ d ¼ ½ I 0 u1 ; y2 ¼ m; y3 ¼ l

ð3:239Þ

one easily gets 3 2 3 2 y1 0 u1 4 u2 5 4 y2 5 ¼ 4 I ¼ M 13 0 u3 e ¼ 0 y3 v¼0 2

0 0 0

32 3 0 y1 0 54 y2 5 0 y3

ð3:240Þ

3.7 H2 Design of 1DOF Systems

155

and 2

GM13

G1 ¼4 0 0

0 G2 0

32 0 0 0 54 I 0 G3

0 0 0

3 2 0 0 05 ¼ 4F 0 0

3 0 0 0 0 5: 0 0

ð3:241Þ

Hence, De ¼ gðdet TÞdet ðI  GM13 Þ ¼ gðdet T1 Þðdet T2 Þðdet T3 Þ ¼ DG1 DG2 DG3 G ¼ ho wG1 wG2 wG3 ð3:242Þ where ho is a polynomial accounting for any hidden poles. Now because P is a submatrix of G1, wG1 ¼ h1 wP where h1 is a polynomial. Moreover, wG2 ¼ wF and wG3 ¼ wL . Thus, De G

we

G 22

¼

ho h1 wF wP wL wFP

ð3:243Þ

and the 1DOF system is admissible iff ho, h1, wL , and wF wP =wFP are all stable polynomials which is assumed to be the case. It would then also be necessary to assume that F is stable in order to invoke (3.36) of Theorem 3.1: it is only under this circumstance from Lemma 2.10 that 1 A1 1 Ap1 ¼ A1 Ao1

ð3:244Þ

would be stable. Moreover, in some circumstances (see Example 3.12) it is necessary that ðTd F  IÞP be stable. It then follows from Remark 2.5 that when the 1DOF system is admissible and both Td and Td1 are stable that F again must be stable. So attention is restricted here to admissible 1DOF systems with F stable. It is important to note as well that wL is a stable polynomial iff L is stable and the necessity for this assumption is again clear. Next, the modeling of the exogenous input w is considered. It is assumed that d and r are modeled as 1 d ¼ A1 d Bd rd þ Xd ld ; r ¼ Ar Br rr þ Xr lr

ð3:245Þ

where Xd and Xr are stable left Wiener–Hopf factors associated with Udd and Urr , the power spectral density matrices for the stochastic components of the signals d and r, respectively; where the elements of the vectors ld and lr are zero-mean independent white-noise processes with unit variance; where Ad ; Bd and Ar ; Br are

3 H2 Design of Multivariable Control Systems

156

left-coprime polynomial matrix pairs; and where rd and rr are vectors whose elements are random variables satisfying 

 rd ½ r0d rr

r0r 

¼ I:

ð3:246Þ

The input n is modeled as purely stochastic with a power spectral density matrix Unn ¼ Xn Xn with Xn a stable left Wiener–Hopf factor. Hence, one can write ð3:247Þ

n ¼ Xn ln

where the elements of ln are zero-mean independent white-noise processes with unit variance. Moreover, Ull ¼ I where l ¼ ½ l0d

l0r

l0n 0 :

ð3:248Þ

Thus, −1 d

⎡d ⎤ ⎡ A Bd ⎢ w = ⎢r⎥ = ⎢ 0 ⎢ ⎥ ⎢⎣ n ⎥⎦ ⎢⎣ 0

0 A Br

Ωd 0

0

0

−1 r

0 Ωr 0

⎡σ d ⎤ 0 ⎤ ⎢σ r ⎥ ~ ~ ⎡σ ⎤ ~ ⎥⎢ ⎥ 0 ⎥ ⎢ μ d ⎥ = [Gw1 Gw 2 ]⎢ ⎥ = Gw q . ⎢ ⎥ ⎣μ ⎦ Ω n ⎥⎦ ⎢ μ r ⎥ ⎢⎣ μ n ⎥⎦

e w1 ¼ A1 Bw one can take Hence, for G w Aw ¼ diag fAd ; Ar ; Ig 2 3 Bd 0 Bw ¼ 4 0 Br 5 0 0

ð3:249Þ ð3:250Þ ð3:251Þ

e w2 is stable, it follows for the left-coprime polynomial matrix pair Aw, Bw. Since G  pffiffiffiffiffi pffiffiffiffiffi e  e for any constants a1 > 0 and a2 0 that Gw ¼ a2 G w2 is accepta1 G w1 e w1 is acceptable. able iff G In so far as optimization of the performance of the 1DOF system is concerned, it should be clear that the most design freedom is available when no restriction on the controller is imposed other than that it be a stabilizing one. Moreover, this is the case in Youla et al. (1976b) and results obtained here can then be compared with e w1 ¼ A1 Bw in those obtained there. So in accordance with Remark 3.5 and with G w place of Gw in Theorem 3.1, it is natural to consider the case in which

3.7 H2 Design of 1DOF Systems

157

 1 Cg ¼ AG21 A1 w ¼ APo Ad

AA1 r

A



ð3:252Þ

is a stable matrix. Clearly, Cg is stable iff  e g ¼ APo A1 C d

AA1 r



ð3:253Þ

is stable. (In Example 2.13, some situations are considered where this is not the e w1 is case.) Under these circumstances, it follows from Theorem 3.1 that G acceptable for all stabilizing controllers iff Cw and Co are also stable. Now Cw ¼ Ao G11 A1 w ¼



Ap Pd A1 d 0

Ap Td A1 r 0

0 0

 ð3:254Þ

is stable iff  e w ¼ Ap Pd A1 C d

Ap Td A1 r



ð3:255Þ

is stable. A number of important necessary conditions follow from the imposition of the e w be stable. Obviously, C e w is stable iff Ap Pd A1 and e g and C requirement that C d 1 e Ap Td A1 r are stable and C g is stable iff AAr and 1 1 APo A1 d ¼ AFPd Ad þ ALAd

ð3:256Þ

are stable. Clearly, since AFA1 p Bp ¼ B is a stable matrix, one gets when F is stable is stable. Hence, that AFA1 p 1 1 AFPd A1 d ¼ ðAFAp ÞðAp Pd Ad Þ

ð3:257Þ

1 is stable and it follows from (3.256) that APo A1 d is stable iff ALAd is stable. In fact, it is established in the sequel in the proof of Lemma 3.4 that LA1 d must be stable for Co to be stable. There is one more key necessary condition to establish. One gets from Fig. 3.3 that the shape-deterministic component of u is given by

u1 ¼ r1  ðL þ FPd Þd1  FPCu1

ð3:258Þ

u1 ¼ ðI þ FPCÞ1 ½r1  ðL þ FPd Þd1 

ð3:259Þ

or

3 H2 Design of Multivariable Control Systems

158

1 1 where r1 ¼ A1 r Br rr and d1 ¼ Ad Bd rd . Setting C ¼ ðY þ A1 KÞðX  B1 KÞ which represents all stabilizing controller transfer matrices it readily follows that

u1 ¼ r1  Fy1 ¼ r1  Fðe1  Td r1 Þ ¼ ðI  FTd ÞA1 r Br rr þ Fe1 :

ð3:260Þ

When F, Td, and e1 are stable, it then follows that u1 is stable for all rr iff ðI  FTd ÞA1 r is stable. Indeed, Lemma 3.4 When the 1DOF system is admissible, then (1) G ¼ ½Gij  is compatible iff Td is stable and (2) Gw is acceptable for all stabilizing controllers, e w2 are stable, iff Ap Pd A1 , LA1 , Ap Td A1 , and ðI  FTd ÞA1 provided F, Td, and G d d r r are stable. Proof That (1) is true is an immediate consequence of Eqs. (3.224) and (3.234) through (3.236). Except for LA1 d , the necessity for the conditions given in (2) has already been established. Clearly, the conditions are sufficient for Cw stable and for stable. With M1 ¼ Ap Td A1 and M2 ¼ ðI  FTd ÞA1 stable matrices it APo A1 d r r follows that 1 1 AA1 r ¼ AðM2 þ FTd Ar Þ ¼ AM2 þ AFAp M1

ð3:261Þ

is stable. Hence, Cg is stable. It only remains to show that Co is stable and that for this to be the case it is necessary that LA1 d be stable. The matrix Co is defined in (3.32) and with the aid of (3.29) and (3.252) can be written as Co ¼ A1 o1 ðYo Cw  YCg ÞAg1 :

ð3:262Þ

It readily follows from (3.231) and (3.233) when Yp Ap ¼ Ap1 Yp1 is recalled that

Co ¼ A1 p1 ½ Yp

  0 Cw  YCg Ag1 ¼ Ca

Cb

 A1 p1 YA Ag1

ð3:263Þ

where 1 1 Ca ¼ Yp1 Pd A1 d þ Ap1 YAPo Ad

ð3:264Þ

1 1 Cb ¼ Yp1 Td A1 r  Ap1 YAAr :

ð3:265Þ

and

3.7 H2 Design of 1DOF Systems

159

Now 1 A1 p1 YA ¼ Ap1 A1 Y1

ð3:266Þ

is stable because admissibility requires PA1 ¼ Bp1 A1 p1 A1 stable and this is the case 1 1 1 iff Ap1 A1 is stable. Since Cg ¼ Bg1 Ag1 is stable iff Ag1 is stable, it is clear that when the 1DOF system is admissible and Cg is stable then Co is stable iff Ca and Cb are stable. Now 1 1 1 1 1 Ca ¼ ðYp1 A1 p þ Ap1 YAFAp ÞðAp Pd Ad Þ þ Ap1 YALAd e a Ap Pd A1 þ A1 YALA1 ¼C d p1 d

ð3:267Þ

where e a ¼ Yp1 A1 þ A1 YAFA1 C p p1 p ¼ A1 p1 ½Yp þ YAFðXp þ PYp Þ ¼ A1 p1 ½A1 Y1 FXp þ ðYB  IÞYp  ¼ A1 p1 A1 ðY1 FXp  X1 Yp 

ð3:268aÞ

1 is stable because F is stable and PA1 stable implies A1 p1 A1 stable. Since Ap Pd Ad is 1 1 stable, it follows that Ca is stable iff Ap1 A1 Y1 LAd is stable. This is the case iff 1 Y1 LA1 d is stable because A1 Ap1 is stable by Lemma 2.10. Hence, it is necessary that 1 1 1 B1 Y1 LA1 d ¼ ðI  XAÞLAd ¼ LAd  XALAd

ð3:268bÞ

1 be stable. Clearly, XALA1 d is stable because ALAd stable follows from (3.256). So 1 it is necessary that LAd be stable and when this is the case one gets from (3.267) that Ca is stable. Next it is established that 1 1 1 1 Cb ¼ ðYp1 Td A1  A1 p1 YÞAAr ¼ Yp1 Td Ar  Ap1 YAAr 1 1 1 1 ¼ Yp1 Td A1 r  ðXp1 þ Yp1 PÞYAAr ¼ Yp1 ðTd Ar  PYAAr Þ  Xp1 YAAr 1 1 ¼ Yp1 ðA1 p M1  PYAM2  PYAFAp M1 Þ  Xp1 YAAr 1 ¼ Yp1 ðI  PA1 Y1 FÞA1 p M1  Yp1 PA1 Y1 M2  Xp1 YAAr

ð3:269Þ

3 H2 Design of Multivariable Control Systems

160

is stable. Since ðI  PA1 Y1 FÞ is stable, ðI  PA1 Y1 FÞA1 p is stable iff ðI  PA1 Y1 FÞA1 p Bp ¼ ðI  PYAFÞP ¼ PðI  YAFPÞ ¼ PðI  YBÞ ¼ PA1 X1 ð3:270Þ is stable which is the case. It immediately follows from (3.269) that Cb and, therefore, Co are stable. This completes the proof of the lemma. Clearly, under the conditions of Lemma 3.4, it follows from Theorem 3.1 and e w1 ¼ A1 Bw , and, therefore, Gw is Remark 3.5 for the Aw given by (3.250) that G w acceptable for the 1DOF system with any stabilizing controller. It is important to note in addition that when Sr ¼ AA1 and Sd ¼ Ap Pd A1 are stable, then r ¼ r d 1 1 e Ar Br rr ¼ A S r where e S r ¼ Sr Br rr is a stable vector and Pd d ¼ Pd A1 d Bd rd ¼ 1 e S d ¼ Sd Bd rd is a stable vector. Hence, any unstable poles of r must A S d where e p

be unstable poles of A−1 or, equivalently, unstable poles of FP which when F is stable can only be unstable poles of P. Also, any unstable poles of Pd d must be poles of A1 p or, equivalently, poles of P. Typically, L = L1L2 where L2 is the transfer matrix of the disturbance sensors and L1 represents any associated cascade compensation that is introduced. Since L1 and L2 are physical blocks outside the feedback loop both must be stable. The is stable imposes a constraint on L1. In particular, with assumption that LA1 d 1 1 is L2 Ad ¼ A B a left coprime polynomial matrix fraction description, LA1 d stable iff L1 A1 ¼ e L 1 is stable. That is, one must choose L1 so that e L 1 is a stable matrix. Attention is now turned to the H2 design of a digital 1DOF system with F and Td e w2 is stable, and for all stable. It is assumed that the system is admissible, G stabilizing controllers, Gw is acceptable for G. That is, r ¼ W½G11 þ G12 ðY þ A1 KÞAG21 Gw ¼ WGd Gw

ð3:271Þ

is stable for all stable K when W is stable. Setting rb ¼ WG12 A1 ¼ W rc ¼ AG21 Gw ¼

 pffiffiffiffiffi a1 C g B w

  P A ; I 1

 pffiffiffiffiffi e w2 ¼ A½ Po a2 AG21 G

ð3:272Þ I

I Gw ; ð3:273Þ

and ra ¼ WG11 Gw  rb A1 1 Yrc

ð3:274Þ

3.7 H2 Design of 1DOF Systems

161

yields r ¼ ra  rb K rc :

ð3:275Þ

Since ∇ is stable for all stable K which includes K  0, it follows that ∇a is stable. e w2 are stable for the Clearly, ∇b and ∇c are stable because W, PA1, Cg , AG21, and G assumptions made here. It is of interest to consider the choice  W¼

I 0 0 Wq

 ð3:276aÞ

for which one gets from (3.223) that 

I Wyd ¼ 0

0 Wq



   e y  Td r ¼ : Wq v v

ð3:276bÞ

Hence, the matrix Wq weights the relative importance of v with respect to the error e. For example, the choice Wq = 0 eliminates v from having any impact on the performance functional and the emphasis is solely on keeping be ðkÞ small. On the other hand, other choices for Wq can be considered to tradeoff error against the size of the control signal ^vðkÞ. It is also useful for comparison with what is obtained for analog 1DOF systems in Youla et al. (1976b) to consider the choice (3.276a) with Wq Wq ¼ kQ

ð3:277Þ

where k is a positive constant. Clearly, Q is good because Wq is stable and also Q is a parahermitian-positive matrix. Then rb rb ¼ A1 ðP P þ kQÞA1 :

ð3:278Þ

It is also convenient to introduce Gt ¼ Po Gd Po þ Gr þ Gn

ð3:279Þ

9 1 Gd ¼ a1 A1 d Bd Bd Ad þ a2 Xd Xd = 1 Gr ¼ a1 A1 r Br Br Ar þ a2 Xr Xr ;: Gn ¼ a2 Xn Xn

ð3:280Þ

where

3 H2 Design of Multivariable Control Systems

162

Then rc rc ¼ AGt A :

ð3:281Þ

det ½A1 ðP P þ kQÞA1  6¼ 0; jkj ¼ 1

ð3:282Þ

det ½AGt A 6¼ 0; jkj ¼ 1;

ð3:283Þ

Hence, when

and

Theorem 3.2 can be invoked with K in place of Km and with Wiener–Hopf spectral factors K and X that satisfy A1 ðP P þ kQÞA1 ¼ K K;

AGt A ¼ XX :

ð3:284Þ

In particular, the performance functional J given by (3.126) is finite for all stable K and all such K are given by 1 e e K ¼ K1 ðZ þ f Cg þ þ f Cg1 ÞX

ð3:285Þ

where Z is a stable matrix and where e ¼ K1 rb ra rc X1 : C  

ð3:286Þ

Moreover, the minimum J is realized with the choice Z  0. Substituting (3.274) for ∇a into (3.286) and introducing Ci ¼ rb WG11 Gw rc

ð3:287Þ

e ¼ K1 Ci X1  KA1 YX C   1

ð3:288Þ

one gets

where

Ci ¼ A1 ½ P

 I Wq  0

0 Wq



Pd 0

Td 0

2  Gd 0 6 4 0 0 0

0 Gr 0

32 3 Po 0 76 7 0 54 I 5A Gn I

¼ A1 P ðTd Gr þ Pd Gd Po ÞA : ð3:289Þ

3.7 H2 Design of 1DOF Systems

163

Using (3.288) in (3.285) and recognizing that 1 1 1 fKA1 1 YXg þ þ fKA1 YXg1 ¼ KA1 YX  fKA1 YXg

ð3:290Þ

gives finally 1 1 1 1 1 K ¼ K1 ðZ þ fK1  A1  Ci X g þ þ fK Ci X g1 þ fKA1 YXg ÞX 1 Y:

ð3:291Þ One can expect typically that det ðX1  KBÞ 6 0 in accordance with Remark 3.3 so that the required controller with transfer matrix C ¼ ðX1  KBÞ1 ðY1 þ KAÞ is 1 realizable. Also, except for the fK1  Ci X g1 term, the expression for K in (3.291) is identical in form with the one given in Youla et al. (1976b) for the analog case. For insights regarding the computation of the controller transfer matrix C see Example 3.18. When the 1DOF system is analog instead of digital, additional assumptions are needed. In particular, attention is turned to the analog case under the practical 1 assumptions that F, L, W, Td, and Pd are proper and that A1 r Br , Ad Bd , Xr , and Xd are strictly proper. It then follows from (3.226) and (3.249) that WG11 Gw  0ðs1 Þ because  P A1 B e WG11 G w ¼ W d d d 0

Td A1 r Br 0

Pd Xd 0

Td Xr 0

 0  0ðs1 Þ: ð3:292Þ 0

Making the same choices for W, ∇b, and ∇c as in the digital case one gets from (3.271) that r ¼ WG11 Gw  rb ðA1 1 Y þ KÞrc

ð3:293Þ

where W, ∇b, and ∇c are again stable for the assumptions made here. Hence, when for finite x det ðrb rb Þ ¼ det ðK KÞ ¼ det ðA1 ðP P þ kQÞA1 Þ 6¼ 0; s ¼ jx

ð3:294Þ

det ðrc rc Þ ¼ det ðXX Þ ¼ det ðAGt A Þ 6¼ 0; s ¼ jx

ð3:295Þ

and

are assumed, the Wiener–Hopf factors K and X together with their inverses are stable. Moreover, one has that rb K1 and X1 rc are paraunitary matrices. Writing (3.293) as

3 H2 Design of Multivariable Control Systems

164

1 r ¼ WG11 Gw  rb K1 ðKA1 1 YX þ KKXÞX rc

ð3:296Þ

and recognizing that any stable matrix K can be written as 1 e m  K1 fKA1 K¼K 1 YXg1 X

ð3:297Þ

e m is a stable matrix then yields where K e m rc r ¼ ra  rb K

ð3:298Þ

where 1 1 ra ¼ WG11 Gw  rb K1 ðfKA1 1 YXg þ þ fKA1 YXg ÞX rc :

ð3:299Þ

e m  0. Hence, ∇a is Again ∇ is stable for all stable K and in particular for K stable. In addition, it is evident from (3.292) and (3.299) that ∇a is strictly proper. e m is given Theorem 3.4 then yields that the performance functional J is finite iff K by (3.173) or, equivalently, 1 1 e K ¼ K1 ðZ þ f Cg þ  fKA1 YXg1 ÞX

ð3:300Þ

where Z is stable and strictly proper and now e ¼ K1 rb ra rc X1 ¼ K1 Ci X1  fKA1 YXg  fKA1 YXg ð3:301Þ C þ      1 1 with Ci again given by the formula (3.289). Substituting (3.301) into (3.300), recognizing that 1 1 ffKA1 1 YXg þ þ fKA1 YXg g þ ¼ fKA1 YXg þ ;

ð3:302Þ

and recalling (3.290) gives 1 1 1 K ¼ K1 ðZ þ fK1  A1  Ci X g þ þ fKA1 YXg ÞX 1 Y

ð3:303Þ

1 which except for the fK1  Ci X g1 term in (3.291) is identical in form with the result obtained for the digital 1DOF system. The expression for K is also identical in form with the one obtained in Youla et al. (1976b) for a 1DOF analog system. For insights regarding the computation of the controller transfer matrix C ¼ ðX1  KBÞ1 ðY1 þ KAÞ see Example 3.18. A controller transfer matrix C ¼ ðX1  KBÞ1 ðY1 þ KAÞ associated with the matrix K given by (3.303) exists iff det ðX1  KBÞ 6 0. Similar reasoning to that used in connection with (3.181) through (3.197) can be employed here (see

3.7 H2 Design of 1DOF Systems

165

Example 3.13) to confirm that this is the case when the order conditions in (3.203) are met with ðk1 þ k2 þ k3 Þ  0. For the 1DOF system,  G12 W WG12 ¼ ½ P

Wq 

P Wq

 ¼ P P þ kQ:

ð3:304Þ

Typically, P  0ðs1 Þ and Q is biproper. Thus, with k > 0 ðG12 W WG12 Þ1 ¼ ðP P þ kQÞ1

ð3:305Þ

is proper and k1 = 0. Also, 1 ðG21 Gw Gw G21 Þ1 ¼ G1 t ¼ ðPo Gd Po þ Gr þ Gn Þ

ð3:306Þ

is typically proper because Gn is biproper and with F and L proper the other terms in Gt are strictly proper. Hence, k2 = 0. Finally, G22 ¼ FP  0ðs1 Þ is typical and one gets k3 = –1. Thus, ðk1 þ k2 þ k3 Þ ¼ 1  0 and C is realizable for the K given by (3.303). Moreover, C is strictly proper on account of (3.197). A summary of the key assumptions made and their implications is now presented for easy reference and clarification. The key assumptions are Assumption 3.1 The 1DOF system is admissible. 1 1 1 Assumption 3.2 The matrices F, Td, LA1 d , Ap Pd Ad , Ap Td Ar and ðI  FTd ÞAr are stable.

Assumption 3.3 The matrices Xd , Xr , Xn , and W ¼ diag fI; Wq g are stable. Assumption 3.4 On jkj ¼ 1 for the digital case and on the finite s = jx axis for the analog case, det ½A1 ðP P þ kQÞA1  6¼ 0 and det ½AGt A  6¼ 0 where kQ ¼ Wq Wq . Assumption 3.5 In the analog case, F, L, Wq, Td, and Pd are proper and A1 r Br , A1 B , X , and X are strictly proper. In addition, the order conditions in (3.203) d r d d or, equivalently, ðP P þ kQÞ1  0ðs2k1 Þ, ðPo Gd Po þ Gr þ Gn Þ1  0ðs2k2 Þ, and FP  0ðsk3 Þ are met with k1 þ k2 þ k3  0. Assumption 3.1 assures that the 1DOF system can be stabilized and that all stabilizing controller transfer matrices are given by (3.11) with K a stable matrix. It 1 must be holds iff the polynomial (3.243) is stable; hence, w1 L and wFP ðwF wP Þ stable. It follows that L must be stable and for F stable A1 1 Ap1 is stable because of e 12 A1 , A G e 21 , and Lemma 2.10. It also follows from Lemma 2.3 that the matrices G e e e G 11 þ G 12 A1 Y1 G 21 must be stable; hence, PA1, AFPd, and ðI  PA1 Y1 FÞPd must be stable by Lemma 3.3.

3 H2 Design of Multivariable Control Systems

166

Assumption 3.2 assures under Lemma 3.4 that the G ¼ ½Gij  associated with the  pffiffiffiffiffi ffie  e w1 pffiffiffiffi 1DOF system is compatible. It also assures that Gw ¼ a2 G a1 G w2 , e w1 and G e w2 are defined by (3.249) and G e w2 is stable because of where G Assumption 3.3, is acceptable for every stable K. Collectively, Assumptions 3.1, 3.2, and 3.3 lead to r ¼ WGd Gw ¼ ra  rb K rc stable for all stable K with ∇a, ∇b, and ∇c stable. Then for the digital case, the additional Assumption 3.4 together with the choice (3.276a) and (3.277) for W leads by Theorem 3.2 to the expression (3.291) for K which for Z  0 yields the minimum value of the J given by (3.105). For the analog case, ∇a is strictly proper with the additional Assumption 3.5 which together with the choice (3.276a) and (3.277) for W leads by Theorem 3.4 to the expression (3.303) for K which for Z  0 yields the minimum value of the J given by (3.144). Finally, the order restrictions on ðP P þ kQÞ1 , ðPo Gd Po þ Gr þ Gn Þ1 , and FP assure that the controller is realizable. The design of an analog 1DOF optimal H2 multivariable system for the control of submarine pitch and depth is presented in Example 3.23 to illustrate the application of the theory developed in this section. The model used for the submarine motion dynamics is the same as the one given in Sect. 10.4.4 of Grimble (2006) which provides an H1 solution. It is important to recognize that significant simplification in the calculation of the controller transfer matrix C for the K given by (3.291) or (3.303) occurs when any unstable poles of FP lie only on jkj ¼ 1 in the digital case or on the finite jx-axis in the analog case. For details in this regard see Example 3.18(b) and (c). It is also important to note that imposition of the requirement that Gw be acceptable for all 1DOF stabilizing controllers led to ym1 ¼ u1 in (3.259) being stable. That is, Gw is actually strictly acceptable for this requirement. Finally, when the transfer matrix F for the feedback sensors is the identity matrix, a clever modification of the standard spectral separation process in connection with partial fraction expansions and the use of nonstandard matrix spectral factorizations in Stefanovski (2007) yield a significant simplification of the formula for the optimal controller in almost all cases.

3.8

H2 Design of 3DOF Systems

The 3DOF system shown in Fig. 1.1 is considered and it is assumed that the system is admissible as defined in Sect. 2.5. The exogenous inputs are modeled as 2

3 2 3 2 3 d1 d d2 6 r 7 6 r1 7 6 r2 7 6 7 6 7 6 7 7 6 7 6 7 e¼6 6 nr 7 ¼ 6 0 7 þ 6 nr 7 ¼ e 1 þ e 2 4 nm 5 4 0 5 4 nm 5 0 nl nl

ð3:307Þ

3.8 H2 Design of 3DOF Systems

167

where d1 and r1 are the shape-deterministic components of d and r and where d2, r2, nr, nm, and nl are purely stochastic. In order to be more descriptive and to be consistent with the notation in Sect. 3.7, nr is used here for the noise n in Fig. 1.1. Specifically, 1 d ¼ A1 d Bd rd þ Xd ld ; r ¼ Ar Br rr þ Xr lr

ð3:308Þ

where Xd and Xr are stable left Wiener–Hopf factors associated with Udd and Urr , the power spectral density matrices for the stochastic components of the signals d and r, respectively; where the elements of the vectors ld and lr are zero-mean independent white-noise processes with unit variance; where Ad, Bd and Ar, Br are left-coprime polynomial matrix pairs; and where r ¼ ½ r0d r0r 0 is a vector whose elements are random variables satisfying hrr0 i ¼



 rd ½ r0d rr

r0r  ¼ I:

ð3:309Þ

It is also assumed that the power spectral density matrices Unr nr ; Unm nm , and Unl nl for nr, nm, and nl, respectively, are good. Then in accordance with Theorem C.1 and Theorem C.2 one can write Unr nr ¼ Xnr Xnr  ; Unm nm ¼ Xnm Xnm  ; Unl nl ¼ Xnl Xnl 

ð3:310Þ

where Xnr ; Xnm , and Xnl are stable left Wiener–Hopf factors. Equivalently, nr ¼ Xnr lnr ; nm ¼ Xnm lnm ; nl ¼ Xnl lnl

ð3:311Þ

b nr , b l nm , and b l nl are zero-mean independent white-noise where the elements of l processes with unit variance. Setting  l ¼ l0d

l0r

l0nr

q ¼ ½ r0

l0nm

0 l0nl ;

ð3:312Þ

l0 0 ;

ð3:313Þ

Xs ¼ diagfXd ; Xr ; Xnr ; Xnm ; Xnl g;

ð3:314Þ

and 

Ad Ae ¼ 0

 0 ; Ar



Bd Be ¼ 0

0 Br

 ð3:315Þ

3 H2 Design of Multivariable Control Systems

168

one can write 

   r e eq ¼G l

e e1 e¼ G

e e2 G

2

3 0 I7 7 1 07 7Ae Be ; 05 0

ð3:316Þ

where

e e1 G

I 60 6 ¼6 60 40 0

e e2 ¼ Xs : G

ð3:317Þ

Clearly, Ae, Be is a left-coprime polynomial matrix pair. The design objective is to keep ^vðkÞ and be ðkÞ small for the digital case and ^vðtÞ and be ðtÞ small for the analog case. Thus,

⎡ y − Td r ⎤ ⎢ v ⎥ ⎡ yδ ⎤ ⎢ w ⎥ ⎡G11 G12 ⎤ ⎡ e ⎤ ⎥=⎢ ⎢y ⎥ = ⎢ ⎥⎢ ⎥ ⎣ m ⎦ ⎢ z ⎥ ⎣G21 G22 ⎦ ⎣v ⎦ ⎥ ⎢ ⎣ u ⎦

ð3:318Þ

is of interest and the 3DOF system corresponds to the generalized persistent input standard configuration of Fig. 3.2 with w = e,     ew ¼ G e e; e e1 G e w1 G e w2 ¼ G e e2 ¼ G G ð3:319Þ   Pd Td 0 0 0 G11 ¼ ; ð3:320Þ 0 0 0 0 0 G12 2

G21

FPd ¼4 L 0

  P ¼ ; I 0 0 I

0 0 I

ð3:321Þ I 0 0

3 0 I 5; 0

ð3:322Þ

and 2

G22

3 FP ¼ 4 0 5: 0

ð3:323Þ

3.8 H2 Design of 3DOF Systems

169

The Gij are easily confirmed by (also see Example 1.1) y  Td r v ¼ ½0 w ¼ ½I z ¼ ½0 u ¼ ½0

9 ¼ ½ I 0 ðG11 e þ G12 vÞ ¼ Pd d  Td r þ Pv > > > > I ðG11 e þ G12 vÞ ¼ 0 þ v = 0 0 ðG21 e þ G22 vÞ ¼ FPd d þ nm þ FPv : > > I 0 ðG21 e þ G22 vÞ ¼ Ld þ nl > > ; 0 I ðG21 e þ G22 vÞ ¼ r þ nr

ð3:324Þ

The matrix G22 is the same as the one in (3.54) of Sect. 3.4 and so (3.58) through (3.66) are again applicable. It is then easy to verify that G ¼ ½Gij  is compatible when the 3DOF system is admissible and Td is stable. In particular,     e 1 ¼ P A1 ¼ PA1 ; G12 A I A1 2

e 21 AG

A ¼40 0

32 0 0 FPd I 0 54 L 0 0 I

0 0 I

0 0 I

ð3:325Þ

3 2 0 AFPd I5¼4 L 0 0

I 0 0

0 0 I

0 0 I

A 0 0

3 0 I 5; 0 ð3:326Þ

and e1 Y e1 G21 ¼ G11 þ G12 A

 

¼

Pd

Td

0

0

0

0

0

0

0

0



 þ

PA1 A1

2

 ½ Y1

0

ðI  PA1 Y1 FÞPd

Td

0

PA1 Y1

0

YAFPd

0

0

A1 Y1

0

FPd

6 04 L 

0

0

3

0

0

I

0

0

0

7 I5

I

I

0

0

ð3:327Þ are all stable on account of Lemma 2.7. e w2 ¼ G e e2 ¼ Xs is a stable matrix, it is clear that attention must be Since G e e1 defined in (3.317) that are acceptable or, equivalently, e restricted to the G w1 ¼ G to those for which e 1ðY e 21 G e w1 ¼ G11 G e w1 þ G12 A e w1 e1 þ K AÞG Gd G

ð3:328Þ

e 1ðY e 21 G e w1 ¼ A e w1 e1 þ K AÞG Gv G

ð3:329Þ

and

3 H2 Design of Multivariable Control Systems

170

are stable. Now in light of (3.320) and (3.321), ½0

e 1ðY e 21 G e w1 ¼ A e w1 ¼ Gv G e w1 : e1 þ K AÞG I Gd G

e w1 and Gv G e w1 are stable iff Gd G e w1 is stable. That is, Hence, Gd G 8 2    FPd 0 one can expect the formulas in Park and Bongiorno (1990) to fail. When and/or A1 the finite unstable poles of A1 r n are confined to the jx-axis, it is true never-the-less that the formulas in Park and Bongiorno (1990) are valid for the assumptions there assure that the matrix K will be stable even though the restriction (3.333) is not explicitly imposed. A summary of the key assumptions made and their implications is now presented for easy reference and clarification. Assumption 3.6 The 3DOF system is admissible. 1 Assumption 3.7 The matrices F, Td, Ap Td A1 r , and Ap Pd Ad are stable.

Assumption 3.8 The matrices Xd , Xr , Xnr , Xnm , Xnl and W ¼ diagfI; Wq g are stable. Assumption 3.9 On jkj ¼ 1 for the digital case and on the finite s = jx axis for the analog case, det ½A1 ðP P þ kQÞA1  6¼ 0 and det ½At G21 Gs G21 At  6¼ 0 where kQ ¼ Wq Wq . 1 Assumption 3.10 In the analog case, Wq, Td, and Pd are proper and A1 r Br , Ad Bd , 1 1 2k1 2k2 and Xs are strictly proper. In addition, ðP P þ kQÞ  0ðs Þ, U  0ðs Þ, and FP  0ðsk3 Þ with k1 þ k2 þ k3  0.

Assumption 3.6 assures that the 3DOF system can be stabilized and that all stabilizing controller transfer matrices are given by (2.208) with K ¼ ½ K1 H2 H1  a stable matrix. It also follows from Lemma 2.8 that w1 L and 1 1 wFP ðwF wP Þ must be stable; hence, L must be stable and with F stable, A1 Ap1 is stable because of Lemma 2.10. Moreover, PA1, AFPd, and ðI  PA1 Y1 FÞPd must be stable from Lemma 2.7. Assumption 3.7 then assures that the G ¼ ½Gij  associated with the 3DOF system  pffiffiffiffiffi ffie  e e e w1 pffiffiffiffi is compatible. It also assures that Gw ¼ a2 G a1 G w2 , where G w1 and G w2

e w2 is stable because of Assumption are defined by (3.317) and (3.319) and where G 3.8, is acceptable for every stable K given by (3.333) iff Kt ¼ ½ Km Kn Kg2  a stable matrix. Collectively, Assumptions 3.6, 3.7, and 3.8 lead to r ¼ WGd Gw ¼ ra  rb Kt rc stable for all stable Kt in the digital case and r ¼ WGd Gw ¼ ra  e t in the analog case with ∇a, ∇b, and ∇c stable. Then e t rc stable for all stable K rb K the additional Assumption 3.9 together with the choice (3.276a) and (3.277) for W leads by Theorem 3.2 to the expressions (3.364) and (3.371) for the submatrices of K ¼ ½ K1 H2 H1  which in the digital case guarantees that the performance functional J defined in (3.105) is finite. With Z ¼ ½ Zm Zn Zg2   0 in these submatrices, the minimum value is obtained for the performance functional

3.8 H2 Design of 3DOF Systems

185

J. For the analog case, the additional Assumption 3.10 assures that WG11Gw and, therefore, ∇a are strictly proper. With the choice (3.276a) and (3.277) for W, Theorem 3.4 then leads to the expressions (3.3) and (3.9) for the submatrices of K ¼ ½ K1 H2 H1  which guarantee that the performance functional J defined in (3.144) is finite. With Z ¼ ½ Zm Zn Zg2   0 in these submatrices, the minimum value is obtained for the performance functional J. Finally, the order restrictions on ðP P þ kQÞ1 , U1 , and FP in Assumption 3.10 assure that the controller is realizable. It is also important to remember that when the unstable poles of r1 and d1 lie only on jkj ¼ 1 for the digital case and on s = jx for the analog case, then the required factorizations of Gt and Go can be carried out in accordance with (3.378) through (3.381) leading to the reduced formulas (3.383) through (3.387) for the digital case and (3.432) through (3.436) for the analog case. It should also be pointed out that when ðTd F  IÞP is stable and the other assumptions in Theorems 2.5 and 2.7 are satisfied then instead of beginning with (3.333) for the subset of suitable K one could take instead K ¼ ½ K1

H2

H1  ¼ ½ Km

Kn An

ðY1 þ Kr Ar Þ 

ð3:437Þ

1 where LA1 d ¼ An Bn and Km, Kn, and Kr are stable matrices. Then the need to find a coprime polynomial matrix fraction description for P can be avoided. For brevity this approach is not considered here since the one presented is more general. The design of an analog 2DOF optimal H2 multivariable system for the control of the Rosenbrock process which exhibits a severe coupling phenomenon (Åström et al. 2002) is presented in Example 3.24 to illustrate the application of the theory developed in this section.

3.9 3.9.1

Trade-off of Optimal Performance for Reduced Sensitivity to Plant Uncertainty Overview

The performance functionals employed in Sects. 3.5 and 3.6 of this chapter take into account tracking error and saturation for nominal plant models. The stabilizing controllers for which the performance functional is finite are parameterized in terms of a real rational stable matrix parameter Z. The performance functional is minimized when Z  0. When non-null choices are made for the matrix parameter, the value of the performance function increases. It follows from Theorem 3.3 and (3.129) and from Theorem 3.5 and (3.175), respectively, that the increase is given simply by

3 H2 Design of Multivariable Control Systems

186

DJ ¼ J  Jo ¼

1 ð H  Tr(ZZ Þdk=k 2pj

ð3:438Þ

jkj¼1

in the digital case and by 1 DJ ¼ J  Jo ¼ 2pj

Zj1 TrðZZ Þds

ð3:439Þ

j1

in the analog case. Clearly, the possibility exists to trade off optimal nominal system performance for improvement in stability margin and reduced sensitivity to plant parameter variations. That is, the possibility exists for choosing a Z 6 0 for which J  N and for which the non-optimal nominal design obtained is more robust with respect to uncertainty in the nominal plant model than the optimal design. Attention is restricted to the more difficult analog case: One can easily apply the same methodology for the digital case. A straightforward Wiener–Hopf methodology for achieving this trade-off is described in the sequel. The methodology begins with the determination of two functionals: one that penalizes designs that are too sensitive to small plant parameter changes and one that penalizes designs for stability margins that are too small. The focus on the increase in the value of the quadratic performance functional here contrasts with the one taken in the loop transfer recovery (LTR) approach (Maciejowski 1989) and/or the modification of weighting functions approach (Grimble 2006). In particular, the matrix parameter Z is used to establish a more transparent methodology for trading-off optimality for more robustness; especially in multivariable cases where modification of weighting functions is not as clear as in the scalar (single variable) case.

3.9.2

Sensitivity to Small Plant Parameter Changes

It easily follows from Fig. 3.2 that v ¼ CG21 w þ CG22 v , v ¼ ðI  CG22 Þ1 CG21 w ¼ RG21 w

ð3:440Þ

and yd ¼ G11 w þ G12 v ¼ G11 w þ G12 ðI  CG22 Þ1 CG21 w ¼ ðG11 þ G12 RG21 Þw ð3:441Þ

3.9 Trade-off of Optimal Performance for Reduced Sensitivity …

187

where R ¼ SC; S ¼ ðI  CG22 Þ1 :

ð3:442Þ

These equations characterize the nominal system. The interest here is in characterizing the changes Δyd and Δv in yd and v, respectively, when G ¼ ½Gij  is actually G þ DG ¼ ½Gij þ DGij  while C and w remain unchanged. Clearly, DR ¼ ðDSÞC. Moreover, when only first order changes are retained, one gets DðSS1 Þ ¼ ðDS)S1 þ S½DðS1 Þ ¼ DðIÞ ¼ 0 , DS ¼ S½DðS1 ÞS:

ð3:443Þ

Hence, DR ¼ ðDSÞC ¼ S½DðI  CG22 ÞSC ¼ SCðDG22 ÞSC ¼ RðDG22 ÞR:

ð3:444Þ

It now follows from (3.440) and (3.441) that Dv ¼ ½ðDRÞG21 þ RðDG21 Þw ¼ ½RðDG22 ÞRG21 þ RðDG21 Þw ¼ RðDG22 Þv þ RðDG21 Þw ¼ R½ DG21 DG22 ½ w0 v0 0

ð3:445Þ

Similarly,  Dyd ¼ ðDG11 Þw þ ðDG12 Þv þ G12 ðDvÞ ¼ f½ DG11 DG12  þ G12 R½ DG21      DG11 DG12 w w ¼ ½ I G12 R DG : ¼ ½ I G12 R  v DG21 DG22 v

DG22 g

w



v

ð3:446Þ In (3.446), v is the nominal controller output; hence, the weighted matrix multiplier  S¼

W11 0

0 W22



 I G12 R ðDGÞ ¼ W1 RðDGÞ; 0 R

ð3:447Þ

where the weighting matrices W11 and W22 can be chosen based on the relative importance of plant uncertainty on deviations of the controlled variables v and yd from their nominal values. (The weighting matrices W11 and W22 are introduced to allow for more generality; however, W11 = k1I and W22 = k2I should be the primary choice with k1 and k2 nonnegative constants.) It is then consistent with the Wiener–Hopf approach to use this indicator to measure sensitivity with the functional

3 H2 Design of Multivariable Control Systems

188

1 Js1 ¼ 2pj

Zj1 j1

1 hTrðSS  Þids ¼ 2pj

Zj1 hTrðW1 RGs R W1 Þids

ð3:448Þ

j1

where Gs ¼ hðDGÞðDGÞ i

ð3:449Þ

is a plant-uncertainty spectral density matrix. (Js2 is used in the sequel for the stability margin functional.) Clearly, Gs is a parahermetian-positive matrix and it follows from Appendix C (Theorem C.1) that there exists a left Wiener–Hopf factorization Gs ¼ Xs Xs where Xs and its left inverse are analytic in Re > 0. Here it is assumed that Gs and W1 satisfy Assumption 3.11 The plant uncertainty matrix Gs is analytic and nonsingular on the finite jx-axis and it is strictly proper. The weighting matrix W1 is stable and proper. The properties imposed on Gs in Assumption 3.11 insure that Xs is square, strictly proper, and it and its inverse are analytic in 0  Re s\1. The assumption that Gs is analytic on the jx-axis is consistent with kDGðjxÞk1 being bounded in H1 approaches (Maciejowski 1989; Grimble 2006). Attention is also restricted here to those R that are given by Theorem 3.5. Specifically, one gets for the analog case that R ¼ A1 ðY1 þ KAÞ ¼ Ao1 K1 ZX1 Ag A þ Ro

ð3:450Þ

where Z is stable and strictly proper and where 1 1 1 Ro ¼ Ao1 K1 ½fK1  rb WG11 Gw rc X g þ þ fKAo1 Ya Xg X Ag A

ð3:451Þ

is the expression for R in the optimal nominal case: i.e., Z  0. Since G is compatible, G12A1 is stable; hence, so is G12R and it follows that  rs1 ¼ W1

I 0

  G12 R I Xs ¼ W1 R 0

G12 R R



Xs1 Xs2

 ¼ rsa1  rsb1 Zrsc1 ð3:452Þ

where 9 W11 ðXs1 þ G12 Ro Xs2 Þ > > > ¼ > > W22 Ro Xs2 > > = " # 1 W11 G12 Ao1 K ¼ > > > W22 Ao1 K1 > > > > ; 1 ¼ X A AX 

rsa1 rsb1 rsc1

g

s2

ð3:453Þ

3.9 Trade-off of Optimal Performance for Reduced Sensitivity …

189

are all stable. The matrix ∇s1 is also strictly proper when G12R is proper. Recalling (3.186), (3.189), and Theorem 3.5, one easily gets from (3.450) and (3.451) that R is strictly proper when k1 þ k2  0 and this inequality is usually satisfied. Hence, when G is proper, G12R is usually strictly proper and it is not unreasonable to impose. Assumption 3.12 R and G12R are strictly proper. Under Assumptions 3.11 and 3.12, ∇s1 is stable and strictly proper which assures that Js1 is finite. Moreover, ∇sa1 is strictly proper. Clearly, R is linear in Z and one can consider choosing a Z 6 0 for which Js1 is minimized subject to the constraint DJ  N. Problems of this type have been addressed by Newton et al. (1957) and Chang (1961) for the scalar case. The same methodology was used for the multivariable case in Bongiorno et al. (1997). In particular, a Lagrange multiplier a2 is introduced and the functional Zj1

Zj1 2pjðJs1 þ a DJÞ ¼

Tr ðrs1 rs1 Þds þ a

2

j1

Tr ðZZ Þds

2

ð3:454Þ

j1

is minimized with respect to Z as a function of a2; then a2 is chosen so that 2pjDJ ¼ N. However, it would be premature to do so without first including an additional term in (3.454) to account for stability margin.

