nonlinear output regulation

Nonlinear OutputRegulation

Advances in Design and Control

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Series VolumesHuang, J., Nonlinear Output Regulation: Theory and ApplicationsHaslinger, J. and Makinen, R. A. E., Introduction to Shape Optimization: Theory,

Approximation, and ComputationAntoulas, A. C., Lectures on the Approximation of Linear Dynamical SystemsGunzburger, Max D., Perspectives in Flow Control and OptimizationDelfour, M. C. and Zolesio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and

OptimizationBetts, John T., Practical Methods for Optimal Control Using Nonlinear ProgrammingEl Ghaoui, Laurent and Niculescu, Silviu-lulian, eds., Advances in Linear Matrix Inequality

Methods in ControlHelton, J. William and James, Matthew R., Extending H Control to Nonlinear Systems:

Control of Nonlinear Systems to Achieve Performance Objectives

Nonlinear OutputRegulationTheory and Applications

Jie HuangThe Chinese University of Hong KongHong Kong

siamSociety for Industrial and Applied Mathematics

Philadelphia

Copyright © 2004 by the Society for Industrial and Applied Mathematics.

1 0 9 8 7 6 5 4 3 2 1

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Library of Congress Cataloging-in-Publication Data

Huang, Jie, 1955-Nonlinear output regulation : theory and applications / Jie Huang.

p. cm. — (Advances in design and control)Includes bibliographical references and index.ISBN 0-89871-562-8

1. Servomechanisms—Design and construction. 2. Nonlinear functional analysis. I. Title.II. Series.

TJ214.H83 2004629.8'323-dc22 2004052533

B ^^

Siam is a registered trademark.

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Contents

List of Figures vii

List of Tables ix

Notation xi

Preface xiii

1 Linear Output Regulation 11.1 Introduction 11.2 Linear Output Regulation 31.3 Linear Robust Output Regulation 151.4 The Internal Model Principle 261.5 Output Regulation for Discrete-Time Linear Systems 291.6 Robust Output Regulation for Discrete-Time Linear Systems 31

2 Introduction to Nonlinear Systems 352.1 Nonlinear Systems 352.2 Stability Concepts for Nonlinear Systems 372.3 Input-to-State Stability 402.4 Center Manifold Theory 452.5 Discrete-Time Nonlinear Systems and Center Manifold Theory for Maps 472.6 Normal Form and Zero Dynamics of SISO Nonlinear Systems 502.7 Normal Form and Zero Dynamics of MIMO Nonlinear Systems . . . . 592.8 Examples of Nonlinear Control Systems 66

3 Nonlinear Output Regulation 733.1 Introduction 733.2 Problem Description 753.3 Solvability of the Nonlinear Output Regulation Problem 793.4 Solvability of the Regulator Equations 893.5 Output Regulation of Nonlinear Systems with Nonhyperbolic Zero

Dynamics 1013.6 Disturbance Rejection of the RTAC System 106

4 Approximation Method for the Nonlinear Output Regulation 1134.1 kth-Order Approximate Solution of Nonlinear Output Regulation

Problem 113

v

vi

5

6

7

8

9

A

B

Contents

4.2 Power Series Approach to Solving Regulator Equations4.3 Power Series Approach to Solving Invariant Manifold Equation . . .4.4 Asymptotic Tracking of the Inverted Pendulum on a Cart

Nonlinear Robust Output Regulation5.1 Problem Description5.2 Two Case Studies5.3 Solvability of the kth-Order Robust Output Regulation Problem . . .5.4 Solvability of the Robust Output Regulation Problem5.5 Computational Issues5.6 The Ball and Beam System Example

From Output Regulation to Stabilization6.1 A New Design Framework6.2 Existence of the Steady-State Generator and the Internal Model . . .6.3 Robust Output Regulation with the Nonlinear Internal Model . . . .6.4 Robust Asymptotic Disturbance Rejection of the RTAC System . . .

Global Robust Output Regulation7.1 Problem Description7.2 Stabilization of Systems in Lower Triangular Form7.3 Global Robust Output Regulation for Output Feedback Systems . . .7.4 Global Robust Output Regulation for Nonlinear Systems in Lower

Triangular Form

Output Regulation for Singular Nonlinear Systems8.1 Problem Formulation8.2 Preliminaries of Singular Linear Systems8.3 Output Regulation by State Feedback and Singular Output Feedback8.4 Output Regulation via Normal Output Feedback Control8.5 Approximate Solution of the Output Regulation Problem for Singular

Systems8.6 Robust Output Regulation of Uncertain Singular Nonlinear Systems

Output Regulation for Discrete-Time Nonlinear Systems9.1 Discrete-Time Output Regulation9.2 Approximation Method for the Discrete-Time Output Regulation . .9.3 Robust Output Regulation for Discrete-Time Uncertain Nonlinear

Systems9.4 The Inverted Pendulum on a Cart Example

Kronecker Product and Sylvester Equation

ITAE Prototype Design

Notes and References

Bibliography

Index

117. 125

127

133133138

. 140145151153

159160

. 166175

. 179

187187192

. 201

216

229229232240246

253255

265265

. 272

279290

297

301

303

307

315

List of Figures

1.1

2.12.22.3

3.13.23.3

3.43.5

4.1

4.2

4.3

5.15.2

6.1

6.2

6.3

9.19.2

Unity feedback control

Rotational/translational actuator.Inverted pendulum on a cartBall and beam system

Nonlinear output regulation problemThe profile of the displacement x1 with = 0.2, w = 3, and Am — 0.5.The profiles of the state variables (x2, x3, x4) with € = 0.2, a) = 3, andAm = 0.5The profile of the control input u with € = 0.2, w = 3, and Am = 0.5. .The profiles of the displacement x\ when undergoes perturbation. . .

The profile of the tracking performance of the closed-loop system underthe nonlinear controller with w = 1 .5 and Am = 1The profile of the tracking performance of the closed-loop system underthe linear controller with w = 1.5 and Am = 1Comparison of the output responses of the closed-loop system under thenonlinear and linear controllers with w = 1.5 and Am = 4

Tracking performance: Nominal case Am = 5 and w =Tracking performance: Perturbed system with Am = 5 and w = . . .

The profiles of the displacement x1 with = 0. 1 8, 0.2, 0.22, w = 3, andAm = 0.5The profiles of the state variables (x2, x3, x4) with = 0.2, w = 3, andAm = 0.5The profile of the control input u with = 0.2, w = 3, and Am = 0.5. .

Tracking performance: Nominal case Am = 1.25 and w = 0.05 . . . .Tracking performance: Perturbed system with Am = 1.25, w = 0.05 ,and b - 1.0.

2

666971

74. 110

111. 1ll. 112

.131

.131

132

158. 158

.184

184. 185

?94

. 295

VII

This page intentionally left blank

List of Tables

4.1

5.15.2

9.19.2

B.1

Maximal steady-state tracking error with Am = 1

Maximal steady-state tracking error of nominal system with a) = . . .Maximal steady-state tracking error of the perturbed system with Am = 5and w = ••

The maximal steady-state tracking errors of the nominal systemThe maximal steady-state tracking errors of the perturbed system withAm = 1.25 and w = 0.057

Pole locations of ITAE prototype design

130

. 157

.157

.296

.296

.301

IX

Notation

Symbol

deg(.)dim(.)rank

Usage

deg( ( ))dim(K)rank A

Meaning

the 2-norm of a vector xthe induced 2-norm of a matrix An-dimensional Euclidean spaceThe set of all n x m matrix with elements in R1

n x n identity matrixspectrum of matrix A

is a member of (A)X is not a member of (A)Kronecker producta (A) divides ( )open left half-complex planeopen right half-complex planeclosed left half-complex planeclosed right half-complex planedegree of polynomial a (X)dimension of Krank of matrix A

XI

Preface

The output regulation problem, or alternatively, the servomechanism problem, addressesdesign of a feedback controller to achieve asymptotic tracking for a class of reference inputsand disturbance rejection for a class of disturbances in an uncertain system while maintainingclosed-loop stability. This is a general mathematical formulation applicable to many controlproblems encountered in our daily life, for example, cruise control of automobiles, aircraftlanding and taking-off, manipulation of robot arms, orbiting of satellites, motor speedregulation, and so forth. Study of the output regulation problem can be traced as far backas 1769, when James Watt devised a speed regulator for a steam engine. Yet rigorousformulation of this problem in a modern state-space framework was not available until the1970s. In contrast to similar problems, such as trajectory tracking, where the trajectory tobe tracked is assumed to be completely known, a distinctive feature of the output regulationproblem is that the reference inputs and disturbances do not have to be known exactly solong as they are generated by a known, autonomous differential equation. In this book, theterm "exogenous signals" will be used to refer to both reference inputs and disturbanceswhen there is no need to distinguish them. The autonomous differential equation generatingexogenous signals will be called the exosystem.

The output regulation problem was first studied for the class of linear systems undervarious names, such as the robust servomechanism problem (Davison) or the structurallystable output regulation problem (Francis and Wonham). It was completely solved bythe collective efforts of several researchers, including Davison, Francis, and Wonham, toname just a few. Solvability conditions for the output regulation problem were workedout either in terms of the location of the transmission zeros of the system or in terms of thesolvability of a set of Sylvester equations. A salient outcome of this research was the internalmodel principle, which includes classical PID (proportional-integral-derivative) control asa special case. From the control theoretic point of view, the significance of the internalmodel principle is that it enables the conversion of the output regulation problem into thewell-known stabilization problem for an augmented linear system.

At almost the same time that research on the linear output regulation problem reachedits peak, in the mid 1970s, Francis and Wonham considered the output regulation problemfor a class of nonlinear systems for the special case when exogenous signals are constant.They showed that a linear regulator design based on the linearized plant can solve the robustoutput regulation problem for a weakly nonlinear plant while maintaining the local stabilityof the closed-loop system. In the late 1980s, Huang and Rugh further studied this problemfor general nonlinear systems using a gain scheduling approach and related the solvabilityof this problem to solvability of a set of nonlinear algebraic equations.

xiii

xiv Preface

To establish a general theory for the output regulation problem for uncertain nonlinearsystems subject to time-varying exogenous signals, one must address three important issues:how to define and guarantee existence of the steady state of the system, and hence charac-terize the solvability of the problem; how to handle plant uncertainty when it is known thatthe linear internal model principle does not work for nonlinear systems in the general case;and how to achieve asymptotic tracking and disturbance rejection in a nonlinear system witharbitrarily large initial states of the plant, the exosystem, and the controller, in the presenceof uncertain parameters that lie in an arbitrarily prescribed, bounded set.

None of these three issues can be dealt with by a simple extension of the existinglinear output regulation theory. Because of these challenges, the output regulation problemfor nonlinear systems has become one of the most exciting research areas since the 1990s.As a result of extensive work, these three issues have now been successfully addressed to acertain degree.

The difficulty associated with the first issue, existence of steady state, lies in the factthat the solution of a nonlinear system is not available. Isidori and Byrnes first addressedthis issue for the case when the plant is assumed to be known exactly. By introducing centermanifold theory, Isidori and Byrnes found that it is possible to use a set of mixed nonlinearpartial differential and algebraic equations, called regulator equations in what follows, tocharacterize the steady state of the system. This discovery coupled with the zero dynamicstheory of nonlinear systems leads to a solvability condition for the output regulation problemin terms of solvability of the regulator equations. It turns out that the regulator equations area generalization of the Sylvester equations mentioned above. The solution of the regulatorequations provided a feedforward control to cancel the steady-state tracking error. Basedon the solution of the regulator equations, both state feedback and error feedback controllaws can be readily synthesized to achieve asymptotic tracking and disturbance rejectionfor an exactly known plant while maintaining local stability of the closed-loop system.

The second issue is concerned with the plant uncertainty characterized by a set ofunknown parameters. The feedforward control approach mentioned in the last paragraphcannot handle this case due to the presence of the unknown parameters. A design approachbased on the linear internal model principle does not work either, as shown by a counterex-ample due to Isidori and Byrnes. Huang first revealed in 1991 that the linear internal modelprinciple failed because, unlike the linear case, the steady-state tracking error in a nonlinearsystem is a nonlinear function of the exogenous signals. Based on this observation, Huangfound that if the solution of the regulator equations is a polynomial in the exogenous signals,then it is possible to solve the output regulation problem for uncertain nonlinear systemsby both state feedback and output feedback control. This approach effectively leads to anonlinear version of the internal model principle. The robust output regulation problemwas further pursued by Byrnes and Isidori, Delli Priscoli, and Khalil, generating varioustechniques and insights on this important issue.

While the first two issues have been intensively addressed since the 1990s, the investi-gation of the third issue, the output regulation problem with global stability, has just startedand is rapidly unfolding. In the original formulation of the output regulation problem, asgiven by Isidori and Byrnes, only local stability is required for the closed-loop system.For this case, the stability issue can be easily handled by Lyapunov's linearization method.When a global stability requirement is imposed on the closed-loop system, the situationbecomes much more complicated. Khalil studied the semiglobal robust output regulation

Preface xv

problem for the class of feedback linearizable systems in 1994. His work was further ex-tended to the class of lower triangular systems by Isidori in 1997. The output regulationproblem with global stability was solved for the class of output strict feedback systems bySerrani and Isidori in 2000. Up to this point, the problem of output regulation with nonlocalstability was handled on a case-by-case basis, and only limited results were obtained. Re-cently, Huang and Chen have established a new framework that converts the robust outputregulation problem for nonlinear systems into a robust stabilization problem. This newframework has offered greater flexibility to incorporate recent stabilization techniques, thushaving set a stage for systematically tackling robust output regulation with global stability.This new framework has been successfully applied to solve the output regulation problemwith global stability for several important classes of nonlinear systems.

The scope of research on the output regulation problem is constantly expanding,and the topic is made richer and more interesting with the injections of new ideas andtechniques from other research areas such as stabilization, adaptive control, neural networks,and numerical mathematics. For example, the output regulation problem with uncertainexosystems was studied recently by Chen and Huang, Nikiforov, Serrani, Marconi andIsidori, and Ye and Huang, respectively. This scenario had not been studied previously,even for linear systems.

The output regulation problem arises from formulating daily engineering control prob-lems. Therefore, in addition to the theoretical issues mentioned above, the application ofthis theory to practical design should be adequately addressed. A key issue critical to theapplicability of the output regulation theory is the solvability of the regulator equations.Being a set of mixed nonlinear partial differential and algebraic equations, the solution ofthe regulator equations is usually unavailable. Thus it is necessary to develop approximationapproaches to solving these equations. An approximation method based on Taylor seriesexpansion was developed by Huang and Rugh in 1991 and was also considered by Krener in1992. The effectiveness of these approximation methods has been demonstrated by manycase studies, including benchmark nonlinear systems such as the ball and beam, the invertedpendulum on a cart, and the rotational/translational actuator.

This book will give a comprehensive and up-to-date treatment of the output regulationproblem in a self-contained fashion. The book begins with an introduction to the linearoutput regulation theory in Chapter 1. Then a review of fundamental nonlinear controltheory is given in Chapter 2. Chapters 3 and 4 are devoted to the output regulation problemand the approximate output regulation problem for continuous-time nonlinear systems,respectively. The robust output regulation problem for uncertain continuous-time nonlinearsystems is presented in Chapters 5 and 6. In Chapter 7, the global robust output regulation isformulated and studied for uncertain continuous-time nonlinear systems. Chapter 8 presentsboth the output regulation problem and the robust output regulation problem for singularnonlinear systems. Finally, in Chapter 9, results on the output regulation problem and therobust output regulation problem are extended to discrete-time nonlinear systems. Theauthor seeks to strike a balance between the theoretical foundations of the output regulationproblem and practical applications of the theory. The treatment is accompanied by manyexamples, including practical case studies with numerical simulations based on the softwareplatform MATLAB®.

This book can be used as a reference for graduate students, scientists, and engineers inthe area of systems and control. Readers are assumed to have some fundamental knowledge

xvi Preface

of linear algebra, advanced calculus, and linear systems. Knowledge needed of nonlinearsystems is summarized in Chapter 2. Some of the present chapters were used in the work-shops of the 1999 IEEE Conference on Decision and Control, the 2004 World Congresson Intelligent Control and Automation, and graduate seminars at the Chinese University ofHong Kong.

The development of this book would not have been possible without the support andhelp of many people, including the author's master's thesis supervisor, Professor XiangqiuZeng; Ph.D. supervisor, Professor Wilson J. Rugh; and numerous colleagues and students.Professor Rugh not only guided the author into the area of nonlinear control, but alsopersonally made substantial contributions to many results covered in Chapters 3 and 4.Some sections from Chapters 6-9 are adapted from joint publications of the author andsome of his past and current students, including Zhiyong Chen, Guoqiang Hu, Weiyao Lan,Dan Wang, and Jin Wang. Three current students, Zhiyong Chen, Guoqiang Hu, and WeiyaoLan, have painstakingly proofread the manuscript several times and checked many exampleswith computer simulations. Professors Zhong-Ping Jiang, Zongli Lin, and Wilson J. Rughhave provided the author with valuable comments and suggestions. Professor Frank Lewisnot only inspired and encouraged the author to embark on this project, but also introducedhim to the SIAM acquisitions editor, Dr. Linda Thiel, who has been extremely helpful andenthusiastic. The SIAM Developmental Editor Simon Dickey and Production Editor LisaBriggeman have done excellent work. The author is greatly indebted to Professor AlbertoIsidori, whose seminal work on the output regulation problem with his coauthors has laidthe foundation for this book.

The bulk of this research was supported by the Hong Kong Research Grants Councilunder grants CUHK 4316 /02E and CUHK 4168 ABE, and by National Natural ScienceFoundations of China under grant 60374038.

Jie Huang

Chapter 1

Linear OutputRegulation

In this chapter, a concise but self-contained treatment of the subject of the output regulationproblem for linear time-invariant systems is given. The output regulation problem was one ofthe central research topics in linear control theory in the 1970s. This research has generateda salient controller synthesis technique known as the Internal Model Principle. The purposeof this chapter is mainly to provide the background for understanding the nonlinear outputregulation problem, and the chapter is organized as follows. In Section 1.1, a typical scenariothat leads to the formulation of the problem is described. In Section 1.2, the precise definitionof the output regulation problem is given and the solvability of the problem via both statefeedback control and measurement output feedback control is presented. In Section 1.3, wefurther take into account model uncertainties, which leads to the formulation of the robustoutput regulation problem. We give the solution of this problem by both state feedbackand error output feedback control. The robust output regulation problem is an enhancedversion of the output regulation problem in the sense that it achieves the same objectives asthe former even in the presence of model uncertainties. In Section 1.4, the solvability of thelinear robust output regulation problem is further examined by introducing what is calledthe internal model principle. While the first four sections are devoted to continuous-timelinear systems, results on the output regulation problem and on the robust output regulationproblem for discrete-time linear systems are established in Sections 1.5 and 1.6.

1.1 IntroductionMany practical control problems such as trajectory planning of a robot manipulator, guidanceof a tactic missile toward a moving target, attitude control of spacecraft subject to torquedisturbance, weapon system pointing under firing disturbances, and so on, fall into thedomain of the problem depicted in Figure 1.1. Here a plant is given that is subject toa disturbance d(t), and a controller is to be designed so that the closed-loop system isexponentially stable, in the sense to be defined precisely later, and the output of the planty(t) asymptotically tracks a given reference input r(t) in the following sense:

1

Chapter 1. Linear Output Regulation

Figure 1.1. Unity feedback control.

This problem is conveniently called asymptotic tracking and disturbance rejection of theoutput. In the particular case where r(t) — 0, the problem is simply called asymptoticregulation.

A linear plant subject to a disturbance d(t) can be modelled as follows:

Thus, the tracking error is given by

The controller can generally be modelled as follows:

This controller must guarantee the stability of the closed-loop system composed of (1.2)and (1.4) while assuring asymptotic tracking of y ( t ) to r(t) in the presence of the distur-bance d(t).

In practice, the reference input to be tracked and the disturbance to be rejected usuallyare not exactly known signals; for example, a disturbance in the form of a sinusoidal functioncan have any amplitudes and initial phases, or even any frequencies, and a reference inputin the form of a step function can have arbitrary magnitudes. It is desirable that a singlecontroller be able to handle a class of prescribed reference inputs and/or a class of prescribeddisturbances. In this chapter, both the reference inputs and the disturbances are assumed tobe generated by linear autonomous differential equations as follows:

where r0 and do are arbitrary initial states. The above autonomous equations can generatea large class of functions; for example, a combination of step functions of arbitrary magni-tudes, ramp functions of arbitrary slopes, and sinusoidal functions of arbitrary amplitudesand initial phases.

2

1.2. Linear Output Regulation 3

Let

Then the reference inputs and the disturbances can be lumped together as follows:

Thus, the plant state and the tracking error can be put into the following form:

where

Now the problem of asymptotic tracking of y(t) to r(t) can be treated as the problemof asymptotic regulation of e(t) to the origin when e(t) is viewed as the output of (1.6).Therefore, it suffices to study the regulation problem described by (1.6) while keeping inmind that the system (1.5), called the exosystem in what follows, can generate either thereference inputs or the disturbances or both. Thus, the problem of asymptotic trackingand disturbance rejection can be called simply the output regulation problem when thedisturbances and the reference inputs are generated by (1.5). Alternatively, the outputregulation problem is called a servomechanism problem.

In (1.6), the plant is defined by six finite-dimensional constant matrices A, B, E, C,D, and F. These matrices are usually obtained by linearizing a nonlinear system aroundan operating condition or by using a certain system identification approach. Due to thevariations in the operating point or the limitations of system identification techniques, thesematrices are invariably inaccurate. Typically, each entry of the matrices A, B, E, C, D, andF can take arbitrary values in an open neighborhood of its nominal value. Therefore, it isdesirable to further require that the controller be able to maintain the property of asymptotictracking and disturbance rejection in the closed-loop system regardless of small variations ofthe entries in the matrices A, B, E, C, D, and F. The problem of designing such controllersis called the robust output regulation problem or the robust servomechanism problem.

The discussion so far has exemplified a scenario of what is called the output regulationproblem and its enhanced version the robust output regulation problem. The solvability ofthese two problems will be established in the remaining sections of this chapter.

1 .2 Linear Output RegulationConsider a class of linear time-invariant systems described by

4 Chapter 1. Linear Output Regulation

where x(t) is the n-dimensional plant state, u(t) the m-dimensional plant input, e(t) thep-dimensional plant output representing the tracking error, and v(t) the q-dimensionalexogenous signal representing the reference inputs and/or the disturbances. The exogenoussignal is generated by an exosystem of the form

For convenience, we put the plant (1.7) and the exosystem (1.8) together into thefollowing form:

and call (1.9) a composite system with col(x, v) as the composite state.Two classes of feedback control laws will be considered in this section, namely,

1. Static State Feedback:

where Kx e Kmxn and Kv e nmxq are constant matrices.

2. Dynamic Measurement Output Feedback:

where z e Rnz with nz to be specified later, ym e Rpm for some positive integer pm

is the measurement output, and K e Rmxn, e RnXn, Q2 e R" tXpm are constantmatrices. It is assumed that ym takes the following form:

where Cm e R p m X n , Dm € R p m X m , and Fm e Rpmxq. A special case of the dynamicmeasurement output feedback control is the dynamic error output feedback controlwhen Cm = C, Dm = D, Fm = F, that is, ym = e. In many cases, the erroroutput e is not the only measurable variable available for feedback control. Usingthe measurement output feedback control allows us to solve the output regulationproblem for some systems that cannot be solved by the error output feedback control.

Denote the closed-loop system consisting of the plant (1.7), the exosystem (1.8), andthe control law (1.10) or (1.11) as follows:

where, under the static state feedback, xc = x and


and, under the dynamic measurement output feedback, xc = col(x, z) and

To describe the requirements on the closed-loop system (1.13), we first introduce thefollowing definition.

Definition 1.1. The closed-loop system (1. 13) is said to be exponentially stable if we havethe following.

Property 1.1. The matrix Ac is Hurwitz, that is, all the eigenvalues of Ac have negative realparts.

The closed-loop system is said to have output regulation property if the following holds.

Property 1.2. For all xco and v0, the trajectories of (1.13) satisfy

Linear Output Regulation Problem (LORP): Design a control law of the form ( 1 . 10) or(1.11) such that the closed-loop system satisfies Properties 1.1 and 1.2.

Remark 1.2. In what follows, a control law that solves the linear output regulation problemwill be called a servoregulator. In particular, if the control law is described by (1.10) or(1.11), then the controller will be called a static state feedback servoregulator or dynamicmeasurement output feedback servoregulator, respectively.

At the outset, we list various assumptions needed for solving the linear output regu-lation problem.

Assumption 1.1. A\ has no eigenvalues with negative real parts.

Assumption 1.2. The pair (A, B) is stabilizable.

Assumption 1.3. The pair

is detectable.

Remark 1.3. Assumption 1.1 is made only for convenience and loses no generality. In fact,if the linear output regulation problem is solvable by any controller under Assumption 1.1,then it is also solvable by the same controller even if Assumption 1.1 is violated. This isbecause Property 1.1 is simply a property of the plant data (A, B, C, D) and has nothing


to do with the exosystem, and because Property 1.2 is only concerned with the asymptotproperty of the closed-loop system. More specifically, the components of the exogenoisignals corresponding to the modes associated with the eigenvalues of AI with negative reparts will exponentially decay to zero and will in no way affect the asymptotic behavior <the closed-loop system so long as the closed-loop system has Property 1.1. Assumption 1is made so that Property 1.1, that is, the exponential stability of Ac, can be achieved ta state feedback. Assumption 1.3, together with Assumption 1.2, renders the exponentistability of Ac by the measurement output feedback

Lemma 1.4. Under Assumption 1.1, consider the controller (1.10) or (1.11). Assume theclosed-loop system (1.13) has Property 1.1. Then the following statements are equivalent:

(i) The closed-loop system has Property 1.2.(ii) The controller solves the linear output regulation problem,

(ni) There exists a unique matrix Xc that satisfies the following matrix equations:

Proof, (i) (ii). This is self-evident.(ii) (iii). The first equation of (1.16) is a Sylvester equation, which has a unique

solution Xc if A1 and Ac have no common eigenvalues (Appendix A). Since the closed-loop system satisfies Property 1.1, Ac is exponentially stable. Thus Assumption 1.1 andthe exponential stability of Ac guarantee the existence of Xc, satisfying the first equation of(1.16). Let =xc- Xcv. Then,

Since Ac is exponentially stable, To show (ii) (iii), assume the matrixXc also satisfies the second equation of (1.16); then

that is, the controller solves the linear output regulation problem. On the other hand, tshow (ii) (iii), assume the controller solves the linear output regulation problem; then,

for all u(t) = v(0) with any v(0) E. Rq. Due to Assumption 1.1, v(t) does not decay tozero for v(0) 0. Therefore, necessarily, CCXC + Dc = 0.

Remark 1.5.

(i) Lemma 1.4 gives a characterization of Property 1.2 in terms of the solvability of a setof linear matrix equations. This characterization allows the linear output regulationproblem to be studied using the familiar mathematic tool of linear algebra. Further, itwill be seen later that this lemma will render a natural translation of the requirements


on the closed-loop system into the requirements on the controller, thus leading to thesynthesis of the various controllers.

(ii) It is seen from the proof of Lemma 1.4 that if the output regulation problem is solvable,then there exists a subspace in Rn+n +q defined by the hyperplane Ccxc + Dcv = 0such that the trajectories xc(t) of the closed-loop system will approach this subspaceasymptotically.

Now let us first consider the static state feedback case where the controller is definedby two constant matrices Kx and Kv such that the closed-loop system is described by

That is,

Ac = A + BKX, Bc = E + BKV,

CC = C + DKX, DC = F + DKV.

The two matrices Kx and Kv will be called the feedback gain and the feedforward gain,respectively. The basic idea of designing the static state feedback controller is to usethe feedback gain to make the closed-loop system satisfy Property 1.1 while using thefeedforward gain to drive trajectories of the closed-loop system toward a subspace of Rn+q

defined by the hyperplane (C + DKx)x + (F + DKv)v = 0. This idea is best illustratedby the following result.

Lemma 1.6. Under Assumptions 1.1 and 1.2, let Kx render the exponential stability of( A + B KX). Then the linear output regulation problem is solvable by a static state feedbackcontroller (1.10) if and only if there exist two constant matrices Xc and Kv that satisfy thefollowing matrix equations:

Proof. Under Assumption 1.2, there exists Kx such that (A + BKX) is exponentially stable.Since equation (1.16) is exactly the same as equation (1.18) except that in (1.18) Kv,, is tobe determined, if Xc and Kv satisfy (1.18), Xc also satisfies (1.16) for the two particularmatrices Kx and Kv. On the other hand, if for some Kx and Kv, Xc satisfies (1.16), then Xc

and Kv also satisfy (1.18). The proof thus follows from Lemma 1 .4.

Lemma 1.6 immediately suggests the following way of synthesizing the desired staticstate feedback controller.

Step 1. Find a feedback gain Kx such that (A + BKX) is stable.

Step 2. Solve for both Xc and Kv from the set of linear equations (1.18). Then the staticstate feedback controller is given by

8 Chapter 1 . Linear Output Regulation

This approach, though straightforward to apply, has a drawback in that Xc and Kv

depend on the feedback gain Kx. Thus, every time, a redesign of the feedback gain neces-sitates a recomputation of Xc and Kv. A better approach can be obtained by making thefollowing linear transformation:

in equation (1.18), which leads to another set of linear matrix equations in unknown matricesX and U as follows:

These equations are completely determined by the plant data A, B, E, C, D, F, and AI. Itis clear that there exist X and U satisfying (1.21) if and only if, for any Kx Rmxn, thereexist Xc and Kv satisfying (1.18). Moreover, (X, U) and (Xc, Kv) are related to each otherby equation (1.20).

Equations (1.21), known as the regulator equations, are instrumental to establishingthe linear output regulation theory. In fact, in terms of the regulator equations, the abovediscussion can be summarized to yield the following result.

Theorem 1.7. Under Assumptions 1.1 and 1.2, let the feedback gain Kx be such that(A + BKx) is exponentially stable. Then, the linear output regulation problem is solvableby a static state feedback control of the form

if and only if there exist two matrices X and U that satisfy the linear matrix equations (1.21),with the feedforward gain Kv being given by

Remark 1.8. A systemic interpretation to the solution of (1.21) is given as follows. Firstconsider the special case where the exogenous signal is constant. Since AI = 0, equations(1.18) and (1.21) become

and, respectively,

Equations (1.22) mean, for each constant v, that Xcv is an equilibrium point of the closed-loop system at which the output is zero. Moreover,


Thus, for each constant v, Xcv is the steady-state state of the closed-loop system at whichthe output is zero. On the other hand, equations (1.23) mean, for each constant v, that Uv isthe input under which the open-loop plant has an equilibrium state Xv at which the outputis zero. Moreover, since Xc = X, and

for each constant v, whether or not the closed-loop system can be made to satisfy theoutput regulation property depends on the solvability of the regulator equations. The aboveinterpretation can be extended to the general case. Under any controller that solves thelinear output regulation problem, the trajectories of the closed-loop system from any initialstate xc(0) and v (0) satisfy

Correspondingly, the control input satisfies

Thus, if the linear output regulation problem is solvable at all, necessarily, all trajectoriesof the closed-loop system approach Xv(t), and the corresponding controls approach Uv(t).Thus, the steady-state behavior of the closed-loop system is completely characterized bythe solution of the regulator equations. For convenience, in what follows, Xv(t) and Uv(t)are called zero-error constrained state and zero-error constrained control, respectively. Inparticular, when v is constant, Xv is called zero-error constrained equilibrium.

An easily testable condition can be given with regard to the solvability of the regulatorequations as shown below.

Theorem 1.9. For any matrices E and F, the regulator equations (1.21) are solvable if andonly if the following holds:

Assumption 1.4. For all (A1), where a(A1) denotes the spectrum of AI,

Proof. The regulator equations (1.21) can be put into the following form:

Using the properties of the Kronecker product, which can be found in Appendix A, we cantransform (1.25) into a standard linear algebraic equation of the form

Qx = b,


where

Here the notation vec(.) denotes a vector-valued function of a matrix such that, for any

where for i = 1,. . . , m, Xi is the ith column of X. Thus, equation (1.25) is solvable for anymatrices E and F if and only if Q has full row rank. To obtain the condition under which Qhas full row rank, we assume, without loss of generality, that A 1 is in the following Jordanform:

where Ji has dimension ni such that n1 + n2 + + nk = q and is given by

A simple calculation shows that Q is a block lower triangular matrix of k blocks with itsith, 1 i k, diagonal block having the form

where

Clearly, Q has full row rank if and only if Assumption 1.4 holds.


In conjunction with Theorem 1.7, Theorem 1.9 immediately leads to the followingsufficient conditions for the solvability of the output regulation problem by the static statefeedback control of the form (1.10).

Corollary 1.10. Under Assumptions 1.1, 1.2, and 1.4, the linear output regulation problemis solvable by the static state feedback control (1.10).

Remark 1.11. If the pair (A, B) is controllable and the pair (C, A) is observable, then thosevalues of 1 at which the matrix

is not full rank are called the transmission zeros of the system. It is a generalization ofthe notion of zeros of the single-input, single-output systems to multi-input, multi-outputsystems. Thus Assumption 1.4 can be paraphrased by saying that the transmission zeros ofthe plant (1.7) do not coincide with the eigenvalues of the exosystem, and it is often simplycalled the transmission zeros condition. The plant (1.7) is called a minimum phase systemif all of its transmission zeros are on the open left-half complex plane. Thus a minimumphase system always satisfies the transmission zeros condition. •

Remark 1.12. A systemic interpretation of Assumption 1.4 can also be given in the samespirit as Remark 1.8. First consider the special case where A\ = 0. For this case, equation(1.24) actually takes the form

as AI =0. Correspondingly, the regulator equations are given by (1.23). Thus, (1.26) isboth necessary and sufficient for the plant to have a pair of zero-error constrained equilibriumand input for any E and F. A similar interpretation can be given to the case where AI 0.For every X (A1), let v be the eigenvector of AI associated with X. Then the solutionof the exosystem starting from v(0) = V is v(t) = v et . Thus, if the closed-loop systemhas Properties 1.1 and 1.2, there exist x Rn and u = Rm such that

Therefore, x and u must satisfy the following equations:

or, equivalently,


Clearly, equation (1.27) has a solution x and u for any E and F if and only if Assump-tion 1.4 holds. It should be noted that, for a particular pair of (E, F), the regulator equationsmay still have a solution even if Assumption 1.4 fails. This happens when

However, this case is not interesting since even arbitrarily small variations in (A, B, E,C,D,F) may fail (1.28).

When the state x and the exogenous signal v are not available for feedback butAssumption 1.3 holds, the measurement output feedback control of the form (1.11) canbe used to solve the linear output regulation problem. In this case,

Due to Lemma 1.4, we need to find a triple (K, g\, g2) such that Ac is exponentiallystable and (1.16) is solvable for Xc. To this end, we first translate the requirements on theclosed-loop system as given by (1.16) into the requirements on the controller (K, g\, g2) asgiven by the following result.

Lemma 1.13. Under Assumption 1.1, suppose there exists a dynamic measurement outputfeedback controller ( K , g \ , g2) such that the closed-loop system has Property 1.1. Then thefollowing are equivalent:

(i) The linear output regulation problem is solvable by the measurement output feedbackcontroller (K, gl ,g2).

(ii) There exists a matrix Xc that satisfies the following matrix equations:

(iii) There exist matrices (X, U, Z) such that X and U are the solution of the regulatorequations

and Z is the solution of the Sylvester equation

which satisfies


Proof. (i) (ii). This is actually Lemma 1.4 specialized to the measurement outputfeedback case.

(ii) «->• (iii). Assume (ii) holds. Partition Xc as

where X Rnxq and Z Rnzxq .Then (1.30) is the same as

which is the same as

Letting U = KZ in (1.35) shows that X and U satisfy the regulator equations (1.31), and Uand Z satisfy (1.32) and (1.33). This completes (ii) -> (iii). On the other hand, assume (iii)holds. We will show that X and Z satisfy (1.34) or equivalently (1.35). Indeed, substitutingU = KZ into equation (1.31) shows that X and Z satisfy the first and third equations of(1.35), and substituting U = KZ into (1.32) shows that Z satisfies the second equation of(1.35).

Now we turn to the construction of the triple (K, g\, g2)- Since we have alreadyknown how to synthesize a static state feedback controller which takes the plant state x andthe exosystem state v as its inputs, we naturally seek to synthesize a measurement outputfeedback controller by estimating the state x and the exogenous signal v. To this end, lumpthe state x and exogenous signals v together to obtain the following system:

Employing the well-known Luenburger observer theory suggests the following observer:

where L is an observer gain matrix of dimension (n + q) by pm.


Clearly, (1.37) can be put into the form

with

Theorem 1.14. Under Assumptions 1.1, 1.2, and 1.3, the linear output regulation problemis solvable by a measurement output feedback controller (Kx , Kv , L) given by (1.37) (equiv-alently, (K, g\, g2) given by (1.38)) if and only if there exists a pair of matrices (X, U) thatsatisfies the regulator equations

Proof. The "only if part is a consequence of part (iii) of Lemma 1.13. To show the "ifpart, first note that, by Assumption 1.2, there exists a state feedback gain Kx such that(A + BKX) is exponentially stable, and, by Assumption 1.3, there exist matrices L\ and L2

such that

is exponentially stable. Now let (X, U) satisfy the regulator equations, and let Kv =U - KXX, K = [Kx, Kv], and

A simple calculation gives

In (1.40), subtracting the first row from the second row and adding the second column tothe first column shows that Ac is equivalent to the following matrix

1.3. Linear Robust Output Regulation 15

Thus (Ac) = (A + BKX) U (AL}; that is, we have shown that the triple (Kx, Kv, L)(equivalently, (K, g\, g2)) renders the closed-loop system Property 1.1. To show that theclosed-loop system also satisfies Property 1.2, let

We will show that the triple (X, U, Z) satisfies the conditions of part (iii) of Lemma 1.13.Since the pair (X, U) satisfies the regulator equations by assumption, it suffices to show that

Indeed, using the definition of g\ given by (1.38) yields

Using U = KXX + Kv in (1.44) gives

upon noting that X and U satisfy the regulator equations. The proof is completed by theequivalence of (i) and (iii) of Lemma 1.13.

By Theorem 1 .9, the solvability of the regulator equations is guaranteed by the satis-faction of the Assumption 1.4. Thus we have the following corollary.

Corollary 1.15. Under Assumptions 1.1 to 1.4, the linear output regulation problem is solv-able by a measurement output feedback controller (Kx , Kv, L) given by (1.37) (equivalently,( K , g 1 , g 2 ) given by (1.38)).

1 .3 Linear Robust Output RegulationIn this section, we will further consider the linear robust output regulation problem in whicha controller has to be able to tolerate certain plant uncertainty. When the plant uncertaintyis taken into consideration, the class of linear time-invariant systems is described by


where x(t), u(t), and e(t) are the same as what are described in Section 1.2, and v(t) isagain generated by the same exosystem (1.8).

In (1.45), the matrices A, B, E, C, D and F represent the nominal part of the plantwhile A, B, and so forth represent the uncertain part. The entries of ( A, 5, E,

C, D, F) are allowed to take arbitrary values.It is convenient to identify the system uncertainties with a vector w in the Euclidean

space Rnw with w = vec ([ ]) and nw = (n + p) x (n + m + q). Thus, we canadopt the following convenient notation:

with

As a result, (1.45) can be written as follows:

For convenience of reference, the plant (1.46) and the exosystem (1.8) can be put togetherinto the following:

and (1.47) will be called the composite system.We consider two classes of feedback control laws which are somehow different from

those considered in the last section.

3. Dynamic State Feedback:

where z Rnz with nz to be specified later, and (K 1 , K 2 ,g 1 , g2) are constant matricesof appropriate dimensions.

4. Dynamic Output Feedback:

where, again, z Rnz with nz to be specified later, and (K, g1t g2) are constantmatrices of appropriate dimensions.


Remark 1.16. Due to the presence of the uncertain parameter w, the robust output regulationproblem that will be formulated shortly cannot be handled via the approach for solving theoutput regulation problem described in Section 1.2. It will be handled by a celebrateddesign methodology called the internal model principle. As a result, there exist no staticstate feedback control laws that can solve the robust output regulation problem, as will beshown in Lemma 1.21. On the other hand, as pointed out before, the measurement outputfeedback control is more general than the error output feedback case. However, in orderto better illustrate the mechanism of the internal model principle, we will focus on theerror feedback case when it comes to the robust output regulation problem. Remark 1.29will give a clue on how to synthesize a measurement output feedback controller under someadditional condition. To save the notation, we use the same notation z, g\, and g2 to describethe dynamic compensator in various controllers (1.11), (1.48), and (1.49). However, thedimension of z and the specific structure of the matrices g\ and g2 are totally differentamong these three different controllers.

Denote the closed-loop system consisting of the plant (1.46), the exosystem (1.8), andthe control law (1.48) or (1.49) as follows:

where, under the dynamic state feedback, xc = col(x, z) and

and under the dynamic output feedback, xc = col(x, z) and

Correspondingly, we use (Aco, BC0, CC0, Dco) or simply (Ac, Bc, Cc, Dc) to denote theclosed-loop system composed of the nominal plant and the control laws.

To describe the requirements on the closed-loop system (1 .50), we first introduce thefollowing definition.

Definition 1.17. The closed-loop system (1.50) is said to be exponentially stable at w = 0if the following property holds:

Property 1.3. The matrix ACO is Hurwitz, that is, all the eigenvalues of ACO have negativereal parts.


The closed-loop system is said to have robust output regulation property at w = 0 if thefollowing holds:

Property 1.4. There exists an open neighborhood W of w = 0 such that, for all XC0 and V0and for all w W, the trajectories of (1.50) satisfy

Remark 1.18. The set W does not have to be small in the statement of Property 1.4. It canbe shown later in Lemma 1.4 that if the closed-loop system (1.50) satisfies Properties 1.3and 1.4 for some open set W, then it also satisfies Property 1.4 for arbitrary set W in whichAcw is exponentially stable. In the following, we implicitly assume that W is an open setof w in which Acw is exponentially stable.

Now we are ready to state the problem precisely as follows.

Linear Robust Output Regulation Problem (LRORP): Design a control law of the form(1.48) or (1.49) such that the closed-loop system satisfies Properties 1.3 and 1.4.

Remark 1.19. Since Property 1.2 is clearly a particular case of Property 1.4, any controllerthat solves the linear robust output regulation problem also solves the linear output regulationproblem. In what follows, a control law that solves the linear robust output regulationproblem will be called a robust servoregulator. In particular, if the control law is describedby (1.48) or (1.49), then the controller is called a dynamic state feedback servoregulator,or dynamic output feedback servoregulator. It is noted that the dynamic output feedbackcontrol law (1.49) is a special case of the dynamic measurement output feedback controllaw (1.11).

In addition to Assumptions 1.1, 1.2, and 1.4 introduced in the last section, we needone more assumption in this section.

Assumption 1.5. The pair (C, A) is detectable.

This assumption is made so that Property 1.2 can be achieved by a dynamic outputfeedback control.

A result similar to Lemma 1.4 is given as follows.

Lemma 1.20. Under Assumption 1.1, consider the controller (1.48) or (1.49). Assume theclosed-loop system (1.50) has Property 1.3. Then the following statements are equivalent:

(i) The closed-loop system has Property 1.4.(ii) The controller solves the linear robust output regulation problem.

(iii) For each w W, where W is an open neighborhood of w = 0 such that Acw isexponentially stable, there exists a unique matrix Xcw that satisfies the followingmatrix equations:


Proof. (i) -> (ii). This is self-evident.(ii) «->• (iii). Since the closed-loop system satisfies Property 1.3, there exists an open

neighborhood W of w = 0 such that, for each w W, Acw is exponentially stable. Notethat, for each w W, the first equation of (1.53) is a Sylvester equation, which has a uniquesolution Xcw if and only if the spectra of AI and Acw do not coincide. Thus Assumption 1.1and the fact that Acw is exponentially stable for w W guarantee the existence of Xcw

satisfying the first equation of (1.53) for w W. Let x = xc — Xcwv. Then,

Since Acw is exponentially stable for each w; W, limt-+ x(t) = 0. Now if the matrixXcw also satisfies the second equation of (1.53) for w W, then

that is, the controller solves the linear robust output regulation problem. On the other hand,assume the controller solves the linear robust output regulation problem; then, for eachw W, such that Acw is exponentially stable,

for all v(t) = eAl tv(0) with any v(0) Rq. Due to Assumption 1.1, v(t) does not decay tozero for v(0) 0. Therefore, necessarily,

Similar to Lemma 1.4, Lemma 1.20 gives a characterization of Property 1.4 in terms ofthe solvability of a set of linear matrix equations that depend on the uncertain parameter w.This characterization also allows a natural translation of the requirements on the closed-loopsystem into the requirements on the controller, thus leading to the synthesis of the variouscontrollers. Nevertheless, the presence of the uncertain parameter w makes the solvabilityof the robust output regulation problem more difficult than the output regulation problem.In fact, let us first point out that the approach used in the last section cannot be carried overto the current case. As manifested by Lemma 1.6, under the static state feedback controller,the output regulation is achieved by appropriately designing a feedforward gain Kv thatis able to annihilate the steady-state tracking error. However, the feedforward gain, as asolution of equations (1.18), is dependent on the plant parameters. As the plant parameters(Aw, Bw, Ew, Cw, Dw, Fw) vary, the desired feedforward gain has to vary as a function ofw, too. As a result, there exists no fixed-gain static feedback controller that solves thelinear robust output regulation problem. The above argument can be formally stated in thefollowing lemma.

Lemma 1.21. There exists no static state feedback robust servoregulator for the linearrobust output regulation problem.

Proof. Assume there exists a static state feedback controller u = Kxx + Kvv that solvesthe linear robust output regulation problem. We will lead to a contradiction by using


Lemma 1.20. To this end, note that since Lemma 1.20(iii) applies to an open neighbor-hood W of w = 0, it also applies to any subset of W. Now fix W, and define a subset ofW, denoted by Ws, as follows:

By part (iii) of Lemma 1.20, for each w Ws (hence, for each Fw and Ew), there must exista matrix Xw such that

Therefore, equations (1.55) define a surjective linear mapping f : Rnxq -> R(n+p)xq. Butthis is impossible since n < (n + p).

As a result, we have to employ other techniques to synthesize controllers that do notrely on the solution of the regulator equations. Again, our starting point is Lemma 1 .20.In particular, part (iii) of Lemma 1.20 lends itself to the following idea of constructing acontroller for the linear robust output regulation problem. Find a compensator (g\ , g2) suchthat the following augmented plant:

has two properties:

(i) (1.56) can be stabilized by a state feedback control u = K\x + K2z or by a partialstate feedback control u = Kz.

(ii) For any state feedback control u = K\x + K2z or any partial state feedback controlu = Kz that makes Ac exponentially stable, the unique solution of the first equationof (1.53) also satisfies the second equation of (1.53) so long as Acw is exponentiallystable.

In this section, we will show that, under Assumptions 1.1 to 1.3, such a compensatorindeed exists. Further insights into the solvability of the linear robust output regulationproblem will be provided in the next section.