3.9.3

Stability Margin

Also important with regard to plant uncertainty is the question of stability. Specifically, it is important that a margin of safety exits so that the system remains stable for the expected range of plant parameter values and for other plant modelling errors. When C in Fig. 3.2 is a stabilizing controller for the nominal e 22 ¼ A1 B ¼ B1 A1 , it follows from (2.33) and(2.60) that G22 ¼ G 1 / ¼ det ðAc A1 þ Bc B1 Þ ¼ det ðLX1 A1  MBA1 þ LY1 B1 þ MAB1 Þ ¼ det L ð3:455Þ is a stable polynomial. When G22 is actually given by the coprime polynomial matrix fraction description G22 þ DG22 ¼ ðB1 þ DB1 ÞðA1 þ DA1 Þ1 ;

ð3:456Þ

3 H2 Design of Multivariable Control Systems

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where ΔA1 and ΔB1 are polynomial matrices whose coefficient matrices are the changes in the coefficient matrices of A1 and B1 from their nominal values, respectively, then / becomes 

     DA1 DA1 ¼ det L þ ½ð LX1  MBÞ ðLY1 þ MA)  Ac A1 þ Bc B1 þ ½ Ac Bc  DB1 DB1    DA1 ¼ ðdet LÞdet I þ ½ ðX1  KBÞ ðY1 þ KAÞ  ¼ ðdet LÞ /D ¼ //D : DB1

/a ¼ det

ð3:457Þ Thus, the system remains stable iff it remains admissible and /D is free of zeros in Re s 0. It is also clear from (3.11), (3.37), and (3.442) that  /D ¼ det I þ A1 1 ½S



DA1 R DB1

 ð3:458Þ

and S ¼ I þ RG22 :

ð3:459Þ

One can always write 

DA1 DB1

 ¼ CD SD

ð3:460Þ

where CΔ contains the elements of the coefficient matrices in ΔA1 and ΔB1 and the elements of SΔ are all powers of s with highest power equal to the largest degree of the elements in B1(s) and A1(s). It now follows from (3.458) that /D ¼ det ðI þ SD A1 1 ½ S R CD Þ ¼ det ðI þ M CD Þ

ð3:461Þ

M ¼ SD A1 1 ½S

ð3:462Þ

where R  ¼ SD ½ ðX1  KBÞ ðY1 þ KAÞ :

Clearly, M is stable. Moreover, when a weighting matrix W2 is introduced and when Assumption 3.13 G22 is proper; R and SD A1 1 are strictly proper; and W2 is proper and stable;

3.9 Trade-off of Optimal Performance for Reduced Sensitivity …

191

holds, it easily follows that rs2 ¼ W2 M XD is strictly proper and stable where GD ¼ hCD CD i ¼ XD XD

ð3:463Þ

is the parameter-uncertainty covariance matrix for CΔ whose elements are modeled as zero-mean random variables. Hence, 1 Js2 ¼ 2pj

Zj1 j1

1 hTrðrs2 rs2 Þids ¼ 2pj

Zj1 TrðW2 M GD M  W2 Þds

ð3:464Þ

j1

is finite. In addition, Js2 is a quadratic functional on Z because it follows from (3.450), (3.451), and (3.459) that rs2 ¼ W2 M XD ¼ rsa2  rsb2 Zrsc2

ð3:465Þ

rsa2 ¼ W2 SD A1 1 ½ ðI þ Ro G22 Þ Ro XD ;

ð3:466aÞ

1 rsb2 ¼ W2 SD A1 1 Ao1 K ;

ð3:466bÞ

where

and rsc2 ¼ X1 Ag A½ G22

I XD :

ð3:466cÞ

Moreover, it follows that rsa2 ; rsb2 ; and rsc2 are stable and rsa2 is strictly proper 1 since A1 1 Ao1 stable is assumed in Theorem 3.5 and SD A1 is strictly proper. The 1 assumption that SD A1 is strictly proper is not overly restrictive. Indeed, when G22 is strictly proper and A1 is column reduced it is not difficult to establish (Bongiorno et al. 1997) that practical cases exist in which SD A1 1 is strictly proper. 1 With M ¼ Am Bm a left coprime matrix fraction description, it follows from (3.461) that /D ¼ detðI þ M CD Þ ¼ detðAm þ Bm CD Þ=ðdet Am Þ ¼ nðsÞ=dðsÞ:

ð3:467Þ

Since M CD is stable, dðsÞ ¼ det Am ðsÞ is a stable polynomial. Since M CD is strictly proper, /D has no zeros at s ¼ 1 and its zeros are those of the polynomial nðsÞ ¼ detðAm þ Bm CD Þ

ð3:468Þ

Clearly, when CΔ = 0 the zeros of n(s) are those of d(s) and all lie in Re s < 0. Moreover, the zeros of n(s) are continuous functions of the elements of CΔ. So if the

3 H2 Design of Multivariable Control Systems

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elements of CΔ are confined to a hypersphere of sufficiently small radius in the Euclidean space whose axes correspond to the elements of CΔ then the zeros of n(s) remain in Re s < 0. The only way zeros of n(s) can occur in Re s 0 is if the radius of the hypersphere is first increased enough for zeros of n(s) to exist on the finite jx-axis. This is only possible from (3.463) if M ðjxÞCD has an eigenvalue equal to −1 for some 0  x < ∞. Any condition on CΔ that assures this is not possible is then sufficient to ensure that no zeros of /D ðsÞ lie in Re s 0 and the system remains stable. It is not difficult to show (see Example 3.26) that the eigenvalue of M ðjxÞCD with largest magnitude has a magnitude which is less than or equal to the largest singular value of M ðjxÞCD or, equivalently, is less than or equal to the positive square root of the largest eigenvalue of ½M ðjxÞCD ½M ðjxÞCD  . Hence, /D is free of zeros in Re s 0 when kM ðjxÞCD k1  kM ðjxÞk1 kCD k1 \1

ð3:469Þ

where k k1 is the usual designation for the H1 -norm of a real rational stable matrix; i.e., the least upper bound for all the singular values of the matrix on the imaginary axis. Following the H1 approach, one might consider selecting Z so that the weighted norm krs2 k1 ¼ kW2 M XD k1 ¼ krsa2  rsb2 Zrsc2 k1

ð3:470Þ

is minimized. In the H1 approach, however, a proper Z is allowed and no attention is paid to the value of ΔJ given by (3.439). Thus, the Z that minimizes krs2 k1 might not be strictly proper and ΔJ could even be infinite. Unfortunately, minimizing the H1 -norm subject to a constraint on the H2 -norm increment is a difficult one for which to find an analytical solution. The difficulties are circumvented here by using Js2 ¼ krs2 k22 instead as an approximate measure of stability margin: The smaller Js2, the better is the expected stability margin. This is in keeping with the use of an H2 -norm as an approximate measure for plant-input saturation. Additional justification for the use of the H2 -norm Js2 as an approximate measure of stability margin is that an optimal H1 problem can be viewed as a weighted Wiener–Hopf problem (Sideris 1990). Thus, Js2 is a performance functional that takes into account stability margin and similarly to minimizing (3.454) one could minimize instead Zj1

Zj1 2pjðJs2 þ a D JÞ ¼

Trðrs2 rs2 Þds þ a

2

j1

TrðZZ Þds

2

ð3:471Þ

j1

with respect to Z and then choose a2 so that ΔJ = N. In the next subsection this is done for sensitivity and stability margin combined.

3.9 Trade-off of Optimal Performance for Reduced Sensitivity …

3.9.4

193

The Analytical Solution for Sensitvity and Stability Margin Combined

Obviously, Js ¼ Js1 þ Js2 is a performance functional that takes sensitivity and stability margin into account. Hence, the general problem of interest is the minimization of Zj1 2pjðJs1 þ Js2 þ a DJÞ ¼

Zj1

Zj1 Trðrs1 rs1 Þds þ

2

j1

Trðrs2 rs2 Þds þ a

TrðZZ Þds

2

j1

j1

ð3:472Þ with respect to Z and choosing a2 so that ΔJ = N. More general problems of this type are addressed in Khargonekar and Rotea (1991) for the multivariable case; the methodology presented here yields the desired solution directly from Theorem 3.4. The performance functional in (3.472) is a special case of the more general one 2pje Js ¼

Zj1 q X k¼1

Trðrsk rsk Þds; rsk ¼ rsak  rsbk Zrsck

ð3:473Þ

j1

where rsak ; rsbk ; and rsck are stable for k ¼ 1 ! q and rsak is strictly proper for k ¼ 1 ! q. As already noted in the next to last paragraph of Sect. 1.4, the Kronecker product G  R is the matrix whose ij-block is gij R and vec G ¼ ½ g02 g02 g02 0 is the vector obtained by stacking the n columns gj ; j ¼ 1 ! n of G. Thus, Trðrsk rsk Þ ¼ Trðrsk rsk Þ ¼ ðvec rsk Þ ðvec rsk Þ ¼ vsk vsk

ð3:474Þ

and 2pje Js ¼

Zj1 X q j1

Zj1 vsk vsk ds ¼

k¼1

Zj1 vs vs ds ¼

j1

Trðvs vs Þds

ð3:475Þ

j1

where vsk ¼ vec rsk and  v0s ¼ v0s1

v0s2

 v0sq :

ð3:476Þ

3 H2 Design of Multivariable Control Systems

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Clearly, one gets from (3.473) that vsk ¼ vec rsk ¼ vec rsak  vec ð rsbk Zrsck Þ:

ð3:477Þ

Two useful identities which are not difficult to verify are vecðAVDÞ ¼ ðD0  AÞvec V ðG1  G2 Þ ¼ G1  G2

:

ð3:478Þ

It follows from the first identity and (3.477) that vsk ¼ vsak  Usk ðvec ZÞ

ð3:479Þ

vsak ¼ vec rsak : Usk ¼ r0sck  rsbk

ð3:480Þ

where

Introducing  v0sa ¼ v0sa1

v0sa2



v0saq

 U0s ¼ U0s1

U0s2



U0sq



ð3:481Þ

and 

ð3:482Þ

yields 2pje Js ¼

Zj1

Zj1 Tr½ðvs vs ds ¼

j1

Tr½ðvsa  Us vec ZÞðvsa  Us vec ZÞ ds: ð3:483Þ j1

Equation (3.483) has the same form as (3.174) when the identifications e m ¼ vec Z ra ¼ vsa ; rb ¼ Us ; rc ¼ 1; K

ð3:484Þ

are made. Since rsak ; rsbk ; and rsck are stable for k ¼ 1 ! q and ∇sak is strictly proper for k ¼ 1 ! q, it is not difficult to confirm from (3.480) that ra and rb are stable and it is obvious that ∇c = 1 is also. Clearly, ∇a is strictly proper and det ðrc rc Þ ¼ 1 6¼ 0 on the finite jx-axis. Moreover, rb rb ¼ Us Us ¼

q X k¼1

Usk Usk

ð3:485Þ

3.9 Trade-off of Optimal Performance for Reduced Sensitivity …

195

is analytic on the finite jx-axis. When Us Us is also nonsingular on the finite jxaxis, it should be clear that the four assumptions in Theorem 3.4 are satisfied. In this e and X e that are square and case, one can construct Wiener–Hopf spectral factors K together with their inverses are analytic in Re s 0 such that e K e rb rb ¼ Us Us ¼ K

ð3:486Þ

eX e ¼ 1 , X e ¼ 1: rc rc ¼ X

ð3:487Þ

and

It then follows from (3.173) that the choice for vec Z which minimizes e J s is given by e 1 f Cg e e mo ¼ K vec Zo ¼ K þ

ð3:488Þ

e 1 Us vsa : e 1 rb ra rc X1 ¼ K e¼K C   

ð3:489Þ

where

e m which minimizes J in Theorem 3.4 is obtained This is so because the choice for K by setting the matrix parameter Z in (3.173) to zero. That Us Us is nonsingular on the finite jx-axis is established next when in (3.473), as is the case in (3.472), rsq ¼ aZ. Clearly, rsq ¼ aZ corresponds to rsaq ¼ 0; rsbq ¼ aI; rscq ¼ I:

ð3:490Þ

Usq ¼ r0scq  rsbq ¼ I  aI ¼ aI

ð3:491Þ

Then from (3.480)

where the identity matrices are appropriately sized. It immediately follows that Us has full column rank at all points in the finite s-plane; so, Us Us is indeed nonsingular on the finite jx-axis. Thus, Theorem 3.6 Under the assumptions (1) rsak ; rsbk ; and rsck are stable for k ¼ 1 ! q; (2) rsak is strictly proper for k ¼ 1 ! q; and (3) rsq ¼ aZ; the performance functional e J s in (3.473) has the minimum value e J so when vec Z is 1 e f Cg e . Specifically, (3.483) yields given by vec Zo ¼ K þ

3 H2 Design of Multivariable Control Systems

196

2pje J so ¼

Zj1

e 1 f Cg e 1 f Cg e Þðvsa  Us K e Þ ds Tr½ðvsa  Us K þ þ 

ð3:492Þ

j1

which simplifies to Zj1

2pje J so ¼

e e ðvsa vsa  f Cg þ  f Cg þ Þds:

ð3:493Þ

j1

Moreover, vec Zo is strictly proper and stable assuring that ΔJ in (3.439) is finite for Z = Zo. Proof It only remains to show that vec Zo is strictly proper and stable and that e is (3.492) simplifies to (3.493). Since K1 , Us , and vsa are stable, it follows that C e good; hence, f Cg is stable and it follows from (3.488) that so is vec Zo. Then, þ

e 1 f Cg e rso ¼ vsa  Us vec Zo ¼ vsa  Us K þ

ð3:494Þ

is also stable. Moreover, from e 1 Us Us K e 1 ¼ I K 

ð3:495Þ

e 1 and K e 1 ¼ a K e 1 Us are proper. Hence, from Usq K e 1 it is it follows that Us K  1 e is proper. So vec Zo and ∇so are not only stable, they are also strictly clear that K proper; this assures that both e J so and ΔJ with Z = Zo are finite. Since vsa is strictly e 1 Us vsa is strictly proper. Thus, e¼K proper, it is also true that C  e e ¼ f Cg e C þ þ f Cg

ð3:496Þ

2 e e e e f Cg  f Cg þ ¼ ½f Cg þ  f Cg   0ðs Þ

ð3:497Þ

and

is analytic in Re s 0 . It then follows that Zj1 j1

e e e e ðf Cg  f Cg þ þ f Cg þ  f Cg Þds ¼ 0;

ð3:498Þ

3.9 Trade-off of Optimal Performance for Reduced Sensitivity …

197

hence, 2pje J so ¼

Zj1 j1

Zj1 ¼ j1

Zj1 ¼ j1

9 > > > > e 1 Us Þðvsa  Us K e 1 f Cg e e Þds > K ðvsa  f Cg > þ  þ > > > > > > > > > = e e e e e e ðvsa vsa  C  f Cg þ  f Cg þ  C þ f Cg þ  f Cg þ Þds > > > > > > > > > > > > e e > ðvsa vsa  f Cg þ  f Cg þ Þds > > ;

ð3:499Þ

and the proof is complete. Clearly, both ΔJ and the sum of the first q – 1 terms on the right hand side of (3.473) given by e J uo ¼ e J so  a2 DJ

ð3:500Þ

are functions of a2 when Z = Zo. Using the same arguments as in Appendix A of Bongiorno et al. (1997), it easily follows that when ΔJ  N for all a2 a2n , the minimum Juo subject to the constraint ΔJ  N is realized with the choice a2 ¼ a2n . It is possible that a2n ¼ 0; this is the case when Z = Zo minimizes Ju (Z) and DJðZo Þ  N.

3.10

Historical Perspective and Commentary

The methodology presented in this chapter is based on Park and Bongiorno (2009) which begins with the identification of the acceptable persistent exogenous inputs for which there exists a parameterization of the subset of stabilizing controllers such that the error transform is stable and the actual plant inputs remain bounded with time. The optimal H2 solution is then obtained from within the subset of free parameters for which the cost functional is finite. A key element in this approach which distinguishes it from the earlier works is the explicit recognition and exploitation of the requirement that the controller output should be stable. Specifically, physical considerations are invoked to justify imposition of the requirement that the deterministic components of both the regulated variable yd and the controller output v be stable. A necessary and sufficient condition for only the regulated variable yd to be stable is derived in Cheng and Pearson (1978, 1981) for the standard configuration and in Wolovich and Ferreira (1979) and Saeks and Murray (1981) for the conventional feedback configuration. So if one uses the results in these papers, there

3 H2 Design of Multivariable Control Systems

198

would be no guarantee that v is stable. To assure that this was the case, the elements of v would have to be included in yd. This would impose restrictions on the generalized plant that were not explicitly exploited in these works where, moreover, no attempt to find optimal H2 solutions was made. Here, the requirement that v be stable is dealt with directly instead.2 The analog 2DOF standard configuration with persistent inputs is treated in Corrêa and Da Silveira (1995), Xie et al. (2000), and Park et al. (2002). The first two papers allow persistent deterministic exogenous inputs with poles in Re s 0 where the plant has none and insist on asymptotic tracking with zero steady-state error. In this case, the plant input must have poles in Re s 0 and the H2 -norm of the plant input is not finite. Corrêa and Da Silveira (1995) circumvent this issue by only incorporating the stable part of the plant input into the system performance measure. Xie et al. (2000) circumvent this issue by only incorporating the H2 -norm of the transfer matrix between the reference input and the plant input into the system performance measure. As a consequence of these approaches, however, plant input saturation can become a problem. In Park et al. (2002) on the other hand, the only persistent deterministic exogenous inputs considered are ones with poles on the imaginary axis that the plant is capable of tracking with a stable plant input. In this case, the H2 -norm of the plant input is finite and can be included in the system performance measure to avoid plant input saturation and, moreover, it is possible to make a connection between state-space and Wiener–Hopf design formulas. In the papers by Doyle et al. (1989), Khargonekar and Rotea (1991), and Hunt et al. (1994), the standard configuration exogenous input is assumed to be a white noise or simply an indeterminate to indicate an input channel. In this setting, the dynamics of the persistent inputs are absorbed into the generalized plant block. However, in these papers certain stabilizability assumptions are made which lead to a distinct disadvantage: as explained in Remark 3.2, the results are not applicable to 2DOF and 3DOF systems with unstable reference inputs. In Park and Bongiorno (1989), this problem is avoided by keeping the model of the persistent inputs separate from the generalized plant; however, persistent inputs with unstable poles only on the stability boundary can be accommodated. Specifically, a parameterization of all stabilizing controllers that yield finite cost in these cases is established and the one for which the cost is minimized is identified. However, for shape-deterministic inputs with poles in Re s > 0, the regulated variables even for a stabilized system can have poles in Re s > 0. In this case, Parseval’s formula does not hold; i.e., when z(s) has poles in Re s > 0, Z1 0

1 Tr½zðtÞz ðtÞdt 6¼ 2pj 

Zj1 Tr½zðsÞz ðsÞds:

ð3:501Þ

j1

© 2009 Taylor & Francis Ltd, http://www.tandfonline.com. This paragraph is reprinted, with permission, from Park and Bongiorno (2009).

2

3.10

Historical Perspective and Commentary

199

So the frequency-domain cost functional that was minimized in Park and Bongiorno (1989) has no meaningful physical significance in these cases.3 Additional insights regarding the material presented in this chapter are contained in the reply by Bongiorno and Park (2010) to comments by Aliev and Larin (2010).

3.11

Examples

Example 3.1 Confirm that the K given by (3.36) in Theorem 3.1 is stable for all stable Kg when Cw , Co , and A1 1 Ao1 are stable and G is compatible. Moreover, for all such K show that the matrices W and U given by (3.20) and (3.21) are stable and, therefore, so are Gv Gw ¼ WBw and Gd Gw ¼ UBw . Solution Since A1 1 Ao1 and Co are stable and since Yg1 and Ag are polynomial matrices, the matrix K given by (3.36) is obviously stable for all stable Kg. Using (3.36) in (3.20) one gets W ¼ ðY þ Ao1 Co Yg1 þ A1 Kg Ag ÞA1 g Bg ¼ ðYBg1 þ Ao1 Co Yg1 Bg1 ÞA1 g1 þ A1 Kg Bg ¼ ½YBg1 þ ðYo Cw Ag1  YBg1 ÞðI  Xg1 Ag1 ÞA1 g1 þ A1 Kg Bg ¼ ½Yo Cw  ðYo Cw Ag1  YBg1 ÞXg1  þ A1 Kg Bg ¼ Yo Cw  Ao1 Co Xg1 þ A1 Kg Bg which is stable for all stable Kg when Cw and Co are stable. Clearly, (3.21) then yields 1 1 U ¼ G11 A1 w þ G12 W ¼ Ao Cw  Bo1 Ao1 ðYo Cw  Ao1 Co Xg1 þ A1 Kg Bg Þ 1 1 ¼ A1 o ðI  Bo Yo ÞCw þ Bo1 ðCo Xg1  Ao1 A1 Kg Bg Þ ¼ Xo Cw þ Bo1 ðCo Xg1  Ao1 A1 Kg Bg Þ:

1 Since G is compatible, G12 A1 ¼ Bo1 A1 o1 A1 is stable or, equivalently, Ao1 A1 is stable. Hence, U is stable when Kg, Cw , and Co are stable. It is also of interest to note with Kg ¼ A1 1 Ao1 Km that

U ¼ Xo Cw  Bo1 ðCo Xg1 þ Km Bg Þ ¼ Xo Cw  Bo1 Kw when (3.34) is recalled which is in agreement with (3.26).

© 2009 Taylor & Francis Ltd, http://www.tandfonline.com. This paragraph is reprinted, with permission, from Park and Bongiorno (2009).

3

3 H2 Design of Multivariable Control Systems

200

Example* 3.2 Using (3.14) and (3.15), show directly that GvGw and GdGw are stable for all stable K when G is compatible and Cw , Co , and AG21 A1 w are stable. That is, confirm Remark 3.5. Solution Since Gv and Gd are stable for all stable K when G is compatible, it 1 1 suffices to show that Gv A1 w and Gd Aw are stable. It is obvious from Gv Aw ¼ 1 1 ðY þ A1 KÞAG21 A1 w that Gv Aw is stable for all stable K when AG21 Aw is stable. Now the last term on the right-hand side of 1 1 1 Gd A1 w ¼ G11 Aw þ G12 A1 Y1 G21 Aw þ G12 A1 KAG21 Aw

is stable for all stable K when AG21 A1 w is stable since G12A1 is stable when G is compatible. So with the help of (3.32) one gets that Gd A1 w is stable provided 1 1 1 Q ¼ G11 A1 w þ G12 A1 Y1 G21 Aw ¼ Ao Cw þ G12 YAG21 Aw 1 1 1 ¼ A1 o Cw þ G12 YBg1 Ag1 ¼ Ao Cw þ G12 ðYo Cw Ag1  Ao1 Co ÞAg1 1 1 ¼ A1 o ðI  Bo Yo ÞCw þ Bo1 Co Ag1 ¼ Xo Cw þ Bo1 Co Ag1 1 1 is stable. This is indeed the case because AG21 A1 w ¼ Bg1 Ag1 is stable iff Ag1 is stable.

Example* 3.3 Establish that Cw ¼ Ao G11 A1 w is stable when G is compatible iff Cw Bw ¼ Ao G11 Gw is stable. Show that Co is stable when Cw is stable iff Cz ¼ ðG11 þ G12 A1 Y1 G21 ÞA1 w Ag1 is stable. Solution Since G is compatible, M ¼ G11 þ G12 A1 Y1 G21 ¼ G11 þ G12 YAG21 ¼ G11  A1 o Bo YAG21 is stable. Hence, Ao G11 ¼ Ao M þ Bo YAG21 is stable because AG21 is stable. Since Aw, Bw is a left-coprime polynomial matrix pair, it immediately follows that Cw ¼ 1 Ao G11 A1 w is stable iff Ao G11 Aw Bw ¼ Ao G11 Gw is stable. So one can check whether or not Cw is stable without the need to calculate Aw and Bw. When Cw is stable, e e Co ¼ A1 o1 C o where C o ¼ Yo Cw Ag1  YBg1 is stable. Hence, Co is stable iff Bo1 Co ¼ G12 ðYo Cw Ag1  YBg1 Þ ¼ G12 ðYo Cw  YAG21 A1 w ÞAg1 1 1 ¼ G12 ðYo Ao G11  YAG21 ÞA1 w Ag1 ¼ ðAo Bo Yo Ao G11 þ G12 YAG21 ÞAw Ag1 1 1 ¼ ½A1 o ðI  Ao Xo ÞAo G11 þ G12 YAG21 Aw Ag1 ¼ ðG11 þ G12 A1 Y1 G21  Xo Ao G11 ÞAw Ag1

¼ Cz  Xo Cw Ag1

is stable. Clearly, this is the case iff Cz is stable.

3.11

Examples

201

Example 3.4 Confirm that the pair U; W given by (3.26) satisfies (3.21). Solution It suffices to show that the U given in (3.26) is obtained from (3.21) when the expression for W given in (3.26) is used in (3.21). Clearly, this is the case from 1 1 U ¼ G12 W þ G11 A1 w ¼ Bo1 Ao1 ðYo Cw þ Ao1 Kw Þ þ Ao Cw 1 1 1 ¼ Bo1 Yo1 A1 o Cw  Bo1 Kw þ Ao Cw ¼ ðXo Ao  IÞAo Cw  Bo1 Kw þ Ao Cw ¼ Xo Cw  Bo1 Kw :

Example 3.5 A matrix Gw is said to be matched to G if the ratio of characteristic denominators wG22 =wG11 Gw is stable. Establish that a necessary condition for Gw to be acceptable for G when G is compatible is that Gw be matched to G. Hint: Begin with Gd Gw ¼ G11 Gw þ G12 Gv Gw and use the fact that when G is compatible then G12 ¼ MA1 1 where M is a stable matrix. Solution When G is compatible and Gw is acceptable, there exists a stable K for which GvGw and Gd Gw ¼ G11 Gw þ G12 Gv Gw are stable. Moreover, from G11 Gw ¼ Gd Gw  G12 Gv Gw ¼ Gd Gw  MA1 1 Gv Gw it follows for any finite po in the unstable region of the complex plane that dðG11 Gw ; po Þ  dðA1 1 ; po Þ because GdGw, GvGw, and M are stable. That is, wG22 =wG11 Gw is stable since to within a nonzero multiplicative constant wG22 is given by det A1. So Gw is matched to G. Example* 3.6 Establish that the polynomial matrix pair  Aq ¼

Am 0

   0 Bm ; Bq ¼ An Bn

is left-coprime when Am, Bm and An, Bn are left-coprime pairs and det Am and det An have no common zeros. Solution At all points where det Am 6¼ 0 and det An 6¼ 0,

[ Aq

⎡A Bq ] = ⎢ m ⎣0

0 An

Bm ⎤ Bn ⎥⎦

has full row rank. When det Am 6¼ 0 and det An = 0,

3 H2 Design of Multivariable Control Systems

202

[ xm′

⎡A xn′ ]⎢ m ⎣0

0 An

Bm ⎤ = 0′ ⇒ xm′ Am = 0′ ⇒ xm′ = 0′ Bn ⎥⎦

and x0n ½ An

Bn  ¼ 00

) x0n ¼ 00

because An, Bn is left-coprime. Hence, ½ Aq Bq  has full row rank at points where det Am 6¼ 0 and det An = 0. Similarly, when det Am = 0 and det An 6¼ 0, ½ Aq Bq  has full row rank because Am, Bm is left-coprime. Since det Am = 0 and det An = 0 simultaneously is ruled out, the pair Aq, Bq is left-coprime. Typically, for the Am and An defined by (3.89), det Am and det An have no common zeros for the following reasons. First, A1 d usually has only unstable poles. 1 But M in (3.89) is stable so none of these poles are poles of both A1 m and An . Second, it would almost always be true that L and AFPd have no stable poles in 1 common. Thus, A1 m and An would have no poles in common and, consequently, det Am and det An would have no zeros in common. Example* 3.7 Confirm Remark 3.6. e e1 one gets from (3.3) and (3.4) that e d1 ¼ T ed G Solution With l = 0 and G e e1 r þ G12 v e 11  T e e1 r þ G e 12 v ¼ G11 G ed Þ G yd ¼ ð G e e1 r þ G e e1 r þ G22 v e 21 G e 22 v ¼ G21 G ym ¼ G which is identical in form with the final expressions in (3.3) and (3.4) except that e e1 r is in place of G e w q and instead of (3.5) one has (3.45). It now follows in now G straightforward fashion that (3.12) and (3.13) become, respectively, e e1 r; yd ¼ Gd G e e1 r v ¼ Gv G where Gv and Gd are given by (3.14) and (3.15) with the Gij given by (3.45). e is admissible and T ed is stable, Moreover, as pointed out in Remark 3.6, when G e e1 in place of Gw the then AG21, G12A1, G11 þ G12 A1 Y1 G21 are all stable. So with G same arguments used to establish the circumstances under which GvGw and GdGw e e1 ¼ Gw , T ed is stable, and the Gij are given are stable again apply. That is, when G by (3.45), Theorem 3.1 still holds. Example* 3.8 Establish directly that the ∇a given by (3.160) is stable when G is compatible and Gw is acceptable for G. Solution From (3.160),

3.11

Examples

203

e o Xg þ fK C e o Xg ÞUc ra ¼ WG11 Gw  Ub ðfK C þ  or, equivalently, e o X  fK C e o Xg ÞUc : ra ¼ WG11 Gw  Ub ðK C 1 e o Xg are stable, it follows that ∇a is stable iff Since Ub, Uc, and fK C 1 e o XUc rea ¼ WG11 Gw  Ub K C is stable. Using (3.148), (3.152), and (3.153) one gets 1 rea ¼ WG11 Gw  WBo1 A1 o1 ðY þ Ao1 Co Yg1 ÞAg Bg Bw 1 1 1 ¼ WðG11 A1 w  Bo1 Ao1 YAg Bg  Bo1 Co Yg1 Ag Bg ÞBw :

Recalling (3.25) then yields 1 1 1 rea ¼ WðA1 o Cw  Bo1 Ao1 YBg1 Ag1  Bo1 Co Ag1 Yg Bg ÞBw :

Introducing 1 1 A1 o ¼ Xo þ Ao Bo Yo ¼ Xo þ Bo1 Ao1 Yo

one gets 1 1 rea ¼ W½Xo Cw þ Bo1 A1 o1 ðYo Cw Ag1  YBg1 ÞAg1  Bo1 Co Ag1 Yg Bg Bw :

With the aid of (3.32) it follows that rea ¼ W½Xo Cw þ Bo1 Co A1 g1 ðI  Yg Bg ÞBw ¼ WðXo Cw þ Bo1 Co Xg1 ÞBw : Since Cw and Co are stable when G is compatible and Gw is acceptable for G, it is clear that rea and, therefore, ∇a are stable in this case. Example* 3.9 In Sects. 3.5 and 3.6 on the H2 design of digital and analog multivariable control systems, it is assumed that Gw is acceptable and Theorem 3.1 is e e invoked with Gw ¼ A1 w Bw . When G w2 is stable, however, Gw is acceptable iff G w1 1 e is acceptable. Hence, one can use Theorem 3.1 with G w1 ¼ Aw Bw instead. (a) With A1 1 Ao1 stable derive the alternative expressions to (3.107) for digital systems and to (3.150) and (3.160) for analog systems which one now gets for ∇a, ∇b, and ∇c. Confirm that ∇a, ∇b, and ∇c are stable and give conditions that assure ∇a is strictly

3 H2 Design of Multivariable Control Systems

204

proper in the analog case. (b) When W is a polynomial matrix, so are the ∇b and ∇c given by (3.107) and (3.150). Is this still true for the ∇b and ∇c found in (a)? What other implications are there? Solution (a) All of the results in Sect. 3.3 still hold, but with Aw, Bw now the e w1 . Again left-coprime polynomial matrix pair associated with G r ¼ WGd Gw ¼ WGd

 pffiffiffiffiffi e w1 a1 G

pffiffiffiffiffi e  a2 G w2

where Gd ¼ G11 þ G12 A1 ðY1 þ KAÞG21 : Using (3.36) for K yields Gd ¼ Ga þ G12 Ao1 Km Ag AG21 where Ga ¼ G11 þ G12 A1 Y1 G21 þ G12 Ao1 Co Yg1 AG21 : Thus, r ¼ WGd Gw ¼ WGa Gw þ WG12 Ao1 Km Ag AG21 Gw  pffiffiffiffiffi  ffi e w1 pffiffiffiffi e w2 : ¼ WGa Gw  WBo1 Km a1 Ag AG21 G a2 Ag AG21 G Now e w1 ¼ Ag ðAG21 A1 ÞBw ¼ Ag ðA1 Bg ÞBw ¼ Bg Bw Ag AG21 G w g and e w2 ¼ Ag ðAG21 A1 ÞAw G e w2 ¼ Ag ðA1 Bg ÞAw G e w2 ¼ Bg Aw G e w2 : Ag AG21 G w g Hence, for the digital case r ¼ ra  rb K m rc where ra ¼ WGa Gw ; rb ¼ WBo1 ; rc ¼

 pffiffiffiffiffi a1 Bg Bw

 pffiffiffiffiffi e w2 : a2 Bg Aw G

3.11

Examples

205

e w2 are stable. Also, since ∇ is stable for Clearly, ∇b and ∇c are stable when W and G Km  0, it follows that ∇a is stable and Theorem 3.2 can be applied. For the analog case, one can again begin with (3.146) which leads to e o þ Km Þrc r ¼ WG11 Gw  rb ð C e o is given by (3.148). Proceeding as previously done in Sect. 3.6 one now where C e m rc where easily gets r ¼ ra  rb K rb ¼ WBo1 ; rc ¼

 pffiffiffiffiffi a1 Bg Bw

 pffiffiffiffiffi e w2 ; a2 Bg Aw G

e w2 are and ∇a is now defined by (3.160). Clearly, ∇b and ∇c are stable when W and G 1 stable. Moreover, ∇a is strictly proper when WG11 Gw  0ðs Þ and is stable because e m  0. Hence, Theorem 3.4 can be applied. (b) Obviously, ∇ is stable when K rb ¼ WBo1 is still a polynomial matrix when W is. However, ∇c may no longer be e w2 may not be: G e w2 is usually a proper rational matrix. Thus, the because Bg Aw G spectral factorization rc rc ¼ XX now typically involves factoring a rational matrix instead of a polynomial matrix. Moreover, the expression for ∇c which was rc ¼ Bg Bw in Sects. 3.5 and 3.6 is now the more complicated one  pffiffiffiffiffi  pffiffiffiffiffi e w2 . On the other hand, there is a significant benefit rc ¼ a2 Bg Aw G a1 Bg Bw from the fact that finding literal expressions for the coprime polynomial matrix Aw, Bw is possible as in Sects. 3.7 and 3.8. Example* 3.10 The purpose of this example is to provide additional insights regarding H2 optimization for the analog case. The obvious modifications for the digital case are omitted. The performance functional considered is

1 J¼ 2pj

Zj1 j1

1 qds ¼ 2pj

Zj1 Tr½ðra  rb Qrc Þðra  rb Qrc Þ ds j1

where ∇a, ∇b, and ∇c are real rational matrices and no stability requirement on r ¼ ra  rb Qrc is imposed initially. It is assumed that Gb ¼ rb rb and Gc ¼ rc rc are full rank matrices. (a) Determine the necessary and sufficient condition for which there exists a parameterization of all marginally stable (analytic in Re s > 0) Q for which J is finite or, equivalently, r ¼ ra  rb Qrc is strictly proper and good and give the parameterization for Q. (b) Under the condition found in (a), which choice for Q minimizes J? (c) When Gb and Gc in addition to being of full 1 rank are such that G1 b and Gc are good and when ∇a is strictly proper and good, establish that the necessary and sufficient condition found in (a) is satisfied and all the Q given by the parameterization are stable. (d) When the assumptions of Theorem 3.4 are satisfied, show that ∇ is strictly proper and stable and Q is stable as e m. an alternative proof of Theorem 3.4 with Q in place of K

3 H2 Design of Multivariable Control Systems

206

Solution (a) Since Gb ¼ rb rb and Gc ¼ rc rc are parahermitian-positive and have full rank, there exist by Theorem C.1 spectral factorizations rb rb ¼ K K and rc rc ¼ XX such that the square matrices K and X together with their inverses are marginally stable. Moreover, the paraunitary matrices Ub ¼ rb K1 e 1 is marginally stable and Uc ¼ X1 rc are good and proper. Clearly, Q ¼ K1 QX e is and one can write iff Q e 1 rc ¼ ra  Ub QU e c: r ¼ ra  rb K1 QX Hence, with e ¼ K1 rb ra rc X1 ¼ Ub ra Uc C   it follows for finite J that eQ e ¼C e Ub rUc ¼ Ub ra Uc  Q must be strictly proper and good because ∇ must be strictly proper and good. Now e one can always write for any marginally stable Q e e e ¼ Z þ f Cg Q 1 þ f Cg þ where Z is marginally stable. Clearly, eQ e  Z  f Cg e e e e ¼C C 1  f Cg þ ¼ f Cg  Z is strictly proper and good iff Z is. Since Z must be good in addition to being marginally stable, Z must be stable. Thus, for finite J it is necessary that Q be given by 1 e e e  f Cg e þ ZÞX1 e 1 ¼ K1 ðf Cg Q ¼ K1 QX ¼ K1 ð C þ þ f Cg1 þ ZÞX 

where Z is strictly proper and stable (i.e., admissible). With such a Q, e  f Cg e þ ZÞX1 rc ¼ ro þ rb K1 ðf Cg e  ZÞX1 rc r ¼ ra  rb K1 ð C   is strictly proper and good or, equivalently, J is finite iff e 1 rc ro ¼ ra  rb K1 CX

3.11

Examples

207

is strictly proper and good: the paraunitary matrices Ub ¼ rb K1 and Uc ¼ X1 rc e  Z is strictly proper and good. That is a e ¼ f Cg are proper and good and Z  parameterization of all marginally stable Q exists for which J is finite iff ∇o is strictly proper and good and the parameterization is given by Q ¼ 1 e e where Z is a strictly proper and stable matrix. K1 ðf Cg þ þ f Cg1 þ ZÞX (b) The choice of Q or, equivalently, Z for which J is minimized is obtained from the following considerations. Clearly,

e X1 rc : r ¼ ro þ rb K1 Z Thus, e X1 rc ½ro þ rc X1 e 1 q ¼ Tr ðrr Þ ¼ Tr f½ro þ rb K1 Z  Z  K rb g 1 1 e 1 1 e e X1 rc rc X1 e 1 ¼ Tr ðro ro þ ro rc X Z  K rb þ rb K Z X rc ro þ rb K1 Z  Z  K rb Þ 1 1 1 1 e 1 e e 1 e 1 ¼ Tr ðro ro Þ þ Tr ðK1  rb ro rc X Z  Þ þ Tr ð Z X rc ro rb K Þ þ Tr ð Z X rc rc X Z  K rb rb K Þ eZ e Þ ¼ Tr ðro ro Þ þ Tr ð Z

1 1 because X1 rc rc X1 ¼ I, and  ¼ I, K rb rb K 1 1 1 1 1 e e e K1  rb ro rc X ¼ ðX rc ro rb K Þ ¼ K rb ra rc X  C ¼ C  C ¼ 0:

Moreover, e 1 rc Þðra  rc X1 C e  K1 rb Þ qo ¼ Tr ðro ro Þ ¼ Tr½ðra  rb K1 CX   1 e 1 1 1 e e 1 rc rc X1 C e  K1 rb Þ ¼ Tr ðra ra  rb K CX rc ra  ra rc X C  K rb þ rb K1 CX 







e 1 rc ra rb K1 Þ  Tr ðK1 rb ra rc X1 C e  Þ þ Tr ðK1 rb rb K1 CX e 1 rc rc X1 C e Þ ¼ Tr ðra ra Þ  Tr ð CX     eC e  Þ  Tr ð C eC e  Þ þ Tr ð C eC e  Þ ¼ Tr ðra ra Þ  Tr ð C eC e  Þ: ¼ Tr ðra ra Þ  Tr ð C

In addition, e  ZÞðf Cg e  ZÞ  eZ e Þ ¼ Tr ½ðf Cg Tr ð Z    e e e e f Cg ¼ Tr ðf Cg   Þ  Tr ðf Cg Z Þ  Tr ðZf Cg Þ þ Tr ðZZ Þ: e Z  0ðs2 Þ and analytic in Re s  0 and since Also, since f Cg  2 e Zf Cg   0ðs Þ and analytic in Re s 0, it follows that

3 H2 Design of Multivariable Control Systems

208

1 J¼ 2pj

Zj1 j1

e Z Þds ¼ 1 Trðf Cg  2pj

Zj1

e TrðZf Cg  Þds ¼ 0:

j1

Hence, 1 J¼ 2pj ¼

1 2pj

Zj1 j1

Zj1

1 qds ¼ 2pj

Zj1 j1

1 Trðrr Þds ¼ 2pj

Zj1

eZ e Þds ½qo þ Trð Z

j1

eC e  Þ þ Trðf Cg e f Cg e ½Trðra ra Þ  Trð C   Þ þ TrðZZ Þds:

j1

Since Tr½ZðjxÞZ ðjxÞ [ 0 for all ZðjxÞ 6 0, the minimum J is obtained iff Z(s)  0 and is given by 1 Jo ¼ 2pj

Zj1

eC e  Þ þ Trðf Cg e f Cg e ½Trðra ra Þ  Trð C   Þds

j1

and the corresponding choice for Q is 1 e e e  f Cg e ÞX1 : Qo ¼ K1 ðf Cg ¼ K1 ð C þ þ f Cg1 ÞX 

e and ∇o because the paraunitary (c) When ∇a is good and strictly proper, so are C 1 1 matrices Ub ¼ rb K and Uc ¼ X rc are proper and good. In addition, when Gb 1 1 and Gc are of full rank and such that G1 b and Gc are good then from Gb ¼ 1 1 1 it follows that the square matrices K1 and X1 are K1 K1  and Gc ¼ X X good and, therefore, stable. Thus, one gets from Q¼ 1 e 1 e e e is K ðf Cg þ þ f Cg1 þ ZÞX that Q is stable since f Cg þ is stable when C 1 e e good and since Z is stable. Moreover, f Cg 1 ¼ 0 because C  0ðs Þ and one gets 1 e 1 the simplified expression Q ¼ K ðf Cg þ þ ZÞX . (d) Under the assumptions in Theorem 3.4, ∇a is strictly proper and stable, ∇b and e is strictly proper and good, and K1 and X1 are stable. Hence, ∇c are stable, C when Z is admissible,

3.11

Examples

209

1 1 e e e Q ¼ K1 ðf Cg ¼ K1 ðf Cg þ þ f Cg1 þ ZÞX þ þ ZÞX

and r ¼ ra  rb Qrc are stable. Also, 1 e e r ¼ ra  rb K1 ðf Cg þ þ ZÞX rc ¼ ra  Ub ðf Cg þ þ ZÞUc

e , and is strictly proper because Ub and Uc are paraunitary matrices and ∇a, f Cg þ Z are strictly proper. Example 3.11 Establish for the 1DOF system of Fig. 3.3 using the notation of Sect. 3.7 that when 

 I 0 r ¼ WGd Gw ¼ G G 0 Wq d w one gets in the integrand of the cost functional J Trðrr Þ ¼ Tr½ðTd  PRÞGr ðTd  PRÞ þ ðPd  PRPo ÞGd ðPd  PRPo Þ þ ðPRÞGn ðPRÞ  þ TrðkQRGt R Þ where R ¼ ðY þ A1 KÞA ¼ A1 ðY1 þ KAÞ. Aside from notational differences, this expression is the same as the one used in Youla et al. (1976b). Solution

For the 1DOF system, Eqs. (3.225) and (3.226) lead to e 12 R G e 21 Gd ¼ G11 þ G12 A1 ðY1 þ KAÞG21 ¼ G11 þ G     Pd Td 0 P ¼ þ R½ Po I I  0 0 0 I

and  WGd ¼

I

0



 Gd ¼

Pd

Td

0



 þ

P

0 Wq Wq 0 0 0   ðPd  PRPo Þ ðPR  Td Þ PR ¼ : Wq RPo Wq R Wq R

 R½ Po

I

I 

3 H2 Design of Multivariable Control Systems

210

 pffiffiffiffiffi ffie  e w1 pffiffiffiffi Also, using (3.248), (3.280), and Gw ¼ yields a2 G a1 G w2 Gw Gw ¼ diagfGd ; Gr ; Gn g. Hence, Trðrr Þ ¼ Tr½ðWGd ÞðGw Gw ÞðWGd Þ  leads to 8 >

Wq RPo Gd :

ðPR  Td ÞGr Wq RGr

2  ðPd  PRPo Þ 6 4 ðPR  Td Þ Wq RGn ðPRÞ PRGn

39 ðWq RPo Þ > = 7 ðWq RÞ 5 > ; ðWq RÞ

¼ Tr½ðPd  PRPo ÞGd ðPd  PRPo Þ þ ðPR  Td ÞGr ðPR  Td Þ þ PRGn ðPRÞ  þ Tr½Wq RðPo Gd Po þ Gr þ Gn ÞðWq RÞ  ¼ Tr½ðPd  PRPo ÞGd ðPd  PRPo Þ þ ðPR  Td ÞGr ðPR  Td Þ þ PRGn ðPRÞ  þ Tr½kQRGt R 

when (3.279) is recalled. Example* 3.12 (a) When the 1DOF system shown in Fig. 3.3 is admissible, a stabilizing controller is employed, and both d and n are zero, establish that e ¼ ðTd A1  PY  PA1 KÞAr: S r where e S r is any stable vector, (b) When F and Td are stable and when r ¼ A1 e prove that e is stable iff ðTd F  IÞP is stable. Hint: Establish the fact that MPY is stable when M is stable iff MP is stable using YB ¼ I  A1 X1 . (c) For the reference inputs r of part (b), show that u and v are also stable for all stable K. (d) When 1 r ¼ A1 r br is a left coprime polynomial matrix fraction description and AAr and Td 1 are stable, prove that e is stable iff ðTd  PA1 Y1 ÞAr is stable. (e) When F and Td 1 are stable, establish that the pair AA1 is stable iff the pair r , ðTd  PA1 Y1 ÞAr 1 1 Ap Td Ar , ðI  FTd ÞAr is stable which is consistent with Lemma 3.4. Note: M ¼ ðTd  PA1 Y1 ÞA1 r is stable iff Td ¼ MAr þ PA1 Y1 where M is a stable matrix and this provides a parameterization of all suitable stable Td. Solution (a) With the usual coprime polynomial matrix fraction descriptions for FP given by (3.58) and (3.59), any stabilizing controller is given by C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1 : It then follows from u ¼ r  FPCu ) u ¼ ðI þ FPCÞ1 r that u ¼ ½I þ A1 BðY þ A1 KÞðX  B1 KÞ1 1 r ¼ ðX  B1 KÞ½AðX  B1 KÞ þ BðY þ A1 KÞ1 Ar ¼ ðX  B1 KÞAr:

3.11

Examples

211

Hence, v ¼ Cu ¼ ðY þ A1 KÞAr; y ¼ PCu ¼ PðY þ A1 KÞAr; and it follows that e ¼ Td r  y ¼ ðTd A1  PY  PA1 KÞAr: (b) Since PA1 is stable because the system is admissible and K is stable because C is a stabilizing controller, it follows that e is stable for all stable Ar ¼ e S r iff Td A1  PY ¼ Td ðX þ FPYÞ  PY ¼ Td X þ ðTd F  IÞPY is stable. Since TdX is stable, this is the case iff ðTd F  IÞPY is stable. Clearly, ðTd F  IÞP stable is sufficient. It is also necessary. For if ðTd F  IÞPY is stable then ðTd F  IÞPYB ¼ ðTd F  IÞPðI  A1 X1 Þ ¼ ðTd F  IÞP  ðTd F  IÞPA1 X1 must be stable. Thus, ðTd F  IÞP must be stable because Td, F, and PA1 are stable. S r and v ¼ ðY þ A1 KÞe S r are stable for all stable K. (d) (c) Clearly, u ¼ ðX  B1 KÞe The error e is stable iff 1 1 1 e ¼ ðTd A1  PYÞAA1 r br  PA1 KAAr br ¼ ðTd  PA1 Y1 ÞAr br  PA1 KAAr br

is stable. Since PA1 KAA1 r br is stable and ðTd  PA1 Y1 Þ is stable, e is stable iff ðTd  PA1 Y1 ÞA1 is stable. (e) When AA1 and ðTd  PA1 Y1 ÞA1 are stable, it r r r easily follows from 1 1 Ap ðTd  PA1 Y1 ÞA1 r ¼ Ap Td Ar  Bp YAAr

that Ap Td A1 r must be stable. Also, from 1 1 FðTd  PA1 Y1 ÞA1 r ¼ ðFTd  B1 Y1 ÞAr ¼ ðFTd  I þ XAÞAr 1 1 ¼ ðFTd  IÞAr þ XAAr

it follows that ðFTd  IÞA1 must be stable. The converse that AA1 and ðTd  r r 1 1 PA1 Y1 ÞAr are stable when M1 ¼ Ap Td A1 and M ¼ ðFT  IÞA are stable is 2 d r r

3 H2 Design of Multivariable Control Systems

212

established next. First, it immediately follows from (3.261) that AA1 r is stable. It also follows from (3.261) that 1 1 1 1 ðTd  PA1 Y1 ÞA1 r ¼ Ap M1  PYAAr ¼ Ap M1  PYAðM2 þ FAp M1 Þ 1 ¼ ðI  PYAFÞA1 p M1  PA1 Y1 M2 ¼ ðI  PA1 Y1 FÞAp M1  PA1 Y1 M2 :

Since ðI  PYAFÞ ¼ ðI  PA1 Y1 FÞ is stable and ðI  PYAFÞA1 p Bp ¼ ðI  PYAFÞP ¼ PðI  YAFPÞ ¼ PðI  YBÞ ¼ PA1 X1 is stable, it follows that ðI  PYAFÞA1 is stable and, therefore, so is p 1 ðTd  PA1 Y1 ÞAr . Example* 3.13 (a) Establish for the analog 1DOF system treated in Sect. 3.7 that detðX1  KBÞ 6 0 when K is given by (3.303) and when ðP P þ kQÞ1  0ðs2k1 Þ, 2k2 k3 G1 t  0ðs Þ, and FP  0ðs Þ with k1 þ k2 þ k3  0. (b) In this case, also establish that C  0ðsk1 þ k2 1 Þ. Solution (a) It suffices to establish that A1 ðX1  KBÞ ! I for s ! 1. From (3.303), K ¼ K1 Ca X1  A1 1 Y where 1 1 Ca ¼ Z þ fK1  Ci X g þ þ fKA1 YXg

is strictly proper. Thus, A1 ðX1  KBÞ ¼ A1 X1  A1 K1 Ca X1 B þ YB ¼ I  A1 K1 Ca X1 B: Now (recall Lemma 3.2) 1 A1 K1 K1  0ðs2k1 Þ , A1 K1  0ðsk1 Þ  A1 ¼ ðP P þ kQÞ

and 1 1 1 1 1 1 2k2 þ 2k3 B X1 Þ  X B ¼ B A A X X AA B ¼ ðFPÞ Gt ðFPÞ  0ðs 1 k2 þ k3 Þ: , X B  0ðs

3.11

Examples

213

Hence, A1 K1 Ca X1 B  0ðsk1 þ k2 þ k3 1 Þ is strictly proper when k1 þ k2 þ k3  0 and in this case A1 ðX1  KBÞ ! I for s ! 1. That is, detðX1  KBÞ 6 0. (b) It easily follows from C ¼ ðX1  KBÞ1 A1 1 A1 ðY1 þ KAÞ that C ¼ ðI  A1 K1 Ca X1 BÞ1 ðA1 Y1 þ A1 K1 Ca X1 A  YAÞ ¼ ðI  A1 K1 Ca X1 BÞ1 A1 K1 Ca X1 A: Now 1 1 1 2k2 k2 G1 t ¼ A X X A  0ðs Þ , X A  0ðs Þ

and therefore A1 K1 Ca X1 A  0ðsk1 þ k2 1 Þ. Since A1 K1 Ca X1 B  0ðsk1 þ k2 þ k3 1 Þ is strictly proper when k1 þ k2 þ k3  0 it follows from the expression for C that C  0ðsk1 þ k2 1 Þ. Example 3.14 Suppose for the 1DOF system treated in Sect. 3.7 that instead of choosing yd as in (3.223) one took 

   y  Td r Pd d þ Pv  Td r e 12 v e 11  T ed Þw þ P es G ¼ ¼ ðG yd ¼ Ps v Ps v e s ¼ diagfI; Ps g. Then G11, G21, and G22 remained unchanged, but now where P e 12 ¼ es G G12 ¼ P



P Ps



1 ¼ Bo1 A1 o1 ¼ Ao Bo :

Invoke (3.58) through (3.61) to establish that when Ps is stable that the system is still compatible and that A1 1 Ao1 is still stable so that (3.36) of Theorem 3.1 again holds. e 12 A1 is stable. Since Ps is stable, Solution Since the 1DOF system is admissible, G e 12 A1 is stable. Also, AG21 ¼ e s is stable. Hence, one again gets that G12 A1 ¼ P esG P e 11 ¼ e esG A G 21 is unchanged and is, therefore, stable. In addition, it follows from P e 11 and P ed ¼ T es T ed that G e 12 A1 Y1 G e 21  T e 21  T e 11 þ P e 11 þ G e 12 A1 Y1 G es G ed ¼ P e sðG ed Þ G11 þ G12 A1 Y1 G21 ¼ G

214

3 H2 Design of Multivariable Control Systems

e 21 are stable. Thus, with the new e 11 þ G e 12 A1 Y1 G e s, T ed , and G is stable because P e is admissible by assumption and choice for yd the system is compatible: G e G22 ¼ G 22 . Also, from     PAo1 Bp1 A1 p1 Ao1 Bo1 ¼ G12 Ao1 ¼ ¼ Ps Ao1 Ps Ao1 1 it follows that Bp1 A1 p1 Ao1 and, therefore, Ap1 Ao1 are stable because Bo1 , a polynomial matrix, is stable. Hence, Ao1 ¼ Ap1 M where M is a stable matrix. It then 1 1 follows that A1 1 Ao1 ¼ A1 Ap1 M is stable by Lemma 2.10: A1 Ap1 is stable since F is stable and it is necessary for admissibility that wFP ðwF wP Þ1 be stable.