>\

Definition 1.22. Given any square matrix A\, a pair of matrices (g\, g2) is said to incor-porate a p-copy internal model of the matrix A\ if the pair (g\, g2) admits the followingform:

where (S1 , S2, S3) are arbitrary constant matrices of any dimensions so long as theirdimensions are compatible, T is any nonsingular matrix with the same dimension as g\,and (G\, G2) is described as follows:


where for i = 1 , . . . , p, B a constant square matrix of dimension di for some integer di,and is a constant column vector of dimension di such that

(i) Bi and a, are controllable.(ii) The minimal polynomial of A1 divides the characteristic polynomial of Bi.

Remark 1.23. Given any matrix A1 and any integer p > 0, it is always possible to find ap-copy internal model for the matrix A1. In fact, let

be the minimal polynomial of A]

Then, clearly, the pair (G1, G2) satisfies the conditions (i) and (ii) of Definition 1.22.Throughout this chapter, we will always assume A1 = AI. It is clear that, under As-sumptions 1.1 and 1.4, the matrix GI with B1 being described by (1.60) has the followingproperty.

Property 1.5. For all (G1),

Remark 1.24. We allow the dimensions of the matrices S1, S2, S3 to be zero and T be anidentity matrix. Therefore, the pair (Gi, G2) itself incorporates a p-copy internal model ofthe matrix AI. In the following, we will call the pair (G1, G2) a minimal p-copy internalmodel of A1 if the minimal polynomial of B1,, the characteristic polynomial of B1, and theminimal polynomial of A1 are the same for all i — 1,..., p.

Definition 1.25. A dynamic compensator of the form

is said to incorporate a p-copy internal model of the composite system (1.47) if the pair(G1

, G2) incorporates a p-copy internal model of the matrix A1. In particular, the dynamiccompensator

is called a p-copy internal model of the composite system (1.47).


Lemma 1.26. Under Assumptions 1.1 and 1.2, if the pair (Gi, GI) incorporates a p-copyinternal model of the matrix A1 with G2 satisfying Property 1.5, then the pair

is stabilizable.

Proof. Let

By the well-known PBH test, the pair (1.64) is stabilizable if and only if

Since (A, 5) is stabilizable, rank = n for all +. Also, det ( ) ^ 0for all . Thus

Write , where

Since (Gi, G2) is controllable, for all has rank n + nz. Since GI satisfiesProperty 1.5, has rank n-n 2 +p for all (Gi). Hence, by Sylvester's inequality,1

Combining (1.66) and (1.68) gives

Thus the pair (1.64) is stabilizable.

Lemma 1.27. Under Assumption 1.1, assume (G1, G2) incorporates a p-copy internal modelof A1. Let

1 rank A + rank B - n rankAB min{rankA, rankB} for any matrices and

rank and


A. A yv A

be exponentially stable, where A, B,C, D are any matrices with appropriate dimensions.Then, for any matrices E and F of appropriate dimensions, the following matrix equations:

have a unique solution X and Z. Moreover, X and Z satisfy

Proof. Since Ac is exponentially stable, by Assumption l.l, (A1) (Ac) = . Therefore,there exist unique matrices X and Z that satisfy equation (1.70). We need to show that theyalso satisfy (1.71). To this end, let

and

where 0 has as many rows as those of GI . Then (1.70) implies

Due to the block diagonal structure of GI and G2, we can assume p = 1 without loss ofgenerality. In this case, GI = \ and G2 = \. Since (G1, G2) is controllable, it can alwaysbe put into the following form:

where

Equating the first (nk — 1) rows of (1.76) gives

Let , j — 1,. . . , nk, denote the jth row of . Then expanding (1.74) gives


Substituting (1.77) into the last row of (1.76) gives

Thus we have y = 0 since the characteristic polynomial of GI is divisible by the minimalpolynomial of A\. As a result, X and Z must satisfy (1.71).

Remark 1.28. Assume the compensator z = g1Z + g2e incorporates a p-copy internalmodel of (1 .47). Define an augmented system as follows:

Suppose a state feedback controller of the form u = K\x + K2z stabilizes the augmentedsystem (1.79). Then the closed-loop system matrix Ac takes the form (1.69) with A =A + BK 1 , B = BK2, C = C + DK1 , D = DK2, E = E, and F = F. Since Ac isexponentially stable, by Lemma 1.27, the matrix equations (1.70) and (1.71) have a uniquesolution for any E and F. But equations (1.70) and (1.71) can be put into the form

with

The solvability of the above equations means the solvability of equation (1.53) for any w inan open neighborhood of w = 0. By Lemma 1.20, the dynamic state feedback controller(1.48) solves the robust output regulation problem of the given system. Similarly, if anoutput feedback control of the form u = Kz can stabilize the augmented system (1.79),then the output feedback control law (1.49) also solves the robust output regulation problemof the given system. The role of the internal model is to define the augmented system (1.79)whose stabilization solution leads to the solution of the robust output regulation problem ofthe original plant.

Remark 1.29. Assume, instead of the error output feedback, that we consider the measure-ment output feedback. Then the augmented system would become

From the proof of Lemma 1.27, it is not difficult to see that, if CmX + DmZ + Fm = 0implies CX + DZ + F = 0 (or, what is the same, that there exists a matrix T such thatC = TCm, D = TDm, F = TFm), then the stabilization solution of the augmented system(1.80) would still lead to the solution of the robust output regulation problem of the originalplant.


Combining Lemmas 1.20, 1.26, and 1.27 leads to the solvability conditions for thelinear robust output regulation problem by a dynamic state feedback control as follows.

Theorem 130. Under Assumptions 1.1 and 1.2, the following are equivalent:

(i) The transmission zeros condition (1.24) holds.(ii) The linear robust output regulation problem is solvable by a dynamic state feedback

controller (K1 K2, g\, g2).(iii) There exists an open neighborhood W of w = 0 such that for each w . W, the

following regulator equations:

have a solution (Xw, Uw).

Proof. (i) -» (ii). Due to Assumption 1.1 and the satisfaction of condition (1.24), thereexists a pair (G1, G2) that is the minimal p-copy internal model of the composite system,for example, the pair described in Remark 1.23. Let (g\, g2} = (Gi, G2). Since the pair(G1, G2) is the minimal p-copy internal model of A\ and GI satisfies Property 1.5, byLemma 1.26, (1.64) is stabilizable. Thus, there exists (K\, K2) such that

is exponentially stable. It follows from Lemma 1.27 that there exists Xc that satisfiesequations (1.70) and (1.71) with A = A + BK1 , B = BK2,C = C+DK1, D - DK2, E -E, and F = F. By Remark 1.28, the dynamic state feedback controller (K 1 , K2, GI, G2)solves the linear robust output regulation problem.

(ii) —> (iii). Assume that (K\, K2, GI, G2) solves the linear robust output regulationproblem; then by the equivalence of (i) and (iii) of Lemma 1.20, there exists an openneighborhood W of w = 0 such that for each w W, equation (1.53) has a solutionXcw = [Xw, Zw with Xw Rn. Let Uw = K 1 X w + K2ZW; then, clearly, Xw and Uw

satisfy (1.81).(iii) ->• (i). Since (iii) holds for w W, it also holds for w Ws. This is the same as

saying that the regulator equations have a solution for any (E, F). Thus, by Theorem 1.9 ,(i) must hold.

When the state is not available for feedback, it is possible to construct an outputfeedback servoregulator on the basis of the state feedback regulator, as shown below.

Theorem 131. Under Assumptions 1.1, 1.2, and 1.5, the following are equivalent:

(i) The transmission zeros condition (1.24) holds.(ii) The linear robust output regulation problem is solvable by a dynamic output feedback

controller ( K , g 1 , g2).


(iii) There exists an open neighborhood W of w = 0 such that for each w W, thefollowing regulator equations:


Proof. We only need to show (i) -» (ii) since the rest follows straightforwardly fromthe proof of Theorem 1.30. Due to Assumptions 1.1 and 1.2 and the satisfaction of con-dition (i), Theorem 1 .30 guarantees the existence of a dynamic state feedback controller(K 1 , K2 ,G1 , G2) that solves the linear robust output regulation problem. Thus (1.82) isexponentially stable. Also, by Assumption 1.5, there exists a constant matrix L Rnxp

such that A — LC is exponentially stable. Let K = (K\, K2), and let

Then, clearly, the pair (g\, g2) incorporates a p-copy internal model of the compositesystem. By Lemma 1.27, it suffices to show that

is exponentially stable. Indeed, a simple calculation gives

In (1.86), subtracting the first row from the second row and adding the second column tothe first column gives

Thus the spectrum of (1.87) is given by those of (1.82) and A — LC. That is, Ac as defined by(1.86) is exponentially stable. Thus, by Lemma 1.27 and Remark 1.28, (K, g1, g2) solvesthe linear robust output regulation problem.

1.4 The Internal Model PrincipleIn the previous section, we first showed that there exists no static state feedback controllerthat solves the linear robust output regulation problem. Then we constructed both dynamicstate feedback and output feedback controllers to solve the linear robust output regulationproblem. One may wonder what the underlying idea is for suggesting the controllers of theform given by Theorem 1.30, and what the minimal order of the controller is. This sectionis aimed to respond to these questions. In fact, we will show that the controllers given inTheorem 1.30 are of the minimal order.

1.4. The Internal Model Principle 27

Lemma 1.32. Under Assumption 1.1, assume that (K 1 , K2, G1 , G2) is any dynamic statefeedback controller that solves the linear robust output regulation problem. LetS :

Let K, be the kernel of S, that is,

Then

Proof. Assume that the dynamic state feedback control ( K 1 , K2 ,G1 , G2) solves the linearrobust output regulation problem. Then, by part (iii) of Lemma 1.20, (1.53) holds in an openneighborhood W of w; = 0, and hence holds in the subset Ws of W as defined in (1.54).Now partition Xcw as follows:

Then we can expand (1.53) for w E Ws as follows:

Equations (1.91) can be viewed as a linear mapping such that

Clearly (1.92) has a solution Xw and Zw for any Ew and Fw only if

That is,

Thus, necessarily,

Theorem 1.33 (Internal Model Principle). Under Assumption 1.1, assume that a dynamicstate feedback control (K1 ,K2 ,G1 , G2) solves the linear robust output regulation problem,and the pair (G1, G2) is controllable. Then G1 must have exactly p invariant factors, eachof which is divisible by the minimal polynomial of A1.

Proof. Let { ,, i = 1 , . . . ,« i} and (Ej, j = 1,. . . , n2} be the lists of invariant factors of G1

and A1, respectively, such that

be a Sylvester mapping such that


where , means divides ,. Let , i = 1 , . . . , n1, j = 1, . . . , n2, be the greatestcommon divisor of , and . By the result on the kernel of the Sylvester map (Appendix A),

Thus, using Lemma 1.32 gives

Combining (1.95) and (1.96) gives p n\. On the other hand, controllability of (g\, g2)implies n\ p. Thus we have n\ = p; that is, the matrix g\ has exactly p invariant factors.As a result, we can write (1.96) as


Since

equation (1.98) is possible only if

Since 1 is the minimal polynomial of AI, equation (1.100) means that the minimal poly-nomial of A1 divides each of p invariant factors of the matrix g\.

Remark 1.34. Since g\ must have exactly p invariant factors, each of which is divisible bythe minimal polynomial of AI , and since (g\, g2) is controllable, the pair (g\, g2) necessarilytakes the form given by (1.58) modulo coordinate transformations. Moreover, by Theorem1.33, the minimal dimension of the matrix g\ is greater than or equal to pnk, where nk isthe degree of the minimal polynomial of AI. On the other hand, Theorem 1.30 has givena pair (g l , g2) that defines a pnk dimensional compensator. Thus, it is concluded that theminimal order dynamic state feedback control law is equal to pnk , which is the degree ofthe minimal polynomial of AI multiplied by the dimension of the output e.

Since deg( ) deg( ), we have

1.5. Output Regulation for Discrete-Time Linear Systems 29

1.5 Output Regulation for Discrete-Time Linear SystemsThe discrete-time counterpart of system (1.7) is described by

where x(t) is the w-dimensional plant state, «(?) the m-dimensional plant input, e(t) thep-dimensional plant output representing the tracking error, and v(t) the q-dimensionalexogenous signal representing the reference inputs and/or disturbance, and is generated byan exosystem of the form

For convenience of reference, we can put the plant (1.101) and the exosystem (1.102)together as follows:

and call (1.103) the composite system.In this section, we will formulate the output regulation problem for discrete-time linear

systems of the form (1.103) and present the solvability conditions for the problem. For thispurpose, let us first describe two classes of feedback control laws as follows.



where with nz to be specified later, (K, G1 G2) are constant matrices with ap-propriate dimensions, and for some positive integer pm is the measurableoutput. It is assumed that

where i are constant matrices.

Clearly, controllers (1.104) and (1.105) are discrete counterparts of (1.10) and (1.11),respectively.

where and are constant matrices.


Denote the closed-loop system consisting of the plant (1.101), exosystem (1.102), andcontrol law (1.104) or (1.105) as follows:

where the four matrices Ac, Bc, Cc, and Dc corresponding to various control laws are definedby exactly the same equations given in (1.14) and (1.15).

We can define the output regulation problem for discrete-time linear systems asfollows.

Discrete-Time Linear Output Regulation Problem (DLORP): Design a control law ofthe form (1.104) or (1.105) such that the closed-loop system (1.106) satisfies the followingtwo properties.

Property 1.6. The matrix Ac is Schur; that is, all the eigenvalues of Ac have modulus smallerthan 1, and

Property 1.7. For all xc(0) and v(0), the trajectories of (1.106) satisfy

At the outset, we list the various assumptions needed for solving the above twoproblems.

Assumption 1.6. A\ has no eigenvalues with modulus smaller than 1.

Assumption 1.7. The pair (A, B) is stabilizable.


is detectable.

The solvability conditions for the discrete-time output regulation problem can beobtained in the same way as those for the continuous-time output regulation problem, andare thereby stated below without proof.

Theorem 1.35.

(i) Under Assumptions 1.6 and 1.7, the discrete-time linear output regulation problem issolvable by a static state feedback controller of the form

1 .6. Robust Output Regulation for Discrete-Time Linear Systems _ 31

if and only if there exist two matrices X and U that satisfy the following linear matrixequations:

(ii) Under Assumptions 1.6, 1.7, and 1.8, the discrete-time linear output regulation prob-lem is solvable by a measurement output feedback controller of the form (1.105) with

if and only if there exist two matrices X and U that satisfy (1.108).

Remark 1.36. Equations (1.108) take exactly the same form as the regulator equations(1.21) for continuous-time linear systems, and they also play the same role in studying thediscrete-time output regulation problem as equations (1.21) do in studying the continuous-time output regulation problem. Thus we will call (1.108) discrete-time regulator equations.Clearly, under Assumption 1.4, the discrete-time regulator equations are also solvable. In(1.107), the feedback gain Kx is such that (A + BKX) is Schur, and the feedforward gainKv is given by Kv = U — KXX. In (1.109), L is such that the matrix

is Schur.

1.6 Robust Output Regulation for Discrete-TimeLinear Systems

The discrete-time counterpart of the uncertain linear system (1.45) is described by

where x(t), u(t), e(t) are described as in equation (1.101) and v(t) is also generated by thesame exosystem (1.102).

As in (1.45), the matrices A, B, E, C, D, and F in (1.110) represent the nominal partof the plant, while A, B, and so forth the uncertain part of the plant. The entries of ( A,

B, E, C, D, F) are allowed to take arbitrary values. Let w = vec ([ ]).Then w Rnw with nw = (n+p)x(n+m+q). We will also use the following convenientnotation:


with

A0 = A, B0 = B, E. = E,

c0 = c, DO = D, FO = F.As a result, (1.110) can be written as follows:

For convenience of reference, we can put the plant (1.111) and the exosystem (1.102)together as follows:

and call (1.112) the composite system.As in the continuous-time case, we consider two classes of feedback control laws as

follows.

7. Dynamic State Feedback:

where z Rnz with nz to be specified later, and (K1 ,K 2 , g \ , g 2 ) are constant matricesof appropriate dimensions.

8. Dynamic Output Feedback:

where, again, z € Rnz with nz to be specified later, and (K, g1, g2) are constantmatrices with appropriate dimensions.

Denote the closed-loop system consisting of the plant (1.111), exosystem (1.102), andcontrol law (1.113) or (1.114) as follows:

where the four matrices Acw, Bcw, Ccw, and Dcw corresponding to various control lawsare defined by exactly the same equations given in (1.51) and (1.52), respectively. Also,we use (Aco, BCO, Cco, DCO), or simply (Ac, Bc, Cc, Dc), to denote the closed-loop systemcomposed of the nominal plant and the control laws.

1.6. Robust Output Regulation for Discrete-Time Linear Systems 33

We can define the robust output regulation problem for discrete-time linear systemsas follows.

Discrete-Time Linear Robust Output Regulation Problem (DLRORP): Design a controllaw of the form (1.113) or (1.114) such that the closed-loop system (1.115) satisfies thefollowing two properties.

Property 1.8. The matrix Aco is Schur.

Property 1.9. There exists an open neighborhood W of w = 0 such that, for all XC0 and V0

and for all w W, the trajectories of (1.115) satisfy

In addition to Assumptions 1.6 to 1.8, we need one more assumption as follows.

Assumption 1.9. The pair (C, A) is detectable.

To study the solvability conditions for the robust output regulation problem for discrete-time linear systems (1.112), we first note that the concept of the internal model as definedin Definition 1.22 also applies to the discrete-time linear systems (1.112) with the pair ofmatrices (G1, G2) given by (1.58) and (1.60). Moreover, under Assumptions 1.4 and 1.6,the matrix GI with i being described by (1.60) has Property 1.5. Thus we can readily ob-tain the following discrete-time counterparts of Lemmas 1.26 and 1.27 and Theorems 1.30and 1.31.

Lemma 1.37. Under Assumptions 1.6 and 1.7, if the pair (G\, G2) incorporates a p-copyinternal model of the matrix A\, and G\ satisfies Property 1.5, then the pair

is stabilizable.

Lemma 1.38. Under Assumption 1.6, assume (g\, g2) incorporates a p-copy internal modelof A1. Let

be Schur, where A, B, C, D are any matrices of appropriate dimensions. Then, for anymatrices E and F of appropriate dimensions, the following matrix equations:

have a unique solution X and Z. Moreover, X and Z satisfy


Theorem 1.39. Under Assumptions 1.6 and 1.7, the following are equivalent:

(i) The transmission zeros condition (1.24) holds.(ii) The discrete-time linear robust output regulation problem is solvable by a dynamic

state feedback controller (K\, K2, g1, g2).(iii) There exists an open neighborhood W of w = 0 such that for each w € W, the

following regulator equations:


Theorem 1.40. Under Assumptions 1.6, 1.7, and 1.9, the following are equivalent:

(i) The transmission zeros condition (1.24) holds.(ii) The discrete-time linear robust output regulation problem is solvable by a dynamic

output feedback controller ( K , g \ , g2).(iii) There exists an open neighborhood W of w = 0 such that for each w W, the

regulator equations (1.120) have a solution (Xw, Uw).

Remark 1.41. Both the dynamic state and the dynamic output feedback controllers for thediscrete-time linear systems can be constructed in the same way as those for continuous-timelinear systems. In particular, under Assumptions 1.6 and 1.7, and the transmission zeroscondition (1.24), there exists a pair of matrices (G1, G2) that incorporates a p-copy internalmodel of AI with GI satisfying Property 1.5. By Lemma 1.37, the pair

is stabilizable. Thus there exist feedback gains K\ and K2 such that the matrix

is Schur. Therefore, the dynamic state feedback control law of the form (1.113) solvesthe discrete-time robust output regulation problem. Under the additional Assumption 1.9,there exists an L such that A — LC is Schur. Let (K 1 , K2, G\, G2) be the dynamic statefeedback control law that solves the discrete-time robust output regulation problem. LetK = ( K 1 , K 2 ) ,

Then, by exactly the same argument as in the continuous-time case, the dynamic outputfeedback control law of the form (1.114) solves the discrete-time robust output regulationproblem.

Chapter 2

Introduction toNonlinear Systems

In this chapter, we review some fundamental concepts and results on nonlinear controlsystems that will be referred to in subsequent chapters. In Section 2.1, we present thedescriptions of various nonlinear systems. In Section 2.2, we summarize the Lyapunovstability results for both autonomous and nonautonomous nonlinear systems. Section 2.3introduces the input-to-state stability of a nonlinear control system. Section 2.4 reviewsthe center manifold theory. Section 2.5 reviews the discrete-time nonlinear systems andsummarizes the center manifold theory for maps. In Sections 2.6 and 2.7, we study thenormal form and zero dynamics for single-input, single-output and multi-input, multi-outputnonlinear systems, respectively. Finally, in Section 2.8, we close this chapter by introducingsome typical nonlinear systems.

The materials presented in this chapter are well known and can be found in manytextbooks on nonlinear systems. Thus proofs of almost all results are omitted. For an in-depth treatment of the nonlinear system theory, the reader is referred to books by Carr [7],Khalil [74], Isidori [63], [64], and Nijmeijer and Van der Schaft [88].

2.1 Nonlinear SystemsA general nonlinear dynamic system is described by

where x Rn, t [t0 , ), and f : Rn x R Rn. x is called the state of the system,X0 Rn the initial state, and to e R the initial time. The components of x and / aredenoted, respectively, by

35

36 Chapter 2. Introduction to Nonlinear Systems

If the function f(x,t) does not explicitly depend on the time t, then (2.1) can besimplified as follows:

A dynamic system of the form (2.1) is called a nonautonomous system, while (2.2) iscalled an autonomous system.

A general multivariable nonlinear control system is described by the following twoequations:

where x Rn is the plant state, u Rn the plant input, y Rp the plant output, andf : Rn x Rm x R -> Rn, h : Rn x Rm x R -+ Rp. The components of x, u, y, f, h aredenoted, respectively, by

If neither f(x, u, t) nor h(x, u, t) explicitly depends on the time t, then the system(2.3) and (2.4) can be simplified as follows:

We call the system (2.3) and (2.4) a nonautonomous nonlinear control system and the system(2.5) and (2.6) an autonomous nonlinear control system.

For many autonomous nonlinear control systems, the function f(x,u) is linear in theinput u, and the function h(x,u) does not depend on the input u explicitly. In this case, wecan write, with some abuse of the notation, h(x, u) = h(x) and f(x, u) = f ( x ) + g(x)ufor some functions / : Rn Rn, g : Rn Rnxm, and h : Rn Rp. Therefore, (2.5)and (2.6) can be further simplified as follows:

We call (2.7) an affine nonlinear control system.Note that g(x) can be expanded as g(x) = [ g 1 ( x ) , . . . , gm(x)], where gi : Rn ->• Rn

for i = 1,... ,m.

2.2. Stability Concepts for Nonlinear Systems 37

The class of nonlinear state feedback control laws takes the following form:

where : Rn x R ->• Rm. The composition of the control system (2.3) and the controllaw (2.8) gives x = f(x, k(x, t), t), which is a nonautonomous system of the form (2.1).In particular, when neither the function f(x, u, t) nor the function k(x, t) depends on texplicitly, we obtain an autonomous system of the form (2.2). Other types of nonlinearcontrol laws will be introduced in the subsequent chapters.

2.2 Stability Concepts for Nonlinear SystemsIn this section, we review the stability concepts for the system described by (2.1) whileviewing (2.2) as a special case of (2.1). Throughout this section, we assume that / :Rn x [to, ) —> Rn is piecewise continuous in f and locally Lipschitz in x; that is, thereexists a constant L such that

for all (x, t) and (y, t) in some open neighborhood of ( X 0 , to). Under this assumption, givenXQ, there exists some t1 > t0 and a unique continuous function x : [to, t1] Rn thatsatisfies (2.1). This time function x(t) is called a (local) solution of (2.1) over the interval[to, t1]. The solution x(t) is also called the state trajectory or simply the state of (2.1).

A constant vector xe € Rn is said to be an equilibrium point of the system (2.1) if

If a nonzero vector xe is an equilibrium point of (2.1), then we can always introducea new state variable z = x — xe and define a new system z = f(z + xe,t) which has z — 0as its equilibrium point. Thus, without loss of generality, we can always assume that theorigin of H" is an equilibrium point of the system (2.1) in this chapter.

Definition 2.1. The equilibrium point xe = 0 of the system (2.1) is

(i) Lyapunov stable at to if for any R > 0, there exists an r(R, ?o) > 0 such that, for all

(ii) unstable at to if it is not stable at to.(iii) asymptotically stable at to if it is stable at to, and there exists a 8 (to) > 0 such that

(iv) globally asymptotically stable at t0 if it is stable at to and \\x(t)\\ —>• 0 as t forall x(t0) Rn.

Definition 2.2. The equilibrium point xe = 0 of the system (2.1) is

(i) uniformly stable if for any R > 0, there exists r(R) > 0, independent of to, such that,for all


(ii) uniformly asymptotically stable if it is uniformly stable, and there exists a 8 > 0,independent of to, such that, for all uniformlyin to, that is, for any € > 0, there exists a T > 0, independent of to, such that, for all

whenever t >t0 + T.(iii) uniformly globally asymptotically stable if it is uniformly stable, and for any € > 0,

and any S > 0, there exists a T > 0, independent of to, such that, for all ,whenever t > t0 + T.

A typical nonlinear system whose equilibrium point is globally asymptotically stablebut not uniformly asymptotically stable is given as follows.

Example 2.3.

It can be verified that, for any initial state x(to) with any initial time to, the solution of(2.11) is

It can be seen that the equilibrium point is uniformly stable and globally asymptoticallystable. But, given e > 0 and S > 0, in order to make for allmust be greater than . Since this T cannot be made independent of to, theequilibrium point is not uniformly asymptotically stable. •

For the autonomous system (2.2), if x(t) is the solution of (2.2) satisfying the initialcondition x(to) = XQ, then x(t) = x(t + to) is the solution of (2.2) satisfying the initialcondition . Thus, we can always assume tQ = 0 for the autonomous system(2.2). Moreover, for the autonomous system (2.2), if the equilibrium point is stable (asymp-totically stable, globally asymptotically stable) at to, it is also uniformly stable (uniformlyasymptotically stable, uniformly globally asymptotically stable).

We now introduce the Lyapunov stability theory to determine the stability of theequilibrium point of the nonlinear systems (2.1) and (2.2), respectively. Let us first focuson the autonomous system (2.2). Assume that f(x) is C1 (continuously differentiable) inan open neighborhood of the origin of Rn. Define the Jacobian matrix of f(x) at the originas . Then we have the following theorem.

Theorem 2.4. The equilibrium point 0 of the system (2.2) is locally asymptotically stable ifall the eigenvalues of the matrix F have negative real parts, and is unstable if at least oneeigenvalue of the matrix F has positive real parts.

Now consider the control system (2.5) and (2.6). Assume f(x, u) and h(x,u) are C1

in an open neighborhood of (x, u) = (0,0) satisfying f(O, 0) = 0 and h(0,0) = 0. Let

2.2. Stability Concepts for Nonlinear Systems 39

Then the system

is a linear approximation of the system (2.5) and (2.6) and is called the Jacobian linearizationof system (2.5) and (2.6) at (x, u) = (0,0).

Suppose the pair (A, B) is stabilizable. Then there exists an m x n constant matrixK such that all the eigenvalues of the matrix A + BK have negative real parts. Applying alinear state feedback controller

to the system (2.5) results in an autonomous system

with x = 0 as an equilibrium point. Clearly, the Jacobian matrix of f(x, Kx) at the originis given by A + BK. Thus Theorem 2.4 concludes that a linear state feedback control is ableto (locally) stabilize the control system (2.5) provided that the Jacobian linearization of thesystem (2.5) at (x, u) = (0, 0) is stabilizable. If, in addition, the pair (C, A) is detectable,there exists a linear output feedback controller of the form

such that the equilibrium point of the closed-loop system composed of (2.5), (2.6), and(2.14) is locally asymptotically stable.

Remark 2.5. The case in which none of the eigenvalues of the matrix A has positive realparts, but at least one of them has zero real parts, is called the critical case. It can beshown that, in the critical case, the equilibrium point of the system (2.2) can be stable,asymptotically stable, or unstable. Thus, the Lyapunov linearization method cannot handlethe critical case. But the Lyapunov direct method to be introduced below or the centermanifold theory to be introduced in Section 2.4 is sometimes applicable to the criticalcase.

Definition 2.6. Let V : X R be a C1 function with X an open neighborhood of theorigin of Rn. V is said to be a (local) Lyapunov function of (2.2) if V(x) is positive definitein X, and

is (locally) negative semi-definite. If X = Rn, and V(x) is negative semi-definite for allx Rn, then V(x) is said to be a global Lyapunov function for (2.2).

Theorem 2.7. If the system (2.2) has a Lyapunov function V(x), then the equilibrium pointxe = 0 is Lyapunov stable. If, in addition, V(x) is locally negative definite in an openneighborhood of xe — 0, then the equilibrium point xe = 0 is asymptotically stable.


Theorem 2.8. Suppose the system (2.2) has a global Lyapunov Junction V(x), which isradially unbounded, that is,

and further, that V(x) is globally negative definite. Then the equilibrium point xe = 0 isglobally asymptotically stable.

Theorem 2.9. Consider an autonomous system of the form

where and . Suppose the equilibrium pointis asymptotically stable, and the equilibrium point of

is Lyapunov stable. Then the equilibrium point (x1 x2) = (0, 0) of (2.16) isLyapunov stable.

To describe the Lyapunov stability theory for the nonautonomous system (2.1), weintroduce the class 1C and class functions.

Definition 2.10. A continuous Junction a : [0, a) R+ is said to belong to class 1C if it isstrictly increasing and satisfies a(0) = 0, and is said to belong to class if, in addition,a = and a(r) as r —> .

Theorem 2.11. Let V : Rn x R R+ be a Cl function such that, for some class 1Cfunctions a( ) and defined on [0, d),

Then the origin is uniformly stable. If (ii) is replaced by

(iii) and allwhere a( ) is some class 1C function defined on [0, d),

then the origin is uniformly asymptotically stable.If d = and a( ) and are class functions, then the origin is uniformly

globally asymptotically stable.

2.3 Input-to-State StabilityIn this section, we will review the concept of the input-to-state stability for the systemdescribed by (2.3). This concept was introduced by Sontag in 1989 [100] and has rapidlybecome an effective tool in the analysis and design of nonlinear control systems. At thebeginning, we assume that f : Rn x Rm x [0, ) Rn is piecewise continuous in t andlocally Lipschitz in x and satisfies f(0, 0,t) = 0 for all t > t0 > 0.

2.3. Input-to-State Stability 41

Definition 2.12. A continuous junction : [0, a) x [0, ) -> is saw? to belong toclass KL if, for each fixed s, the function ( - , s ) is a class K function defined on [0, a) and,for each fixed r, the function ft(r, •) : [0, ) -» [0, ) is decreasing and (r, s) -> 0 ass .

While the stability of an equilibrium point is a property of the solution of a dynamicsystem of the form (2.1) excited by an initial state X0 , the input-to-state stability is concernedwith a relation between the trajectory of equation (2.3) and the initial state x (t0) and theinput u (t) of (2.3). We will use the notation to denote the set of all piecewise continuousbounded functions u : [t0 ) —» Rm with the supremum norm

Definition 2.13. The system (2.3) is said to be input-to-state stable (ISS) if there exist aclass KL function and a class 1C function y such that for any initial state x(t0) and anyinput function u(t) e L , the solution x(t) exists and satisfies

For an ISS system, the solution x(t) is bounded for all initial states x(t0) and all inputfunctions u(t) e L . In particular, when the input u is held at zero, the solution of (2.3)starting from any initial state x(t0) for any initial time to satisfies

Thus, the equilibrium point 0 of the unforced system x = f(x, 0, t) is uniformly globallyasymptotically stable.

On the other hand, for any x(t0) and any to, x(to) \\, t — t0) ->• 0 as t -> . Thus

That is, as t goes to , the solution x(t) will ultimately be bounded by a class K functionof ||u(-)||. Thus, the class K function y will be called a gain function of (2.3).

Remark 2.14. Since max{ y} < + y < max{2 , 2y] for any pair > 0, y > 0, anequivalent way to characterize the input-to-state stability of (2.3) is that there exist a classKL function ft and a class K function y such that for any initial state x(t0) and any inputfunction u(-) € L , the solution x(t) exists and satisfies

Definition 2.15. Let V : Rn x R R + be a C1 function. It is called an ISS-Lyapunovfunction for system (2.3) if there exist class K functions (•), (•) and (•) and a classK junction x( . ) such that

for all and


Theorem 2.16. If the system (2.3) has an ISS-Lyapunov function, then it is ISS with a gainfunction a-1 o a o x; that is, there exist a class JCC function and a class 1C functiony = l o a o x such that for any initial state x(to) and any input function M(-) 6 L^,, thesolution x(t) of (2.3) exists and satisfies (2.21).

Now assume V : H" x -» + is a C1 function and satisfies

for all x e ,", , and t > to, where a(•) is some class K function and (-) is someclass /C function. Let

with 0 < e < 1. Noting the fact that

and using (2.22) gives

for all \\x\\ > x(||u||), x e R n, u e L , and t > t0. Thus, V(x) is an ISS-Lyapunovfunction of (2.3). As a result, we obtain the following theorem.

Theorem 2.17. Let V : R n x R -> R + be a C1 function satisfying, for some class KJunctions a(-), (•) anda(-) and a class 1C function (.),

for a// x e Rn u , and t > t0. Then the system (2.3) is ISS with a gain function

a-1 oa oa r - 1 o ( )/0ra/ry 0 < 6 < 1.

Theorem 2.18 (Small Gain Theorem). Consider the following system:

where, for i — 1,2 and xt e Rni, fi are locally Lipschitz in col(x1, X2, u) and piecewisecontinuous in t, u e Rm, and, for all t >t0>0, f1(0,0, 0, f) = 0 and f2(0, 0,0, t) = 0.

Assume that the subsystem (2.25) is ISS viewing x\ as state and colfo, u) as input,and the subsystem (2.26) is ISS viewing xi as state and col(x1, u) as input; that is, thereexist class KL functions (-, •), (-, •) and class K functions Y 1 ( . ' ) >

2.3. Input-to-State Stability 43

such that for any initial state x1(to), and any input functions x2(.) , thesolution of (2.25) exists and satisfies, for all t > to > 0,

and for any initial state X2(to), and any inputfunctions , the solutionof (2.26) exists and satisfies, for all t > t0 > 0,

Further assume that

Then the system (2.25) and (2.26) is ISS viewing col(x1, x2) as state and u as input; thatis, there exist class KL Junctions B( ) and class K junctions y( ) such that for any initialstate x(to) and any input function u( ) , the solution of (2.25) and (2.26) exists andsatisfies, for all t > to > 0,

with the gain Junction given by any class k function satisfying

Two special cases of Theorem 2.18 are worth mentioning, namely, the case where f1does not depend on x2 explicitly and the case where f1 does not depend on u explicitly.Specializing Theorem 2.18 to these two cases gives the following corollary.

Corollary 2.19. Consider the following system:

Assume that the subsystem (2.32) is ISS viewing x1 as state and x2 as input, and that thesubsystem (2.33) is ISS viewing x2 as state and col(x1, u) as input; that is, there exist classKL Junctions B1( ), B2( ) and class K Junctions such that, for allt > to > 0,


Further assume that (2.29) holds. Then the system (2.32) and (2.33) is ISS viewingcol(x1, x2) as state and u as input, with its gain function given by any class 1C functionsatisfying

Corollary 2.20. Consider the following system:

Assume that the subsystem (2.37) is ISS viewing x1 as state and u as input, and the subsystem(2.38) is ISS viewing x2 as state and col(x1 ,u)as input; that is, there exist class ICC, functionsB 1 ( - , 0. B2( ) and class K. functions such that, for all t > f0 > 0,

Then the system (2.37) and (2.38) is ISS viewing col(x1, x2) as state and u as input with itsgain junction given by any class 1C function satisfying

Proof. If the inequality (2.39) holds, then the inequality (2.27) also holds for any class 1C

functions In particular, when = min (2.29) holds, and(2.31) becomes (2.41).

Remark 2.21. In Chapter 7, we will encounter systems of the following form:

where x e R" is the state, u e Rm is input, and : [to, ) -> is a piecewisecontinuous function with being a prescribed compact set of . The function is usedto model the system's uncertainty or disturbance. It is assumed that the function / is locallyLipschitz with respect to x and satisfies f(0, 0, ) = 0 for all . For each given

, system (2.42) can be viewed as a special form of (2.3). Thus we can still apply theinput-to-state stability concept to system (2.42). Moreover, if we let be the class ofpiecewise functions from [to, ) to with , being a prescribed compact set of , it ispossible to define the concept of robust input-to-state stability on (2.42) as follows.

Definition 2.22. Given , the system (2.42) is said to be robust input-to-state stable (RISS)with respect to if there exist a class function B and a class K function y, both of whichare independent of any such that for any initial state x(t0), any input functionu(t) e , and any piecewise continuous function , the solution x(t) exists andsatisfies inequality (2.18) or, equivalently, inequality (2.21).

2.4. Center Manifold Theory 45

The ISS-Lyapunov function for (2.3) defined in Definition 2.15 can also be extendedto the RISS-Lyapunov function for (2.42) if, in Definition 2.15, f ( x , u , f) is viewed as/(x, u, ), and all the class functions and the class K function x (•)are assumed to be independent of . Similarly, Theorems 2.16 to 2.18 also applyto system (2.42) if input-to-state stability is replaced everywhere by robust input-to-statestability, and all class KL, functions, all class functions, and class K functions in thesetheorems are assumed to be independent of . I

2.4 Center Manifold TheoryThe center manifold theory will play a crucial role in establishing the solvability of thenonlinear output regulation problem. In this section, we will present a few results from thecenter manifold theory for the autonomous system (2.2) with the assumption that /(•) isa locally defined sufficiently smooth function vanishing at the origin; that is, f(•) is a Ck

function for some sufficiently large integer k defined in an open neighborhood of the originof Rn and f(0) = 0. Readers are referred to Carr [7] for the proofs of these results.

Definition 2.23. Let X be an open set of Rn. A set of the form

where H : Rn -> Rn1 is a sufficiently smooth Junction and rank for all is called an (n — n 1)dimensional hypersurface in Rn.

A hypersurface is a special type of a manifold in Rn. A set M as described in (2.43)is called a (locally) invariant manifold of (2.2) if the solution of (2.2) starting from X0 e Mremains in M for sufficiently small t > 0.

Remark 2.24. If the system (2.2) has an invariant manifold M which contains the origin,then by the Implicit Function Theorem [93], there exist some partition x = col(x,x2)

with x1 e Rn1 and Rn2 with «2 = n — n1 and a locally defined sufficiently smoothfunction x1 = a(x2) satisfying a(0) = 0 such that H(a(x2), x2) = 0. Corresponding tothe partition x = col(x1,x2), we can decompose the system (2.2) as follows:

Let col(xl (t), x2(t)) be a solution of (2.44) starting from an initial state col(x1 (0), x2 (0)) €M. Then the fact that M is an invariant manifold for (2.2) implies that xl(t) = a(x2(t))for sufficiently small f > 0. Differentiating xl(t) = (x2(t)) with respect to t gives

The function a(-) must satisfy (2.45) for all solutions of (2.44) contained in M. Thus thefunction a (•) must satisfy the following partial differential equation:


In what follows, (2.46) will be called an invariant manifold equation. On the other hand,suppose (2.44) is a decomposition of (2.2) with x1 e Rn1 andx2 e Rn2. Let a : Rn2 -> Rn1

be any sufficiently smooth function satisfying (2.46) for all x2 in an open neighborhood ofthe origin of Rn2. Then it can be easily verified that the solution (xl(t), x 2 ( t ) of (2.44)starting from any sufficiently small initial state (x1(0) , x2(0)) satisfying x1(0) = (x 2(0))will satisfy x*(t) = (*2(t) for sufficiently small t > 0. Thus, the hypersurface H(x) =x1 — (x2} = 0 defines an invariant manifold for (2.2). •

Now consider the nonlinear system (2.2), and let F e Rn*n be the Jacobian matrixof f ( x ) at the origin. Assume F has 0 < n1 < n eigenvalues with nonzero real parts andn2 = « — n1 eigenvalues with zero real parts. Then there exists a nonsingular matrix Tsuch that, in the new coordinates col(y, z) = Tx where y e Rn1 and z Rn2, (2.2) can bewritten as follows:

with

where all the eigenvalues of A have nonzero real parts and all the eigenvalues of A1 havezero real parts.

Theorem 2.25 (Center Manifold Theorem). Consider the system (2.47). There exist anopen neighborhood Z c Rn2 of z = 0 and a C k - l function y : Z Rn1 with y(0) = 0,such that, for all z e Z,

Let

By Remark 2.24, M is an H2-dimensional invariant manifold for (2.47) passing through theorigin. Moreover, (2.48) implies that || (0) satisfies the following Sylvester equation:

which yields

That is, the tangent space to the manifold y = y(z) at the origin is the invariant subspaceof the linear mapping F spanned by all generalized eigenvectors of F associated with alleigenvalues of F with zero real parts. For this reason, the manifold M is called a centermanifold for (2.47) at the origin.

2.5. Discrete-Time Nonlinear Systems and Center Manifold Theory for Maps 47

Theorem 2.26. Consider the system (2.47). Let y(/) : Rn2 Rn1 be a C1 function withy(1)(0) = 0 and

where O (\\z\\1) '• Rn2 Rn1 is a sufficiently smooth function such that

is a finite constant for some integer l > 1. Then

where y(z) is any solution of equation (2.48) satisfying y(0) = 0.

Theorem 2.27 (Reduction Theorem). Consider the system (2.47). Suppose all the eigen-values of the matrix A have negative real parts. Let y(z) be a solution of equation (2.48)satisfying y(0) = 0. Then the equilibrium point of the system (2.47) at the origin is Lya-punov stable (asymptotically stable) (unstable) if and only if the equilibrium point v = 0 ofthe system

is Lyapunov stable (asymptotically stable) (unstable).

Theorem 2.28. Consider the system (2.47). Suppose all the eigenvalues of the matrix Ahave negative real parts and the equilibrium point of the system (2.52) at v = 0 is stable.Let col(y(f), z(t)) be a solution of equation (2.47) with col(y (0), z(0)) sufficiently small.Then, there exist positive constants and . such that, for all t > 0,

The center manifold described in Theorem 2.28 is called a stable center manifold.

2.5 Discrete-Time Nonlinear Systems and CenterManifold Theory for Maps

A discrete-time autonomous nonlinear dynamic system is described by the followingequation:

where x e Rn is called the state of the system, f : Rn Rn, X0 e Rn is the initial state, t0

is an integer called the initial time, and t = to, to + 1, to + 2, Without loss of generality,we assume t0 = 0 throughout this book.

48 Chapter 2. Introduction to Nonlinear System;

A constant vector xe e Rn is said to be an equilibrium point of the system (2.54) if

If a nonzero vector xe is an equilibrium point of (2.54), then we can always introducea new state variable z = x — xe and a new system z(t + 1) = f ( z ( t ) + xe) — f(xe) thathas ze = 0 as its equilibrium point. Thus, without loss of generality, we can always assumethat the origin of Rn is an equilibrium point of the system (2.54).

Definition 2.29. The equilibrium point xe = 0 of the system (2,54) is

(i) Lyapunov stable if for any R > 0, there exists an r(R) > 0 such that, for all ||x(0)|| <r(R), R for all t>0.

(ii) unstable if it is not stable.(iii) asymptotically stable if it is stable, and there exists a 8 > 0 such that as

t-> for all .(iv) globally asymptotically stable if it is stable and for all

Theorem 2.30. Assume that the function f ( x ) is C1 in an open neighborhood of the originof Rn and f(O) = 0. Let F e Rnxn be the Jacobian matrix of f(x) at the origin. Theequilibrium point 0 of the system (2.54) is asymptotically stable if all the eigenvalues ofthe matrix F have modulus smaller than 1 and is unstable if at least one eigenvalue of thematrix F has modulus greater than 1.

In the following, we will introduce four basic theorems of the center manifold theoryfor maps that are parallel to Theorems 2.25 to 2.28. These theorems will play the samerole to discrete-time nonlinear systems as Theorems 2.25 to 2.28 do to continuous-timenonlinear systems. We will assume that the function / that defines the nonlinear system(2.54) is Ck for some integer k > 2, and vanishes at the origin. Also assume the Jacobianmatrix F of f(x) at the origin has 0 < n1 < n eigenvalues with modulus not equal to 1 andn2 = n — n1 eigenvalues with modulus equal to . Then there exists a nonsingular matrixT such that, in the new coordinates col(y, z) = Tx, where y e Rn1 and z € Rn2, (2.54)can be written as follows:

with

where all the eigenvalues of A have modulus not equal to 1, and all the eigenvalues of A\have modulus equal to 1.

2.5. Discrete-Time Nonlinear Systems and Center Manifold Theory for Maps 49

Theorem 231 (Center Manifold Theorem for Maps). Consider the system (2.56). Thereexist an open neighborhood Z C Rn2 of z = O and a C k- l with k > 2 function y : Z -> Rn1

with y(0) = 0, such that, for all z € Z,

It can be easily verified that the function y has the property that the solution col(y (t), z (0)of (2.56) starting from any sufficiently small initial state col(y(0), z(0)) satisfying y(0) =y(z(0)) will satisfy y(t) = y(z(t) for all t as long as z(t) € Z. In other words, let

Then M is a (locally) invariant manifold of (2.56) in the sense that the solution of (2.56)starting from this manifold will remain in this manifold for all t as long as z(t) e Z.Moreover, a relation similar to equation (2.49) holds. For this reason, we call Md a centermanifold at the origin of the map col( , or a center manifold of (2.56)passing through the origin.

Theorem 232. Consider the system (2.56). Let be a Cl map withy(1)(0)=0 and

Then

where y(z) is any solution of equation (2.57) satisfying y(0) = 0.

Theorem 2.33 (Reduction Theorem). Consider the system (2.56). Suppose all the eigen-values of the matrix A have modulus smaller than 1. Let y(z) be a solution of equation(2.57) satisfying y(0) = 0. Then the equilibrium point of the system (2.56) at the originis Lyapunov stable (asymptotically stable) (unstable) if and only if the equilibrium pointv = 0 of the following system:

is Lyapunov stable (asymptotically stable) (unstable).

Theorem 2.34. Consider the system (2.56). Suppose all the eigenvalues of the matrix Ahave modulus smaller than 1 and the equilibrium point of the system (2.60) at v = 0 15stable. Let col(y(t), z ( t ) be a solution of equation (2.56) with col(;y(0), z(0)) sufficientlysmall. Then, there exists a solution v(t) of the system (2.60) such that, for all t = 0, 1,...,

where 8 and X are positive constants with X < 1.

The center manifold described in Theorem 2.34 is called a stable center manifold.


2.6 Normal Form and Zero Dynamics of SISONonlinear Systems

In this and subsequent sections, we will review the concepts of the normal form and zerodynamics for the class of affine nonlinear systems (2.7). This section will focus on thesingle-input, single-output (SISO) systems while multiple-input, multiple-output (MIMO)systems will be covered in the next section. Normal form and zero dynamics providestructural information on the nonlinear systems and will be used in Chapter 3 for studyingthe solvability of the nonlinear regulator equations. We will use a rather casual manner topresent these concepts while referring readers to Isidori [63] for all the proofs.