Example* 3.15 Establish that G = MM* is analytic on Cs iff M is analytic on Cs where for the digital case Cs is jKj ¼ 1 and for the analog case Cs is the finite jxaxis.  Solution For the digital case, GðkÞ ¼ MðkÞM ðkÞ where M ðkÞ ¼ M  ð1 kÞ ¼ M 0 ð1=kÞ. Thus, M ðkÞ can only have poles at k ¼ k1 o where ko is a pole of MðkÞ. Hence, if MðkÞ is free of poles on jkj ¼ 1, so is M ðkÞ and GðkÞ. Conversely, if GðkÞ is free of poles on jkj ¼ 1, then when MðkÞ is nr  nc one gets that Tr½Gðejh Þ ¼ Tr½Mðejh ÞM  ðejh Þ ¼

nc nr X X mlm ðejh Þ 2 l¼1 m¼1

must be finite or, equivalently, every element mlm ðejh Þ must be finite. That is, MðkÞ must be free of poles on jkj ¼ 1. Similarly for the analog case GðsÞ ¼ MðsÞM ðsÞ where M ðsÞ ¼ M  ðsÞ ¼ M 0 ðsÞ. Thus, M ðsÞ can only have finite poles at s ¼ so where so is a finite pole of M(s). Hence, if M(s) is free of poles on the finite jxaxis, so is M ðsÞ and G(s). Conversely, when G(s) is free of poles on the finite jxaxis and M(s) is nr  nc , one gets for finite x that Tr ½GðjxÞ ¼ Tr ½MðjxÞM  ðjxÞ ¼

nc nr X X

jmlm ðjxÞj2

l¼1 m¼1

must be finite or, equivalently, every element mlm ðjxÞ must be finite. That is, M (s) must be free of poles on the finite jx-axis. Example 3.16 (a) For the 3DOF system treated in Sect. 3.8 establish that detðrb rb Þ 6¼ 0 on Cs when the parahermitian-positive matrix kQ ¼ Wq Wq is positive definite on Cs where Cs is the contour described in Example 3.15. (b) It is clear from (3.349) that detðrc rc Þ 6¼ 0 on Cs iff detðXt Xt Þ 6¼ 0 and detðXo Xo Þ 6 ¼ 0 on Cs . When ðXr Xr þ Xnr Xnr  Þ [ 0 on Cs and a1 and a2 are positive, establish that detðXo Xo Þ 6¼ 0 on Cs . (c) An often encountered model is one in which

3.11

Examples

215

Pd = P and d is a constant vector whose elements are zero-mean random variables. This corresponds to impulsive disturbances entering at the plant input and one can choose Ad ¼ I and Bd a diagonal matrix whose elements are the standard deviations of the respective elements of d. For this case with a1 and a2 positive establish that detðXt Xt Þ 6¼ 0 when Xd Xd [ 0, Gnm [ 0, and Gnl [ 0 on Cs . Comment: It is important to recognize that under the circumstances of this example confirmation of detðrb rb Þ 6¼ 0 and detðrc rc Þ 6¼ 0 on Cs does not require a determination of a coprime matrix fraction description for FP. Solution (a) Clearly, 1 rb rb ¼ A1 ðP P þ kQÞA1 ¼ A1 A1 p1 ðBp1 Bp1 þ kAp1 QAp1 ÞAp1 A1 :

Hence,  detðrb rb Þ ¼

det A1 det Ap1





 det A1 det ðBp1 Bp1 þ kAp1 QAp1 Þ : det Ap1 

Since the  3DOF  system is admissible, ðwFP =wF wP Þ 6¼ 0 on Cs or, equivalently, det A1 det Ap1 6¼ 0 on Cs because F is stable. Thus, detðrb rb Þ 6¼ 0 on Cs iff  detðBp1 Bp1 þ kAp1 QAp1 Þ ¼ det ½ Bp1



I Ap1  0

0 kQ



Bp1 Ap1



6¼ 0

 0 on Cs . Since Bp1 ; Ap1 is a right coprime polynomial matrix pair, B0p1 A0p1 has column rank throughout the complex plane and in particular on Cs . Hence, detðrb rb Þ 6¼ 0 on Cs if 

I 0

 0 [0 kQ

on Cs . This is the case when kQ > 0 on Cs . (b) From (3.347) and (3.351), Go ¼ Xo Xo ¼ Ar ðGr þ Gnr ÞAr ¼ a1 Br Br þ a2 Ar ðXr Xr þ Xnr Xnr  ÞAr    0 a1 I Br ¼ ½ Br Ar  : 0 a2 ðXr Xr þ Xnr Xnr  Þ Ar Since ½ Br Ar  has row rank throughout the complex plane and in particular on Cs , it readily follows when ðXr Xr þ Xnr Xnr  Þ [ 0 on Cs and a1 and a2 are positive that Go > 0 and detðXo Xo Þ 6¼ 0 on Cs . (c) From (3.350),  Gt ¼ Xt Xt ¼

A 0

0 An



FPd Gd L ðFPd Gd Pd F þ Gnm Þ LGd Pd F LGd L þ Gnl



A 0

 0 : An

3 H2 Design of Multivariable Control Systems

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It is easy to verify that  Gt ¼ Xt Xt ¼

 AFPd Gd ½ Pd F A An L

 L An  þ

AGnm A 0

0 An Gnl An



which is of the form  Gt ¼ MGtd M þ NGtn N ¼ ½ M

N

0 Gtn

Gtd 0



M N



where 

     AFPd 1 AFPd A1 AFPd A1 d d M¼ Ad ¼ ¼ ; An L An LA1 Bn d 

A N¼ 0

  G nm 0 ; Gtn ¼ 0 An

 0 ; Gnl

and Gtd ¼ a1 Bd Bd þ a2 Ad Xd Xd Ad : By similar reasoning as for Go it should be clear that Gtd > 0 on Cs since a1 and a2 are positive and Xd Xd [ 0 on Cs . Obviously, Gtn > 0 on Cs because Gnm [ 0 and Gnl [ 0 on Cs . Now Gt > 0 on Cs iff x = 0 is the only solution for x Gt x ¼ 0 on Cs . This is the case iff on Cs the matrix 

½M

AFPd A1 d N¼ Bn

A 0

0 An



has row rank or, equivalently, x = 0 is the only solution of 

x ½M

N¼x





AFPd A1 d Bn

A 0

0 An



¼ 00 :

1 When Ad = I and Pd = P one gets the following. First, since LA1 d ¼ L ¼ An Bn 0 0 0 and L is stable, det An 6¼ 0 on Cs . Hence, with x ¼ ½ x1 x2  it follows from the requirement that x2 An ¼ 0 that x2 = 0. Then, x1 must satisfy

 x1 AFPd A1 d

 A ¼ x1 ½ AFP

A  ¼ x1 ½ B

A  ¼ 00 :

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Examples

217

But A, B is a left coprime polynomial matrix pair and so x1 = 0 is the only solution. It immediately follows that Gt > 0 and detðXt Xt Þ 6¼ 0 on Cs . Example 3.17 (a) For the 1DOF system treated in Sect. 3.7 establish that detðrb rb Þ 6¼ 0 on Cs when the parahermitian-positive matrix kQ ¼ Wq Wq is positive definite on Cs where Cs is the contour described in Example 3.15. (b) Establish that for (3.281) detðrc rc Þ 6¼ 0 on Cs when Xd Xd [ 0, Xr Xr [ 0, and Xn Xn [ 0 on Cs ; a1 and a2 are positive; and detðAA1 r Þ 6¼ 0 on Cs . Comment: It is important to recognize that under the circumstances of this example confirmation of detðrb rb Þ 6¼ 0 and detðrc rc Þ 6¼ 0 on Cs does not require a determination of a coprime matrix fraction description for FP since 1 detðAA1 ¼ gwFP ðdet Ar Þ1 r Þ ¼ ðdet AÞðdet Ar Þ

where g is a constant and wFP can be determined directly from the minors of FP. Solution (a) The solution is the same as for Example 3.16(a). (b) For (3.281), rc rc ¼ AGt A ¼ AðPo Gd Po þ Gr þ Gn ÞA 2 32 2 3 Po 0 Gd 0 Gtd 6 76 6 7 ¼ A½ Po I I 4 0 Gr 0 54 I 5A ¼ M 4 0 0

0

Gn

I

0

0 Gtr 0

3 0 7 0 5M Gn

where   M ¼ APo A1 AA1 A ; d r Gtd ¼ a1 Bd Bd þ a2 Ad Xd Xd Ad ; Gtr ¼ a1 Br Br þ a2 Ar Xr Xr Ar ; and Gn ¼ a2 Xn Xn : It readily follows from the same reasoning as in the solution to Example 3.16(b) that Gtd > 0 and Gtr > 0 on Cs when Xd Xd [ 0 and Xr Xr [ 0 on Cs and a1 and a2 are positive. Also, Gn > 0 on Cs when Xn Xn [ 0 on Cs and a2 > 0. Hence, AGt A [ 0 on Cs iff x M ¼ 0 on Cs implies that x = 0. Clearly, 1 x M ¼ 0 requires x AA1 r ¼ 0 and, therefore, x = 0 when detðAAr Þ 6¼ 0. That is, AGt A [ 0 on Cs is indeed the case and it follows that detðrc rc Þ 6¼ 0 on Cs . Example* 3.18 (a) It is clear from (3.291) and (3.303) that for the H2 design of 1DOF systems the formula for K is of the form 1 1 K ¼ K1 ðZa þ fK1  A1  Ci X g1 ÞX 1 Y

3 H2 Design of Multivariable Control Systems

218

for the digital case and K ¼ K1 Za X1  A1 1 Y for the analog case where 1 1 Za ¼ Z þ fK1  Ci X g þ þ fKA1 YXg

Establish for the analog case that C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðI  A1 K1 Za X1 BÞ1 A1 K1 Za X1 A and show that one gets the same formula, but with Za replaced by 1 Zd ¼ Za þ fK1  Ci X g1 , for the digital case. (b) Establish that when PA1 is stable; the unstable poles of FP ¼ A1 B ¼ B1 A1 1 all lie on the boundary Cs described in Example 3.15; det½A1 ðP P þ kQÞA1  6¼ 0 and det AGt A 6¼ 0 on Cs ; Q and AGtA are e and X e are square Wiener–Hopf factors parahermitian-positive and good; and K e and Gt ¼ X eK e  X; e then the Wiener–Hopf factor K satsatisfying P P þ kQ ¼ K isfying (3.284) and (3.344) and the Wiener–Hopf factor X satisfying (3.284) can be e 1 and X ¼ A X. e (c) When the conditions of part (b) prevail, chosen as K ¼ KA establish that in the digital case the formula for C reduces to e 1 Zd X e 1 Zd X e 1 FPÞ1 K e 1 C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðI  K e i ¼ P ðTd Gr þ Pd Gd Po Þ one has where with C e 1 C e 1 C e 1 g þ f K e 1 g : eiX eiX Zd ¼ Z þ f K þ 1     Also, establish that the same formula holds for the analog case with Zd replaced by e 1 C e 1 g . Comment: It is important to note that these reduced eiX Za ¼ Z þ f K þ   formulas for C do not require a determination of a coprime matrix fraction description for FP and the conditions detðrb rb Þ 6¼ 0 and detðrc rc Þ 6¼ 0 on Cs are automatically satisfied when the assumptions in Example 3.17 prevail. Solution (a) For the analog case, 1 1 Y1 þ KA ¼ Y1 þ K1 Za X1 A  A1 1 YA ¼ K Za X A

and 1 1 1 X1  KB ¼ X1  K1 Za X1 B þ A1 1 YB ¼ X1  K Za X B þ A1 ðI  A1 X1 Þ 1 1 1 1 1 ¼ A1 1  K Za X B ¼ A1 ðI  A1 K Za X BÞ:

Hence, C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðI  A1 K1 Za X1 BÞ1 A1 K1 Za X1 A:

3.11

Examples

219

Obviously, for the digital case the only thing that changes in the above is that Za is replaced by Zd. (b) Since PA1 is stable and Q is good, e  KA e 1 A1 ðP P þ kQÞA1 ¼ A1 P PA1 þ kA1 QA1 ¼ A1 K e 1 is good or, equivalently, is analytic is good. Hence (see Example 3.15), K ¼ KA on Cs . Since det½A1 ðP P þ kQÞA1  6¼ 0 on Cs , it follows that detðP P þ kQÞ 6¼ 0 e such that on Cs . Thus, there does indeed exist a square Wiener–Hopf factor K e e e e e K  K ¼ P P þ kQ and from det (A1 K  KAÞ 6¼ 0 on Cs one has det ( KAÞ 6¼ 0 on e 1 are good. Moreover, because K e 1 and K1 ¼ A1 K e is Cs . It follows that K ¼ KA 1 e a Wiener–Hopf factor, K and its inverse are analytic in jkj\1 for the digital case and in Re s [ 0 for the analog case. The same is true for A1 1 because all the 1 1 e 1 e unstable poles of FP lie on Cs . Thus, K ¼ KA1 and K ¼ A1 K are analytic in e 1 is an jkj  1 for the digital case and in Re s 0 for the analog case: i.e., K ¼ KA acceptable Wiener–Hopf factor for A1 ðP P þ kQÞA1 ¼ K K. Similarly, from eX e  A it follows that A X e is good and detðA XÞ e 6¼ 0 on Cs . That is, AGt A ¼ A X 1 1 1 e e e is a Wiener–Hopf X ¼ A X and X ¼ X A are good. Moreover, because X e and its inverse are analytic in jkj\1 for the digital case and in Re s [ 0 factor, X for the analog case. The same is true for A1 because all the unstable poles of FP lie e and X1 ¼ X e 1 A1 are analytic in jkj  1 for the digital case on Cs . Thus, X ¼ A X e is an acceptable Wiener–Hopf and in Re s 0 for the analog case: i.e., X ¼ A X factor for AGt A ¼ XX . (c) It readily follows from the solution for part (a) that in the digital case C ¼ ðI  A1 K1 Zd X1 BÞ1 A1 K1 Zd X1 A: So when the conditions of part (b) prevail, e 1 Zd X e 1 Zd X e 1 Zd X e 1 Zd X e 1 A1 BÞ1 K e 1 ¼ ðI  K e 1 FPÞ1 K e 1 C ¼ ðI  K where 1 1 1 1 1 Zd ¼ Za þ fK1  Ci X g1 ¼ Z þ fK Ci X g þ þ fK Ci X g1 1 because fK A1 1 Y Xg ¼ 0 when all the unstable poles of A1 lie on Cs which is the case when all the unstable poles of FP lie on Cs . Moreover, from (3.289), Ci ¼ e i A and one gets A1 C 1 1 1 e e Zd ¼ Z þ fK1  A1 C i A X g þ þ fK A1 C i A X g1 1 1 1 1 e C e C e g þ fK e g : eiX eiX ¼ Z þ fK þ 1    

3 H2 Design of Multivariable Control Systems

220

Example* 3.19 When K is given by (3.36), establish that (3.11) yields C ¼ ðXs1  Kg Ag BÞ1 ðYs1 þ Kg Ag AÞ ¼ ðYs þ A1 Kg Ag ÞðXs  B1 Kg Ag Þ1 where Ks ¼ A1 1 Ao1 Co Yg1 ; Xs ¼ X  B1 Ks ; Xs1 ¼ X1  Ks B; and Ys ¼ Y þ A1 Ks ; Ys1 ¼ Y1 þ Ks A: Moreover, AXs þ BYs ¼ I; Xs1 A1 þ Ys1 B1 ¼ I: Solution From (3.11), C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1 : From (3.36), K ¼ A1 1 Ao1 Co Yg1 þ Kg Ag ¼ Ks þ Kg Ag : Hence, C ¼ ðX1  Ks B  Kg Ag BÞ1 ðY1 þ Ks A þ Kg Ag AÞ ¼ ðXs1  Kg Ag BÞ1 ðYs1 þ Kg Ag AÞ and also C ¼ ðY þ A1 Ks þ A1 Kg Ag ÞðX  B1 Ks  B1 Kg Ag Þ1 ¼ ðYs þ A1 Kg Ag ÞðXs  B1 Kg Ag Þ1 : Moreover, AXs þ BYs ¼ AðX  B1 Ks Þ þ BðY þ A1 Ks Þ ¼ AX þ BY þ ðBA1  AB1 ÞKs ¼ I

3.11

Examples

221

and Xs1 A1 þ Ys1 B1 ¼ ðX1  Ks BÞA1 þ ðY1 þ Ks AÞB1 ¼ X1 A1 þ Y1 B1  Ks ðBA1  AB1 Þ ¼ I: Example* 3.20 In Sects. 3.5 and 3.6, the first expression for K in (3.36) was used. The purpose of this example is to investigate for the analog case the use of the second expression for K instead. In particular, show that rb ¼ WBo1 A1 o1 A1 ¼ WG12 A1 is then the appropriate choice for ∇b instead of rbo ¼ WBo1 . Moreover, verify when A1 1 Ao1 is stable and G is compatible that on the finite jx-axis, rank rb ¼ rank rbo . In addition, establish that in place of (3.205) and (3.206) in Theorem 3.5 one gets K ¼ K1 Ca X1 Ag  A1 1 Y where 1 1 1 Ca ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKA1 ðY þ Ao1 Co Yg1 ÞAg Xg

with K now the Wiener–Hopf spectral factor satisfying 1 rb rb ¼ A1 A1 o1 Bo1 W WBo1 Ao1 A1 ¼ K K:

Moreover, K ¼ Ko A1 o1 A1 where Ko is the Wiener–Hopf spectral factor satisfying rbo rbo ¼ Bo1 W WBo1 ¼ Ko Ko : In addition, confirm that in place of (3.210) one gets C ¼ ðI  A1 K1 Ca X1 Ag BÞ1 A1 K1 Ca X1 Ag A: Solution Using A1 1 Ao1 Km ¼ Kg in (3.146) yields r ¼ WG11 Gw þ WG12 ðY þ Ao1 Co Yg1 ÞAG21 Gw þ WG12 A1 Kg Ag AG21 Gw 1 1 1 ¼ WG11 Gw  WBo1 A1 o1 ðY þ Ao1 Co Yg1 ÞAg Bg Bw  WBo1 Ao1 A1 Kg Ag Ag Bg Bw

e o Bg Bw  WBo1 A1 A1 Kg Bg Bw : ¼ WG11 Gw  WBo1 C o1 This expression is identical with (3.147) except that WBo1 Km Bg Bw ¼ rbo Km Bg Bw 1 is replaced with rb Kg Bg Bw where rb ¼ WG12 A1 ¼ WBo1 A1 o1 A1 ¼ rbo Ao1 A1 . 1 1 Since Ao1 A1 is stable when G is compatible and A1 Ao1 is stable by assumption, it follows that A1 o1 A1 is nonsingular on the finite jx-axis. Thus, rank rb ¼ rank rbo

3 H2 Design of Multivariable Control Systems

222

on the finite jx-axis. Moreover, in terms of the spectral factorization rbo rbo ¼ Bo1 W WBo1 ¼ Ko Ko one gets 1 1 1 rb rb ¼ A1 A1 o1 Bo1 W WBo1 Ao1 A1 ¼ A1 Ao1 Ko Ko Ao1 A1

and it follows that now rb rb ¼ K K where K ¼ Ko A1 o1 A1 is a spectral factor 1 because A1 A and and A A are stable. Since in Theorem 3.5, ∇b is actually o1 o1 1 1 1 1 rbo ¼ rb A1 Ao1 and K is actually Ko ¼ K A1 Ao1 , it follows from (3.205) and (3.206) that 1 1 1 1 1 K ¼ A1 1 ðAo1 Ko Ca X Ag  YÞ ¼ K Ca X Ag  A1 Y

where, with Ya ¼ Yo Cw Yg þ YXg , 1 1 Ca ¼ Z þ fK1 o rbo WG11 Gw rc X g þ þ fKo Ao1 Ya Xg 1 1 1 ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKA1 ðYo Cw Yg Ag þ YXg Ag ÞAg Xg 1 1 1 ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKA1 ðYo Cw Ag1 Yg1 þ YXg Ag ÞAg Xg 1 1 1 ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKA1 ðYo Cw Ag1 Yg1 þ Y  YBg1 Yg1 ÞAg Xg 1 1 1 ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKA1 ðY þ Ao1 Co Yg1 ÞAg Xg :

Finally, from (3.210) one gets 1 1 1 1 C ¼ ðI  Ao1 K1 o Ca X Ag BÞ Ao1 Ko Ca X Ag A 1 1 1 1 1 ¼ ðI  A1 K Ca X Ag BÞ A1 K Ca X Ag A:

Example 3.21 (a) When C in Fig. 3.2 is given by (3.11), establish that the transfer matrix from w to ym is given by Gm ¼ ðI  G22 CÞ1 G21 ¼ ðX  B1 KÞAG21 : Use this result together with (3.14) and (3.15) to prove when G is compatible and Aw ; Bw is a left coprime polynomial matrix pair that Gw ¼ A1 w Bw is strictly acceptable (see Definition 3.3) iff Cg ¼ AG21 A1 and Ch ¼ w 1 1 ðG11 þ G12 A1 Y1 G21 ÞAw are stable. (b) Establish when A1 Ao1 is stable that an 1 acceptable Gw is strictly acceptable iff Cg ¼ AG21 A1 w is stable. (c) When A1 Ao1 is 1 stable, show that Cg and Ch are stable iff Cw ¼ Ao G11 Aw and Cv ¼ ðG21 þ G22 Yo Ao G11 ÞA1 w are stable.

3.11

Examples

223

Solution (a) It readily follows from ym ¼ G21 w þ G22 v ¼ G21 w þ G22 Cym that ym ¼ ðI  G22 CÞ1 G21 w ¼ ½I þ A1 BðY þ A1 KÞðX  B1 KÞ1 1 G21 w ¼ ðX  B1 KÞ½AðX  B1 KÞ þ BðY þ A1 KÞ1 AG21 w ¼ ðX  B1 KÞAG21 w ¼ Gm w:

Since Gd , Gv , and Gm are stable, Gw ¼ A1 w Bw is strictly acceptable iff 2

3 2 3 2 3 ðG11 þ G12 A1 Y1 G21 Þ G12 A1 Gd A1 w 5¼4 5A1 þ 4 A1 5KAG21 A1 Tw ¼ 4 Gv A1 YAG21 w w w 1 XAG21 B1 Gm Aw is stable. Hence, it is necessary that 

½0 B

A Tw ¼ ½ B ¼ Cg

Gv A1 w A Gm A1 w



¼ ½ðAX þ BYÞ þ ðBA1  AB1 ÞKAG21 A1 w

be stable. It is also necessary that 1 1 Gd A1 w ¼ ðG11 þ G12 A1 Y1 G21 ÞAw þ G12 A1 KAG21 Aw ¼ Ch þ G12 A1 K Cg

be stable. Since G12 A1 and K are stable, it follows that Ch must be stable. Obviously, Cg and Ch stable is sufficient for stable Tw . That is, Gw is strictly acceptable iff Cg and Ch are stable. (b) From Theorem 3.1, Gw is acceptable iff 1 Cw ¼ Ao G11 A1 w and Co ¼ Ao1 ðYo Cw Ag1  YBg1 Þ are stable. It is established in Example 3.3 when Cw is stable that Co is stable iff Cz ¼ Ch Ag1 is stable. So if 1 1 Cg ¼ A1 g Bg ¼ Bg1 Ag1 is stable, Ch ¼ Cz Ag1 is stable and Gw is strictly acceptable. That Cg stable is necessary was established in part (a) of this example. (c) Using G11 ¼ Ch Aw  G12 A1 Y1 G21 ¼ Ch Aw  G12 YAG21 yields 1 1 Cv ¼ ðG21 þ G22 Yo Ao G11 ÞA1 w ¼ G21 Aw þ G22 Yo Ao ðCh Aw  G12 YAG21 ÞAw 1 1 ¼ A1 ðCg  BYo Ao ðCh þ A1 o Bo Y Cg ÞÞ ¼ A ðI  BYo Bo YÞCg  A BYo Ao Ch

¼ A1 ðI  BðI  Ao1 Xo1 ÞYÞ Cg  B1 A1 1 Ao1 Yo1 Ch ¼ A1 ðI  BY þ BAo1 Xo1 YÞ Cg  B1 A1 1 Ao1 Yo1 Ch 1 ¼ ðX þ B1 A1 1 Ao1 Xo1 YÞ Cg  B1 A1 Ao1 Yo1 Ch

3 H2 Design of Multivariable Control Systems

224

which is stable when Cg , Ch , and A1 1 Ao1 are stable. Also, from 1 Ao Ch ¼ Ao G11 A1 w þ Ao G12 YAG21 Aw ¼ Cw  Bo Y Cg

one gets that Cw is stable when Cg and Ch are stable. That is, Cw and Cv are stable when Cg and Ch are stable. Conversely, when Cw and Cv are stable it follows from 1 ACv ¼ AG21 A1 w þ AG22 Yo Ao G11 Aw ¼ Cg  BYo Cw

that Cg is stable. Moreover, using G21 ¼ Cv Aw  G22 Yo Ao G11 yields 1 1 Ch ¼ ðG11 þ G12 YAG21 ÞA1 w ¼ G11 Aw þ G12 YAðCv  G22 Yo Ao G11 Aw Þ 1 ¼ ðI  G12 YAG22 Yo Ao ÞG11 A1 w þ G12 A1 Y1 Cv ¼ Ao ðI  Bo YBYo ÞCw þ G12 A1 Y1 Cv

¼ A1 o ½I  Bo ðI  A1 X1 ÞYo Cw þ G12 A1 Y1 Cv ¼ ðXo  G12 A1 X1 Yo ÞCw þ G12 A1 Y1 Cv

which is stable when Cw and Cv are stable because G12 A1 is stable when G is compatible. So Cg and Ch are stable when Cw and Cv are stable. That is, Cg and Ch are stable iff Cw and Cv are stable. Example 3.22 The case in which the generalized persistent input standard configuration of Fig. 3.2 represents an analog system for which Gw is strictly acceptable is considered in this example. Prove the following parallel theorem to Theorem 3.5. Theorem 3.5′ When G is compatible, Gw is strictly acceptable for G, and W is stable; when on the finite jx-axis rb ¼ WG12 A1 ¼ WBo1 A1 o1 A1 has full column rank and rc ¼ AG21 Gw ¼ A1 g Bg Bw has full row rank; and when the order relationships WG11 Gw  0ðs1 Þ; ðG12 W WG12 Þ1  0ðs2k1 Þ; ðG21 Gw Gw G21 Þ1  0ðs2k2 Þ; G22  0ðsk3 Þ

are satisfied with k1 þ k2 þ k3  0, then all controller transfer matrices C given by (3.11) for which Gv Gw , Gd Gw , and Gm Gw are stable and the performance functional 1 J¼ 2pj

Zj1 Tr ½ðWGd Gw ÞðWGd Gw Þ  ds j1

3.11

Examples

225

is finite are the ones for which K ¼ K1 Ca X1  A1 1 Y is chosen where 1 1 Ca ¼ Z þ fK1  rb WG11 Gw rc X g þ þ fKA1 Y Xg

with Z a strictly proper stable matrix. In the formulas for K and Ca , the matrices K and X are the square stable Wiener–Hopf spectral factors with stable inverses satisfying rb rb ¼ K K; rc rc ¼ XX : In this case, the controller transfer matrix is given by C ¼ ðI  A1 K1 Ca X1 BÞ1 A1 K1 Ca X1 A and satisfies C  0ðsk1 þ k2 1 Þ: In addition, 1 J¼ 2pj

Zj1

e f Cg e fTr ½ZZ þ f Cg    þ qo gds

j1

where eC e Þ qo ¼ Tr ðra ra Þ  Tr ð C and the minimum value for J is obtained when Z  0 is chosen. In the formulas for J and qo , 1 1 ra ¼ WG11 Gw  rb K1 ðfKA1 1 YXg þ þ fKA1 YXg ÞX rc

and e ¼ K1 rb WG11 Gw rc X1  fKA1 YXg  fKA1 YXg : C þ   1 1 

3 H2 Design of Multivariable Control Systems

226

Solution

From (3.15) and Example 3.21,

r ¼ WGd Gw ¼ WðG11 þ G12 A1 Y1 G21 ÞGw þ WG12 A1 KAG21 Gw ¼ WCh Bw þ WG12 A1 K Cg Bw ¼ WG11 Gw  rb ðK þ Y1 A1 Þrc where G12 A1 is stable because G is compatible and, since Gw is strictly acceptable, Cg and Ch are stable. Hence, ∇ is stable for all stable K because W stable is assumed. In addition, rb and rc are stable. Moreover, on account of the rank assumptions on rb and rc , one can construct square Wiener–Hopf spectral factors K and X which together with their inverses are analytic in Re s 0 and satisfy rb rb ¼ K K and rc rc ¼ X X . (Clearly, A1 g and its inverse are stable. When it is also assumed that A1 Ao1 is stable, then A1 o1 A1 and its inverse are stable. Hence, 1 A1 and A A are nonsingular on the finite jx-axis and the rank assumptions on g o1 1 rb and rc are satisfied iff WBo1 has full column rank and Bg Bw has full row rank on the finite jx-axis.) Since A1 Y1 ¼ YA, 1 r ¼ WG11 Gw  rb K1 ðKK X þ KA1 1 Y X ÞX rc

and any stable K can be expressed as 1 e  K1 fKA1 K¼K 1 Y Xg1 X

e rc where e is a stable matrix. Hence, r ¼ ra  rb K where K 1 1 ra ¼ WG11 Gw  rb K1 ðfKA1 1 YXg þ þ fKA1 YXg ÞX rc :

Since rb K1 and X1 rc are proper because they are paraunitary matrices and since WG11 Gw  0ðs1 Þ is assumed, it immediately follows that ra is strictly proper. Moreover, since ∇ is stable for any stable K or, equivalently for any stable e , one gets from r ¼ ra when K e ¼ 0 that ra is stable. In addition, on account of K the rank assumptions on rb and rc , it is true that det (rb rb Þ 6¼ 0 and det (rc rc Þ 6¼ 0 on the finite jx-axis. Moreover, rb ¼ WG12 A1 and rc ¼ e e A1 g Bg Bw ¼ Cg Bw are stable. Thus, Theorem 3.4 applies with K in place of K m and it follows that J is finite iff

1 1 1 e e  K1 fKA1 K¼K ¼ K1 ðZ þ f Cg þ  fKA1 Y Xg1 ÞX 1 Y Xg1 X

where Z is a strictly-proper stable matrix and e ¼ K1 rb ra rc X1 C   1 1 ¼ K1 r WG G rc X1 b 11 w   fKA1 Y Xg þ  fKA1 Y Xg 

3.11

Examples

227

is strictly proper and good because ra is and because rb K1 and X1 rc are proper and stable. Clearly, 1 1 1 e f Cg þ ¼ fK rb WG11 Gw rc X g þ  fKA1 Y Xg þ

and it readily follows that 1 1 1 1 K ¼ K1 ðZ þ fK1  rb WG11 Gw rc X g þ  fKA1 Y Xg þ  fKA1 Y Xg1 ÞX 1 1 1 1 ¼ K1 ðZ þ fK1  rb WG11 Gw rc X g þ þ fKA1 Y Xg  KA1 YXÞX

¼ K1 Ca X1  A1 1 Y Also from Theorem 3.4, 1 J¼ 2pj

Zj1

e f Cg e fTr ½ZZ þ f Cg    þ qo g ds

j1

where eC e Þ qo ¼ Tr ðra ra Þ  Tr ð C and the minimum value for J is obtained when Z  0 is chosen. It remains to show that all controller transfer matrices C given by (3.11) exist when K ¼ K1 Ca X1  A1 1 Y. From (3.11), C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ½A1 ðX1  KBÞ1 A1 ðY1 þ KAÞ ¼ ðA1 X1  A1 K1 Ca X1 B  YBÞ1 ðA1 Y1 þ A1 K1 Ca X1 A  YAÞ ¼ ðI  A1 K1 Ca X1 BÞ1 A1 K1 Ca X1 A: Hence, the controllers exist provided detðI  A1 K1 Ca X1 BÞ 6 0. From the order assumptions, 1 1 A1 K1 K1  0ðs2k1 Þ  A1 ¼ A1 ðrb rb Þ A1 ¼ ðG12 W WG12 Þ

and 1 1 1 A X1  0ðs2k2 Þ:  X A ¼ A ðrc rc Þ A ¼ ðG21 Gw Gw G21 Þ

3 H2 Design of Multivariable Control Systems

228

Invoking Lemma 3.2 yields A1 K1  0ðsk1 Þ, X1 A  0ðsk2 Þ, and X1 B ¼ X1 AG22  0ðsk2 þ k3 Þ: Clearly, Ca is strictly proper. It then follows that A1 K1 Ca X1 B  0ðsk1 þ k2 þ k3 1 Þ is strictly proper when k1 þ k2 þ k3  0. Thus, detðI  A1 K1 Ca X1 BÞ 6 0 is true and A1 K1 Ca X1 A  0ðk1 þ k2  1Þ ) C  0ðsk1 þ k2 1 Þ: Example* 3.23 (An example of the optimal 1DOF H2 design for a strictly acceptable input) Consider the 1DOF system in Fig. 3.3 in Sect. 3.7. The linearized plant model for submarine motion at 30 knots speed is represented in the form x_ ¼ Rx þ Nv; v ¼ ½ v1 v2 0 ; 2 0:19003 4:4802 6 0:0085526 0:45988 R¼6 4 1 0 0 1 2

0:1855 6 0:043308 N¼6 4 0 0 

0 H¼ 0

y2 0 ¼ Hx; 3 0:0014673 0:0056095 7 7; 15:433 5 0

y ¼ ½ y1 0 0 0 0

3 0:57149 0:055543 7 7; 5 0 0

0 1 0 0

 0 ; 1

where the outputs y1 ¼ dD and y2 ¼ dh indicate the changes of depth and pitch from the nominal values, respectively, and the control variables v1 ¼ db and v2 ¼ ds indicate the changes in bow and stern plane angles, respectively (Grimble 2006, Sect. 10.4). That is, the transfer functions of the plant PðsÞ in Fig. 3.3 are given by dD 0:1855ðs þ 2:81906Þðs þ 0:19793Þ ¼ ; db sðs þ 0:55476Þðs þ 0:066663Þðs þ 0:028485Þ

3.11

Examples

229

dD 0:57149ðs  1:0113Þðs þ 0:40668Þ ¼ ; ds sðs þ 0:55476Þðs þ 0:066663Þðs þ 0:028485Þ dh 0:043308ðs þ 0:153397Þ ¼ ; db ðs þ 0:55476Þðs þ 0:066663Þðs þ 0:028485Þ dh 0:055543ðs þ 0:27803Þ ¼ : ds ðs þ 0:55476Þðs þ 0:066663Þðs þ 0:028485Þ The disturbance dðsÞ, reference input rðsÞ and measurement noise nðsÞ in Fig. 3.3 are modeled as the following ones based on Sect. 10.4 of Grimble (2006); dðsÞ ¼ Xd ðsÞld ; r ¼ A1 r Br rr þ Xr lr ; n ¼ Xn ðsÞln Here l ¼ ½ l0d l0r l0n 0 is a column whose elements are independent zero-mean unit variance white-noise and rr is a vector whose elements are real random variables with zero mean and identity covariance matrix. The stable matrices Xd , Xn and Xr are given by 2 3 0:4s 0 6 ðs þ 0:1Þ 7 7; Xd ðsÞ ¼ dw1 ðsÞ6 2 4 5 0:4s 0 ðs þ 0:1Þðs þ 0:04Þ   0:02 0 4s2 =dw 0 Xn ¼ ; 0 0:02 0 0 dw ¼ s4 þ 8:555s3 þ 2:44s2 þ 1:884s þ 0:15;   0:02 1 0 : Xr ¼ s þ 0:0001 0 1 The shape deterministic part of rðsÞ is described by  Ar ¼

s 0

  0 1 ; Br ¼ s þ 0:02 0

 0 ; 1

where the −1 in Br is chosen so that with a positive step increase command in depth the pitch command is negative. The effects of ocean currents, second-order wave and suction in Grimble (2006) are not considered here; Design the optimal H2 controller of the 1DOF configuration in Fig. 3.3 in Sect. 3.7 with appropriate choices of the weighting matrices WðsÞ and Gw ðsÞ (Assume that L ¼ 0 and F ¼ Pd ¼ Td ¼ I in the figure). Solution The 1DOF configuration in Fig. 3.3 in Sect. 3.7 is considered with L ¼ 0 and F ¼ I. The matrices Td and Pd in the figure are set as the identity ones; hence, Po ¼ I. The 1DOF system is admissible since F ¼ I and it is also compatible by

3 H2 Design of Multivariable Control Systems

230

Lemma 3.4 since Td ¼ I. A coprime polynomial matrix fraction description of the plant PðsÞ ¼ HðsI  RÞ1 N ¼ A1 ðsÞBðsÞ is given by 

 sðs þ 0:19003Þ 10:95278s  2:93127 AðsÞ ¼ ; 0:008553s s2  0:45988s þ 0:12638 

0:1855 BðsÞ ¼ 0:04331

 0:57149 : 0:055543

This fractional description is obtained by using the command ss2lmf of the MATLAB Polynomial Toolbox. Invoking the standard model in Fig. 3.2 in Sect. 3.3 with the setup w ¼ ½ d0

r0

n0 0 ; yd ¼ ½ e0

v0 0 ; ym ¼ u; v ¼ v;

(the left-hand side variables are the ones in Fig. 3.2 and the right-hand side variables are the ones in Fig. 3.3.), one obtains the generalized matrices Gij ðsÞ in Fig. 3.2 as  G11 ¼

I2 0

I2 0

   PðsÞ 0 ; G12 ¼ ; G21 ¼ ½ I2 I2 0

I2

I2 ; G22 ¼ PðsÞ:

The weighting constants a1 and a2 are chosen as a1 ¼ a2 ¼ 1 and it follows that 2

0 Gw ¼ 4 0 0

0

0 0 0

A1 r Br 0

Xd 0 0

0 Xr 0

3 0 0 5; Xn

with the setup q ¼ ½r0d r0r r0n l0d l0r l0n 0 . The reason of taking the element at (2,2) position of Ar as s þ 0:02, instead of s, is to meet the assumption that the matrix rc ¼ AG21 Gw has full row rank on the finite jx-axis. It can be shown that a pair of left coprime polynomial matrices for Gw ¼ A1 w Bw is given by 2

0 Aw ¼ diagfDd ; dr Ar ; Dn g; Bw ¼ 4 0 0

0 dr Br 0

0 Nd 0 0 0 0

0 0:02Ar 0

3 0 0 5 Nn

where Dd ¼ diagfðs þ 0:1Þdw ; ðs þ 0:1Þðs þ 0:04Þdw g; Dn ¼ diagfdw ; 1g; dr ¼ s þ 0:0001;

3.11

Examples

231



0:02dw Nd ¼ diagf0:4s; 0:4s g; Nn ¼ 0 2

0 4s2 0:02 0

 0 : 0

  1 Since AG21 A1 is stable, the input weighting dr1 AA1 AD1 w ¼ ADd r n matrix Gw is strictly acceptable from the results in (b) of Example 3.21. In this case, it is convenient to exploit the optimal H2 controller formula in Theorem 3.5′ in Example 3.22 if the assumptions for the theorem are met. Performance of the optimal H2 design is greatly affected by the choice of the weighting matrix WðsÞ. The matrix WðsÞ ¼ diagfWe ; Wv g in this example is chosen as 2 3 2 3 300ðs þ 1Þ 30ðs þ 1Þ   0 0 0 We1 6 7 6 s þ 30 7 We ¼ ;W ¼ : ¼ 4 s þ 30 ðs þ 1Þ 5 v 4 30ðs þ 1Þ 5 0 We2 0 0 s þ 0:01 s þ 30 The weighting functions We1 is chosen as a high-pass filter in order to suppress the overshoot of the output y1 ¼ dD stimulated by the actuator step input of db. An efficient way to remove the steady-state errors for step inputs is to insert the integrators as a part of the controller CðsÞ. Such approach is not taken here. Instead, the weighting function We2 is chosen as a low-pass filter in order for the output y2 ¼ dh to track the step reference input ds with a small steady-state error. The output y1 ¼ dD is to track the step reference input of db or ds with zero steady-state error since the transfer functions dD=db and dD=ds have integrator terms. The weighting function Wv is chosen as a high-pass filter to suppress the measurement noise. The filter gains are determined by observing the simulation results. It is easy to verify that all assumptions in Theorem 3.5′ are satisfied. In fact, it can be shown that Ao1 ¼ A1 and the matrix rb ¼ WG12 A1 has full column rank on the finite jx-axis. The matrix rc ¼ AG21 Gw has full row rank on the finite jxaxis. Specifically, since 2 3 0 0 0 Nd 0 0   6 7 rc ¼ AD1 dr1 AA1 AD1 4 0 dr Br 0 0 0:02Ar 0 5 d r n 0 0 0 0 0 Nn   ¼ 0 AA1 0 AXd 0:02dr1 A AXn ; r Br it has full row rank on the finite jx-axis when AA1 r Br does. It is not difficult to show 1 that detðAAr Br Þ has no zeros on the finite jx-axis, which implies that AA1 r Br has full row rank on the finite jx-axis. The order relationships in Theorem 3.5′ are satisfied with k1 ¼ k2 ¼ 0 and k3 ¼ 1. The optimal H2 controller is obtained by the formula C ¼ ðI  A1 K1 Ca X1 BÞ1 A1 K1 Ca X1 A;

232

3 H2 Design of Multivariable Control Systems

where 1 1 Ca ¼ fK1  rb WG11 Gw rc X g þ þ fKA1 Y Xg ;

and the matrices K and X are the square stable Wiener–Hopf spectral factors with stable inverses satisfying rb rb ¼ K K; rc rc ¼ XX : 1 1 has It is noted here that the term fKA1 1 Y Xg ¼ fKY1 A Xg is zero since A no poles in Re s\0 and thus the matrices X, Y, X1 , and Y1 in (3.10) are not needed to obtain the optimal solution. For a strictly acceptable Gw , the optimal solution can be more conveniently obtained from the state-space formulas in Chap. 5. In fact, state-space formulas of the rational matrices A1 K1 , X1 A, Ca , and

Fig. 3.4 Unit step responses of the outputs to reference changes in depth and pitch ðy1 ¼ dD; y2 ¼ dhÞ

Fig. 3.5 Control activities due to reference changes in depth and pitch ðv1 ¼ db; v2 ¼ dsÞ

3.11

Examples

233

C ¼ ðI  A1 K1 Ca X1 BÞ1 A1 K1 Ca X1 A can be found in (5.66), (5.80), (5.83), and (5.89), respectively. The optimal H2 controller is successfully obtained by using the formula in (5.89). The number of the states of the optimal controller is 26 and is not presented here. The unit step responses of y1 and y2 are shown in Fig. 3.4. The step reference inputs for y1 and y2 are exerted at t ¼ 0 and t ¼ 200, respectively. The actuating controls for these reference inputs are shown in Fig. 3.5. The actuating signals seem to change rather rapidly at t ¼ 200 in Fig. 3.5. However, it is unlikely that a step input is a reference to the system in real operations; hence, the rapid actuating would not arise. The steady-state error of y2 with respect to its step reference input is not exactly zero due to lack of an integrator in the corresponding loop and if one tries to make the error small, the overshoot of y2 tends to increase. Example* 3.24 (An example of the optimal 2DOF H2 design for an acceptable input) The Rosenbrock’s process, whose transfer matrix is given below, is a typical multivariable system which shows a severe coupling phenomenon (Åström et al. 2002); 

y1 y2



2 1  v1 6sþ1 ¼ P0 ðsÞ ; P0 ðsÞ ¼ 4 1 v2 sþ1 

3 2 sþ37 1 5: sþ1

The transfer matrix has a non-minimum phase zero at s ¼ 1 and this yields fundamental limitations on control system performance. Design the optimal 2DOF H2 controller for the configuration in Fig. 1.1 in Sect. 1.2 with appropriate choices of the weighting matrices WðsÞ and Gw ðsÞ (Remove the feedforward part and assume F ¼ I in the figure). Solution This example concerns the design of an analog 2DOF control system which is the special case of the 3DOF system shown in Fig. 1.1 with L ¼ 0; nl ¼ 0; and Xnl ¼ 0. Hence, the design formulas for the 2DOF system can be obtained from those for the 3DOF system by removing from all matrices any block rows and block columns that contain the variables L; nl ; and Xnl . In particular, for the 2DOF system one gets in place of (3.348), (3.350), (3.407), (3.408), (3.412), and (3.413) the reductions

3 H2 Design of Multivariable Control Systems

234

Gs ¼ diagfGd ; Gr ; Gnr ; Gnm g; Gt ¼ Xt Xt ¼ AðFPd Gd Pd F þ Gnm ÞA ; e w ¼ K1 A1 P Pd Gd Pd F A X1 ; C 

t

e a ¼ KY1 A1 Xt ; C e wg þ fC e a g ÞX1 A; Rw ¼ A1 K1 ðZm þ f C t þ  and C ¼ ½ Cw

Cu  ¼ ðX1  K1 BÞ1 ½ ðY1 þ K1 AÞ

H1 :

It is assumed that Pd ðsÞ ¼ FðsÞ ¼ Td ðsÞ ¼ I for the 2DOF system; thus, it is admissible since F ¼ I and it is compatible since Td ¼ I. In order to eliminate the steady state errors for step reference inputs, the integrators s1 I2 are inserted in front of the plant and the augmented plant 2

1 6 sðs þ 1Þ Paug ðsÞ ¼ PðsÞ ¼ 6 4 1 sðs þ 1Þ

3 2 sðs þ 3Þ 7 7 1 5 sðs þ 1Þ

is considered as the plant PðsÞ in Fig. 1.1. The integrators are implemented as a part of the controller after the design of the optimal controller is carried out. The 2DOF system with the augmented plant is also admissible and compatible since F ¼ I and Td ¼ I. Coprime polynomial matrix fraction descriptions for the augmented plant are given by  AðsÞ ¼

   s þ 3 2ðs þ 1Þ ; BðsÞ ¼ ; sðs þ 1Þ 1 1    0 1 2ðs þ 1Þ ; B1 ðsÞ ¼ : sðs þ 1Þðs þ 3Þ 1 sþ3

sðs þ 1Þðs þ 3Þ

0 sðs þ 1Þ A1 ðsÞ ¼ 0 

0

With the choice of a1 ¼ a2 ¼ 1, the input weighting matrix Gw becomes 2

A1 d Bd 6 0 Gw ¼ 6 4 0 0

0

A1 r Br 0 0

Xd 0 0 0

0 Xr 0 0

0 0 Xnr 0

3 0 0 7 7; 0 5 Xnm

3.11

Examples

235

and its element values are selected as Ar ¼ Ad ¼ sI2 ; Br ¼ Bd ¼ I2 ; Xr ¼ Xd ¼ 0; Xnr ¼ Xnm ¼ 0:1 I2 : The weighting matrices to the control variable are simply chosen as kQ ¼ I2 . Then it is not difficult to verify that Assumptions 3.6–3.10 in Sect. 3.8 are satisfied. Thus, Gw is acceptable (note that Ap ¼ A since F ¼ I). As for the optimal solution formulas, since the shape deterministic inputs in Gw only have a pole at s ¼ 0, the more convenient formulas in (3.432) and (3.435) can be used instead of the ones in e ag ¼ fC e b g ¼ 0 since A1 has no poles in (3.404) and (3.411). Furthermore, f C   1 e t for the 2DOF case, that Re s [ 0. It then follows, since Xt ¼ X 1 e e 1 e 1 e rg X Ru ¼ A1 K1 f C þ o ; Rw ¼ A1 K f C w g þ X t A; e 1 ; e 1 ; e r ¼ K1 A1 P A1 Br Br A1 X e w ¼ K1 A1 PA1 Bd Bd A1 A X C C o t   r r d d

e t are Wiener–Hopf spectral factors of the equations e o and X where K, X 1 e e A1 ðP P þ kQÞA1 ¼ K K; A1 r Br Br Ar þ Xnr Xnr  ¼ X o X o ; etX e t : AðA1 Bd Bd A1 þ Gn ÞA ¼ X d

d

m

The computational results are given by  K¼

0:5s2 þ 1:0576s þ 1:036

0:40925s2 þ 2:0396s þ 2:1088

0:40925s þ 0:96262 0:5s3 þ 2:6104s2 þ 4:4816s þ 2:9246  0 1 e o ¼ 1 þ 0:1s ; X 0 1 þ 0:1s s   3 2 0 e t ¼ 0:1s þ 1:4s þ 4:3s þ 3 X : 0 0:1s2 þ 1:1s þ 1

 ;



;

e 1 A, and it therefore follows that e 1 ¼ X This example is a special case in that X o t e r g ¼ fC e w g , Ru ¼ Rw , and Cu ¼ Cw . The computational results are given by fC þ þ  1 0:99933 e e fCr g þ ¼ fCwg þ ¼ s 0:036708   r11 ðsÞ r12 ðsÞ 1 Rw ¼ Ru ¼ ; dr ðsÞ r21 ðsÞ r22 ðsÞ

0:036708 0:99933

 ;

3 H2 Design of Multivariable Control Systems

236

r11 ¼ 19:987sðs þ 2:6979Þðs2 þ 2:5529s þ 2:2254Þ; r12 ¼ 0:73416sðs  21:941Þðs þ 3:4322Þðs þ 1:447Þ; r21 ¼ 0:73416sðs þ 21:911Þðs þ 3Þðs þ 2:4866Þ; r22 ¼ 19:987sðs þ 3Þðs2 þ 2:0852s þ 2:0013Þ; dr ¼ ðs þ 10Þðs þ 2:6853Þðs þ 0:36635Þðs2 þ 3:2843s þ 4:066Þ:

1.6

y1 y2

1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0

20

40

sec

60

80

100

Fig. 3.6 Unit step responses of the outputs to reference changes

5 v1 v2

4 3 2 1 0 -1 -2 -3 -4 -5

0

20

40

sec

Fig. 3.7 Control activities due to reference changes

60

80

100

3.11

Examples

237

The controller matrices Cw and Cu are obtained by the formulas Cw ¼ Cu ¼ ðI  Rw PÞ1 Rw and the computational results are given by Cw ðsÞ ¼ Cu ðsÞ ¼

 1 c11 ðsÞ dc ðsÞ c21 ðsÞ

 c12 ðsÞ ; c22 ðsÞ

c11 ¼ 19:987ðs þ 9:7345Þðs þ 1Þðs2 þ 5:5164s þ 7:9224Þ; c12 ¼ 0:73416ðs  20:476Þðs + 13:266Þðs þ 1Þðs þ 0:14786Þ; c21 ¼ 0:73416ðs þ 3Þðs þ 1Þðs2 þ 34:398s þ 298:46Þ; c22 ¼ 19:987ðs þ 9:7385Þðs þ 3Þðs + 2:3467Þðs þ 1Þ; dc ¼ ðs þ 10:135Þðs þ 9:1359Þðs þ 4:5579Þðs2 þ 3:5067s þ 4:132Þ: The McMillan degree of the controller CðsÞ ¼ ½ Cw Cu  is 5 and hence the final controller CðsÞ1s can be implemented as a state-space model with 7 states. The above optimal H2 controller can also be obtained by using the state-space formula in Park et al. (2002). It is confirmed that the formula in Park et al. (2002) gives the same results as the above ones. The unit step responses of y1 and y2 are shown in Fig. 3.6. The step reference inputs for y1 and y2 are exerted at t ¼ 0 and t ¼ 40, respectively. The actuating controls for this reference inputs are shown in Fig. 3.7. If the augmented plant Paug ðsÞ is regarded as the plant PðsÞ in Fig. 1.1 and the control variable vðsÞ is regarded as the input to the inserted integrators, the cost J ¼ kWGd Gw k22 for the H2 design in this example is 21.869. Example* 3.25 Establish that the expressions for the functional J given by (3.129) and (3.130) in the digital case are equivalent. Also, show the same is true for (3.175) and (3.176) in the analog case. Solution From Theorem 3.2 for the digital case, the expression for J in (3.129) is given by J¼

1 ð dk eC e  Þ þ Tr ½f Cg e f Cg e H  fTr ½ZZ  þ Trðra ra Þ  Tr ð C   g 2pj k jkj¼1

Thus, it only remains to show that 1 ð dk eC e  Þ  Tr ½f Cg e f Cg e e J¼ H  fTr ð C   g 2pj k jkj¼1

¼

1 ð dk e e e e H  Tr ½ðf Cg þ þ f Cg1 Þðf Cg þ þ f Cg1 Þ g 2pj k jkj¼1

3 H2 Design of Multivariable Control Systems

238

e is good, f Cg e e e e e Since C þ is stable. Moreover, C ¼ f Cg þ þ f Cg þ f Cg1 . Hence, ð

2pje J¼

e e e e e e e e fTr ½ðf Cg þ þ f Cg þ f Cg1 Þðf Cg þ þ f Cg þ f Cg1 Þ   Tr ½f Cg f Cg g

H 

dk k

jkj¼1

ð

¼

e f Cg e e e e e fTr ½f Cg þ þ  þ f Cg þ f Cg þ f Cg þ f Cg1 g

H 

dk k

jkj¼1

ð

þ

H 

dk e f Cg e e e fTr ½f Cg  þ  þ f Cg f Cg1 g k

jkj¼1

ð

þ

H 

dk e f Cg e e e e e : fTr ½ðf Cg 1 þ  þ f Cg1 f Cg þ f Cg1 f Cg1 g k

jkj¼1

¼

ð

e f Cg e e e e e fTr ½f Cg þ þ  þ f Cg þ f Cg þ f Cg f Cg þ  g

H 

jkj¼1

ð

þ

dk e e e fTr ½ðf Cg þ þ f Cg Þf Cg1 g k

H 

jkj¼1

ð

þ

e ðf Cg e e fTr ½f Cg 1 þ  þ f Cg Þg

H 

dk k

ð dk dk e f Cg e þ H fTr ½f Cg : 1 1 g k k

jkj¼1

jkj¼1

e Now it follows from (3.121) with Z ¼ f Cg þ that ð H 

e f Cg e e e fTr ½f Cg þ  þ f Cg f Cg þ  g

dk ¼ 0: k

jkj¼1

Moreover, one gets from (3.119) and (3.120) that ð H 

jkj¼1

dk e Þf Cg e ¼ fTr ½f Cg  1 g k

ð H 

dk e f Cg e ¼0 fTr ½f Cg 1  g k

jkj¼1

1 e e since f Cg 1 and k f Cg are analytic in jkj  1. Thus,

2pje J¼

ð

dk e f Cg e e e e e e e fTr ½f Cg þ þ  þ f Cg þ f Cg1 þ f Cg1 f Cg þ  þ f Cg1 f Cg1 g k

H 

jkj¼1

¼

ð H 

jkj¼1

dk e e e e : fTr ½ðf Cg þ þ f Cg1 Þðf Cg þ þ f Cg1 Þ g k

3.11

Examples

239

This completes the proof of Theorem 3.2. Similarly, it follows from (3.175) and (3.176) for the analog case that 1 J¼ 2pj

Zj1

eC e  Þ þ Tr ½f Cg e f Cg e fTr ½ ZZ  þ Tr ðra ra Þ  Tr ð C   gds:

j1

Thus, it only remains to show that 1 e J¼ 2pj

Zj1 j1

1 eC e  Þ  Tr ½f Cg e f Cg e fTr ð C    gds ¼ 2pj

Zj1

e f Cg e Tr ½f Cg þ þ   gds:

j1

e is good and strictly proper, C e ¼ f Cg e e e Since C þ þ f Cg . Moreover, f Cg þ is stable. Hence, 1 e J¼ 2pj ¼

1 2pj

Zj1

e e e e e e Tr ½ðf Cg þ þ f Cg Þðf Cg þ  þ f Cg Þ  Tr ½f Cg f Cg  gds

j1

Zj1

e f Cg e e e e e Tr ½f Cg þ þ  þ f Cg þ f Cg þ f Cg þ  f Cg  gds ¼

j1

1 2pj

Zj1

e f Cg e Tr ½f Cg þ þ   gds

j1

e because it follows from (3.169) with Z ¼ f Cg þ that 1 2pj

Zj1

e f Cg e e e Tr ½f Cg þ  þ f Cg þ  f Cg  gds ¼ 0:

j1

This completes the proof of Theorem 3.4. Example* 3.26 For a square complex matrix M of order q with eigenvalues k1 ; k2 ; ; kq establish that when jkk j jki j, i ¼ 1 ! q, then jkk j is less than or equal to the largest singular value of M (i.e., the positive square root of the largest eigenvalue of the matrix H ¼ M M  ). Solution The matrix H ¼ M M  is hermitian and nonnegative definite. Hence, its eigenvalues are real and nonnegative. Since kk is an eigenvalue of M , it follows from

 ½detðkk I  M Þ ¼ det kk I  M  ¼ 0

3 H2 Design of Multivariable Control Systems

240

that kk is an eigenvalue of M  . When xk is an eigenvector associated with kk , then M  xk ¼ kk xk , xk M ¼ kk xk : Thus, xk Hxk ¼ xk MM  xk ¼ kk kk kxk k2 ¼ jkk j2 kxk k2 : Moreover, since H is hermitian, there exists a unitary matrix U satisfying UU  ¼ U  U ¼ I for which n o H ¼ U  RU; R ¼ diag r21 ; r22 ; ; r2q : where the ri are the singular values for M . Clearly, with y ¼ Uxk , xk Hxk ¼ y Ry ¼

Xq i¼1

r2i jyi j2  rmax2 k yk2 ¼ rmax2 kxk k2 :

Hence, xk Hxk ¼ jkk j2 kxk k2  rmax2 kxk k2 , jkk j  rmax So when rmax \1; all eigenvalues of M are less than unity in magnitude.

Chapter 4

H2 Design of Multivariable Control Systems with Decoupling

4.1

Overview

In some linear multivariable control system designs, it is required for some set of plant outputs that each output in the set be controlled by only one of an equal number of reference inputs. In this case, the system is said to be decoupled and the transfer matrix from the reference inputs to the plant outputs, denoted by T, is a diagonal matrix. In this chapter, all stabilizing controllers for which T is diagonal are established and the H2 design of decoupled systems is treated for the standard model. Any T, whether it is diagonal or not, that can be obtained using a stabilizing controller is called realizable. Further characterization of T is also needed to treat H2 design of decoupled systems. When T can be realized with a stabilizing controller that makes the regulated variables (yd in Fig. 3.1) and the control variables (v in Fig. 3.1) stable for a specific set of persistent inputs, then T and the inputs are called acceptable. The characterization of all realizable and acceptable T is treated in Sect. 4.2. When decoupling design is required, the structure of T is confined to be diagonal and this property is used in Sect. 4.3 to parameterize all realizable and acceptable T for decoupled systems. In Sect. 4.4, minimization of a given cost functional is carried out over all diagonal acceptable T and the formula for all controllers, including the optimal one, that yield a finite cost functional is presented. In Sects. 4.5 and 4.6, H2 controller formulas for 1DOF and 3DOF decoupled systems are obtained from the general formulas presented in Sect. 4.4.

© Springer Nature Switzerland AG 2020 J. J. Bongiorno Jr. and K. Park, Design of Linear Multivariable Feedback Control Systems, https://doi.org/10.1007/978-3-030-44356-6_4

241

242

4.2

4

H2 Design of Multivariable Control …

Characterization of Realizable and Acceptable T

The standard model for decoupling design is shown in Fig. 4.1. The variable r represents the reference inputs and eo represents the exogenous inputs other than the reference inputs. With regard to Fig. 3.1, this corresponds to the choice e0 ¼ ½ r 0 e0o : The variable v represents the control variables. The variables y, yz , and ym denote the plant outputs, other controlled variables, and the measured variables, respectively. With regard to Fig. 3.1, this corresponds to the choice y0c ¼ ½ y0 y0z : Dimensions of the vectors r and y are the same. The objective of decoupling design is to find stabilizing controllers that make Tyr , the transfer matrix from r to y, diagonal and invertible. e of the generalized plant follows from The transfer matrix P 2

3 2 2 3 y r P00 4 yz 5 ¼ P e 4 eo 5 ; P e ¼ 4 P10 ym v P20

P01 P11 P21

3 P02 P12 5 P22

ð4:1Þ

when the input–output relations are given by y ¼ P00 r þ P01 eo þ P02 v

ð4:2Þ

yz ¼ P10 r þ P11 eo þ P12 v

ð4:3Þ

ym ¼ P20 r þ P21 eo þ P22 v:

ð4:4Þ

and

e is assumed to be admissible which in accordance with The generalized plant P Definition 2.2(i) in Sect. 2.3 is equivalent to e Assumption 4.1 The generalized plant PðpÞ is free of hidden poles in Re p  0 for the analog case and in j pj  1 for the digital case and the polynomial we =wP22 is P stable.

Fig. 4.1 The standard model for decoupling design (© 2009 IEEE. Reprinted with permission, from Park (2009))

y

r eo

P

yz ym

v

C

4.2 Characterization of Realizable and Acceptable T

243

Equation (4.2) accounts for the contributions from each input to the plant output y. In most cases, the plant output y is not impacted directly by the reference input r; hence, it is reasonable to assume that P00 is a null matrix. It is also typical that e y, where F e is the stable transfer matrix of a sensor or measuring device. It ym ¼ F e P02 . These observations are the basis then follows from (4.2) to (4.4) that P22 ¼ F for the following assumption. e P02 with F e stable. Assumption 4.2 P00 ¼ 0 and P22 is of the form P22 ¼ F With this assumption the target transfer matrix to be diagonalized is given by Tyr ¼ P02 CðI  P22 CÞ1 P20 ¼ P02 RP20

ð4:5Þ

R ¼ CðI  P22 CÞ1 :

ð4:6Þ

with

Since the decoupling design requires that Tyr be invertible, it must have full rank (see Sect. B.2). It then follows from (4.5) that the next assumption is necessary. Assumption 4.3 P02 has full row rank and P20 has full column rank. Since the decoupling requirement is directly embodied in the imposition of the diagonal structure on the transfer matrix Tyr , it is convenient to define first all Tyr that can be obtained with stabilizing controllers. Definition 4.1 A rational matrix T, whether diagonal or not, is called realizable for e if there exists a stabilizing controller C that realizes the transfer matrix the plant P Tyr as the matrix T. The characterization of all realizable T is established next. It is always possible 1 1 to set P22 ¼ A1 B ¼ B1 A1 1 , where A B and B1 A1 are coprime polynomial matrix fraction descriptions for which there always exist polynomial matrices X; Y; X1 , and Y1 such that 

A Y1

B X1



X Y

B1 A1



 ¼

X Y

B1 A1



A Y1

B X1



 ¼

I 0

0 I

 ð4:7Þ

with det X det X1 6 0: (The coprime polynomial matrix fraction descriptions A1 B and B1 A1 1 are used to represent G, G22 , or P22 in different sections of this book. It is always made clear which is intended so that no confusion arises.) The class of e is characterized by the formula (see Lemma all stabilizing controllers for the plant P 2.2, Sect. 2.3) CðpÞ ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1 ;

ð4:8Þ

244

4

H2 Design of Multivariable Control …

where K is an arbitrary real rational stable matrix such that detðX  B1 KÞ  detðX1  KBÞ 6 0. Inserting C in (4.8) into (4.6), one obtains ð4:9Þ

R ¼ ðY þ A1 KÞA; hence, the transfer matrix Tyr becomes Tyr ¼ P02 YAP20 þ P02 A1 KAP20 :

ð4:10Þ

By Assumption 4.1 and Lemma 2.3, the matrices P02 YAP20 , P02 A1 , and AP20 are e iff it is of stable. In view of (4.10), a rational matrix T is realizable for the plant P the form T ¼ P02 YAP20 þ P02 A1 KAP20 :

ð4:11Þ

Since the matrices P02 and AP20 have row rank and column rank, respectively, there exist unimodular matrices Va and Vb such that (see Theorem B.1) 

P02 Va ¼ ½ Pa

0  and Vb AP20

Pb ¼ 0

 ð4:12Þ

with Pa and Pb square and invertible. Clearly, Pb is stable since AP20 is stable. One way to obtain the unimodular matrix Va in (4.12) is to find a coprime polynomial matrix fraction description P02 ¼ A1 02 B02 and to construct a row-Hermite form for B02 . Similarly, Vb can be obtained by introducing a coprime polynomial matrix fraction description of AP20 ¼ B120 A1 120 and constructing a column-Hermite form for B120 . However, this is obviously more than is necessary. For the system in Fig. 4.1, the tracking error is defined as e ¼ Td r  y;

ð4:13Þ

where Td r can be regarded as the desired output. (In the previous chapters, the tracking error is defined by e ¼ y  Td r. This discordance, however, does not affect the H2 design results in the sequel). The following assumption is critical in order to develop a tractable formula for realizable T. Assumption 4.4 The matrix Td is stable and there exists a stable matrix Qo such that P02 YAP20 ¼ Td þ Qo Pb :

ð4:14Þ

Assumption 4.4 can be justified with regard to the practical requirement that the tracking error e ¼ Td r  y ¼ ðTd  Tyr Þr be stable. First, it is natural to require that e be stable for any stable r. Hence, Te r ¼ ðTd  Tyr Þ must be stable which, since Tyr must be stable, requires that Td be stable. Second, it is also natural from the

4.2 Characterization of Realizable and Acceptable T

245

standpoint of tracking persistent inputs when P1 b is unstable to require that e be 1 stable for any stable ro when r ¼ Pb ro . It then follows that 1 Te r P1 b ¼ ðTd  P02 YAP20  P02 A1 KAP20 ÞPb   1 I ¼ ðTd  P02 YAP20 ÞP1 b  P02 A1 KVb 0

ð4:15Þ

and, therefore, Qo ¼ ðTd  P02 YAP20 ÞP1 b

ð4:16Þ

must be stable. That is, Eq. (4.14) must hold. Now it is possible to state the following theorem. The proof of the theorem is not given here since it can be obtained simply from the proof of Theorem 4.2 (see Remark 4.3 after the proof of Theorem 4.2). Theorem 4.1 Suppose that Assumptions 4.1 through 4.4 are satisfied. A rational 1 1 e iff T, ðTd  TÞP1 matrix T is realizable for the plant P b and Pa TPb are stable and there exists a stable matrix Q for which ð4:17Þ For a realizable T, the corresponding R in (4.5) is given by R ¼ ½ Rro

Rvo  Vb A

ð4:18Þ

   1 P1 1 0 a TPb þ A1 Q; ; Rvo ¼ YVb I M21

ð4:19Þ

with  Rro ¼ Va

where M21 is an arbitrary stable matrix. For a realizable T, the corresponding stable K in (4.8) is given by K ¼ A1 1 ð ½ Rro

Rvo  Vb  YÞ:

ð4:20Þ

Remark 4.1 When P20 is square, Vb ¼ I and Pb ¼ AP20 . In this case, the formulas in (4.17) and (4.18) reduce to e TP1 detðI þ F 20 Þ 6 0; R ¼ Va



 1 P1 a TP20 : M21 A

ð4:21Þ

246

4

H2 Design of Multivariable Control …

When P02 is square, Va ¼ I and Pa ¼ P02 . In this case, the formula in (4.17) remains the same and R in (4.18) reduces to ð4:22Þ When P20 and P02 are square, the formulas in (4.17) and (4.18) reduce to 1 1 e TP1 detðI þ F 20 Þ 6 0; R ¼ P02 TP20 :

ð4:23Þ

The characterization of all realizable T in Theorem 4.1 is enough to derive a parameterized formula of all realizable T in decoupled systems (see Theorem 4.3 in Sect. 4.3). However, further characterization of T is needed to treat H2 design of decoupled systems. In particular, when T is realizable and there exists a stabilizing controller for it that makes the regulated variables and the control variables stable for a specified set of persistent inputs, then T and the inputs are called acceptable. In order to be more precise with the definition of an acceptable T and acceptable inputs, it is first necessary to model the exogenous inputs which are treated in the H2 designs. For the H2 design of the system in Fig. 4.1, the exogenous inputs r and eo are modeled as done in Sect. 3.2. Moreover, as in Sects. 3.7 and 3.8, the H2 design problem of interest here is modeled in the form of Fig. 3.2 with the choices w0 ¼ e0 ¼ ½ r 0 e0o  and yd ¼ ½ e yz : Specifically,  w¼e¼

r eo



e w1 ¼½G

    e w q; q ¼ r ; e w2  r ¼ G G l l

ð4:24Þ

where r is a vector whose elements are real random variables with zero mean and covariance matrix \rr0 [ ¼ I and l is a stochastic vector whose power spectral density matrix is given by Ull ¼ I: It is assumed that r and l are independent e w2 is assumed to be stable; hence, it follows for any processes. The matrix G constants a1 [ 0 and a2  0 that according to Definition 3.2 ew Gw ¼ G

 pffiffiffiffiffi a1 I 0

0 pffiffiffiffiffi a2 I



pffiffiffiffiffi e ¼ ½ a1 G w1

pffiffiffiffiffi e a2 G w2 

ð4:25Þ

e w1 is acceptable. As in Sect. 3.3, the matrix Gw is introduced here is acceptable iff G pffiffiffiffiffi pffiffiffiffiffi in anticipation of the optimization problem, where a1 I and a2 I can be viewed as the weightings on r and l, respectively. In many cases, some elements of the exogenous input e do not have shape-deterministic components and in this case one can assume with no loss in generality that

4.2 Characterization of Realizable and Acceptable T

247

  Ge e G w1 ¼ : 0

ð4:26Þ

It should be noted that Gw is acceptable iff Gv1 Ge and Gd1 Ge are both stable where Gv1 ¼ Gv ½ I 0 0 and Gd1 ¼ Gd ½ I 0 0 : The regulated variable for H2 design considered here is 



e yd ¼ yz



Td r  y ¼ yz

 ð4:27Þ

and the following generalized plant model for H2 design is used to exploit the results developed in Chap. 3 for Fig. 3.2: 

yd ym

 ¼G

     e w G11 ¼G ;G¼ G21 v v

 G12 ; G22

ð4:28Þ

and  G11 ¼

Td P10

   P01 P02 ; G12 ¼ ; G21 ¼ ½P20 P21 ; G22 ¼ P22 : P11 P12

ð4:29Þ

e in (4.1) is admissible and Td is stable under the assumptions made here, Since P it is important to note that the matrices AG21 , G12 A1 , and G11 þ G12 YAG21 are stable; hence, the model G is compatible and this is implicit in the sequel. The vector yz in (4.27) is the regulated variable other than the tracking error and often is the control variable itself which in view of the Eq. (4.3) corresponds to the case P12 ¼ I; a stable matrix. It is reasonable, therefore, to make the following assumption. Assumption 4.5 P12 is stable. In order to characterize the acceptable Gw , the coprime polynomial matrix fraction descriptions  G12 ¼

P02 P12



1 ¼ A1 o Bo ¼ Bo1 Ao1

ð4:30Þ

with 

Ao Yo1

Bo Xo1



Xo Yo

Bo1 Ao1



 ¼

Xo Yo

Bo1 Ao1



Ao Yo1

Bo Xo1



 ¼

I 0

0 I

 ð4:31Þ

248

4

H2 Design of Multivariable Control …

are needed. It can be easily shown that A1 1 Ao1 is stable. In fact, pre-multiplying e 0  G12 ¼ F e P02 ¼ P22 ¼ G22 e (4.30) by ½ F 0  yields, by Assumption 4.2, that ½  F 1 1 e 0  Bo1 Ao1 ¼ B1 A1 . Clearly, ½  F e 0 Bo1 ¼ B1 A1 and, therefore, ½  F 1 Ao1 is e stable since F is stable by Assumption 4.2. Then it can be concluded that A1 1 Ao1 is stable since ðA1 ; B1 Þ is a coprime pair. In the sequel, Theorem 3.1 is applied to the model G in (4.28) to characterize the acceptable input Gw ¼ A1 w Bw , where ðAw ; Bw Þ is a coprime polynomial pair. In Theorem 3.1, the acceptability of Gw is expressed in terms of A1 w . In this chapter, however, the acceptability condition is stated in terms of Ge which reduces computational efforts to check the acceptability of Gw . Recall that a matrix Gw is acceptable for G if there exists a stabilizing controller such that Gv Gw and Gd Gw are both stable, where Gv and Gd denote the stable transfer matrices from w ¼ e to v and yd , respectively. It then holds that Gw is acceptable for G when there exists a e w1 ¼ Gv ½ G0e 00 0 and Gd G e w1 ¼ Gd ½ G0e 00 0 stabilizing controller such that Gv G are both stable. Since Gv ¼ ðY þ A1 KÞAG21 and Gd ¼ G11 þ G12 Gv , the partitions G11 ¼ ½ G111

G112 ; G21 ¼ ½ G211

G212 

ð4:32Þ

that are compatible with the multiplications G11 ½ G0e 00 0 ¼ G111 Ge and G21 ½ G0e 00 0 ¼ G211 Ge are introduced. With Ge ¼ A1 e Be a coprime polynomial matrix fraction description, Gw is acceptable for G iff  W ¼ Gv

A1 e 0



¼ ðY þ A1 KÞAG211 A1 e

ð4:33Þ

and  U ¼ Gd

A1 e 0



¼ G111 A1 e þ G12 W

ð4:34Þ

are stable. These equations have the same form as the ones in (3.20) and (3.21); hence, the same results are obtained as in Theorem 3.1 with the replacements 1 G11 ! G111 , G21 ! G211 , and A1 w ! A e . This leads to the acceptability condition of Gw and the resulting K as follows. One can always set 1 1 AG211 A1 e ¼ Ag Bg ¼ Bg1 Ag1

ð4:35Þ

with 

Ag Yg1

Bg Xg1



Xg Yg

Bg1 Ag1



 ¼

Xg Yg

Bg1 Ag1



Ag Yg1

Bg Xg1



 ¼

 I 0 : ð4:36Þ 0 I

4.2 Characterization of Realizable and Acceptable T

249

In this case, a matrix Gw ¼ A1 w Bw having the form shown in (4.25) and satis1 fying (4.26) with Ge ¼ Ae Be is acceptable for G iff Cw ¼ Ao G111 A1 e and Co ¼ A1 ðY C A  YB Þ are both stable and the associated controller can be realized o w g1 g1 o1 in the form of (4.8) with the stable parameter K constrained by K ¼ A1 1 Ao1 Co Yg1 þ Kg Ag ¼ Ks þ Kg Ag ;

ð4:37Þ

Ks ¼ A1 1 Ao1 Co Yg1

ð4:38Þ

where

is stable and Kg is any stable rational matrix for which detðX  B1 KÞ  det ðX1  KBÞ 6 0 It is convenient to introduce the following definitions for developing H2 design problems. Definition 4.2 A stabilizing controller CðpÞ in (4.8) is called acceptable for G and a given acceptable Gw if Gd Gw and Gv Gw are stable with this controller. Definition 4.3 A realizable T is called acceptable for G and a given acceptable Gw if there exists an acceptable controller that realizes the transfer matrix Tyr as the matrix T. e is used to formulate the decoupling As can be observed, the plant model P problem and the model G is used to develop H2 design problems with an acceptable Gw . It is important to recognize that when the model G is given, the acceptable Gw and the acceptable C are completely identified. Specifically, all acceptable inputs Gw are implicitly identified as the ones for which Cw ¼ Ao G111 A1 e and Co ¼ A1 ðY C A  YB Þ are stable and for which there exist stable matrices Kg such o w g1 g1 o1 that detðX  B1 KÞ  detðX1  KBÞ 6 0 when K is given by (4.37). All acceptable C for a given acceptable Gw are identified as in (4.8) with K being in the form of (4.37). The characterization of all acceptable T is now sought. It is known that a realizable T always has the form T ¼ P02 YAP20 þ P02 A1 KAP20 in (4.11). An acceptable T can be obtained by inserting the constrained formula of K in (4.37) into this equation. Hence, it can be concluded that a rational matrix T is acceptable for a given acceptable Gw iff it is of the form T ¼ P02 ðY þ A1 Ks ÞAP20 þ P02 A1 Kg Ag AP20 :

ð4:39Þ

A useful equality for later use is Y þ A1 Ks ¼ Y þ Ao1 Co Yg1 ¼ Y þ Yo Cw Ag1 Yg1  YBg1 Yg1

ð4:40Þ

250

4

H2 Design of Multivariable Control …

¼ Y þ Yo Cw Yg Ag  YðI  Xg Ag Þ

ð4:41Þ

¼ Yo Cw Yg Ag þ YXg Ag ¼ Cy Ag ;

ð4:42Þ

Cy ¼ Yo Cw Yg þ YXg

ð4:43Þ

where

is a stable matrix. Since the matrix Ag AP20 has full column rank, there exists a unimodular matrix Vg such that  Vg Ag AP20 ¼

Pg 0

 ð4:44Þ

with Pg square and invertible. Clearly, Pg is stable since AP20 is stable. The following assumption is an extension of Assumption 4.4 which assures that a tractable formula for an acceptable T can be developed. Assumption 4.4′ The matrix Td is stable and there exists a stable matrix Qg such that P02 ðY þ A1 Ks ÞAP20 ¼ Td þ Qg Pg :

ð4:45Þ

As in the case of Assumption 4.4, the above assumption can be justified with regard to the practical requirement that the tracking error e ¼ Td r  y ¼ ðTd  Tyr Þr be stable. It is natural from the standpoint of tracking persistent inputs 1 when P1 g is unstable to require that e be stable for any stable ro when r ¼ Pg ro . From this reasoning it follows that 1 Te r P1 g ¼ fTd  P02 ðY þ A1 Ks ÞAP20  P02 A1 Kg Ag AP20 gPg 1 ¼ fTd  P02 ðY þ A1 Ks ÞAP20 gP1 g  P02 A1 Kg Vg

  I 0

ð4:46Þ ð4:47Þ

and, therefore, Qg ¼ fTd  P02 ðY þ A1 Ks ÞAP20 gP1 g

ð4:48Þ

must be stable. That is, Eq. (4.45) must hold. The matrix P20 is usually a minimum phase stable matrix; hence, the unstable poles of P1 g are a subset of the unstable poles of the plant and Gw . In particular, when P20 ¼ I and A1 g is stable, then

4.2 Characterization of Realizable and Acceptable T

251

1 1 1 r o ; where er o is any stable vector. Pg ¼ Ag A and r ¼ P1 g ro ¼ A A g ro ¼ A e e is stable and the system is admissible, the unstable poles of A1 are the Since F same as the unstable poles of P02 which in most cases of interest corresponds to the actual plant transfer matrix. That is, the usual expectation that feedback systems can track persistent inputs with unstable poles the same as those of the plant is fulfilled in this case. It should be noticed that when Assumption 4.4′ holds, then Assumption 4.4 is satisfied. In fact, it follows from (4.12) and (4.44) that

Vg Ag AP20 ¼ Vg Ag Vb1



Pb 0



 ¼

 Pg : 0

ð4:49aÞ

Introducing the partition of the polynomial matrix Vg Ag Vb1 ¼



L11 L21

 L12 : L22

ð4:49bÞ

Then yields Vg Ag Vb1



Pb 0





L11 Pb ¼ L21 Pb





 Pg ; ¼ 0

ð4:49cÞ

hence, L11 Pb ¼ Pg ;

ð4:49dÞ

where L11 is a polynomial matrix. When there exists a stable matrix Qg such that P02 ðY þ A1 Ks ÞAP20 ¼ Td þ Qg Pg , then P02 YAP20 ¼ Td  P02 A1 Ks AP20 þ Qg Pg ¼ Td  P02 A1 Ks Vb1



 Pb þ Qg Pg : 0 ð4:50Þ

Thus, from (4.49d) and the partitioned stable matrix P02 A1 Ks Vb1 ¼ ½ Lb1 Lb2  one gets P02 YAP20 ¼ Td  Lb1 Pb þ Qg L11 Pb ¼ Td þ ðQg L11  Lb1 ÞPb ; which shows that Assumption 4.4 is satisfied. It is now possible to state and prove the following theorem.

ð4:51Þ

252

4

H2 Design of Multivariable Control …

Theorem 4.2 Suppose that Assumptions 4.1, 4.2, 4.3, 4.4′ and 4.5 are satisfied. A rational matrix T is acceptable for a given acceptable Gw iff T, ðTd  TÞP1 g and 1 1 Pa TPg are stable and there exists a stable matrix Q for which ð4:52Þ

For any such T, the corresponding R in (4.5) is given by R ¼ ½ Rr

Rv  Vg Ag A

ð4:53Þ

   1 0 P1 a TPg þ A1 Q; ; Rv ¼ Cy Vg1 I M21

ð4:54Þ

with  Rr ¼ Va

where M21 is an arbitrary stable matrix. For an acceptable T, the corresponding stable Kg in (4.37) is given by Kg ¼ A1 1 ð ½ Rr

Rv  Vg  Cy Þ:

ð4:55Þ

Remark 4.2 When P20 is square, Vg ¼ I and Pg ¼ Ag AP20 . The formulas in (4.52) and (4.53) then reduce to e TP1 detðI þ F 20 Þ 6 0; R ¼ Va



 1 P1 a TP20 : M21 Ag A

ð4:56Þ

When P02 is square, Va ¼ I and Pa ¼ P02 . In this case, the formula in (4.52) remains the same and R in (4.53) reduces to ð4:57Þ When P20 and P02 are square, the formulas in (4.52) and (4.53) reduce to 1 1 e TP1 detðI þ F 20 Þ 6 0; R ¼ P02 TP20 :

ð4:58Þ

Proof Necessity is considered first. Clearly, T in (4.39) must be stable since P02 YAP20 , P02 A1 , Ks and AP20 are stable. It follows from (4.39) and (4.12) that

4.2 Characterization of Realizable and Acceptable T

T ¼ ½ Pa

253

0 Va1 ðY þ A1 Ks ÞAP20 þ ½ Pa

0 Va1 A1 Kg Ag AP20 :

ð4:59Þ

0 Va1 ðY þ A1 Ks ÞAP20 þ ½ I

0 Va1 A1 Kg Ag AP20

ð4:60Þ

Hence, P1 a T ¼ ½I

must be stable. Next, it also follows from (4.39) and (4.44) that  0 T  P02 ðY þ A1 Ks ÞAP20 ¼ P02 A1 Kg Vg1 P0g 00 ; hence, 1 fT  P02 ðY þ A1 Ks ÞAP20 gP1 g ¼ P02 A1 Kg Vg

  I : 0

ð4:61Þ

Invoking Assumption 4.4′ then yields that 1 ðTd  TÞP1 g ¼ P02 A1 Kg Vg

  I  Qg 0

ð4:62Þ

and this must be stable. Finally, rewriting (4.60) yields P1 a T ¼ ½I

0 Va1 ðY þ A1 Ks þ A1 Kg Ag ÞAP20 :

ð4:63Þ

Hence, it follows from (4.42) to (4.63) that P1 a T ¼ ½I ¼ ½I

0 Va1 ðCy þ A1 Kg ÞAg AP20

0 Va1 ðCy

þ A1 Kg ÞVg1



ð4:64Þ

 Pg : 0

ð4:65Þ

1 It can now be seen that P1 a TPg must be stable. It is important to recognize that if T is acceptable for a given acceptable Gw there must exist an R ¼ CðI  P22 CÞ1 which has the structure of R ¼ ðY þ A1 KÞA with K of the form in (4.37). That is, there must exist an

R ¼ ðY þ A1 KÞA ¼ ðY þ A1 Ks ÞA þ A1 Kg Ag A

ð4:66Þ

¼ Cy Ag A þ A1 Kg Ag A ¼ ðCy þ A1 Kg ÞAg A:

ð4:67Þ

Clearly, T ¼ P02 RP20 ¼ P02 RðAg AÞ1 Ag AP20 ¼ ½ Pa

0 Va1 RðAg AÞ1 Vg1



 Pg ; ð4:68Þ 0

254

4

1 hence, P1 a TPg ¼ ½ I

0  Va1 RðAg AÞ1 Vg1 ½ I

H2 Design of Multivariable Control …

0 0 must be true. Therefore,

Va1 RðAg AÞ1 Vg1 must be of the form

Va1 RðAg AÞ1 Vg1 ¼



 M12 : M22

1 P1 a TPg M21

ð4:69Þ

In view of (4.67), Va1 RðAg AÞ1 Vg1 is stable; hence, M12 ; M21 , and M22 must be stable. Thus, for an acceptable T the matrix R must be of the form  R ¼ Va

1 P1 a TPg M21

   1 1  M12 P TPg V A A ¼ Va a M22 g g M21

 Va

M12 M22

 Vg Ag A: ð4:70Þ

The stable matrices M12 and M22 in (4.70) are not completely arbitrary since R is constrained by the structure in (4.67). That is  R ¼ Va



1 P1 a TPg M21



 Va

M12 M22

 Vg Ag A ¼ ðCy þ A1 Kg ÞAg A

ð4:71Þ

is necessary. Multiplying (4.71) on the right by ðAg AÞ1 Vg1 ½ 0 I 0 yields  Va

M12 M22



¼ ðCy þ A1 Kg ÞVg1

¼ Cy Vg1

  0 I

ð4:72Þ

  0 þ A1 Q; I

ð4:73Þ

where Q ¼ Kg Vg1 ½ 0 I 0 is a stable matrix. Hence, R must be of the form described by (4.53) and (4.54). It remains to prove sufficiency. Solving for Kg in (4.67) and using (4.53), one obtains Kg ¼ A1 1 ð ½ Rr

Rv  Vg  Cy Þ:

ð4:74Þ

It has to be shown that this Kg is stable and that detðX  B1 KÞ 6 0 when (4.52) is satisfied. Clearly, Kg is stable iff ½ A01 B01 0 Kg is stable. Obviously A1 Kg is stable; hence, Kg is stable iff B1 Kg is stable. It follows from (4.74) that B1 Kg ¼ P22 ð ½ Rr

(

Rv  Vg  Cy Þ

ð4:75Þ

)

ð4:76Þ

4.2 Characterization of Realizable and Acceptable T

255

(

) (

)

(

ð4:78Þ

)

(

ð4:79Þ

) (

)

ð4:77Þ

ð4:80Þ

ð4:81Þ

Since 1 Ag AP20 P1 g ¼ Vg

  I ; 0

ð4:82Þ

it follows from (4.81) that ð4:83Þ Using the equality in (4.42) and the equality (4.45) in Assumption 4.4′, one gets ð4:84Þ ð4:85Þ ð4:86Þ which is stable. Hence, Kg is stable. Finally, it is established that (4.52) is equivalent to det½ðX  B1 KÞA 6 0 which is the case iff det½ðX  B1 KÞ 6 0: Clearly, from the expression for Kg in (4.74), ðX  B1 KÞA ¼ XA  B1 Ks A  B1 Kg Ag A

ð4:87Þ

256

4

¼ XA  B1 Ks A  B1 A1 1 ð½ Rr e P02 ½ Rr ¼ XA  B1 Ks A þ F e P02 ½ Rr ¼ XA  B1 Ks A þ F

H2 Design of Multivariable Control …

Rv Vg  Cy Þ Ag A

ð4:88Þ

Rv Vg Ag A þ B1 A1 1 Cy Ag A

ð4:89Þ

Rv Vg Ag A þ B1 A1 1 ðY þ A1 Ks ÞA

ð4:90Þ

e P02 ½ Rr ¼ XA þ F

Rv Vg Ag A þ B1 A1 1 YA

e P02 ½ Rr ¼IþF

Rv Vg Ag A

ð4:91Þ ð4:92Þ ð4:93Þ

ð4:94Þ

ð4:95Þ

ð4:96Þ

ð4:97Þ and the proof of Theorem 4.2 is complete. Remark 4.3 A comparison of the Eqs. (4.11) and (4.12) with (4.39) and (4.44) easily yields the proof of Theorem 4.2 from that of Theorem 4.1 using the substitutions of Vg ! Vb , Pg ! Pb , Ag ! I, and Y þ A1 Ks ! Y (hence Cy ! YÞ.