Throughout this section, we will call a sufficiently smooth function f : vector field in Rn. We begin by introducing some notations and terminology.

Definition 2.35. Let h : Rn —> R be a sufficiently smooth scalar function, and f : Rn a vector field. Then

Also, let g : Rn -> Rn be a vector field; then, for k = 0 , l , . . . ,

We will call the gradient of h(x) and Lfh(x) the Lie derivative of the function halong the vector field f.

Definition 2.36. The system (2.7) is said to have a relative degree r at x° if

(i)

for all k < r — 1 and for all x in an open neighborhood of x°, and

(ii)

2.6. Normal Form and Zero Dynamics of SISO Nonlinear Systems 51

Example 237. Consider a three-dimensional system of the form (2.7) with x = col(x1,x2,x3), and

where 0 is any real number. Simple calculation gives

Thus, by Definition 2.36, this system has a relative degree 2 at any point

Remark 238. The system may not have a well-defined relative degree at some point x°when there exists a positive integer r such that

(i)

for all k < r — 1 and for all x in an open neighborhood of x°, and

(ii)

However, there exists no open neighborhood of x° such that

in this neighborhood. For instance, in Example 2.37, if the function exp(x2) is replacedby sin x2, then the system does not have a well-defined relative degree at x° = 0. It willbe seen later that the ball and beam system to be described in Section 2.8 does not have awell-defined relative degree at x° = 0, either.

Assume the system has a relative degree r at x°. Then it can be verified that thetrajectory of the system starting from any x(0) sufficiently close to x° is such that, forsufficiently small t,

with

Solving the equation


where e R is viewed as a new input to the system (2.7), gives a state feedback controllerof the form

Applying (2.66) to system (2.7) results in a new system whose input-output relation obeys,for all sufficiently small t,

Returning to Example 2.37, a direct calculation gives

So the controller

gives the relationship

Remark 2.39. The control law (2.66) is called the input-output linearizing control law, as itresults in a linear input-output relation between the new input u and the output y. A furtherlinear feedback control of the form

where a0, a1 , . . . , cr-1 are such that

is a Hurwitz polynomial, will make the output y satisfy a stable linear differential equationas follows:

Thus, the output y(t) will approach 0 as t . The composition of (2.66) and (2.67)yields a state feedback control law of the form

which will be called an input-output linearization-based control law. It should be noted thatsuch a control law may not guarantee the asymptotic stability of the equilibrium point ofthe closed-loop system. In fact, the closed-loop system composed of (2.64) and the controllaw (2.68) with r = 2 is


The Jacobian matrix of (2.69) at the origin is given by

which has a characteristic polynomial

By the Lyapunov linearization method, when 0 > 0, the equilibrium point of the closed-loop system is unstable regardless of the choice of a0 and a1. It will be seen later that when0 > 0, the system is a nonminimum phase system, and the input-output linearization-basedcontrol law can only stabilize a minimum phase system. •

Next we will introduce the normal form and the zero dynamics for the system (2.7)with m = p = 1.

Definition 2.40. Let T(x) be a sufficiently smooth vector field defined on some open setX C Rn. T(x) is said to be a local diffeomorphism on X C Rn if there exists a sufficientlysmooth vector field T - 1 ( z ) define donXsuchtha tT - l (T(x) ) = x for all x X. IfX = Rn,then T(x) is said to be a global diffeomorphism on Rn.

If T(x) is a diffeomorphism on X C Rn, then we can define a coordinate transforma-tion z = T(x) for (2.7). Under the new state vector z, the system (2.7) can be expressed asfollows:

Moreover, if T(0) = 0, then T-1 (0) = 0. Thus, the origin z = 0 is also an equilibrium pointof (2.70) when T(0) = 0. We will say that (2.7) is diffeomorphic to (2.70) on X C Rn.

Remark 2.41. It can be shown that, if the system (2.7) has a relative degree r at x°, thenthe following row vectors:

are linearly independent [63]. As a result, if, at a point x°, the relative degree r of (2.7) iswell defined, then r < n. For convenience, let

We will call H(x) the H-vector of (2.7). Clearly, if the system (2.7) has a relative degree rat x°, then the rows of (x°) are linearly independent.


Now assume that the system (2.7) has a relative degree r at x° = 0. Let

By Remark 2.41, there exist n — r sufficiently smooth functions such that the vector field

is a diffeomorphism on an open neighborhood X of 0 and satisfies T(Q) = 0. LetZi = T i(x),i = 1, • • • , n. Then zi's satisfy the following equations:

We call (2.73) the normal form of the system (2.7).

Remark 2.42. By the Frobenius Theorem, it is possible to choosesuch that T(x) is locally invertible and

for x in an open neighborhood of x°. It is clear from (2.73) that this set of choices willrender the equations (2.73) a more special expression as follows:


Next, we will introduce the notion of a (local) output zeroing manifold for the generalnonlinear system described by (2.5) and (2.6).

Definition 2.43. Let M be a manifold containing the origin o f R n . M is called a (local)control invariant manifold of the system described by (2.5) if there exists a sufficiently smoothstate feedback control of the form u = k(x) with k(0) = 0 such that M is a (local) invariantmanifold ofx = f(x, k(x)), and it is called a (local) output zeroing manifold of the system(2.5) and (2.6) if it is a (local) control invariant manifold of (2.5) and is contained in thekernel of the mapping h(x, k(x)); that is, for all x € M, h(x, k(x)) = 0.

Returning to the affine nonlinear system (2.7), assume that the system (2.7) has arelative degree r at x° = 0 and let the function H(x) be defined as in (2.71). Then thereexists an open neighborhood X of the origin of Rn such that M = {x X \ H(x) = 0}is a manifold of dimension n — r. We will show that the set M is a (local) output zeroingmanifold of (2.7). In fact, by the definition of H(x), M is contained in the kernel of theoutput mapping h(x). Now assume that the normal form of (2.7) is given by (2.73). Definea state feedback control law as follows:

Then the closed-loop system is given by


Then it is clear from (2.77) that, under the state feedback control u = 0, for all initialstates z(0) = col( ) satisfying z1(0) = z2(0) = ••• = zr(0) = 0, thefirst r components of the solution z(t) of (2.77) starting from z(0) are identically zero forsufficiently small t. This is the same as saying that, in the original coordinates x, underthe state feedback control u = (—Lr

fh(x))/(LgLrjTlh(x}), for all sufficiently small t, the

solution x(t) of (2.7) starting from any initial state x(0) M belongs to M. Thus, M is aninvariant manifold of (2.7). Moreover, by the definition of M, h(x) = 0 for all x e M, andthus M is an output zeroing manifold of (2.7).

Remark 2.44. A system may have several output zeroing manifolds of different dimensions.An output zeroing manifold M is locally maximal if, for any other (local) output zeroingmanifold M', there exists an open neighborhood X of the origin of Rn such that X M' cX M. It can be shown that if the system (2.7) has a relative degree r at the origin, thenthe manifold defined by the hypersurface H(x) = 0 with H(x) being given by (2.71) is the(locally) maximal output zeroing manifold of (2.7). In fact, assume that M' is any other(local) output zeroing manifold of (2.7) under a sufficiently smooth state feedback controlu = k'(x); then, the closed-loop system has the property that, for any sufficiently smallx(0) M', the solution x(t) of the closed-loop system starting from any x(0) sufficientlyclose to x° is such that y(t) = h(x(t)) = 0 for all sufficiently small t > 0. Therefore, thederivatives of y(t) up to any orders are identically zero for all sufficiently small t > 0. Itfollows from (2.65) that x(t) M for all sufficiently small t > 0. •

We can put equation (2.77) into a more compact form. To this end, let

Then equation (2.77) becomes the following:

where


From (2.78), we can define an (n — r)-dimensional subsystem as follows:

which has an equilibrium point at — 0. This system is precisely the system that governs themotion of the last n — r components of z when the motion of the system (2.77) is restrictedto the manifold M. For this reason we call the subsystem (2.79) the zero dynamics of (2.7).

Remark 2.45.

(i) If a feedback control u = k(x) is required to render the output y(t) of the system(2.7) zero for all sufficiently small t, then, necessarily, the solution of the system (2.7)must be on the manifold M and the feedback control u = k(x) must take the form(2.76) with = 0. Thus, requiring the output y(t) of the system (2.7) to be zerofor all sufficiently small t > 0 uniquely identifies the zero dynamics (2.79) modulecoordinate transformations.

(ii) The subsystem (2.79) is identified from the normal form (2.73). Thus the represen-tation of the function also depends on the way that T(x) is chosen. Nevertheless,for different choices of T(x), the resulting zero dynamics are locally diffeomorphicto each other.

(iii) Let the Jacobian linearization of system (2.7) be

Then the transfer function of (2.80) is

On the other hand, it can be verified that the transfer function of (2.80) is also given by

where Q is the Jacobian matrix of (0, , 0) at = 0. Thus, if the triple (A, B, C)is controllable and observable, then the eigenvalues of Q coincide with the zeros of(2.80). Therefore, naturally, we call the system (2.7) minimum phase if all the eigen-values of Q have negative real parts or nonminimum phase if at least one eigenvalueof Q has positive real parts. In the critical case when none of the eigenvalues of Qhave positive real parts but at least one eigenvalue of Q has zero real parts, we define(2.7) to be minimum phase if the equilibrium point = 0 of the zero dynamics (2.79)is asymptotically stable and nonminimum phase if the equilibrium point = 0 of thezero dynamics (2.79) is unstable. Returning to Example 2.37, it can be verified thatthe zero dynamics of the system is = 0x3. Therefore, the system is nonminumumphase when 6 > 0.

(iv) It can also be verified that the matrix Q is unaffected under the class of input-outputlinearization-based control laws (2.68). Therefore, the input-output linearization-based control laws can only stabilize a minimum phase nonlinear system.


(v) The equilibrium point of the zero dynamics is called hyperbolic if all the eigenvaluesof Q have nonzero real parts. Otherwise it is called nonhyperbolic. In Chapter 3, wewill see that nonlinear systems whose zero dynamics has a nonhyperbolic equilibriumpoint present a hurdle to the solvability of the output regulation problem.

Remark 2.46. We can always choose the functions to be somen — rcomponents of x. In this case, the zero dynamics of (2.7) can be represented using thesen — r components of x. This procedure can be detailed as follows:

(i) By Remark 2.41, there exist r components of x denoted by X j 1 , . . . , Xjr such that

Denote the remaining n — r components of x by X j r + l , . . . , Xjn; then, by the ImplicitFunction Theorem, there exist an open neighborhood X0 of the origin of R(n-r) anda function a : XQ -> R+ satisfying (0) = 0 such that

Clearly, the function defined by

is invertible in an open neighborhood of the origin of x° = 0.

(ii) Let = col(z1, z2, • • •, zr) = col(h(*),..., Lrj~lh(x)), n = col(zr+i,..., zn) =col(j:yH.1,...,jrjfJ1),and

2.7. Normal Form and Zero Dynamics of MIMO Nonlinear Systems 39

Then the zero dynamics as defined in (2.79) has the following representation:

where, for j = 1 , . . . , n , gj is the jth component of g. It is noted that, in derivingthe above representation of the zero dynamics, there is no need to resort to the normalform of system (2.7).

2.7 Normal Form and Zero Dynamics of MIMONonlinear Systems

In this section, we will further extend such notions as the relative degree, normal form, andzero dynamics to MIMO affine nonlinear systems (2.7) with m > p > 1.

Definition 2.47. For each i = 1 , . . . ,p , the ith output yi of the system (2.7) is said to havea relative degree r( at a point x° if

(i)

for all k <ri — 1, and if for all x in an open neighborhood ofx°,

(")

The system (2.7) is said to have a vector relative degree [ri,..., rp] at a point x° if

(i) for all I <i < p, the ith output ht (x) has a relative degree r,- at x°, and(ii) the p x m matrix

has full row rank at x = x°.


Suppose, for each i = 1,. . . ,p, the output yi of the system (2.7) has a scalar relativedegree ri, at x = x°. Then the trajectory starting from any x(0) sufficiently close to x° issuch that

where

Let

Then y(r) and the input u can be related by the following equation:

Further, if the system has vector relative degree at x°, then D(x°) has full row rank; hence(D(x)DT(x)) is invertible in an open neighborhood of x°. Thus, the following equation:

is solvable for u. When p = m, the solution of (2.88) is unique. When p < m, the solutionof (2.88) is not unique. One of the solutions of (2.88) is given by

Under this control law, the trajectory starting from any x(0) sufficiently close to x° is suchthat, for all sufficiently small t > 0,

Thus the control law (2.89), which is an extension of (2.66), achieves the input-outputlinearization for the system (2.7) for the general case when m > p > 1.

Remark 2.48. For convenience of later reference, we will call D(x) and E(x) the decouplingmatrix and the £-vector of (2.7), respectively. Also, we extend the H-vector defined in (2.71)


for single-output systems to multi-output systems as follows:

We will still call this vector H-vector of (2.7). Again, it can be shown that if the system(2.7) has a vector relative degree {r1,..., rp] at x°, then the rows of (x0) are linearlyindependent [63].

Example 2.49. Consider a two-input, two-output system of the form (2.7) with x =C O l ( X 1 , X 2 , X 3 , X 4 ) :

Simple calculation gives

hi(x) = xi + x3 +x4, Lfhi(x) = xi, Lghi(x) = [1 exp(*2)Lh2(x) = x2, Lfh2(x) = x3+ x4, L2

fh2(x) = xi,L8h2(x) = [0 0], LgLfh2(x) = [1 0].

Thus, the system has well-defined scalar relative degree {n, r2} = {1,2} at any x°. Also,we have

Since rank D(x) = 2 for all x°, the system has a vector relative degree at any x°. Using(2.89) gives an input-output linearizing controller

which results in

To describe the normal form and zero dynamics for MIMO systems, assume thesystem (2.7) has a vector relative degree [ r 1 , . . . , rp] at jc° = 0, and by Remark 2.48, if


defr = r1+ + rp is less than n, then there exist n — r scalar functions Tr+1 (x),..., Tn (x)such that

is invertible in an open neighborhood of x° = 0 and satisfies T(0) = 0. Consider thecoordinates transformation

where z is an n-dimensional vector whose components are denoted by

In terms of z, (2.7) can be represented as follows:

Equation (2.94) can be viewed as an extension of (2.73) to the MIMO system andis called the normal form of the MIMO system (2.7). If can


be chosen such that L8Tj(x) = 0, j = r + 1,...,n, then the last n — r equations of(2.94) can be made independent of u. Unfortunately, for MIMO systems, it is in generalimpossible to make LgTj(x) = 0, j = r + 1,..., n. Nevertheless, it is possible to showthe existence of n — r sufficiently smooth functions such thatLgTj(x) = 0, j = r+1,...,«, under the assumption that the distribution span{g1,..., gm}is involutive near x = O.2

Next, let k(x, «) be any solution of (2.88), for example,

Then, applying the input transformation u = k(x, ) to (2.94) gives

It can be seen that system (2.96) exhibits a linear input-output relation.From system (2.96), it can be seen that, under the state feedback control = 0, for

all initial states z(0) = (z 10), . . . , zn(0)) satisfying thefirst r components of the solution z(t) of (2.96) starting from z(0) are identically zero forsufficiently small t. This is the same as saying that, in the original coordinates x, under thestate feedback control u = ue(x) = k(x, 0), for all sufficiently small t, the solution x(t) of

(2.7) starting from any initial state x(0) € M belongs to M, where M = {x e X \H(x) =0} with X an open neighborhood of the origin of Rn. Thus, M is an output zeroing manifoldof the MIMO system (2.7). Note that though ue(x) may not be unique when p < m, thismanifold is uniquely defined by H(x) = 0.

Next, we can define the zero dynamics of the MIMO system (2.7) similarly to that ofthe SISO system. Let z = col( , ), where

2See Chapter 5 of [63] for details.


Then the n — r equations of (2.94) governing zr+i , . . .,zn can be put into the followingcompact form:

From (2.97), we can identify an (n — r)-dimensional subsystem

Similar to the SISO case, this subsystem can be viewed as being induced by the requirementof rendering the output y ( t ) = 0 for all sufficiently small t > 0 under the state feedbackcontrol M = ue(x), and is thus called the zero dynamics of (2.7).

Remark 2.50. When p = m, ue(x) is uniquely defined by . Hence,the zero dynamics (2.98) is also unique within the coordinate transformations. When p < m,the zero dynamics (2.98) is not unique because ue(x) is not. In particular, the stabilityproperty of the equilibrium of (2.98) at the origin may depend on the particular functionue(x). To better illustrate this point, perform a partition u = col(ul, u2) with u1 e Rp,u2 € Rm - p . Then there exists a function ku : Hn+m-P -+ nm such that

regardless of the values of u2. Substituting u = ku(x, u2) into (2.97) gives

Let 0( , ) be any sufficiently smooth function satisfying 0(0, 0) = 0. It can be seen that,under the state feedback control u = ku(x, (0, ))U=T-1(o, ) when col( (0), (0)) M,

(t) will be identically 0 for sufficiently small t > 0, and (t) will be governed by thesystem

Thus, (2.100) can be viewed as a family of the zero dynamics of (2.7) parameterized byfunction 9 (0,77). It is interesting to note that 9 (0,;;) can be used to modify the zero dynamicsof system (2.7). I

Example 2.51. To find the normal form and zero dynamics of the system in Example 2.49,note that

Choose z4 = x4. Then


is invertible for all x R4. The inverse mapping of T(x) can be obtained as follows:

In terms of z, we can obtain the normal form of the system as follows:

Further, let

Then equation (2.96) takes the form

From equation (2.102), it is clear that (2.97) becomes

so that the zero dynamics of (2.102) is given by, according to (2.98),

Remark 2.52. Let Q = (0, 0, 0). Then, if the Jacobian linearization of system (2.7) iscontrollable and observable, the eigenvalues of Q coincide with the transmission zeros ofthe Jacobian linearization of system (2.7). As in the SISO case, we will call system (2.7)minimum phase if all the eigenvalues of Q have negative real parts, and nonminimum phaseif at least one of the eigenvalues of Q has positive real part. The critical case can also beclassified in a way similar to the SISO case.


Figure 2.1. Rotational/translational actuator.

2.8 Examples of Nonlinear Control SystemsIn this section, we introduce three well-known nonlinear systems, namely, the rotational/translational actuator (RTAC) system, the inverted pendulum on a cart system, and theball and beam system. It is well known that the asymptotic tracking and/or disturbancerejection problem associated with these systems present challenges to conventional input-output linearization-based method since, as will be seen shortly, all these three systems arenonminimum phase. Nevertheless, we will further show in later chapters that the outputregulation theory introduced in this book can practically solve the asymptotic tracking and/ordisturbance rejection problem associated with these systems.

The RTAC [2], [3]. The RTAC, depicted in Figure 2.1, consists of a cart of mass Mconnected to a fixed wall by a linear spring of stiffness k. The cart is constrained to haveone-dimensional travel. The proof-mass actuator attached to the cart has mass m and momentof inertia / about its center of mass, which is located at a distance e from the point aboutwhich the proof-mass rotates. Its motion occurs in a horizontal plane so that no gravitationalforces need to be considered. The motion of RTAC is described as follows:

where is the one-dimensional displacement of the cart, the angular position of the proofbody, F the disturbance, and u the control input. The coupling between the translational

2.8. Examples of Nonlinear Control Systems 67

and rotational motion is captured by the parameter , which is defined by

where 0 < < 1 is the eccentricity of the proof body.Letting x = col(x1 x2x3 x4) = col( ) and y = £ yields the following

state-space representation of (2.103):

where

where 1 — for all x3 and < 1 .When the disturbance F is zero, the RTAC system takes the standard form of (2.7).

Let us consider the problem of finding the normal form and the zero dynamics of the RTACsystem with F = 0. To this end, note that the relative degree of the system at the originis 2. Define the coordinates transformation z = T(x) as follows:

whose inverse transformation is given by

Under the new coordinates, the system can be described by its normal form as follows:


The zero dynamics of the RTAC system can be identified from (2.105) and (2.106), whichcan be put into the form

where = col(z1, z2) and = col(z3,z4). The zero dynamics of the system is defined by= (0, ) or, what is the same,

The Jacobian matrix of the zero dynamics at (0, 0) is

Since both of the eigenvalues of J are at the origin, we cannot determine the stability of theequilibrium point of the zero dynamics of the system based on the matrix j. Nevertheless, itcan be verified that the solution of this equation is given by sin( )and . Clearly, the equilibrium point of the zero dynamic is unstable. Therefore,the system is nonminimum phase.

The zero dynamics of the RTAC system with F = 0 can also be identified using thealgorithm described in Remark 2.46. As a matter of fact, simple calculation gives

Thus, applying the algorithm described in Remark 2.46 gives the partition xwith x1 = col(x1, x2) and x2 = col(x3,x4), and the following mapping:

as well as the zero dynamics of the RTAC system as follows:

It can be easily verified that the two representations (2.107) and (2.108) of the zero dynamicsare locally diffeomorphic to each other under the coordinate transformation z3 = x3 and

Inverted Pendulum on a Cart [31]. Shown in Figure 2.2 is a system known as the invertedpendulum on a cart. The pendulum is freely hinged to the cart, which is free to move on a


Figure 2.2. Inverted pendulum on a cart.

horizontal plane. The control available is a force applicable to the cart. The motion of thesystem can be described by

where M is the mass of the cart, m the mass of the block on the pendulum, / the length of thependulum, g the acceleration due to gravity, b the coefficient of viscous friction for motionof the cart, 0 the angle the pendulum makes with vertical, x the position of the cart, and u theapplied force. With the choice of the state variables xi = x, KI = x, xj = 9, x$ = 0,the state-space equations of the system are


We can put the system (2.109) into the following standard form:

where

and h(x) = x1.We can now see that the relative degree of (2.109) is 2, and simple calculation gives

Thus, applying the algorithm described in Remark 2.46 gives the partition x = col(x1, x2),with xl = col(x1, X2) and x2 = col(x3, x4), and the following mapping:

as well as the zero dynamics of the system (2.109):

Simple calculation shows that the Jacobian matrix of the zero dynamics has two eigenvaluesat the origin given by . Thus the system is nonminimum phase.

Ball and Beam System. Shown in Figure 2.3 is the ball and beam system. The motionequation of the system can be derived as follows:

where 6 and r are the beam angle and the ball position, respectively; r is the torque appliedto the beam; J is the moment of inertia of the beam; M and Jb are the mass and moment ofinertia of the ball, respectively; R is the radius of the ball; and G is the acceleration of gravity.


Figure 2.3. Ball and beam system.

Letting x = col( )state-space equations:

and y = r yields the following

where H

An input transformation of the form

will further simplify the system into the following:


which is in the standard form of (2.7) with

It can be verified that

and

Since there exists no open neighborhood of x° = 0 in which LgL2^h(x) = 2Hx\x$ is

identically zero, the relative degree of the ball and beam system is not well defined atx° = 0.

Chapter 3

Nonlinear OutputRegulation

3.1 IntroductionBeginning with this chapter, we turn to the nonlinear output regulation problem, a nonlinearanalog of the linear output regulation problem studied in Chapter 1. The typical scenariostudied by the nonlinear output regulation problem is shown in Figure 3.1, where we havea nonlinear plant described by

where x(t) is the plant state, u(t) the plant input, y(t) the plant output, and d(t) the distur-bance signal generated by an exogenous system described by

In addition, there is a reference input also generated by an exogenous system

The tracking error is defined by

To handle the nonlinear system described in (3.1), we need to go beyond the classof linear control laws described in Chapter 1 and resort to the class of nonlinear feedbackcontrol laws. A typical nonlinear feedback control law takes the following form:

where k and g are some nonlinear functions. This control law can be viewed as a nonlinearanalog of the linear dynamic output feedback control law (1.49) described in Chapter 1.

73

74 Chapter 3. Nonlinear Output Regulation

Figure 3.1. Nonlinear output regulation problem.

The objective of the control law is that the closed-loop system be stable in the sense to bedescribed later and that the output be able to track the reference input asymptotically in thefollowing sense:

The control systems as described in Section 2.8 are all nonlinear. To achieve bettersystem performance, it is desirable to design the control system based on the nonlinearmodel, thus leading to the nonlinear output regulation problem.

As in the linear case, we can combine the reference input r(t) and disturbance d(t)into a single exogenous signal vector v = col(r, d), thereby leading to a more compactnotation,

As a result, the plant with the tracking error e(t) as the output takes the following form:

Thus, we can focus on the problem of driving the output e of the system of the form (3.7) tozero asymptotically. It should be noted that the plant (3.7) can be viewed as a nonautonomousnonlinear system with x as the state, u as the input, and e as the output. On the other hand,we can put the plant (3.7) and the exosystem (3.6) together as follows:

Then the system (3.8), which is called a composite system, can be viewed as an autonomousnonlinear system with col(x, u) as the state, u as the input, and e as the output.

3.2. Problem Description 75

Since the plant inevitably contains uncertainties, it is desirable to further require thecontroller to be able to maintain the property of asymptotic tracking and disturbance rejectionin the closed-loop system regardless of model uncertainties. The problem of designing suchcontrollers for nonlinear systems is called the robust nonlinear output regulation problem,which will be studied in Chapters 5 to 1. In this chapter, we will focus only on the casewhere no uncertainty is present. The results are basically extensions of those of Section 1 .2to the nonlinear setting.

In the reminder of this chapter, we first give a precise description of the nonlinear outputregulation problem in Section 3.2. In Section 3.3, we study the solvability of the nonlinearoutput regulation problem. In analogy to the linear case, we give the characterization ofthe solvability conditions for the problem in terms of a set of constrained nonlinear partialdifferential equations, which are an extension of the regulator equations given in Chapter1 and are called the nonlinear regulator equations. In Section 3.4, we study the solvabilityof the nonlinear regulator equations, through the zero dynamics algorithm, for the classof nonlinear systems whose zero dynamics have a hyperbolic equilibrium. In Section 3.5,we study the output regulation problem of nonlinear systems whose zero dynamics is nothyperbolic. Finally, we study the problem of asymptotic disturbance rejection for the RTACsystem in Section 3.6.

3.2 Problem DescriptionWe consider a nonlinear plant described by

where x ( t ) is the w-dimensional plant state, u(t) the m-dimensional plant input, e(t) the p-dimensional plant output representing tracking error, and v(t) the q -dimensional disturbancesignal which can represent either disturbance signal or the reference input or both. It isassumed that v(t) is generated by a q-dimensional autonomous differential equation

We will consider two classes of control laws as follows.


where the function k(•, •) satisfies k(0,0) = 0.


where z(t) is the compensator state of dimension nz to be specified later, ym(t) =hm(x(t), u(t), v(t)) is the measurement output of dimension pm for some integer pm,and the functions and satisfy , andg(0,0) = 0.


The two control laws (3.1 1) and (3.12) are obviously nonlinear analogs of the linearstatic state feedback control law (1.10) and the linear dynamic measurement output feedbackcontrol law (1.11) described in Chapter 1 . It is noted that the dynamic measurement outputfeedback control law (3.12) is more general than the dynamic error output feedback controlas described in (3.5) because it always includes the error output feedback control as aspecial case by letting hm(x, u, v) = h(x, u, v). In Section 3.6, we will see that the outputregulation problem for the RTAC system is solvable by a measurement output feedbackcontrol but not any error output feedback control.

Our requirements will be imposed on the closed-loop composite system, that is, thesystem consisting of the plant (3.9), the exosystem (3.10), and the controller (3 . 1 1 ) or (3. 1 as follows:

where, under the static state feedback control, xc = x and

and under the dynamic measurement output feedback control, xc = col(x, z) and

For simplicity, all the functions involved in this setup are assumed to be sufficientlysmooth and defined globally on the appropriate Euclidean spaces, with the value zero atthe respective origins. Our results will be stated locally in terms of an open neighborhoodV of the origin in Rq, and we implicitly permit V to be made smaller to accommodatesubsequent local arguments.

Nonlinear Output Regulation Problem (NORP): Design a controller of the form (3.11)or (3.12) such that the closed-loop system has the following two properties.

Property 3.1. For all sufficiently small XC0 and V0, the trajectories col(xc(t), v(t) of theclosed-loop composite system (3.13) exist and are bounded for all t > 0, and

Property 3.2. For all sufficiently small xc0 and VQ, the trajectory col(xc(t), v(t)) of theclosed-loop composite system (3.13) satisfies

Remark 3.1. By Definition 2.2, Property 3.1 is guaranteed if the equilibrium point ofthe closed-loop composite system (3.13) at col(xc, v) = col(0,0) is stable in the senseof Lyapunov. Moreover, by Theorem 2.27 and Assumption 3.1, to be introduced later,


the equilibrium point of the closed-loop composite system (3.13) at col(xc, v) = col(0,0)is stable in the sense of Lyapunov if the closed-loop composite system has the followingproperty.

Property 3.3. All the eigenvalues of the matrix

have negative real parts.

As it is quite straightforward to achieve Property 3.3 by using a linear feedback controlunder Assumptions 3.2 and/or 3.3 to be given below, we often impose Property 3.3 insteadof Property 3.1 on the closed-loop system. We will say that a controller of the form (3.11)or (3.12) solves the output regulation problem with exponential stability if it makes theclosed-loop composite system (3.13) satisfy Properties 3.2 and 3.3.

The output regulation problem that has just been described is of local nature in thesense that the desirable properties imposed on the closed-loop system hold only for suf-ficiently small initial states of the closed-loop composite system (3.13). Thus the aboveproblem can be more precisely called the local nonlinear output regulation problem. Later,we will further study the global nonlinear output regulation problem in the sense to bedescribed in Chapter 7.

If there exists a controller such that the closed-loop system satisfies Properties 3.1and 3.2, we say that the (local) nonlinear output regulation problem is solvable, and thecontroller is called a nonlinear servoregulator. In particular, the controller in the form of(3.11) is called a state feedback servoregulator, and the controller in the form of (3.12)is called a measurement output feedback servoregulator. Alternatively, we say that thecontroller achieves asymptotic tracking and disturbance rejection in the plant.

Various assumptions needed for the solvability of the above problem are listed below.

Assumption 3.1. The equilibrium of exosystem (3.10) at v = 0 is Lyapunov stable, and allthe eigenvalues of |2(0) have zero real parts.

Assumption 3.1'. The equilibrium of the exosystem (3.10) at v = 0 is Lyapunov stable,and there is an open neighborhood of v = 0 in which every point is Poisson stable in thesense to be described in Remark 3.2.


is stabilizable.

Assumption 33. The pair

is detectable.


Remark 3.2. A point v° Rq is said to be Poisson stable if the solution v(t, u°) existsfor all t e 'R, and for each open neighborhood V° of v° and if for any real number T >0, there exists a time t1 > T such that v(t1 , u°) € V° and a time t2 < — T such that

Remark 3.3. Assumption 3.1 is more restrictive than its linear counterpart Assumption.1. For example, it does not accommodate the ramp function. This is because we require

that all trajectories of the closed-loop composite system (3.13) starting from sufficientlysmall initial states be bounded. Thus, we have to exclude any unbounded signals suchas the ramp signal. Assumption 3.1' is a somewhat strengthened version of Assumption3.1. It always implies Assumption 3.1. Assumption 3.1' is only used for establishing thenecessary condition for the solvability of the output regulation problem and is not essentialfor our development. Assumption 3.2 guarantees that the plant can be locally stabilizedby a state feedback control, and Assumption 3.2 together with Assumption 3.3 guaranteesthat the plant can be locally stabilized by a measurement output feedback control basedon an estimation of the composite state col(x, u). It is noted that the error output e isalways measurable, but the measurement output ym does not have to be the error output e.Thus, in some cases, for example, the RTAC system to be studied in Section 3.6, the outputregulation problem may be solvable by the measurement output feedback but not the erroroutput feedback control. •

Example 3.4 (RTAC). Consider the RTAC system described in Section 2.8. Our objectiveis to design a state or measurement output feedback controller such that, despite the pres-ence of a sinusoidal disturbance of the form F(t) = Am sin t, the closed-loop system isasymptotically stable, and the position of the cart can asymptotically approach the origin.For this purpose, let us introduce the following exosystem:

with

Let h(x,v) = x1. Then the disturbance rejection problem can be formulated as an outputregulation problem of the following composite system:

Assuming that the position of the cart x] and the angular position of the proof-mass x3

are measurable, then as will be shown in Section 3.6, the above output regulation prob-lem is solvable by a dynamic measurement output feedback control with hm(x,u,v) =Col( ).

Example 3.5 (Asymptotic Tracking of Inverted Pendulum on a Cart). Consider theproblem of designing a state or output feedback controller for the inverted pendulum on a

3.3. Solvability of the Nonlinear Output Regulation Problem 79

cart system described in Section 2.8 such that the position of the cart can asymptoticallytrack a sinusoidal input y (t) = Am sin t. For this purpose, we need to design a feedbackcontroller to locally stabilize the closed-loop system and to achieve

To this end, again we can introduce the same exosystem as described in (3.18) and (3.19).Then, clearly, yd(t) = v1(t)- Let h(x, v) = x1 — v1. Then, the above asymptotic trackingproblem can be formulated as the output regulation problem of the following compositesystem:

We will show in Chapter 4 that the output regulation problem for this system is solvable byeither state feedback control or error output feedback control. •

3.3 Solvability of the Nonlinear OutputRegulation Problem

The idea of synthesizing a controller to solve the nonlinear output regulation problem issimilar to what has been used to solve the linear output regulation problem, that is, using afeedback control to achieve Property 3.3 and a feedforward control to achieve Property 3.2.Since Property 3.3 is a property of the linearization of the plant, it can be achieved by thesame control techniques as used in Chapter 1 based on Lyapunov's linearization method.However, in the present case, the feedforward control is much more difficult to find since, aswill be seen shortly, it is determined by a set of nonlinear partial differential and algebraicequations, which is a nonlinear analog of the regulator equations encountered in Chapter 1.In this section, we will focus on relating the solvability of the nonlinear output regulationproblem to that of the nonlinear regulator equations. Solvability of the nonlinear regulatorequations will be given only for the special case where the exogenous signals are constant.The more general case will be studied in the next section.

We first establish a result parallel to Lemma 1.4.

Lemma 3.6. Under Assumption 3.1', suppose the closed-loop composite system (3.13)resulting from the controller (3.11) or (3.12) has Property 3.3. Then, it also has Property3.2 if and only if there exists a sufficiently smooth function Xc (v) with xc (0) = 0 that satisfies,for v e V, where V is an open neighborhood 0/0 6 Rq, the following partial differentialequations:

Proof. First note that Assumption 3.1' implies Assumption 3.1; thus the exosystem has astable equilibrium point at the origin and all the eigenvalues of its Jacobian matrix have zero


real parts. Since the closed-loop composite system (3.13) has Property 3.3, by Theorem2.25, there exists a center manifold for the closed-loop composite system (3.13). That is,there exists a sufficiently smooth function xc (v ) with Xc(0) = 0 that satisfies (3.23) forv e V. Moreover, by Theorem 2.27, the equilibrium point of the closed-loop system (3.13)at the origin is Lyapunov stable. Thus, the solution of the closed-loop composite system(3.13) starting from any sufficiently small initial state exists for all t > 0.

(If part): Since the function Xc(u) with Xc(0) — 0 that satisfies (3.23) for v e V definesa center manifold xc — Xc(v) for the closed-loop composite system (3.13), by Theorem 2.28,there exist positive constants 8 and A such that for all sufficiently small xc(0) and u(0), thetrajectories col(xc(t), u(f)) of the closed-loop composite system (3.13) satisfy

Furthermore, there exists a compact set Sc in Rn+nz+q such that, for t > 0,col( Sc, coi(xc(v(f)), w(0) e Sc. Also, there exists a finite constant L such that

for (xc, u) € Sc. Thus, if the function Xc(v) also satisfies (3.24), then we have

that is, the closed-loop system also has Property 3.2.(Only if part): Assume the closed-loop system has both Property 3.2 and Property 3.3,

yet (3.24) is not true. Then there exists a sufficiently small vO e V such that the solutionof the closed-loop system (3.13) satisfying col(xc(0), u(0)) = col(Xc(vo), V0), denoted bycol(xc(f, XC(VQ)), v(t, i>0)), exists for all t > 0 and satisfies

yet

Thus there exists an open neighborhood Vb C V of VQ and some real number R > 0 suchthat

for all v e V0. Clearly, xc(t, X c (V 0 ) ) = Xc(v(t, u0)), since xc(0, Xc(vo)) = Xc(v0) =Xc(u(0, UQ)), and (3.23) implies


But, since the exosystem satisfies Assumption 3.1', we can assume that v0 is small enoughso that it is Poisson stable. Therefore, given any T > 0, there exists t\ > T such thatV(t1 , VQ) € VQ. Thus,

which contradicts (3.28).

In what follows, we call the manifold xc = xc(v) , where Xc(v) satisfies (3.23) and(3.24), a zero error center manifold for (3.13).

Remark 3.7. A systemic interpretation to Lemma 3.6 can be given as follows. First considerthe special case where the exogenous signals are constant. Then, (3.23) and (3.24) reduceto the following algebraic equations:

since a(v) = 0 in this case. Thus, the solution Xc(v) of (3.29) defines an equilibriummanifold of the closed-loop compositesystem on which the output is identically zero. For the general case, the existence of thesufficiently smooth function Xc(u) satisfying (3.23) simply says that the manifold Mc isa stable center manifold of the closed-loop composite system (3.13). Thus the trajectorycol(x(t), v>(0) of the closed-loop composite system (3.13) starting from any sufficientlysmall initial state col(x(0), v(0)) will approach this manifold asymptotically. The fact thatXc(v) also satisfies (3.24) means that the center manifold Mc is contained in the kernelof the output mapping hc(xc, v). Thus, as the trajectory approaches the center manifold,the output e will approach zero asymptotically. Lemma 3.6 has also led to an equivalentcharacterization of Property 3.2 in terms of a set of partial differential and algebraic equationsresulting from the center manifold theory. Thus the asymptotic property of the system canbe addressed using the center manifold theory. Also, we emphasize that Assumption 3.1'is only used for establishing the necessary condition. It suffices to use Assumption 3.1 toestablish the sufficient condition.

Next we will establish the solvability of the state feedback output regulation problemin terms of the given plant.

Theorem 3.8. Under Assumptions 3.1' and 3.2, the nonlinear output regulation problemwith exponential stability is solvable by a static state feedback control of the form (3.11) ifand only if there exist two sufficiently smooth functions x(u) and u(v) defined for v Vsatisfying x(0) = 0 and u(0) = 0 such that

Proof. Assume a controller of the form u = k(x, v) solves the nonlinear output regulationproblem. Then, by Lemma 3.6, there exists a sufficiently smooth function Xc(u) that satisfies


(3.23) and (3.24) for v e V. Let x(v) = xc(u) and u(u) = k(x(u), v). Then, x(u) andu(v) satisfy (3.30). On the other hand, assume x(y) and u(v) satisfy (3.30) for v V. LetKx e Rmxn be any constant matrix such that the eigenvalues of the following matrix:

have negative real parts. Due to Assumption 3.2, Kx always exists. Let

Then, the closed-loop system (3.13) under k(x, v) satisfies Property 3.3. Moreover, lettingXc(u) = x(v) leads to

as x(v) and u(v) satisfy the regulator equations (3.30). By Lemma 3.6, the controller solvesthe nonlinear output regulation problem.

Remark 3.9. Equations (3.30) are clearly a nonlinear analogue of the linear regulatorequations (1.21) encountered in Chapter 1. In fact, suppose equations (3.9) and (3.10) arelinear, that is,

where A, B, E, C, D, F, and AI are constant matrices of appropriate dimensions. Letx(u) = Xv and u(u) = Uv for some matrices X and U. Then equations (3.30) become

Since equations (3.33) hold for all v € V, they are equivalent to the following:

which are exactly the linear regulator equations (1.21).

Remark 3.10. We can also give a systemic interpretation to Theorem 3.8. First consider thespecial case where the exogenous signals are constant. The nonlinear regulator equations(3.30) are reduced to the following algebraic equations:


The solution of (3.35) gives the desired control u(u) under which the plant has an equilibriumstate x(v) at which the output is identically zero. For the general case, the solvability ofthe regulator equations (3.30) simply means that the composite system (3.8) has an outputzeroing manifold characterized by

In fact, the first equation of (3.30) means that M is a control invariant manifold of thecomposite system (3.8) rendered by the state feedback control u = u(v), and the secondequation of (3.30) means that this manifold is contained in the kernel of the output mappingh(x, u(u), v). Thus, Theorem 3.8 can be interpreted as follows: if the composite systemhas an output zeroing manifold as defined by the solution of the regulator equations (3.30),and the plant satisfies Assumption 3.2, then there exists a state feedback control u — k(x, v)such that the output zeroing manifold M is also a stable center manifold Mc of the closed-loop composite system (3.13) which is contained in the kernel of the mapping hc(xc, u).Note that x(u) can be viewed as the steady-state state of the closed-loop system since thetrajectory xc(t) of the closed-loop system starting from any sufficiently small initial state(xc(G), v(0)) necessarily satisfies, by (3.25),

Correspondingly, the control input also approaches its steady state in the following sense:

By the same token as Remark 1.8, we will call the functions u(t>) and x(u) the zero-error constrained input and zero-error constrained state for the plant and the exosystem,respectively.

In the linear case, the solvability of the regulator equations can be related to thelocations of the system's transmission zeros. For the nonlinear case, a similar condition canalso be established. Here we only study the special case when the exogenous signals areconstant. The general case will be studied in Section 3.4.

Proposition 3.11. Under the assumption that the exogenous signals are constant, thereexist sufficiently smooth Junctions u(v) and x(v) satisfying equations (3.35) if

Proof. The conclusion is a straightforward application of the Implicit Function Theorem.


Remark 3.12. Let

Then

It can easily be verified that and satisfy

Thus, if any state feedback controller of the form u = k(x, u) with fc(0, v) = 0 stabilizesthe equilibrium point at the origin of the system jc = f(x,u,v), then the state feedbackcontroller u = u(v) + k(x — x(v), u) solves the output regulation problem of the originalsystem. Therefore, the solution of the regulator equations provides a coordinate and inputtransformation such that the stabilization solution of the transformed system (3.39) leads tothe solution of the output regulation problem of the original plant. •

Remark 3.13. Once the solution of equations (3.30) is available, there are a variety of waysto synthesize a state feedback servoregulator k(x, u). In fact, it can be verified that any

defcontroller of the form u = k(x, u) satisfying k(x(v), v) = u(u) will make Xc(v) = x(u)satisfy equations (3.23) and (3.24). If, in addition, the controller also renders all eigenvaluesof the matrix (3.31) negative real parts, then the controller solves the state feedback outputregulation problem. Clearly, the controller given in (3.32) satisfies the above conditions.A more general controller is given as follows:

Awhere k(x, v) is any state feedback control such that the closed-loop system satisfies Property3.3. For example, let k(v) be a sufficiently smooth function such that all the eigenvaluesof the matrix

are fixed complex numbers with negative real parts for all v in an open neighborhood V ofthe origin of Rq. Let k(x, v) = K(v)x. Then (3.41) gives

This controller can uniformly place the eigenvalues of the linearization of the closed-loopsystem to be fixed complex numbers for all v e V and is expected to be able to accommodatelarger exogenous signals.

Example 3.14. To illustrate the mechanism of the design process, consider the followingexample:


where the disturbance signal v1 and reference input V2 are generated by the followingexosystem:

For this simple system, the regulator equations (3.35) can easily be solved to give thefollowing solution:

The Jacobian linearization of this system along the output zeroing manifold is given by

Given a Hurwitz polynomial, for example,

we can compute a feedback gain K(v 1 , v2) such that the eigenvalues of the matrix (3.42)are given by the roots of the above polynomial for all v1 and v2. Doing so yields

Then a state feedback controller of the form (3.43) is given by

If, instead of controller (3.45), a controller of the form (3.32), that is,

is adopted, then the Jacobian matrix of the closed-loop system on the manifold {(x, v) | x =x(v) } is

which is unstable for all (v1, v2) such that 5 - e ( V 2 - 2 v 1 ) < 0.

When the plant state and/or disturbance state is not available, one can consider usingthe measurement output feedback controller to solve the output regulation problem. Thebasic idea is similar to what has been used in Chapter 1 and is described as follows. Considera dynamic controller of the form


where co\(z1, z2) = z with z\ Rn and z2 Rq, and g(z, ym) and k(z1, z2) are suchthat the solution of the closed-loop composite system composed of the composite system(3.8) and the controller (3.47) and (3.48) satisfies, for all sufficiently small initial statescol(x(0), v(0), z(0)),

In other words, the dynamic system (3.48) can be considered as a (local) asymptotic observerof the composite system (3.8).

To implement the above idea, we first establish a result that translates the requirementon the closed-loop system as given by Lemma 3.6 into the requirements on the controller(3.12).

Lemma 3.15. Under Assumption 3.1', suppose there exists a dynamic measurement outputfeedback control law of the form (3.12) such that the closed-loop system (3.13) has Property3.3. Then the following are equivalent:

(i) The nonlinear output regulation problem is solvable by the dynamic measurementoutput feedback controller (3.12).

(ii) There exists a sufficiently smooth function xc(v) with xc(0) = 0 such that

(iii) There exist sufficiently smooth functions (x(u), u(v), z(v)) with (x(0), u(0), z(0)) =(0, 0, 0) such that x(v) and u(u) are the solution of the nonlinear regulator equations(3.30) and z(u) is the solution of the nonlinear partial differential equation

which satisfies

Proof, (i) <->• (ii). This has actually been done by the proof of Lemma 3.6.(ii) «-> (iii). Assume (ii) holds. Partition Xc(u) as

where x(i>) e 7£" andz(u) e ll"1. Since (fc(xc, v), hc(xc, u)) is given by (3.15), expanding(3.49) gives


Letting u(v) = k(z(v)) gives (3.51), and using (3.51) in the second equation of (3.53) gives(3.50). Finally, using (3.51) in the first and the third equations of (3.53) shows that x(u)and u(v) satisfy the regulator equations (3.30). On the other hand, assume (iii) holds with(x(v), u(v)) being the solution of the regulator equations (3.30). Letz(v) be the solution of(3.50) that satisfies (3.51). We need to show that (x(v), z(u)) satisfies (3.53). To this end,using (3.51) in (3.50) gives the second equation of (3.53), and using (3.51) in (3.30) showsthat (x(u), z(u)) satisfy the first and third equations of (3.53). Let xc(v) be given by (3.52).Then clearly (3.53) implies that xc(v) satisfies (3.49).

Theorem 3.16. Under Assumptions 3.1', 3.2, and 3.3, the nonlinear output regulationproblem with exponential stability is solvable by a dynamic measurement output feedbackcontrol law of the form (3.12) if and only if there exist two sufficiently smooth functions x(v)and u(v) with x(0) = 0 and u(0) = 0 that satisfy the nonlinear regulator equations (3.30).

Proof. The necessity part is actually implied by the equivalence of parts (i) and (iii) ofLemma 3.15. To show the sufficiency part, note that, by Theorem 3.8, under Assumptions3.1 and 3.2 and the assumption that there exist sufficiently smooth functions x(v) and u(v)with x(0) = 0 and u(0) = 0 that satisfy regulator equations (3.30), there exists a staticstate feedback control law of the form u = k(x, v) satisfying u(v) = k(x(v), v) that solvesthe state feedback nonlinear output regulation problem. By Assumption 3.3, there existconstant matrices L1 and L2 such that all the eigenvalues of the matrix

have negative real parts. Let z — col(z1, z2) with z1 Rn and Z2 Rq, and

This controller yields a closed-loop system with

xc = col(x, z1, z2), hc(xc, v) = h(x, k(z 1 , Z 2 ) , v),

and

We first show that the closed-loop system has Property 3.3. For convenience of the notation,let


Then, a simple calculation gives

As in the proof of Theorem 1.14, in (3.56), subtracting the first row from the second rowand adding the second column to the first column shows that Ac is equivalent to

ThusNext we show that there exists a sufficiently smooth function z(v) with z(0) = 0 that

satisfies equations (3.50) and (3.51). Indeed, let (x(v), u(v)) be the solution of the regulatorequations (3.30), let z1(v) = x(v) and z2(v) = v, and let

Then,

and

Remark 3.17. From the statement of Lemma 3.15, we can see that a measurement outputfeedback servoregulator of the form (3.12) can be characterized as follows:

(i) It makes the closed-loop system satisfy Property 3.3.(ii) It is such that the following equation:

has a local solution z(v) satisfying z(0) = 0 and

3.4. Solvability of the Regulator Equations 89

As a result, the controller given in (3.54) is not unique. In particular, similar to Remark3.13, the observer gains (L1, L2) in (3.54) need not be constant. We can choose sufficientlysmooth functions L1(v) and L2 (v) such that all the eigenvalues of the matrix

are fixed complex numbers with negative real parts for all v in an open neighborhood ofRq. Then let

This control law is also expected to be capable of accommodating larger exogenoussignals.

3.4 Solvability of the Regulator EquationsAs we have seen in the last section, the key condition to the solvability of the nonlinearoutput regulation problem is the solvability of the regulator equations. By Remark 3.10,the solvability of the regulator equations is related to the existence of a particular type ofthe output zeroing manifold M of the composite system (3.8) described in (3.36). Thismanifold must be contained in the maximal output zeroing manifold of (3.8). Thus, we willbegin this section by introducing the following assumption.

Assumption 3.4. There exists a (locally) maximal output zeroing manifold Me for compositesystem (3.8), which is characterized by

where e is an open neighborhood of the origin of Rn+q and He(x, v) : Rn+q Rr forsome integer r is a sufficiently smooth function satisfying He(0, 0) = 0 and

Remark 3.18. By condition (3.62), there exist some partitionx = col(x1, x2) withx1 Rr

and x2 RN-r and a locally defined sufficiently smooth function x1 = (x2, v) satisfying(0,0) = 0 such that He( (x2, v),x2,v) = 0. Moreover, by the definition given in

Section 2.6, the fact that Me is an output zeroing manifold for (3.8) implies the existence ofa locally defined sufficiently smooth feedback control ue(x, v) satisfying ue(0, 0) = 0 suchthat, under the control u = ue(x, v), Me is an invariant manifold of system (3.8), which is


contained in the kernel of the mapping h(x, ue(x, v), v). More specifically, correspondingto the partition x = col(xl,x2), we can rewrite system (3.8) as follows:

Then the fact that Me is an output zeroing manifold for (3.8) means the existence of (x2, v)and ue(x, v) such that

Furthermore, the two functions (x2, v) and ue(x, v) will induce a subsystem from system(3.63) as follows:

which is the zero dynamics of the composite system (3.63).

Proposition 3.19. Under Assumption 3.4, there exist sufficiently smooth functions x(v) andu(v) defined for v V with x(0) = 0 and u(0) = 0 satisfying the regulator equations ifthere exists a sufficiently smooth function x2 : V Rn-r with x2(0) = 0 such that

Proof. Assume (3.67) has a solution x2(v). Letx1(v) — (x2(v), v),x(v) = c o l ( x 1 ( v ) ,and u(v) = ue(x(v), v). Then combining (3.64), (3.66), and (3.67) gives


and using (3.65) gives

That is, the two functions x(v) and u(v) satisfy the regulator equations associated with(3.63).

By Theorem 2.25, if all the eigenvalues of the matrix

have nonzero real parts, then there exists a sufficiently smooth function x2 : V Rn-r

with x2(0) = 0, which satisfies (3.67). Thus we have reached the following corollary.

Corollary 3.20. Under Assumption 3.4, suppose all the eigenvalues of the matrix

have nonzero real parts. Then there exist locally defined sufficiently smooth junctions x(v)and u(v) with x(0) = 0 and u(0) = 0 satisfying the regulator equations.

As described in Remark 2.24, the equation of the form (3.67) is an invariant manifoldequation. In what follows, we will further call (3.67) a center manifold equation if all theeigenvalues of the matrix

have nonzero real parts and all the eigenvalues of (0) have zero real parts. Note that itis the special form of the zero dynamics (3.66) of (3.8), which contains the exosystem as asubsystem, that reduces the solvability of the regulator equations into that of the invariantequation (3.67). Also note that the mere existence of an output zeroing manifold for (3.8)is not enough to make the zero dynamics (3.66) of (3.8) satisfy the desired form. Theadditional condition (3.62) has to be imposed on the output zeroing manifold.

From the above discussion, we need to find out the conditions under which the com-posite system has a maximal output zeroing manifold satisfying condition (3.62). Thisissue can be addressed by the concepts of the normal form and zero dynamics described inChapter 2. For convenience of notation, we will focus on the class of nonlinear systemsdescribed as follows:

where x Rn, uj, j = 1,. . . , m, are m scalar plant inputs; ej, j = 1,..., p, are p scalarplant outputs; f : Rn+q Rn and gj : Rn+q Rn, j = 1,. . . , m, are sufficiently


smooth functions; and hj : Rn+q R1, j = 1, . . . , p, are sufficiently smooth scalarfunctions. Let g(x, v) = [g 1 ( x , v ) , . . . , gm(x, v)], and

Then system (3.68) can be put into the following compact form:

The composite system composed of (3.69) and the exosystem i) = a(v) can be putinto the standard form of the nonlinear affine system as follows:

where xa = col(x, v), fa(xa) = col(f(x, v), a(v)), and ga(xa) = col(g(x, v), 0qxm).Also, we can define another nonlinear affine system out of (3.69) as follows:

where f0(x) = f(x, 0), go(x) = g(x, 0), and h0(x) = h(x, 0).The regulator equations associated with (3.69) can be written as follows:

Remark 3.21. Assume (3.70) has a (vector) relative degree {r1, r2, . . . , rp} at xa = 0, thatis, that there exist integers ri, i = 1, . . . , p, such that

(i) for each i = 1, . . . ,p ,

for all 0 k ri — 1 and for all xa in an open neighborhood of the origin of R,n+q ;and

(ii) the p x m matrix

has full row rank at xa = 0.

Then, from Chapter 2, the hypersurface Ha(x, v) = 0 defines the maximal output zeroingmanifold of the composite system (3.70), where Ha(x, v) is the H vector of (3.70) defined


in Chapter 2 and is described as follows:

and the corresponding state feedback control ue (x, v) is governed by the following equation:

where Da (x, v) is given by (3.73) and Ea(x, v) is the E vector of the system (3.70) definedin Chapter 2 and described as follows:

We will call the restriction of the flow of the composite system (3.70) to the manifoldHa(x, v) = 0 the zero dynamics of the composite system (3.70). If the vector Ha(x, v)further satisfies condition (3.62), then the zero dynamics of (3.70) will admit a form of(3.66); that is, the zero dynamics of (3.70) will include the exosystem as a subsystem.

In what follows, we will show that if the composite system (3.70) has a vector relativedegree at the origin, then the vector Ha(x, v) indeed satisfies condition (3.62).

Lemma 3.22. For i = 1,..., p, the functions hi, (x, v), the vector field fa, and the mappingga associated with (3.70) satisfy

where, with some abuse of the notation,

and k : Rn+q Rqx1 and yk : Rn+q Rqxm are sufficiently smooth Junctions.


Proof. By definition,

with and

where y1(x, v) is some sufficiently smooth function. Thus both (3.76) and (3.77) hold fork=1.

Next assume that both (3.76) and (3.77) hold for all positive integers less than or equalto some integer k > 0. Then

where k+1(x, v) is some sufficiently smooth function. Also,

with Yk+1(x, v) some sufficiently smooth function.

Corollary 3.23. If system (3.70) has a relative degree {r1, r2 , . . . , rp] at xa =0, then system(3.71) also has a relative degree


Proof. Due to (3.76) and (3.77), for i = 1, . . . , p, and k = 1, 2, . . . ,

Using induction on k, it can be easily verified that

where hoi(x) is the ith component of h 0 (x ) . Thus, we have

As a result, if Lga Lkfa hi(x, v) = 0 in an open neighborhood of xa = 0, then Lgo L

kfohoi (x) =

0 in an open neighborhood of x = 0. Moreover, let D 0 ( X ) be the decoupling matrix of(3.71). Then

Da(x,0) = A>(x),

and therefore rank Da(0,0) = rank D0(0).

Due to (3.76), let H0(x) and E0(x) be the H and E vectors of (3.71), respectively.Then

Moreover, by Remark 2.48 of Chapter 2, the fact that (3.71) has a relative degree {r1, r2, . . . , rp]at x = 0 implies

Thus (3.78) implies

Thus, we have reached the following result.

Proposition 3.24. Assume (3.70) has a relative degree {r1 , r2 , . . . , rp] at xa = 0 withr1+ r2 +...+ rp = r. Then, the vector Ha(x, v) satisfies condition (3.62).

By Propositions 3.19 and 3.24, if the system (3.70) has a relative degree at the origin,then it will induce a subsystem of the form (3.66) such that the solvability of the regulatorequations is reduced to the solvability of the center manifold equation (3.67). If the equi-librium point of the system x2 = (x2,0) is hyperbolic, then the center manifold equation


is always solvable. As discussed in Remark 2.50, when p = m, the subsystem (3.66) isuniquely determined within the coordinate transformations. However, when p < m, thesubsystem (3.66) is not uniquely determined. Thus, it is of interest to further characterizethe normal form of the system (3.70). For this purpose, let r = r1 + r2 +...+ rp. Then,there are r components of x denoted by x1 = col(x j l,..., x j r ) such that

Let x2 — co\(Xjr+1 ,... ,Xjn). By the Implicit Function Theorem, there exists a sufficientlysmooth function a : Rn+q Rr satisfying (0) = 0 such that, for sufficiently smallRr,

Lemma 3.25. Assume (3.70) has a relative degree [r 1 , r2, ..., rp] at xa = 0 with r1+r2++ rp = r. Then, (3.70) is locally diffeomorphic to the following system:

where

and

where gji, i = r + 1, . . . , n, is the jith row ofg.


Proof. Let

Clearly Ta is a local diffeomorphism in an open neighborhood of the origin of Rn+q into itsimage.

Nowlezij = Lj-1

fa h i(x, v),i = 1,... , pandj = 1,..., ri,andletza = col( , x2, v).Then the components of za satisfy

Clearly, (3.84) is in the form of (3.82) with x2 = col(xjr+1,..., .xjn) and ( , x2, v, u) beinggiven by (3.83).

Since Da is of full row rank at xa = 0, there exists a function ue : Rn+q Rm sat-isfying (3.75). Letting (x2, v) = (0, x2, v) shows Ha(x, v)\x\= ( x

2, v ) = 0, and defining

8(x2, v) = (0, x2, v, ue( (x2, v), x2, u)) gives the zero dynamics of (3.70) as follows:

Now applying Proposition 3.19 and Corollary 3.20 gives the main result of this sectionas follows.

Theorem 3.26. Suppose the composite system (3.70) has a relative degree {r1 , r2,... ,rp}at (x, v) = (0,0) with r1 + r2 +...+ rp = r. Assume, for some sufficiently smooth functionue(x, v) satisfying (3.75), that there exists a sufficiently smooth Junction x2 : Rq Rn-r

with x2(0) = 0 such that

Then the two Junctions x(v) = (x1(v),x2(v)) and u(v) = u e(x(v) , v), where x1(v) =(x2(v), v), are the solution of the regulator equations (3.72).

Corollary 3.27. Suppose the composite system (3.70) has a relative degree {r1 , r2 , . . . , rp]at (x,v) = (0,0) with r1 + r2 +...+rp = r. Then there exist locally defined sufficiently


smooth functions x(v) and u(v) with x(0) = 0 and u(0) = 0 satisfying the regulatorequations (3.72) if there exists some sufficiently smooth feedback control u = ue(x,v)satisfying ue(0, 0) = 0 such that all the eigenvalues of the matrix

have nonzero real parts.

Remark 3.28. As discussed in Remark 2.50, if p = m, the feedback control ue(x, v)is uniquely determined by ue(x, v) = — D - 1 ( x , v)Ea(x, v). Thus, the zero dynamics of(3.70) is also unique within coordinate transformations. The fact that all the eigenvalues ofthe matrix (3.87) evaluated at x2 = 0 have nonzero real parts simply means that the plant(3.71) has a hyperbolic zero dynamics. If p < m, there exist a partition u = col(u l, u2)with u1 Rp, u2 Rm-p and a function ku : Rn+q+m-p Rm such that

regardless of the values of u2. Letting k( , x2, v, u2) = ku(x, v, u2) x1= ( ,x2,u) and substi-tuting u = k( , x2, v, u2) into x2 = ( , x2, v, u) gives

Thus, for any sufficiently smooth feedback control u2 = (x2) satisfying 0(0) = 0, thefollowing system:

where (x2, v) = (0, x2, v, k(0, x2, v, (x2))), is the zero dynamics of (3.70). If, forsome (x2), all the eigenvalues of the matrix

have nonzero real parts, then the regulator equations are solvable. Therefore, one can takethe advantage of the m — p extra control components to modify the zero dynamics of system(3.70).

Remark 3.29. Though the identification of the zero dynamics of (3.70) involves a coordinatetransformation, there is no need to perform the coordinate transformation in order to solvethe regulator equations. Indeed, similar to the zero dynamics algorithm described in Remark2.46, we can reduce the regulator equations to an invariant manifold equation of the form(3.86) through a simple algorithm summarized below.

(i) Solve the equation

for r components of x in terms of the remaining n — r components of x and v.By property (3.81) and the Implicit Function Theorem, there exist a partition x =


col(x1, x2), with xl = col(xji ,...,Xjr) and x2 = col(xjr+1,..., x j n ) , and a mapping: R(n-r+q) Rr such that

(ii) Solve ue(x, v) from the equation Ea(x, v) + Da(x, v)ue(x, v) = 0.(iii) Solve the invariant manifold equation associated with the following system

and denote the solution by x2(v). Let x1(v) = (x2(v), v). Then the solution of theregulator equations is given by x(v) = col(x1(v), x2(v)) and u(v) = ue(x(v), v).

Remark 3.30. It can be verified that, in the special case in which f(x, v) = f(x),g(x, v) =g(x), and h(x, v) = h(x) — d(v), if (3.71) has a relative degree {r1, r2, ..., rp] at x = 0,then (3.70) also has a relative degree {r1, r2,..., rp] at (x, v) = (0,0). Thus, a somehowsimpler algorithm can be obtained. For this purpose, let D(x), E(x), and H(x) be thedecoupling matrix, E vector, and H vector of the system x = f ( x ) + g(x)u and e = h(x),and let

Then we can simplify the algorithm described in Remark 3.29 as follows.

(i) Solve the equation

for r components of x in terms of the rest n — r components of x and v. By property(3.81) and the Implicit Function Theorem, there exists a partition x = col(x1,x2), withx1 = col(xj , , . . . , Xjr) and x2 = col(x j r+l,..., xjn), and a mapping a : R(n-r+q)

Rr such that

(ii) Solve ue(x, v) from the equation Ed(v) = E(x) + D(x)ue(x, v).(iii) Solve the invariant manifold equation associated with the following system:

and denote the solution by x2(v). Let x1(v) = (x2(v), v). Then the solution of theregulator equations is given by x(v) = col(x1 (v), X2(v)) and u(v) = ue(x(v), v).


Example 3.31. Consider the following system:

It is easy to verify that the system has a relative degree {1,2} at the origin with

Furthermore, using the algorithm described in Remark 3.30 gives the zero dynamics of(3.71) as follows:

Thus, by Corollary 3.27, the regulator equations associated with (3.91) are solvable. Asa matter of fact, applying the algorithm described in Remark 3.29 gives the partition x =col(xl,x2) with x1 = col(X1+, X2, x4) and x2 = x3 and the following functions:

as well as the zero dynamics of (3.91):

As a result, x$(v) can be obtained by solving the following center manifold equation:

3.5. Output Regulation of Systems with Nonhyperbolic Zero Dynamics 101

Therefore, the solution of the regulator equations is given by

Example 332. The RTAC system described in Section 2.8 is also in the form of (3.68). Thecomposite system has a relative degree 2 at the origin. We have already shown in Section2.8 that the zero dynamics of the system (3.71) is as follows:

The Jacobian matrix of this system at the origin has two eigenvalues at the origin. Therefore,Corollary 3.27 cannot tell whether or not the regulator equations (3.72) have a solution.Nevertheless, it is still possible to show, in the last section of this chapter, that the regulatorequations of the RTAC system will admit a solution. •

3.5 Output Regulation of Nonlinear Systems withNonhyperbolic Zero Dynamics

As shown in Section 3.4, if the composite system (3.8) satisfies Assumption 3.4, then thesolvability of the regulator equations associated with (3.8) can be reduced to the solvabilityof an invariant manifold equation of the form (3.67). In the case when the equilibrium of(3.66) is not hyperbolic, we cannot guarantee the solvability of the regulator equations, andhence we cannot guarantee the solvability of the output regulation problem. Nevertheless,under certain conditions, it is still possible to solve the output regulation problem for systemswith nonhyperbolic zero dynamics. In this section, we will develop a procedure to handlethis case which involves a reduction of the plant dynamics and an augmentation of theexosystem.

We assume that the system (3.8) satisfies Assumption 3.4. To save the notation,we can start from the system (3.63) and assume that the zero dynamics of (3.63), that is,x2 = (x2, v), are described by (3.66). Now assume that the equilibrium of (x2,0) isnot hyperbolic; then, without loss of generality, we can decompose x2 = (x2, v) into thefollowing:

where x2 Rn, x2 Rn2 with n1 + n2 = n — r, all the eigenvalues of the matrix A havenonzero real parts, all the eigenvalues of the matrix B have zero real parts, and g and g2.

102 Chapter3. Nonlinear Output Regulation

are sufficiently smooth functions satisfying

Otherwise, we can always find a coordinate transformation matrix T such that, under thenew coordinate z = Tx2, the system x2 = (x2, v) can be decomposed as in (3.92).

Since A is hyperbolic, by the Center Manifold Theorem, there exists a locally definedfunction x2(x2, v) satisfying x2(0,0) = 0 such that

In terms of the partition x = col(x1, x2, x2), we can write the composite system (3.8) asfollows

Note that in conjunction with (3.92), the notation used in (3.94) to (3.98) implies

where the functions (x2, v)andue(x, v) are defined in (3.64) and (3.65). Now if f (xl, x2,x2,, u, v) does not depend on col (x 1 , x 2 , u),then we may beable to solve theoutput regulationproblem for the plant (3.94) to (3.98) by considering (3.94) and (3.95) as the plant and (3.96)and (3.97) as the exosystem. However, what makes this problem interesting is that it may besolved under much less restrictive conditions. Indeed, it suffices to assume the following.

Assumption 3.5. The input u does not appear in the function f2; that is

Assumption 3.6.

Remark 3.33. Assumption 3.5 is made so that the dynamics of (3.100) is not affected byany feedback control. This assumption is not as restrictive as it might appear. In fact, it


is satisfied for a large class of nonlinear systems. For example, the affine SISO nonlinearsystem with well-defined relative degree at the origin always has a normal form described inRemark 2.42. Clearly, Assumption 3.5 is satisfied for this class of systems. Assumption 3.6is made for invoking the Center Manifold Theorem (Theorem 2.25) later in the proof ofTheorem 3.34. •

Theorem 3.34. Under Assumptions 3.1 and 3.4 to 3.6, suppose that the pair

is stabilizable and the equilibrium point at the origin of the following system:

is stable in the sense ofLyapunov. Then there exists a state feedback control law of the formu = k(xl, x2, x2, v) such that the equilibrium of the composite system (3.94) to (3.98) at(x,v) = (0, 0) is stable in the sense ofLyapunov, and for all sufficiently small initial statesX0 and vo, the tracking error e(t) satisfies

Proof. Let

Then, combining (3.64), (3.65), and (3.93) shows that xr(x , v) and ur(x , v) satisfy

Also, by the stabilizability assumption, there exists a matrix Kr such that all the eigenvaluesof the matrix

have negative real parts. Define a state feedback controller as follows:


We now show that this controller solves the output regulation problem for the compositesystem (3.94) to (3.98). To this end, consider the closed-loop system composed of thecomposite system (3.94) to (3.98) and the controller (3.106):

which has the following properties:

(i) Due to (3. 105), all the eigenvalues of the Jacobian matrix at the origin of the reduced-order closed-loop system composed of (3.94), (3.95), and (3.106) have negative realparts.

(ii) Due to Assumption 3.6 and the decomposition (3.92), we have

where all the eigenvalues of B and A1 have zero real parts by assumption, and thefunction g vanishes at (0,0, 0, 0) together with its first-order partial derivatives withrespect to x.

(iii)

These facts, togetherwith (3. 103), show that col(x1, x2) = x r (x 2 , v) is a center manifold for(3. 107). Since the equilibrium of the augmented exosystem (3.101) is stable by assumption,it follows from Theorem 2.27 that the equilibrium of the closed-loop system (3.107) isalso stable. Thus system (3.107) satisfies Property 3.1. Moreover, by Theorem 2.28, forsufficiently small x(0) and v(0), there exist real numbers > 0 and > 0 such that thesolution of (3.107) satisfies

We now show that (3.107) also satisfies Property 3.2. In fact, from (3.104) and (3.109), wehave

It follows from the continuous differentiability of h and k and (3.110) that


Remark 3.35. A reduced-order plant can be defined out of the original plant as follows:

where col(v, v) is generated by system (3.101), which can be viewed as an augmentedexosystem. Then clearly the two functions x r(v, v) and ur(v, u) are the solution of theregulator equations associated with the reduced-order plant (3.111) and the augmentedexosystem (3.101). Thus, basically, Theorem 3.34 says that if the state feedback controllaw u = k(xl

t x2, v, v) is the solution of the state feedback output regulation problem

with exponential stability for the reduced-order plant (3.111) and the augmented exosystem(3.101), then the state feedback control law u = k(xl, x2, x2, v) is the solution of the statefeedback output regulation problem of the composite system composed of the original plantand original exosystem.

Remark 3.36. By Theorem 2.9, if the equilibrium point v = 0 of the system

is asymptotically stable, then the equilibrium point of the origin of (3.101) is also Lyapunovstable since the exosystem satisfies Assumption 3.1.

Example 3.37. Consider the nonlinear system

where a(., ., ., .) and b(., ., ., .) are sufficiently smooth scalar functions, a(0,0,0,0) = 0,and b(0, 0,0,0) 0. The system is in the form (3.70). Using the approach given inSection 3.4, we can obtain the zero dynamics and the associated control as follows:

and


Clearly the equilibrium of the subsystem x3=x3, x4 = — (x4)3 is not hyperbolic. Never-

theless, the subsystem

admits the form given by (3.92). Thus (3.113) has a center manifold denoted by x3 =x3(x4, v1, v2). Now let the reduced-order plant be

and the augmented exosystem be

It can be verified that the linearization of the reduced plant is controllable. Moreover,since the equilibrium of v = — (v)3 is asymptotically stable and the equilibrium of v1 =v2, v2 = — v1 is Lyapunov stable, by Theorem 2.9, the equilibrium of (3.117) is alsoLyapunov stable. Therefore, Theorem 3.34 concludes that, for system (3.112), the out-put regulation with Lyapunov stability can be achieved using the state feedback control(3.106). •

3.6 Disturbance Rejection of the RTAC SystemNow we turn our attention to the disturbance attenuation problem of the RTAC systemformulated in Section 3.2. Let us first consider the solvability of the regulator equationsassociated with the RTAC system. As pointed out in Example 3.32, Corollary 3.27 cannottell whether or not the regulator equations (3.72) have a solution, since the zero dynamicsof the system with the disturbance being set to zero is not hyperbolic. Nevertheless, we willshow that the regulator equations of the RTAC system admit a solution.

For this purpose, consider the composite system consisting of the RTAC system andthe exosystem as follows:

3.6. Disturbance Rejection of the RTAC System 107

Differentiating the error output e twice gives

Thus the composite system has a well-defined relative degree 2 at the origin with

Applying the algorithm described in Remark 3.29 gives the partition x = col(x1, x2) withxl = col(x1, x2) and x2 — col(x3, x4) and the following functions:

as well as the zero dynamics of (3.118)

Therefore, the solution of the regulator equations is given by

with x3(v) and x4(v) satisfying

where (x3 , x4, v) = x tan x3 + • Equations (3.121) can be viewed as the invariantmanifold equation associated with the zero dynamics (3.119).


It is usually impossible to obtain an analytic solution for a nonlinear partial differentialequation of the form (3.121). However, by taking advantage of the special structure of(3.121), it is possible to solve (3.121) as follows. First note that equations (3.121) hold ifand only if, for all sufficiently small trajectories v(t) of the exosystem,

Equation (3.123) can be written as

Using the identity

which further yields, upon noting X4(0) = 0,


which further yields, upon noting x3(0) = 0,

or equivalently,

Substituting (3.127) into (3.125) gives

where


Once we obtain the solution of the regulator equations, we can obtain a state feedbackcontroller as follows:

u = u(v) + Kx(x-x(v)),

where Kx is such that (0) + g1(0)Kx is Hurwitz. A simple calculation gives

Clearly, the pair is controllable for all However, it can be verified thatthe pair is not detectable. Thus the problem cannot be solved byan error output feedback controller.

Nevertheless, since the angular position of the proof-mass actuator X3, is also measur-able, we can define the measurement output as ym = hm(x, u, v) = col(x1, x3). Let

and

Then it can be verified that the following pair:

is detectable. Thus the problem can be solvable by the dynamic measurement outputfeedback control.

Let L = col(L1, L2) with L1 R4x2 and L2 R2x2 be such that

is Hurwitz, and z = col(z1, z2) with z1 R4 and Z2 R2. Then a dynamic measurementoutput feedback controller that solves the output regulation problem for the RTAC systemcan be given as follows:


Figure 3.2. The profile of the displacement x1 with = 0.2, = 3, and Am = 0.5.

To evaluate the performance of this controller by computer simulation, let us givethe specific gains Kx and L for the case where = 0.20 and = 3. First, letting Kx =[-16.52 -83.52 -15.4 - 20.7] places the eigenvalues of (0)+g1(0)kx at [(-0.848 ±2.52j), (-1.25 ± 0.828j)]. The above eigenvalues are based on the ITAE (integral of thetime multiplied by the absolute value of the error) prototype design with cutoff frequencyequal to 1 (described in Appendix B).

Next, letting the eigenvalues of (3.129) be given by

[-0.1871 ±73.0918 -0.7065 ± j1.1866 -1.3627 -12.6325]

gives

Simulation has been run for the initial state x(0) = col(0.1, 0,0, 0), z(0) = 0, andvarious values of the amplitude Am. With = 3, Figure 3.2 shows the profile of thedisplacement x1 of the closed-loop system, Figure 3.3 shows the profile of the other threestate variables, x2, x3, x4, and Figure 3.4 shows the profile of the control input u(t).


Figure 3.3. The profiles of the state variables (x2, x3, x4) with = 0.2, = 3,and Am = 0.5.

Figure 3.4. The profile of the control input u with € = 0.2, = 3, and Am = 0.5.


Figure 3.5. The profiles of the displacement x1 when € undergoes perturbation.

It is known that the feedforward part of the controller depends on the solution ofthe regulator equations, and thus demands precise knowledge of the plant. It is interestingto know what will happen if some parameters of the plant undergo some perturbations.Figure 3.5 shows the profiles of the displacement x1 of the closed-loop system under thesame controller with the parameter € being equal to 0.18,0.20, and 0.22, respectively. It canbe seen that when the parameter deviates from its nominal value 0.20, the displacementx1 displays a sizable nondecaying oscillation. Thus we have seen that the performance ofthis controller is not robust with respect to parameter variations. It is desirable to have aregulator that can maintain its performance in the presence of small parameter variations.Such a regulator is called a robust regulator and will be introduced in Chapter 5. A robustregulator for the same RTAC system will be designed in Chapter 6.

Chapter 4

ApproximationMethod for theNonlinear OutputRegulation

As we have seen in Chapter 3, the construction of the control laws for solving the outputregulation problem relies on the solution of the nonlinear regulator equations (3.30), whichare repeated below for convenience:

Since (4.1) are a set of nonlinear partial differential and algebraic equations, it is rarely pos-sible to find the closed-form solution for them. Therefore, it is desirable to have a numericalapproach that can solve (4.1) approximately. This chapter will present an approximationapproach to the solution of the nonlinear output regulation problem that is based on theapproximate solution of (4.1) in terms of power series. The chapter is organized as fol-lows. Section 4.1 introduces the fcth-order nonlinear output regulation problem and givesits solvability conditions by both state feedback and measurement output feedback controls.Section 4.2 presents an approximate solution of the regulator equations in terms of powerseries. Section 4.3 further gives an approximation solution of the center manifold equationsin terms of the power series. Finally, the approximation approach developed in this chapteris applied, in Section 4.4, to design a state feedback control law to approximately solve theasymptotic tracking problem of the inverted pendulum on a cart system.

4.1 kth-Order Approximate Solution of Nonlinear OutputRegulation Problem

In this chapter, we will study the same class of nonlinear plants, exosystems, and controllaws as those described in Chapter 3. All assumptions introduced in Chapter 3 will beadopted. We will first introduce another property for the closed-loop system described by(3.13) as follows.

Definition 4.1. Let V be an open neighborhood ofthe origin ofR,q. Afunctionoks : V Rs

is said to be zero up to kth order if it is sufficiently smooth and vanishes at the origin together

113

114 Chapter 4. Approximation Method for the Nonlinear Output Regulation

with all partial derivatives of order less than or equal to k. The notation ok(v) will be usedto denote a generic function ofv which is zero up to kth order regardless of the dimensionof its range space.

kth-Order Nonlinear Output Regulation Problem (KNORP): Design a control law ofthe form (3.11) or (3.12) such that the closed-loop composite system (3.13) has Property3.3 as well as the following property.

Property 4.1. For all sufficiently small xco and v0, the trajectories col(xc(t), v(t)) of theclosed-loop composite system (3.13) satisfy

If the closed-loop composite system has Properties 3.3 and 4.1, then we say thatthe steady-state tracking error of the closed-loop system is zero up to kth order. In whatfollows, a controller that solves the kth-order nonlinear output regulation problem will becalled kth-order servoregulator. In particular, (3.11) and (3.12) are called, respectively,the kth-order state feedback servoregulator and the kth-order measurement output feedbackservoregulator.

To study the solvability of the kth-order nonlinear output regulation problem, wefirst establish an equivalent characterization of Property 4.1 for the closed-loop compositesystem.

Lemma 4.2. Under Assumption 3.1', suppose the closed-loop composite system (3.13) hasProperty 3.3. Then the following are equivalent:

(i) The closed-loop composite system (3.13) has Property 4.1.(ii) There exists a sufficiently smooth function xc(v) with xc(0) = 0 that satisfies, for

v V, the following partial differential and algebraic equations:

(iii) There exists a sufficiently smooth function x(k)(v) with x(k)(0) = 0 that satisfies, forv V, the following partial differential and algebraic equations:

Proof, (i) (ii). Define another system as follows:

4.1. kth-Order Approximate Solution of Nonlinear Output Regulation Problem 115

where

Clearly, the system (3.13) has Property 4.1 if and only if (4.5) has Property 3.2. ByLemma 3.6 of Chapter 3, if (4.5) has Property 3.3, then it also has Property 3.2 if andonly if there exists a sufficiently smooth function xc(v) with xc(0) = 0 that satisfies, forv V,

or, equivalently, the function xc(v) satisfies (4.3).(ii) (iii). (ii) trivially implies (iii) by letting x ( k ) (v) = x c(v) . To show that (iii) also

implies (ii), let x(k)(v) satisfy, for v V, (4.4). Since (3.13) has Property 3.3, by Theorem2.26, there exists a sufficiently smooth function x c(v) with xc(0) = 0 that satisfies the firstequation of (4.3). Moreover,

We need to show that xc(v) also satisfies the second equation of (4.3). Indeed,

Lemma 4.2 leads to the following characterization of the control law that solves thekth-order nonlinear output regulation problem.

Theorem 4.3. Under Assumptions 3.1' and 3.2, the kth-order nonlinear output regulationproblem is solvable by a static state feedback controller

if and only if there exist two sufficiently smooth junctions x(k)(v) and u(k)(v) satisfyingx(k) (0) = 0 and u(k) (0) = 0 such that

Proof. Assume that the controller (4.10) solves the kth-order nonlinear output regulationproblem. Then, by Lemma 4.2, there exists a sufficiently smooth function x(k)(v) thatsatisfies (4.4) for v V. Let x(k)(v) = x(k)(v) and u(k)(v) = k(x(k)(v), v). Then, clearly,x(k)(v) and u(k)(v) satisfy (4.11). On the other hand, let x(k)(v) and u(k}(v) satisfy (4.11).Using the same argument as used in the proof of Theorem 3.8, there exists a state feedbackcontroller k(x, v) with k(0, 0) = 0 such that the closed-loop system has Property 3.3.Furthermore, if k(x, v) satisfies


for example,

where Kx is some constant feedback gain, then, clearly, this controller is such that theclosed-loop system

still has Property 3.3. Moreover, letting x(k)(v) = x(k)(v) leads to

Thus, by Lemma 4.2, the controller solves the kth-order nonlinear output regulationproblem.

Analogous to Lemma 3.15, we can also establish the following result on the solvabilityof the kth-order nonlinear output regulation problem via a measurement output feedbackcontroller of the form

Lemma 4.4. Under Assumption 3.1', assume that there exists a measurement output feed-back control law of the form (4.13) such that the closed-loop composite system (3.13) hasProperty 3.3. Then the following are equivalent:

(i) The kth-order nonlinear output regulation problem is solvable by the measurementoutput feedback controller (4.13).

(ii) There exists a sufficiently smooth function x(k)(v) with x(k) (0) = 0 such that

(iii) There exist sufficiently smooth Junctions (x(k)(v), u(k)(v), z(k)(v)) with (x(k)(0),u ( k ) (0) , z(k}(0)) = (0, 0, 0) such thatx(k)(v) andu(k)(v) satisfy equations (4.11) andz(k) (v) satisfies

Moreover,

4.2. Power Series Approach to Solving Regulator Equations 117

Proof. (i) (ii). The proof is similar to that of Lemma 4.2 and is thus omitted.(ii) (iii). Assume (ii) holds. Partition x(k)(v) as

where x(k)(v) Rn and z(k)(v) Rnz. Since (fc(xc, v), hc, v)) is given by (3.15),expanding (4.14) gives

Letting u(k)(v) = k(z(k)(v)) gives (4.16), and using (4.16), in the second equation of (4.18)gives (4.15). Finally, using (4.16) in the first and third equations of (4.18) shows that x(k) (v)and u ( k ) ( v ) satisfy (4.11). On the other hand, assume (iii) holds. Let (x(k)(v), u(k)(v)) bethe solution of (4.11). Let z(k}(v) satisfy (4.15) and (4.16). We will show that x ( k )(v) andz(k) (v) satisfy (4.18). To this end, using (4.16) in (4.15) gives the second equation of (4.18),and using (4.16) in (4.11) shows that ( x ( k ) ( v ) , z(k)(v)) satisfies the first and third equationsof (4.18). Thus, letting x(k) (v) be given by (4.17) shows that x(k) (v) satisfies (4.14).

Theorem 4.5. Under Assumptions 3.1 to 3.3, suppose there exist two sufficiently smoothfunctions x(k)(v) and u(k)(v) with x(k)(0) = 0 and u(k)(0) = 0 that satisfy (4.11). Then,there exists a measurement output feedback control law that solves the kth-order nonlinearoutput regulation problem.

Proof. Under the assumptions of Theorem 4.5, there exists a state feedback control law ofthe form k(x, v) that solves the kth-order nonlinear output regulation problem. By Assump-tion 3.3, there exist constant matrices L1 and L2 such that all the eigenvalues of the matrix

have negative real parts. Now let

Then, in a fashion similar to the proof of Theorem 3.16, it can be verified that the closed-loopsystem under this controller has Properties 3.3 and 4.1. Details are left to the reader.

4.2 Power Series Approach to Solving Regulator Equations

By Theorems 4.3 and 4.5, the key to the solvability of the kth-order nonlinear output regula-tion problem is to find the solution of the nonlinear regulator equations (4.1) up to kth order.


In this section, we will consider a power series approximation approach to solving (4.1). Ourconsideration will involve power series representations for the unknown functions x(v) andu(v), and this entails the following notation. For any matrix K, we will use the Kroneckerproduct notation

Then we can write the problem description in terms of the series expansions

To obtain unique representations for the coefficients in series expansions of the unknownfunctions x(v) and u(v), the following notation will be used. For the q x 1 vector v =col(v1,..., vq), let v[l] denote the vector

Then the Taylor series of the functions x(v) and u(v) can be uniquely expressed as follows:

where X1 and U1 are constant coefficient matrices. We need to find these matrices suchthat equations (4.1) are satisfied formally. Note that the dimensions of u[1] and v(1) are,respectively,

and that there exist matrices M1 and N1 of appropriate dimensions such that

For example, with q = 2, v(2) and v[2] are given by, respectively,


and M2 and N2 are given by, respectively,

Although M1 is not unique, it is easy to check that M1N1 is an identity matrix regardless ofthe specific form of M1.

Our purpose is to derive explicit equations that generate all matrices X1 and U1,/ = 1,2, — To this end, we first list some useful identities involving the Kroneckerproduct as follows.

Lemma 4.6.

(i) Forl 1,

(ii) For any integers i, j, k 0, and any matrix T of dimension q by qk,

(iii) For k, l 1, and any matrix T of dimension q by qk,

Proof. Equation (4.26) follows straightforwardly from the definition of the Kroneckerproduct. Equation (4.27) can be proved as follows:

Note that in deriving equation (4.27), we have repeatedly utilized the identity

which can be found in Appendix A.


To show (4.28) using (4.25), (4.26), (4.27), and (4.25) sequentially gives

Substituting equations (4.21) and (4.23) into equations (4.1), expanding equations(4.1) into the Taylor series, and identifying the coefficients of v[l], l = 1,2, . . . , yields thefollowing result.

Lemma 4.7. The power series (4.23) formally satisfies equations (4.1) if and only if thefollowing linear equations are satisfied for l = 1,2, . . . :

where

and, for l = 2 ,3 , . . . ,


Proof. Substituting equations (4.21) and (4.23) into equations (4.1) yields the followingequations:

The left-hand side of (4.36) can be written as

Thus using (4.28) in (4.38) gives

Also, we can write

and


where i,l and i,l are given by equations (4.34) and (4.35). Then

where G m is given by equation (4.33). This permits the right-hand sides of equations (4.36)and (4.37) to be written as

and

respectively. Using (4.39), (4.43), and (4.44) in (4.36) and (4.37) and equating the coeffi-cients of v[l] on both sides of the rewritten (4.36) and (4.37) gives, for l 1,

Finally, using G = 1,l = XlMl, G = 1,l = UlMl, G = 1, G = 0, / > 1, alongwith the fact that M1N1 is an identity matrix, completes the proof.

Note that E1 and F1 depend only on X 1 , . . . , Xl_1 and U 1 , . . . , Ui-\, so that equations(4.29) provide a sequence of linear matrix equations. The following result establishes thesolvability condition for these equations.


Lemma 4.8. There exists a solution (unique ifp = m) of equations (4.29) for any eE1andF l,l = 1,2,... , if and only if the plant satisfies the following assumption.

Assumption 4.1.

rank

for all l, where

with 1 , . . . , q being the eigenvalues of the matrix

Proof. For a given /, equations (4.29) actually take the same form as the linear regulatorequations (1.21). Thus, by Theorem 1.9, equations (4.29) have a solution for any E1 and F1

if and only if equation (4.45) holds for all in the spectrum of

We now show that the eigenvalues of A[1] are precisely those given by (4.46). To this end,using (4.28) with T = A1 and k = 1 gives

Note that the components of v[1] consist of all products of the variables v1 , . . . , vq taken / ata time. Therefore, if we define Pl as the vector space of all homogeneous polynomials inv1 , . . . , vq of degree /, then the components of v[1] give a basis of Pl. Now define a linearmapping L A1v : P

l Pl such that, for each Pl,

Then, using (4.48) shows

Thus, (A[1])r is the matrix of the linear mapping under the ordered basis

Thus, the spectrum of A[1] is the same as that of the linear mapping (4.49).


Now let the Jordan canonical form of A \ be

where

is a ni x ni Jordan block with eigenvalue i. Suppose the generalized row eigenvectors ofA1 are

satisfying

Clearly,

also constitutes a basis for Pl. Furthermore,

Now define an order on (4.54) in the following "lexicographic" way:

if and only if there exist positive integers i0 and j0 nio such that

if i < io , j ni or i = io, j < jo. Then (4.54) constitutes an ordered basis of Pl. Using(4.55) gives

and

+ terms greater than

4.3. Power Series Approach to Solving Invariant Manifold Equation 1 25

Thus, the matrix of the linear mapping LA1v on Pl under the ordered basis (4.54) with theorder (4.56) is upper triangular with the diagonal elements being

Therefore, the eigenvalues of LAlV on Pl are exactly given by equation (4.46).

Remark 4.9. In the case when the solution of equations (4.29) is such that (4.23) has apositive convergent radius, then (4.23) is an exact solution of equations (4.1) in power seriesform. In particular, if the solution of equation (4.1) is a polynomial in v[1], then Lemma 4.7gives an approach to exactly solving equations (4.1). Note that equation (4.45) representsthe constraints on the transmission zeros of the Jacobian linearization of the plant which canbe viewed as an extension of the transmission zeros condition for the linear output regulationproblem as described in Remark 1.11.

Remark 4.10. Assume that the transmission zeros condition in equation (4.45) holds up tosome positive integer k. Let

Then, it is not difficult to see from the proof of Lemma 4.7 that x(k)(v) and u(k)(v) aresuch that

In conjunction with Theorems 4.3 and 4.5, this observation immediately leads to the fol-lowing sufficient conditions for the solvability of the kth-order nonlinear output regulationproblem.

Theorem 4.11.

(i) Under Assumptions 3.1, 3.2, and4.1, the kth-order nonlinear output regulation prob-lem is solvable by the state feedback control law of the form (3,11).

(ii) Under the additional Assumption 3.3, the kth-order nonlinear output regulationproblem is solvable by the measurement output feedback control law of the form(3.12).

4.3 Power Series Approach to Solving InvariantManifold Equation

As we have seen in Section 3.4, when the composite system (3.8) satisfies Assumption 3.4,we can reduce the solvability of the regulator equations into the solvability of an invariant


manifold equation of the form (3.67), which is associated with the zero dynamics (3.66)of the composite system (3.8). Since the dimension of the invariant manifold equation issmaller than that of the regulator equations, it is more convenient to solve the invariantmanifold equation. To put our technical development in a more general context, in thissection, we will consider a general nonlinear system of the form

where x Rn, v Rq, and F : Rn+q Rn is a sufficiently smooth function satisfyingF(0,0) = 0. Associated with (4.59) and the exosystem (3.10) is a partial differentialequation of the form

This equation can be viewed as a special case of the regulator equations when p = m = 0.Recall from Chapter 2 that an equation of the form (4.60) is called an invariant manifoldequation. In particular, when the equilibrium point of x = F(x, 0) at x = 0 is hyperbolicand all the eigenvalues of have zero real parts, (4.60) is called a center manifoldequation.

Similar to the last section, we will seek series of the form

such that (4.60) is satisfied. For this purpose, we can again write F(x, v) and a(v) in termsof Taylor series as follows:

Analogous to Lemmas 4.7 and 4.8, we can obtain the following two lemmas. Theproofs of these two lemmas are omitted since they can be directly deduced from the proofsof Lemma 4.7 and Lemma 4.8, respectively.

Lemma 4.12. The power series (4.61) formally satisfies equation (4.60) if and only if thefollowing linear equation is satisfied for l = 1, 2, . . . :

where

4.4. Asymptotic Tracking of the Inverted Pendulum on a Cart 127

and, for l = 2,3,...,

Lemma 4.13. There exists a unique solution of equation (4.63) for any E1, l = 1, 2, . . . , ifand only if none of the eigenvalues of the matrix (0,0) coincides with any , l, where

l = { | = l1 1 + • • • + lq 1 l1 + • • • + lq = l , l 1 , . . . , lq = 0, 1 , . . . , / }

and where 1 , . . . , q are eigenvalues of the matrix (0).

Remark 4.14. In the case that the solution of equation (4.60) is such that (4.61) has a positiveconvergent radius, then (4.61) is an exact solution of equation (4.60) in power series form.In particular, if the solution of equation (4.60) is a polynomial in v[l], then Lemma 4.12gives an approach to solving equation (4.60) exactly. I

Remark 4.15. When the system (4.59) represents the zero dynamics of the system x =f(x, u, 0), e = h(x, u, 0), the eigenvalues of the matrix (0,0) are precisely the trans-mission zeros of the linearization of the system x = f(x, u, 0), e = h(x, u, 0). Thusthe condition of Lemma 4.13 is consistent with the transmission zero condition given inAssumption 4.1. Note that this condition is much less stringent than the hyperbolicity as-sumption of the matrix because it only prohibits the eigenvalues of the matrixfrom belonging to a countable set. Moreover, the solvability of (4.60) in power series doesnot have to rely on the assumption that the eigenvalues of are on the imaginary axis.In the next section, we will see that the invariant manifold equation associated with theinverted pendulum on a cart system admits a formal power series solution. •

4.4 Asymptotic Tracking of the Inverted Pendulumon a Cart

We now return to the problem of the asymptotic tracking of the inverted pendulum on a cartformulated in Section 3.2. Let us first note that the Jacobian linearization of the invertedpendulum on a cart system at the origin is as follows:

which is controllable. Thus the system satisfies Assumption 3.2.


Recall from Section 2.8 that the relative degree of (2.109) is 2, and the zero dynamicsof (2.109) is given by

which has a hyperbolic equilibrium as the eigenvalues of the Jacobian matrix of (4.67) atthe origin are given by . Thus the solution of the regulator equations associated withthe inverted pendulum system exists. As a matter of fact, it can be further verified that, forsystem (2.109),

Thus, applying the algorithm described in Remark 3.30 gives the partition x = col(x1, x2)with x1 = col(x1, x2) and x2 = col(x3, x4) and the following functions:

as well as the zero dynamics of the composite system (3.22):

We can put (4.68) in the following form:

with

The center manifold equation associated with (4.69) is given by the following partial dif-ferential equation:


Since does not have eigenvalues on the imaginary axis, the Center Manifold Theoremguarantees the existence of the solution of (4.70). Let

be the solution of (4.70). Then the solution of the regulator equations associated with(3.22) is

It should be noted that even though the solvability of the equation (4.70) is guaranteed byTheorem 2.25, due to the nonlinear nature of equation (4.68) we are not able to find anexplicit solution for the center manifold equation (4.70). Nevertheless, we will show thatthe Taylor series solution of the center manifold equation studied in Section 4.3 will givean approximate solution to (4.70).