4.3

Characterization of Realizable Diagonal T and Acceptable Diagonal T

As a consequence of Theorem 4.1, a necessary condition for any rational matrix 1 1 e is that T, ðTd  TÞP1 T to be realizable for the plant P b and Pa TPb be stable. As a consequence of Theorem 4.2, a necessary condition for any rational matrix T to be 1 1 acceptable is that T, ðTd  TÞP1 g and Pa TPg be stable. These necessary conditions apply whether T is diagonal or not. When decoupling design is considered, the

4.3 Characterization of Realizable Diagonal T and Acceptable Diagonal T

257

structure of T is confined to be diagonal and this property is used in the sequel to parameterize all realizable T and all acceptable T for a decoupled system. In a decoupled system, it is natural to choose a diagonal Td in defining the tracking error e ¼ Td r  y: It is also convenient to impose the restriction that Td1 is stable. Thus, these additional restrictions on Td are included in Assumption 4.6 Td is diagonal and Td1 is stable. Let T ¼ diagfti g; ti ¼

hi ; gi

ð4:98Þ

where hi and gi are polynomials. Since T must be stable, the polynomial gi must be stable (that is, it must be Hurwitz for the analog case and Schur for digital case). 1 1 1 Since Pb and Pg are stable, the stability of P1 a TPb and Pa TPg implies the 1 1 stability of Pa T: For the stability of Pa T; it is required that hi cancel all unstable poles of i-th column of P1 a . Let hi denote the unique monic polynomial of the minimal degree such that f i-th column of P1 a g  hi is stable and define Dh ¼ diagfhi g. Then hi must be of the form hi ¼ mi hi , mi polynomial. That is, ti must be of the form ti ¼ hi d1i , where d1i ¼ mi =gi is an arbitrary stable rational function. This leads to T ¼ Dh D1 ;

ð4:99Þ

where D1 ¼ diagfd1i g is an arbitrary stable diagonal matrix. For the same reason, the stability of ðTd  TÞP1 b for a realizable T requires that Td  T ¼ Du D2 ;

ð4:100Þ

where ui denotes the unique monic polynomial of the minimal degree such that ui  fi-th row of P1 b g is stable, Du ¼ diagfui g, and D2 is an arbitrary stable diagonal matrix. It follows from (4.99) and (4.100) that Td ¼ Dh D1 þ Du D2 :

ð4:101Þ

Notice that any common zero of hi and ui , being a zero in the unstable region, is a zero of the i-th diagonal element of Td but this violates the assumption that Td1 is stable. Thus, it is necessary that the pair hi , ui be coprime. When the pair hi , ui is coprime, Eq. (4.101) has the same form as (2.242). Hence, the reasoning in Sect. 2.6 can be used to obtain the general solution formulas for D1 and D2 in (4.101) which are

258

4

H2 Design of Multivariable Control …

D1 ¼ Da Td þ DDu

ð4:102Þ

D2 ¼ Db Td  DDh :

ð4:103Þ

and

Here D is an arbitrary stable diagonal matrix, and Da and Db are diagonal matrices defined by Da ¼ diagf^ai g; Db ¼ diagfbi g;

ð4:104Þ

where ^ai and bi are unique polynomials of minimal degree satisfying (see Sect. B.9) ^ai hi þ bi ui ¼ 1:

ð4:105Þ

It now follows from (4.99) to (4.102) that necessarily any realizable diagonal T must have the form T ¼ Dh ðDa Td þ DDu Þ:

ð4:106Þ

Moreover, 1 1 1 1 1 1 1 P1 a TPb ¼ Pa Dh ðDa Td þ DDu ÞPb ¼ Pa Dh Da Td Pb þ Pa Dh DDu Pb

ð4:107Þ must be a stable matrix. Since the second term in (4.107) is stable by the definitions 1 of hi and ui , it is necessary that R ¼ P1 a Dh Da Td Pb be stable. Conversely, when Assumptions 4.1 through 4.4 and Assumption 4.6 are satisfied it is easy to show 1 1 that the T given by (4.106) satisfies T, ðTd  TÞP1 b , and Pa TPb stable if R is stable. It then follows from Theorem 4.1 that Theorem 4.3 When Assumptions 4.1 through 4.4 and 4.6 are satisfied, a diagonal rational T is realizable (equivalently, a decoupling controller C exists) iff (1) hi and ui are coprime (equivalently, no column of P1 a and the corresponding number row in P1 possess coincident poles in the unstable region); (2) the data b 1 construct R ¼ P1 D D T P is stable; and (3) there exists a diagonal stable h a d b a rational matrix D and a stable matrix Q for which (4.17) holds with T ¼ Dh Da Td þ Dh DDu :

ð4:108Þ

The results in Theorem 4.3 are very important in the sense that the conditions for the existence of a decoupling controller are described and the characterization of all realizable T in decoupled systems is explicitly expressed. Also,

4.3 Characterization of Realizable Diagonal T and Acceptable Diagonal T

259

Remark 4.4 Assumption 4.4 is critical to derive the parameterized formula for T in (4.108) of Theorem 4.3. When Assumption 4.4 is not met, a different method to parameterize the acceptable and diagonal T is possible (Park 2012). This method, however, requires vector operations which lead to a dimensional inflation that complicates design calculations. Attention is now turned to the characterization of acceptable T for H2 design of decoupled systems with acceptable persistent inputs. A necessary condition for a rational matrix T to be acceptable for a given acceptable Gw is that T, ðTd  TÞP1 g , 1 and P1 TP be stable. These conditions are the same as the ones required for a g realizable T except for the replacement of Pb by Pg ; hence, by slight modification of the derivations leading to Theorem 4.3 for realizable T one gets Theorem 4.4 below. In this theorem, hi is the same one defined previously and ugi denotes the unique monic polynomial of the minimal degree such that ugi  fi-th row of P1 g g is stable, where Pg is defined as in (4.44). Theorem 4.4 When Assumptions 4.1, 4.2, 4.3, 4.4′, 4.5, and 4.6 are satisfied, a diagonal rational T is acceptable (equivalently, an acceptable decoupling controller C exists) for a given acceptable Gw iff (1) hi and ugi are coprime (equivand the corresponding number row in P1 possess alently, no column of P1 a g coincident poles in the unstable region); (2) the data construct Rg ¼ P1 a Dh Dag 1 Td Pg is stable, where Dag ¼ diagfagi g and agi , bgi are unique polynomials of minimal degrees satisfying agi hi þ bgi ugi ¼ 1;

ð4:109Þ

and (3) there exists a diagonal stable rational matrix D and a stable matrix Q for which (4.52) holds with T ¼ Dh Da g Td þ Dh DDu g ;

ð4:110Þ

where Dug ¼ diagfugi g. When a diagonal T is determined by Theorem 4.3 or Theorem 4.4, the corresponding controller C can be obtained from the formula C ¼ ðI þ RP22 Þ1 R;

ð4:111Þ

where the matrix R is given by (4.18) for Theorem 4.3 and by (4.53) for Theorem 4.4, respectively.

260

4

H2 Design of Multivariable Control …

Remark 4.5 When a diagonal T exits which is acceptable in accordance with Theorem 4.4, it is likely that a diagonal T exits which is realizable in accordance with Theorem 4.3. The following derivation shows that this is indeed the case. First 1 it is shown that ui in Theorem 4.3 divides ugi in Theorem 4.4. Since P1 b ¼ Pg L11 from (4.49d) and L11 is a polynomial matrix, the highest multiplicity of any unstable pole in the i-th row of P1 g is greater than or equal to its highest multiplicity 1 in the i-th row of Pb . Hence, ui divides ugi and ugi ¼ ui li , where li is a polynomial. That is, if hi , ugi is a coprime pair, then so is hi , ui . It follows from (4.109) that agi hi þ bgi ui li ¼ 1 and, therefore, agi hi þ bgi ui li  1 ¼ agi hi þ bgi ui li  ðai hi þ bi ui Þ ¼ 0 which leads to the equalities agi hi ¼ ai hi þ bi ui  bgi ui li

ð4:112Þ

agi  ai ¼ h1 i ðbi  bgi li Þui :

ð4:113Þ

and

Since hi ; ui is a coprime pair, no cancelation happens between the product of 1 hi and ui ; hence, h1 i ðbi  bgi li Þ must be a polynomial. Next suppose that Rg ¼ 1 1 1 1 P1 D D T P ¼ P is stable (L11 is invertible since h ag d g a a Dh Dag Td Pb L11 1 Pg ¼ L11 Pb Þ. Then, Rg L11 ¼ Pa Dh Dag Td P1 is stable. The equality in (4.112) b

yields that Dag Dh ¼ Da Dh þ Db Du  Db g Du Dl , with Dl ¼ diagfli g, and it follows from 1 1 1 1 1 Rg L11 ¼ P1 a Dh Dag Td Pb ¼ Pa Da Dh Td Pb þ Pa ðDb Du  Dbg Du Dl ÞTd Pb

ð4:114Þ that 1 1 1 1 Rg L11 ¼ P1 a Da Dh Td Pb þ Pa Dh Dh ðDb  Dbg Dl ÞD/ Td Pb :

ð4:115Þ

1 1 1 Since P1 a Dh , Du Td Pb ¼ Td Du Pb , and Dh ðDb  Dbg Dl Þ are all stable, the 1 stability of Rg L11 implies the stability of Pa Da Dh Td P1 b . Therefore, if a diagonal T is acceptable in accordance with Theorem 4.4, then it is realizable in accordance with Theorem 4.3 provided there exists a stable matrix Q for which (4.17) holds. It is also useful to scrutinize the case when a decoupling controller exists by Theorem 4.3, but the conditions of Theorem 4.4 are not all met. When one encounters this situation, H2 design cannot proceed unless it is practical to consider an alternative Gw for which an acceptable diagonal T exists. (It is not difficult to show as in Example 4.9 that when a realizable diagonal T exists, it is also an acceptable diagonal T for any strictly acceptable Gw .) In order to avoid a trial and error approach to find an acceptable Gw , one might try to identify all the matrices Gwd for which there exists a

4.3 Characterization of Realizable Diagonal T and Acceptable Diagonal T

261

realizable and diagonal T such that Gd Gwd and Gv Gwd are stable and to parameterize such T. This would be similar to the approach taken in Sect. 3.3 to differentiate acceptable controllers from stabilizing controllers. Although this may be possible, it is expected that for decoupled systems little or no advantage over the trial and error method would be gained. Hence, Theorem 4.4 provides a practical way to find an acceptable T and acceptable Gw for the decoupled G.

4.4

H2 Design of Decoupled Systems for Acceptable Inputs

In this section, minimization of a given cost functional over the diagonal acceptable T in Theorem 4.4 is carried out when decoupling controllers exist. In the plant model G in (4.28), yd is the regulated variable and the transfer matrix from e to yd is Gd . As explained in Sects. 3.5 and 3.6, a cost functional which incorporates the transient responses and the steady-state stochastic performance for the generalized plant in (4.28) is given by J¼

kWGd Gw k22 ¼

1 2p j

Zj1 Tr ½ðWGd Gw Þ ðWGd Gw Þ ds;

ð4:116Þ

1 ð dk ; H

Tr ½ðWGd Gw ÞðWGd Gw Þ  2p j k

ð4:117Þ

j1

in the analog case and J ¼ kWGd Gw k22 ¼

jkj¼1

in the digital case. The weighting matrix W is stable and Gw is an acceptable matrix for which a diagonal acceptable T in Theorem 4.4 exists. The transfer matrix Gd is given by Gd ¼ G11 þ G12 CðI  G22 CÞ1 G21 ¼ G11 þ G12 RG21 :

ð4:118Þ

The H2 decoupling problem for the system in Fig. 4.1 is to find the decoupling controller which minimizes the quadratic cost functional J. Employing the formula in (4.53) for R, one obtains E ¼ WGd Gw ¼ WG11 Gw þ WG12 ½ Rr

Rv Vg Ag AG21 Gw :

ð4:119Þ

Introducing the compatible partition pffiffiffiffiffi Vg Ag AG21 Gw ¼ Vg ½ a1 Bg Be

  pffiffiffiffiffi e w2  ¼ Ng1 a2 Ag AG21 G Ng2

ð4:120Þ

262

4

H2 Design of Multivariable Control …

leads to E ¼ WG11 Gw þ WG12 ðRr Ng1 þ Rv Ng2 Þ     1 1   0 P TPg ¼ WG11 Gw þ WG12 Va a þ A1 QÞNg2 Ng1 þ ðCy Vg1 I M21

ð4:121Þ ð4:122Þ

    1 1  P02 Pa TPg 1 0 ¼ WG11 Gw þ W N þ WG12 A1 QNg2 Va Ng1 þ WG12 Cy Vg I g2 P12 M21 

ð4:123Þ ¼ WG11 Gw þ WG12 Cy Vg1

     1 0 ½ Pa 0  P1 a TPg Ng1 þ WG12 A1 QNg2 : Ng2 þ W I P12 Va M21

ð4:124Þ One can now introduce the compatible partition P12 Va ¼ ½ Na1

Na2 

ð4:125Þ

to obtain

(

) ð4:126Þ

     

Pa 1 1 0 0 Ng2 þ ¼ W G11 Gw þ G12 Cy Vg1 Pa TPg Ng1 þ M21 Ng1 þ G12 A1 QNg2 : Na2 I Na1

ð4:127Þ Inserting the diagonal acceptable T of (4.110) into (4.127) then yields  E ¼ U1 þ W

   Dh 0 1 P N þ W DD M21 Ng1 þ WG12 A1 QNg2 ; ug g g1 Na1 P1 Na2 a Dh ð4:128Þ

where U1 ¼

WG11 Gw þ WG12 Cy Vg1

    Pa 1 0 N þW Pa Dh Dag Td P1 g Ng1 : ð4:129Þ I g2 Na1

It easily follows from (4.120) and (4.125) that Na1 , Na2 , Ng1 , and Ng2 are stable. Hence, U1 is a stable matrix since E ¼ WGd Gw is stable and the last three terms in (4.128) are all stable. It is also important to note for the structure of E in (4.128) that

4.4 H2 Design of Decoupled Systems …

263

the free design parameters available to minimize the cost functional J ¼ kEk22 are D; M21 ; and Q. Among them, D is restricted to be diagonal and this requirement prevents one from directly applying the method developed in Chap. 3 to obtain the optimal H2 solution. However, the structure of E can be transformed into the solvable standard form treated in Chap. 3 with the aid of the following vector-matrix operations and notations. As already noted in the next to last paragraph of Sect. 1.4, the Kronecker product G  R is the matrix whose ij-block is gij R and vec G ¼ ½ g01 g02 . . . g0n 0 is the vector obtained by stacking the n columns gj ; j ¼ 1 ! n of G. Also needed is the Khatri-Rao product G R which is the matrix whose i-th column is given by gi  ri , where gi and ri are the respective i-th columns of G and R. In addition, for a diagonal matrix G, the vector formed by stacking all the diagonal elements of G is denoted by vecd G ¼ ½ g11 g22 . . . gnn 0 . Moreover, vec ðAVDÞ ¼ ðD0  AÞvec V and when V is a diagonal matrix then vec ðAVDÞ ¼ ðD0 AÞvecd V is easily verified. Finally (see Example 4.1), kE k2 ¼ kvec E k2 : i.e., the vec operation does not alter the value of the H2 norm. It now follows from (4.128) that 

 Dh ðW vec E ¼ /1 þ Þ e d Na1 P1 a Dh    n o 0 0 0  ðW  ðWG12 A1 Þ e q; Þ m21 þ Ng2 þ Ng1 Na2 0 ðDug P1 g Ng1 Þ



ð4:130Þ

where e d ¼ vecdD; m21 ¼ vec M21 ; e q ¼ vec Q;

ð4:131Þ

and  

n o 0 1 1 0 /1 ¼ vec U1 ¼ vec ðWG11 Gw Þ þ Ng2  ðWG12 A1 Þ vec A1 Cy Vg 1     D h 0 þ ðDug P1 vecdðDag Td D1 ug Þ: g Ng1 Þ W Na1 P1 D h a ð4:132Þ Rearranging (4.130), one obtains vec E ¼ /1 þ P qL ; where qL ¼ ½ de0

m021

0 e q 0  is an arbitrary stable vector and

ð4:133Þ

264

4

H2 Design of Multivariable Control …

ð4:134Þ is a stable matrix. Actually, qL is not completely arbitrary because (4.52) must be satisfied. Relaxing this requirement initially, however, makes the optimization problem treated in the sequel tractable. Moreover, optimum solutions obtained in practice usually lead to controllers with a K matrix that satisfies detðX  B1 KÞ  detðX1  KBÞ 6 0 That is, the solutions are ones which can be achieved with a stabilizing controller. The form in (4.133) fits the standard one treated in H2 problems and the procedures developed in Sects. 3.5 and 3.6 to obtain the optimal solution can now be used again here. The Wiener–Hopf spectral factor of P P plays an important role in the sequel and as in Sects. 3.5 and 3.6 it is assumed that P has full column rank on jkj ¼ 1 for the digital case and on the finite part of the s ¼ jx axis for the analog case. In order to treat the digital and analog cases together within a unified frame, the terminology of “the stability boundary” is used to denote the unit circle of jkj ¼ 1 for the digital case and the finite part of the s ¼ jx axis for the analog case. Some reasonable assumptions guaranteeing that P has full column rank are established in the sequel with the aid of the following lemmas and identities. Lemma 4.1 For the set of matrices Gi ; i ¼ 1 ! n; whose elements are complex numbers it is true that if G1 has column rank, then rankðG1 G2 Þ ¼ rankðG2 Þ and if Gi has full column rank for i ¼ 1 ! n  1; then rankðG1 G2    Gn Þ ¼ rankðGn Þ. (See Example 4.2 for the proof.) Lemma 4.2 M and N are matrices whose elements are complex numbers. If M and N each have full column rank, then M  N has full column rank. (See Example 4.3 for the proof.) Also useful are the identities ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ, ðA  BÞ ðC DÞ ¼ ðAC BDÞ, and Starting with (4.134) one obtains that

( ð4:135Þ

ð4:136Þ

4.4 H2 Design of Decoupled Systems …

265

ð4:137Þ

ð4:138Þ

ð4:139Þ

(

)(

) ð4:140Þ

¼ P1 P2 P3 ;

ð4:141Þ

where ð4:142Þ

and

ð4:143Þ

266

4

H2 Design of Multivariable Control …

Hence, P has full column rank on the stability boundary if P1 , P2 , and P3 each have full column rank on the stability boundary (Lemma 4.1). The following assumptions assure that this is the case. Assumption 4.7 The stable weighting matrix W has full column rank on the stability boundary. Assumption 4.8 Ag AG21 Gw ¼ Ag A½ P20 P21  Gw has full row rank on the stability boundary. Assumption 4.9 P12 has full column rank on the stability boundary.   P02 Assumption 4.10 G12 A1 ¼ A1 has full column rank on the stability P12 boundary.   Dh Assumption 4.11 has full column rank on the stability boundary. P1 a Dh has row rank (i.e., rank of 1) on the Assumption 4.12 Each row of Dug P1 g stability boundary (or, each row of Dug P1 does not vanish on the stability g boundary). Assumptions 4.7 and 4.8 assure, by Lemma 4.2, that P1 has full column rank on the stability boundary. Assumption 4.9 assures that P2 has full column rank on the stability boundary. In fact, there exist constant permutation matrices E1 , E2 , and E3 (Brewer 1978) such that

(

)

(

) ð4:144Þ

ð4:145Þ

ð4:146Þ

ð4:147Þ

This shows that if P12 has full column rank on the stability boundary, then P2 has full column rank on the stability boundary. Assumptions 4.10, 4.11, and 4.12

4.4 H2 Design of Decoupled Systems …

267

assure that P3 has full column rank on the stability boundary. In fact, when G12 A1 satisfies Assumption 4.10, P3 has full column rank on the stability boundary if 0 P4 ¼ ðDug P1 g Þ



Dh P1 a Dh



¼

 I

Dh P1 a Dh



0 fðDug P1 g Þ Ig

ð4:148Þ

has full column rank on the stability boundary. When Assumption 4.11 holds, 0 1 0 1 0 then rankðP4 Þ ¼ rankfðDug P1 g Þ Ig ¼ rankfE4 ðI ðDug Pg Þ Þg ¼ rankfðI ðDug Pg Þ g; where E4 is a permutation matrix. When Assumption 4.12 is satisfied, then 0 I ðDug P1 g Þ has full column rank on the stability boundary. Thus, P4 and, therefore, P have full column rank on the stability boundary. With H a Wiener–Hopf spectral factor in the equation P P ¼ H H

ð4:149Þ

U ¼ PH1 :

ð4:150Þ

it is convenient to define

By the property of Wiener–Hopf factors, H and H1 are analytic in jkj\1 for the digital case and in Re s [ 0 for the analog case. The matrix H is good since P is stable. Moreover, since P has full column rank on the stability boundary, H1 is analytic on the stability boundary. Hence, H and H1 are stable. 1 ¼ IÞ, stable The matrix U ¼ PH1 is paraunitary (i.e., U U ¼ H1 P PH and proper (Lemma 3.2). Since U U ¼ I; it is clear that UU U ¼ U and U  UU U ¼ ðI  UU ÞU ¼ 0: Thus, ðI  UU ÞP ¼ 0 and it readily follows from (4.134) that   0 ðI  UU Þ ðDug P1 N Þ ðW g1 g

Dh Na1 P1 a Dh

 ¼0

ð4:151Þ

and 0  ðWG12 A1 Þg ¼ 0: ðI  UU ÞfNg2

ð4:152Þ

Since ðI  UU Þ ¼ ðI  UU Þ; it follows from (4.151) and (4.152) that 

and

0 ðDug P1 g Ng1 Þ ðW



Dh Na1 P1 a Dh



ðI  UU Þ ¼ 0

ð4:153Þ

268

4

H2 Design of Multivariable Control …

0 fNg2  ðWG12 A1 Þg ðI  UU Þ ¼ 0:

4.4.1

ð4:154Þ

H2 Design for the Digital Case

It is convenient at this point to treat H2 designs for the digital and analog cases separately. The digital case is treated first in this section and the analog case is treated in Sect. 4.4.2. The objective is to find all qL that yield a finite cost functional J ¼ kvec Ek22 . It follows from (4.133) and (4.150) that ðvec EÞ vec E ¼ ð/1 þ PqL Þ ð/1 þ PqL Þ ¼ ð/1 þ UH qL Þ ð/1 þ UH qL Þ ð4:155Þ ¼ ðU /1 þ H qL Þ ðU /1 þ H qL Þ þ qo ;

ð4:156Þ

where qo ¼ /1 /1  /1 UU /1 : It can be shown from (4.132), and (4.151) to (4.154) that q0 ¼ /1 ðI  UU Þ/1 ¼ ðvec ðWG11 Gw ÞÞ ðI  UU Þvec ðWG11 Gw Þ:

ð4:157Þ

Since fU /1 g þ , fU /1 g1 , H, and H1 are stable and qL is an arbitrary stable vector, it follows that df ¼ fU / 1 g þ þ fU / 1 g1 þ H qL

ð4:158Þ

is an arbitrary stable vector. Using (4.158) in (4.156) then yields ðvec EÞ vec E ¼ ðdf þ fU /1 g Þ ðdf þ fU /1 g Þ þ qo :

ð4:159Þ

By the same reasoning as in Sect. 3.5 ((3.118)–(3.122)), it follows that  2 1 ð dk J ¼ kvec E k22 ¼ fU /1 g þ df 2 þ H

qo 2p j k

ð4:160Þ

 2  2 1 ð dk ¼ df 2 þ fU /1 g 2 þ H

qo 2p j k

ð4:161Þ

jkj¼1

jkj¼1

It now becomes clear that all qL ¼ H1 ðfU /1 g þ fU /1 g1 þ df Þ yield finite cost for J and minimization of J is achieved when df ¼ 0: So with no further proof necessary the following theorem can be stated.

4.4 H2 Design of Decoupled Systems …

269

Theorem 4.5 When Assumptions 4.1, 4.2, 4.3, 4.4′, and 4.5 through 4.12 hold and when the existence condition for a decoupling controller in Theorem 4.4 is met, the T and R matrices associated with the decoupling acceptable controllers that yield finite cost J are given as in (4.110) and (4.53), respectively, with qL ¼ H1 ðfU /1 g1 þ fU /1 g þ  df Þ;

ð4:162Þ

where df is an arbitrary stable rational vector. The optimal qL minimizing the cost functional is given by qopt ¼ H1 ðfU /1 g1 þ fU /1 g þ Þ:

ð4:163Þ

When Jopt denotes the cost functional associated with the optimal solution qopt , the cost functional J yielded by qL in (4.162) is given by  2  2 1 ð dk : J ¼ Jopt þ df 2 ; Jopt ¼ fU /1 g 2 þ H

qo 2p j k

ð4:164Þ

jkj¼1

At this point there is no guarantee that using (4.162) for qL in (4.53) yields a K determined by (4.37) and (4.55) such that detðX1  KBÞ 6 0: However, this is typically the case on account of Remark 3.3. Remark 4.6 For the stable paraunitary matrix U in (4.150), it is possible to find a left Wiener–Hopf factor U? such that I  UU ¼ U? U? (see Example 4.4). In this case, the integral term in (4.164) is given by 1 ð dk ¼ kvec ðU? WG11 Gw Þk22 : H

qo 2p j k

ð4:165Þ

jkj¼1

4.4.2

H2 Design for the Analog Case

The feature that distinguishes H2 design for the analog case from that for the digital case is the fact that for the cost functional J ¼ kvec E k22 to be finite in the analog case, it is required that vec E  Oðs1 Þ: Then U vec E  Oðs1 Þ and it follows from U vec E ¼ U /1 þ U PqL ¼ U /1 þ HqL ¼ fU /1 g1 þ fU /1 g þ þ fU /1 g þ HqL that the matrix HqL must have the form HqL ¼ fU /1 g1 þ dz ;

ð4:166Þ

270

4

H2 Design of Multivariable Control …

where dz is an arbitrary stable and strictly proper vector. Inserting (4.166) into vec E ¼ /1 þ P qL ¼ /1 þ UH qL , one obtains vec E ¼ /1  U fU /1 g1 þ Udz ¼ /1  UðU /1  fU /1 g þ fU /1 g Þ þ Udz ð4:167Þ ¼ ðI  UU Þ/1 þ U fU /1 g þ þ U fU /1 g þ Udz :

ð4:168Þ

It follows from (4.132), (4.151), and (4.152) that vec E ¼ ðI  UU Þvec ðWG11 Gw Þ þ U fU /1 g þ þ U fU /1 g þ Udz :

ð4:169Þ

Since U and /1 are stable, the vector fU /1 g þ is also stable. Hence, it is possible to parameterize dz as dz ¼ fU /1 g þ þ df , where df is an arbitrary stable and strictly proper vector. With this parameterization, it follows from (4.169) that vec E ¼ ðI  UU Þvec ðWG11 Gw Þ þ UðfU /1 g þ df Þ

ð4:170Þ

and from (4.166) that qL ¼ H1 ðfU /1 g1 þ dz Þ ¼ H1 ðfU /1 g1 þ fU /1 g þ df Þ:

ð4:171Þ

In view of (4.170), vec E is strictly proper when the following assumption is met.   Td P01 Assumption 4.13 The matrix WG11 Gw ¼ W Gw is strictly proper. P10 P11 It is also clear from (4.168) that vec E is good; hence, J is finite. Moreover, it is not difficult to show using (4.170) that ðvec EÞ ðvec EÞ ¼ vec ðWG11 Gw Þ ðI  UU Þvec ðWG11 Gw Þ þ ðfU /1 g þ df Þ ðfU /1 g þ df Þ:

ð4:172Þ

Thus, J¼

kvec E k22

1 ¼ 2pj þ

Zj1 vec ðWG11 Gw Þ ðI  UU Þvec ðWG11 Gw Þds j1

1 2pj

ð4:173Þ

Zj1 ðfU /1 g þ df Þ ðfU /1 g þ df Þds: j1

4.4 H2 Design of Decoupled Systems …

271

The same reasoning as in Sect. 3.6 ((3.168)–(3.171)) now leads to  2 J ¼ kE k22 ¼ kvec E k22 ¼ Jopt þ df  ;

ð4:174Þ

where Jopt

 2 1 ¼ fU /1 g 2 þ 2pj

Zj1 vec ðWG11 Gw Þ ðI  UU Þvec ðWG11 Gw Þds: j1

ð4:175Þ Therefore, minimization of the cost functional J can be achieved by taking df ¼ 0 and the cost functional associated with the optimal solution qopt ¼ H1 ðfU /1 g1 þ fU /1 g þ Þ is given by Jopt (see Remark 4.6 for another expression for the integral term in (4.175)). When Assumption 4.13 holds, fU vec ðWG11 Gw Þg1 ¼ 0: It then follows from (4.129) and (4.132) that

fU / 1 g1 ¼ U / g 1 ; where /g ¼ vec ðWG12 Cy Vg1 ¼

0 fNg2



W



    Pa 1 0 Ng2 þ W Pa Dh Dag Td P1 g Ng1 Þ I Na1

1 ðWG12 A1 Þgvec ðA1 1 Cy Vg

ð4:176Þ

ð4:177Þ

  0 0 Þ þ fðDug P1 g Ng1 Þ I

 Dh gvecdðDag Td D1 ug Þ: Na1 P1 a Dh

ð4:178Þ

It is next established that the controller C associated with the qL in (4.171) is strictly proper when an order assumption is imposed on the model G. It has been shown that the stable vector qL in (4.171) yields finite cost J ¼ kWGd Gw k2 . Thus, WGd Gw ¼ WG11 Gw þ WG12 CðI  G22 CÞ1 G21 Gw ¼ WG11 Gw þ WG12 RG21 Gw ð4:179Þ is strictly proper when qL in (4.171) is used. Since WG11 Gw is strictly proper, WGd Gw is strictly proper iff WG12 RG21 Gw is also. The implication of this requirement on the controller transfer matrix C is explored next under Assumption 4.14 P22 ¼ G22 is proper. The matrices WG12 and G21 Gw behave as WG12 ! E12 sk and G21 Gw ! E21 sl as s ! 1 with the condition that the constant matrices E12 and E21 have full column rank and full row rank, respectively, and k þ l  0: (In most applications, k ¼ l ¼ 0:)

272

4

H2 Design of Multivariable Control …

Since WG12 RG21 Gw is strictly proper, it follows from Assumption 4.14 that lim WG12 RG21 Gw ¼ lim ðWG12 sk Þðsk Rsl ÞðG21 Gw sl Þ ¼ E12 lim ðsk Rsl ÞE21 ¼ 0:

s!1

s!1

s!1

This implies that R  Oðs1 Þ since E12 and E21 have full column and full row rank, respectively, and k þ l  0: Since R  Oðs1 Þ and P22 ¼ G22  Oðs0 Þ; the controller C ¼ ðI þ RP22 Þ1 R is well defined and is strictly proper. In addition, since S  RP22 ¼ I with S ¼ ðI  CP22 Þ1 ¼ A1 ðX1  KBÞ; it follows that S ! I; s ! 1. Hence, for any K meeting this requirement one gets detS ¼ ðdetA1 Þ detðX1  KBÞ ! 1; s ! 1 and detðX1  KBÞ 6 0: It is now possible to state without the need of further proof Theorem 4.6 When Assumptions 4.1, 4.2, 4.3, 4.4′, and 4.5 through4.14 hold and when the existence condition for a decoupling controller in Theorem 4.4 is met, the T and R matrices associated with the decoupling acceptable controllers that yield finite cost J are given as in (4.110) and in (4.52), respectively, with qL ¼ H1 ðfU /g g1 þ fU /1 g þ  df Þ;

ð4:180Þ

where df is an arbitrary stable strictly proper vector. The optimal qL that minimizes the cost functional is given by

qopt ¼ H1 ð U /g 1 þ fU /1 g þ Þ:

ð4:181Þ

The cost functional J yielded by qL in (4.180) is given by  2 J ¼ Jopt þ df  ;

ð4:182Þ

where Jopt in (4.175) is the cost functional associated with the optimal solution qopt . The associated controller CðsÞ with qL in (4.180) is strictly proper.

4.4.3

H2 Design of Decoupled Systems for Strictly Acceptable Inputs

A matrix Gw ¼ A1 w Bw , ðAw ; Bw Þ being a coprime polynomial pair, is called strictly acceptable for the plant G in (4.28) if there exists a stabilizing controller C that 1 1 makes Gd A1 w , Gv Aw , and Gm Aw stable, where Gd , Gv , and Gm are the transfer 0 0 0 matrices from e ¼ ½r eo  to yd , v and ym , respectively (see Definition 3.3). Note that a stable Gw is always strictly acceptable. A stabilizing controller C for which 1 1 Gd A1 w , Gv Aw , and Gm Aw are stable is called strictly acceptable for the plant G. It can be shown that Gw is strictly acceptable for a compatible G iff AG21 A1 w and ðG11 þ G12 YAG21 ÞA1 are stable (see Example 3.21(a)) and the class of all strictly w

4.4 H2 Design of Decoupled Systems …

273

acceptable controllers is given by the formula in (4.8) with K an arbitrary real rational stable matrix such that detðX  B1 KÞ  detðX1  KBÞ 6 0. That is, the class of all strictly acceptable controllers is equivalent to the class of all stabilizing controllers (this follows indirectly from the result in Remark 3.5 or directly from the formula for Tw in Solution 3.21(a)). In other words, when Gw is strictly acceptable 1 1 and C is any stabilizing controller, the three matrices Gd A1 w , Gv Aw , and Gm Aw are stable. It is important to recognize that the R matrices associated with decoupling controllers that yield finite cost J for a given strictly acceptable Gw are given as in (4.18) of Theorem 4.1 with the diagonal T in (4.108) of Theorem 4.3. In this section, minimization of a given cost functional over the diagonal realizable T in Theorem 4.3 is treated when decoupling controllers exist. The cost functional to be minimized is J ¼ kWGd Gw k22 , where W is a stable weighting matrix and Gw is a strictly acceptable matrix. The transfer matrix Gd is given by Gd ¼ G11 þ G12 CðI  G22 CÞ1 G21 ¼ G11 þ G12 RG21 :

ð4:183Þ

Employing the formula in (4.18) for R, one gets WGd Gw ¼ WG11 Gw þ WG12 ½ Rro

Rvo  Vb AG21 Gw :

ð4:184Þ

The steps needed to find H2 solutions here are the same as the ones in the previous section with minor modifications and the approach is simply outlined as follows. First, the compatible partition of the stable matrix  Vb AG21 Gw ¼ Vb A½ P20

P21  Gw ¼

Nb1 Nb2

 ð4:185Þ

is introduced. Clearly, (4.184) and (4.185) follow from (4.119) and (4.120) with the substitutions Vg Ag ! Vb , Ng1 ! Nb1 , Ng2 ! Nb2 , Rr ! Rro , and Rv ! Rvo . The latter two are obtained with the substitutions Pg ! Pb and Cy Vg1 ! YVb1 . Moreover, the expression for T needed here is given by (4.108) which is obtained from (4.110) with the substitutions Dag ! Da and Dug ! Du . Hence, in place of (4.129) one gets Uo1 ¼

WG11 Gw þ WG12 YVb1

    0 Pa 1 N þW Pa Dh Da Td P1 b Nb1 I b2 Na1

ð4:186Þ

and it is convenient to introduce /o1 ¼ vec Uo1 ; and

ð4:187Þ

274

4

/ob ¼

vec ðWG12 YVb1

H2 Design of Multivariable Control …

    0 Pa 1 N þW Pa Dh Da Td P1 b Nb1 Þ: I b2 Na1

ð4:188Þ

Moreover, in place of (4.134) one has

)

(

(

) ð4:189Þ

where all variables are defined as in Sects. 4.2 and 4.3. So for the case of interest here it follows that in place of (4.133) one has vec E ¼ /o1 þ Po qL . Also, analogous to (4.149) and (4.150), the Wiener–Hopf spectral factor Ho of the equation Po Po ¼ Ho Ho

ð4:190Þ

Uo ¼ Po H1 o

ð4:191Þ

is introduced and

is defined. Finally, it is necessary that Assumptions 4.8 and 4.12, respectively, be replaced by the following two: Assumption 4.8′ AG21 Gw ¼ A½ P20 boundary.

P21  Gw has full row rank on the stability

and has row rank (i.e., rank of 1) on the Assumption 4.12′ Each row of Du P1 b stability boundary (or, each row of Du P1 does not vanish on the stability b boundary). It now follows in straightforward fashion with the assumptions in Theorems 4.1 and 4.3 and those introduced in Sect. 4.4 that the theorems corresponding to Theorems 4.5 and 4.6 for the case in which Gw is strictly acceptable rather than acceptable are the two that follow: Theorem 4.7 (Digital case) When Assumptions 4.1 through 4.4, 4.6, 4.7, 4.8′, 4.9, 4.10, 4.11, and 4.12′ hold and when the existence condition for a decoupling controller in Theorem 4.3 is met, and the T and R matrices associated with decoupling controllers that yield finite cost J for a given strictly acceptable Gw are given as in (4.108) and in (4.18), respectively, with qL ¼ H1 o ðfUo /o1 g1 þ fUo /o1 g þ  df Þ;

ð4:192Þ

where df is an arbitrary stable rational vector. The optimal qL minimizing the cost functional is given by

4.4 H2 Design of Decoupled Systems …

275

qopt ¼ H1 o ðfUo /o1 g1 þ fUo /o1 g þ Þ:

ð4:193Þ

When Jopt denotes the cost functional associated with the optimal solution qopt , the cost functional J yielded by qL in (4.192) is given by  2  2 1 ð dk ; J ¼ Jopt þ df 2 ; Jopt ¼ fUo /o1 g 2 þ H

qo 2p j k

ð4:194Þ

jkj¼1

where qo ¼ /o1 ðI  Uo Uo Þ/o1 ¼ ðvec ðWG11 Gw ÞÞ ðI  Uo Uo Þvec ðWG11 Gw Þ: ð4:195Þ Theorem 4.8 (Analog case) When Assumptions 4.1 through 4.4, 4.6, 4.7, 4.8′, 4.9, 4.10, 4.11, 4.12′, 4.13, and 4.14 hold and when the existence condition for a decoupling controller in Theorem 4.3 is met, the T and R matrices associated with decoupling controllers that yield finite cost J for a given strictly acceptable Gw are given as in (4.108) and in (4.18), respectively, with qL ¼ H1 o ðfUo /ob g1 þ fUo /o1 g þ  df Þ;

ð4:196Þ

where df is an arbitrary stable strictly proper rational vector. The optimal qL that minimizes the cost functional is given by qopt ¼ H1 o ðfUo /ob g1 þ fUo /o1 g þ Þ:

ð4:197Þ

When Jopt denotes the cost functional associated with the optimal solution qopt , the cost functional J yielded by qL in (4.196) is given by  2 J ¼ Jopt þ df  ;

ð4:198Þ

where Jopt

 2 1 ¼ fUo /o1 g 2 þ 2pj

Zj1 vec ðWG11 Gw Þ ðI  Uo Uo Þvec ðWG11 Gw Þds: j1

ð4:199Þ The controller C associated with qL in (4.196) is strictly proper. In the next two sections, the solution formulas for the H2 design of decoupled systems with 1DOF and 3DOF controller configurations are obtained from the results derived in this section by simple substitutions. This shows the merit of

276

4

H2 Design of Multivariable Control …

developing a theory for a generalized model when several configurations, which are special cases of this model are of interest.

4.5

H2 Design of 1DOF Decoupled Systems

In this section, H2 design of decoupled systems for the 1DOF system in Fig. 3.3 is treated for strictly acceptable inputs (it is explained later in Remark 4.8 that the acceptable inputs Gw considered in Sect. 3.7 for H2 design of 1DOF systems are actually strictly acceptable when F in Fig. 3.3 is stable). For the 1DOF system in Fig. 3.3, the design objective is to find stabilizing controllers for which the transfer matrix from r to y is diagonal and the H2 cost functional penalized on the error e ¼ Td r  y and the control variable v is finite. The H2 solution for 1DOF decoupled systems is obtained by using the formulas derived in Sect. 4.4 for the standard model in Fig. 4.1. Specifically, the results in Theorems 4.7 and 4.8 are e in (4.1) and G in (4.28) for the 1DOF used. For this purpose, the plant model P system in Fig. 3.3 are obtained first. When r ¼ r; eo ¼ ½ d 0 n0 0 , v ¼ v; y ¼ y; yz ¼ v; and ym ¼ u; where the left-hand side variables are the ones in Fig. 4.1 and the right-hand side variables are the ones in Fig. 3.3, it then follows that

ð4:200Þ

The regulated variable yd and the exogenous input e for H2 design are set as yd ¼ ½e0 v0 0 and e ¼ ½r 0 e0o 0 : One then gets from (4.29) that the plant G in (4.28) is given by 

Td ¼ 0

Pd 0

   0 P ; G12 ¼ ; 0 I

ð4:201Þ

G21 ¼ ½ I

Po

I ; G22 ¼ FP:

ð4:202Þ

G11

The exogenous input e ¼ ½r 0 e0o 0 is modeled by 

r e¼ eo



2 3 2 3 2 3 r2 r r1 ¼ 4 d 5 ¼ 4 d1 5 þ 4 d2 5 ¼ e 1 þ e 2 ; 0 n n

ð4:203Þ

4.5 H2 Design of 1DOF Decoupled Systems

277

where r1 and d1 are the shape-deterministic components of r and d and where r2 , d2 , and n are purely stochastic. Consider the fractional descriptions for the 1 shape-deterministic signals r1 ¼ A1 r Br rr , d1 ¼ Ad Bd rd , where Ar ; Br and Ad ; Bd are left coprime polynomial matrix pairs. Then e w1 e ¼ e1 þ e2 ¼ ½ G

    r Ge e e e w2 l; e ¼ G w1 r þ G w2 l ¼ rþG G w2  l 0 r0d 0 ; l ¼ ½ l0r l0d l0n 0 ;

r ¼ ½ r0r Ge ¼

A1 e Be ;



Ar Ae ¼ 0

ð4:204Þ ð4:205Þ

  0 Br ; Be ¼ Ad 0

 0 ; Bd

ð4:206Þ

and e w2 ¼ Xs ; Xs ¼ diagfXr G

Xd

Xn g;

ð4:207Þ

where r ¼ ½ r0r r0d 0 is a vector whose elements are random variables satisfying hrr0 i ¼ I; the elements of l are in the temporal domain zero-mean independent white-noise processes with unit variance; and Xr , Xd , and Xn are stable left Wiener–Hopf factors associated with the power spectral density matrices for the stochastic signals r2 , d2 , and n, respectively. Clearly, Ae ; Be is a left coprime polynomial matrix pair and the matrices G111 and G211 in (4.32) are given by  G111 ¼

 Pd ; G211 ¼ ½ I 0

Td 0

Po :

ð4:208Þ

Finally, the coprime polynomial matrix fraction description for FP used in the 1DOF system of Sect. 3.7 is used again. Specifically, G22 ¼ FP ¼ A1 B ¼ B1 A1 1

ð4:209Þ

and 

A Y1

B X1



X Y

B1 A1





X ¼ Y

B1 A1



A Y1

B X1





I ¼ 0

 0 ; I

ð4:210Þ

where all the submatrices in (4.210) are polynomial matrices. The cost functional to be minimized is J ¼ kWGd Gw k22 ; where Gd is the transfer pffiffiffiffiffi e pffiffiffiffiffi e matrix from e to yd , W is a stable weighting matrix, and Gw ¼ ½ a1 G a2 G w2  w1 with a1 [ 0 and a2  0: The solution formula for the optimization problem is contained in Theorems 4.7 and 4.8 under Assumptions 4.1 through 4.4,

278

4

H2 Design of Multivariable Control …

Assumptions 4.6, 4.7, 4.8′, 4.9, 4.10, 4.11, 4.12′, 4.13, and 4.14 for the standard model. The corresponding assumptions for the 1DOF system in Fig. 3.3 are now enumerated. In place of Assumption 4.1 one naturally gets Assumption 4.15 The 1DOF system in Fig. 3.3 is admissible or, equivalently, (1) The subsystems L, ½ Pd P ; and F have no unstable hidden poles (i.e., ho in (3.241) is a stable polynomial), (2) w½ Pd P  =wP is a stable polynomial (i.e., h1 in (3.242) is a stable polynomial), (3) L is a stable matrix, and (4) wF wP =wFP is a stable polynomial. When the 1DOF system in Fig. 3.3 is admissible, it easily follows from Lemma 3.3 that APo , PA1 , and Pd  PA1 Y1 Po are stable. It also follows from Lemma 3.4 that G ¼ ½Gij  in (4.201) and (4.202) is compatible when the 1DOF system is admissible and Td is stable (the stability of Td is assumed later). e ¼ F and It is clear from (4.200) that P02 ¼ P and P22 ¼ FP; hence, F Assumption 4.2 is equivalent Assumption 4.16 F is stable. It also follows from (4.200) that P20 ¼ I; thus, Assumption 4.3 is equivalent to Assumption 4.17 The plant P has full row rank. When P has full row rank, there exists as in Sect. 4.2 a unimodular matrix Va such that PVa ¼ ½ Pa

0 :

ð4:211Þ

The conditions which assure that Assumption 4.4 is met are developed next. Since AP20 ¼ A; it follows from (4.12) that Vb ¼ I and Pb ¼ A: Hence, Assumption 4.4 is satisfied when Td is stable and there exists a stable matrix Qo such that PYA ¼ Td þ Qo A: The latter condition is satisfied iff ðTd  PYAÞA1 is stable. Since ðTd  PYAÞ ¼ ðTd  PA1 Y1 Þ is stable, this is the case iff ðTd  PYAÞA1 B is stable. Now ðTd  PYAÞA1 B ¼ ðTd  PYAÞFP ¼ ¼ ðTd F  IÞP þ PðI  YBÞ ¼ ðTd F  IÞP þ PA1 X1 and the matrix ðTd  PYAÞA1 B is stable iff ðTd F  IÞP is stable. Thus, Assumption 4.4 holds when the following assumption is made. Assumption 4.18 Td is stable and ðTd F  IÞP is stable. Assumption 4.5 is trivially satisfied and Assumption 4.6 is retained as Assumption 4.19 Td is diagonal and Td1 is stable. Now in order to proceed with the H2 decoupling design for strictly acceptable inputs, a decoupling controller for the 1DOF system is assumed to exist. In this regard, it is convenient to introduce the set fhi g of unique monic polynomials of minimal degree such that the f i-th column of P1 a g  hi is stable for all columns

4.5 H2 Design of 1DOF Decoupled Systems

279

of P1 a and the set fui g of unique monic polynomials of minimal degree such that 1 ui  fi-th row of A1 g is stable for all rows of P1 b ¼ A . It is also convenient to introduce Dh ¼ diagfhi g, Du ¼ diagfui g, Da ¼ diagf^ ai g, and Db ¼ diagfbi g; where ^ai and bi are unique polynomials of minimal degrees satisfying ^ ai hi þ bi ui ¼ 1 when hi ; ui are coprime. Then (see Theorem 4.3), the existence of a decoupling controller requires Assumption 4.20 (1) hi and ui are coprime (equivalently, no column of P1 a and the corresponding number row in A1 possess coincident poles in the unstable 1 is stable. region) and (2) the data construct R ¼ P1 a Dh Da Td A The satisfaction of Assumption 4.7 is guaranteed by Assumption 4.21 The weighting matrix W is of the form W ¼ diagfWe ; Wv g. The matrices We and Wv are stable and have full column rank on the stability boundary. Assumption 4.8′ is changed to the following one since a1 [ 0 and pffiffiffiffiffi G21 Gw ¼ ½ a1 A1 r Br



pffiffiffiffiffi a1 Po A1 d Bd

pffiffiffiffiffi pffiffiffiffiffi a2 X r  a2 P o X d

pffiffiffiffiffi  a2 Xn : ð4:212Þ

pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi Assumption 4.22 A ½ A1 a2 Xr a2 Po Xd a2 Xn  has full Po A1 r Br d Bd row rank on the stability boundary. Assumption 4.9 is trivially met since P12 ¼ I: Assumptions 4.10, 4.11, and 4.12′ are, respectively, simply replaced with the following three assumptions:   P Assumption 4.23 A1 has full column rank on the stability boundary. I   Dh Assumption 4.24 has full column rank on the stability boundary. P1 a Dh Assumption 4.25 No row of Du A1 vanishes on the stability boundary. Since  pffiffiffiffiffi a1 We Td A1 r Br WG11 Gw ¼ 0

pffiffiffiffiffi  a1 We Pd A1 d Bd 0

pffiffiffiffiffi a2 We Td Xr 0

pffiffiffiffiffi  a 2 We P d X d 0

 0 ; 0

ð4:213Þ Assumption 4.13 is equivalent to 1 Assumption 4.26 The matrices We ½ Td A1 r Br Pd Ad Bd  and Pd Xd  are strictly proper.