To this end, expand equation (4.70) as follows:

We have already known that the equilibrium point of the zero dynamics of a plant with v = 0is hyperbolic. Therefore, by Lemma 4.13, equation (4.73) admits a power series solutionof the form (4.61). Now let us proceed to find an approximate solution to equation (4.73),and then an approximate controller based on the approximate solution to equation (4.73).Though we can use the general method given in Section 4.3 to obtain an approximate solutionto equation (4.73) with the help of a computer program, it is possible to obtain a lower orderapproximate solution to equation (4.73) using hand calculation. For this purpose, assumethat the power series expansion of x3(v) and x 4 (v) takes the following form:

Then, substituting (4.74) and (4.75) into (4.73) and identifying the coefficients gives athird-order approximation of x 3 ( v ) and x4(v) as follows:


where

Using the expressions (4.71) and (4.72), we can obtain a third-order approximation of thesolution of the regulator equations associated with (4.68) as follows:

Based on x(3)(v) and u(3)(v), an approximate controller is given as follows:

where Kx is such that the matrix + g(0)K x is Hurwitz. Let b = 12.98 kg/sec,M - 1.378 kg, / = 0.325m, g = 9.8m/sec2, m = 0.051 kg, and let the eigenvaluesof the matrix + g(0)^ be [(-0.848 ± 2.52j), (.1.25 ± 0.828j)]. Then kx =[0.0457 13.16 16.7 1.85]. The above eigenvalues are based on the ITAE prototypedesign with cutoff frequency equal to 1.

Frequency(o= 1.0

= 1.5(0 = 2.0

Nonlinear controller0.000760.00450.0210

Linear controller0.0400.0650.0825

Table 4.1. Maximal steady-state tracking error with Am = 1.

The performance of the controller has been evaluated by computer simulation withvarious values of the frequency and fixed amplitude Am = 1. Table 4.1 lists the maximalsteady-state tracking errors of the closed-loop system for several different frequencies withAm = 1. For comparison, we also give the maximal steady-state tracking errors resultingfrom a linear controller of the following form:


Figure 4.1. The profile of the tracking performance of the closed-loop systemunder the nonlinear controller with = 1.5 and Am = 1.

Figure 4.2. The profile of the tracking performance of the closed-loop systemunder the linear controller with = 1.5 and Am — 1.


Figure 4.3. Comparison of the output responses of the closed-loop system underthe nonlinear and linear controllers with — 1.5 and Am = 4.

where

That is, this linear controller is a linear approximation of the third-order controller. It is seenthat, in all cases, the third-order nonlinear controller performs much better than the linearcontroller. Figures 4.1 and 4.2 show the profiles of the tracking performance of the closed-loop system under the nonlinear controller and the linear controller, respectively, for the case

= 1.5 and Am = 1. It can be seen that, under the nonlinear controller, no steady-statetracking error is visible, while, under the linear controller, a sizable steady-state trackingerror is present. Figure 4.3 further compares the output responses of the closed-loop systemunder the nonlinear controller and linear controller with = 1.5 and Am = 4.

Remark 4.16. We have seen that the coefficients of v[2] of X3(v) and x4(v) are zero. This isnot a coincidence. In fact, it can be seen that if the power series expansion of x 3 (v) and x4(v)only contains such terms as v[1] with / an odd integer, so does the power series expansion ofthe expressions on both sides of (4.73). Thus, we can conclude that the power series solutionof equation (4.73) does not contain such terms as v[l] where / is an even integer.

Chapter 5

Nonlinear RobustOutput Regulation

We now turn our attention to the nonlinear robust output regulation problem in which the sameobjectives as described in Chapters 3 and 4 must be achieved via either dynamic state feed-back or output feedback control in the presence of appropriately defined model uncertainties.

Two robust control problems will be defined for a class of general nonlinear systemsin this chapter, namely, the robust output regulation problem and the kth-order robust outputregulation problem. They are, respectively, the robust enhancement of the output regulationproblem studied in Chapter 3 and the kth-order output regulation problem studied in Chap-ter 4. The chapter is organized as follows. Section 5.1 gives precise descriptions of the twoproblems and lists some standard assumptions. An equivalent characterization of the robustoutput regulation property in terms of the solvability of a set of partial differential equationswill also be given. Section 5.2 introduces two examples. The first example shows that,when the exogenous signal is constant, the nonlinear robust output regulation problem canstill be solved by a linear controller that solves the linear robust output regulation problemof the linearized system of the given nonlinear system. However, this technique does notwork when the exogenous signal is time varying, as illustrated by the second example. InSection 5.3, we first reveal why the design method for the linear systems fails to work for thenonlinear system and then proceed to establish the solvability conditions for the kth-orderrobust output regulation problem. In Section 5.4, we pass to the robust output regulationproblem. It is shown that if the solution of the regulator equations is a degree k polynomialin the exogenous signal v, then a controller that solves the kth-order robust output regulationproblem also solves the robust output regulation problem. Moreover, by incorporating thefeedforward control technique, it is possible to solve the robust output regulation problemfor some cases where the solution of the regulator equations is not polynomial. In Sec-tion 5.5, we address some computation issues. In Section 5.6, the ball and beam exampleis used to illustrate the design approach.

5.1 Problem DescriptionIn analogy to the description of the uncertain linear plant given in (1.46) of Chapter 1, wedescribe an uncertain nonlinear plant as follows:

133

134 Chapter 5. Nonlinear Robust Output Regulation

with the same exosystem

where x(t) is the n-dimensional plant state, u(t) the m-dimensional plant input, e(t) the p-dimensional plant output representing the tracking error, v(t) the q-dimensional exogenoussignal representing the disturbance and/or the reference input, and w the nw -dimensionalvector representing the unknown plant parameter. It is assumed that 0 is the nominal value ofthe uncertain parameter w and that f(0,0, 0, w) = 0 and h(0, 0,0, w) = 0 for all w Rnw.

The class of control laws are described by

where z(t) is the compensator state vector of dimension nz to be specified later. With anabuse of notation, the above controller encompasses three cases:

1. Dynamic State Feedback Controller: When v(t) does not appear in (5.3), that is,

2. Dynamic Output Feedback Controller: When x(t) and v(t) do not appear in (5.3),that is,

3. Dynamic Output Feedback with Feedforward Controller: When x(t) does not ap-pear in (5.3), that is,

Letting xc = col(x, z), the resulting closed-loop system can be written as

where

For simplicity, all the functions involved in this setup are assumed to be sufficientlysmooth and defined globally on the appropriate Euclidean spaces, with the value zero at


the respective origins. Throughout this chapter, we use V and W to denote some openneighborhoods of the origins of Rq and R"w, respectively. For convenience of presentation,we allow V and W to be made arbitrarily small.

For convenience, let us lump the closed-loop system (5.7) and the exosystem (5.2)together as follows:

and call (5.9) the closed-loop composite system. It is clear that, for all w, the statecol(xc v) = col(0,0) is an equilibrium point of the composite system.

Robust Output Regulation Problem (RORP): Find a controller of the form (5.3) suchthat the closed-loop composite system (5.9) satisfies the following two properties.

Property 5.1. For all sufficiently small xc(0), v(0), and w;, the trajectory col(xc(t), v(t)) ofthe closed-loop composite system (5.9) exists and is bounded for all t 0, and

Property 5.2. For all sufficiently small xc(0), u(0), and w, the trajectory col(.xc(t), v(t)) ofthe closed-loop composite system (5.9) satisfies

kth-Order Robust Output Regulation Problem (KRORP): Find a controller of the form(5.3) such that the closed-loop composite system (5.9) satisfies Property 5.1 and the follow-ing property.

Property 5.3. For all sufficiently small xc(0), y(0), and w, the trajectory col(xc(t), v(t)) ofthe closed-loop composite system (5.9) satisfies

where k is some given positive integer.

Remark 5.1. It is clear that the robust output regulation problem and the kth-order robustoutput regulation problem are extensions of the output regulation problem described inChapter 3 and the kth-order output regulation problem described in Chapter 4, respectively,by further taking into account the uncertain parameter w. On the other hand, the descriptionof the plant (5.1) includes the linear uncertain plant as described in Chapter 1 as a specialcase. Thus the robust output regulation problem described here is an extension of the linearrobust output regulation studied in Chapter 1. Moreover, noting that, for the class of linearsystems, Property 5.3 is the same as Property 1.4, the kth-order robust output regulationproblem described here is also an extension of the linear robust output regulation studied inChapter 1.


Remark 5.2. The constant parameter w can be viewed as being produced by an autonomoussystem = 0, w(0) = w. Combining this system with the closed-loop composite systemgives

This system takes exactly the same form as (3.13), viewing = a(v) and = 0 as theexosystem.- Thus, using the same argument as in Remark 3.1, Property 5.1 is guaranteedif the equilibrium point of the system (5.12) at col(xc, v, w) = col(0,0, 0) is stable in thesense of Lyapunov. Moreover, by Theorem 2.27 and Assumption 3.1, the equilibrium pointof the system (5.12) at col(xc, v, 0) = col(0, 0,0) is stable in the sense of Lyapunov if theclosed-loop system has the following property.


have negative real parts.

Thus, in this chapter, we will directly impose Property 5.4 instead of Property 5.1 onthe closed-loop composite system.

The reason for studying the ton-order robust output regulation problem is at leasttwofold. First, from a practical point of view, it suffices to require that the steady-statetracking error be sufficiently small, and Property 5.3 is a reasonable measure of smallnessof the steady-state tracking error. Second, as will be shown in Section 5.4, under someadditional assumption on the solution of the regulator equations, a controller that solves thefcth-order robust output regulation problem also solves the robust output regulation problem.In what follows, a controller that solves the robust output regulation problem or the kth-orderrobust output regulation problem will be called a robust servoregulator or kth-order robustservoregulator. In particular, (5.4) and (5.5) are called the (kth-order) state feedback robustservoregulator, and the (kth-order) output feedback robust servoregulator, respectively.

The following result is an extension of Lemmas 3.6 and 4.2 to the case where themodel uncertainty is taken into account. The proof of this lemma is exactly the same asthose of Lemmas 3.6 and 4.2, viewing w as produced by w = 0, and is thus omitted.

Lemma 5.3. Under Assumption 3.1', suppose the closed-loop system (5.7) has Property 5.4.Then

(i) The closed-loop composite system (5.9) has Property 5.2 if and only if it has thefollowing two properties:


Property 5.5. There exists a sufficiently smooth function xc(v, w) with xc(0,0) = 0that satisfies, for v € V and w W, the following partial differential equations:

N ,514 a(v) = fc(\c(v, w), v, w), (5.14)

(ii) The closed-loop composite system (5.9) has Property 5.3 if and only if

Property 5.6. There exists a sufficiently smooth functionx (v, w) withx (0,0) = 0that satisfies, for v e V and w W, the following partial differential equations:

Various assumptions needed for the solvability of the above two problems are listedas follows.

Assumption 5.1. There exist sufficiently smooth functions x(v, w) and u(u, w) withx(0,0) = 0 and u(0, 0) = 0 satisfying, for v € V and w W, the following equations:

Asssumption 5.2. The pair ( (0,0,0, 0), (0,0, 0,0)) is stabilizable.

Assumption 5.3. The pair ( (0, 0,0,0), (0,0,0,0)) is detectable.

Assumption 5.4. For / = 1 , 2, . . . ,

for all A given by

where 1, . . . , 9 are the eigenvalues of the matrix (0).

Remark 5.4. Clearly, equations (5.18) are the extension of equations (3.30) and are thuscalled the regulator equations of the uncertain nonlinear systems (5.1). Using the sameargument as that in Theorem 3.8, it can be shown that, under Assumption 3.1', the solvabilityof the regulator equations is necessary for the solvability of the robust output regulation


problem for the uncertain system However, the solvability of the regulator equationsdoes not guarantee the solvability of the robust output regulation problem for the uncertainsystem (5.1). As will be shown in Section 5.3, an additional condition has to be imposedon the solution of the regulator equations (5.18). Assumption 5.4 is made to guaranteethe existence of the formal Taylor series solution of the regulator equations (5.18). It isnoted that this assumption does not guarantee the existence of the solution of the regulatorequations (5.18).

Remark 5.5. If the functions x(v, w) and u(v, u) described in Assumption 5.1 are definedfor all v Rq and all w R"w and satisfy equations (5.18) for all v Rq and allw Rn w , then the functions x(v, w) and u(v, w) are called the global solution of theregulator equations.

5.2 Two Case StudiesBy Lemma 5.3, a controller that solves the robust output regulation problem must be ableto induce a center manifold defined by the solution Xc(v, w) of (5.14), and, on the centermanifold, the output of the system is identically zero; that is, Xc(v, w) also satisfies equation(5.15). For the class of linear systems, (5.14) reduces to the Sylvester equation given inthe first equation of (1.53). A controller that incorporates a p-copy internal model of theexosystem can solve the robust output regulation problem because the employment of theinternal model guarantees that the solution of the first equation of (1.53) also satisfies thesecond equation of (1.53). For the class of nonlinear systems, due to the Center ManifoldTheorem, if a controller can make the closed-loop system satisfy Property 5.4, then equation(5.14) is solvable for a sufficiently smooth function Xc(u, w) with Xc(0, 0) = 0. The issue iswhether or not X c (v , w) also satisfies (5.15) if the controller is such that it solves the robustoutput regulation problem for the linearization of the nonlinear plant (5.1).

Case 1: Let us first take a look at a special case where the exogenous signal v is constant,that is, where v is generated by the exosystem — 0. Assume that a linear state feedbackcontroller of the form

makes the closed-loop system (5.7) satisfy Property 5.4. Under this controller, equations(5.14) and (5.15) become

Since fc has Property 5.4, the existence of a sufficiently smooth function Xc(v, w) withXc(0, 0) = 0 satisfying (5.22) is guaranteed by the Implicit Function Theorem. Sinceg(z, e) = e, satisfaction of equation (5.22) by Xc(u, w;) implies the satisfaction of equation(5.23) by Xc(u, w). That is, the controller also solves the robust output regulation problemfor the nonlinear system.

Clearly, the controller given by (5.21) is simply a linear robust controller based on theJacobian linearization of the nonlinear system (5.1). The robustness is achieved by having

5.2. Two Case Studies 139

the controller incorporate the p-copy internal model of the exosystem. Unfortunately, sucha technique only works for the spatial case where the exogenous signal is constant. Thefollowing example shows that the above technique is no longer effective for nonlinearsystems subject to time-varying exogenous signals.

Case 2: Consider the following one-dimensional plant:

where the exosystem is given by

and Wi and w2 are two unknown parameters with their nominal values being zero.The Jacobian linearization of (5.24) is given by

It can be verified that the robust output regulation problem for the linear system (5.26) issolvable by either state or output feedback control. A simple output feedback controller isgiven by

However, this controller does not solve the robust output regulation problem for (5.24). Tosee this point, it suffices to show that there exists no sufficiently smooth function Xc(v, w)that satisfies both (5.14) and (5.15). In fact, assume x(v, w), z1(v, w), and Z 2(v , W) satisfy(5.14) and (5.15), that is,

and


Then, necessarily, we have, from equation (5.29) and the first two equations of (5.28),

However, the left-hand and right-hand sides of the third equation of (5.28) are given by

and

respectively, so that the third equation in (5.28) does not hold. This gives a contradiction.Nevertheless, since the controller (5.27) renders the closed-loop system composed

of (5.24) and (5.27) into Property 5.4, the center manifold equation (5.14) associated withthe closed-loop system has a solution Xc(u, w). Moreover, it is possible to show that thissolution will annihilate the linear term of the right-hand side of (5.15) for all sufficientlysmall w. However, the solution of (5.14) may not satisfy equation (5.15), as the right-handside of (5.15) is in general a nonlinear function of v.

5.3 Solvability of the kth-Order Robust OutputRegulation Problem

To pursue the problem a little further, let us first introduce the following notations:

where A\ = (0) and A(w), B(w), and so forth are

For convenience, in what follows, we will use the shorthand notation A, B, and so forth todenote A(0), B(0), and so forth.

Now assume that a control law of the form (5.3) with g(z, e) = G1z + G2e rendersthe closed-loop system (5.7) into Property 5.4. Then Theorem 2.25 ensures the existenceof a locally defined sufficiently smooth function c(v,w) with xc(0,0) = 0 such that, forv V, w W,

5.3. Solvability of the /rth-Order Robust Output Regulation Problem 141

By partitioning x(v,w) = col(x(v, u;), z(v,w)) with (u, w) Rn, (5.31) becomes

where

For any k 1, x(v, w), (v, w), and e(v,w) can be uniquely expressed as

where (X l w , Z lw, yiW) are constant matrices of appropriate dimensions depending, perhaps,on w. In analogy to the derivation of equation (4.29), substituting (5.34) into (5.32) and(5.33), expanding (5.32) and (5.33) into power series in v[/1, and identifying the coefficientsof v[l] yield, for l = 1,2,...,*,X

and

where A{1] is as defined in (4.47) and is repeated below:

(Eiw„,, Fiw,) = (E(W), F(w)), and, for / = 2, 3,..., (Elw„,, Flw,) depend only on X(l-l),...,X(I-1)W andZ l w , . . . , Z(i-l)W.

Now we can invoke Lemma 1.27 to yield the following result.

Lemma 5.6. Under Assumption 3.1, assume that a control law of the form (5.3) withg(z, e) — G1z +G2e makes the closed-loop system (5.7) satisfy Property 5.4. Then

(i) For some integer I > 0, let Ylu,v[l] be the Ith-order term of the Taylor series expansionof hc(xc(v, w), u, w) as a Junction ofv. Then Ylw = O for all w W if the pair(G1, G2) incorporates a p-copy internal model of the matrix A[l].


(ii) The kth-order robust output regulation problem is solvable if the pair (G1} G2) incor-porates a p-copy internal model of the matrix Akf, where

Proof, (i) Since, for the given /, equations (5.35) and (5.36) take the same form as (1.70)and (1.71), the fact that the closed-loop system has Property 5.4 means that the matrix

is Hurwitz. Thus, by Lemma 1.27, YIW = 0 for all w W if the pair (G1, G2) incorporatesa p-copy internal model of the matrix A[l].

(ii) By the definition of Akf, if the pair (G1, G2) incorporates a p-copy internal modelof the matrix Akf, it also incorporates a p-copy internal model of all the matrices A[l] for/ = l , . . . ,£ . Therefore, the control law makes Ytw = 0 for all / = ! , . . . ,&, therebysolving the kth-order robust output regulation problem. D

As pointed out in Remark 1.23, given any matrix Akf, it is always possible to find apair of matrices (G\, G2) such that it is a p-copy internal model of the matrix Akf- Thus wecan define an augmented system as follows:

where the pair (G1, G2) incorporates a p-copy internal model of the matrix Akf. By Lemma5.6, the kth-order robust output regulation problem is solvable by a control law of the form(5.3) with g(z, e) = G1Z +G2e if the static feedback control law of the form u = k(x, z, v)can exponentially stabilize the equilibrium point of the augmented system (5.38). Indeed,such control laws can be found in the linear form under the assumptions listed in Section 5.1.

Theorem 5.7.

(i) Under Assumptions 3.1, 5.2, and 5.4, for any positive integer k, the kth-order robustoutput regulation problem is solvable by a linear state feedback control of the form

where (G1, G2) incorporates a p-copy internal model of the matrix Akf with G1satisfying Property 1.5, i.e.,

forall (G1).

5.3. Solvability of the fcth-Order Robust Output Regulation Problem 143

(ii) Under Assumptions 3.1 and 5.2 to 5.4, for any positive integer k, the kth-order robustoutput regulation problem is solvable by a linear output feedback control of the form

where (G1, G2) incorporates a p-copy internal model of the matrix Akf, where (G1,G2)takes the form (1.57) with G1 satisfying Property 1.5.

Proof, (i) Recall from Chapter 4 that the eigenvalues of the matrix Al/] are given by

where .I , . . . , q are eigenvalues of A1. Therefore, Assumption 5.4 guarantees that, for anyfixed integer k > 0, there exists a pair (Gi, G2) that incorporates a p-copy internal modelof Akf with GI satisfying Property 1.5. By Lemma 1.26, under Assumptions 3.1 and 5.2,the pair

is stabilizable. Thus there exist feedback gains KI and K2 such that the eigenvalues of thematrix

have negative real parts. Thus, under the control law (5.39), the closed-loop system satisfiesProperty 5.4. It follows from part (ii) of Lemma 5.6 that the control law (5.39) solves thekth-order robust output regulation problem.

(ii)Let (K\, K2, GI, G2) be what was obtained from part (i). Under Assumption 5.3,there exists L such that A — LC is stable. Let K = [Ki, K2],

Clearly, the pair (G1, G2) incorporates a p-copy internal model of the matrix Akf- Moreover,under the control law (5.40), the Jacobian matrix of the closed-loop system is given by

Subtracting the first row from the second row and then adding the second column to the firstcolumn shows that the spectrum of (5.43) is given by those of (5.42) and A — LC. Thus,the closed-loop system satisfies Property 5.4. Again, it follows from Lemma 5.6 that thecontrol law (5.40) solves the kth-order robust output regulation problem. D

144 Chapters. Nonlinear Robust Output Regulation

Remark 5.8.

(i) It is interesting to know that if v satisfies = A\v, then v[/] satisfies = All]v[l].Let

Then the matrix Akf is such that

The system (5.45) can be considered as a generalized exosystem which generates notonly the exogenous signal v (when a(v) — A\v), but also the higher order terms ofthe exogenous signal v up to order k. For convenience, we will call the system (5.45)a fc-fold exosystem.

(ii) For linear systems, the right-hand side of equation (5.15) is a linear function of v.Thus, in order to solve the robust output regulation problem, it suffices to requirea linear control law to incorporate a p-copy internal model of the matrix A\. Fornonlinear systems, the right-hand side of equation (5.15) is a nonlinear function of v.A linear control law that incorporates the p-copy internal model of the matrix Akf isable to render the right-hand side zero up to order k in v. But the control law cannotsolve the robust output regulation problem in general.

(iii) Effectively, Lemma 5.6 asserts that designing a kth-order robust controller for a non-linear system (5.1) is equivalent to designing a linear robust servoregulator for thelinear system consisting of the linear approximation of (5.1) and the k-fold exosystem(5.45).

Remark 5.9. The solvability conditions of the kth-order output regulation problem studiedin Chapter 4 and the jtth-order robust output regulation problem are basically the same,but the design philosophy of the control laws are completely different. The controller thatsolves the former problem relies on the approximate solution of the regulator equations, thusdemanding the complete knowledge of the plant. On the other hand, the kth-order robustservoregulator is designed completely based on the linearization of the given nonlinear plantat the origin. Regardless of the variations of the uncertain parameter w, the controller canguarantee the zero steady-state tracking error up to order k of the exogenous signal v.

Remark 5.10. Assumption 5.4 is an extension of the transmission zero assumption, that is,Assumption 1 .4. For linear systems, the solvability of the linear robust output regulationproblem will necessitate the condition Assumption 1.4. However, Assumption 5.4 may notbe necessary for the solvability of the kth-order robust output regulation problem. This isbecause our description of the plant uncertainty does not make the matrices EIW and FIW

change arbitrarily in an open neighborhood of E/0 and F/o as w varies arbitrarily in W.Thus, even though Assumption 5.4 fails to hold, the linear equation (5.35) may still have asolution.

5.4. Solvabilty of the Robust Output Regulation Problem 145

5.4 Solvability of the Robust Output Regulation ProblemIn this section, we will further show that, under some additional assumptions on the solutionof the regulator equations, a controller that solves the kth-order robust output regulationproblem for the composite system (5.1) and (5.2) also solves the robust output regulationproblem for the same system. Let us begin by characterizing the control law of the form(5.3) that solves the robust output regulation problem.

Lemma 5.11. Under Assumption 3.1', assume a control law of the form (5.3) is such thatthe closed-loop system has Property 5.4; then the control law also solves the robust outputregulation problem if and only if there exist sufficiently smooth functions (x(u, w), u(v, w),z(u, w)) locally defined in v V, w W with (x(0,0), u(0, 0), z(0,0)) = (0,0,0) suchthat x(V, w) and u(v, w) are the solution of the nonlinear regulator equations (5.18), andz(v, w) satisfies

Proof. Necessity. By Lemma 5.3, there exists a sufficiently smooth function \c(v, w) withXc(0,0) = 0 that satisfies (5.14) and (5.15). Partition Xc(v, w) as

where x(v, w) R". Since (fc(xc, v, w), hc(xc, v, w)) is given by (5.8), expanding (5.14)and (5.15) gives

Letting u(u, w) = k(x(v, w), v, z(v, w)) gives (5.46), and using (5.46) in the first and thirdequations of (5.49) shows that x(v,w) and u(v, w) satisfy the regulator equations (5.18).Finally, using the third equation of (5.49) in the second equation of (5.49) gives (5.47).

Sufficiency. By Lemma 5.3, we only need to show that there exists a sufficientlysmooth function Xc(v, w) with Xc(0,0) = 0 that satisfies (5.14) and (5.15). To this end,define Xc(v, w) = col(x(u, w), z(v, w)). Using (5.8) yields

Using (5.46) in (5.50) and (5.51) gives

146 Chapters. Nonlinear Robust Output Regulation

Using the second equation of (5.18) in (5.52) gives

hc(Xc(v, w), v, w) = 0,

that is, equations (5.15) hold. Using (5.15) in (5.53) gives

Finally, using the first equation of (5.18) and (5.47) in (5.54) gives

that is, equations (5.14) hold. D

To solve the robust output regulation problem, we need to impose an additional re-striction on the exosystem (5.2).

Assumption 5.5. The exosystem (5.2) is linear, that is, u = A1v, for some matrix AI.Further, all the eigenvalues of AI are simple with zero real parts.

Theorem 5.12.

(i) Under Assumptions 5.1, 5.2, 5.4, and 5.5, assume the solutions x(v, w) and u(v, w)of the regulator equations are degree k polynomials in v. Then if the state feedbackcontroller (5.39) solves the kth-order robust output regulation problem, it also solvesthe robust output regulation problem.

(ii) Under Assumptions 5.1 to 5.5, assume the solution u(u, w) of the regulator equationsis degree k polynomial in v. Then if the output feedback controller (5.40) solves thekth-order robust output regulation problem, it also solves the robust output regulationproblem.

Proof, (i) Assume that the controller (5.39) solves the kth-order robust output regulationproblem. By Lemma 5.11, we need to show the existence of a sufficiently smooth functionz(v, w) that satisfies

To this end, note that since the closed-loop system has Property 5.4, there exist suffi-ciently smooth functions x(y, w) and z(v, w) satisfying (5.32). Let e(v, w) be as defined in(5.33). Again, express x(v, w), z(v, w), and e(u, w) as in (5.34). Then, since the controller(5.39) solves the kth-order robust output regulation problem for / = 1,. . . , k, Xlw and Zlw

must satisfy (5.35) and (5.36), where Ac(w) — A(w) + B(w)Ki(w),Bc(w) = B(w)K2(w),

5.4. Solvability of the Robust Output Regulation Problem 147

Cc(w) = C(w) + D(w)Ki(w), Dc(w) = D(w)K2(w). Let Ulw = KiX[w + K2Z lw. Then(5.35) and (5.36) imply, for / = 1, ...,K,

By Lemma 4.7, there exist sufficiently smooth functions xk(v, tu) = ok(v) and uk(u, w) =Ok(v) such that

But, by assumption of this theorem, x(u, u;) and u(v, w) are degree k polynomials in u, thus

Let

Clearly (5.55) is satisfied. Now using (5.35) and (5.36) and noting that Ylw = 0 for / =!,. . . ,£ gives

Multiplying both sides of (5.61) from the right by vl/1 and then summarizing from / = 1 tok gives

Using

in (5.62) gives

which is the same as (5.56) upon using (5.60).


(ii) The proof of part (ii) is almost the same as that of part (i). Assume that a controllerof the form (5.40) solves the kth-order robust output regulation problem. By Lemma 5.11,we need to show the existence of a sufficiently smooth function z(u, w) with z(0, 0) = 0that satisfies

Let x(u, w) and z(v, w) be sufficiently smooth functions satisfying (5.32), and e(v, u) beas defined in (5.33). Again, express x(u, w), z(v, if), and e(v, u) as in (5.34). Then, for/ = 1 , . . . ,k , Xiw and Z/u, satisfy (5.35) and (5.36), where Ac(w) = A(w), Bc(w) =B(w)K(w), Cc(w) = C(w), Dc(w) = D(w)K(w). Let Utw = KZtw. Then (5.35) and(5.36) imply, for / = 1,.. . , k,

By Lemma 4.7, there exist sufficiently smooth functions xk(u, w) = ok(v) andUk(v, iy) = ok(v) such that

But, by assumption of this theorem, u(u, u;) is a degree k polynomial in v\ thus

Let

Clearly (5.63) is satisfied. The proof of the satisfaction of (5.64) is exactly the same as thatof (5.56) and is omitted.

Example 5.13. Consider

5.4. Solvability of the Robust Output Regulation Problem 149

With w = (wi, W2), we have

It is clear that the system satisfies Assumptions 5.1 to 5.5. Moreover, both x(u, w) andu(u, w) are polynomials in v with k = 2. By Theorem 5.12, the robust output regulationproblem for this system is solvable by either state feedback or output feedback. As a matterof fact, a simple calculation gives

Also, the minimal polynomials of A f l ] and A[2] are ( 2 + 1) and .( 2 + 4), respectively.Thus the minimal polynomial of the matrix A is

The minimal p-copy internal model for the matrix A2f is thus given by

This pair of matrices together with a pair of feedback gains K\ R2, K2 R5 that makesthe matrix (5.42) Hurwitz constitutes a state feedback robust servoregulator. I

The polynomial requirement on the solution of the regulator equations is obviouslytoo restrictive. It is possible to somehow relax this requirement if the exogenous signal v isavailable for control.

Theorem 5.14.

(i) Under Assumptions 5.1, 5.2, 5.4 and 5.5, suppose there exists some integer k > 0such that the solution of the regulator equations takes the following form:

where xw(v, w) and u[l](v, w) are degree k polynomials in v with coefficients de-pending on w, and xhk(u) and UM(U) are some sufficiently smooth functions of v,independent of w, vanishing at the origin together with their derivatives up to order


k. Then if a state feedback controller of the form (5.39) solves the kth-order robustoutput regulation problem, then the following controller

solves the robust output regulation problem.(ii) Under Assumptions 5.1 to 5.5, suppose there exists some integer k > 0 such that the

solution u(v, w) of the regulator equations takes the following form:

where u[k](v, w) is a degree k polynomial in v with coefficients depending on w, andUhk(v) is some sufficiently smooth function ofv, independent of w, vanishing at theorigin together with its derivatives up to order k. Then if an output feedback controllerof the form (5.40) solves the kth-order robust output regulation problem, the followingcontroller:

solves the robust output regulation problem.

Proof. We will only prove part (i) since the proof of part (ii) is almost the same as the proofof part (i). Let x(u, tu) and u(v, w) be the solution of the regulator equations associatedwith (5.1). Let xhk(u) and Uh k(U) be as defined in (5.69). Applying a state and inputtransformation x — x + Xhk(v), u = u hk(v) to (5.1) gives

It can be verified that x(v, w) — Xhk(U) andu(u, w)—Uhk(v) are the solution of the regulatorequations associated with the system (5.73). System (5.73) is still in the form of (5.1) andsatisfies Assumptions 5.2, 5.4, and 5.5, and x(u, w) = X(v, w) — xhk(u) and u(v, u;) =u(v, w) — U h k ( v ) are degree k polynomials in v. By Theorem 5.12, there exists a statefeedback controller of the form (5.39) that solves the robust output regulation problem forsystem (5.73). Thus, a controller of the form (5.70) solves the robust output regulationproblem for system (5.1).

Example 5.15. Consider the system

5.5. Computational Issues 151

where

and w = ( f l i , . . . , a/fc,), that is, ai's are the only uncertain parameters. Simple computationgives

Clearly, the solution of the regulator equations satisfies the condition (5.69). I

5.5 Computational IssuesTo synthesize a kth-order robust servoregulator, we need to compute the minimal polynomialof the matrix Akf. Thanks to Assumption 5.5, this seemingly tedious work can be easilyhandled due to the following result.

Theorem 5.16. Under Assumption 5.5, the matrix Akf is similar to a diagonal matrix.Therefore, the roots of the minimal polynomial of Akf are precisely given by all the distinctmembers of the following set:

where \,..., q are eigenvalues of the matrix A1.

Proof. As pointed out in the proof of Lemma 4.8, (AI/])r is the matrix of the linearmapping LA :Pl Pl as defined in (4.49) under the ordered basis given by (4.50).Therefore, we only need to show that this linear mapping has Cl

q+l-l linearly independenteigenvectors since the dimension of A[l is Cl q+ 1. To this end, let the row eigenvectors of A icorresponding to eigenvalues ., be ,, i = 1,..., q. By Assumption 5.5, /, i = 1, . . . , q,are linearly independent. Therefore, the following set:


consists of Clq+l_l linearly independent vectors. Moreover, noting LAlV(( v) = s i( v)s

gives

Thus, ( iv)11 ( 2y) l 2 • • • ( qV)lg is the eigenvector of LAIV associated with the eigenvalue= li.i + --+lq q. D

Theorem 5.16 leads to a straightforward way to calculate the minimal polynomial of Akf

as follows. Consider the following two cases:

(i) The total number of the distinct members in k is an even number. Then there exista positive integer nk and positive distinct real numbers 1 , . . . ,w n k such that

where j = —\. Thus the minimal polynomial of block diag{ A [ 1 J ] . . . , A[k]} is givenby

Let

G\ = block diag G2 — block diag

where ( , r,), i = 1, . . . , /? , is any controllable pair with , a column vector anda ( ) = |A7 — |. For example,

i = block diag

Clearly, ( ,-, ) is controllable and the minimal polynomial of i is equal to ( ).Thus the pair (G1, G2) is the minimal p-copy internal model of Akf.

(ii) The total number of the distinct values of (5.77) is an odd number; then there exist apositive integer n^ and positive distinct real numbers w 1 , . . . , wnt such that

Then the minimal polynomial of block diag{A[1],..., A[k] is given by

5.6. The Ball and Beam System Example 153

Thus, letting

leads to a minimal p-copy internal model of At/.

Example 5.17. Let

Then the minimal polynomial of block diag{A[1], A[2], A[3]} is

Note that the degree of the minimal polynomial of block diag{Afl], A[2], Af31} is 7, whilethe degree of the characteristic polynomial of block diag{A[l], A[2], A[3]} is 19. I

5.6 The Ball and Beam System ExampleWe will consider the approximate asymptotic tracking problem for the ball and beam systemdescribed in Section 2.8. For convenience, let us duplicate equation (2.113) as follows:

where x = col(x1, x2, x3, x4) = col(r, r, , ), y = r, H — M/(Jb/R2 + M).The objective is to design a state-feedback controller such that the position r of the

ball asymptotically tracks a sinusoidal reference input Amsinwt, where w is fixed.As before, we first define the exosystem as follows:

which yields i>i(t) = Amsinwt. Thus the error equation is given by


Assume the ball mass M and the moment inertia of the beam J in (5.79) are uncertainparameters. Let us write

where J0 and MO denote the nominal values of J and M, and / and AM the perturbedvalues of J and M. Perturbation of M will also cause the variation of H, which can bewritten as H = Ho + H, with H O being the nominal value and H the perturbed value.Let w — ( M, J). Then clearly, (5.79) is in the form of (5.1). Our design will be basedon the nominal plant, that is, the plant (5.79) with A7 = 0 and AM = 0. For this nominalplant, we can simplify the system by performing an input transformation

which leads to the following:

Recall from Section 2.8 that the system (5.79) does not have a well-defined relativedegree at the origin; therefore we cannot assure the existence of the solution of the regulatorequations. Nevertheless, it is easy to verify that this system satisfies Assumptions 5.2 and5.4. Therefore, for any integer k > 0, the kth-order robust output regulation problem forthis system is solvable. Since the kth-order output regulation problem is the special case ofthe kth-order robust output regulation problem, the kth-order output regulation problem forthis system is also solvable for any integer k assuming all the plant parameters are preciselyknown. In what follows, we will design both a third-order state feedback servoregulatorand a third-order state feedback robust servoregulator for this system.

A third-order controller for this plant can be designed as follows. First, let us use theapproach described in Chapter 4 to obtain a third-order solution of the regulator equationsassociated with the ball and beam system. The scalar form of the regulator equationsassociated with the above tracking problem takes the following form:


By inspection, we can obtain the partial solution as follows:

with two undetermined functions X(v) and X4(v) satisfying

Again, by the reason given in Remark 4.16, we can assume that the Taylor seriessolution of (5.85) and (5.86) can be expressed as follows:

Substituting (5.87) and (5.88) into (5.85) and (5.86) and identifying the coefficients gives athird-order approximation of \v) and X4(u) as follows:

where

Using the last equation of (5.84) gives the third-order approximation of u(u) as follows:


Thus the third-order approximation of the solution of the regulator equations of the ball andbeam system is given by (5.90) and

The Jacobian linearization of the nominal plant is given by

It can be verified that the pair (A, 5) is controllable. Thus a feedback gain Kx thatrenders the matrix A + BKX Hurwitz can be found. To be more specific, letting Kx =[-0.2826, -1.1604, 6.8783, 3.1500] will place the eigenvalues of A + BKX at

which is based on the ITAE criterion with the cutoff frequency equal to 1.5.Next we consider the design of a third-order robust servoregulator. For this purpose,

we need to find a pair of matrices (Gi, G2) that incorporates a minimal p-copy (p = 1)internal model of A f. But as pointed out above, since the solution of the regulator equationsdoes not contain the second-order term, the output equation hc(xc(v, W), v) = x(u, tu) — v\of the closed-loop system for any state feedback control law of the form (5.39) will notcontain the second-order term either. Thus, it suffices to find a pair of matrices (Gi, GZ)that incorporates a minimal 1-copy internal model of A[1] and A[3]. The minimal polynomialsof A[3] and A[3] are computed as follows:

The minimal polynomial of block diag{A[1], A[3]) is thus

Therefore,

The compensator together with the plant forms an eight-dimensional system. The feedbackgain (K 1 , K2) is chosen such that the eigenvalues of the linearized closed-loop system are


Amp First order Third order Third-order robust3.0000 " 0.0180 " 0.0001 0.00005.0000 0.0877 0.0021 0.00036.0000 I 0.1585 I 0.0058 | 0.0008

Table 5.1. Maximal steady-state tracking error of nominal system with w = .

Case M J First order Third order Third-order robust1 0 " 0 0.0877 0.0021 0.00032 0.0100 0.0100 2.6586 2.6882 0.03333 0.0150 0.0100 6.4298 6.6523 0.05274 -0.0200 0.0100 2.9178 2.8305 0.0417T I -0.0250 I 0.0100 I Unstable | Unstable | 0.0484

Table 5.2. Maximal steady-state tracking error of the perturbed system with Am =5 and w =

which again are obtained based on the ITAE prototype design with the cutoff frequencyequal to 1.5 rad/sec. The resulting feedback gains are

Ki = [-4.4018, -6.0091, 24.8522,7.8000],

K2 = [1.1226, -1.4605,0.0144, 2.6865].

Computer simulation is conducted to compare the performance of the two controllers.The nominal values of the various system parameters are given as follows: MO = 0.05 kg,R = 0.01 m, Jo = 0.02 kg m2, Jb = 2 x 10~6 kg m2, and G = 9.81 m/s2. As a result,HQ = 0.7134. It is assumed that the initial states of the plant and compensator are zero. Thefrequency of the reference input is fixed at W = , while the amplitude Am of the referenceinput takes the values of 3,5, and 6. Five cases are presented:

• Nominal case: AM = 0.0 kg, and J = 0.0 kg m2.• AM = 0.010 kg, J = 0.01 kg m2.• M = 0.015 kg, J = 0.01 kg m2.• M = -0.02 kg, J = 0.01 kg m2.• M = -0.025 kg, J = 0.01 kg m2.

Comparison is first made for the nominal case. Table 5.1 shows the maximal steady-state tracking errors of the closed-loop systems under the linear controller, third-order con-troller, and third-order robust controller for W = and Am = 3, 5, 6. It is seen that, inevery case, the performance of the various controllers is quite good, though the third-orderrobust controller is superior to the third-order controller, while the third-order controlleris superior to the linear controller. Next, we compare the performance of the various con-trollers in the presence of the parameter uncertainty with Am = 5 and W = |. As shownin Table 5.2, the third-order robust controller is quite capable of tolerating the parametricuncertainties. In various cases of the parametric uncertainty, the maximal steady-state track-ing errors are kept within the order of 10-2. In contrast, the tracking performance of both thelinear and the third-order controller severely deteriorates when the parametric uncertainties


Figure 5.1. Tracking performance: Nominal case Am = 5 and w = .

Figure 5.2. Tracking performance: Perturbed system with Am = 5 andw = .

are present. For example, in case 3, the maximal steady-state tracking errors of the lin-ear and third-order controllers are over 100 times that of the third-order robust controller.Moreover, in case 5, neither the linear controller nor the third-order controller can stabilizethe system. Also note that while the third-order controller performs much better than thelinear controller in the nominal case, it has no advantage over the linear controller whenthe parameter uncertainties are present. Figures 5.1 and 5.2 show the tracking performanceof the closed-loop system resulting from the third-order controller and third-order robustcontrollers with W = and Am = 5.

Chapter 6

From OutputRegulation toStabilization

The approach described in Chapter 5 employs an extended version of the internal modelprinciple introduced in Chapter 1 to handle the robust output regulation problem for nonlin-ear systems. The design approach consists of two steps. First, augment the given plant by alinear dynamic system that incorporates a p-copy internal model of the K-fold exosystem ofthe given system. Second, stabilize the linear approximation of the augmented system. Thisdesign method has two fundamental limitations. First, the linearity of the internal modelis incapable of handling nonlinear systems whose regulator equations have nonpolynomialsolution. Second, the linear stabilization method employed is incapable of achieving globalstability of the closed-loop system. In this chapter, we introduce a new design frameworkto deal with the robust output regulation problem. This design framework aims to sys-tematically convert the robust output regulation problem for a given system into a robuststabilization problem for an appropriately augmented system.

This new framework, on one hand, removes the polynomial assumption on the solutionof the regulator equations, and on the other hand, offers greater flexibility in incorporatingglobal stabilization techniques, thus setting the stage for studying a robust output regulationproblem with global stability in Chapter 7.

This chapter is organized as follows. In Section 6.1, the notion of the steady-stategenerator is introduced which is a dynamic system that can reproduce the solution or partialsolution of the regulator equations of the given plant. The notion of the steady-state generatorleads to a new definition of the internal model. The composition of the given plant and theinternal model is called the augmented system. It is shown that the stabilizing solution ofthe augmented system will lead to the solution of the robust output regulation problem ofthe original system. In Section 6.2, the existence conditions of the steady-state generatorare established. These conditions in turn lead to the construction of a nonlinear internalmodel. Section 6.3 shows that, due to the employment of the nonlinear internal model, it ispossible to design a dynamic output feedback controller to solve the robust output regulationproblem for a nonlinear system whose regulator equations admit a nonpolynomial solution.In Section 6.4, the new framework is applied to solve the robust disturbance rejectionproblem of the RTAC system.

159

160 Chapter 6. From Output Regulation to Stabilization

The notation defined in Chapter 5 will be used freely in this chapter. In particular, wedefine

Also, for convenience, we will lump the plant (5.1) and the exosystem (5.2) together asfollows:

We will refer to (6.1) as a composite system.

6.1 A New Design FrameworkAs pointed out in Remark 3.12, the output regulation problem can be viewed as a stabilizationproblem about an invariant manifold defined by the solution of the regulator equations. Whenthe solution of the regulator equations is available for feedback control, one can convert theoutput regulation problem into a stabilization problem about the equilibrium point at theorigin of a translated system, as was done in Chapter 3. However, when the plant containsunknown parameters, the solution of the regulator equations cannot be used for feedback.One wonders if the solution of the regulator equations can be obtained by some other meansso that the robust output regulation problem can also be converted into the stabilizationproblem of some related system. This idea motivates a new design framework to tacklethe robust output regulation problem. This framework includes the following three steps.First, introduce the concept of the steady-state generator for the system (6.1), which is somedynamic system that can produce a partial or whole solution of the regulator equations.Second, define a generalized internal model based on the steady-state generator which,together with the plant, is called the augmented system. Third, show that, after a suitablecoordinate and input transformation, the stabilizability of the equilibrium at the origin ofthe augmented system implies the solvability of the robust output regulation problem of theoriginal system.

Definition 6.1. Let F : V x W Rl, where V and W are some open neighborhoods of theorigins ofR,q and R , respectively, and I is some integer, be a smooth function vanishingat the origin. The function F is said to have a generator if, for some integer s, there exists atriple { , , ], where : V x W Rs, :Rs Rs, and : Rs Rl are sufficientlysmooth functions vanishing at the origin, such that, for all trajectories v(t) V of theexosystem (5.2) and all w W,

// V = Rq, W = U , then the triple [ , , } is called a global generator ofF(v, w).

6.1. A New Design Framework 161

Let the triple {0, a, ft] be a (global) generator of F(v, w). If, in addition, the lin-

Definition 6.2. Let g0 : 'R,n+m Rl be a mapping for some integer 1 l n + m.Under Assumptions 3.1 and 5.1, the composite system (6.1) is said to have a (global)steady-state generator with output g0(x, u) if the junction g0 (x (v, w) , u (v,W)) has a(global) generator. The system (6.1) is said to have a (global) steady-state generator withoutput g0(x, u) with linear observability if the Junction g0 (x (v, w), u (v, w)) hasa(global)generator with linear observability.

Remark 6.3. Existence of a steady-state generator with output g0(x, w) means that somefunction of the solution of the regulator equations can be reproduced by an autonomoussystem of the form

which is independent of the model uncertainty w and exogenous signal v. As will be seerlater, it is possible to use the information provided by g 0 ( X ( v ( t ) , u(v(t), w)) to designa controller. In particular, when g0(x, u) = col(x, M), the steady-state generator reproducesthe whole solution x(w, w;) and u(v,w) of the regulator equations, and when g0(x, u) = u,the steady-state generator reproduces the partial solution u(u, w) of the regulator equationsIn what follows, we will assume that g0 (x, u) = col(Xi1,X,i2,Xi3,.........U), where 1 i1 </2 ... id n for some integer d satisfying 0 d n. Without loss of generality,we can always assume ij= j for j = 1,..., d, since the index of the state variable can berelabelled to have this assumption satisfied. •

Remark 6.4. The motivation of introducing the notion of the steady-state generator willbe briefly elucidated in Remark 6.11. Here let us first connect this notion to the previousresults obtained in Chapter 5. By Lemma 5.11, under Assumptions 3.1' and 5.2, if thereis an output feedback control law of the form (5.5) that solves the robust output regulationproblem for system (6.1), then there exists a sufficiently smooth function z(v, w) definedfor v V, w W with z(0,0) = 0 such that z(v, u;) satisfies

Let0(v, w) = z(v, w), ( ) = g( , 0), ( ) = k( ). Then clearly, the triple { , ( ), ( )}is a steady-state generator of system (6.1) with output g 0 (x ,u ) = u. Moreover, denote thelinearization at the origin of the control law (5.5) by the triple (AT, \, i). Then

The fact that the matrix (0,0,0) is Hurwitz implies that the pair

earization of the pair at the origin is observable, then the triple iscalled a linearly observable (global) generator of


is detectable. Hence, the following decomposition:

further shows that the pair (K, G1} is detectable, too. In particular, when all the eigenvaluesof G1 have zero real parts, then the pair (K, G1) is observable. Thus, if the robust outputregulation for the system (6.1) is solvable by an output feedback controller of the form (5.5),then the system (6.1) must have a steady-state generator whose linearization at the origin isdetectable.

Remark 6.5. It is known that, under Assumptions 5.1 to 5.5, if the solution of the regulatorequations of the system (6.1) is a degree k polynomial in v, then the robust output regulationfor the system (6.1) is solvable by a linear output feedback control law of the form u =Kz, z = G1z + G2&, where the pair (G1, G2) incorporates a p-copy internal model of thematrix Akf. Moreover, by Lemma 5.11, there exists a sufficiently smooth function z(u,w)locally defined in v e V, w W with z(0, 0) — 0 such that z(u, W) satisfies

Let <9(u, w) = z(u, w), (G1 ) = G1 , ( ) = K9. Then, clearly, the triple is a steady-stategenerator of (6.1) with output g0(x, u) = u. Nevertheless, the polynomial assumption onthe solution of the regulator equations is restrictive. We will show in the next section thatthe steady-state generator may exist even when the solution of the regulator equations isnot polynomial. Before doing this, let us first give a more general characterization of theconcept of the internal model as follows.

Definition 6.6. Under Assumptions 3.1 and 5.1, suppose the composite system (6.1) hasa (global) steady-state generator with output g0(x, u). Let y : Rs+c{+m+p -+ Rs be asufficiently smooth function vanishing at the origin. Then we call the following system:

an internal model of (6.1) with output g0(x, u) if

For convenience at the price of the abuse of the notation, in what follows, we willalways use the notation y(r), x, u, e) to stand for y(rj, g0(x, u), e).

Remark 6.7. The reason for defining the internal model this way will be given in Remark6.11. At this stage, let us first note that the characterization of the internal model here,on one hand, contains the one described in Chapter 5 as a special case. As pointed out in


Remark 6.5, if a linear output feedback control law of the form u — Kz, z = G\z + g2esolves the robust output regulation problem for system (6.1), then there exists a func-tion z(v, u ) such that u(u, w) = Kz(v, w) and A1v = G1z(v,w). Moreover, let

(v, w) = z(v, w), ( ) = G1 , ( ) = K . Then the triple { , ( ), ( )} is a steady-state generator of (6.1) with output g0(x, u) — u. Now, let y (rj, x,u,e) = G1t]+G2e. Theny( (v, w), x(u, u;), u(u, u;), 0) = G1 (v, w) — ( (v, w)). Thus, the internal model de-scribed here is an extension of what is described in Chapter 5. On the other hand, thischaracterization is much more general than the existing one in two aspects. First, it can bea nonlinear system, and second, the system dynamics can be coupled to the given systemnot only through the tracking error e, but also through the state x and input u. We willsee in Section 6.3 that this generality can be used to construct a particularly useful nonlin-ear internal model. For the time being, we will first show that an internal model definedthis way leads to an augmented system with the property that the stabilizability of this aug-mented system implies the solvability of the robust output regulation problem of the originalsystem (6.1).

Attaching the internal model (6.6) to the given plant yields the following augmentedsystem:

Performing on (6.8) the following coordinate and input transformation:

gives a new system denoted by

where

The system (6.10) has the following property.


Proposition 6.8. Suppose the composite plant (6.1) satisfies Assumptions 3.1 and 5.1 andhas a steady-state generator with output g0(x,u) = col (xI, .. . , xd, u) and an internalmodel described by (6.6). Then the augmented system in the new coordinates and inputdescribed by (6.10) has the property that, for all trajectories v(t) V of the exosystem andall w W, where V and W are some open neighborhoods of the origins ofR,q and R ,respectively,

Proof. Consider the augmented system (6.8). Since x(v, w) and u(u, u;) are the solutionof the regulator equations, and (v, w) satisfies (6.2), the hypersurface {(x, n, v) | x =x(u, w), = 9(v, w)} is an output zeroing manifold of the composite system consistingof (6.8) and the exosystem (5.2) rendered by the feedback control u = u(v, w). Therefore,the hypersurface {( , , ) \ x = 0, = 0} is the output zeroing manifold of the compositesystem (6.10) and (5.2) rendered by the feedback control u = 0. This is, the origin (x, ) =(0, 0) is the equilibrium point of the unforced augmented system for all trajectories v(t) Vof the exosystem, and any w W, and the error output equation is identically zero at(x, , u) = (0,0, 0) for all trajectories v(t) V of the exosystem and for any w W.Thus the proof is completed. D

Consider a controller of the form

where R for some integer n , and k and g are sufficiently smooth functions vanishingat their respective origins. Let xc = co\(x, r), ) be the state of the closed-loop systemcomposed of the augmented system (6.10) and the controller (6.12). Then this closed-loopsystem takes the following form:

It is possible to show that if (6.13) satisfies Property 5.4, then the following controller:

solves the robust output regulation problem for the original plant (6.1).

Corollary 6.9. If the controller (6.12) is such that the closed-loop system (6.13) satisfiesProperty 5.4, then the controller (6.14) solves the robust output regulation problem for theoriginal system (6. 1).


Proof. Consider the closed-loop system composed of the plant (6.1) and the controller(6.14) and denote its state by xc = col(, , ). Then

Thus, when v = 0 and w = 0, the state xc of the closed-loop system (6.13) and thestate xc of the closed-loop system composed of (6.1) and (6.14) are related by a dif-feomorphism xc = xc + col( ( ),..., ( ), 0,. . . , 0, 0). Thus the closed-loop sys-tem composed of (6.1) and (6.14) also satisfies Property 5.4. Next let Xc(u, w) =col(x(v, w), (v, w), 0). Then it can be verified that Xc(v, w) is a zero error center manifoldof the closed-loop system (6.1) and (6.14). The proof is thus completed by using part (i) ofLemma 5.3. D

Remark 6.10. Corollary 6.9 concludes that if a controller solves the stabilization problemfor system (6.10), then this controller together with the internal model solves the outputregulation problem for the original system (6.1). Thus the robust output regulation prob-lem for (6.1) is converted into a robust stabilization problem for (6.10). In particular, thecontroller (6.14) can take the output feedback form when d — 0 and the function y isindependent of x, or the full state feedback form when d — n, or the partial state feedbackform when 0 < d < n. The number d depends on how many components of the state xcan be reproduced by the steady-state generator and /or how many components of the statex are needed for feedback. I

Remark 6.11. Having established Proposition 6.8 and Corollary 6.9, it is possible to fur-ther elaborate the concepts of the steady-state generator and the internal model as wellas their relationship. For convenience, first assume g0(xt u) — u. Then, from Proposi-tion 6.8 and Corollary 6.9, it can be seen that, in order to convert a robust output regu-lation problem for the composite system (6.1) into a robust stabilization problem for theaugmented system (6.8), the dynamic compensator as defined by (6.6) should have twoproperties:

(i) The augmented system (6.8) together with the exosystem (5.2) has an output zeroingmanifold {(x, ,v) \ x = x(v, w), r) = (v, w)} rendered by the feedback controlu = u(u, w).

(ii) The output zeroing manifold of the augmented system (6.8) and the exosystem (5.2)can be made attractive by a feedback control independent of jc, v, and w.

Once the coordinate and input transformation (6.9) are introduced, then the secondproperty can be translated into saying that the equilibrium point of the augmented system(6.10) with d = 0 can be stabilized by an output feedback controller.

In other words, the first property of the dynamic compensator (6.6) guarantees thatthe robust output regulation problem for the original system (6.1) can be converted into arobust stabilization problem of the equilibrium point of the augmented system (6.10), and


the second property of the dynamic compensator (6.6) guarantees that the equilibrium pointof the augmented system (6.10) is stabilizable by an output feedback controller.

Clearly, the internal model as defined in Definition 6.6 renders the dynamic com-pensator (6.6) the first property explicitly. However, Definition 6.6 does not say anythingabout the second property of the dynamic compensator (6.6). The reason is that there isno uniform concept of the stabilizability for nonlinear systems due to the varieties of thestability concepts and the complexity of nonlinear systems. The construction of the internalmodel (6.6) depends not only on the systems under consideration, but also on the specificstability requirements on the augmented system (6.10). Nevertheless, the generality of Def-inition 6.6 has offered the functional flexibility for constructing an internal model with thesecond property for a given class of nonlinear systems with a specific stability requirement.When it comes to local asymptotic stabilizability of the equilibrium point of the augmentedsystem, it is possible to synthesize a generic internal model having this property. This modelwill be shown in Section 6.3. In Chapter 7, we will further address the global robust outputregulation problem. In this case, the stabilizability of nonlinear systems is intractable ingeneral. We have to address this issue on a case by case basis when the specific form of thenonlinear systems is available.

It is quite clear that the steady-state generator itself can be viewed as a dynamiccompensator of the form (6.6) with property (i); that is, it can be viewed as an internalmodel. However, the steady-state generator can never have property (ii) since the dynamics

= a(r) of the steady-state generator is not coupled with the plant (6.1), and the equilibriumpoint of = a(n) at the origin is not asymptotically stable, as will be shown in the nextsection. Thus a more general characterization of the internal model has to be introduced inDefinition 6.6.

The above description also applies to the case where g0(x, u) also depends on the statex or part of the components of the state x. In this case, the state x or part of the componentsof the state x is assumed to be available for feedback control. •

6.2 Existence of the Steady-State Generator and theInternal Model

Let us first show that the steady-state generator exists when the solution of the regulatorequations satisfies certain differential equations.

Proposition 6.12. Assume the exosystem satisfies Assumption 5. 5, and let : V x W Rbe a sufficiently smooth function vanishing at the origin. Then (v,w) has a generator withlinear observability if there exists some set ofr real numbers a\,ai,...,ar such that

for all trajectories v(t) V of the exosystem and all w W.

6.2. Existence of the Steady-State Generator and the Internal Model 167

Proof. Let T be any nonsingular matrix of dimension r,

where

It can be readily verified that the triple defined by (6.16) is a generator of (v, w) with linearobservability. D

Corollary 6.13. Let g0 : Rn+m Rl for some positive integer 1 / n + m be asufficiently smooth function vanishing at the origin. Under Assumptions 5.1 and 5.5, fori = 1, . . . , / , let (v, w) = goi(x(v, u;), u(u, w)). Then the system (6. J) has a steady-stategenerator with output g0(x,u) with linear observability if, for each i = 1,. . . , / , there existpositive integers r, and real numbers a,,i,..., a,,r. such that

Proof. For each z, let { ,-, ,-, } be a generator of (u,w) with linear observability. Let

Then it is possible to verify that the triple { , , } is a steady-state generator withlinear observability of the system (6.1) with output g0(x, u) = co\(g0i(x, u),...,goi(x, M)).


Equation (6.15) is a linear differential equation. It is interesting to find the class offunctions that satisfy this equation. For this purpose, without loss of generality, we assumethat the dimension q of the matrix A i is an odd integer and AI takes the following form:

where So = 0, and

Proposition 6.14. Assume that the exosy'stem satisfies Assumption 5.5, and let n : V W7£ be an analytic function vanishing at the origin. Then the following are equivalent:

(i) There exists some set ofr real numbers a1, a 2 , . . . , ar such that

for all trajectories v(t) V of the exosvstem and all w W.(ii) Let Then, there exist

and \for some finite integer nk such that

where and, for and where C is theconjugate complex ofC1 .

(iii) There exist some integer nt and real numbers (w, V 0), / = 1,. . . ,n i•, such that

Proof. For i — 1,..., k, let

Then the solution of the exosy stem i) = A\v is v\(t) = vi(0), and for i = 1,... , k,


(i) (ii). Since n(v, w) is an analytic function of v, we can expand TT(U, w) intoa power series in v as follows:

where (w) is some real number depending on w and the notation v^ is as defined inChapter 4.

Substituting v 1 ( t ) = v1(0) and (6.25) into the power series expansion of (v(t), w)gives

where and Using (6.26) gives

Thus equation (6.22) implies

where

Since p( ) can only have r roots, there must exist an integer nk such that C/(w, Vo) = 0 forall |/| > nk. Thus n(v(t), w) must take the form (6.23).

(ii) (i). Let r = 2nk + 1 and a\, a 2 , . . . , ar be such that

Then n(v(t), w) as given by (6.23) satisfies

which shows that (v(t), w) satisfies (6.22) upon using (6.29).

1 70 Chapter 6. From Output Regulation to Stabilization

(ii) —> (iii). Note that

Also note that there exist integers /i, ...,lk such that w/ = l1w1)\ + • • • + lkwk. For conve-nience, assume the integers l\,..., lk are nonnegative. Then,

Thus, (v(t), w) must be a polynomial in v(t) with real coefficients depending on both theuncertain parameter w and the initial state Vo. Clearly, the above derivation can be slightlymodified to suit the case where some of the integers i1-, 1 i k, are negative,

(iii) (ii). It follows straightforwardly from (6.24) and (6.25). D

Remark 6.15.

(i) Let (v, w) be any sufficiently smooth function in v and w. We call a monic poly-nomial P( ) = — a1 — a2"k — ••• — ark

r~l a zeroing polynomial of (U, w) if,along all trajectories v(t) of the exosystem v = \v, (v(t), w) satisfies a differ-ential equation of the form (6.22). By Proposition 6.14, if n(v, w) has a zeroingpolynomial P( .), then (v(t), w) must be a polynomial in the trajectory v(t) ofthe exosystem or a trigonometric polynomial of the form (6.23). But (v, w) it-self does not have to be a polynomial in v. In fact, consider a function of the form

( | + v\) (v, w), where (v, w) satisfies (6.22) and (•) is any sufficiently smoothscalar function. Since >|(0 + ( ) is actually a constant equal to (0) + (0),

( ( )+i>3(0)*MO, ) = (U2 (0)+ (0)) ( ( , w). Clearly, (u|+v|)7r(v, w)also satisfies (6.22).

(ii) If 7 ( , u;) is a degree polynomial in v, we can write

Let P ( ) = .r — ai — a2 — ar) be the minimal polynomial of the matrixAkf. Then it follows from the Cay ley-Hamilton theorem and equation (5.45) thatP(X) is a zeroing polynomial of (v, w).

(iii) A monic polynomial P( .) is called a minimal zeroing polynomial of (v, w) if P(A,)is a zeroing polynomial of (I;, w) of least degree. Now assume that (v(t) , w) takesthe form (6.23) and C/ 0, / = 0, 1, 2 nk; then, clearly, P( ) = (A.2 +ft)/2) is the minimal zeroing polynomial of I(V, w). It is noted that, if P( ) is theminimal zeroing polynomial of (v, w), then all the zeros of P( .) are simple andpure imaginary. This property will be useful later when the stabilizability of theaugmented system (6.10) is considered. •


By Propositions 6.12 and 6.14, the existence of the steady-state generator of the form (6.16)requires that the solution of the regulator equations be a polynomial function of v(t) whichis still quite restrictive as it essentially requires that the nonlinear systems contain onlypolynomial nonlinearity. We now propose a more general steady-state generator as follows.

Definition 6.16. Let , (v(t), w), i = 1 , . . . , , for some positive integer I, be I trigono-metric polynomials of t or polynomials in v(t). They are called pairwise coprime if theirminimal zeroing polynomials Pi(X),. . . , P/( ) are pairwise coprime.

Lemma 6.17. Let g0 : Rn+m Rd+m for some integer 0 d n be a sufficientlysmooth function vanishing at the origin. Under Assumptions 5.1 and 5.5, assume, fori = 1,. . . , d + m, that there exist pairwise coprime polynomials (u, w) , . . . , (u, w),with , . . . , being the degrees of their minimal zeroing polynomials P ( .),..., P/' (X)

and a sufficiently smooth function , : Rr' +R ri' R. vanishing at the origin such that,for all trajectories v(f) V of the exosystem, and w W,

Then

(i) For i = 1, . . . , d + m, j = 1,... ,/, , let

block diag( ,..., ), and ( ) = (T ))- Then the system (6.1) has a steady-state generator [6, a, } with output g0(x, u) = col(gol(x,u)), ..., g0(d+m)(x, M)) asfollows:

where = block diag and T = block diag

with Ti being any nonsingular matrix of dimension the companion

matrix of satisfying with


(ii) For i = 1, . . . , d + m, let ty; = [ , . . . , /'] be the Jacobian ofTi at the origin,

where / . Then the pair ( , ,) is observable, hence the generator (6.32)is linearly observable if

the pair ( , /) is observable, i = 1, . . . , d + m, j = 1 , . . . , / / . (6.33)

Proof. (i) The triple ( ,, , •) is clearly a steady-state generator of (6.1) with outputgoi(x, M). Thus, the triple defined in (6.32) satisfies ( = T T = (( )and ( (u, iy)) =g0(x(u, w), u(u, iy)); that is, it is a steady-state generator of (6.1) with output g0(x, u).

(ii) To verify the observability of ( ,), it suffices to show, by the PBH test, that,for any A,

rank

= rank

It is clear that (6.34) holds for any A ( ). For any A, cr( ,), there exists 1 k 7,such that A. cr(4 ) and A ( ), and 1 , since, for any j K:, P ( .)and Pf (A) are coprime. Thus

.rank

since ( , ) is observable. Moreover, the linearization of ( ) and ( ) at the originis T T~ and T"1 with = diag( ,..., m). As a result, the generator (6.32) islinearly observable. The proof is thus completed. D

Remark 6.18. Denote the eigenvalues of A\ by ..., q. Then, by Proposition 6.14, thecollection of the zeros of all P (X) or, what is the same, the eigenvalues of take the form

Thus, all the eigenvalues of the matrix are semisimple with zero real parts.

Remark 6.19. From the proof of part (ii) of Lemma 6.17, it can be seen that it is necessary torequire that, for each i, the zeroing polynomials P/( ), j = 1,. . . , / ,- , be pairwise coprimeto guarantee the observability of the pair ( ,, ). But, for i k, the polynomials P ( ),


j = 1 , . - . , / , , / = 1, . . . ,Ik , do not have to be different. As we will see in Section6.4, for some class of systems, the functions F, and Tk, i £, may rely on the same set ofpolynomials. In this case, one can synthesize a reduced-order steady-state generator. I

Remark 6.20. Suppose a function n(v, w) satisfies equation (6.15), with r as the degree ofits minimal zeroing polynomial. Then defining T : R -> R. as a linear function such that,for all x € nr, T (x ) = [1,0,..., 0]x shows that

Thus, the class of functions satisfying (6.31) includes the class of polynomial or trigonomet-ric polynomial functions. Moreover, this class of functions is much larger than the class ofthe polynomial or trigonometric polynomial functions; for example, the regulator equationsof the system in Example 6.25 to be introduced later admit the following solution:

Assume I. Let Then

are the minimal zeroing polynomials ofand: respectively. Thus, defining

gives

Thus, the system has a steady-state generator with outputis easy to see that

Moreover, it

and Thus this steady-state generator is linearly observable. Note that theselection of 'v, w) and T is not unique. Taking the same example as above with andletting snows that


which in turn gives

and * = [1, 0, 0, 0 ]. Thus, the function u(i>, w) has another generator with linear observ-ability.

Corresponding to a steady-state generator of the form (6.32), we can construct a nonlinearinternal model as follows. Let Mi e ft('/+-+'/i>><('-/+-+'/>'> and Nt e K('}+"+rh*1, i =1,. . . , d -f m, be such that M, is Hurwitz, and (M,, TV,) is controllable. Since, for eachi = 1,.. .,d + m, the spectra of the matrices and M, are disjoint and the pair ( ,-, )is observable, there exists a unique, nonsingular matrix 7 Ti e 7 R(r/+"+ril)x(ri +-+r,-') thatsatisfies the Sylvester equation (Appendix A)

Let

is an internal model of the system (6.1) with output g0(x, u).

Proof. Let

Putting these equations together with = c o l ( r j i , . . . , rjd+m) gives

Thus (6.36) is an internal model of (6.1) with output g0(x, M), where g0 = col(g0i,...,go(d+m))-

6.3. Robust Output Regulation with the Nonlinear Internal Model 175

Remark 6.22. In the next section, we will give conditions for the composite system (6.1)under which the internal model given by (6.36) will render the augmented system (6.10)the local asymptotic stabilizability property by the output feedback. We also note that,in the special case where the solution of the regulator equations is polynomial in the so-lution v(t) of the exosystem v = A\v, the function (3(0) is linear with B(0) = T - 1 O .Thus the internal model (6.36) reduces to a linear internal model of the form = M +Ng0(x, u). I

6.3 Robust Output Regulation with the NonlinearInternal Model

In this section, we will apply the framework described in Section 6.1 to establish the solv-ability of the robust output regulation problem without assuming that the solution of theregulator equations is a polynomial.

Theorem 6.23. Consider composite system (6.1). Let Assumptions 5.1 to 5.3 and 5.5 holdand the conditions (6.31) and (6.33) be satisfied with g0(x, u) = u. Further, assume

for all A. such that PtJ (A.) = Ofor some i = 1,..., m, and some j = 1,...,/,.

Then, the robust output regulation problem is solvable by an output feedback con-trol law.

Proof. Under the assumptions of this theorem, the system (6.1) has a linearly observablesteady-state generator of the form (6.32) with output g0(x, u) = u. Corresponding to thissteady-state generator, define the internal model as given by (6.36) with output g0(x,u) = u,and a transformation of the form (6.9) with d = 0. This transformation converts theaugmented system (6.8) into the form (6.10), where

By Corollary 6.9, it suffices to (locally) stabilize the equilibrium point at the origin of (6.10)with v = 0 and w = 0. To this end, linearizing (6.10) at x = 0, = 0, u = 0 with v and w


being set to zero gives

Consider the decomposition

From Assumption 5.2 as well as the fact that M is Hurwitz, we conclude that (6.40) is stabi-lizable using the PBH test. To show that (6.40) is detectable, first note that M + N T'-1 =T T~-1 and all the eigenvalues of have zero real parts. Thus, under Assumption 5.3, thefollowing matrix:

has full rank for all A. a ( ) and Re{.} > 0. Next, using the decomposition

and condition (6.39), we conclude that the matrix also has full rank for all A € &( ). Thedetectability of (6.40) then follows from the PBH test. As a result, let K and L be such that

and

are Hurwitz. Then, system (6.40) can be stabilized by a linear feedback control law asfollows:

6.3. Robust Output Regulation with the Nonlinear Internal Model 177

where £ = col( 2) with £1 € 7Rn and 2 e Rs- Note that the variable e1 in thecontrol law (6.41) is not the true error output of the original plant and may not be mea-surable. Nevertheless, replacing el in (6.41) by the true error output of the plant e —h (x + x, u + B( ) + 0), v, w) gives an output feedback control law as follows:

Clearly, the linear approximation of the closed-loop system composed of the compositesystem (6.1) and this control law at the origin is the same as that of the closed-loop systemcomposed of the composite system (6.1) and the control law (6.41). Thus (6.42) also solvesthe robust output regulation problem of the composite system (6.1).

The control law (6.42) can be written as follows:

where

Finally, using (6.14) with d = 0 shows that the following output feedback control law:

solves the robust output regulation problem of the original system (6.1). D

Remark 6.24. In the special case where the system (6.1) is linear, the solvability conditionsof Theorem 6.23 are basically the same as those given in Theorem 1.31, and the controlleralso takes a linear form. However, the design method illustrated here is quite different fromthat described in Chapter 1. In particular, the dimension of the output feedback controllergiven in Chapter 1 is nq x m + n (assuming m = p), where nq is the degree of the minimalpolynomial of A 1, but the dimension of the output feedback controller (6.44) is 2nq x m+n.This difference is caused by the need to estimate the state col(Jc, rj) of the system (6.40).Next consider the nonlinear system (6.1) and assume the solution of the regulator equationsof (6.1) is a degree k polynomial in v. In this case, the dimension of the output feedbackcontroller given in Chapter 5 is nk x m + n (assuming m = p\ where nk is the degree of theminimal polynomial of the matrix Akf, while the dimension of the output feedback controller(6.44) is 2 x K + n(assuming m = p), where K is the dimension of the matrix . K can bemuch smaller than nk. For example, given some hypothetical nonlinear system with m = 1,nw = 1, and q = 2, suppose, for some k > 1, that u(u, w) = £^j+/2_.i a?,/2(iyi>i)/l(u>i>2)'2,


where v1 =• V2, v2 = —v1, and al1,/2 are known real scalars. Let n(v, w) = wv1. Then(v, w) = wv2. Defining r(n(v,w),x(v,w)) = E/i+/2=ifl/1/2 w))ll(n(v, u>))/2

gives u(y, w) = ( (v, w), (v, if)). Thus, for this system K = 2 regardless of k, butnk = 2k + 1 whenfc > 1. I

Example 6.25. Consider the following nonlinear system:

where d and yd are produced by

with a)i ^ w2- The robust output regulation problem of system (6.45) reflects the objectiveof asymptotic tracking of a sinusoidal reference input yd and rejection of a sinusoidaldisturbance d. The system is clearly nonminimum phase. Therefore, none of the inversion-based control approaches can handle this problem.

By inspection, the solution of the regulator equations is

Two different steady-state generators with linear observability with output g0(x, u) = uhave been constructed in Remark 6.20. Here we will further construct an output feedbackcontroller to solve the robust output regulation problem. To give a specific solution, wesuppose o>i — 1, 0)2 = 2. As described in Remark 6.20, letting

gives

and Thus, we can obtain a generator for i withand, for any nonsingular

matrix T and

block diag ( block diag

6.4. Robust Asymptotic Disturbance Rejection of the RTAC System 179

To obtain an internal model, let

M = block diag

Solving the Sylvester equations gives

T = block diag (7i, T2)

= block diag

The matrices that define the Jacobian linearization of the system (6.45) at the origin are

It can be verified that the pair (A, B) is controllable, and (C, A) observable. Also, thesystem (A, B, C, D) has only one transmission zero, which is equal to 1, and thus doesnot coincide with the eigenvalues of . Hence, it is possible to achieve the robust outputregulation for this system by Theorem 6.23. I

6.4 Robust Asymptotic Disturbance Rejection of theRTAC System

In Chapter 3, we have formulated the disturbance rejection problem of the RTAC systemas an output regulation problem and solved the problem with both the static state feedbackand the dynamic measurement output feedback controllers. It is seen that while the con-troller can completely eliminate the effect of the sinusoidal disturbance on the output of thesystem asymptotically for the nominal case, its performance deteriorates when the system'sparameter € is perturbed. In this section, we will further apply the approach introduced inthis chapter to design a robust output feedback controller for the asymptotic disturbancerejection of the RTAC system.

Let us write € = €Q + u>, where €Q is the nominal value of € and w is the perturbation.Thus the regulator equations of the system can be written as follows:


We note that the solution of the regulation equations is not polynomial in v, andtherefore the approach given in Chapter 5 cannot solve the robust output regulation problemof the RTAC system. Nevertheless, assuming that the displacement x\ of the cart and theangular position x3 of the proof-mass are measurable output variables, it is possible to designa measurement output feedback control law to solve the asymptotic disturbance rejectionproblem of the RTAC system in the presence of the variations of the parameter €. Indeed,let, Then

Thus letting

shows that the solution of the regulation equations satisfies condition (6.31).Since , we have that is, is the minimal

zeroing polynomial ofiig l>uiyliuiiiiai ui Ji {v, vu ).

It is ready to verify that the RTAC system admits a steady-state generatorwhere

where T e 7£2x2 is any nonsingular matrix. Clearly, the steady-state generator is linearlyobservable since the pair (4>u, ) is observable, where / — [1 0] is the Jacobian of Fu

at the origin. Thus, condition (6.33) is also satisfiedCorresponding to the above steady-state generator, we can obtain an internal model

as follows:

where with and and T is the solution of the

Sylvester equation T4> — MT — N . Since M is Hurwitz and (M, N) is controllable,the Sylvester equation has a unique nonsingular solution T as follows:


Performing the following coordinate and input transformation:

on the augmented system consisting of the RTAC system and the internal model (6.46)gives


By Corollary 6.9, it suffices to (locally) stabilize the equilibrium point at the originof (6.47) with v = 0 and w = 0 by a controller depending on x1 and x3 only. To thisend, linearizing the augmented system (6.47) with v and w being set to zero and noting

= [1,0] gives

The above system can be put into the following matrix form as follows:

where

Moreover, let

where

Then it can be verified that the linear system with col(jt, fj) as the state, u as the input, andym as the output is both stabilizable and detectable.


Now let K and L be such that the two matrices

and

are Hurwitz. Then a linear output feedback controller that stabilizes (6.47) can be given asfollows:

By Corollary 6.9, the controller that solves the robust output regulation problem ofthe original system is given as follows:

A specific controller has been synthesized with the various parameters as follows:

Also,

A" = [5.9374 -3.4198 -0.9555 -2.5082 5.9333 -1.7874],

which is such that the eigenvalues of the matrix (6.49) are

and

which is such that the eigenvalues of the matrix (6.50) are given by

[ -1.50 ±j 1.50 -2.25 -3.75 -4.50 -5.25 ].

h184 Chapter 6. From Output Regulation to Stabilization

Figure 6.1. The profiles of the displacement xi with € = 0.18,0.2,0.22, w = 3,and Am = 0.5.

Figure 6.2. The profiles of the state variables fa, x3, x4) with € = 0.2, w = 3,and Am = 0.5.


Figure 6.3. The profile of the control input u with € = 0.2, a> = 3, and Am = 0.5.

Computer simulation has been used to evaluate the performance of the closed-loopsystem with the initial state being x(0) = col (0.1,0,0,0), rj (0) = 0, and (0) = 0.Figure 6.1 shows the displacement x1 of the cart under a sinusoidal disturbance v1 =0.5 sin for cases where = 0.18, 0.2, 0.22. As expected, the parameter variations donot affect the steady-state response of the output. This is in sharp contrast with the nonlinearservoregulator designed in Chapter 3, where the same parameter variations significantlyaffect the steady-state response of the output. Figure 6.2 shows the profile of the other threestate variables x2, x3, x4, and Figure 6.3 shows the profile of the control input u(t).

Chapter 7

Global RobustOutput Regulation

The robust output regulation problem that we studied in previous chapters is local in thesense that Property 5.1 only guarantees the boundedness of the trajectories of the closed-loop system, and Property 5.2 only ensures the asymptotic regulation of the error output ofthe closed-loop system when the initial state of the plant, the controller, and the exosystem,and the uncertain parameter w are all sufficiently small. In practice, it is desirable to designcontrollers that render the global boundedness of the trajectories of the closed-loop system,asymptotic regulation of the error output of the closed-loop system for any initial state of theplant, the controller, arbitrarily large exogenous signals, and an arbitrarily large uncertainparameter w. A formal formulation of such a problem is called the global robust outputregulation problem and is the topic of this chapter.

We have already known from Chapter 6 that, under some suitable assumptions, therobust output regulation problem for a given plant can be converted into a robust stabilizationproblem for an augmented system. This design philosophy can also be used to handle theglobal robust output regulation problem. However, the global robust stabilization problemitself is a challenging topic. Only some limited results are available for handling certainclasses of nonlinear systems with special structures. Two such classes of nonlinear systemsare called nonlinear systems in output feedback form and nonlinear systems in lower tri-angular form, respectively. This chapter will give the solvability conditions of the globalrobust output regulation problem for both of these classes of nonlinear systems.

This chapter is organized as follows. Section 7.1 describes the problem. Section 7.2presents some stabilization results for nonlinear systems in lower triangular form. Sections7.3 and 7.4 establish the solvability conditions of the global robust output regulation problemfor nonlinear systems in output feedback form and for nonlinear systems in lower triangularform, respectively.

7.1 Problem DescriptionThe plant and exosystem considered in this chapter is described by

187

188 Chapter 7. Global Robust Output Regulation

where x(t) is the n-dimensional plant state, u(t) the m-dimensional plant input, e(t) the p-dimensional plant output representing the tracking error, v(t) the q-dimensional exogenoussignal representing the disturbance and/or the reference input, and w the nw -dimensionalplant uncertain parameter. The plant is somehow simpler than (5.1) in that the error outputequation does not depend on u explicitly. Again, we assume that all the eigenvalues of thematrix A\ are simple with zero real parts.

The class of control laws considered here is described by

where is the compensator state vector of dimension n to be specified later. The abovecontrol law is called the dynamic state feedback control law. When x does not explicitlyappear in (7.2), that is,

the control law is called the dynamic output feedback control law. With xc = col(x, ), theclosed-loop system can be written as

where

Again, all the functions involved in this setup are assumed to be sufficiently smooth anddefined globally on the appropriate Euclidean spaces, with the value zero at the respectiveorigins. Also it is assumed that 0 is the nominal value of the uncertain parameter w, and/(O, 0,0, w) = 0 and h(0,0, w) = 0 for all . Note that in (7.2), the feedbackcontrol is allowed to rely on the error output e explicitly.

In terms of the closed-loop system, we can describe the problem as follows.

Global Robust Output Regulation Problem (GRORP): For any compact set V0 Rq

with a known bound and any compact set W e R,nw with a known bound, find a controllerof the form (7.2) such that the closed-loop system (7.3) has the following two properties.

Property 7.1. For all v(0) € V0 and the trajectory of the closed-loop system (7.3)starting from any initial states xc(0) exists and is bounded for all t > 0.

Property 7.2.

A few remarks are in order.


Remark 7.1.

(i) By saying the bound of a compact set X e Rn is known, we mean that there exists aknown number c > 0 such that X

(ii) Since v(t) is generated by a stable linear system with v(0) e V0, where V0 is somecompact set of Rq with a known bound, there exists a compact set V e Rq with aknown bound such that v(t) € V for all t > 0.

(iii) Unlike the local case, Property 7.1 cannot be guaranteed by requiring the globalasymptotic stability of the equilibrium point of the system

at xc = 0. For example, consider the following system:

The solution of the system is given by

It can be seen that the equilibrium point of this system is globally asymptotically stablewhen v = 0. Nevertheless, when v 0, for example, v = 1, and col(x1(0), x2(0)) =col(l, 1),x2(0 = e°'5t approaches infinity.Thus, as will be seen later, in order to guarantee the satisfaction of Property 7.1by the closed-loop system, we need go farther than rendering xc = 0 a globallyasymptotically stable equilibrium point of xc = fc(xc, 0,0).

When dealing with the (local) robust output regulation problem, it suffices to assumethat the solution of the regulator equations exists in an arbitrarily small open neighborhoodof the origin of Rq x R"w. To handle the global robust output regulation problem, werequire that the solution of the regulator equations exist globally. Thus, Assumption 5.1 ismodified as follows.

Assumption 7.1. There exist sufficiently smooth functions x(v, w) and u(v, w) withx(0,0) = 0 and u(0,0) = 0 satisfying, for all v Rq and w Rnw, the followingequations:

Remark 7.2. Let x(v, w) and u(v, w) be a global solution of the regulator equations(7.6). Assume that system (7.1) has a global steady-state generator and an internal modelcharacterized in (6.6). Then the coordinate transformation (6.9) and the augmented system(6.10) are defined globally. As a result, we can obtain a global version of Proposition 6.8as follows. •


Proposition 7.3. Suppose Assumption 7.1 and assume that system (7.1) has a global steady-state generator with output g0(x, u) = col(x1,..., xd, u) and an internal model describedby (6.6). Then the augmented system in the new coordinates and input described by (6.10)has the property that, for all trajectories v(t) R,q of the exosystem, and all w Rnw,

Using this proposition, it is also possible to convert the global robust output regulationproblem for the given plant (7.1) into a global robust stabilization problem of the equilibriumpoint (n, x) = (0,0) of the augmented system (6.10) for any v(t) V and w W by theclass of controllers of the form (6.12). To this end, recall that the closed-loop systemcomposed of the augmented system (6.10) and the controller (6.12) is denoted by (6.13) andis repeated as follows:

where xc = col(x, n, ).

Global Robust Stabilization Problem (GRSP). For any compact set V0 Rq with aknown bound and any compact set W R.nw with a known bound, find a controller of theform (6.12) such that, for any xc(0), any v(0) € V0, and any w W, the trajectory of theclosed-loop system (7.8) exists for all t > 0 and satisfies

for some class KL function k l(., •) independent of v and w.

Corollary 7.4. Suppose Assumptions 7.1 and 5.5 hold. Given any compact set V0 Rq

with a known bound and any compact set W € Rnw with a known bound, assume thatcontroller (6.12) solves the global robust stabilization problem for the augmented system(6.10). Then a controller of the form (7.2), where = col(n, ),

solves the global robust output regulation problem for the original system (7.1).

Proof. Assume that the controller (6.12) solves the global robust stabilization problem ofsystem (6.10) for some given compact sets Vb e Rq and W R"w. Denote the stateof the closed-loop system composed of the plant (7.1) and the controller (7.10) by xc =col(x, n, ). Then


Let Xc(v, w) = col(x(u, w), (v, w), 0). Then

Using inequality (7.9) gives, for all t > 0,

Note that, for all xc(0), all v(0) V0, and all w W, n( t ) and 0(v(t), w) arebounded for all t > 0. Therefore, the fact that the functions and are C1 and vanish attheir respective origins guarantees the existence of constants L; > 0, i = 1,..., d, suchthat, for t > 0,

Then we further have, for t > 0,

for some positive constant L. As a result, for all xc(0), all v(0) V0, and all e W, xc(t)is bounded for all t > 0 and

Similarly, since the function h is C1, there exists a constant L0 > 0 such that

Thus, using (7.11) gives

Due to this corollary, we have also converted the global robust output regulationproblem for the given plant (7.1) into a global robust stabilization problem of the augmentedsystem (6.10).


7.2 Stabilization of Systems in Lower Triangular FormIn this section, we will study the class of nonlinear systems in the following form:

where x — col(x1,..., xr) and z = col(zo, z1 , . . • ,zr) are the states with xt R, i =1,. . . , r, and zi Rni, i = 0, . . . , r, u R is the input, and : [to, ) -> is apiecewise continuous function with S a prescribed compact set of . The function :

and functions Qt :R,, and bi : R for i = 1,. . . , r are C1 satisfying f0(0, 0, ) = 0, Qi (0,..., 0, ) =0, and f i(0,..., 0, ) = 0 for all .

In (7.12), the vector represents a set of unknown parameters and/or disturbancesand is called the static uncertainty. On the other hand, the functions Qi may not be knownprecisely and/or the state zi may not be available for feedback control. Thus the dynamicsgoverning zi, i = 1, . . . , r, are called the dynamic uncertainty of system (7.12) as opposedto the static uncertainty (t). In the special case where system (7.12) involves no dynamicuncertainty, that is, ni = 0, i = 1,..., r, the system reduces to the following:

System (7.13) is called a (strictly feedback) lower triangular system. In the morespecial case where (f) does not appear on the right side of (7.13), the subsystem zo =fo ( Z 0 , 0) is the zero dynamics of system (7.13) viewing x1 as the output.

In what follows, we will consider the global robust stabilization problem for system(7.12) with respect to both static and dynamic uncertainties using a sufficiently smoothpartial state feedback control of the form u = k ( x 1 , . . . , xr) with k(0,..., 0) = 0. For thispurpose, let us list a few assumptions as follows.

Assumption 7.2. For all and all

Assumption 7.3. The system Z0 = f (zo, x1, ( t ) ) , t > t0» is RISS with respect to withstate Z0 and input x1 and has a known C1 gain function K 0 ( . ) .

Assumption 7.4. For all i = 1,. . . , r, the system z, = i

t > to > 0, is RISS with respect to with state zi, and input col(zo,Z1,... ,Zi-1,x\,. . . , X 1 )and has a known C1 gain function ki, (•).

Remark 7.5. By the definition of RISS for systems of the form (7.12), Assumptions 7.3and 7.4 mean the existence of some class KL function (., •), some known C1 class

7.2. Stabilization of Systems in Lower Triangular Form 193

K functions Ki(•), i = 0,1, . . . , r, which are independent of , such that, for all :[to, ) ->• R , the solution zo(t) of system zo = f ( Z 0 , x1, (t)) and the solutionszi(t) of zi,- = Qi (zo, zi, • • •, Zi, x1, • •., xi, (t)), i = 1,.. . , r, exist and satisfy, for allt > to > 0,

and

for all zi,(t0) e Rni, and all col(z0,..., z i - 1 , x1, . . . ,x i) . Also note that,under Assumption 7.2, for any compact set , there exist real numbers bMi, bmi,i = 1,..., r, such that > bMi > bi( ) > bmi > 0 for all

The main result of this section is given as follows.

Theorem 7.6. Under Assumptions 7.2 to 7.4, there exists a sufficiently smooth state feedbackcontroller of the form u = k ( x 1 , . . . , xr) satisfying k(0,..., 0) = 0 such that the equilibriumpoint of the closed-loop system at the origin is globally asymptotically stable for all :

We will use a recursive approach to synthesize a state feedback controller to globallystabilize (7.12). The recursive approach will be based on the following proposition, whichhandles a special case of (7.12) with r = 1 and n1 = 0.

Proposition 7.7. Consider the system

in which (z, x) is piecewise continuous witha prescribed compact set of R,"*, and are C1 Junctions satisfying

for . Suppose the following:

(i) The upper subsystem in (7.16) is RISS with respect to with state z and input x, andhas a known C1 class gain Junction k(•)•

(ii) For all

Then, there exists a smooth Junction such that, under the controller

the closed-loop system (7.16) and (7.17) is RISS with respect to with state Z = col (z, x)and input u and has a known C1 class gain Junction k(.).


Proof. Consider the system composed of the lower subsystem of (7.16) and controller(7.17):

If p(x) can be chosen such that system (7.18) is RISS with respect to , with state x andinput col (z, u), in particular, the solution of system (7.18) exists and satisfies, for someclass KL function (-, •). some known class function y z(.), and some known C1 class

function yu(-)» independent of

for all x(t0) and : [rfl, oo) -» . Further, if p(x) issuch that

then the proof is completed upon using Corollary 2.19 (The Small Gain Theorem) with theC1 gain function k(s) being any C1 class K function satisfying

To complete the proof of Proposition 7.7, we need to establish two more lemmas asfollows.

Lemma 7.8. Let f : Rm x Rn x Rp -» R be a C1 Junction satisfying /(0,0, ) = 0 for all, with being a compact set of RP. Then there exist smooth functions F1 : Rm —> R,

and F2 : Rn -> Rsatisfying F1 (0) = 0 and F2(0) = 0 such that

Proof. Let

and

Then, |f(x, y, )\ < fi(x) for all when ||y|| < ||x||, and |f(x, y, )\ < f2(y) forall when ||x|| < ||y||. Thus, for all (x, y, ) Rm xRn x ,

Clearly f1(0) = 0 and f2(0) = 0. Moreover, since f(x,y, ) is C1 and is compact,there exists a constant L > 0, independent of , such that \ f ( x , y, )| < L(||JC|| + \\y\\)

7.2. Stabilization of Systems in Lower Triangular Form 1 95

for all sufficiently small x Rm, y Rn, and all . Thus, for all sufficiently smallx Rm,

that is, f1(x) is linear locally. Similarly, f2(y) is also linear locally. Therefore, there existsmooth functions FI(x) and F2(y) with F1(0) = 0 and F2(0) = 0 such that f1(x) F1(x)for all x Rm, and f2(y) F2(y) for all y Rm.

Lemma 7.9. There exists a smooth function p : R [0, ), such that, under the controller(7. 1 7), system (7. 18) is RISS with respect to with state x and input col (z, ). In particular,for any given C1 class K Junction K(•), p(x) can be chosen such that the solution of (7. 18)satisfies the inequality (7.19) with a known class K function yz(.) satisfying the small gaincondition (7.20), and a known C1 class K function yu(.).

Proof. By assumption (ii) of Proposition 7.7, there exist bM bm > 0 such that bM

( ) bm for all . Also, since (z, x, u) is a C1 function satisfying (0, 0, u) = 0for all u Rnu, by Lemma 7.8, there exist smooth functions F1(x) with F1(0) = 0and F2(y) with F2(0) = 0 such that

Moreover, by Taylor's theorem, there exist smooth functions o(.) 1 and 1(.) 1 suchthat FI(x) x (x) for all x R and F2(z) z 1(z) for all z Rm. Thus,

As a result, the function V(x) = x2 satisfies

for all x R, all z , all L , and all .Now, given any smooth function ao : R [0, ), letting p(x) 0 be any smooth

function satisfying

gives


for all x Rm, all z L , all u L , and all u . In particular, assume a0 (X) is evenand nondecreasing in [0, +00), and let a( x ) = x2ao(x) for all x R. Then a(•) is a classK function.

Since 1(z) 1 for all z Rm, there exists a smooth nondecreasing function c :[0, ) [0, ) satisfying c( z ) (z). Letting (s) — s2(1 + c(s)), which is aclass K function, gives

for all x R, z L , u L , and u . Also letting a(s) = a(s) = s2, which areclass K functions, gives ( x ) V(x) ( x ) for all x R. Thus, by Theorem 2. 17as well as Remark 2.21, the closed-loop system (7.18) is RISS with respect to u, with statex and input col (z, u).

To obtain an estimation of the form (7.19), let z(s) = s2c(s), u(s) = s2, x z ( s ) =

a-1( z(s)),and X u ( s ) = a- l ( u(s)), where 8 > 2. Then the inequality x max{xz( z ),Xu( u )} implies

which in turn implies

for all x R, z L , u L , and u, . By Theorem 2.16, an inequality ofthe form (7.19) holds, where yz(s) = .l o a o Xz(s) and yu(s) = a-1 o a o Xu(s).Since a(s) = a(s) = s2, we have yz(s) = Xz(s) = a-l( z(s)) = a- l( s2c(s)) andyu(s) = Xu(s) = a-l( u ( s ) ) = a-l( s2).

Clearly, yu (•) is a class K function. It remains to show that, the function a(.), hencep( . ) , can be chosen to satisfy the small gain condition (7.20) and yu(.) is C1. To this end,for the given C1 class K function k(•), let a(s) = z ( k (2 s ) ) = K 2(2s)c(k(2s)) , whichis a class K function, and satisfies, for all s 0,

Since k(.) is C1 and K(0) = 0, there exists a C° function a(x) such that a( x ) = x2 (x)for all x. Letting ao(x) a(x) shows that a (s) a(s), hence,

Thus, the small gain condition (7.20) is satisfied. To show that yu(.) is C1, note thaty (s) = s a 0 (s) / . Thus yu

-1(s) is C1 and its derivative with respect to s is greaterthan 0 for all s 0. By the Inverse Function Theorem, yu (•) is also C1.

Remark 7.10. In summary, the function p(x) can be obtained as follows:

(i) Obtain 0 ( x ) and 1 (z) from (7.23).


(ii) Obtain the function a(s) = k2(2s)c(k(2s)), where > 2 and c(.) is a nondecreasingsmooth function such that c( z ) (z).