Assumption 4.14 is simply changed to

pffiffiffiffiffi a2 We ½ Td Xr

280

4

H2 Design of Multivariable Control …

Assumption  4.27 FP is proper. In addition, as s ! 1, the following behavior pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi We P 1 holds: ! E12 sk and ½ A1 a2 X r a2 P o X d a2 X n  ! r Br Po Ad Bd Wv E21 sl ; where E12 and E21 have full column rank and full row rank, respectively, and k þ l  0: Having established assumptions for the 1DOF case which assure that the assumptions made in Theorems 4.7 and 4.8 are met, one can invoke these theorems to find the solution for the 1DOF case. In particular, the formula qL ¼ H1 o ðfUo /o1 g1 þ fUo /o1 g þ  df Þ can be used. Calculation of Po Po is done first to obtain Ho in (4.190). Since P20 ¼ I is a square matrix, the matrix R is given by (4.21) in Remark 4.1. Thus, ½ Rro Rvo  in (4.184) is replaced with Rro and there is no need to introduce the partition (4.185) for the 1DOF case. Instead, one can use Vb AG21 Gw ¼ A½ I Po I Gw ¼ Nb1 . That is, for the 1DOF case one can set Nb2 ¼ 0 in (4.185) and the equations following it. In particular, the third block 0 column of Po in (4.189) disappears and accordingly, qL ¼ ½ e d 0 m021  : Moreover, Nb1 ¼ AG21 Gw pffiffiffiffiffi pffiffiffiffiffi 1 ¼ A ½ a1 A1 r Br  a1 Po Ad Bd

pffiffiffiffiffi a2 X r



pffiffiffiffiffi a2 Po Xd

pffiffiffiffiffi  a2 Xn : ð4:214Þ

When Va is partitioned as Va ¼ ½ Va1 Va2  corresponding to the column sizes of the partition ½Pa 0 ; the partition matrices Na1 and Na2 in (4.125) are given by the pair Na1 ¼ Va1 ; Na2 ¼ Va2 :

ð4:215Þ

A number of helpful matrix formulas for the calculation of Po Po are now presented. (For easy reference the definition of the Schur product stated in Sect. 1.4 is repeated here: the Schur product of two equi-size matrices G and R is denoted by G R and is the matrix whose i-row, j-column entry is gijrij). It is not difficult to confirm that 

ðABÞ ¼ B A ; ½A

 A B  ¼ ; ðA  BÞ ¼ A  B ; B

ðA1 Þ ¼ ðA Þ1 ; ðA BÞ ¼ ðA0 B0 Þ0 ;

ð4:216Þ ð4:217Þ

ðF  GÞðL MÞ ¼ ðFLÞ ðGMÞ; ðF GÞ0 ðL MÞ ¼ ðF 0 LÞ ðG0 MÞ;

ð4:218Þ

ðF GÞ ðF GÞ ¼ ðF FÞ ðG GÞ; ðF GÞ ¼ F G :

ð4:219Þ

A straightforward calculation now yields

4.5 H2 Design of 1DOF Decoupled Systems

Po Po "

W ðDu Grd A Þ0 ðVa2 Wv Wv Va1 P1 a Dh Þ

¼

281

# ; fðDu Grd A Þ0 ðVa2 Wv Wv Va1 P1 a Dh Þg ðAGrd A Þ0  ðVa2 Wv Wv Va2 Þ

ð4:220Þ where 1 W ¼ ðDu Grd Du Þ0 fDh ðWe We þ P1 a Va1 Wv Wv Va1 Pa ÞDh g;

ð4:221aÞ

and 1 1 1 Grd ¼ a1 ðA1 r Br Br Ar þ Po Ad Bd Bd Ad Po Þ þ a2 ðXr Xr þ Po Xd Xd Po þ Xn Xn Þ:

ð4:221bÞ The Wiener–Hopf factor Ho in (4.190) is usually calculated numerically after the values of the variables are inserted into the expression for Po Po given by (4.220). 1 Formulas for Uo /o1 ¼ H1 o Po /o1 and Uo /ob ¼ Ho Po /ob in the 1DOF case are easy (although tedious) to derive beginning with (4.186)–(4.188). The vectors /o1 ¼ vec Uo1 and /ob become /o1 ¼ vec ðWG11 Gw Þ þ /ob

ð4:222Þ

and  /ob ¼

  We ðDu G21 Gw Þ ð Dh vec dðDa Td D1 u Þ; Wv Va1 P1 a 0



ð4:223Þ

where WG11 Gw and G21 Gw are given in (4.213) and (4.212), respectively. Straightforward calculations then yield Uo /o1 ¼ H1 o



W vecdðDa Td D1 u Þ  ðDu Dh Þ vec ðWe We Gtp Þ vec ðVa2 Wv Wv Va1 P1 a Dh Da Td Grd A Þ

 ð4:224Þ

with 1 1 1 Gtp ¼ a1 ðTd A1 r Br Br Ar þ Pd Ad Bd Bd Ad Po Þ þ a2 ðTd Xr Xr þ Pd Xd Xd Po Þ:

ð4:225Þ When Assumption 4.26 is satisfied, Uo vec ðWG11 Gw Þ is strictly proper and in this case, fUo /o1 g1 ¼ fUo /ob g1 and Uo /ob is given by

282

4

Uo /ob ¼

H1 o



H2 Design of Multivariable Control …

 W vecdðDa Td D1 u Þ : vec ðVa2 Wv Wv Va1 P1 a Dh Da Td Grd A Þ

ð4:226Þ

The preparation needed for the application of Theorems 4.7 and 4.8 to the 1DOF system in Fig. 3.3 is now complete. Before stating the resulting theorems, it is convenient to identify the strictly acceptable inputs Gw (or, equivalently, Ge ) for the 1DOF system in Fig. 3.3. The strictly acceptable inputs in the standard model are ones for which AG21 Gw and ðG11 þ G12 YAG21 ÞGw (equivalently, AG211 Ge and ðG111 þ G12 YAG211 ÞGe ) are stable. When F is stable, it is shown later in Remark 4.7 that these conditions are met and a persistent input Ge is strictly acceptable for the 1DOF system iff 1 1 1 Ap Pd A1 d , LAd , Ap Td Ar , and ðI  FTd ÞAr are stable, where Ap is the denominator matrix of the coprime polynomial matrix fraction description P ¼ A1 p Bp :

ð4:227Þ

These conditions are the same as the ones in Sect. 3.7 (see the conditions in Lemma 3.4). Theorem 4.9 (Digital case) When Assumptions 4.15 through 4.25 hold and 1 1 1 Ap Pd A1 d , LAd , Ap Td Ar , and ðI  FTd ÞAr are stable, decoupling controllers for the 1DOF system in Fig. 3.3 exist. Moreover, the set of all loop transfer matrices R ¼ CðI  P22 CÞ1 associated with the ones for which the cost J ¼ kWGd Gw k22 is finite is given by  R ¼ Va

  1  P1 Pa Dh ðDa Td þ DDu Þ a T ¼ V ; a M21 A M21 A

ð4:228Þ

e d ¼ vecdD; m21 ¼ vec M21 ;

ð4:229Þ

and 

e d m21



¼ H1 o ðfUo /o1 g þ þ fUo /o1 g1  df Þ;

ð4:230Þ

where df is an arbitrary stable vector and Uo /o1 is obtained from (4.224). All finite values of the cost functional J associated with any stable choice for df in (4.230) are given by  2  2 1 ð dk ; J ¼ Jopt þ df 2 ; Jopt ¼ fUo /o1 g 2 þ H

qo 2p j k jkj¼1

ð4:231Þ

4.5 H2 Design of 1DOF Decoupled Systems

283

where WG11 Gw is given by (4.213) and qo ¼ ðvec ðWG11 Gw ÞÞ ðI  Uo Uo Þvec ðWG11 Gw Þ:

ð4:232Þ

The choice for df which yields the minimum value of J is called the optimal choice and is given by df ¼ 0: Theorem 4.10 (Analog case) When Assumptions 4.15 through 4.27 hold and 1 1 1 Ap Pd A1 d , LAd , Ap Td Ar , and ðI  FTd ÞAr are stable, decoupling controllers for the 1DOF system in Fig. 3.3 exist. Moreover, the set of all loop transfer matrices R ¼ CðI  P22 CÞ1 associated with the ones for which the cost J ¼ kWGd Gw k22 is finite is given by  R ¼ Va

  1  P1 Pa Dh ðDa Td þ DDu Þ a T ¼ V ; a M21 A M21 A

ð4:233Þ

e d ¼ vecdD; m21 ¼ vec M21 ;

ð4:234Þ

and 

e d m21



¼ H1 o ðfUo /ob g1 þ fUo /o1 g þ  df Þ;

ð4:235Þ

where df is an arbitrary stable strictly proper vector and Uo /o1 and Uo /ob are given by (4.224) and (4.226), respectively. All finite values of the cost functional J associated with any stable strictly proper choice for df in (4.230) are given by  2 J ¼ Jopt þ df 2 ;

ð4:236Þ

where

Jopt

 2 1 ¼ fUo /o1 g 2 þ 2p j

Zj1 vec ðWG11 Gw Þ ðI  Uo Uo Þvec ðWG11 Gw Þds j1

ð4:237Þ and WG11 Gw is given by (4.213). The choice for df which yields the minimum value of J is called the optimal choice and is given by df ¼ 0: The controller C associated with R in (4.233) is strictly proper.

284

4

H2 Design of Multivariable Control …

Remark 4.7 It remains to establish for the 1DOF system that when F is stable the 1 1 strictly acceptable Ge are those for which Ap Pd A1 d , LAd , Ap Td Ar , and ðI  FTd ÞA1 are stable. In order to identify strictly acceptable inputs, coprime pairs r 1 Ao ; Bo and Ao1 ; Bo1 for G12 ¼ ½ P0  I 0 ¼ A1 o Bo ¼ Bo1 Ao1 in (4.30) are obtained first. The coprime polynomial matrix fraction description for P given by (3.60) and (3.61) is used again for the 1DOF system: 1 P ¼ A1 p Bp ¼ Bp1 Ap1

ð4:238Þ

with 

Ap Yp1

Bp Xp1



Xp Yp

Bp1 Ap1



 ¼

Xp Yp

Bp1 Ap1



Ap Yp1

Bp Xp1



 ¼

 I 0 : ð4:239Þ 0 I

Then the desired coprime polynomial pairs Ao ; Bo and Ao1 ; Bo1 and associated polynomial matrices Xo , Yo , Xo1 , and Yo1 satisfying Ao Xo þ Bo Yo ¼ I; Xo1 Ao1 þ Yo1 Bo1 ¼ I; and Yo1 Xo ¼ Xo1 Yo are given by 

Ap Ao ¼ 0

      Xp 0 Bp Bp Bo ¼ ; Bo ¼ ; Xo ¼ Yp I I I

 0 ; Yo ¼ ½ Yp 0  I ð4:240Þ

and  Ao1 ¼ Ap1 ; Bo1 ¼

 Bp1 ; Xo1 ¼ Xp1 ; Yo1 ¼ ½ Yp1 0 : Ap1

ð4:241Þ

As explained in the statement after Eq. (4.31), A1 1 Ao1 is stable since F is stable. In Example 3.21(c), it is proven for the system in (4.28) that when A1 1 Ao1 is stable 1 Gw ¼ Aw Bw , ðAw ; Bw Þ being a coprime polynomial pair, is strictly acceptable iff e w ¼ Ao G111 A1 ) Cw ¼ Ao G11 A1 C and Cv ¼ w (equivalently e 1 1 e ðG21 þ G22 Yo Ao G11 ÞAw (equivalently C v ¼ ðG211 þ G22 Yo Ao G111 ÞAe ) are stable.   e w is e w ¼ Ao G111 A1 ¼ Ap Td A1 Ap Pd A1 ; hence C In the 1DOF system, C e r d 1 1 stable iff Ap Pd Ad and Ap Td Ar are stable. Next, e v ¼ ðG211 þ G22 Yo Ao G111 ÞA1 C e

ð4:242Þ

ð4:243Þ

4.5 H2 Design of 1DOF Decoupled Systems

285

ð4:244Þ ð4:245Þ 1 1 1 is stable iff ðI  FTd ÞA1 r and LAd are stable since F, Ap Pd Ad , and Ap Td Ar are stable. In summary, when F is stable a persistent input Ge in the 1DOF system is 1 1 1 strictly acceptable iff Ap Pd A1 d , LAd , Ap Td Ar , and ðI  FTd ÞAr are stable.

Remark 4.8 The persistent input Gw considered in Sect. 3.7 for the H2 design of the 1DOF system shown in Fig. 3.3 is acceptable for all stabilizing controllers. It is proven in Lemma 3.4 that Gw is acceptable for all stabilizing controllers, provided e w2 are stable, iff Ap Pd A1 , LA1 , Ap Td A1 , and ðI  FTd ÞA1 are F, Td , and G d d r r stable. As can be seen in Remark 4.7, this condition is the same as the one for strictly acceptable inputs. In fact, it is proven in Example 3.21(b) for the system in Fig. 3.2 that when A1 1 Ao1 is stable an acceptable Gw is strictly acceptable iff Cg ¼ AG21 Gw is stable. Since the Gw considered in Sect. 3.7 satisfies this condition, it is strictly acceptable when F is stable. Remark 4.9 It is important to note that for the 1DOF system explicit necessary and sufficient conditions for Gw to be strictly acceptable are easily derived as in Remark 4.7. This is not the case, however, for acceptable Gw . For the standard model shown in Fig. 3.2, all the acceptable inputs Gw are ones for which Cw ¼ 1 1 Ao G11 A1 w and Co ¼ Ao1 ðYo Cw Ag1  YBg1 Þ (or Cz ¼ ðG11 þ G12 A1 Y1 G21 ÞAw Ag1 , 1 1 see Example 3.3) are stable, where AG21 A w ¼ Bg1 Ag1 is a coprime polynomial matrix fraction description. In the case of the 1DOF system shown in Fig. 3.3, this condition is difficult to express explicitly. It is, however, possible to find a sufficient condition for an acceptable Gw which has a rather simple form (see Example 4.5 (a)). In Example 2.12(c), it is shown that there can exist an acceptable Gw that is not strictly acceptable for a specific 1DOF system. The following is another illustration of this phenomenon in connection with decoupling design. Consider the system in Fig. 3.3 with  P¼

1 s sþ1 s1

sþ2 s 3 s1

  0 ; F ¼ Td ¼ Pd ¼ I2 ; L ¼ 0

 1 ; 0

ð4:246Þ

286

4

H2 Design of Multivariable Control …

and

ð4:247Þ

1 Since LA1 d (equivalently, LAd Bd ) is not stable, the matrix Ge is not strictly acceptable but it can be shown that Ge is acceptable (see Example 4.5(b)). In this case, the formula in Theorem 4.10 cannot be used; however, the general formula in Theorem 4.6 can be used to obtain the H2 optimal decoupling solution.

4.6

H2 Design of 3DOF Decoupled Systems

In this section, H2 design of decoupled systems for the 3DOF system in Fig. 1.1 is treated for acceptable inputs. For the 3DOF system in Fig. 1.1, the design objective is to find stabilizing controllers for which the transfer matrix from r to y is diagonal and the H2 cost functional penalized on the error e ¼ Td r  y and the control variable v is finite. The H2 solution for 3DOF decoupled systems is obtained by using the formulas derived in Sect. 4.4 for the standard model in Fig. 4.1. Specifically, the results in Theorems 4.5 and 4.6 are used. For this purpose, the plant e in (4.1) and G in (4.28) for the 3DOF system in Fig. 1.1 are obtained first. model P When r ¼ r; eo ¼ ½ d 0 n0r n0m n0l 0 ; v ¼ v; y ¼ y; yz ¼ v; and ym ¼ ½ w0 z0 u0 0 ; where the left-hand side variables are the ones in Fig. 4.1 and the right-hand side variables are the ones in Fig. 1.1 (for consistency with the notation in Sect. 3.8, nr is used here for the noise n in Fig. 1.1), then

ð4:248Þ

The regulated variable yd and the exogenous input e for H2 design are set as yd ¼ ½ e0 v0 0 and e ¼ ½r 0 e0o 0 : One then gets from (4.29) that the plant G in (4.28) is given by

4.6 H2 Design of 3DOF Decoupled Systems



G11

2

G21

Pd 0

Td ¼ 0 0 ¼ 40 I

287

0 0 0 0 I

FPd L 0

   0 P ; G12 ¼ ; 0 I

ð4:249Þ

3 2 3 I 0 FP 0 I 5; G22 ¼ 4 0 5: 0 0 0

ð4:250Þ

0 0

As in Sect. 3.8, the model for the exogenous input e ¼ ½r 0 e0o 0 is 3 2 3 2 3 r1 r2 r   6 d 7 6 d1 7 6 d2 7 6 7 6 7 6 7 r 7 6 7 6 7 e¼ ¼6 6 nr 7 ¼ 6 0 7 þ 6 nr 7 ¼ e 1 þ e 2 ; eo 4 nm 5 4 0 5 4 nm 5 0 nl nl 2

ð4:251Þ

where r1 and d1 are the shape-deterministic components of r and d and where r2 , d2 , nr , nm , and nl are purely stochastic. The fractional descriptions for the 1 shape-deterministic signals are r1 ¼ A1 r Br rr ; d1 ¼ Ad Bd rd ; where Ar ; Br and Ad ; Bd are left coprime polynomial matrix pairs. Then e w1 e ¼ e1 þ e2 ¼ ½ G r ¼ ½ r0r

    r Ge e e e w2 l; e  ¼ G r þ G l ¼ rþG G w2 w1 w2 l 0 r0d 0 ; l ¼ ½ l0r

Ge ¼ A1 e Be ; Ae ¼



Ar 0

e w2 ¼ Xs ; Xs ¼ diag f Xr G

l0d

l0nr

l0nl 0 ;

l0nm

  Br 0 ; Be ¼ Ad 0 Xd

Xnr

Xnm

 0 ; Bd

Xnl g;

ð4:252Þ ð4:253Þ ð4:254Þ ð4:255Þ

where r ¼ ½ r0r r0d 0 is a vector whose elements are random variables satisfying ^r , l ^d , l ^ nr , l ^nm , and l ^nl are zero-mean independent hrr0 i ¼ I; where the elements l white-noise processes with unit variance; and where Xd , Xr , Xnr , Xnm , and Xnl are stable left Wiener–Hopf factors associated with the power spectral density matrices for the stochastic signals r2 , d2 , nr , nm , and nl , respectively. Clearly, Ae ; Be is a left coprime polynomial matrix pair and the matrices G111 and G211 in (4.32) are given by 

G111

Td ¼ 0

2  0 Pd ; G211 ¼ 4 0 0 I

3 FPd L 5: 0

ð4:256Þ

The coprime polynomial matrix fraction description for FP used in the 3DOF system of Sect. 3.8 is employed again here. Specifically,

288

4

H2 Design of Multivariable Control …

FP ¼ A1 B ¼ B1 A1 1

ð4:257Þ

with polynomial submatrices satisfying 

A Y1

B X1



X Y

B1 A1



 ¼

X Y

B1 A1



A Y1

B X1



 ¼

I 0

 0 : I

ð4:258Þ

e 1 B e 1 coprime polynomial matrix fraction descriptions for e¼B e1 A With A 1 e B; e 1 , and B e A e 1 are used for conG22 ¼ P22 ¼ ½ ðFPÞ0 00 00 0 (here A; sistency with the notation in Sects. 3.4 and 3.8) it follows that e B; e 1; B e 1; X e; Y e; X e 1 ; and Y e1 satisfying (4.7) are given by e A A; 2

A e ¼40 A 0

0 I 0

3 2 3 2 0 B X e ¼ 4 0 5; X e ¼40 0 5; B I 0 0

0 I 0

3 0 e ¼ ½ Y 0 5; Y I

0

0 ð4:259Þ

and 2

3 B1 e 1 ¼ A1 ; B e 1 ¼ 4 0 5; X e 1 ¼ X1 ; Y e1 ¼ ½ Y1 A 0

0

0 :

ð4:260Þ

The cost functional to be minimized is J ¼ kWGd Gw k22 , where Gd is the transfer matrix from e to yd , W is a stable weighting matrix, and pffiffiffiffiffi e pffiffiffiffiffi e G w ¼ ½ a1 G a2 G w2  with a1 [ 0 and a2  0: The solution formula for this w1 optimization problem is contained in Theorems 4.5 and 4.6 under Assumptions 4.1 through 4.14 for the standard model. The corresponding assumptions for the 3DOF system are now enumerated. Exploiting the results of Lemma 2.8 yields in place of Assumption 4.1 (here, Pa in Fig. 2.1 is assumed to be I) Assumption 4.28 The 3DOF system in Fig. 1.1 is admissible or, equivalently, (1) (2) (3) (4)

The subsystems L, ½ Pd P ; and F have no unstable hidden poles, w½ Pd P  =wP is a stable polynomial, L is a stable matrix, and wF wP =wFP is a stable polynomial.

When the 3DOF system in Fig. 1.1 is admissible, it easily follows from Lemma 2.7 that AFPd , L, PA1 , and ðI  PA1 Y1 FÞPd are all stable. It is also not difficult to verify that G ¼ ½Gij  in (4.249) and (4.250) is compatible when the 3DOF system is admissible and Td is stable (the stability of Td is assumed later).

4.6 H2 Design of 3DOF Decoupled Systems

289

e P02 ; where F e ¼ ½ F 0 0 0 0 ; hence, It is obvious from (4.248) that P22 ¼ F Assumption 4.2 is equivalent to Assumption 4.29 F is stable. It is also clear from (4.248) that P02 ¼ P and P20 ¼ ½ 0 0 I 0 ; hence, Assumption 4.3 is equivalent to Assumption 4.30 The plant P has full row rank. When P has full row rank, there exists as in Sect. 4.2 a unimodular matrix Va such that PVa ¼ ½ Pa

0 :

ð4:261Þ

It is established next that a stable Qg satisfying (4.45) in Assumption 4.4′ always exists for the 3DOF system when Td is stable. In this regard, the polynomial matrices Ao ; Bo ; Ag , and Bg are needed. First, the coprime polynomial matrix pairs Ag ; Bg and Ag1 ; Bg1 for which 2

e 211 A1 AG e

0 ¼4 0 A1 r

3 AFPd A1 d 5 ¼ A1 Bg ¼ Bg1 A1 LA1 g g1 d 0

ð4:262Þ

are determined. This can be done simply (as in Sect. 3.4) in terms of the coprime polynomial matrix fraction descriptions associated with 

AFPd A1 d LA1 d



1 ¼ A1 q Bq ¼ Bq1 Aq1 ;

ð4:263Þ

where the polynomial matrices satisfy 

Aq Yq1

Bq Xq1



Bq1 Aq1

Xq Yq





Xq ¼ Yq

Bq1 Aq1



Aq Yq1

Bq Xq1





 I 0 ¼ : ð4:264Þ 0 I

Clearly, 1    0 A1 0 Aq 0 q Bq ¼ 0 A I A1 0 r  r  1 0 0 Bq1 Ar ¼ : 0 Aq1 I 0

e 211 A1 ¼ AG e



Bq 0



It is easy to confirm that the equalities in (4.36) are satisfied with

ð4:265Þ

290

4

  0 0 ; Bg ¼ Ar I



Aq Ag ¼ 0

H2 Design of Multivariable Control …

  Ar Bq ; Ag1 ¼ 0 0

  0 0 ; Bg1 ¼ Aq1 I

Bq1 0



  I 0 ; Xg1 ¼ 0 0

  0 0 ; Yg1 ¼ Xq1 Yq1

 I : 0

ð4:266Þ

and  Xg ¼

  0 0 ; Yg ¼ Yq 0

Xq 0

ð4:267Þ

Next, coprime pairs Ao ; Bo and Ao1 ; Bo1 are derived for which 1 G12 ¼ ½ P0  I 0 ¼ A1 o Bo ¼ Bo1 Ao1 . Using the coprime polynomial matrix fraction description for P in the 3DOF system one has 1 P ¼ A1 p Bp ¼ Bp1 Ap1

ð4:268Þ

with 

Bp Xp1

Ap Yp1



Xp Yp

Bp1 Ap1





Xp ¼ Yp

Bp1 Ap1



Ap Yp1

Bp Xp1





 I 0 ¼ : ð4:269Þ 0 I

It then easily follows that  Ao ¼

Ap 0

    Xp 0 Bp ; Bo ¼ ; Xo ¼ Yp I I

 0 ; Yo ¼ ½ Yp 0  I

ð4:270Þ

and  Ao1 ¼ Ap1 ; Bo1 ¼

 Bp1 ; Xo1 ¼ Xp1 ; Yo1 ¼ ½ Yp1 0  Ap1

ð4:271Þ

satisfy (4.30) and (4.31). It is established in Sect. 3.4 that when F and Td are stable and the 3DOF system 1 1 is admissible then A1 e Be is acceptable iff Ap Pd Ad and Ap Td Ar are both stable. Also, hence, (4.44) is satisfied with 2

0 Vg ¼ 4 I 0

3 0 I 0 0 5 and Pg ¼ Ar . Clearly, I 0 Cw ¼ Ao G111 A1 e ¼



Ap Td A1 r 0

Ap Pd A1 d 0

 ð4:272Þ

4.6 H2 Design of 3DOF Decoupled Systems

291

and (4.43) yields ð4:273Þ Thus, ð4:274aÞ

Since Ap Xp þ Bp Yp ¼ I implies Xp þ PYp ¼ A1 p and Xp Ap Td þ PYp Ap Td ¼ Td , it follows that PYp Ap Td ¼ Td  Xp Ap Td ¼ Td  Xp Ap Td A1 r Ar . Hence, (4.274a) yields e 1 Ko Þ AP e 20  Td P1 ¼ ðPYp Ap Td  Td ÞA1 ¼ Xp Ap Td A1 eþA Qg ¼ ½P02 ð Y g r r ð4:274bÞ which is stable for acceptable A1 e Be , the only case of interest. That is, Assumption 4.4′ simply becomes Assumption 4.31 The matrix Td is stable. Assumption 4.5 is met since P12 ¼ I and Assumption 4.6 remains the same as Assumption 4.32 Td is diagonal and Td1 is stable. It is of interest to examine whether or not the 3DOF system has a decoupling controller before attempting an H2 design. The existence of a decoupling controller can be checked by Theorem 4.3. Since 2

it follows that

3 2 3 Pb 0 0 I e 20 ¼ 4 0 5 is satisfied with Vb ¼ 4 I 0 0 5 and Pb ¼ I: Moreover, Vb AP 0 0 I 0 e 20 ¼ 0: Therefore, (4.14) is satisfied with Qo ¼ Td , a stable matrix, and e AP P02 Y Assumption 4.4 is met. In addition, for the 3DOF system, ui ¼ 1 for each i since Pb ¼ I; hence, the pair hi ; ui is coprime and it follows that ^ ai ¼ 0; bi ¼ 1 in 1 D D T P ¼ 0 is stable and it is not difficult (4.105) and Da ¼ 0: Clearly, R ¼ P1 h a d b a e1 to show that with Q ¼ 0 Eq. (4.17) is satisfied when B1 and Y are replaced with B e : Therefore, under Theorem 4.3 the 3DOF system always admits a decoupling and Y controller.

292

4

H2 Design of Multivariable Control …

Now in order to proceed with the H2 decoupling design for acceptable inputs, an acceptable decoupling controller for the 3DOF system is assumed to exist. In this regard it is convenient to introduce the set fhi g of unique monic polynomials of minimal degree such that the f i-th column of P1 a g  hi is stable for all columns 1 of Pa and the set fugi g of unique monic polynomials of minimal degree such that 1 1 ugi  fi-th row of A1 r g is stable for all rows of Pg ¼ Ar . That Pg ¼ Ar can be used is clear from the expressions for Vg and Pg given in the paragraph containing (4.272). It is also convenient to introduce Dh ¼ diagfhi g, Dug ¼ diagfugi g, Dag ¼ diagfagi g, and Dbg ¼ diagfbgi g; where agi and bgi are unique polynomials of minimal degrees satisfying agi hi þ bgi ugi ¼ 1 when hi , ugi are coprime. Then (see Theorem 4.4), the existence of an acceptable decoupling controller requires Assumption 4.33 (1) hi and ugi are coprime (equivalently, no column of P1 a and 1 the corresponding number row in Ar possess coincident poles in the unstable 1 region) and (2) the data construct Rg ¼ P1 a Dh Dag Td Ar is stable. As for Assumption 4.7, the weighting matrix W is specified as follows: Assumption 4.34 The weighting matrix W is of the form W ¼ diagfI; Wv g and Wv is stable and has full column rank on the stability boundary. Since ffie e 21 Gw ¼ Ag AG e 21 ½ pffiffiffiffi Ag AG a1 G w1  0 ¼ pffiffiffiffiffi a 1 Br

pffiffiffiffiffi a1 Bq Bd 0

pffiffiffiffiffi e a2 G w2  0 pffiffiffiffiffi a 2 A r Xr

pffiffiffiffiffi a2 Bq Ad Xd 0

0 pffiffiffiffiffi a2 Ar Xnl

pffiffiffiffiffi  a2 Aq diagfAXnm ; Xnl g 0

;

ð4:275Þ Assumption 4.8 is replaced by pffiffiffiffiffi pffiffiffiffiffi a2 Bq Ad Xd a2 Aq diagfAXnm ; Xnl g  Assumption 4.35 The matrices ½ Bq Bd pffiffiffiffiffi pffiffiffiffiffi and ½Br a2 Ar Xr a2 Ar Xnl  have full row rank on the stability boundary (see Example 4.8 for an equivalent condition). Assumption 4.9 is trivially met since P12 ¼ I: Assumptions 4.10 through 4.12 are, respectively, simply replaced with the following three assumptions:   P Assumption 4.36 A1 has full column rank on the stability boundary. I   Dh Assumption 4.37 has full column rank on the stability boundary. P1 a Dh vanishes on the stability boundary. Assumption 4.38 No row of Dug A1 r

4.6 H2 Design of 3DOF Decoupled Systems

293

Since WG11 Gw ¼

 pffiffiffiffiffi a1 Td A1 r Br 0

pffiffiffiffiffi  a1 Pd A1 d Bd 0

pffiffiffiffiffi a2 Td Xr 0

pffiffiffiffiffi  a2 Pd Xd 0

 0 0 0 ; ð4:276Þ 0 0 0

Assumption 4.13 is equivalent to pffiffiffiffiffi pffiffiffiffiffi 1 Assumption 4.39 The matrices Td A1 a2 Td Xr , and a2 Pd Xd are r Br , Pd Ad Bd , strictly proper. The assumption corresponding to Assumption 4.14 is simply   P Assumption 4.40 FP is proper and when s ! 1, then ! E12 sk Wv and 2

pffiffiffiffiffi A1 a2 X r r Br 4 0 0 0 0 ! E21 sl ;

0 FPd A1 d Bd LA1 d Bd

0 pffiffiffiffiffi a2 FPd Xd pffiffiffiffiffi a2 LXd

pffiffiffiffiffi a2 Xnr 0 0

0 pffiffiffiffiffi a2 Xnm 0

3 0 0 5 pffiffiffiffiffi a2 Xnl

where E12 and E21 have full column rank and full row rank, respectively, and k þ l  0: Having established assumptions for the 3DOF case which assure that the assumptions made in Theorems 4.5 and 4.6 are met, one can invoke these theorems to find the solution for the 3DOF case. In particular, the formula qL ¼ H1 ðfU /1 g1 þ fU /1 g þ  df Þ can be used. For the 3DOF system, it follows from (4.53) that the acceptable R ¼ CðI  P22 CÞ1 is of the form ð4:277aÞ

ð4:277bÞ The minus sign in the definition of Rw and Rz in (4.277b) is for consistency with the definition in (2.228). The separate roles of ½ Rw Rz  and Ru in feedback design are well explained in Youla and Bongiorno (1985) and Park and Bongiorno (1990). In decoupling design, only Ru is relevant since it follows from (4.5) and (4.277b) that

294

H2 Design of Multivariable Control …

4

Tyr ¼ P02 RP20 ¼ P½ Rw

Rz

Ru ½ 0 0

I 0 ¼ PRu ¼ PRr Ar :

ð4:278Þ

In view of the expressions for Rv and Rr in (4.54), ½ Rw Rz  in the 3DOF decoupled system is determined by the selection of the free parameter Q (equivalently, e q Þ and Ru is determined by the selection of the free parameters of T (equivalently, e dÞ and M21 (equivalently, m21 ). In the theorems of this section, solution formulas are expressed in terms of Ru and ½ Rw Rz : The evaluation of qL ¼ H1 ðfU /1 g1 þ fU /1 g þ  df Þ begins with the calculation of P P so that H in (4.149) can be obtained. The submatrices Ng1 and Ng2 are determined by (4.120) and (4.275) which yield pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi Ng1 ¼ ½ a1 Br 0 a2 Ar Xr 0 a2 Ar Xnl 0

0

ð4:279Þ

and Ng2 ¼ ½ 0

pffiffiffiffiffi pffiffiffiffiffi a1 Bq Bd 0 a2 Bq Ad Xd

pffiffiffiffiffi a2 Aq diagfAXnm ; Xnl g : ð4:280Þ

0

When Va is partitioned as Va ¼ ½ Va1 Va2  corresponding to the column sizes of the partition ½Pa 0 ; the partition matrices Na1 and Na2 in (4.125) are given by the pair Na1 ¼ Va1 ; Na2 ¼ Va2 :

ð4:281Þ

It now easily follows from (4.134) that P ¼ ½ Pg1 Pg2  with

(

)

(

)

ð4:282Þ

ð4:283Þ and 0 Pg2 ¼ Ng2  ðW



   P P 0 A1 Þ ¼ Ng2 ð A Þ: I Wv 1

ð4:284Þ

It can be readily shown that Ng1 Ng2 ¼ 0: It then follows with the aid of the identities given after Lemma 4.2 and those contained in (4.216)–(4.218) that Pg1 Pg2 ¼ 0: Hence,  P P ¼

 Pg1 ½ Pg1 Pg2

 Pg2  ¼

Pg1 Pg1 0

 0 : Pg2 Pg2

ð4:285Þ

4.6 H2 Design of 3DOF Decoupled Systems

295

Straightforward calculations using the properties in (4.216)–(4.219) yield Pg1 Pg1 " ¼

# ; fðDug Grl Ar Þ0 ðVa2 Wv Wv Va1 P1 Wg a Dh Þg 0 ðDug Grl Ar Þ0 ðVa2 Wv Wv Va1 P1 D Þ ðA G A Þ  ðV W W V Þ r rl r a2 v v a2 h a

ð4:286Þ where 1 Wg ¼ ðDug Grl Dug Þ0 fDh ðI þ P1 a Va1 Wv Wv Va1 Pa ÞDh g

ð4:287aÞ

and 1 Grl ¼ a1 A1 r Br Br Ar þ a2 ðXr Xr þ Xnr Xnr Þ:

ð4:287bÞ

In view of the diagonal form in (4.285), it is natural to introduce Wiener–Hopf factors Hg1 ; Hg2 satisfying Pg1 Pg1 ¼ Hg1 Hg1 ; Pg2 Pg2 ¼ Hg2 Hg2 :

ð4:288Þ

Now it follows from (4.285) and (4.288) that  H¼

Hg1 0

 0 ; U ¼ PH1 ¼ ½ Ug1 Ug2  Hg2

ð4:289Þ

with 1 Ug1 ¼ Pg1 H1 g1 ; Ug2 ¼ Pg2 Hg2 :

ð4:290Þ

Clearly, Ug1 Ug1 ¼ I and Ug2 Ug2 ¼ I; hence, Ug1  Oðs0 Þ and Ug2  Oðs0 Þ by Lemma 3.2.   Ug1 /1 For the calculation of U /1 ¼ it is convenient to denote U1 in Ug2 /1 (4.129) as U1 ¼ U11 þ Ug1 þ Ug2 ;

ð4:291Þ

where  U11 ¼ WG11 Gw ; Ug1 ¼ W and

 Pa 1 Pa Dh Dag Td A1 r Ng1 ; Va1

ð4:292Þ

296

4

H2 Design of Multivariable Control …

   P 1 0 N : ¼ Cy Vg I g2 Wv 

Ug2

ð4:293Þ

Then, /1 ¼ vec U1 ¼ /11 þ /g1 þ /g2 with /11 ¼ vec ðWG11 Gw Þ; /g2 ¼ vec Ug2 ¼

0 fNg2

   P 1 1 0 ð Þ; A Þgvec ðA1 Cy Vg Wv 1 I 

ð4:294Þ and 0 /g1 ¼ vec Ug1 ¼ fðDug A1 r Ng1 Þ ðW

0 0 ¼ ðNg1  WÞfðDug A1 r Þ





 Dh ÞgvecdðDag Td D1 ug Þ Va1 P1 a Dh

 Dh gvecdðDag Td D1 ug Þ: Va1 P1 a Dh

ð4:295Þ ð4:296Þ

In the expression for /g2 in (4.294) the term in braces is Pg2 and in the expression for /g1 in (4.295) the term in braces is Pg1 ½ I 0 0 : It then easily follows from Pg1 Pg2 ¼ 0; that Ug1 /1 ¼ Ug1 ð/11 þ /g1 Þ and Ug2 /1 ¼ Ug2 ð/11 þ /g2 Þ: That is, the solution for qL ¼ ½ e d 0 m0 e q 0 0 given by 21

"

H1 g1 qL ¼  0

0 H1 g2

# 

Ug1 /1 Ug2 /1



 þ 1

Ug1 /1 Ug2 /1





d  f1 d f2 þ



ð4:297Þ

with df ¼ ½ d0f 1 d0f 2 0 yields 

e d m21



¼ H1 g1 ðf Ug1 ð/11 þ /g1 Þ g1 þ f Ug1 ð/11 þ /g1 Þ g þ  df 1 Þ ð4:298Þ

and e q ¼ H1 g2 ðf Ug2 ð/11 þ /g2 Þ g1 þ f Ug2 ð/11 þ /g2 Þ g þ  df 2 Þ:

ð4:299Þ

when it is recognized that the square matrix Hg2 has the same number of columns as q : Straightforward calculations (see Pg2 which in turn equals the number of rows of e Example 4.6) now lead to

4.6 H2 Design of 3DOF Decoupled Systems

" Ug1 ð/11 þ /g1 Þ ¼

H1 g1

297

# Wg vecdðDag Td D1 ug Þ  ðDug Dh Þ vecðTd Gr Þ ; vecðVa2 Wv Wv Va1 P1 a Dh Dag Td Grl Ar Þ ð4:300Þ

where 1 Gr ¼ a1 A1 r Br Br Ar þ a2 Xr Xr :

ð4:301Þ

When Assumption 4.39 is satisfied, Ug1 /11 is strictly proper and hence



Ug1 /11 1 ¼ 0: In this case, Ug1 ð/11 þ /g1 Þ 1 ¼ Ug1 /g1 1 and Ug1 /g1 is given by

" Ug1 /g1 ¼

H1 g1

# Wg vecdðDag Td D1 ug Þ : vec ðVa2 Wv Wv Va1 P1 a Dh Dag Td Grl Ar Þ

ð4:302Þ

Simplification of the formula for e q in (4.299) is also possible. To begin with, 0 0  ðPw A1 Þg fNg2  ðPw A1 Þ g ¼ ðNg2 Ng2 Þ0  fðPw A1 Þ ðPw A1 Þ g; Pg2 Pg2 ¼ fNg2

ð4:303Þ where  Pw ¼

 P : Wv

ð4:304Þ

It is also convenient to introduce the Wiener–Hopf factors Xqd and K which satisfy 

Ng2 Ng2

   AFPd AFPd ¼ Aq ð þ diagfAGnm A ; Gnl g ÞAq ¼ Xqd Xqd Gd L L ð4:305Þ

and ðPw A1 Þ ðPw A1 Þ ¼ A1 ðP P þ Wv Wv ÞA1 ¼ K K;

ð4:306Þ

where 1 Gd ¼ a1 A1 d Bd Bd Ad þ a2 Xd Xd ; Gnm ¼ a2 Xnm Xnm ; Gnl ¼ a2 Xnl Xnl : ð4:307Þ

298

4

H2 Design of Multivariable Control …

Assumption 4.35 assures that X1 qd and Assumptions 4.34 and 4.36 assure that K1 are analytic on the stability boundary and, therefore, stable. Clearly, Pg2 Pg2 ¼ ðX0qd X0qd Þ  ðK KÞ ¼ ðX0qd  K ÞðX0qd  KÞ:

ð4:308Þ

Now for two square nonsingular matrices M and N it is not difficult to confirm that ðM  NÞ1 is given by M 1  N 1 . Hence, Hg2 ¼ X0qd  K

ð4:309Þ

is a stable Wiener–Hopf factor for (4.288) whose inverse is also stable. Moreover, 0 1 1 01 1 0 Ug2 ¼ Pg2 H1 g2 ¼ fNg2  ðPw A1 Þ g ðXqd  K Þ ¼ ðXqd Ng2 Þ  ðPw A1 K Þ

ð4:310Þ and 0 1 Ug2 /1 ¼ Ug2 ð/11 þ /g2 Þ ¼ fðX1 qd Ng2 Þ  ðPw A1 K Þ gð/11 þ /g2 Þ 0 1 ¼ fðX1 qd Ng2 Þ  ðPw A1 K Þ g vec ðU11 þ Ug2 Þ

ð4:311Þ ð4:312Þ

e e ¼ vec fðPw A1 K1 Þ ðU11 þ Ug2 Þ ðX1 qd Ng2 Þ g ¼ vec ð U 11 þ U g2 Þ; ð4:313Þ where e 11 ¼ ðPw A1 K1 Þ U11 ðX1 Ng2 Þ U qd

ð4:314Þ

1 ¼ K1 A1 P Pd Gd Ad Bq Xqd

ð4:315Þ

e g2 ¼ ðPw A1 K1 Þ Ug2 ðX1 Ng2 Þ U qd

ð4:316Þ

and

¼

1 KA1 1 Cy Vg

  0 Xqd : I

ð4:317Þ

It then follows from (4.299) that e q ¼ vec Q ¼ H1 g2 ðf Ug2 ð/11 þ /g2 Þ g1 þ f Ug2 ð/11 þ /g2 Þ g þ  df 2 Þ ð4:318Þ 1 e e ¼ ðX01 qd  K Þvec ðf U 11 þ U g2 g1; þ  Zf 2 Þ

ð4:319Þ

4.6 H2 Design of 3DOF Decoupled Systems

299

1 e 11 þ U e g2 g ¼ vec ½K1 ð f U 1; þ  Zf 2 Þ Xqd ;

ð4:320Þ

where Zf 2 is uniquely determined by its size from vec Zf 2 ¼ df 2 and fGg1; þ denotes fGg1 þ fGg þ . The formula (4.320) for e q represents a significant simplification since no Kronecker products are involved. Obviously, 1 e 11 þ U e g2 g Q ¼ K1 ð f U 1; þ  Zf 2 Þ Xqd

ð4:321Þ

  1 e 11 g e e ¼ K1 f U 1; þ þ U g2  f U g2 g  Zf 2 Xqd

ð4:322Þ

  e g2 g  Zf 2 X1  K1 U e g2 X1 e 11 g ¼ K1 f U  f U qd qd 1; þ 

ð4:323Þ

    e 11 g e g2 g  Zf 2 X1  A1 Cy V 1 0 : ¼ K1 f U  f U 1; þ  qd 1 g I

ð4:324Þ

Hence, (4.277b) and (4.53) yield  ½ Rw

Rz  ¼ Rv Aq

¼ A1 K

1



A 0

    0 A 1 0 ¼ ðCy Vg þ A1 QÞAq I I 0

  A 1 e e f U 11 g1; þ  f U g2 g  Zf 2 Xqd Aq 0

 0 : I

0 I

 ð4:325Þ ð4:326Þ

 2 Finally, in the cost functional given by (4.164), one can use df 2 ¼  2  2 df 1  þ Zf 2  since df ¼ ½ d0 d0 0 and vec Zf 2 ¼ df 2 . One can also use f1 f2 2 2     





 2 fU /1 g 2 ¼  Ug1 /1 2 þ  Ug2 /1 2 ¼  U ð/ þ / Þ þ  g1 11 g1    2  2  2 2  2

  0 0 0  Ug2 ð/11 þ /g2 Þ   since U /1 ¼ ½ ðUg1 /1 Þ j ðUg2 /1 Þ  : 2

The solution of the digital H2 design problem for the decoupled 3DOF system in Fig. 1.1 now easily follows from Theorem 4.5. Specifically, Theorem 4.11 (Digital case) When Assumptions 4.28 through 4.38 hold and both 1 Ap Pd A1 d and Ap Td Ar are stable, decoupling controllers for the 3DOF system in Fig. 1.1 exist. Moreover, the set of all loop transfer matrices R ¼ C ðI  P22 C Þ1 ¼ ½Rw  Rz Ru  associated with the ones for which J ¼ kWGd Gw k22 is finite is given by  Ru ¼ Rr Ar ¼ Va

P1 a T M21 Ar



 ¼ Va

  P1 a Dh Dag Td þ DDug ; M21 Ar

ð4:327Þ

300

4

e d ¼ vecdD; m21 ¼ vec M21 ;



e d m21



¼ H1 g1

H2 Design of Multivariable Control …



   Ug1 /11 þ /g1 1; þ df 1 ; ð4:328Þ

and ½Rw Rz  ¼ A1 K

1

n

e 11 U

o

n

1; þ

e g2  U

o



A 1 Zf 2 Xqd Aq 0 

 0 ; I

ð4:329Þ

where df 1 is an arbitrary stable vector and Zf 2 is an arbitrary stable rational   e 11 and U e g2 are given by (4.300), (4.315), matrix. The matrices Ug1 /11 þ /g1 ; U and (4.317), respectively. The optimal Ru and ½Rw Rz  that minimize the cost functional are the ones given by (4.327) and (4.329), respectively, with df 1 and Zf 2 zero. When Jopt denotes the cost functional associated with the optimal solution, the cost functional J yielded by Ru and ½Rw Rz  in (4.327) and (4.329) is given by  2  2 J ¼ Jopt þ df 1 2 þ Zf 2 2 ;

ð4:330Þ

where Jopt ¼

  1 ð dk H

ðvec ðWG11 Gw ÞÞ I  Ug1 Ug1  Ug2 Ug2 ðvec ðWG11 Gw ÞÞ 2pj k jkj¼1

          2  2 þ  Ug1 /11 þ /g1   þ  Ug2 /11 þ /g2   ; 2

2

ð4:331Þ

  with Ug2 /11 þ /g2 and WG11 Gw given by (4.313) and (4.276), respectively. The solution of the analog H2 design problem for the decoupled 3DOF system in Fig. 1.1 is obtained from Theorem 4.6. Specifically, Theorem 4.12 (Analog case) When Assumptions 4.28 through 4.40 hold and both 1 Ap Pd A1 d and Ap Td Ar are stable, decoupling controllers for the 3DOF system in Fig. 1.1 exist. Moreover, the set of all loop transfer matrices R ¼ C ðI  P22 C Þ1 ¼ ½Rw  Rz Ru  associated with the ones for which J ¼ kWGd Gw k22 is finite is given by  Ru ¼ Rr Ar ¼ Va

P1 a T M21 Ar



 ¼ Va

  P1 a Dh Dag Td þ DDug ; M21 Ar

ð4:332Þ

4.6 H2 Design of 3DOF Decoupled Systems

301

e d ¼ vecd D; m21 ¼ vec M21 ; 

e d m21



ð4:333Þ

  df 1 ; ¼ H1 g1 ðfUg1 /g1 Þg1 þ Ug1 /11 þ /g1 þ

ð4:334Þ

and ½Rw Rz  ¼ A1 K1

n

e 11 U

o



n o e g2 Zf 2 X1 Aq A  U qd 0 þ 

 0 ; I

ð4:335Þ

where df 1 is an arbitrary stable strictly proper vector and Zf 2 is an arbitrary stable   strictly proper rational matrix. The matrices Ug1 /11 þ /g1 ; Ug1 /g1 ; e g2 are given by (4.300), (4.302), (4.315), and (4.317), respectively. The e 11 ; and U U optimal Ru and ½Rw Rz  that minimize the cost functional are the ones given by (4.332) and (4.335) with df 1 and Zf 2 zero, respectively. When Jopt denotes the cost functional associated with the optimal solution, the cost functional J yielded by Ru and ½Rw Rz  in (4.332) and (4.335) is given by  2  2 J ¼ Jopt þ df 1 2 þ Zf 2 2 ;

ð4:336Þ

where

Jopt

1 ¼ 2pj

Zj1

  vec ðWG11 Gw Þ I  Ug1 Ug1  Ug2 Ug2 vec ðWG11 Gw Þds

j1

          2  2 þ  Ug1 /11 þ /g1   þ  Ug2 /11 þ /g2   ; 2

ð4:337Þ

2

  with Ug2 /11 þ /g2 and WG11 Gw given by (4.313) and (4.276), respectively. The controller C associated with Ru and ½Rw Rz  in (4.332) and (4.335) is strictly proper. Remark 4.10 As stated previously, ½Rw Rz  is irrelevant to the decoupling design and its H2 solution is not affected by the decoupling constraint. In fact it can be shown that if Wv Wv in (4.306) is replaced with kQ as in (3.344), the H2 solution formula for ½Rw Rz  in (4.329) is the same as the one for ½Rw Rz  in (3.376) in Sect. 3.8, where the ordinary H2 problem is treated (see Example 4.7). That is, 1

½Rw Rz  ¼ A1 K

n

e 11 U

o 1; þ

n

e g2  U



o 

Zf 2

X1 qd Aq



A 0

0 I

 ð4:338Þ

302

4

ew ¼ A1 K ðZmn þ f½ C 1

ew e z g þ f½ C C þ

H2 Design of Multivariable Control …

ea e z g þ f½ C C 1

 1 A e C b g ÞXt 0

0 An



ð4:339Þ e w; C e z; C e a; C e b , and Xt are defined as in Sect. 3.8. It can be where the variables C shown similarly that the formula in (4.335) is the same as the one in (3.411).