(iii) Let a(s) be a C° function such that a( x ) = x 2 a(x) , and let a 0 ( X ) be a smooth andeven function nondecreasing in [0, ) such that a 0 ( X ) a(x) for all x R..

(iv) Obtain p(x) from (7.24). •

Lemma 7.11. Consider the system

in which x Rn, u Rm, u : [t0, ) piecewise continuous with a compact setof Rnu, and /(0, 0, /u) = 0 for all u . Suppose system (7.28) is RISS with respect to uwith x as state and u as input and has a known C1 class K gain Junction K( . ) . Then, forany square matrix G(u) of dimension m with its entries being sufficiently smooth functionsof u, the system

is also RISS with respect to u, with x as state and u as input and has a known Cl class K gain Junction y(s) = K(SC(S)), where c : [0, ) [0, ) is some Cl nondecreasing

Junction such that c( u ) G(u) for all u Rm.

Proof. By the assumption, there exist some class KL function k1(., •) and some knownC1 class K function k(.), independent of u(.), such that the solution of (7.28) exists andsatisfies, for all x(t0) Rn, all u L ,, all u and all t t0 0,

Let y(s) = sc(s). Then y(s) is a C1 class K function satisfying x( u ) G(u) u .It is now possible to verify that the solution of (7.29) satisfies, for all t > to > 0,

where y(s) = K(y(s)) = K(SC(S}) is a known C1 class k function. The proof is thuscompleted.


We are now ready to complete the proof of Theorem 7.6. For this purpose, we considerthe following transformation:

for some integer 1 j r, where a j ( x j ) = — x J P J ( X J ) , with P j ( X J ) 0 some smoothscalar function. Then, for any 1 j r, under the transformation (7.30), system (7.12)can be put into the following form:

where Zj = col (zo, z1, x 1 , . . . , Zj, xj) and the other functions are defined recursively asfollows:

with a j;(x j) = for 1 j r. It is clear that system (7.12) itself is also in the form(7.31) with j= 0 upon defining zo = Z0, F0(z0, x1, u) = f0(zo, x1, u), and a0(xo) = 0.

Lemma 7.12. Under Assumptions 7.2 to 7.4, for any 0 j r, there exist smoothJunctions p 1 ( . ) , . . . , pj(.) such that, with «i(^i) = x1p1(x1),..., aj(xj) = - X j P J ( X J ) ,system (7.31) satisfies the following property.

Property 7.3. The subsystem Zj = F J ( Z J , X J + 1 , u), t to 0, is RISS with respect to uwith state Zj and input xj+i, and has a known C1 class k gain function kj(•)•

Proof. We will prove it by using mathematical induction. When 7 = 0, Property 7.3is implied by Assumption 7.3 with the known C1 class k gain function being given byKO(.) = K0(.).


Now assume that, for some integer 0 J r, there exist smooth functionsP1(.) • • •. pj(.) such that, with a1(x1) .= x1p1(x1), • • aj,(xj) = - x j p j ( x j ) , system(7.31) with j = J satisfies Property 7.3.

First note that applying a coordinate transformation xj+2 = xj+2 — a j+i(x j+i) tosystem (7.31) with j = J yields a system of the same form as (7.31) with 7 = J + 1. Thenwe will show that system (7.31) with 7 = 7 + 1 satisfies Property 7.3. Consider the systemcomposed of the following three equations:

By induction assumption, system (7.32) is RISS with respect to /z, with zj as state andxj+1 as input, and has a known class k gain function j(•). Consider system (7.33). Bydefinition,

Let uj+1 = co1(zj, xj+1). Then there exists a square matrix Gj+1 (uj+i) of dimensionn0 + nj + J + 1 with its entries a smooth function of uj+1 such that

CO\(Z0 ,Z1 , • • • , z j , x 1 , x 2 + a 2 (x i ) , . . . , x j + 1 + a j ( x j ) ) = Gj+1(uj+i)uj+1.

Therefore, (7.33) can be written as

By Lemma 7.11, Assumption 7.4 implies that system (7.36), hence (7.33), is RISS withrespect to u with z j+1 as state and col(zj, xj+1) as input, and has some known C1 gainfunction y j + 1(S) . Thus by Corollary 2.20, the system consisting of (7.32) and (7.33) isRISS with co\(zj, zj+1) as state and xj+i as input and has some known C1 gain functionkj+1 (s). Finally, note that equations (7.32) and (7.33) can be viewed as the upper subsystemof (7.16), and equation (7.34) can be viewed as the lower subsystem of (7.16). ApplyingProposition 7.7 to system (7.32) to (7.34) shows the existence of a smooth function pj+1(.)such that the following system:

where aj+1(xj+1) = —xj+1pj+1(xj+1) is RISS with respect to u with zj+1 as state andxj+2 as input and has some known C1 class k gain function Kj+1(.). The induction iscompleted upon noting that the system zj+i = F j+1(Z j+1, xj+2, u) is nothing but (7.37)in a compact form.

When 7 = r, system (7.31) becomes zr = Fr (zr, xr+1, u), it is RISS with respect to /nwith state zr and input J r+1, and it has a known C1 class k gain function kr (•). In particular,

116 4. GRADIENT MAPPINGS AND MINIMIZATION

provided that f ( s , t, r, p, q) is again strictly convex in r, p, q for each s, t\ seeE 4.4-12. However, Stepleman [1969] has shown that the following is also true:Suppose that f ( s , t, r, p, q) is convex in r, p, q for each fixed s, t, and strictlyconvex in p and q. Assume further that f ( s , t, r, p, q) -* +00 as p2 -f- q2 -*• ooand that the matrix H of (8), (12) has rank n. Then the functional

with y, > 0, i = \,...,M, has a unique minimizer. (Note that this holdsregardless of the f tf and |i;.)

More generally, Stepleman has also given results when / is not convex in r,as well as a treatment of the "nonlinear" discretization (1.5.17).

EXERCISES

E 4.4-1. Conclude that 4.4.1 remains valid provided that there is a constantc > —A, where A is the minimal eigenvalue of A, such that either (a)<f> is continuously differentiable and <f>'(x) — cl is symmetric, positive semidefinitefor all x 6 R", or (b) <f> is continuous and diagonal and </>-cl is isotone.

E 4.4-2. Let/: [0,1] X R1—>• R1 have a continuous partial derivative d^fwhich satisfies &2f(t, s) ̂ 77 > — •** for all t e [0, 1] and i e R1. Use E 4.4-1 andE 2.3-4 to conclude that the system (2) has a unique solution for sufficientlysmall h = (n -f I)"1. Apply this result to the pendulum problem (1.1.1) withI c \< **.

E 4.4-3. Consider the boundary value problem

where a is a continuous function on [0, 1] and/satisfies the hypotheses of 4.4.2.Set a, = a(ih), i = 1 n, and show, by applying 4.4.1 and E 2.3-5, that, forall h ̂ k0 < (max | at \)~l, the system of equations

has a unique solution.

E 4.4-4. Let BeL(R") be symmetric, negative definite and suppose that<£: Rn —»• /?" is continuously differentiable and that <£'(#) is symmetric, positivesemidefinite for all x. For any b e Rn, show that the equation x — B<f>x + b hasa unique solution.

7.3. Global Robust Output Regulation for Output Feedback Systems 201

Remark 7.14. As pointed out before, system (7.13) is a special case of (7.12) when n\ =. • • — nr = 0. Under the transformation (7.30), for 0 j r, system (7.13) can be putinto the following form:

where Zj = col (z, x 1 , . . . , xj) and the other functions are defined recursively as follows:

Since system (7.13) satisfies Assumption 7.4 automatically, we have the followingresult on the solvability of the global robust stabilization of the lower triangular system(7.13) as follows.

Corollary 7.15. Under Assumptions 7.2 and 7.3, for any 0 j r, there exist smoothfunctions ai(•), i = 0,.. . , j, such that system (7.40) satisfies the following property.

Property 7.4. The subsystem Zj = FJ(ZJ, xj+1, u) is RISS with respect to u with state Zj

and input Xj+1 and has a known C1 class k gain function kj(•)•

As a result, there exists a smooth state feedback controller u = k(x 1 , . . . , xr) thatsolves the global robust stabilization problem of system (7.13).

Remark 7.16. For system (7.40), for any 0 j r, a C1 class K gain function Kj+1(.)of the subsystem Zj = FJ(ZJ, Xj+1, u) can be more easily obtained from a given C1 gainfunction kj(•) as follows. In fact, applying Proposition 7.7 to the system consisting of thesubsystem zj — F J (Z J , , xj+1, u) and (7.38) immediately concludes that the gain functionK J + I ( . ) is given by any C1 class K function satisfying

where yy+2 (s) is as defined in Remark 7.13.

7.3 Global Robust Output Regulation for OutputFeedback Systems

Consider the class of nonlinear systems described by


where col(x, y) Rn is the state, y R the output, u R the input, w Rnw is a vectorof uncertain parameter, and all the functions are sufficiently smooth.

Systems described by (7.42) are called nonlinear systems in output feedback form.The problem of global robust stabilization of such systems by output feedback control hasbeen well studied in the literature [86]. In this section, we will further study the robustoutput regulation problem for a modified version of (7.42) as follows:

where u is the exogenous signal generated by u = A1v and D1(v, w ) , D 2 ( v , w), and q(v, w)are sufficiently smooth functions satisfying D1(0, w) = 0, D2(0, w) = 0, and q(0, w) = 0for all w; Rnw.

The first step towards solving the robust output regulation problem for system (7.43)is to convert the system into the lower triangular form through a suitable dynamic extensionand coordinate transformation. For this purpose, let us first make the following assumption.

Assumption 7.5. System (7.43) has a uniform relative degree r 2; that is, for all w Rnw,

and

Now define the following dynamic extension:

where

with being positive numbers.Next we perform on the extended system (7.43) and (7.44) the following coordinate

transformation:

which turns the extended system (7.43) and (7.44) into the following:


Clearly, in order to render system (7.46) a lower triangular form, it suffices to chooseD(w) and h(w) such that, for some scalar function b(w),

or, equivalently, for some scalar function b(w),

Let us first obtain D(w). For mis purpose, assume

Substituting (7.48) into the first equation of (7.47) gives

with h(w) and b(w) satisfying

Substituting (7.48) into the second equation of (7.47) gives


It is noted that, when r = 2, the last equation of (7.52) should be understood as d 1 ( w ) =g(w).

It is now possible to verify, using Assumption 7.5, that D(w) as defined in (7.52)indeed satisfies the third equation of (7.47) by letting

Finally, substituting d\(w) into (7.50) gives

where


With D(w), h(w), and b(w) defined as above, the extended system together with theexosystem takes the following form:

where

Finally, we will establish a property regarding the matrix F(w) as follows.

Lemma 7.17. Assume 0 for i = 1, . . . , r — I. Then the eigenvalues of the matrixF(w) have negative real parts for all w Rnw if and only if the following assumption holds.

Assumption 7.6. For all w Rnw, the linear system

with y as output is a minimum phase system.

Proof. The numerator polynomial of the transfer function from u to y of (7.55) is given by

On the other hand, the numerator polynomial of the transfer function from u to y of thefollowing system:


is given by

Thus, under assumption > 0, i = 1,..., r — 1, system (7.57) with y as output isminimum phase if and only if Assumption 7.6 holds.

Now by a mere inspection (refer to Remark 2.46), it can easily be found that the zerodynamics of the following system:

with y as the output, is given by

The proof follows from the fact that the zero dynamics of systems (7.57) and (7.60)with y as the output are the same (modulo the coordinate transformation (7.45)), and, frompart (iii) of Remark 2.45, that the eigenvalues of the matrix F coincide with the roots of thenumerator polynomial of the transfer function from u to y of system (7.57). D

We are now ready to consider the robust output regulation problem of system (7.54).We need two more assumptions.

Assumption 7.7. For all

Assumption 7.8. There exists a sufficiently smooth function z(u, w) with z(0,0) = 0satisfying, for all v e Kq and all w e Rnw,

Under Assumptions 7.7 and 7.8, let


and

Then it can be verified that the regulator equations associated with system (7.54) have a solu-tion given by col (z(u, tu),y (v, u>), (v, w))and u(v, w), where (v, w) = col(Si(v, w),..., r_1(v, w;)).

Remark 7.18. Equation (7.61) is a type of center manifold equation studied in Section 4.4.By Lemma 4.13, if none of the eigenvalues of the matrix F(w) coincide with any . givenby l1 + • • • + lq = I, I = 1, 2 , . . . , l i , . . . , lq = 0, 1,...,l },where , . . . , are eigenvalues of the matrix AI, then (7.61) has a formal power seriessolution of the form

where, for all l = 1, 2 , . . . , Zl (w) satisfies the Sylvester equation of the form

where GI(W) is such that G(q(v, w), v, w)q(v, w) + D1(v, w) = . Inparticular, when q(v, w) and D1 (v, w) are polynomials in v and G(y, v, w) is a polynomialin v and y, then for some integer k, G(q(v, w), v, w)q(v, w) + D1(v, w) is a degree k:polynomial in v. In this case, equation (7.61) has a unique globally defined solution whichis a polynomial of degree k in v. I

Next, we will convert the robust output regulation problem for system (7.54) into arobust stabilization problem for an augmented system. For this purpose, we will followthe procedure detailed in Section 6.2 to obtain the steady-state generator of (7.54) and acorresponding internal model.

Lemma 7.19. Assume that there exist pairwise coprime polynomials (v, w ) , . . . , (v, w),withri,..., r1 being the degrees of their minimal zeroing polynomials P 1 (S ) , ..., PI (s) andsufficiently smooth function FI : Rr+1 +r/ 7£ vanishing at the origin such that, for allv e Rq and all w E H"*,

and

for i = 1 , . . . , I , the pair is observable,


where ) is the gradient of T1 at the origin with , and isthe companion matrix of Pi(s). Then system (7.54) has a linearly observable steady-stategenerator with output = col

Proof. By Lemma 6.17, system (7.54) has a linearly observable steady-state generatorwith output . Specifically, let

where = diag( ) and T is any nonsingular matrix with the appropriate di-mension. Then, 0 = a( ) = and (v, w) = B1( (v, w)) = ).Now, utilizing the relation (7.62) gives a linearly observable steady-state generator withoutput = ) as follows. Let B( (v, w)) =col( ),..., ) where

Then, clearly,

Therefore, is a steady-state generator with output =col ( ). Moreover, since the pair (B1, a) is linearly observable, so is the pairB,a).

Note that in synthesizing the steady-state generator with output col( ),we have taken advantage of the fact that the functions ,(u, u;),i = 1,..., r—l,and u(u, w)rely on the same set of polynomials. Therefore, the dimension of the steady-state generatorwith output col( ) is the same as that of the steady-state generator with output

. As a result, the dimension of the steady-state generator with output col( )is much smaller than what would have been obtained by the general approach given inLemma 6.17.

Taking advantage of the lower dimensional steady-state generator obtained here, wecan also obtain a lower dimensional internal model. Pick any matrices

such that (M, N) is controllable and M Hurwitz. Then there exists a unique nonsingularmatrix T satisfying the Sylvester equation


since the pair ( ) is observable. Let

Then

Thus, (7.67) is an internal model of system (7.54) with output g0( )= col( ). It will be seen later, in Theorem 7.21, that this particular internalmodel will facilitate the solution of the robust stabilization problem of the augmented systemcomposed of the given plant and the internal model.

Remark 7.20. It is known from Remark 6.22 that in the special case where (v, w) isa polynomial in v, the function FI is linear, and therefore Pi(n) = ]. The internalmodel (7.67) becomes

Now attaching the internal model (7.67) to system (7.54) yields the augmented systemwith the state variables ( ). Performing on the augmented system thefollowing coordinate and input transformation:

defines the augmented system in new coordinates and input as follows:

where

and . It is noted that, in deriving equation (7.71), we have used the following identity:


By Corollary 7.4, all we need to do is globally stabilize the transformed augmentedsystem consisting of (7.69) to (7.72). However, this system is not in the familiar lowertriangular form (7.13) yet. Therefore, let us perform on the subsystem (7.72) anothercoordinate transformation as follows:

which yields

Introducing the notation (x) to denote the nonlinear part of B1 (x), that is,

gives


where

withDenoting

puts equations (7.77) and(7.69) to (7.71) into the following form:

where

and, for i = 1,2, . . . , r — 1,


System (7.78) is in the form of the lower triangular systems described in (7.13). Byappealing to Corollary 7.15, we can obtain the solvability conditions for the global robuststabilization problem for (7.78), and hence the solvability conditions for the global robustoutput regulation problem for system (7.54) as follows.

Theorem 7.21. Under Assumptions 7.6 to 7.8, assume that

(i) the function (v, w) satisfies conditions (7.65) and (7.66), and(ii) system (7.77) is RISS with respect to u viewing rj as state and col(z, e) as input with

a known Cl gain function.

Then, the global robust output regulation problem for system (7.54) is solvable.

Proof. By Corollary 7.15, it suffices to show that under condition (ii), the subsystemZ — fo(Z, X1, u) of (7.78) is RISS with respect to u, viewing Z as state and X1 as inputwith a C1 gain function. This can be done by utilizing Corollary 2.20 as follows.

First, let us show that the following system:

is RISS with respect to /LI with a C1 gain function, viewing z as state and e as input. ByLemma 7.17, the matrix F(w) is Hurwitz for all . Therefore, there exists asymmetric positive definite matrix Q(w) continuously depending on w, such that

. Clearly, for all satisfies

for suitable a > 0 and a > 0, and its derivative along (7.79) satisfies

Pick any 0 < e < 1 and let x(.) be a C1 class k function satisfying, for all v E V and aW € W,

Then

Thus, by Theorem 2.16, system (7.79) is RISS with respect to u with state and input eand with a C1 gain function ; in particular, for all

for some class KL function .


Next, note that condition (ii) guarantees the existence of some class KL functionand two known C1 class K functions and such that the solution of system (7.77satisfies, for all ,

Since the subsystem Z = fo(Z, Xi, u) consists of (7.77) and (7.79), applying Corol-lary 2.20 to the subsystem Z = fo(Z, xi, u) shows that this subsystem is RISS with respectto u, viewing Z as state and x1 as input with a C1 gain function, which is any C1 classfunction y(.) satisfying

Since and are C1 functions, it is always possible to choose a C1 class functionsatisfying (7.82). The proof is completed.

Remark 7.22. The controller that solves the robust stabilization problem for the lowertriangular system (7.78) takes the following form:

where the smooth functions ai, i, = 1,..., r, can be obtained by the algorithm describedin Remark 7.10. By Corollary 7.4, the controller that solves the robust output regulationproblem of system (7.54) is

Finally, the controller that solves the global robust output regulation problem of system(7.43) is given by

which only relies on the error output e of system (7.43).

Remark 7.23. Since M is Hurwitz, there exists a symmetric positive definite matrix P suchthat


To guarantee condition (ii) in Theorem 7.21, that is, the RISS property of (7.77), it sufficesto suppose that there exists a positive number r0 < 1 satisfying

for all n, d. This assumption is to restrict the growth of the nonlinear part of the functionB1(.). Indeed, rewrite (7.77) as follows:

where

Let Then isthe maximal (minimal) eigenvalue of P. And the derivative of V(n) along system (7.85)satisfies

Noting that the function is C1 satisfying (0, 0, v, w) = 0 and that (v, w) EV x W, with V x W a compact set, we have

for some smooth function a 1(z , e) 1. And there exists a smooth nondecreasing functiona2(.) satisfying


As a result, we have

for some smooth class function a(s) = sa2(s). Thus, for any 0 < € < 1,

Thus, by Theorem 2.16, choosingshows that the condition (ii) holds for a known C gain function

Remark 7.24. The inequality (7.84) is satisfied in at least two meaningful cases. First,(7.84) holds for some . Thus, (7.84)

holds if B[2] is globally Lipschitz, that is, for some positivenumber L, and the Lipschitz constant L satisfies ... Second, when the solution ofthe regulator equations is a trigonometric polynomial in t, condition (i) of Theorem 7.21 isautomatically satisfied and the function B1(.) is linear. In this case, condition (7.84), andhence condition (ii) of Theorem 7.21, is also automatically satisfied. Thus we obtain thefollowing corollary of Theorem 7.21.

Corollary 7.25. Under Assumptions 7.6 to 7.8, assume the solution of the regulator equa-tions of (7.54) is a polynomial or a trigonometric polynomial in t. Then the global robustoutput regulation problem for system (7.54) is solvable.

Example 7.26. Consider the following system:

and the exosystem

It is assumed that andThis system is in the form (7.43) with


It can be verified that the system has a uniform relative degree r = 2. Using (7.53) and (7.52)gives D(w) = g(w) = 10 and h(w) = 2. Thus, applying the coordinate transformation

gives the following extended system:

This system is clearly in the form (7.54) with

In order to solve the global robust output regulation problem for this system, let usfirst verify, by inspection, that the solution of the regulator equations exists globally and isgiven by

Then the minimal zeroing polynomial of

Thus, the system has a steady-state generator

and is any nonsingular matrix. Since the pair isobservable. Thus the generator is linearly observable.

Choose

which makes a controllable pair. For this pair of matrices, the solution of the Sylvesterequation is given by


which is nonsingular with

Under this design,

and

Using the internal model (7.67) and the coordinate transformation (7.68) gives the followingaugmented system:

A further coordinate transformation of the form (7.73) puts (7.89) into the lowertriangular system of the form (7.78) with r = 2, Z = co\(z, ), and x = col(x\, x2) =o6Ke 1):

where


To verify condition (ii) of Theorem 7.21, we resort to Remark 7.23. Solving the Lyapunovequation (7.83) gives

Simple calculation gives

Thus, the inequality (7.84) holds for 0 < T0 < 0.72. Therefore, condition (ii) of Theorem7.21 also holds.

Thus, by Theorem 7.21, the global output regulation problem for system (7.87) issolvable. Finally, by Remark 7.22, an output feedback controller can be synthesized and isgiven as follows:

7.4 Global Robust Output Regulation for NonlinearSystems in Lower Triangular Form

In this section, we will consider the global robust output regulation problem for the class ofthe lower triangular systems described in Section 7.2. When taking into account the effectof the exogenous signals v, system (7.13) can be modified into the following form:

where z 6 nm, xt e 1R, i = 1, . . . , r, u, y e U, v e Tlq, w e Rnw , and the functions

/, ft, bi, i' = 1,..., r, and qa, are sufficiently smooth functions satisfying /(O, 0,0, w) =0, f i ( 0 , . . . , 0, w) = 0, i = 1,. . . , r, and 0, w) = 0, for all w e Rnw».

7.4. Global Robust Output Regulation for Systems in Lower Triangular Form 217

Again, all the eigenvalues of the matrix A\ are simple with zero real part.At the outset, let us make the following assumptions.

Assumption 7.9. For i = 1 , . . . , r, fy(v, w) > 0 for all v € Hq and w e ft"-'.

Assumption 7.10. There exists a sufficiently smooth function z(u, w) with z(0,0) = 0satisfying the following equation for all v e Tlq and w e TV*:

The solution of the regulator equations will be denoted by z(u, w), x(u, w), u(i>, w) withx(u, u w) = col(xi(v, w) , . . . , xr(u, a;)). Also, for convenience, we define Xr+1(u, w) —u( v, w). •

As before, we need to convert the global robust output regulation problem of system(7.90) into the global robust stabilization problem of an augmented system. For this purpose,we will assume that the solution of the regulator equations satisfies the following assumption.

Assumption 7.11. For i = 1,..., r, there exist pairwise coprime polynomials JT/(U, w),..., TT/'( v, w;) with r/ , . . . , ' being the degrees of their minimal zeroing polynomials

P s ) , . . . , P/'Cs), and sufficiently smooth function F, : Kr +'"+ri' -> K vanishing at theorigin such that, for all trajectories u(0 of the exosystem, and w e T?."Rnw,

and

where 4>, is the gradient of ,- at the origin, and , = block diag ( , . . . , with ,

j = 1, . . . , / , , being the companion matrix of the polynomial P (s).

Remark 7.27. Under Assumptions 7.9 and 7.10, the solution of the regulator equations ofsystem (7.90) exists globally and can be obtained as follows:


By Lemma 6.17, under Assumption 7.11, system (7.90) has a linearly observablesteady-state generator { , or,, B} with output xi,-+i, i = 1, 2 , . . . , r. To be more specific, let

where 7} is any nonsingular matrix with the appropriate dimension. Then, ,- = or/(0/) =T-fc/r,-^, and *+1(u, u;) = B(0,-(i>, «>)) = /(v, w)).

Further, by Proposition 6.21, the following system:

is an internal model of (7.90) with output xi+i, where the pair (M,, JV/) is controllable withM{ Hurwitz, and 7} satisfies the Sylvester equation 7 (- — M,-7} = M - Clearly, puttingthe r systems given by (7.94) with i = 1, . . . , r gives an internal model of system (7.90)with output g0(z, xi, . . . , xr, M) = col(.*2, . . . , xr, u).

Next, define the coordinate and input transformation according to (6.9), which be-comes

This transformation converts the augmented system composed of the original plant (7.90)and the internal model (7.94) into the following form:


where xr+\ = u and

By Corollary 7.4, the global robust output regulation problem for system (7.90) will besolved if we can make the equilibrium point of system (7.95) at (z, x , ) = (0, 0, 0) globallyasymptotically stable for all trajectories v(t ) e V of the exosystem, and all w € W. Aninspection of the structure of (7.95) reveals that (7.95) is in the lower triangular form (7.12)if we identify ZQ with z and zi with ?/,-,/ = 1, . . . , r. However, since M, + N T-1 =T - 1 &iTi and all the eigenvalues of the matrix , have zero real part, the subsystemsdescribed by the second equation of (7.95) does not satisfy Assumption 7.4. Therefore,Theorem 7.6 cannot be directly applied to system (7.95). To circumvent this difficulty,similar to what has been done in Section 7.3, we further perform on (7.95) another coordinatetransformation:

which yields

Using the identity in the above equation gives


Substituting (7.97) into the above equations gives

Let zo = z and = col(v, w;). Then, in terms of the coordinate col(zo, Z1, • • • , zr> x1,. . . ,Jtr), equation (7.95) can be put into the standard lower triangular form (7.12) as follows:

where, for i = 1,. . . , r,

The functions fo, Q{, yi, ft are all sufficiently smooth in their arguments.It is important to note that j/, (ZQ, zi , . . . , z, _i , Jti , . . . , f , , /u,) does not depend on the

variable zi since, from (7.96),

which does not depend on rji .


It can be seen that, under the coordinate transformation (7.97), the transformed aug-mented system (7.98) is still in the lower triangular form (7.12) with the dynamics of theinternal model as the dynamic uncertainty. Moreover, the linear approximation of the func-tion QJ (0, 0, . . . , 0, Zi , 0, . . . , 0, ) is given by Af/z,- , with Af, a Hurwitz matrix. Therefore,as will be seen later in Remark 7.31, in many interesting cases, the subsystems describedby the second equation of (7.98) do satisfy Assumption 7.4. Thus, appealing to Theorem7.6 immediately gives the following solvability conditions of the global robust stabilizationproblem of system (7.98):

Proposition 7.28. Suppose system (7.98) satisfies the following two conditions.

(i) zo = f0 (ZQ, xi, ) is RISS with respect to [i, with ZQ as state and x1 as input and hasa known Cl gain Junction KQ(-).

(ii) For all i = 1 , . . . , r, z,- = Qi (ZQ, Zi , . • . , Zj , *i , . . . , *,• , /*) is RISS with respect to \JL,with Zi as state and col (ZQ, Zi, . . . , Zt-i, x1, . . . , xi) as input, and has a known C1

gain junction Ki(-).

Then, there exists a smooth feedback control u = k(x\, . .. ,xr) with fc(0, . . . , 0) = 0 suchthat the equilibrium point of the closed-loop system at the origin is globally asymptoticallystable for all n e V x W.

Combining Proposition 7.28 and Corollary 7.4 gives the solvability condition of theglobal robust output regulation problem for the original system (7.90) as follows.

Theorem 7.29. Suppose system (7.90) satisfies Assumptions 7.9 to 7.11, and the sameconditions (i) and (ii) of Proposition 7.28. Then the global robust output regulation problemcan be solved by a dynamic state feedback controller of the form

Remark 7 JO. The three Assumptions 7.9 to 7.1 1 of Theorem 7.29 are mainly made for theexistence of the regulator equations and the appropriate nonlinear internal model. Similarassumptions have to be made even for the solvability of the (local) robust output regulationproblem. Conditions (i) and (ii) of Theorem 7.29 are made so that the augmented systemcan be globally robustly stabilized.

Similar to Remark 7.23, we can identify two nontrivial cases where condition (ii) ofTheorem 7.29 is satisfied as follows.

Remark 731. When the solution of the regulator equations, X2(v, w), . . . , XT(U, u>),u(u, w),are polynomial, the equation governing z,-, i = 1 , . . . , r, takes the special form as follows:

Qi(ZQ, Zl, • • • , Zj, *l, • . . , *«•, /i) = MiZi + Yi(ZQ, Zi, . . . , Zi-1, *1, . • . , *,', /*)•

Thus, for this special case, condition (ii) of Theorem 7.29 automatically holds. In the currentcase, condition (ii) of Theorem 7.29 has to be verified. The way that we have already used

222 _ Chapter 7. Global Robust Output Regulation

in Remark 7.23 can be used directly to verify condition (ii) here and, for convenience, isrepeated here. Fort = 1, . . . , r, denoted, = b - 1 ( v , w)N,Jt,+0/. Then the second equationof (7.98) can be written as follows:

where

As in Remark 7.23, let P1 be a symmetric positive definite matrix such that

We will show that condition (ii) is verified if there exists a positive number R1 < 1 satisfying

for all z/, 4. In fact, let VZi(Zi) = %zj PiZi. Then J-^-J^II2 Vz,. ) < 2,where Xmax (A.m,n) is the maximal (minimal) eigenvalue of P{. Further, in exactly the sameway as deriving inequality (7.86), we can show that the derivative of VZi(zt) along system(7. 100) satisfies

Noting that function Yi, is C1 satisfying Yi,(0,. . . , 0, 0 ) = 0 and E, with £ a compactset, we have

for some smooth function a,-i(zo, zi, • • •, Zi-i, x 1 , . . . , x1,-) 1. And there exists a smoothnondecreasing function 0,2(•) satisfying

As a result, we have


for some smooth class £<» function a/ (5) = 50,2(5). Thus, for any 0 < e

Thus, by Theorem 2.16, choosing XiC*) = (s) gives that the condition (ii) holds for

the known C1 gain function

Again, it can be seen that (7.102) holds for some 0 < R1« < 1 when

Thus, (7.102) holds if $2] is globally Lipschitz, that is

for some positive number L,, and if the Lipschitz constant L, satisfies L, <

v eV and w 6 W. •

Example 7.32. Consider the following lower triangular system:

with the exosystem

These equations formulate the control problem of designing a state feedback regulator tohave the output y of system (7. 1 04) asymptotically track a sinusoidal signal of frequency 0.5with arbitrarily large amplitude in the presence of two uncertain parameters uw1 , u 2- Denoteu = XT,. It can be verified that this system satisfies Assumptions 7.9 to 7.11. In particular, theregulator equations associated with this system have a globally defined solution as follows:

for all


Let g0(x,u) — col(x2, M), y, w) = — 0.4v2, and (v, w) = SM^UI v2- Then, theminimal zeroing polynomials of (v, w) and (v, )are 4A.2 + 1 and A.2 + 1 , respectiv Assumption 7.1 1 is satisfied with

and the corresponding gradients and companion matrices are

For each i = 1,2, the steady-state generator with output xi is given by

wherei is any nonsingular matrix. To design an internal model, let

Solving the pertinent Sylvester equation gives

Thus,

Then

Thus, the internal model is as follows:

Using the canonical coordinate and input transformation


and Z0 = z, Zi = n1 — 0.08N1x1, ii = r\i — N2x2 puts the augmented system (7.104) and(7.108) into the following form:

where

Next, we will verify that all the solvability conditions given in Theorem 7.29 aresatisfied. To be specific, we assume that v(t) e V = and ,i = 1,2. Let us first verify the condition (i) of Proposition 7.28. Let Vz0 = . Then

with a = a = . And the derivative of Vz0(zo) along the first equationof (7.109) is

Thus, for any

Thus, choosing gives a gain function

To verify condition (ii), we will resort to Remark 7.31. First, solving the Lyapunov

equation (7.101) gives P1 = P2 = . When i = 1, it is satisfied since (-) = 0.

In particular, let VZl(zi) = 2z1TP1z1. Then, from (7.103), the derivative of V Z l ( z 1 ) along

the trajectories of the second equation of system (7.109) is

Thus, when condition (ii) is satisfied with forsome


When i = 2, the inequality (7.102) becomes

where d2 = N2x2 + 2- Letting z2 = col(z22, z22) leads (7.110) to

Simple manipulation shows that (7.111), hence the inequality (7.102), holds for 0 < R2 <0.773, and we choose R2 = 0.77. In particular, let VZ2(z2) = P2Z2 and note that

Then, from (7.103), the derivative of VZ2(z2) along the trajectories of the fourth equation ofsystem (7.109) is

Thus, when i = 2, condition (ii) is satisfied with

for some 0 < 62 < 1 •


By Proposition 7.28, the global robust stabilization problem of system (7.109) issolvable. In fact, using the procedure described in the proof of Proposition 7.28 shows thatthe following controller:

globally robustly stabilizes system (7.109) for all u(t) € V = and all. Further, by Corollary 7.4, the overall controller for solving the

global robust output regulation problem for system (7.104) is given by the composition ofthe internal model (7.108) and

Chapter 8

Output Regulationfor SingularNonlinear Systems

Singular systems are dynamical systems whose behaviors are governed by both differentialequations and algebraic equations. Such systems arise in electrical networks, power systems,large-scale systems, and so on. In this chapter, we study the output regulation problem forsingular nonlinear systems. In Section 8.1, we give a formulation of the output regulationproblem for singular nonlinear systems. In Section 8.2, we review some basic results onsingular linear systems that will be invoked in subsequent sections. Section 8.3 starts froma generalized version of the Center Manifold Theorem that applies to singular nonlinearsystems and then presents the solvability conditions of the output regulation problem byboth state feedback control and singular output feedback control. In Section 8.4, we furthergive the solvability conditions of the output regulation problem by normal output feedbackcontrol. Section 8.5 studies the approximation of the output regulation problem for singularsystems. Finally, in Section 8.6, we turn to the study of the robust output regulation problemfor uncertain singular systems.

8.1 Problem FormulationConsider the plant described by

and an exosystem described by

where x(t) € Rn is the plant state, u(t) e Rm the plant input, e(t) e Rp the plant outputrepresenting the tracking error, v(t) e Rq the exogenous signal representing the disturbanceand/or the reference input, and 5 e Rnxn a constant matrix. When 5 is an identity matrix,(8.1) is called a normal system, and when 5 is singular, (8.1) is called a singular system.Throughout this chapter, we assume that S is singular and denote rank 5 = ns.

229

230 Chapter 8. Output Regulation for Singular Nonlinear Systems

We will focus on two classes of control laws, namely,

1. Static State Feedback Control Laws:

2. Dynamic Output Feedback Control Laws:

where z(t) is the compensator state vector of dimension nz to be specified later, andSz € 'Rn2xnz is a constant matrix.

Equation (8.4) is said to be a normal controller if Sz is an identity matrix. The closed-loopsystem composed of plant (8.1), (8.2), and control law (8.3) or (8.4) can be put into thefollowing form:

where for the state feedback case, xc = x, Sc = S, f c ( x , v ) = f(x,k(x,v),v), andhc(xc, v) = h(x, u), and for the output feedback case, xc = col(x, z) and

h

Again, all functions involved in this setup are assumed to be sufficiently smooth anddefined globally on the appropriate Euclidean spaces, with the value zero at the respectiveorigins. As in Chapter 3, the results will be stated locally in terms of an open neighborhoodV of the origin in Rq, and we implicitly permit V to be made smaller to accommodatesubsequent local arguments. We denote the dimension of xc by nc with the understandingthat nc = n for the static state feedback case and nc = n+nz for the output feedback case.

Remark 8.1. Unlike the normal systems studied in the previous chapters, the input u doesnot appear on the right-hand side of the second equation of (8.1). This is because, as willbe seen later, we usually need to resort to a dynamic control law of the form (8.4) to controla singular system. Using the simplified output equation can avoid inconsistent feedbackcomposition of the plant and the control law. •

Before stating the objective of control, let us first introduce some notation and ter-minologies. Let 5, A € Rnxn, B Rnxm, and C Rpxn. Letc (S, A) = { . | CC, det( S - A) = 0}, C- = { . | . C, Re( .) < 0}, and <L = { . | . C, Re( ) 0).A complex number . is said to be the eigenvalue of (5, A) if . (S, A). (5, A) is saidto be stable if (S, A) C C-; (S, A, B) is stabilizable if there exists K e m*n such

8.1. Problem Formulation 231

that (5, A + BK) is stable. (S, C, A) is detectable if there exists an L e 1Zn*p such that(S, A - LC) is stable. (5, A) is said to be standard if degdet(A5 - A) = rank 5. (S, A) issaid to be strongly stable if it is both stable and standard. (5, A, B) is strongly stabilizableif there exists a matrix K e nm*" such that (5, A + BK) is strongly stable. (S, C, A) isstrongly detectable if there exists a matrix L € 7£"x/n such that (5, A + LC) is stronglystable. (A, B) is said to be normalizable if there exists a matrix K e 7£mx/l such thatA + BK is nonsingular.

Our objective is to find a controller (static state feedback or dynamic output feedback)such that the closed-loop system (8.5) has the following two properties.

Property 8.1. The pair (Sc, Ac) is strongly stable where

Property 8.2. The trajectory starting from any sufficiently small initial state col (*<*>, VQ)satisfies

Remark 8.2. Property 8.1 is slightly stronger than the stability of (Sc, Ac). The additionalcondition deg(det(A.5c — Ac)) = rank(Sc) guarantees, as will be seen later from the proof ofLemma 8.9, that the closed-loop system (8.5) will induce a stable center manifold passingthe origin of fR,"c+q that is crucial for the fulfillment of Property 8.2. Moreover, it is wellknown that the response of a strongly stable singular linear system is impulse free, a desirableproperty by all practical engineering systems. We will see in Remark 8.10 that this niceproperty will also be retained by nonlinear systems with Property 8.1. Thus, Property 8.1will guarantee that the trajectories of the closed-loop system exist and are bounded for allt > 0 and for all sufficiently small initial states. I

Many of our results will rely on the properties of the linear approximation of the plantand the exosystem. Therefore, we introduce the following familiar notation:

As a result, the linear approximation of the plant and the exosystem at the origin can bedescribed by

where


Now we are ready to list the following assumptions.

Assumption 8.1. The triple (S, A, B) is strongly stabilizable.

Assumption 8.2. The triple

is strongly detectable.

Remark 8.3. Assumptions 8.1 and 8.2 are made to ensure the fulfillment of Property 8.1by state feedback and/or output feedback control. We note that, in the special case whereS = I, Assumptions 8.1 and 8.2 reduce to Assumptions 3.2 and 3.3 made for the solvabilityof the output regulation problem of the normal systems. •

8.2 Preliminaries of Singular Linear SystemsIn this section, we will introduce some properties of a singular linear system of the form(8.9). These properties will be used in the subsequent sections. Let us first note that thereexist two nonsingular matrices T\, TI e K"x" such that

Let

where An e K"<xn°, BI e Kn**m,Ei e n"'*q,Ci e KP*"',^ e Un\ and all othermatrices have appropriate dimensions. Then the coordinate transformation x = T^lx on(8.9) leads to a singular system of the form

From del (7KA.S - A)r2) = det (kS -A), it is clear that (5, A) is standard orstrongly stable if and only if (5, A) is standard or strongly stable. Moreover, system (8.10)will retain the strong stabilizability and detectability properties of (8.9) as shown below.

8.2. Preliminaries of Singular Linear Systems 233

Lemma 8.4.

(i) (5, A) is standard if and only if A22 is nonsingular and is strongly stable if and onlyif A.22 is nonsingular and An — AÂ^î is Hurwitz.

(ii) (5, A, B) is strongly stabilizable if and only if(S, A, B) is, and (S, C, A) is stronglydetectable if and only if(S, C, A) is.

(iii)

is strongly detectable if and only if

is.

Proof, (i) From

(S, A) is standard if and only if A.22 is nonsingular. On the other hand, if Ai2 is nonsingular,then

Thus, (5, A) is standard and is strongly stable if, additionally, AH — AÂ^1 AII is Hurwitz.(ii) The proof follows from

(iii) The proof follows from

where L! = TiL\. D

If (5, A) is standard, we can always define a reduced-order normal system from (8.10)as follows. Let


Substituting (8.11) into the first and third equations of (8.10) gives a reduced-order normalsystem as follows:

where

This normal system has the following property.

Lemma 8.5. Assume (S, A) is standard. Then,

(i) (Ar, Br) is stabilizable if(S, A, B) is strongly stabilizable, and (Cr, Ar) is detectableif (S, C, A) is strongly detectable,

(ii) The pair

is detectable if the triple

is strongly detectable.(iii) For all X C,

Proof, (i) By part (ii) of Lemma 8.4 if (5, A, B) is strongly stabilizable, so is (S, A, B).Thus, there exists a matrix K € Rmxw such that (S, A + 5 tf) is strongly stable.

Denote K = [ KI K2 ] with K1 € R m x n . Then


Using the fact that A22 + B2K2 is nonsingular since (S, A + BK) is standard gives further

Noting that A22 is nonsingular since (S, A) is standard and using the following matrixidentity:

gives

where

Therefore,

which shows that the pair (Ar, Br) is stabilizable. The detectability of (Cr, Ar) follows fromthe fact that (S, C, A) is detectable if and only if (S, AT, CT) is stabilizable, and (Cr, Ar)is detectable if and only if ( ) is stabilizable.

(ii) Recall from part (iii) of Lemma 8.4 that

is strongly detectable if and only if

is. Let


These two matrices are clearly nonsingular. A straightforward calculation shows

Thus the triple

is strongly detectable, which in turn implies the detectability of

(iii) The proof of the first equality follows directly from

To show the second equality, let

Then it can be verified that


Lemma 8.6. Assume that (S, A) is standard. If a linear output feedback control law of theform

stabilizes the reduced normal system (8.12), then it also strongly stabilizes the originalsingular system (8.9).

Proof. Let the closed-loop system composed of (8.12) and (8.14) be denoted by xcr =AcrxCr + Bcrv with xcr = col( , z), and the closed-loop system composed of (8.9) and(8.14) by Scxc - Acxc + Bcv with xc = col(x, z). Then,

and

Let

A simple calculation shows

from which we can verify that the stability of Acr and the nonsingularity of A22 imply thestrong stability of (Sc, Ac). D

When (5, A) is not standard, it is possible to employ an output feedback control toyield a new system that is standard and retains some desirable structural properties of theoriginal system as shown by the following lemma.


Lemma 8.7. Consider a singular linear system of the form (8.9). Assume (S, A) is notstandard but (S, A, B) is strongly stabilizable and (S, C, A) is strongly detectable. Then,there exists a linear output feedback control

such that the following system:

satisfies the following.

(i) (5, A) is standard.(ii) (5, A, B) is strongly stabilizable and (S, C, A) is strongly detectable.

(iii)

is strongly detectable if

is.

Proof, (i) Using the same transformation matrices T1, T2 as those used in Lemma 8.4, wecan convert a system of the form (8.16) into the form given by (8.10), in particular,

Thus,

By part (i) of Lemma 8.4, (5, A) is standard if and only if there exists a matrix Ksuch that det(A22 + B2KeC2) 0.

Since (S, A, B) is strongly stabilizable, there exists a matrix K such that

where [K1, K2] - KT2. By part (i) of Lemma 8.4, det(A22 + B2K2) 0; that is, the pair(A22, B2) is normalizable.


Similarly, since (5, C, A) is strongly detectable, there exists a matrix L such that

where

Hence, det(A22 — L2C2) 0; that is, the Pair ( ) is normalizable.We now show that the normalizability of ( ) and the normalizability of ( )

guarantee the existence of a matrix Ke Rmxp such that det(A22+ B2KeC2 0. Forthis purpose, denote rank A22 = na. lfna = n — ns, A22 is nonsingular and it suffices tolet Ke = 0 to solve the problem. Otherwise, suppose na < n — ns. Then there exist twononsingular matrices P, Q R ( n - n s s ) x ( n - n s ) such that

where B21 € RnaXm, C21 . We now claim that the gain Ke = solves the

problem. In fact,

It follows fromthe above decomposition that the matrix (A22 + is nonsingularif and only_if B22 has full row rank and C22 has full column rank. Now let K2 be such thatA22 + B2K2 is nonsingular and denote K2Q = [K21 K22]- Then

Thus the nonsingularity of the matrix A22 + B2K2 implies that B22 has full row rank.Similarly, we can show that the nonsingularity of the matrix — implies that C22

has full column rank.(ii) To prove part (ii), one only needs to note that, for any matrices K Rmxn and

L € Rn x p ,


and

(iii) The proof follows from the fact that, for any L € n(n+q)*p,

where

Remark 8.8. If there exist matrices L\ and L2 such that

where

then, necessarily, det (A22 — L12C2) 0; that is, the pair ( ) is normalizable. Thuspart (i) still holds if we replace the strong detectability of (5, C, A) by that of

8.3 Output Regulation by State Feedback and SingularOutput Feedback

It is known that the Center Manifold Theorem plays a key role in solving the output regulationproblem for normal nonlinear systems. In this section, we will establish a generalizedversion of the Center Manifold Theorem that applies to the class of singular nonlinearsystems described in (8.5).

Lemma 8.9. Assume that the exosystem (8.2) satisfies Assumption 3.1' and the closed-loopsystem (8.5) has Property 8.1. Then,

(i) there exists a sufficiently smooth function Xc(v) defined for v V satisfying Xc(0) = 0and

8.3. Output Regulation by State Feedback and Singular Output Feedback 241

(ii) for any sufficiently small XC0 and V0, the solution of (8.5) denoted by col(xc(t), v(t))exists and is bounded for allt > 0 and satisfies

(iii) The closed-loop system (8.5) satisfies Property 8.2 if and only if there exists a suffi-ciently smooth Junction Xc(v) locally defined inv V with Xc(0) = 0 such that

Proof. Part (i). Rewrite the first two equations of system (8.5) into the following form:

where (xc, v) and (v) vanish at their origins together with their first-order derivatives.Assume rank Sc = r. Then there exist two nonsingular matrices T\ and 72 such that

Let

where and Then, premultiplying T\ on bothsides of (8.22) gives

where 1 ( v) and ( v) vanish at col( , , v) = 0 together with theirfirst-order derivatives.