4.7

Historical Perspective and Commentary

One of the main characteristics of multivariable systems is interaction between input and output variables. Efforts to eliminate cross-coupling between these variables require controllers that make the transfer matrix from the inputs to the outputs diagonal, or more generally, block diagonal. The decoupling design problem dates back to Voznesenskij (1936), Kavanagh (1957), and Strejc (1960) (see also the references in Wang (2003) and Kučera (2011)). The early approaches are based on transfer matrix methods without imposing the stability constraint. The first meaningful results are presented in Morgan (1964) where decoupling design with static state feedback is treated. Falb and Wolovich (1967) present a necessary and sufficient condition for the existence of a decoupling controller for state-space models with a characterization of the class of feedback matrices which decouple a system. Decoupling with dynamic state feedback is investigated using the geometric approach in Morse and Wonham (1970) where it is claimed that if a linear constant multivariable system can be decoupled at all, then it can always be decoupled using linear dynamic compensation. Hautus and Heymann (1983) exploit the transfer matrix methods for decoupling design and show that dynamic state feedback is equivalent to combined dynamic output feedback and feedforward reference compensation, often referred to as a 2DOF controller. For the 2DOF configuration, Doseor and Gündes (1986) and Lee and Bongiorno (1993a, b) show that the decoupling and stability problems can be treated independently and decoupling design is always possible if the plant is internally stabilizable. On the other hand, decoupling and stability properties are not independent in the 1DOF configuration; hence, if decoupling is a design requirement, it is more restrictive for the 1DOF configuration than for the 2DOF configuration. So when decoupling is required, one might be tempted to exclude 1DOF systems from consideration. However, Howze and Bhattacharyya (1997) point out that the asymptotic tracking property of 2DOF controllers is fragile with respect to the controller parameters in the absence of error feedback. This is not the case for the unity-feedback 1DOF controller configuration. With respect to this issue then a 1DOF system could be preferable and should be considered as well.

4.7 Historical Perspective and Commentary

303

Unlike the 2DOF case, a 1DOF decoupling controller does not always exist. A sufficient condition for the existence of one is suggested in Vardulakis (1987) and necessary and sufficient conditions are established in Wang (1992), Lin (1997), Youla and Bongiorno (2000), and Gómez and Goodwin (2000). An algebraic approach based on coprime factorizations is adopted by Gómez and Goodwin (2000) to treat both triangular and diagonal decoupling design problems. A notable feature in Youla and Bongiorno (2000) is the parameterization of all realizable decoupled transfer matrices in an explicit affine form which makes it very convenient to treat performance and/or robust stability in the design of decoupled systems. In the papers by Wang (1992) and Lin (1997), the conventional model with unity feedback is treated. In Youla and Bongiorno (2000), the conventional model with non-unity feedback is addressed. In Park (2012), a parameterization of diagonal decoupling controllers is presented for the generalized plant model which accommodates 1, 2, and 3DOF configurations with nonsquare plant and non-unity feedback cases. However, the method used to obtain the realizable diagonal T requires vector operations which inflate the dimensionality of the problem. Thus, the benefit gained from the general formulation in Park (2012) lies more in a unified treatment rather than in any simplification of the solution. In this regard, the formula of the realizable diagonal T in Theorem 4.3 is an improved one which provides a unified treatment along with a simpler solution for the generalized plant model. Though the existence and the structure of realizable decoupled systems have been fairly well analyzed, not many papers treat the issue of performance or robust stability. The first to treat the robust stability problem with decoupling controllers is Safonov and Chen (1982). In this paper, decoupling controllers are described for which stability margin in the H1 norm context is maximized and output regulation constraints are taken into consideration. The H2 decoupling controller formulas for analog 2DOF and 3DOF systems appear first in Lee and Bongiorno (1993a, b). In these two papers, the tracking error and plant input are penalized in the quadratic cost functional and persistent inputs are treated whose only unstable poles lie on the jx axis. Corrêa, Soares, and Sales (2001) treat the optimal tracking problem with decoupling constraints for 2DOF analog systems. This paper allows persistent deterministic exogenous inputs with poles in Res  0 where the plant has none and insists on asymptotic tracking with zero steady-state error. In this case, the plant input must have poles in Res  0 and the H2 norm of the plant input is not finite. This difficulty is circumvented by incorporating only the stable part of the plant input into the system performance measure and is the same approach taken in Corrêa and Da Silveira (1995). As a consequence of this approach, however, plant input saturation can become a problem. In Park, Choi, and Kuc (2002), the results contained in Lee and Bongiorno (1993a) are extended to the 2DOF standard

304

4

H2 Design of Multivariable Control …

configuration and computational algorithms based on state-variable representations are also given. Kučera (2011) treats the optimal H2 block decoupling problem by adopting a general setting for the 2DOF controller configuration. The class of all controllers that decouple and stabilize the system is determined in parametric form and the parameter is used to obtain the optimal controller which minimizes the H2 norm of the transfer matrix from the reference input to the error. In formulating the cost functional, however, the plant input is not considered and neither are persistent inputs. The H2 problem for 1DOF systems is considerably more difficult to solve than that of 2DOF systems. Brinsmead and Goodwin (2001) investigate the fundamental limits on a decoupled 1DOF system employing an H2 measure of the tracking error. Youla and Bongiorno (2000) and Bongiorno and Youla (2001) treat H2 decoupling problems in which both tracking error and plant saturation for the 1DOF configuration are considered. Park (2008) treats the H2 decoupling design problem for the generalized plant model, but many restrictive assumptions are made; hence, the merit of the generalized plant model is not fully exploited. The H2 solution formulas for decoupling design derived in Sect. 4.4 are applicable for the most general setting and cover 1DOF, 2DOF, and 3DOF systems with acceptable persistent inputs as special cases. In this regard, Sect. 4.4 represents a generalization of the results in Youla and Bongiorno (2000), Bongiorno and Youla (2001), and Park (2008). Moreover, many of the results contained in this chapter are new ones appearing here for the first time. That the Schur matrix product plays a key role in the Wiener–Hopf design of optimal decoupled systems was first reported in Lee and Bongiorno (1993a). The parameterization of all closed-loop transfer matrices each of which can be realized with a 1DOF stabilizing decoupling controller for a plant with a square transfer matrix first appeared in Youla and Bongiorno (2000). The parameterization for a plant with rectangular transfer matrix appears in Bongiorno and Youla (2001).

4.8

Examples

Example 4.1 Show that the vector operation vec does not alter the H2 norm of a matrix.   Solution Let E ¼ ei j and suppose that the size of E is p  q. It can be readily shown then that TrðE EÞ ¼

q X p X  j¼1 i¼1

hence, kEk2 ¼ kvecEk2 .

 ei j ei j ¼ ðvec EÞ ðvec EÞ;

4.8 Examples

305

Example 4.2 (Lemma 4.1) Prove the following when Gi ; i ¼ 1 ! n; are matrices whose elements are complex numbers. If G1 has column rank, then rankðG1 G2 Þ ¼ rankðG2 Þ. If Gi has full column rank for i ¼ 1 ! n  1 then rank ðG1 G2    Gn Þ ¼ rank ðGn Þ: Solution Employing the Sylvester inequality for G1 and G2 , one obtains rank ðG1 Þ þ rank ðG2 Þ 

the number of the columns of G1

 rank ðG1 G2 Þ  rank ðG2 Þ:

has full column rank it follows that Since G1 rank ðG2 Þ  rank ðG1 G2 Þ  rank ðG2 Þ; hence, rank ðG1 G2 Þ ¼ rank ðG2 Þ. When G2 has full column rank, G1 G2 has full column rank too. When Gi has full column rank, for i ¼ 1 ! n  1, one easily obtains by induction that rank ðG1 G2    Gn Þ ¼ rank ðGn Þ. Example 4.3 (Lemma 4.2) Prove the following when M and N are matrices whose elements are complex numbers. If M and N each have full column rank, then M  N has full column rank. Solution There always exist permutation matrices E5 and E6 such that M  N ¼ E5 ðN  MÞE6 (Brewer 1978). Since permutation matrices are invertible, it follows that rankðM  NÞ ¼ rankðN  MÞ: When the sizes of M and N are n  m and p  q; respectively, the Sylvester inequality and the identity M  N ¼   M  Ip ðIm  N Þ yield   rank M  Ip þ rank ðIm  N Þ  mp  rankðM  NÞ  rank ðIm  N Þ ¼ mq:     Since rank M  Ip ¼ rank Ip  M ¼ mp; it follows that rankðM  NÞ ¼ mq which is the number of the columns of M  N: Example 4.4 A real rational matrix UðsÞ is called inner if it is stable and U ðsÞUðsÞ ¼ I (paraunitary). Show that for an inner matrix UðsÞ there always exists a left Wiener–Hopf factor U? ðsÞ; called complementary inner, such that I  UðsÞU ðsÞ ¼ U? ðsÞU? ðsÞ: In this case I ¼ UðsÞU ðsÞ þ U? ðsÞU? ðsÞ ¼ e ðsÞ U e ðsÞ, where U e ðsÞ ¼ ½UðsÞU? ðsÞ is a square inner matrix so that U e 1 ðsÞ ¼ U e ðsÞ; hence, U e ðsÞ U e ðsÞ ¼ I. An equivalent result for a digital system inner UðkÞ U also holds.

306

4

H2 Design of Multivariable Control …

Solution When the size of UðsÞ is l  m ðl  mÞÞ and when QðsÞ ¼ Il  UðsÞU ðsÞ is introduced, one can first establish that the normal rank of QðsÞ is l  m and then that QðjxÞ  0 for all x. Since U ðsÞUðsÞ ¼ Im , the normal rank of both UðsÞ and UðsÞU ðsÞ is m. Clearly, from the equality Il  UðsÞU ðsÞ þ UðsÞU ðsÞ ¼ Il one gets l ¼ rank ðIl  UðsÞU ðsÞ þ UðsÞU ðsÞÞ  rankðQðsÞÞ þ rank ðUðsÞU ðsÞÞ ¼ rankðQðsÞÞ þ m: Thus, l  m  rankðQðsÞÞ: On the other hand, QðsÞUðsÞ ¼ ðIl  UðsÞU ðsÞÞ UðsÞ ¼ 0 leads to rankðQðsÞÞ þ rankðUðsÞÞ  l ¼ rankðQðsÞÞ þ m  l  0 by the Sylvester inequality. Hence, rankðQðsÞÞ ¼ l  m: Next, QðjxÞ is obviously Hermitian. Moreover, since U ðjxÞUðjxÞ ¼ I; an eigenvalue kðjxÞ of UðjxÞ U ðjxÞ is 0 or 1. Thus, an eigenvalue 1  kðjxÞ of QðjxÞ ¼ Il  UðjxÞU ðjxÞ is 0 or 1. Since x QðjxÞx  kmin ðQðjxÞÞx x for any x, it is concluded that QðjxÞ  0: That is, the matrix QðsÞ is parahermitian-positive and it follows from Theorem C.1 in Appendix C that there exists a left Wiener–Hopf factor U? ðsÞ whose size is l  ðl  mÞ which satisfies QðsÞ ¼ U? ðsÞU? ðsÞ. Following the same procedures for a digital system inner UðkÞ; one obtains identical results for the digital case. Example 4.5 Suppose that F; Td ; Td1 and ðTd F  I ÞP are stable and the 1DOF system described by (4.200)–(4.208) is admissible. (a) Prove that the persistent 1 1 input Ge in (4.206) is acceptable if Ap Td A1 r ; Ap Pd Ad and LAd An1 are stable, where Ap and An1 are the denominator matrices of the coprime polynomial matrix 1 1 fraction descriptions P ¼ A1 p Bp and APo Ad ¼ Bn1 An1 , respectively. (b) Show that the Ge in (4.247) for the example in Remark 4.9 is acceptable. Hint: For part (a) first apply the concepts discussed in the paragraph containing Eqs. (4.32)–(4.36) e w ¼ Ao G111 A1 to Example 3.3. In particular, establish that Ge is acceptable iff C e e z ¼ ðG111 þ G12 A1 Y1 G211 ÞA1 Ag1 are stable. and C e Solution (a) As noted between (4.34) and (4.35), all one has to do is make the 1 replacements G11 ! G111 ; G21 ! G211 , and A1 w ! Ae to get from Example 3.3 1 e z are stable. For the 1DOF system in e w and C that Ge ¼ Ae Be is acceptable iff C   1 e w is stable e ; hence, C (4.200)–(4.208), C w ¼ Ao G111 Ae ¼ Ap Td A1 Ap Pd A1 r d 1 1 iff Ap Pd Ad and Ap Td Ar are stable. In addition,

4.8 Examples

307

1 1 Clearly, when Ap Pd A1 d is stable, ðI  PA1 Y1 F ÞPd Ad ¼ ðI  PA1 Y1 F ÞAp Ap 1 1 1 1 Pd Ad and YAFPd Ad ¼ YAFAp Ap Pd Ad are stable because ðI  PA1 Y1 F ÞA1 p (equivalently, ðI  PA1 Y1 F ÞP ¼ PA1 X1 Þ and AFA1 p (equivalently, AFP ¼ BÞ are e z is stable iff stable. Therefore, when Ap Pd A1 is stable, C d

e zo is stable if is Since Td , PA1 , and ðTd F  I ÞP are stable by assumption, C  stable. 1 AAr 0 Ag1 is stable. It can be shown that the stability of ðTd F  I ÞP 0 LA1 d assures the stability of ðI  FTd ÞA1 when F; Td , and Td1 are stable. In fact, r 1 ðTd F  I ÞP is stable iff ðTd F  I ÞAp is stable. It then follows since Td1 and 1 1 are stable that the matrix Td1 ðTd F  I ÞA1 Ap Td A1 r p Ap Td Ar ¼ Td ðTd F  I Þ 1 is stable. When ðI  FTd ÞA1 and Ap Td A1 are stable, it Td A1 r ¼ ðFTd  I ÞAr r r 1 can be shown that AAr is stable (see (3.261)). Moreover, the denominator matrix Ag1 of the coprime polynomial matrix fraction description AG211 A1 e ¼  1  1 1 1 ¼ B  APo A1 A is easily parameterized. With AA ¼ B A AAr g1 g1 m1 m1 and d r

308

4

H2 Design of Multivariable Control …

1 e g1 ¼ APo A1 coprime polynomial matrix fraction descriptions; A d ¼ Bn1 An     Am1 0 e g1 ¼ ½ Bm1 Bn1 ; it is clear that AA1 APo A1 ¼ ; and B r d 0 An1 1 e e g1 ; B e g1 may not be a e B g1 A g1 is a polynomial matrix fraction description for which A 1 coprime pair. In this case, since Bg1 Ag1 is a left-coprime polynomial matrix fraction   description for AA1 ; there exists a square polynomial matrix Mg1 APo A1 r d e g1 ¼ Ag1 Mg1 and B e g1 ¼ Bg1 Mg1 (see which is not unimodular such that A 1 Definition B.1 and Lemma B.3). Since the matrix AAr is stable, det Am1 is Hurwitz and it can be concluded that det Mg1 is also Hurwitz for the following reason. Suppose that po is a finite unstable pole of

 Hg ¼ AA1 r

  1 ¼ AAr APo A1 d

  0 þ 0

 : APo A1 d

  Then, the equality dðH; po Þ ¼ d APo A1 d ; po holds by the property B.3 of the McMillan degree in Sect. B.7 of Appendix B. This property holds for every finite unstable pole of Hg ; hence, det Ag1 and det An1 possess the same unstable zeros with e g1 ¼ det Am1 det An1 ¼ det Ag1 det Mg1 , it is the same multiplicities. Since det A e g1 M 1 ¼ concluded that det Mg1 must be Hurwitz. Clearly, Ag1 ¼ A g1    1  Am1 0 0 AAr 1 1 Mg1 with Mg1 stable and it immediately follows that 0 A 0 LA1 d  n1 1  0 AAr Am1 1 1 1 Ag1 ¼ Mg1 is stable because AAr and LAd An1 are stable. 0 LA1 d An1 e w and C e z are stable and Ge in (4.206) is acceptable. Thus, C   s 0 ; B¼ (b) For the system in (4.246) and (4.247), one gets A ¼ Ap ¼ 0 s1       1 sþ2 1 1 s 0 Bp ¼ ; Po ¼ ; Ar ¼ Ad ¼ ; Br ¼ Bd ¼ I2 , sþ1 3 0 1 0 s1       s 1 s1 1 0 1 s APo A1 ; and Bn1 ¼ . Now it can be ; An1 ¼ d ¼ 0 0 s1 0 s1 1 1 1 readily shown that Ap Td A1 r , Ap Pd Ad , and LAd An1 are stable; hence, it follows from part (a) that Ge in (4.247) is acceptable.   Example 4.6 Fill in the details of the calculations for Ug1 /11 þ /g1 in (4.300). Solution It is first important to recall the identity vec ðAVDÞ ¼ ðD0  AÞvec V; the useful identities cited after Lemma 4.2; and (4.216)–(4.219). It then follows from (4.282) that

4.8 Examples

309

{

(

}

)

Hence,

It easily follows from (4.276) and (4.279) that 



 n

 o    0 vec W U11 Ng1 þ Ng1  W /g1 Va1 P1 D h a         n  o    0 Dh Dh Td Gr Ar 1 0 0 ¼ Dug Ar þ Dug A1 vec Ng1  W /g1 ; r 1 1 Va1 Pa Dh Va1 Pa Dh 0

e1 ¼ U



0 Dug A1 r

Dh

ðE4:1Þ where

1 Gr ¼ a1 A1 r Br Br Ar þ a2 Xr Xr :

The first term of (E4.1) simplifies to 



 0 Dug A1 r

Dh Va1 P1 a Dh



 vec

Td Gr Ar 0



 ¼

A01  r



I



Va1 P1 a  01 ¼ Dug Dh Ar   

 





Dug Dh I

Va1 P1 a





 vec



vec





Td Gr Ar



0  Td Gr Ar

 Td Gr Ar

0

 A1 r

¼ Dug Dh vec I P1 a Va1 0     ¼ Dug Dh vecðTd Gr Þ ¼  Dug Dh vecðTd Gr Þ:

ðE4:2Þ

310

4

H2 Design of Multivariable Control …

Using the expression for /g1 in (4.295), the second term of (E4.1) reduces to 

  0 Dug A1 r

Dh Va1 P1 a Dh

 n    o  0 Ng0 1  W /g1 ¼ Dug A1 r Ng1 W



Dh Va1 P1 a Dh

  /g1

        0  0 Dh Dh ¼ Dug A1 Dug A1 r Ng1 W r Ng1 W 1 1 Va1 Pa Dh Va1 Pa Dh   vecd Dag Td D1 ug (    ) h 0  0 i Dh Dh 1

¼ Dug A1 N D A N W W g1 ug g1 r r Va1 P1 Va1 P1 a Dh a Dh   1 vecd Dag Td Dug       1 ¼ Dug G0rl Dug Dh I þ P1 a Va1 Wv Wv Va1 Pa Dh   vecd Dag Td D1 ug :

ðE4:3Þ Next,  

 

e 2 ¼ ðI  ½ 0 Va2 Þvec W U11 þ Ug1 Ng1 ¼ vec ½ 0 Va2 W U11 þ Ug1 Ng1 U

  ¼ vec ½ 0 Va2 W Ug1 Ng1 ¼ vec Va2 Wv Wv Va1 P1 a Dh Dag Td Grl Ar :

ðE4:4Þ

Equation (4.300) now easily follows from (E4.2), (E4.3), and (E4.4). Example 4.7 Show that the formula in (4.329) is the same as the one in (3.376). 1 1 1 Solution Let AFPd A1 Bn be coprime polynomial d ¼ Am Bm and LAd ¼ An    0 A Bm m e e matrix fraction descriptions. Define A q ¼ and B q ¼ . Clearly, 0 An Bn   AFPd A1 d e 1 B e q is a polynomial matrix fraction description, where Hd ¼ ¼A q LA1 d e q; B e q may not be a coprime pair. Since A1 A q Bq is a left-coprime fraction description e q ¼ Mq Aq . Since the matrix for Hd , there exists a polynomial matrix Mq such that A

AFPd A1 d is stable (see (3.90)), det Am is Hurwitz and it can be concluded that det Mq is also Hurwitz for the following reasons. Suppose that po is a finite unstable pole of 

AFPd A1 d Hd ¼ LA1 d



   0 AFPd A1 d ¼ : þ LA1 0 d 

  Then, the equality dðHd ; po Þ ¼ d LA1 d ; po holds by the property B.3 of the McMillan degree in Sect. B.7 of Appendix B. This property holds for every finite unstable pole of Hd ; hence, det Aq and det An possess the same unstable zeros with e q ¼ det Am det An ¼ det Mq det Aq , it is conthe same multiplicities. Since det A cluded that det Mq must be Hurwitz. Clearly,

4.8 Examples

311

Aq ¼

eq Mq1 A

¼

Mq1



Am 0

0 I



I 0 0 An

 ðE4:5Þ

with Mq1 stable and it follows from (4.305) that  Ng2 Ng2 ¼ Aq ¼ Mq1



Am 0

0 I



I 0

0 An

   AFPd AFPd þ diagfAGnm A ; Gnl g Aq Gd L L

 

    I AFPd AFPd þ diagfAGnm A ; Gnl g Gd 0 L L

0 An



ðE4:6Þ Am 0

 0 1 ; Mq I

ðE4:7Þ where Gd is given by (4.307). Since Gnm and Gnl are good and since AFPd A1 d ¼ 1 1 1 Am Bm and LAd ¼ An Bn are stable, it follows that there exists a stable Wiener– Hopf factor Xt satisfying 

I 0 0 An ¼ Xt Xt :

Gt ¼

 

  

 AFPd AFPd I þ diagfAGnm A ; Gnl g Gd L L 0

0 An

 ðE4:8Þ

One then gets from (E4.6), (E4.7), and (E4.8) that Ng2 Ng2 ¼ Mq1



Am 0

  0 A Xt Xt m I 0

 0 1 Mq ¼ Xqd Xqd ; I

where Xqd ¼

Mq1



Am 0

 0 Xt I

ðE4:9Þ

is a stable matrix. Moreover, on account of Assumption 4.35, Ng2 has full row rank on the stability boundary so that " #" #   ðdet Am Þðdet Xt Þ ðdet Am Þðdet Xt Þ     det Ng2 Ng2 ¼ 6 0 det Mq det Mq

is analytic on the stability boundary and, on the stability boundary. Hence, X1 t therefore, stable. It then follows that the Xqd given by (E4.9) is a stable Wiener– Hopf factor for Ng2 Ng2 whose inverse is also stable. In addition, it follows from (E4.5) and (E4.9) that

312

4

 Xqd ¼ Aq

I 0

H2 Design of Multivariable Control …

 0 Xt A1 n

ðE4:10Þ

and X1 qd Aq



  0 A ¼ X1 t I 0

A 0

 0 : An

ðE4:11Þ

Equations (3.369), (4.315), and (E4.10) then yield e 11 ¼ K1 A1 P Pd Gd Ad Bq A1 U q ¼

K1 A1 P Pd Gd Ad



I 0

 0 X1 An t

 1 Ad Pd F A

A1 d L An n

e g2 The remaining task is to show that  U

o 

¼





X1 t

ea C



ew ¼ C

eb C

 

 ez : C

ðE4:12Þ

. First notice that if

Wv Wv in (4.306) is taken as kQ in (3.344), then K in (4.306) is equal to K in (3.344 ). It follows from (4.273), (4.317), and (E4.10) that     0 1 0Xq Xqd Xqd ¼ KA1 1 Yp Ap Pd Ad Yq þ ½Y I     I 0 1 A ¼ KA1 Y A P A Y þ ½Y 0X Xt : p p d d q q q 1 0 A1 n

e g2 ¼ KA1 Cy V 1 U 1 g



Since Aq Xq þ Bq Yq ¼ I; it follows that Xq Aq ¼ I  A1 q Bq Yq Aq ¼ I  1 AFPd Ad Yq Aq ; hence, ½ Y 0 Xq Aq ¼ ½ Y 0  þ YAFPd A1 d Yq Aq . That is, LA1 d e g2 ¼ KA1 ½Y U 1

   I 1 P 0Xt þ KA1 YAF  Y A A Y A p p d d q q 1 0

 0 Xt : ðE4:13Þ A1 n

It can be shown that the second term on the right-hand side of (E4.13) is stable. In fact, it follows from the equalities Ap Xp þ Bp Yp ¼ I and AX þ BY ¼ I that FPYp Ap ¼ F  FXp Ap and FPYAF ¼ F  XAF; respectively. Thus,     1 1 FP YAF  Yp Ap ¼ FXp Ap  XAF and B1 A1 1 YAF  Yp Ap Ap ¼ FXp  XAFAp .   1 Since AFA1 and FXp are stable, B1 A1 is stable; hence, p 1 YAF  Yp Ap Ap   1 YAF  Y A is stable. Therefore, the second term on the right-hand side A1 A p p 1 p of (E4.13) is stable and one gets from (3.370) that

4.8 Examples

313

n o e g2 ¼ KA1 ½Y  U 1 

0Xt



¼



ea C

eb C

 

:

ðE4:14Þ

Setting Zf 2 ¼ Zmn and using (E4.11), (E4.12), and (E4.14), one can show that the expression for ½Rw Rz  in (4.329) is equal to the one in (3.376). Example 4.8 Show when A1 q Bq is given by (4.263) that the full rank condition on the stability boundary of the matrix 

Bq Bd

pffiffiffiffiffi a2 Bq Ad Xd

 pffiffiffiffiffi a2 Aq diagfAXnm ; Xnl g

is equivalent to that of the matrix 

AFPd A1 d Bd Bn Bd

pffiffiffiffiffi a AFPd Xd pffiffiffiffi2ffi a2 Bn Ad Xd

pffiffiffiffiffi a2 AXnm 0

 0 pffiffiffiffiffi ; a2 An Xnl

1 where LA1 d ¼ An Bn is a coprime polynomial matrix fraction description.

Solution Using the relationship in (E4.5) of the previous example, one obtains 

 pffiffiffiffiffi pffiffiffiffiffi Bq Bd a2 Bq Ad Xd a2 Aq diagfAXnm ; Xnl g h i pffiffiffiffiffi 1 pffiffiffiffiffi a A a diag AX B B B A X ; X ¼ Aq A1 f g q d 2 q d d 2 n n m l q q # "   pffiffiffiffiffi pffiffiffiffiffi 1 0 I 0 a2 AFPd Xd a2 AXnm 0 AFPd Ad Bd 1 Am ¼ Mq pffiffiffiffiffi pffiffiffiffiffi 0 I 0 An LA1 a2 LXd 0 a2 Xnl d Bd #  " p ffiffiffiffi ffi p ffiffiffiffi ffi Am 0 AFPd A1 a2 AFPd Xd a2 AXnm 0 d Bd ¼ Mq1 : pffiffiffiffiffi pffiffiffiffiffi 0 I Bn Bd a2 B n A d X d 0 a2 An Xnl

Since Mq1 ; Mq ; A1 m , and Am have full rank on the stability boundary, the third matrix in the last expression above has the same rank on the stability boundary as the original one. Example 4.9 Establish that any rational diagonal T which is realizable is also acceptable for any strictly acceptable Gw when G is compatible. Hint: Make use of the results obtained in Example 3.21. Solution When a rational diagonal T is realizable, then there exists a stable K and associated stabilizing controller C given by (4.8) for which Tyr ¼ T. With this same controller it follows from Example 3.21 when G is compatible and Gw is strictly acceptable that Gv Gw ¼ ðY þ A1 K ÞAG21 A1 w Bw and Gd Gw ¼ ½ðG11 þ G12 A1 Y1 G21 Þ þ 1 1 G12 A1 KAG21 A1 B are stable because G w 12 A1 ; AG21 Aw , and ðG11 þ G12 A1 Y1 G21 ÞAw w are stable. Hence, the stabilizing controller C and the diagonal T it realizes are acceptable.

314

4

H2 Design of Multivariable Control …

Example 4.10 (An example of the optimal decoupling 2DOF H2 design for an acceptable input) For the Rosenbrock’s process in Example 3.24, design the optimal decoupling 2DOF H2 controller of the configuration in Fig. 1.1 in Sect. 1.2 with appropriate choices of the weighting matrices WðsÞ and Gw ðsÞ (Remove the feedforward part and assume F ¼ I in the figure). Solution The 3DOF configuration in Fig. 1.1 is considered with the feedforward part removed and Pd ðsÞ ¼ FðsÞ ¼ Td ðsÞ ¼ I. As done in Solution 3.24, the integrators 1s I2 are inserted in front of the plant P0 ðsÞ and the augmented plant Paug ðsÞ ¼ 1s P0 ðsÞ is regarded as the plant PðsÞ in Fig. 1.1. This 2DOF system with the augmented plant is admissible and compatible since F ¼ I and Td ¼ I. Coprime polynomial matrix fraction descriptions for the augmented plant are given in Solution 3.24. With the choice of a1 ¼ a2 ¼ 1; the input weighting matrix Gw becomes 2

A1 d Bd 6 0 Gw ¼ 6 4 0 0

0

A1 r Br 0 0

Xd 0 0 0

0 Xr 0 0

0 0 Xnr 0

3 0 0 7 7; 0 5 Xnm

and its elements values are selected as Ar ¼ Ad ¼ sI2 ; Br ¼ Bd ¼ I2 ; Xr ¼ Xd ¼ 0; Xnr ¼ Xnm ¼ 0:1I2 : It is explained that Gw is acceptable in Solution 3.24. The weighting matrix to the control variable are chosen as Wv ¼ I2 . The inverse of the plant is given by 1

P

  sðs þ 1Þ ðs þ 3Þ 2ðs þ 1Þ ; ¼ sþ3 ðs þ 3Þ s1

and it follows that Dh ¼ ðs  1ÞI2 ; Dug ¼ sI2 ; Dag ¼ I2 ; Dbg ¼ I2 : Then it can be shown that Assumptions 4.28 through 4.40 in Sect. 4.6 hold and 1 both Ap A1 d and Ap Ar are stable (note that Ap ¼ A since F ¼ IÞ. Thus the optimal decoupling solution for the 2DOF system is given by the formulas in Theorem 4.12. Reminding that the matrix L is empty, one obtains   Ru ¼ P1 Dh Dag þ DDug ; 

   e d ¼ vecdD ¼ H1 Ug1 /g1 1 þ Ug1 /11 þ /g1 þ : g1

4.8 Examples

315

Here, Hg1 is a Wiener–Hopf spectral factor of the equation Wg ¼ Hg1 Hg1 with  0   1 Wg ¼ Dug Grl Dug Dh I þ P1 Dh ; P where   1 1 1 B B A þ X X ; U / ¼ H W vecd D D Grl ¼ A1 r r n n g g ag g1 r r 1 g1 ug ; r r         1 Ug1 /11 þ /g1 ¼ H1 g1 Wg vecd Dag Dug  Dug Dh vecðGr Þ 1 Gr ¼ A1 r Br Br Ar :

The computational results obtained by MATLAB Polynomial Toolbox are given by 

Hg1

Hg11 ¼ 0

 0 ; Hg12

Hg11 ¼ 0:14142s4 þ 2:012s3 þ 6:5344s2 þ 5:6638s þ 1; Hg12 ¼ 0:22361s4 þ 2:8777s3 þ 6:934s2 þ 5:28s þ 1;  

ð0:14142s3 þ 2:012s2 þ 6:5344s þ 5:6638Þ Ug1 /g1 1 ¼ ; ð0:22361s3 þ 2:8777s2 þ 6:934s þ 5:28Þ   Ug1 /11 þ /g1 þ ¼ 0; 2

3 s3 þ 14:277s2 þ 46:205s þ 40:049 6 s4 þ 14:277s3 þ 46:205s2 þ 40:049s þ 7:0711 7 6 7 e d ¼ vecdD ¼ 6 7; 4 5 s3 þ 12:869s2 þ 31:01s þ 23:613 s4 þ 12:869s3 þ 31:01s2 þ 23:613s þ 4:4721   1 ru11 ðsÞ ru12 ðsÞ Ru ¼ ; rud ðsÞ ru21 ðsÞ ru22 ðsÞ ru11 ¼ 7:0711sðs þ 3Þðs þ 1:5877Þðs þ 0:28168Þ; ru12 ¼ 8:9443sðs þ 2:9907Þðs þ 1Þðs þ 0:23644Þ;

316

4

H2 Design of Multivariable Control …

ru21 ¼ 7:0711sðs þ 3Þðs þ 1:5877Þðs þ 0:28168Þ; ru22 ¼ 4:4721sðs þ 3Þðs þ 2:9907Þðs þ 0:23644Þ; rud ¼ ðs þ 10Þðs þ 2:9907Þðs þ 1:5877Þðs þ 0:28168Þðs þ 0:23644Þ: The optimal solutions Rw and Cw are the same as the ones in Solution 3.24 by Theorem 4.12. The optimal controller Cu is obtained by the formula Cu ¼ ðI  Rw PÞ1 Ru and it is given by  1 cu11 Cu ðsÞ ¼ cud cu21

 cu12 ; cu22

cu11 ¼ 7:0711ðs þ 8:6187Þðs þ 5:808Þðs þ 2:6158Þðs þ 2:2632Þ    ðs þ 1:5877Þðs þ 0:28168Þ s2 þ 1:0303s þ 0:52012 ; cu12 ¼ 8:9443ðs þ 9:376Þðs þ 3:7468Þðs þ 2:9907Þðs þ 2:8387Þ    ðs þ 0:23644Þðs þ 0:008201Þ s2 þ 2:3663s þ 1:8027 ; cu21 ¼ 7:0711ðs þ 9:3755Þðs þ 3Þðs þ 1:5877Þðs þ 1:1184Þ    ðs þ 0:28168Þðs þ 0:18583Þ s2 þ 6:6563s þ 11:246 ; cu22 ¼ 4:4721ðs þ 8:603Þðs þ 5:8243Þðs þ 3Þðs þ 2:9907Þ    ðs þ 1:5444Þðs þ 0:23644Þ s2 þ 1:3643s þ 0:59025 ; cud ¼ ðs þ 10:135Þðs þ 9:1359Þðs þ 4:5579Þðs þ 2:9907Þðs þ 1:5877Þ    ðs þ 0:28168Þðs þ 0:23644Þ s2 þ 3:5067s þ 4:132 : The McMillan degree of the controller CðsÞ ¼ ½ Cw Cu  is 16 and hence the final controller CðsÞ 1s can be implemented as a state-space model with 18 states. The unit step responses of y1 and y2 are shown in Fig. 4.2. The step reference inputs for y1 and y2 are exerted at t ¼ 0 and t ¼ 40; respectively. The actuating controls for these reference inputs are shown in Fig. 4.3.

4.8 Examples

317

1.6

y1 y2

1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0

20

40

60

80

100

sec

Fig. 4.2 Unit step responses of the outputs to reference changes

5 v1 v2

4 3 2 1 0 -1 -2 -3 -4 -5

0

20

40

60

80

100

sec

Fig. 4.3 Control activities due to reference changes

I If the augmented plant Paug ðsÞ is regarded as the plant PðsÞ in Fig. 1.1 and the control variable vðsÞ is regarded as the input to the inserted integrators, the cost J ¼ kWGd Gw k22 for the H2 design in this example is 23.878. Compared to the non-decoupling H2 design in Example 3.24 where the cost J is 21.869, the decoupling design yields 9.19% increase in its cost.

Chapter 5

Numerical Calculation of Wiener–Hopf Controllers

5.1

Overview

A state-variable representation of the Wiener–Hopf controllers determined in Chap. 3 for strictly acceptable persistent inputs is developed in this chapter for both the analog and digital cases. First, a generalized Bezout identity (GBI) in stable rational matrices is derived from a polynomial GBI. Then the Wiener–Hopf controllers in Chap. 3, which are described in terms of the parameters in a polynomial GBI, are described instead by the parameters in a stable rational GBI. A state-space representation for the stable rational GBI description is then obtained from two algebraic Riccati equations making it possible to find a state-space representation for the Wiener–Hopf controllers. It is shown that the frequency-domain assumptions which guarantee the existence of the Wiener–Hopf controllers correspond to equivalent ones in the state-space domain.

5.2

Basic Machinery for Analog Systems

A collection of useful relationships between state-space and frequency-domain representations needed in the sequel is provided in this section for easy reference and review. These basic properties can be found in Kailath (1980), Chen (1984), and Zhou et al. (1996). A linear time-invariant system described by ^x_ ðtÞ ¼ R^xðtÞ þ N^uðtÞ; ^yðtÞ ¼ H^xðtÞ þ D^ uðtÞ

ð5:1Þ

is considered where the elements of the vectors ^xðtÞ; ^ uðtÞ, and ^yðtÞ are, respectively, the state, the input, and the output variables. The transfer matrix of the system is given by GðsÞ ¼ HðsI  RÞ1 N þ D. A constant matrix R is called stable if all eigenvalues of R lie in Re s\0 and a rational matrix GðsÞ is called stable if it is © Springer Nature Switzerland AG 2020 J. J. Bongiorno Jr. and K. Park, Design of Linear Multivariable Feedback Control Systems, https://doi.org/10.1007/978-3-030-44356-6_5

319

320

5 Numerical Calculation of Wiener–Hopf Controllers

analytic in Re s  0. A triple ðR; N; HÞ is minimal iff it is controllable and observable (Kailath 1980). In this case, the characteristic polynomial DðsÞ ¼ detðsI  RÞ of the system and the characteristic denominator wG ðsÞ of the transfer matrix GðsÞ ¼ HðsI  RÞ1 N þ D are identical. When the triple ðR; N; HÞ is not minimal, some poles of ðsI  RÞ1 are canceled out in the product of HðsI  RÞ1 N. In this case, the polynomials DðsÞ and wG ðsÞ are related by the equation detðsI  RÞ ¼ hðsÞwG ðsÞ;

ð5:2Þ

where the polynomial hðsÞ accounts for the hidden poles of the system. This is easily established from a well-known result on system equivalence. When ðR; N; HÞ is uncontrollable and unobservable, there exists a nonsingular matrix L such that (see Theorem 3.10 in Zhou et al. 1996)

~ HL = [ H 1

~ 0 H3

~ ⎡ Σ11 ⎢~ Σ 0] , L−1ΣL = ⎢ 21 ⎢ 0 ⎢ ⎣ 0

0 ~ Σ 22 0 0

~ Σ13 ~ Σ 23 ~ Σ 33 ~ Σ 43

~ ⎡ N1 ⎤ 0 ⎤ ~ ⎥ ⎢~ ⎥ Σ 24 ⎥ −1 N , L N = ⎢ 2⎥ ⎢0⎥ 0 ⎥ ⎢ ⎥ ~ ⎥ Σ 44 ⎦ ⎣0⎦

ð5:3Þ

e 11 ; N e 1; H e 1 Þ is controllable and observable. Hence, the transfer where the triple ð R matrix of the system is also equivalent to e 11 Þ1 N e 1 þ D ¼ HLðsI  L1 RLÞ1 L1 N þ D: e 1 ðsI  R GðsÞ ¼ H

ð5:4Þ

Clearly, the characteristic polynomial of the system is given by e D ¼ det½L1 ðsI  RÞL ¼ detðsI  RÞ e 22 Þ detðsI  R e 33 Þ detðsI  R e 44 Þ: e 11 Þ detðsI  R ¼ detðsI  R

ð5:5Þ

e 11 ; N e 1; H e 1 Þ is minimal, it can be concluded from (5.4) that Since the triple ð R e 11 Þ and it follows from (5.2) and (5.5) that wG ¼ detðsI  R e 22 Þ detðsI  R e 33 Þ detðsI  R e 44 Þ: hðsÞ ¼ detðsI  R

ð5:6Þ

When ðR; N; HÞ is controllable and unobservable, or when ðR; N; HÞ is uncontrollable and observable, a similar approach can show that the equality in (5.2) still holds.

5.2 Basic Machinery for Analog Systems

321

e 22 , R e 33 , When the triple ðR; N; HÞ is stabilizable and detectable, the matrices R e 44 in (5.3) are stable and the polynomial hðsÞ is Hurwitz. In this case, the and R system in (5.1) is free of unstable hidden poles and its transfer matrix GðsÞ is stable e 11 is stable. In general, one can always write (Lemma B.5) iff R e 11 Þ1 N e 1 þ D ¼ HðsI  RÞ1 N þ D ¼ A1 ðsÞBðsÞ e 1 ðsI  R GðsÞ ¼ H 1 ¼ B1 ðsÞA1 ðsÞ;

ð5:7Þ

where the last two expressions are, respectively, left and right coprime polynomial matrix fraction descriptions for GðsÞ. It then follows that wG ðsÞ ¼ g det AðsÞ ¼ g1 det A1 ðsÞ, where g and g1 are constants chosen so that wG ðsÞ is monic (Lemma B.7). Obviously, detðsI  RÞ ¼ hðsÞwG ðsÞ ¼ ghðsÞ det AðsÞ ¼ g1 hðsÞ det A1 ðsÞ:

ð5:8Þ

When GðsÞ ¼ B1 ðsÞA1 1 ðsÞ denotes a coprime polynomial matrix fraction description, a complex number so is a transmission zero of GðsÞ if the rank of the matrix B1 ðsÞ is less than its normal rank at so . If so is a transmission zero of GðsÞ which is not a pole, then the rank of GðsÞ is less than its normal rank at so . The pair ðR; NÞ is controllable iff the matrix ½ sI  R N  has full row rank for all s. The pair ðR; HÞ is observable iff the matrix ½ sI  R0 H 0 0 has full column rank for all s. The pair ðR; NÞ is stabilizable iff the matrix ½ sI  R N  has full row rank in Res  0. The pair ðR; HÞ is detectable iff the matrix ½ sI  R0 H 0 0 has full column rank in Res  0. Using these facts it is easy to show that if ðR; N1 Þ or ðR; N2 Þ is controllable (stabilizable) then ðR; ½N1 N2 Þ is controllable (stabilizable), and if ðR; H1 Þ or ðR; H2 Þ is observable (detectable) then ðR; ½ H10 H20 0 Þ is observable (detectable). For the linear time-invariant system in (5.1), it is convenient to define the system matrix 

R  sI SðsÞ ¼ H

 N : D

ð5:9Þ

The rank of the system matrix SðsÞ can only be less than its normal rank at those points in the complex plane which are eigenvalues of R corresponding to uncontrollable or unobservable modes of the triple ðR; N; HÞ or transmission zeros of GðsÞ ¼ HðsI  RÞ1 N þ D. When ðR; N; HÞ is minimal, the system matrix SðsÞ is Smith-equivalent to diagfI; B1 ðsÞg. That is, the two matrices SðsÞ and diagfI; B1 ðsÞg have the same Smith form (see Sects. 6.5.3 and 8.3.2 in Kailath 1980 and Example 5.9). When ðR; N; HÞ is stabilizable and detectable (i.e., the region < in Example 5.9 is the half-plane Res  0), then two matrices SðsÞ and diagfI; B1 ðsÞg have the same rank in Res  0.

322

5 Numerical Calculation of Wiener–Hopf Controllers

In Sects. 5.2 and 5.3, it is convenient to represent the transfer matrix GðsÞ associated with the state-space parameters ðR; N; H; DÞ as a block matrix distinguished by dashed partition lines. Specifically, ⎡Σ G ( s ) = H ( sI − Σ) −1 N + D = ⎢ ⎣H

N⎤ . D ⎥⎦

ð5:10Þ

When ⎡Σ G ( s) = ⎢ ⎣H

N⎤ ⎡ Σ1 , G (s) = ⎢ D ⎥⎦ 1 ⎣ H1

N1 ⎤ ⎡ Σ2 , G2 ( s ) = ⎢ D1 ⎥⎦ ⎣H2

N2 ⎤ , D2 ⎥⎦

ð5:11Þ

and D1 denotes both the inverse of D when D is square and nonsingular and the right inverse of D when D has full row rank, it is not difficult to verify that ⎡Σ G (s) = ⎢ ⎣H

L−1 N ⎤ ⎡ −Σ′ H ′⎤ , ⎥ , G∗ ( s ) = ⎢ ′ ′⎥ D ⎥⎦ ⎣− N D ⎦

N ⎤ ⎡ L−1ΣL =⎢ D ⎥⎦ ⎢ HL ⎣

⎡Σ G ( s )G −1 ( s ) = ⎢ ⎣H

⎡ Σ1 ⎢ G1 ( s )G2 ( s ) = ⎢ 0 ⎢H ⎣ 1

N ⎤ ⎡Σ − ND −1 H ⎢ D ⎥⎦ ⎣ − D −1 H

N1 H 2 Σ2 D1 H 2

ND −1 ⎤ ⎥=I, D −1 ⎦

N1 D2 ⎤ ⎡ Σ 2 ⎥ ⎢ N 2 ⎥ = ⎢ N1 H 2 D1 D2 ⎥⎦ ⎢⎣ D1 H 2

0 Σ1 H1

N2 ⎤ ⎥ N1 D2 ⎥ , D1 D2 ⎥⎦

ð5:12Þ

ð5:13Þ

ð5:14Þ

and ⎡ Σ1 ⎢ G1 ( s ) + G2 ( s ) = ⎢ 0 ⎢H ⎣ 1

0 Σ2 H2

⎤ ⎥ N2 ⎥ . D1 + D2 ⎥⎦ N1

ð5:15Þ

In particular, (5.13) can be confirmed using (5.14) with G1 ðsÞ ¼ GðsÞ and G2 ðsÞ ¼ G1 ðsÞ. For a compact description of transfer matrices it is convenient to write  UðsÞ ¼

   U11 ðsÞ U12 ðsÞ V11 ðsÞ V12 ðsÞ ; VðsÞ ¼ ; U21 ðsÞ U22 ðsÞ V21 ðsÞ V22 ðsÞ

ð5:16Þ

and to introduce the following transformations. The linear fractional transformation (LFT) of UðsÞ with respect to /ðsÞ, UðsÞh/ðsÞ, is defined as 

U11 UðsÞh/ðsÞ ¼ U21

 U12 h/ðsÞ ¼ U11 þ U12 /ðI  U22 /Þ1 U21 ; U22

ð5:17Þ

5.2 Basic Machinery for Analog Systems

323

and the homographic transformation of VðsÞ with respect to /ðsÞ, VðsÞ}/ðsÞ, is defined as  VðsÞ}/ðsÞ ¼

V11 V21

 V12 }/ðsÞ ¼ ðV11 / þ V12 ÞðV21 / þ V22 Þ1 : V22

ð5:18Þ

A homographic transformation with respect to /ðsÞ can be obtained from a LFT with respect to /ðsÞ through the relationship VðsÞ}/ðsÞ ¼ UðsÞh/ðsÞ when 

  1 U11 ðsÞ U12 ðsÞ V12 V22 UðsÞ ¼ ¼ 1 U21 ðsÞ U22 ðsÞ V22

 1 V11  V12 V22 V21 ; 1 V22 V21

ð5:19Þ

1 where the existence of V22 is assumed. When the state-space parameters of VðsÞ are given by

⎡Σ ⎡V V ⎤ V = ⎢ 11 12 ⎥ = ⎢⎢ H1 ⎣V21 V22 ⎦ ⎢ H ⎣ 2

N1

N2 ⎤ D12 ⎥⎥ , D22 ⎥⎦

D11 D21

ð5:20Þ

it follows that (see Example 5.10(a)) −1 ⎡ Σ − N 2 D22 H2 ⎡U11 ( s ) U12 ( s ) ⎤ ⎢ −1 = ⎢ H1 − D12 D22 U (s) = ⎢ H2 ⎥ ⎣U 21 ( s ) U 22 ( s ) ⎦ ⎢ − D −1 H 22 2 ⎣

−1 N 2 D22 −1 12 22 −1 22

D D D

−1 N1 − N 2 D22 D21 ⎤ ⎥ −1 D11 − D12 D22 D21 ⎥ −1 ⎥ − D22 D21 ⎦

ð5:21Þ

describes a set of state-space parameters for UðsÞ in (5.19). When the state-space parameters of UðsÞ and /ðsÞ are given by ⎡Σ ⎡U ( s ) U12 ( s ) ⎤ ⎢ U ( s ) = ⎢ 11 = ⎥ ⎢ H1 ⎣U 21 ( s ) U 22 ( s ) ⎦ ⎢ H ⎣ 2

N1 D11 D21

N2 ⎤ ⎡ Σφ D12 ⎥⎥ , φ ( s ) = ⎢ ⎣ Hφ D22 ⎥⎦

Nφ ⎤ , Dφ ⎥⎦

ð5:22Þ

a set of state-space parameters for UðsÞh/ðsÞ is given by (see Example 5.10(b)) ⎡ Σ + N 2 E1−1 Dφ H 2 ⎢ U φ=⎢ Nφ E2−1 H 2 ⎢ H + D D E −1 H 12 φ 2 2 ⎣ 1

N 2 E1−1 H φ −1 φ 2 −1 12 1

Σφ + N E D22 H φ D E Hφ

N1 + N 2 E1−1 Dφ D21 ⎤ ⎥ Nφ E2−1 D21 ⎥ −1 D11 + D12 Dφ E2 D21 ⎥⎦

ð5:23Þ

where E1 ¼ I  D/ D22 ; E2 ¼ I  D22 D/ : 1

ð5:24Þ

For a positive definite matrix E, the notation E2 denotes a symmetric positive 1 1 definite matrix satisfying E 2 E2 ¼ E.

324

5.3

5 Numerical Calculation of Wiener–Hopf Controllers

State-Space Representation of Wiener–Hopf Controllers: The Analog Case

In this section state-space algorithms for one of the Wiener–Hopf controllers in Chap. 3 are developed for the analog case. In particular, the generalized persistent input standard configuration of Fig. 3.2 with G compatible and proper is considered e w , is strictly acceptable and proper. for the case in which Gw , a weighted G As explained in Sect. 3.3, when the coprime polynomial matrix fraction descriptions G22 ¼ A1 B ¼ B1 A1 1

ð5:25Þ

associated with the generalized Bezout identity 

A Y1

B X1



X Y

B1 A1



 ¼

I 0

  0 X ¼ Y I

B1 A1



A Y1

B X1

 ð5:26Þ

are given and G is compatible, the class of all stabilizing controller transfer matrices is characterized by C ¼ ðX1  KBÞ1 ðY1 þ KAÞ ¼ ðY þ A1 KÞðX  B1 KÞ1 ;

ð5:27Þ

where K is a stable rational matrix such that detðX  B1 KÞ  detðX1  KBÞ 6 0. is used to (The coprime polynomial matrix fraction description A1 B ¼ B1 A1 1 represent G, G22 , or P22 in different sections of this book. It is always made clear which is intended so that no confusion arises.) When the system G is compatible, the matrices G12 A1 ; AG21 , and G11 þ G12 A1 Y1 G21 are stable. A matrix Gw is strictly acceptable for a compatible G if there exists a stable matrix K such that Gv Gw , Gd Gw , and Gm Gw are stable, where Gd , Gv , and Gm are the transfer matrices from w to yd , v, and ym , respectively (see Definition 3.3). When G is compatible, a rational matrix Gw is strictly acceptable iff AG21 Gw and ðG11 þ G12 A1 Y1 G21 ÞGw are stable (see Example 3.21). The H2 design problems solved in Chap. 3 with a compatible G in Fig. 3.2 yield all stabilizing controllers that make the performance functional 1 J¼ 2pj

Zj1 Tr½ðWGd Gw ÞðWGd Gw Þ ds j1

ð5:28Þ

5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case

325

finite when the weighting matrix W is stable and Gw is acceptable or strictly acceptable. The results for the case of strictly acceptable Gw are given by Theorem 3.5′ in Example 3.22 under the following assumptions: Assumption 5.1 G is compatible and Gw is strictly acceptable. Assumption 5.2 WG11 Gw is strictly proper and W is stable. Assumption 5.3 The following order relationships are satisfied: G22  Oðs0 Þ;

ð5:29Þ

ðG12 W WG12 Þ1  Oðs0 Þ;

ð5:30Þ

ðG21 Gw Gw G21 Þ1  Oðs0 Þ:

ð5:31Þ

(Attention is restricted here to the particular case of k1 ¼ k2 ¼ k3 ¼ 0 in Theorem 3.5′.) Assumption 5.4 rb ¼ WG12 A1 has full column rank on the finite part of the s ¼ jx axis. Assumption 5.5 rc ¼ AG21 Gw has full row rank on the finite part of the s ¼ jx axis. The following corollary is obtained from Theorem 3.5′ with k1 ¼ k2 ¼ k3 ¼ 0. Corollary 5.1 Suppose that Assumptions 5.1 through 5.5 are satisfied and that the matrices K and Ω are square stable Wiener–Hopf spectral factors with stable inverses satisfying rb rb ¼ A1 G12 W WG12 A1 ¼ K K

ð5:32Þ

rc rc ¼ AG21 Gw Gw G21 A ¼ XX :

ð5:33Þ

and

Then all controller transfer matrices C given by (5.27) for which Gv Gw , Gd Gw , and Gm Gw are stable and the performance functional J in (5.28) is finite are the ones for which K ¼ K1 Ca X1  A1 1 Y

ð5:34Þ

326

5 Numerical Calculation of Wiener–Hopf Controllers

is chosen where 1 1 Ca ¼ Z þ fC/ g þ þ fKA1 1 YXg ; C/ ¼ K rb WG11 Gw rc X

ð5:35Þ

with Z a strictly proper stable matrix. In this case, the controller transfer matrix in (5.27) satisfies C  0ðs1 Þ. Moreover, the performance functional J is minimized when Z ¼ 0. The stabilizing controllers in (5.27) are described in terms of the polynomial matrices in the generalized Bezout identity (GBI) in (5.26); hence, the same is true for the Wiener–Hopf controllers given by Corollary 5.1. In order to obtain state-space representations for these controllers, it is therefore convenient to first establish a stable proper rational GBI associated with G22 using the polynomial GBI in (5.26). The basic ways to do this are explained in Sect. 2.7. Here, the method using the Wiener–Hopf approach (Park et al. 1999) is taken. First, notice that 

X1 Ch X1

0 K



X K1 Ch

0 K1





X ¼I¼ K1 Ch

0 K1



X1 Ch X1

0 K



ð5:36Þ and consider the case    1   1    1 Ch ¼ C/  KA1 1 YX þ  KA1 YX 1 ¼ C/ þ þ KA1 YX  KA1 YX: ð5:37Þ Clearly, the matrix Ch is stable iff fC/  KA1 1 YXg þ is stable. That this is the case follows from (5.32) and (5.33) which give 1 1 KA1 1 ¼ K A1 G12 W WG12 ; X ¼ AG21 Gw Gw G21 A X :

ð5:38Þ

Hence, 1 1 C/  KA1 1 YX ¼ K A1 G12 W WðG11 þ G12 YAG21 ÞGw Gw G21 A X :

ð5:39Þ Since K1 ; G12 A1 ; W; X1 ; ðG11 þ G12 YAG21 ÞGw ; and AG21 Gw are stable, the matrix C/  KA1 1 YX is analytic on the finite part of the jx-axis. Thus, fC/  YXg is indeed stable and it follows that Ch ¼ fC/  KA1 KA1 þ 1 1 YXg þ  1 fKA1 YXg1 is also stable.

5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case

327

Using the equalities in (5.36), one gets from (5.26) that 

X1 Ch X1

0 K



A Y1



B X1

X Y

B1 A1



X K1 Ch

0 K1



 ¼

 0 ; I

ð5:40Þ

 B ; X1

ð5:41Þ

I 0

and 

  I 0 X ¼ Y 0 I

B1 A1





X1 Ch X1

X K1 Ch

0 K1



  e 0 X ¼ e I Y

0 K



A Y1

which leads to 

e A e Y1

e B e1 X



e X e Y

e1 B e1 A



I ¼ 0

e1 B e1 A



e A e Y1

 e B e1 ; X

ð5:42Þ

where e ¼ X1 A; B e 1 ¼ A1 K1 ; B e ¼ X1 B; A e 1 ¼ B1 K1 ; A   e ¼ XX  B1 K1 Ch ¼ A1 X  B1 K1 fC/ g þ þ fKA1 X 1 YXg ;   e ¼ YX þ A1 K1 Ch ¼ A1 K1 fC/ g þ þ fKA1 Y 1 YXg ;   1 1 e 1 ¼ KX1  Ch X1 B ¼ KA1 X 1  fC/ g þ þ fKA1 YXg X B;

ð5:43Þ ð5:44Þ ð5:45Þ ð5:46Þ

and   1 e1 ¼ KY1 þ Ch X1 A ¼ fC/ g þ þ fKA1 Y 1 YXg X A:

ð5:47Þ

It is also clear that e 1 : e 1 B e ¼ B e1 A G22 ¼  A 1

ð5:48Þ

Lemma 5.1 When Assumptions 5.1, 5.3, 5.4, and 5.5 are satisfied, then the expression in (5.42) is a stable proper rational GBI associated with G22 . Proof By Assumptions 5.4 and 5.5, the matrices K1 and X1 are analytic in e 1 ¼ A1 K1 and e B; e 1 ; and B e 1 are stable. The matrices A e A Res  0; therefore, A; e ¼ X1 A are proper. In fact, it follows from (5.32) and (5.33) that A 1 G12 W WG12 ¼ A1 1 K KA1 ;

G21 Gw Gw G21 ¼ A1 XX A1  :

ð5:49Þ

328

5 Numerical Calculation of Wiener–Hopf Controllers

e1 ¼ By the order assumptions in (5.30) and (5.31) it can be concluded that A 1 1 0 0 e ¼ X A  Oðs Þ (Lemma 3.2). The properness of B e1 ¼ A1 K  Oðs Þ and A 1 1 e e e e e B1 K and B ¼ X B is obvious since B 1 ¼ G22 A 1 and B ¼  AG22 , where G22 e1; X e; Y e1 , and Y e is is proper by the assumption in (5.29). The properness of X 1 1 1 1 e e e e obvious since A 1 ¼ A1 K , A ¼ X A, B 1 ¼ B1 K , B ¼ X B, and e e e e fC/ g þ þ fKA1 1 YXg are all proper. That X 1 ; X ; Y 1 , and Y are stable is obvious 1 since K; K ; X; X1 and Ch are all stable. Now it is possible to describe the Wiener–Hopf controllers of Corollary 5.1 in terms of the stable proper rational matrices in (5.42). First, it should be noticed from (5.34) and (5.35) that 1 K ¼ K1 ðCa  KA1 1 YXÞX 1 1 ¼ K1 ðZ þ fC/ g þ þ fKA1 1 YXg  KA1 YXÞX

ð5:50Þ

1 1 ¼ K1 ðZ þ fC/  KA1 ¼ K1 ðZ þ Ch ÞX1 : 1 YXg þ  fKA1 YXg1 ÞX

ð5:51Þ It now follows from (5.27) and (5.34) that   1 C ¼ Y þ A1 K1 ðZ þ Ch ÞX1 X  B1 K1 ðZ þ Ch ÞX1

ð5:52Þ

  1 ¼ YX þ A1 K1 ðZ þ Ch Þ XX  B1 K1 ðZ þ Ch Þ

ð5:53Þ

e 1 ZÞð X eþA eB e 1 ZÞ1 ¼ ¼ ðY



e1 A e1 B

 e Y e }Z: X

ð5:54Þ

An alternative form for the controller transfer matrix in (5.54) is given by e 1 ZÞð X e eþA eB e 1 ZÞ1 ¼ ð X e 1  Z BÞ e 1 ð Y e1 þ Z AÞ: C ¼ ðY

ð5:55Þ

This can be shown by using the method in Lemma 2.2 or can be shown directly by multiplying ½Z I on the left and ½I  Z 0 0 on the right to the first equality in (5.42) to get ½Z

 e A I e Y1

e B e1 X



e X e Y

e1 B e1 A



I Z

 ¼ ½Z

  I I ¼ 0: Z

ð5:56Þ

e 1 ZÞ ¼ 0 from which the eþY eB e 1 ZÞ þ ðZ B eX e 1 Þð Y eþA e1 Þð X This leads to ðZ A equality in (5.55) follows.

5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case

329

The next step is the derivation of a state-space realization for the controller. Specifically, given ⎡Σ ⎡WG G WG12 ⎤ ⎢ H = G N = ⎢ 11 w 1 ⎥ G22 ⎦ ⎢ ⎣ G21Gw ⎣H 2

N1 D11 D21

N2 ⎤ D12 ⎥ ⎥ D22 ⎦

ð5:57Þ

the objective is the determination of a state-space realization free of any unstable hidden poles for the controller transfer matrix in (5.54). It is of course necessary that GN be proper and this is assured when W; G; and Gw are proper. Moreover, since G is compatible, the response to initial conditions associated with the physical components within G is stabilized by C. So there is no need to be concerned about this issue in regard to (5.57) although it makes good sense that the state-space model for GN be consistent with G22 ¼ H2 ðsI  RÞ1 N2 þ D22 being  stabilized by C. Specifically, with DR ¼ detðsI  RÞ, it is assumed that h2 ¼ DR wG22 is a stable polynomial or, equivalently, ðR; N2 Þ is stabilizable and ðR; H2 Þ is detectable. This equivalence is established in the solution of Example 5.9(b). It is shown in the sequel for this case that the derived state-space realization for C is indeed free of unstable hidden poles. In addition, it is shown later that the order of the state-space realization for the optimal Wiener–Hopf controller is equal to that of the matrix R. Therefore, choosing the lowest order possible for R is desirable in this regard. However, the constraint that the controller realization be minimal is not imposed here for more flexibility. Constraints on the realization in (5.57) imposed by Assumptions 5.1 through 5.5 are summarized in the following lemma. Lemma 5.2 The frequency-domain conditions in Assumptions 5.1 through 5.5 impose constraints on the state-space parameters in (5.57). In particular, the following assumptions must be satisfied: Assumption 5.1′ ðR; N2 Þ is stabilizable and ðR; H2 Þ is detectable. Assumption 5.2′ D11 ¼ 0. Assumption 5.3′ D12 and D21 have full column and  R  jxI Assumption 5.4′ The matrix S12 ðjxÞ ¼ H1 for all x.  R  jxI Assumption 5.5′ The matrix S21 ðjxÞ ¼ H2 all x.

full row rank, respectively.  N2 has full column rank D12 N1 D21

 has full row rank for

330

5 Numerical Calculation of Wiener–Hopf Controllers

Proof The need for Assumption 5.1′ has already been discussed. Clearly, Assumption 5.2 requires that WG11 Gw be strictly proper and this is possible in (5.57) iff Assumption 5.2′ holds. Assumption 5.3′ is a consequence of the following considerations. Employing the formulas in (5.12) and (5.14) one gets from WðsÞG12 ðsÞ ¼ H1 ðsI  RÞ1 N2 þ D12 that ⎡ −Σ′ ⎢ G12∗W∗WG12 = ⎢ 0 ⎢− N ′ ⎣ 2

H1′H1 Σ D12′ H1

H1′D12 ⎤ ⎥ N2 ⎥ . D12′ D12 ⎥⎦

ð5:58Þ

Assumption 5.3 implies that ðG12 W WG12 Þ1 exists and is proper. It then follows from (5.13) and (5.58) that this is the case iff the inverse of the matrix D012 D12 exists or, equivalently, the matrix D12 has full column rank. A similar argument can be used to establish that ðG21 Gw Gw G21 Þ1  Oðs0 Þ iff D21 has full row rank. Assumption 5.4′ is considered next. In particular, it is now established under Assumption 5.1′ that Assumption 5.4′ holds iff Assumption 5.4 holds or, equivalently, detðA1 G12 W WG12 A1 Þ 6¼ 0 on the finite part of the s ¼ jx axis. One can always write WðsÞG12 ðsÞ ¼ H1 ðsI  RÞ1 N2 þ D12 ¼ Bp1 A1 p1

ð5:59Þ

G22 ðsÞ ¼ H2 ðsI  RÞ1 N2 þ D22 ¼ B1 A1 1 ;

ð5:60Þ

and

with Ap1 ; Bp1 and A1 ; B1 right coprime polynomial matrix pairs. It then follows that detðsI  RÞ¼ hp1 det Ap1 ¼ h1 det A1 and the polynomial h1 must be Hurwitz by Assumption 5.1′. Hence, detðA1 G12 W WG12 A1 Þ ¼ det A1 detðBp1 Bp1 Þ det A1 =ðdet Ap1 det Ap1 Þ ð5:61Þ ¼ hp1 detðBp1 Bp1 Þhp1 =ðh1 h1 Þ:

ð5:62Þ

Since h1 is Hurwitz, detðA1 G12 W WG12 A1 Þ has no jx-axis zeros iff both hp1 and detðBp1 Bp1 Þ have no jx-axis zeros. Since ðR; N2 Þ is stabilizable, ½ ðR  jxIÞ N2  has full row rank on the s ¼ jx axis. In this case, hp1 has no jx-axis zeros iff ½ ðR0  jxIÞ H10  has full row rank on the jx-axis (follows from Example 5.9(b) with < the jx-axis). Also, detðBp1 Bp1 Þ has no jx-axis zeros iff Bp1 ðsÞ has full column rank on the finite part of the jx-axis or, equivalently, WðsÞG12 ðsÞ has no

5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case

331

transmission zeros on the s ¼ jx axis. It then follows (from Example 5.9(a) with < the jx-axis) that diag fI; Bp1 ðjxÞg and S12 ðjxÞ have the same rank; hence, S12 ðjxÞ has full column rank for all x. That is, under Assumption 5.1′, the Assumptions 5.4 and 5.4′ are equivalent. The proof of the equivalence of Assumptions 5.5 and 5.5′ is similar and omitted. In the next lemma, a state-space representation of the stable proper rational e 1 is developed under Assumptions 5.1′ through 5.5′ (or, equivalently, 5.1 matrix A through 5.5). Lemma 5.3 Suppose that Assumptions 5.1′ through 5.5′ are satisfied and that ðsI  RÞ1 N2 ¼ B2 ðsÞA1 2 ðsÞ

ð5:63Þ

denotes a right coprime polynomial fraction description for ðsI  RÞ1 N2 . Then the Wiener–Hopf factor KðsÞ in (5.32) is given by KðsÞ ¼ E12 ðA2 ðsÞ þ L1 B2 ðsÞÞV11 ðsÞ ¼ E12 ðA1 ðsÞ þ L1 B2 ðsÞV11 ðsÞÞ; 1

1

ð5:64Þ

where V1 ðsÞ ¼ A1 1 ðsÞA2 ðsÞ is a polynomial matrix whose determinant is Hurwitz and E1 and L1 are given below in (5.67). Moreover, state-space parameters for the e 1 ðsÞ ¼ A1 ðsÞK1 ðsÞ are obtained from the formula matrix A

1 1  e 1 ðsÞ ¼ A1 ðsÞK1 ðsÞ ¼ I þ L1 ðsI  RÞ1 N2 E1 2 A

ð5:65Þ



1  ¼ I  L1 ðsI  R þ N2 L1 Þ1 N2 E1 2 ;

ð5:66Þ

E1 ¼ D012 D12 ; L1 ¼ E11 ðD012 H1 þ N20 M1 Þ:

ð5:67Þ

in which

The matrix M1 is the unique positive semi-definite solution of the algebraic Riccati equation (ARE) ðR  N2 E11 D012 H1 Þ0 M1 þ M1 ðR  N2 E11 D012 H1 Þ  M1 N2 E11 N20 M1 þ H10 ðI  D12 E11 D012 ÞH1 ¼ 0 with the property that R  N2 L1 is stable.

ð5:68Þ

332

5 Numerical Calculation of Wiener–Hopf Controllers

1 1 Proof Since G22 ¼ H2 ðsI  RÞ1 N2 þ D22 ¼ B1 A1 1 , ðsI  RÞ N2 ¼ B2 ðsÞA2 ðsÞ, the pair ðR; N2 Þ is stabilizable, and the pair ðR; H2 Þ is detectable, it follows that detðsI  RÞ ¼ h1 ðsÞ det A1 ðsÞ and detðsI  RÞ ¼ h2 ðsÞ det A2 ðsÞ, where h1 ðsÞ and h2 ðsÞ are Hurwitz polynomials. In addition, it is clear that B1 A1 1 A2 ¼ G22 A2 ¼ H2 B2 þ D22 A2 is a polynomial matrix. Hence, 1 1 V1 ðsÞ ¼ A1 1 A2 ¼ ðX1 þ Y1 B1 A1 ÞA2 ¼ X1 A2 þ Y1 B1 A1 A2

ð5:69Þ

is a polynomial matrix and det V1 ¼ det A2 =det A1 ¼ h1 =h2 is a Hurwitz polynomial. Moreover, since ðR; N2 Þ is stabilizable and Assumption 5.4′ is satisfied, the ARE in (5.68) has a unique solution M1  0 with the property that R  N2 L1 is stable (Zhou et al. 1996; Corollary 13.10). A little algebra from (5.68) yields (see Example 5.1)



G12 ðsÞW ðsÞWðsÞG12 ðsÞ ¼ L1 ðsI  RÞ1 N2 þ I D012 D12 L1 ðsI  RÞ1 N2 þ I 

ð5:70Þ   0   1 ¼ L1 B2 A1 2 þ I  D12 D12 L1 B2 A2 þ I

ð5:71Þ

and this leads to A1 G12 W WG12 A1 ¼ ðA1 ðsÞ þ L1 B2 ðsÞV11 Þ D012 D12 ðA1 ðsÞ þ L1 B2 ðsÞV11 Þ ð5:72Þ 1 ¼ V1 ðA2 ðsÞ þ L1 B2 ðsÞÞ D012 D12 ðA2 ðsÞ þ L1 B2 ðsÞÞV11 :

ð5:73Þ

It is easy to prove that ðA2 þ L1 B2 Þ1 is analytic in Res  0. In fact, detðI þ L1 ðsI  RÞ1 N2 Þ ¼ detðI þ ðsI  RÞ1 N2 L1 Þ ¼ detðsI  R þ N2 L1 Þ= detðsI  RÞ and it is also true that detðI þ L1 ðsI  RÞ1 N2 Þ ¼ detðI þ L1 B2 A1 2 Þ ¼ h2 detðA2 þ L1 B2 Þ= detðsI  RÞ. It therefore follows that h2 detðA2 þ L1 B2 Þ ¼ detðsI  R þ N2 L1 Þ; hence, detðA2 þ L1 B2 Þ is a Hurwitz polynomial. It is now clear that ðA2 þ L1 B2 Þ1 1 is analytic in Res  0 and ðD012 D12 Þ2 ðA2 þ L1 B2 ÞV11 is the Wiener–Hopf factor KðsÞ. Finally, e 1 ðsÞ ¼ A1 ðsÞK1 ðsÞ ¼ A1 V1 ðL1 B2 þ A2 Þ1 E2 ¼ A2 ðL1 B2 þ A2 Þ1 E 2 A 1 1

ð5:74Þ



1 1  1 12 1 ¼ ðL1 B2 A1 þ IÞ E ¼ I þ L ðsI  RÞ N E1 2 1 2 2 1

1  ¼ I  L1 ðsI  R þ N2 L1 Þ1 N2 E1 2 :

ð5:75Þ

1

1

5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case

333

e can be obtained under In a similar way, state-space parameters for A Assumptions 5.1′ through 5.5′ and the results are presented in the following lemma whose proof is omitted. Lemma 5.4 Suppose that H2 ðsI  RÞ1 ¼ A1 3 ðsÞB3 ðsÞ

ð5:76Þ

denotes a left coprime polynomial fraction description for H2 ðsI  RÞ1 . Then the Wiener–Hopf factor XðsÞ in (5.33) is given by XðsÞ ¼ V21 ðsÞðA3 ðsÞ þ B3 ðsÞL2 ÞE22 ¼ ðAðsÞ þ V21 ðsÞB3 ðsÞL2 ÞE22 ; 1

1

ð5:77Þ

where E2 and L2 are given below in (5.81) and where V2 ðsÞ is a stable square polynomial matrix satisfying A3 ¼ V2 A:

ð5:78Þ

e The state-space parameters of the rational matrix AðsÞ ¼ X1 ðsÞAðsÞ are obtained from the formula

1 1 e AðsÞ ¼ X1 ðsÞAðsÞ ¼ E2 2 I þ H2 ðsI  RÞ1 L2

ð5:79Þ



1 ¼ E2 2 I  H2 ðsI  R þ L2 H2 Þ1 L2 ;

ð5:80Þ

E2 ¼ D21 D021 ; L2 ¼ ðN1 D021 þ M2 H20 ÞE21 :

ð5:81Þ

where

The matrix M2 is the unique positive semi-definite solution of the ARE ðR  N1 D021 E21 H2 ÞM2 þ M2 ðR  N1 D021 E21 H2 Þ0  M2 H20 E21 H2 M2 þ N1 ðI  D021 E21 D21 ÞN10 ¼ 0

ð5:82Þ

with the property that R  L2 H2 is stable. The existence of such a solution is assured when ðR; H2 Þ is detectable and Assumption 5.5′ is satisfied (see Example 5.11 and its solution). Establishing the formula in the following lemma is the last step needed to find a state-space representation for the stable proper rational GBI in (5.42). Since its proof is rather long, it is given in the solution of Example 5.2.

334

5 Numerical Calculation of Wiener–Hopf Controllers

Lemma 5.5 1 2 2 fC/ g þ þ fKA1 1 YXg ¼ E1 L1 ðsI  RÞ L2 E2 : 1

1

ð5:83Þ

When ðR; N2 ; H2 Þ is minimal, one can always satisfy (5.63) and (5.76) with A2 ¼ A1 and A3 ¼ A, respectively. Then, V1 in Lemma 5.3 and V2 in Lemma 5.4 are identity matrices. In this case, it can be shown that fC/  KA1 1 YXg þ ¼ 0 and

1 1 2 2 fC/ g þ þ fKA1 1 YXg ¼ fKA1 YXgsp ¼ E1 L1 ðsI  RÞ L2 E2 , where fGðsÞgsp denotes the strictly proper part of GðsÞ (see Example 5.3). Moreover, Ch in (5.37) becomes fKA1 1 YXg1 and the GBI in (5.42) is the same as the one in (2.373) in Sect. 2.7. 1

1

e e 1 ðsÞ ¼ A1 ðsÞK1 ðsÞ, AðsÞ ¼ X1 ðsÞAðsÞ, Once the state-space parameters for A 1 and fC/ g þ þ fKA1 YXg are computed, the determination of the state-space parameters for the eight matrices in (5.42) is straightforward. State-space paramee ¼ X1 B and B e 1 ¼ B1 K1 are easily obtained from the identities B e¼ ters for B 1 1 1 1 e 1 ¼ B1 K ¼ G22 ðA1 K Þ. As for Y e; Y e1 , X e , and X e1, X B ¼ ðX AÞG22 and B the identities in (5.44) through (5.47) are used. Employing straightforward algebra and the formulas in (5.12) through (5.15), one gets the following state-space parameters for the eight matrices in (5.42):

⎡Σ − L2 H 2 ⎡ A B ⎤ ⎢ − 12 ⎢ ⎥ = ⎢ − E2 H 2 1 ⎣Y1 − X 1 ⎦ ⎢ 2 ⎣ − E1 L1

L2 − 12 2

E

0

N 2 − L2 D22 ⎤ ⎥ −1 − E2 2 D22 ⎥ , 1 ⎥ − E12 ⎦

⎡ Σ − N 2 L1 − L2 E22 ⎢ ⎤ 1 B1 E22 ⎥ = ⎢ −( H 2 − D22 L1 ) − A1 ⎦ ⎢ L1 0 ⎢⎣ 1

⎡X ⎢ ⎣Y

− N 2 E1 2 ⎤ ⎥ −1 − D22 E1 2 ⎥ . ⎥ −1 − E1 2 ⎥ ⎦

ð5:84Þ

1

ð5:85Þ

In (5.54), the Wiener–Hopf controllers determined by Corollary 5.1 are given in terms of the homographic transformation defined in (5.18). From an implementation point of view, the linear fractional transformation (LFT) defined in (5.17) is more convenient. Invoking the formula in (5.19) one gets from (5.54) that the LFT form for CðsÞ is CðsÞ ¼ Gc ðsÞhZ

ð5:86Þ

with  Gc ðsÞ ¼

eX e 1 Y e X 1

e1 þ Y e1 eX e 1 B A 1 e e X B1



 ¼

e1 e 11 Y X eþB e1 eX e 11 Y A

 e 11 X : eX e 11 B

ð5:87Þ

5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case

335

As explained in Sect. 2.7, the controller in (5.86) can be implemented as a tunable realization with a separate block for ZðsÞ as in Fig. 2.2. It is now possible to state the following theorem. Theorem 5.1 The Wiener–Hopf controllers satisfying Corollary 5.1 are obtained in state-space form from (5.86) with ⎡Σ − N 2 L1 − L2 H 2 + L2 D22 L1 ⎢ Gc ( s ) = ⎢ − L1 ⎢ − 12 − E2 ( H 2 − D22 L1 ) ⎢⎣

L2 0 − 12 2

E

( N 2 − L2 D22 ) E1 2 ⎤ ⎥ −1 ⎥. E1 2 ⎥ − 12 − 12 − E2 D22 E1 ⎥⎦ −1

ð5:88Þ

Implementation of CðsÞ in (5.86) with the realization of Gc ðsÞ in (5.88) is free of unstable hidden poles. The optimal controller minimizing the cost is given by

⎡ Σ − N 2 L1 − L2 H 2 + L2 D22 L1 Copt = Y X −1 = ⎢ − L1 ⎣

L2 ⎤ . 0 ⎥⎦

ð5:89Þ

Proof The state-space parameters for Gc ðsÞ given by (5.88) can be obtained from the state-space parameters in (5.85) by setting 

V11 V¼ V21

V12 V22





e ¼ Ae1 B1

e Y e X





0 ¼ I

I 0



e X e Y

e1 B e1 A



0 I

I 0

 ð5:90aÞ

in (5.54) and using (5.20) and (5.21). The formula in (5.89) is obtained from (5.86) by taking Z ¼ 0. In Sect. 2.7, it is shown that the implementation of (5.86) with (5.87) is free of unstable poles iff the realization of (5.88) is free of unstable hidden poles (it is assumed that ZðsÞ is realized without unstable hidden poles). Hence, it only remains to show that the realization of (5.88) is free of unstable hidden poles. It is first shown that ðRc ; L2 Þ, where Rc ¼ R  N2 L1  L2 H2 þ L2 D22 L1 , is a stabilizable pair. Since R  N2 L1 is a stable matrix, the pair ðR  N2 L1 ; L2 Þ is stabilizable, or equivalently, ½ sI  R þ N2 L1 L2  has full row rank in Res  0. Clearly, 

½ sI  R þ N2 L1 þ L2 H2  L2 D22 L1

L2  ¼ ½ sI  R þ N2 L1

I L2  H2  D22 L1

 0 ; I

ð5:90bÞ and it follows that rank ½ sI  Rc L2  ¼ rank½ sI  R þ N2 L1 L2  in Re s  0. That is, ðRc ; L2 Þ is stabilizable. Similarly, it can be shown that the pair ðRc ; L1 Þ is detectable. The condition that ðRc ; L2 Þ is stabilizable and ðRc ; L1 Þ is detectable 1 assures that ðRc ; ½L2 ðN2  L2 D22 ÞE1 2  Þ is stabilizable and ðRc ; ½L01 ðH2 

336

5 Numerical Calculation of Wiener–Hopf Controllers 1

D22 L1 Þ0 E2 2 0 Þ is detectable, in which case the realization of Gc ðsÞ in (5.88) has no unstable hidden poles (follows from Example (5.9(b)). An alternative proof that the implementation of CðsÞ in (5.86) with the realization of Gc ðsÞ in (5.88) is free of unstable hidden poles is given in Example 5.4. The state-space parameters for the controllers CðsÞ in (5.86) are also determined in the solution of this example. Remark 5.1 The generalized plant G in Fig. 3.2 is assumed to be compatible; hence, compatibility must be checked before applying the formulas in Corollary 5.1 or Theorem 5.1. Verification of the compatibility of a given generalized plant can be carried out in either the state-space or the frequency domains. A state-space method is now described. A generalized plant G is compatible if the physical e within it is admissible, G22 ¼ G e 22 , and the transfer matrices system G G12 A1 ; AG21 , and G11 þ G12 A1 Y1 G21 are stable. The verification of compatibility is carried out in two steps. First, the admissibility of the physical system is considered. Suppose that Σ ⎡ G11 G12 ⎤ ⎢ G=⎢ ⎥ = ⎢ H1 ⎣G21 G22 ⎦ ⎢ H ⎣ 2

N1 D11 D21

N2

⎥ D12 ⎥ D22 ⎥⎦

ð5:91Þ

e denotes an internal description of the physical system block GðsÞ inside of GðsÞ. e Since De ¼ De ¼ DR ¼ detðsI  RÞ it follows that G is admissible (Definition G

G 22

e N e H e 22 is admissible. This is the case iff ð R; e 2 Þ is stabilizable and ð R; e 2 Þ is 2.2) iff G detectable (see Example 5.9(b)). When the physical system is admissible the next e 22 step is to check the stabilities of the three transfer matrices (checking of G22 ¼ G is trivial). Suppose that ⎡Σ ⎡ G G12 ⎤ ⎢ G = ⎢ 11 = ⎥ ⎢ H1 ⎣G21 G22 ⎦ ⎢ H ⎣ 2

N1 D11 D21

N2 ⎤ ⎡Σ D12 ⎥⎥ = ⎢ H D22 ⎥⎦ ⎣

N⎤ D ⎥⎦

ð5:92Þ

is an admissible state-space realization for G which is the case iff ðR; N2 Þ is stabilizable and ðR; H2 Þ is detectable. Then wG wG22 is a stable polynomial and it follows from Lemma 2.3 that G12 A1 , AG21 , and G11 þ G12 A1 Y1 G21 are stable. It is crucial to recognize that there is a basic difference between two state-space descriptions in (5.91) and (5.92). The state-space model in (5.91) is an internal e description of the physical system. Thus, all the states of the sub-blocks inside G must be included when the state-space model in (5.91) is found (a systematic way to find a state-space internal description is presented in Example 5.5(a)). In general, the state-space model in (5.91) is not a minimal one. However, the state-space description in (5.92) is a realization of the transfer matrix GðsÞ; hence, it is always possible to find a minimal one. In this regard, the state-space model for the

5.3 State-Space Representation of Wiener–Hopf Controllers: The Analog Case

337

generalized plant in Doyle et al. (1989) is regarded as an internal description and only internal stabilizability and detectability (equivalent to admissibility here) of the generalized plant is imposed. Rigorously speaking, therefore, H2 design with unstable inputs is not considered in Doyle et al. (1989) because all the requirements for compatibility are not imposed. A frequency-domain method to check the admissibility of a given physical generalized plant can be found in Example 5.5(b). Remark 5.2 As explained in Remark 3.2, it is not desirable to include the unstable input signal model Gw into the generalized plant. When W is stable and Gw is strictly acceptable for a compatible G, however, the matrices WG12 A1 , AG21 Gw , and ðG11 þ G12 A1 Y1 G21 ÞGw are stable. It can then be concluded that the system GN in (5.57) is compatible. This implies that the strictly acceptable Gw is an unstable input model that can be absorbed into a generalized compatible plant to result in a new compatible system.

5.4

Basic Machinery for Digital Systems

As was done for analog systems, a collection of useful relationships between state-space and frequency-domain representations needed in the sequel are provided in this section for digital systems. A linear time-invariant discrete system described by ^xðk þ 1Þ ¼ R^xðkÞ þ N^uðkÞ; ^yðkÞ ¼ H^xðkÞ þ D^ uðkÞ

ð5:93Þ

is considered where the elements of the vectors ^xðkÞ; ^ uðkÞ, and ^yðkÞ are, respectively the state, the input, and the output variables. Applying the k-transform to the difference equations in (5.93) yields the polynomial matrix description  ðI  RkÞxðkÞ ¼ NkuðkÞ : yðkÞ ¼ HxðkÞ þ DuðkÞ

ð5:94Þ

The transfer matrix of the system is GðkÞ ¼ Hðk1 I  RÞ1 N þ D ¼ HðI  RkÞ1 Nk þ D. Clearly, Eq. (5.94) are a special case of the generalized digital plant equations TðkÞxðkÞ ¼ UðkÞuðkÞ, yðkÞ ¼ VðkÞxðkÞ þ WðkÞuðkÞ with system matrix ⎡− T (λ ) U1 (λ ) U 2 (λ ) ⎤ ⎡− T (λ ) U (λ ) ⎤ ⎢ = S (λ ) = ⎢ V1 ( λ ) W11 ( λ ) W12 ( λ ) ⎥ . ⎥ ⎣ V ( λ ) W ( λ )⎦ ⎢⎢ V ( λ ) W ( λ ) W ( λ )⎥⎥ 21 22 ⎣ 2 ⎦

ð5:95Þ

In particular, (5.94) corresponds to the case TðkÞ ¼ ðI  RkÞ, UðkÞ ¼ Nk, VðkÞ ¼ H, and WðkÞ ¼ D. Obviously, Tð0Þ ¼ I is nonsingular; hence, the system

338

5 Numerical Calculation of Wiener–Hopf Controllers

(5.94) is nondegenerate and causal (see Sect. A.5). In addition, the characteristic polynomial for the system is DðkÞ ¼ l det TðkÞ ¼ l detðI  RkÞ; where l is the constant for which DðkÞ is monic; hence, the system is stable iff the polynomial DðkÞ is Schur (that is, all zeros of DðkÞ lie in jkj [ 1). In the discrete state-space model, a constant matrix R is said to be stable if all its eigenvalues lie in jkj\1 or, equivalently, the polynomial detðI  RkÞ is Schur. A state-space model for GðkÞ can be conveniently realized from the z- transform e version of the transfer matrix given by GðzÞ ¼ Gðz1 Þ which is obtained from GðkÞ 1 by setting k ¼ z . A z-transform rational matrix is state-space realizable iff it is proper; hence, a rational matrix GðkÞ is state-space realizable iff GðkÞ is analytic at k ¼ 0. That a rational matrix GðkÞ is not state-space realizable implies that the rational matrix cannot be the transfer matrix of a causal system (see Sect. A.5). Recognizing the similarity of the digital system described here with the analog case treated in Example 5.9 one gets from Example 5.9(b) that DðkÞ ¼ l det TðkÞ ¼ h2 ðkÞwG22 ðkÞ, where h2 ðkÞ is monic and has no zeros in < iff ½ TðkÞ U2 ðkÞ  and ½ T 0 ðkÞ V20 ðkÞ  have full row rank in