It follows from the strong stability of (Sc., Ac) that Ac22 is nonsingular. By the ImplicitFunction Theorem, there exists a unique sufficiently smooth function or( v) defined inan open neighborhood of ( v) = (0,0) that satisfies (0,0) = 0 and

Furthermore,


where vanishes at ( , ) = 0 with its first-order derivative.By part (i) of Lemma 8.4, — is Hurwitz since (Sc, Ac) is strongly

stable by assumption.Now consider the following normal system:

where

and T2 = [T21 T22] with T21 7Rncxr. Since all the eigenvalues of (Ac11 - )have negative real parts and all the eigenvalues of AI have zero real parts by Assumption 3.1',by Theorem 2.25, system (8.27) has a stable center manifold defined in an open neighborhoodof the origin of Rq, or, equivalently, there exists a sufficiently smooth function (v) definedfor v € V that satisfies (0) = 0 and is such that

Moreover, there exist positive constants and . such that, for all sufficiently small (0)and u(0), the solution of (8.27) satisfies

Let

Then it can be readily verified, using (8.26) and (8.29), that (8.31) satisfies (8.18).Part (ii). In terms of the solution of (8.27), we can define

Clearly, for t > 0, xc(t) is bounded and col(xc(t), v(t)) satisfies (8.5). Moreover, by (8.30)and the sufficient smoothness of ( ), we have


Part (iii). Sufficiency. Assume (8.20) and (8.21) hold for some xc(u). Then, by part(ii) of this lemma, for all sufficiently small Xco and v0, the solution of (8.5) satisfies (8.19).It follows from the sufficient smoothness of h( ) as well as (8.21) and (8.19) that

Necessity. Since the closed-loop system (8.5) satisfies Property 8.1, by part (i) of thislemma, there exists some sufficiently smooth function Xc(v) for v V with Xc(0) = 0satisfying (8.20). We will further show that the function Xc(u) also satisfies (8.21) if theclosed-loop system (8.5) satisfies Property 8.2. For this purpose, we first show that thefunction Xc1(u) defined by (8.29) also satisfies

In fact, by (8.32), (8.28), and the assumption that the closed-loop system (8.5) satisfiesProperty 8.2, we have

Thus the reduced normal system (8.27) satisfies Properties 8.1 and 8.2. We now recall fromthe output regulation theory for the normal system as stated in Lemma 3.6 that if, in additionto Property 8.1, (8.27) also satisfies Property 8.2, then Xc1(u) necessarily satisfies (8.35).Now noting that Xc(v) and Xc1(v) are related by (8.31) gives

That is, Xc(u) also satisfies (8.21).

Remark 8.10. A distinct feature of singular linear systems from normal linear systems isthat the zero input response of the system may contain an impulsive function. However,when the system is strongly stable, the zero input response of the system is impulse free.This nice property is also retained for the singular nonlinear system described by (8.5) ifthe linearization of Scxc = fc(xc, 0) is strongly stable. This is evident from the explicitexpression given by (8.32). However, as opposed to the normal system, the response xc(t)may be discontinuous at t = 0. The magnitude of the discontinuity of xc(t) as given by(8.32) can be calculated as follows. Let

and

where Then the magnitude of the discontinuity of xc(t) at t = 0 is

Clearly, this magnitude can be made arbitrarily small by having XC0 and V0 sufficientlysmall.


Remark 8.11. A geometric interpretation of Lemma 8.9 can be given as follows. Letxa = col(xc, v) and rewrite the system (8.5) as follows:

where Sa = block diag (Sc, Iq). Then equations (8.18) and (8.2) can be put into the fol-lowing:

where Xa(v) = col(Xc(v), v). Thus the manifold defined by xa = col(xc(v), v) for v Vis a locally invariant manifold for the singular system (8.38). What is more, we can showthat Xa (v) is actually a center manifold for the system (8.38) in a meaningful sense. In fact,denote the Jacobian matrices of fa (xa) and Xa (v) at their origins by Aa and Xa, respectively.It is not difficult to verify, by linearizing (8.39), that

SaXaAl = AaXa. (8.40)

Since (Sa, Aa) = (Sc, Ac) (Iq, A1) and the matrix A1 has only zero-real-part eigen-values, the eigenspace of (Sa, Aa) associated with the eigenvalues of (Iq, A1) is the tan-gent space to the manifold xa = x«(v) at xa = 0. Thus, the manifold xa = xa(v))can be reasonably called the local center manifold of the system (8.38) passing throughxa = 0.

Having established Lemma 8.9, it is possible to obtain the solvability conditions ofthe output regulation problem for singular systems via both the state feedback controllerand the output feedback controller as given in the following two theorems.

Theorem 8.12. Assume that the exosystem (8.2) satisfies Assumption 3.1' and the singularplant (8.1) satisfies Assumption 8.1. Then the output regulation problem for the singularsystem (8.1) and (8.2) is solvable by a state feedback controller if and only if there existsufficiently smooth Junctions x(v) with x(0) = 0 and u(v) with u(0) = 0, both defined inan open neighborhood V of the origin of Rq, satisfying the following:

Proof. Necessity. Assume the state feedback control u = k(x, u) solves the state feedbackoutput regulation problem. Then, by Lemma 8.9, there exists some sufficiently smoothfunction x c ( V ) for v V with Xc(0) = 0 satisfying (8.20) and (8.21). Define x(v) = Xc(u)and u(y) = k(x(v), v). Then it is straightforward to verify that x(v) and u(v) satisfy (8.41)and (8.42).

Sufficiency. Observe that, by Assumption 8.1, there exists a matrix Kx such that(5, A + BKX) is strongly stable. Suppose equations (8.41) and (8.42) are satisfied


for some x(v) and u(v). Let

This controller yields a closed-loop system with xc = x,Sc = S, fc(xc, v) = f(x, k(x, v), v),and hc(xc, v) = h(x,v). Then, Property 8.1 is satisfied since the Jacobian matrix offc(xc, 0) = f(x, k(x, 0), 0) at the origin is equal to A + BKX. Next, let Xc(v) = x(v).Clearly, k(xc(v), v) = u(v). Thus (8.41) and (8.42) lead to (8.20) and (8.21). It followsfrom Lemma 8.9 that Property 8.2 is also fulfilled.

Theorem 8.13. Assume that the exosystem (8.2) satisfies Assumption 3.1' and the singularplant (8.1) satisfies Assumptions 8.1 and 8.2. Then the output regulation problem for thesingular system (8.1) and (8.2) is solvable by an output feedback controller if and only ifthere exist sufficiently smooth Junctions x(v) with x(0) = 0 and u(v) with u(0) = 0, bothdefined for v V, satisfying equations (8.41) and (8.42).

Proof. Necessity. Assume that the output feedback control u = k(z, e), Szz = g(z, e) solvesthe output regulation problem. Then, by Lemma 8.9, there exists some sufficiently smoothfunction Xc(v) for v V with Xc(0) = 0 satisfying (8.20) and (8.21). Perform the partitionXc(v) = col(x!(v), x2(v)) suchthatxi(v) € Rn. Letx(v) = XI(v)andu(v) = k(x2(v), 0).Then it is possible to verify that x(v) and u(v) satisfy (8.41) and (8.42).

Sufficiency. By Assumptions 8.1 and 8.2, there exist matrices Kx, L1, and L2 suchthat

are strongly stable.Suppose equations (8.41) and (8.42) are satisfied by some sufficiently smooth func-

tions x(v) andu(v) satisfying x(0) = 0 and u(0) = 0. Now let z = col(z1, z2) withz1 Rn

andz2 R9,and

This controller yields a closed-loop system with xc = col(x, Z1 Z2):

and


The Jacobian matrix of fc(xc, 0) at the origin is given as follows:

where Kv = (0, 0). Some elementary transformation shows that

Thus (Sc, Ac) is also strongly stable. That is, Property 8.1 is satisfied.To verify Property 8.2, let xc(v) — col(x(v), x(v), u). Then it is clear that

Using (8.47) and (8.48) and then (8.41) successively in (8.45) gives

That is, (8.20) and (8.21) are satisfied.

Remark 8.14. It is seen that the solvability of the output regulation problem by both statefeedback and output feedback control relies on the same set of equations given by (8.41) and(8.42). Clearly, this set of equations can be viewed as the singular analog of the regulatorequations introduced in Chapter 3. For convenience, we will refer to (8.41) and (8.42) assingular regulator equations in what follows. I

8.4 Output Regulation via Normal OutputFeedback Control

The output feedback controller constructed in Theorem 8.13 is also singular due to thesingularity assumption on S. It is known that singular controllers are of high order, andit is less easy to implement singular controllers physically. Thus, in this section we willconsider how to synthesize normal controllers to solve the output regulation problem forsingular systems. Our approach to studying this problem consists of three steps. In thefirst step, we apply the standard coordinate transformation to the singular plant (8.1) toyield a reduced-order normal system. In the second step, we give the solvability conditionsof the output regulation problem for the reduced-order normal system by a normal outputfeedback controller. Finally, we show that this normal output feedback controller also solvesthe output regulation problem for the original system.

8.4. Output Regulation via Normal Output Feedback Control 247

Before introducing Lemma 8.15, let us note that there exist two nonsingular matricesT1,T2 e Rnxn such that

Let

where A11 RnsXns, B1 Rn*xm, EI e nn**i,Ci e Wx*',xi e Rns and all othermatrices have appropriate dimensions. This coordinate transformation leads to the followingsingular system:

where the notation o(x) denotes higher-order terms in x, and

Lemma 8.15. Assume that the exosystem (8.2) satisfies Assumption 3.1' and the singularplant (8.1) satisfies Assumptions 8.1 and 8.2. Suppose (5, A) is standard. Then, the outputregulation problem of system (8.1) and (8.2) via a normal output feedback controller issolvable if and only if there exist sufficiently smooth junctions \(v) with x(0) = 0 and u(v)with u(0) = 0, both defined in an open neighborhood V of the origin ofRfl, satisfying thesingular regulator equations (8.41) and (8.42).

Proof. The necessity follows trivially from Theorem 8.13.The proof of sufficiency can be divided into three steps. In the first step, we apply

the standard coordinate transformation to the singular plant (8.1) to yield a reduced-ordernormal system.

Step 1. Let us begin with the system (8.49) to (8.51). By Lemma 8.4, the systemdescribed by (8.49) to (8.51) possesses two properties, namely, that (S, A, B) is stronglystabilizable and that

is strongly detectable.


Moreover, A22 is nonsingular since (5, A) is standard.By the Implicit Function Theorem, there exists a unique, sufficiently smooth func-

tion (x 1 , u, v) defined in an open neighborhood of (x1, u, v) = (0, 0, 0) that satisfies(0, 0, 0) = 0 and

It is easy to show that

Substituting x2 = (x1, u, v) into (8.49) and (8.51) gives a reduced-order normal systemas follows:

where Ar, Br, Cr, Er, Fr, Dr are as defined in (8.13).We are now ready to carry out the second step, which will show that the output

regulation problem for the normal system obtained in Step 1 is solvable.Step 2. System (8.53) is a normal system. We will show in this step that the output

regulation problem for this system is solvable. By Lemma 8.5, (Ar, Br) is stabilizable and

is detectable. By Theorem 3.16, it suffices to verify that the regulator equations associatedwith (8.53) are solvable.

In fact, let x(v) = x(i;) and denote x(v) = with Thenu(v) and x(v) satisfy

Also, it is clear from (8.52) that

Substituting (8.57) into (8.54) and (8.56) gives


Thus, the two functions x1 (v) and u(v) are the solution of the regulator equations associatedwith system (8.53).

By Theorem 3.16, the output regulation problem for system (8.53) is solvable by anormal output feedback controller of the following form:

where z Rnz for some integer nz.We are now ready to carry out the third step to show that this normal output feedback

controller also solves the output regulation problem for the original system.Step 3. To show that the controller (8.59) also solves the output regulation problem

for the original system (8.1), we only need to show that the closed-loop system composedof (8.1) and (8.59) satisfies Properties 8.1 and 8.2. To this end, let the linear approximationof the controller (8.59) be given by u = Kzz, z = G1z + G2e, let Acr be the Jacobianmatrix of the closed-loop system composed of (8.53) and (8.59), and let (Sc, Ac) be thelinearization of the closed-loop system composed of (8.1) and (8.59). Then, similar to theproof of Lemma 8.6, we have

and

It follows from Lemma 8.6 that (5C, Ac) is strongly stable.Finally, to verify the satisfaction of Property 8.2, one only needs to note that, for

sufficiently small XQ and VQ,

The assumption that (5, A) is standard is the key to the validity of Lemma 8.15. Thisassumption is of course undesirable and can actually be removed through a linear outputfeedback precompensator, as shown in Lemma 8.7. Thus, combining Lemmas 8.7 and 8.15leads to the main result of this section, as follows.

Theorem 8.16. Assume that the exosystem (8.2) satisfies Assumption 3.1' and the singularplant (8.1) satisfies Assumptions 8.1 and 8.2. Then the output regulation problem of system(8.1) and (8.2) via a normal output feedback controller is solvable if and only if there existsufficiently smooth Junctions x(v) with x(0) = 0 andu(v) with u(0) = 0, both defined in anopen neighborhood V of the origin o fR q , satisfying the singular regulator equations (8.41)and (8.42).


Proof. The necessity follows trivially from Lemma 8.15. To establish the sufficient condi-tion, applying a linear output feedback control

to (8.1) gives a new system, with u as an input:

By Lemma 8.7, under Assumptions 8.1 and 8.2, there exists a gain matrix Ke suchthat (8.61) satisfies the following:

(i) (5, A) is standard, where A is the Jacobian matrix of / (x, 0, 0) at x = 0;(ii) (S, A, B) is strongly stabilizable, and [c F], is strongly detectable.

Now, suppose that x(v) and u(v) are the solution of the regulator equations associatedwith (8.1) and (8.2). Then

that is, x(v) and u(v) are also the solution of the regulator equations associated with (8.61).Thus, system (8.61) satisfies all assumptions of Lemma 8.15. As a result, there exists anormal output feedback controller of the form u = k(z), z = g(z, e) that solves the outputregulation problem for the system (8.61). Therefore, the following normal output feedbackcontroller:

solves the output regulation problem for the original system (8.1) and (8.2).

Example 8.17. Consider the following singular nonlinear system:

with the exosystem


This system is already in the standard form (8.49) to (8.51). Linearizing (8.63) at the origingives

It is easy to verify that the plant and the exosystem satisfy Assumptions 8.1 and 8.2. More-over, the regulator equations of (8.63) admit the following unique solution:

By Theorem 8.16, the output regulation problem for the given plant is solvable by a normaloutput feedback controller.

To actually construct a normal output feedback controller, first note that (S, A) is notstandard. Applying the output feedback compensator u = e + u to plant (8.63) gives

which gives

which clearly renders (5, A) standard. Eliminating x3 and X4 from equation (8.64) givesthe following reduced-order normal system:


This system is in the normal form (8.53) with x1 = col(jci, X2) and

By Theorem 3.16, the robust output regulation problem for this system is solvable by anoutput feedback control of the form (3.54) with ym = e. To be more specific, linearizing(8.65) gives

We are now ready to design a controller to solve the output regulation problem of the normalsystem according to the method described in Chapter 3 as follows. Letting Kx be such thatthe eigenvalues of Ar + Br Kx are

gives

Kx = [ -2.0000 3.4142 ],

and letting L be such that the eigenvalues of

are

gives

Then, by (3.54), the following controller:

where z1 R2, Z2 R2, x1(z2) = col (xi(z2),x2(z2)), solves the output regulationproblem for the normal system. Composition of this controller with the precompensatoru = e + u gives the normal output feedback controller, which solves the output regulationproblem of the original system.

8.5. Approximate Solution of Output Regulation for Singular Systems 253

8.5 Approximate Solution of the Output RegulationProblem for Singular Systems

Like normal systems, the key to the existence of either state feedback or output feedbackcontroller is the solvability of the singular regulator equations (8.41) and (8.42). Due tothe nonlinearity of the plant and the exosystem, it is difficult to obtain the exact solutionx(u) and u(v) for the singular or normal regulator equations. Thus, it is interesting to studythe approximate solution of the singular regulator equations by Taylor series. In fact, byemploying the technique similar to the one detailed in Chapter 4, we can also seek series ofthe form

such that the singular regulator equations are satisfied formally.For this purpose, expand the functions f(x, u, v), h(x, y), and a(v) as follows:

Substituting (8.67) and (8.66) into (8.41) and (8.42) and identifying the coefficientsof u[1], l = 1,2,..., yields the following result.

Lemma 8.18. The power series (8.66) formally satisfy the singular regulator equations(8.41) and (8.42) if and only if the following equations are satisfied for l = 1,2, . . . :

where


and, for l = 2,3,...,

Proof. The proof is quite similar to that given in Lemma 4.7 of Chapter 4 and is thereforeomitted.

Equation (8.68) is an iterative sequence of the singular Sylvester equations. Thefollowing result establishes the solvability condition for these equations.

Theorem 8.19. There exists a solution (unique ifp=m) of (8.68) for any E1 and FI, l =1,2, . . . , if and only if

for all . , where

with being the eigenvalues of A1.

8.6. Robust Output Regulation of Uncertain Singular Nonlinear Systems 255

Remark 8.20. Assume that the transmission zeros condition described in equation (8.73)holds up to some positive integer k. Let

Then, it is not difficult to see from the proof of Lemma 4.7 that x(k) (v) and u(k) (u) aresuch that

Moreover, if we replace x(v) and u(v) in the state feedback controller (8.43), the singularoutput feedback controller (8.44), and the normal output feedback controller (8.62) byx(k)(v) and u(k)(v), then it is not difficult to show that each of these controllers will resultin a closed-loop system that satisfies Property 8.1 and admits a sufficiently smooth function

(v) with (0) = 0 such that

It can be readily shown, using the argument similar to what was used in Lemma 4.7, that theclosed-loop system resulting from these controllers has the property that, for all sufficientlysmall XCo and V0, the trajectories col(xc(t), u(t)) of the closed-loop system satisfy

Therefore, we say that these controllers solve the kth-order output regulation problem forthe singular systems (8.1) and (8.2). I

8.6 Robust Output Regulation of Uncertain SingularNonlinear Systems

In this section, we turn to the problem of the robust output regulation problem for uncertainsingular nonlinear systems described by

where x(t) € Rn is the plant state, u(t) Rm the plant input, e(t) Rmthe plantoutput representing the tracking error, and v(t) Rq the exogenous signal representing thedisturbance and/or the reference input generated by the following exosystem:


In (8.78), w Rnw is the plant unknown parameter and S Rnxn a singular constantmatrix, and rank S = ns < n. Also it is assumed that 0 is the nominal value of the uncertainparameter w.

As in Section 8.4, we will seek a normal dynamic output feedback controller asfollows:

where z(t) is the compensator state vector of dimension nz.The closed-loop composite system composed of the singular plant (8.78), the exosys-

tem (8.79), and the control law (8.80) can be put into the following form:

where

Again, it is assumed that all the functions involved in this setup are sufficientlysmooth and defined globally on the appropriate Euclidean spaces, and f(0, 0, 0, w) = 0and h(0, 0, w) = 0 for any w W, with W an open neighborhood of the origin of Rnw.Our results will be stated locally in terms of V and W, with V an open neighborhood ofthe origin in Rq. In what follows, V and W are implicitly permitted to be made smaller toaccommodate subsequent local arguments.

The linearization of the system (8.78) at (x, u, v) = (0, 0,0) will be frequently used,which entails the following notations:

As a result, the system composed of (8.78) and (8.79) can also be written as

where o(x, u, v, w) (o(x, u, w)) is a sufficiently smooth function vanishing at (x, u, v) =(0, 0, 0) ((x, v) = (0, 0)) together with its first-order derivatives with respect to (x, u, v)((x, v))forany w W. For convenience, let A, B,..., denote A(0), B(0),... respectively.


As in Chapter 5, we can list two desirable properties of the closed-loop system asfollows.

Property 8.3, The linearization of Scxc = fc(xc, 0,0) at xc = 0 is strongly stable.

Property 8.4. The trajectory starting from any sufficiently small initial state (XCQ, UQ) satisfies

The Robust Output Regulation Problem. Find a controller of the form (8.80) such thatthe closed-loop composite system (8.81) satisfies Properties 8.3 and 8.4.

The above problem is clearly the extension of the robust output regulation problem forthe normal systems studied in Chapter 5 to the singular systems. It can also be viewed as anextension of the output regulation problem of singular systems studied in Sections 8.1 to 8.4by taking into account the uncertainty. Viewing w as being generated by an exosystem ofthe form w = 0, a solvability condition can be obtained by slightly modifying Lemma 8.9,as follows.

Lemma 8.21. Assume that the exosystem (8.79) satisfies Assumption 8.3 below and that theclosed-loop system (8.81) has Property 8.3. Then the closed-loop system (8.81) also hasProperty 8.4 if and only if it has the following property.

Property 8.5. There exists a sufficiently smooth function xc(v, w) with Xc(0,0) = 0 thatsatisfies, for v V and w W, the following partial differential equations:

Various assumptions needed for the solvability of the above problem are listed asfollows.

Assumption 83. All the eigenvalues of the matrix A1 are simple and have zero real parts.

Assumption 8.4. The triple (5, A, B) is strongly stabilizable.

Assumption 8.5. The triple (5, C, A) is strongly detectable.

Assumption 8.6. There exist two sufficiently smooth functions x (v, w) and u (v, w) satis-fying x(0,0) = 0 and u(0, 0) = 0 such that, for v V, w W,

Remark 8.22. Assumptions 8.4 and 8.5 guarantee the existence of a linear normal outputfeedback control to achieve Property 8.3, and Assumption 8.3, together with Property 8.3,guarantees the boundedness of the solution of the closed-loop system for sufficiently smallinitial state xc(0) and v(0). I


We will study the above robust output regulation problem by an approach similar towhat has been used to solve the output regulation problem by a normal output feedbackcontrol. For this purpose, let us tentatively assume that (+S, A) is standard. Then we canperform the same coordinate transformation on (8.78) as was done on (8.1) in Section 8.4,which yields a system of the form

where

and HKO, 0,0,0,0) is nonsingular. By the Implicit Function Theorem, there exists aunique, sufficiently smooth function a(xi, u, v, w) defined in an open neighborhood of(Jci, u, v, w) = (0, 0,0,0) that satisfies a(0,0, 0,0) = 0 and

Substituting x2 — <*(*!» «, v, w) into the first and third equations of (8.86) gives a reduced-order normal system

It is now possible to see that the linear approximation of (8.88) at (jq, u, v, w) = (0, 0,0, iy)takes the following form:

where all the matrices in the above two equations are defined in Section 8.2. We will firstestablish the following result.

Lemma 8.23. Assume that the exosystem (8.79) satisfies Assumption 8.3 and the plant(8.78) is standard, that is, (5, A) is standard. Then, if a controller of the form (8.59) solvesthe robust output regulation problem for the normal system (8.88), it also solves the robustoutput regulation problem for the singular system (8.78).

Proof. Assume a controller of the form (8.59) solves the robust output regulation problemfor the normal system (8.88). We need to show that the closed-loop system composed of(8.78) and (8.59) also satisfies Properties 8.3 and 8.4. For this purpose, let the closed-loopsystem composed of (8.88) and (8.59) be denoted by


where xcr = col(x1, z) and

Also, let the linearization of (8.89) with w = 0 be denoted by xcr •= Acrxcr + Bcrv, andthe linearization of the closed-loop system composed of (8.78) and (8.59) with w = 0 byScxc = Acxc + Bcv with xc = co[(x, z). We will first show that the stability of Acr impliesthe strong stability of (Sc, Ac). To this end, let the linearization of the controller (8.59) bedenoted by u = Kzz, z = G\z + G2e. Then

and

It follows from Lemma 8.6 that (Sc, Ac) is strongly stable.Next we will show that the closed-loop system composed of (8.78) and (8.59) satisfies

Property 8.5. Let xcr (v, w) be a sufficiently smooth function with xcr (0,0) = 0 that satisfies

Perform a partition Xcr(v, w) = col(xi(v, w), z(v, w)) with X1(u, w) e Rns Thenusing (8.90) leads to an expansion of (8.91) into the following:

Now letX2(v, w) = (x1(v, w), k(z(v, w)), v, w), where the function a is defined in(8.87). Then equation (8.87) implies

and equation (8.88) implies


Thus combining (8.92), (8.93), and (8.94) shows

Finally, let

Then it is possible to verify, using (8.95), that xc(v, w) satisfies (8.83) and (8.84).

The solvability of the robust output regulation for normal systems of the form (8.88)has been established in Theorem 6.23. Combining Theorem 6.23 and Lemma 8.23 estab-lishes the main result of this section.

Theorem 8.24. Assume that the exosystem (8.79) satisfies Assumption 83, the singularplant (8.78) satisfies Assumptions 8.4 to 8.6, and the Junction u(v, w) satisfies conditions(6.31) and (6.33) with g0(x, u) = u. Further, assume the following assumption.

Assumption 8.7. For all , such that ( ) = 0 for some i = 1,. . . , m and some j =I , . . . , / , ,

Then the robust output regulation problem of the singular system is solvable by a controllerof the form (8.80).

Proof. Let us divide the proof into two steps. In the first step, we assume that (S, A) isstandard, and in the second step, we remove this assumption.

Step 1. By Lemma 8.23, it suffices to show that the robust output regulation problemof the reduced-order normal system (8.88) is solvable. By Theorem 6.23, we need to showthat the reduced-order normal system (8.88) satisfies Assumptions 5.1 to 5.3 and the functionu(v, if) satisfies the conditions (6.31) and (6.33) with g0(x, M) = u; moreover, for all Asuch that P/(A) = 0 for some i = 1,. . . , m and some j — 1, . . . , Ii,

By Lemma 8.5, satisfaction of Assumptions 8.4 and 8.5 by (8.78) implies stabi-lizability of (Ar, Br) and detectability of (Cr, Ar). Next, we verify that (8.88) satisfies


Assumption 5.1; that is, the regulator equations associated with the reduced-order normalsystem (8.88) admit a solution. To this end, let x(v, w) and u(u, w) be the solution of theregulator equations of the singular plant (8.78). Let x(v, w) = (v, w) and denote

with x1(v, w) € Rns. Then u(v, w) and x(v, w) satisfy

Also, (8.87) implies that


Thus, the two functions x1(v, w) and u(v, w) are the solution of the regulator equationsassociated with the normal system (8.88).

Clearly, u(v, w) still satisfies conditions (6.31) and (6.33) with g0(x, u) = u. Finally,it follows from part (iii) of Lemma 8.5 and Assumption 8.7 that the reduced-order normalsystem (8.88) satisfies (8.98).

Step 2. In this step, we will remove the assumption that (5, A) be standard. To thisend, applying a linear output feedback control

to the plant (8.78) gives

Suppose x(v, w) and u(v, w) are the solution of the regulator equations associated with(8.78) and (8.79). Then

that is, x(v, w) and u(v, w) are also the solution of the regulator equations associated with(8.105) and (8.79). Thus, the system (8.105) satisfies Assumption 8.6 and u(v, w;) satisfiesthe conditions (6.31) and (6.33) with g0(x, u) = M.


The linear approximation of (8.105) at (x, u, v, w) = (0, 0, 0, w) can be expressed as

where A(w) = A(w) + B(w)KeC(w) and E(w) = E(w) + B(w)KeF(w).By Lemma 8.7, under Assumptions 8.4 and 8.5, there exists a matrix Ke such that

(a) (5, A) is standard,(b) (S, A, B) is strongly stabilizable and (5, C, A) is strongly detectable.

That is, system (8.105) also satisfies Assumptions 8.4 and 8.5.Finally, note that, for all

Thus, system (8.105) also satisfies condition (8.97). Since (5, A) is standard, by the firststep of the proof of this theorem, the robust output regulation problem for system (8.105)and exosystem (8.79) can be solved by a controller of the form (8.59). Therefore, the robustoutput regulation problem for the original plant (8.78) and exosystem (8.79) can be solvedby the composition of (8.104) and (8.59), that is, by

which is clearly in the form of (8.80).

Example 8.25. Let us slightly modify Example 8.17 by introducing an unknown parameterw in the second equation of (8.63) to yield the following uncertain singular nonlinear system:

with the same exosystem:

Correspondingly, the solution of the regulator equations of (8.108) is modified into thefollowing:


The linearization of (8.108) at the origin with w = 0 is the same as that in Example8.17. Therefore, the plant and the exosystem satisfy Assumptions 8.3 to 8.6. Moreover, let

(v, w) = (1 + w)v1. A simple calculation gives (v, w) + 4 (v, w) = 0. Thus (v, w)has a minimal zeroing polynomial P( ) = 2+4. As a result, there exists a smooth function

: R2 R such that

Also, = [1, 1]and

Thus, conditions (6.31) and (6.33) with g0(x, u) — u are also satisfied. It remains to verifyAssumption 8.7. Note that

which has two roots = 0.5 ± Thus Assumption 8.7 is satisfied. By Theo-rem 8.24, the robust output regulation problem for the given plant is solvable.

The desirable normal controller can be constructed based on the following reduced-order normal system:

which is modified from (8.65) by taking into account the uncertain parameter w.By Theorem 6.23, the robust output regulation problem for this system is solvable

by a normal output feedback control of the form (6.44). To actually construct a controller,linearizing (8.109) gives

Let

Solving the Sylvester equation — MT = N gives

Letting the eigenvalues of the following matrices:


and

be given by-0.4240 ± 1.2630J, -0.6260 ± 0.4141.;

and-1.2720 ± 3.7890j, -1.8780 ± 1.2423;,

respectively, gives the control gain

K = [ -2.2334 2.0667 1.4333 -2.0333 ]

and the observer gain

Finally, the controller is given by

where r\ e 7£2,

and ( ) = [1 0 ]T - 1 n) + sin([0 l ] T - l n ] ) .Composition of this controller with the precompensator u = u + e gives the nor-

mal output feedback controller that solves the output regulation problem of the originalsystem.

Chapter 9

Output Regulationfor Discrete-TimeNonlinear Systems

In this chapter, we will study the output regulation problem for discrete-time nonlinearsystems. The contents of this chapter are basically the discrete-time counterparts of whatare covered in Chapters 3 to 5 for continuous-time systems. Whereas in linear systems, thetechnicalities for dealing with discrete-time and continuous-time systems are quite similar,for nonlinear systems, there are some subtle differences between the discrete-time outputregulation problem and the continuous-time output regulation problem. Most notably, aswe will see in the next section, the regulator equations associated with the discrete-timesystems are a set of algebraic functional equations, in contrast with the regulator equationsassociated with the continuous-time systems.

Technically, the major tool used for handling the output regulation problem forcontinuous-time nonlinear systems is the center manifold theory for differential equationsas summarized in Section 2.4, while the major tool used for handling the output regula-tion problem for discrete-time nonlinear systems is the center manifold theory for mapssummarized in Section 2.5.

The chapter is organized as follows. In Section 9.1, we formulate and solve the outputregulation problem for discrete-time systems without involving uncertain parameters. InSection 9.2, we present an approximation method for a discrete-time output regulationproblem based on Taylor series expansion. In Section 9.3, we study the robust outputregulation problem for a discrete-time systems with uncertain parameters. In Section 9.4,an example is given to illustrate the discrete-time robust output regulation problem.

9.1 Discrete-Time Output RegulationWe consider a class of discrete-time nonlinear systems of the form described by

where x(f) is the w-dimensional plant state, u(t) the m-dimensional plant input, e(f) the p-dimensional plant output representing tracking error, and v (t) the q -dimensional disturbance

265

266 Chapter 9. Output Regulation for Discrete-Time Nonlinear Systems

signal, which can represent either disturbance signal or the reference input or both. v(t) isgenerated by a q-dimensional autonomous difference equation of the following form:

For simplicity, all the functions involved in this setup are assumed to be sufficientlysmooth and defined globally on the appropriate Euclidean spaces, with the value zero atthe respective origins. Our results will be stated locally in terms of an open neighborhoodV of the origin in Rq, and we implicitly permit V to be made smaller to accommodatesubsequent local arguments.

We will also consider two classes of control laws, namely,


where the function k(-, •) is required to be sufficiently smooth and satisfies k(0, 0) = 0.


where z(t) is the compensator state of dimension nz to be specified later; ym(f) =hm(x(t), u(t), v(t)), where hm : Rn+m+q Rpm for some integer pm, and is calledthe measurement output; and the functions k(.) and g(-, •) are required to be suffi-ciently smooth and satisfy k(0) = 0 and g(0, 0) = 0.

To formulate the requirements on the closed-loop system, we denote the closed-loopsystem consisting of the plant (9.1), the exosystem (9.2), and the controller (9.3) or (9.4) asfollows:

where, under the static state feedback control, xc — x, and hc(-, •) and f c ( . , .) are describedas follows:

and, under the dynamic measurement output feedback control, xc = col(x, z) and hc(., •)and fc(-, .) are described as follows:

9.1. Discrete-Time Output Regulation 267

Discrete-Time Nonlinear Output Regulation Problem (DNORP): Design a control law(9.3) or (9.4) such that the closed loop composite system (9.5) has the following properties.

Property 9.1. The equilibrium point of the closed-loop system (9.5) at (xc, v) = (0,0) isstable in the sense of Lyapunov, and

Property 9.2. For all sufficiently small xc(0) and v(0), the trajectory col(xc(t), v(t)) of (9.5)satisfies

Remark 9.1. As a result of Property 9.1, for all sufficiently small xc(0) and v(0), thetrajectories (xc(t), v(f)) of the closed-loop system (9.5) exist and are bounded for all t =0,1, — By Theorem 2.33 and Assumption 9.1, to be introduced later, Property 9.1 isautomatically satisfied if the closed-loop system has the following property:


have modulus smaller than 1.

Like the continuous-time case, it is quite straightforward to achieve Property 9.3 byusing a linear feedback control under Assumption 9.2 and/or 9.3 to be given below. Weoften impose Property 9.3 instead of Property 9.1 on the closed-loop system. In analogy tothe continuous-time case, we will call the problem of synthesizing a feedback control lawsuch that the closed-loop system satisfies Properties 9.2 and 9.3 as the discrete-time outputregulation problem with exponential stability.

If there exists a control law such that the closed-loop system satisfies Properties 9.1and 9.2, we say that the nonlinear output regulation problem is (locally) solvable and thecontrol law is called a nonlinear servoregulator. In particular, the control law given byequation (9.3) is called a state feedback servoregulator; and the control law given by equation(9.4) is called a measurement output feedback servoregulator. Alternatively, we say that thecontrol law achieves asymptotic tracking and disturbance rejection in the plant.

Various assumptions needed for the solvability of the problem are listed below.

Assumption 9.1. The equilibrium point of exosystem (9.2) at v = 0 is Lyapunov stable,and all the eigenvalues of (0) are on the unit circle.

Assumption 9.1'. The equilibrium point of the exosystem (9.2) at v = 0 is Lyapunov stableand there is an open neighborhood of v = 0 in which every point is Poisson stable in thesense to be described in Remark 9.2.


is stabilizable.



is detectable.

Remark 9.2. A point v° Rq is said to be Poisson stable if the solution v(t, v°) of theexosystem (9.2) exists for all t = 0,1, 2 , . . . , and for each open neighborhood V° of v°and for any integer N > 0, there exists an integer n1 > N such that v(n1, v°) € V0, and aninteger n2 < — N such that v(n2, v°) V°.

Remark 93. Assumptions 9.1 to 9.3 are clearly the discrete-time counterparts of As-sumptions 3.1 to 3.3. They will play the same role in dealing with discrete-time systemsas Assumptions 3.1 to 3.3 do to continuous-time systems. Also, Assumption 9.1' is thediscrete-time counterpart of Assumption 3.1'. This assumption is only needed when thenecessary condition of the solvability of the discrete-time output regulation problem isconcerned.

We first establish a result parallel to Lemma 3.6.

Lemma 9.4. Under Assumption 9.1', suppose that the closed-loop system (9.5) resultingfrom the controller (9.3) or (9.4) has Property 9.3. Then, it also has Property 9.2 if and onlyif there exists a sufficiently smooth junction Xc(v) with Xc(0) = 0 that satisfies, for v V,where V is an open neighborhood 0f 0 Rq, the following algebraic equations:

Proof. First note that Assumption 9.1' implies Assumption 9.1, and thus the exosystemhas a stable equilibrium at the origin and all the eigenvalues of its Jacobian matrix havemodulus 1. Since the closed-loop system has Property 9.3, by Theorem 2.31, there exists acenter manifold for the closed-loop system (9.5). That is, there exists a sufficiently smoothfunction xc(u) with Xc(0) = 0 that satisfies (9.10) for v V. Moreover, by Theorem 2.33,the equilibrium of the closed-loop system (9.5) at the origin is Lyapunov stable. Thus, thesolution of the closed-loop system (9.5) starting from sufficiently small initial state existsfor allt = 0, 1,2,. . . .

If part. Since the function Xc(v) with Xc(0) = 0 that satisfies (9.10) for v V definesa center manifold xc = xc(v) for the closed-loop system (9.5), by Theorem 2.34, thereexist positive constants and . < 1 such that, for all sufficiently small Xc(0) and v(0), thetrajectories xc(t) of the closed-loop system (9.5) satisfy

Furthermore, there exists a compact set Sc in Rn+Wz+<?, where nz = 0 for state feedback,such that, for therefore


there exists a finite constant L such that

for col(xc, v) Sc. Thus, if the function xc(v) also satisfies (9.11), then

that is, the closed-loop system also has Property 9.2.Only if part. Assume that the closed-loop system has both Properties 9.2 and 9.3,

yet (9.11) is not true. Then there exists a sufficiently small V0 e V such that the solutionof the closed-loop system (9.5) satisfying col(xc(0), v(0)) = col(xc(vo), V0), denoted bycol(xc(t, Xc(vo)), v(t, VQ)), exists for all t — 0, 1, 2,... and satisfies

yet

Thus there exists a neighborhood VQ C V of VQ and some real number R > 0 such that

for all v € V0- Clearly, xc(t, XC(VQ)) = Xc(u(f, VQ)), since xc(0, Xc(u0)) = XC(VQ) =Xc(y(0, V0)) and (9.10) implies

But, since the exosystem satisfies Assumption 9.1', we can assume that V0 is small enoughso that it is Poisson stable, and therefore, given any integer N > 0, there exists an integern1 > N such that v(n1, V0) € V0. Thus,

which contradicts (9.15).

Next we will establish the solvability of the state feedback output regulation problemin terms of the given plant.

Theorem 9.5. Under Assumptions 9.1' and 9.2, the discrete-time nonlinear output regula-tion problem with exponential stability is solvable by a static state feedback control of theform (9.3) if and only if there exist two sufficiently smooth functions x(v) and u(v) satisfyingx(0) = 0and u(0) = 0 such that


Proof. Assume that a controller of the form u = k(x, v) solves the discrete-time nonlinearoutput regulation problem. Then, by Lemma 9.4, there exists a sufficiently smooth functionXc(u) that satisfies (9.10) and (9.11) for v V. Let x(v) = Xc(v) and u(v) = k(x(u), v).Then, x(u) and u(u) satisfy (9.16). On the other hand, assume that x(u) and u(u) satisfy(9.16) for v e V. Let Kx € 7?.mxn be any constant matrix such that the eigenvalues of thefollowing matrix:

have modulus smaller than 1. Due to Assumption 9.2, Kx always exists. Let

Then, under (9.18), the closed-loop system (9.5) satisfies Property 9.3. Moreover, lettingXc(u) = x(t>) leads to

/c(xc(v), v) = f(Xc(v), k(Xc(v), v), v) = f(x(v), u(v), v) = x(a(v)) = Xc(a(v)),

hc(Xc(v), u) = h(Xc(v), k(xc(u), v), v) = h (x (v ) , u(v), u) = 0

as x(v) and u(v) satisfy equations (9.16). By Lemma 9.4, the controller as defined by (9.18)solves the discrete-time nonlinear output regulation problem.

Remark 9.6. Equations (9.16) play the same role for the discrete-time nonlinear output reg-ulation problem as equations (3.30) do for the continuous-time nonlinear output regulationproblem and are thus called the discrete-time regulator equations. In contrast to the linearcase, in which both the continuous-time and discrete-time regulator equations take exactlythe same form as follows:

the discrete-time regulator equations are a set of nonlinear algebraic equations that aredistinctly different from the continuous-time regulator equations, which are a set of non-linear partial differential and algebraic equations. It is this difference that necessitates anindependent treatment of the nonlinear discrete-time output regulation problem.

By the same token as Remark 1.8, we will call the functions u(v) and x(u) zero-error constrained input and zero-error constrained state for the plant and the exosystem,respectively.

When the plant state and/or disturbance state are not available, one can consider usingthe measurement output feedback to solve the output regulation problem.

Theorem 9.7. Under Assumptions 9.1', 9.2, and 9.3, the discrete-time nonlinear outputregulation problem is solvable by a dynamic measurement output feedback controller ifand only if there exist two sufficiently smooth functions x(u) and u(u) with x(0) = 0 andu(0) = 0 that satisfy the discrete-time nonlinear regulator equations (9.16).


Proof. Necessity. Assume that the output feedback control «(t) = k(z(t)), z(t + 1) =g(z(t), y m ( t ) solves the output regulation problem. Then, by Lemma 9.4, there existssome sufficiently smooth function Xc(v) for v € V with Xc(0) = 0 satisfying (9.10) and(9.11). Partition Xc(v) as

where Xcl(.) Rn and Xc2(v) Rnz. Substituting (9.7) into (9.10) and (9.11) gives

Letting x(u) = Xci(u) and u(u) = £(Xc2(u)) shows that x(v) and u(v) satisfy (9.16).Sufficiency. Note that, under Assumption 9.2, there exists a state feedback gain Kx

such that all the eigenvalues of 0,0, 0) + 0,0,0)KX have modulus smaller than 1.By Assumption 9.3, there exist constant matrices L1 and L2 such that all the eigenvalues ofthe matrix

have modulus smaller than 1.Suppose equations (9.16) are satisfied by some sufficiently smooth functions x(v) and

u(v) satisfying x(0) = 0 and u(0) = 0. Let z = colfa, z2) with z\ 6 Ktt and z2 Rq, and

This controller yields a closed-loop system with

and

The function defined in (9.23) takes a form similar to that given in (3.55). Therefore, theJacobian matrix fc(xc, 0) at the origin takes the same expression as the matrix Ac calculatedin (3.56) and is thus exponentially stable, that is, is a Schur matrix.

To verify that the closed-loop system satisfies Property 9.2, let col(x(v), u(v)) be thesolution of the regulator equations (9.16), z1(v) = x(v) and z2(v) = v, and


Then, from (9.21),

and

Letxc(v) = col(x(v), x(u), v). Then, by (9.24) and (9.25),

That is, (9.10) and (9.11) are satisfied.

9.2 Approximation Method for the Discrete-TimeOutput Regulation

Similar to the continuous-time case, due to the nonlinear nature of the discrete-time reg-ulator equations (9.16), it is usually impossible to obtain the exact solution of regulatorequations (9.16). In this section, the kth-order output regulation problem formulated forthe continuous-time systems will be extended to the discrete-time systems; then an approx-imation method for obtaining the solution of the discrete-time regulator equations by theTaylor series will be presented, which in turn leads to a method to synthesize both the statefeedback and the output feedback control laws to approximately solve the discrete-timenonlinear output regulation problem in a similar way to what was done to continuous-timesystems in Chapter 4.

Discrete-Time kth-Order Nonlinear Output Regulation Problem (DKNORP): Givensome integer k > 1, design a control law of the form (9.3) or (9.4) such that the closed-loopsystem (9.5) has Property 9.3 and the following:

Property 9.4. For all sufficiently small Xc0 and v0, the trajectories col(xc(t), v(t)) of theclosed-loop system (9.5) satisfy

where ok(v) is some sufficiently smooth function of v zero up to kth-order.

Let us first state some results that are discrete counterparts of Lemma 4.2, Theorem 4.3,and Theorem 4.5.

9.2. Approximation Method for the Discrete-Time Output Regulation 273

Lemma 9.8. Under Assumption 9.1', suppose the closed-loop system (9.5) has Property 9.3.Then the closed-loop system (9.5) also has Property 9.4 if and only if there exists a sufficientlysmooth function x(

ck)(v) with x^(0) = 0 that satisfies, for v € V, the following equations:

The proof of Lemma 9.8 is quite similar to that of Lemma 4.2 and is thus omitted.

Theorem 9.9.

(i) Under Assumptions 9.1' and 9.2, the discrete-time kth-order nonlinear output reg-ulation problem is solvable by a static state feedback controller of the form (9.3) ifand only if there exist two sufficiently smooth functions x(k)(v) and u(k)(v) satisfyingx(k)(0) = 0 and u(k)(0) = 0 such that

(ii) Under Assumptions 9.1', 9.2, and 9.3, the discrete-time kth-order nonlinear outputregulation problem is solvable by a measurement output feedback controller of theform (9.4) if and only if there exist two sufficiently smooth functions x(k) (v) and u(k) (v)satisfying x(fc)(0) = 0, u(k}(0) = 0, and (9.29).

Proof. The proof of this theorem can be directly obtained from Lemma 9.8. Here we willonly sketch the sufficient part of the proof. Consider the following state feedback controller:

and the measurement output feedback controller of the form (9.3) with z = col(z1, 22)'

which are obtained by replacing x(-) and u(-) in the state feedback controller (9.18) andthe measurement output feedback controller (9.21) and (9.22) with u(k)() and x(k}(-). It isnot difficult to verify that each of these controllers will result in a closed-loop system thatsatisfies Property 9.3 and induces a sufficiently smooth function x(k)(v) with (0) = 0such that (9.27) and (9.28) hold. Thus, it follows from Lemma 9.8 that (9.30) and (9.31)solve, respectively, the state feedback and the measurement output feedback kth-order outputregulation problem for the discrete-time nonlinear systems (9.1) and (9.2).

As indicated by Theorem 9.9, like the continuous-time kth-order output regulationproblem, the key to solving the discrete-time fcth-order output regulation problem is toobtain a kth-order solution of the discrete-time regulator equations. In what follows, wewill present a method for approximately solving the discrete-time regulator equations byTaylor series. The approach is similar to what was developed in Chapter 4. Therefore, the


same Kronecker product notation as used in Chapter 4 will be adopted. Let us first writethe problem description in terms of the series expansions

Also, for the q x I vector v = [vi,..., vq]T, let u[/1 denote the vector

Then we seek series of the form

such that (9.16) is satisfied formally. Once again, note that there exist matrices M/ and NIof appropriate dimensions such that

Our approach involves substituting equations (9.32), (9.34), and (9.35) into the regulatorequations (9.16) and identifying the coefficients of v['], / = 1 ,2 , . . . , which yields thefollowing result.

Lemma 9.10. The power series (9.34) formally satisfy the regulator equations (9.16) if andonly if the following linear equations are satisfied for I = 1,2, ... :

where

and, for I =2 ,3 , . . . ,


where

Proof. Substituting equations (9.32) and (9.34) into equations (9.16) yields the followingequations:

The left-hand side of (9.43) can be written as

where is given by (9.39).


The right-hand sides of (9.43) and (9.44) are the same as those of (4.36) and (4.37)and hence are given by (4.43) and (4.44). For convenience, they are repeated below:

and

Thus, we have, for / > 1,

Equating the coefficients of v[/1 on both sides of the above two equations, and using= = X1M1, = = U1M1 = 1, = 0, / > 1 along with the fact

that M1N1 is an identity matrix completes the proof.

An examination of equations (9.36) to (9.42) shows that E1 and F1 depend only onX1, . . . , X1-i and U1, . . . , U1-1. Therefore, equation (9.36) provides an iterative sequenceof linear matrix equations.

Lemma 9.11. There exists a solution (unique if p = m) of equations (9.36) for any E1 andF1I, I — 1,2,. . . , if and only if the plant satisfies the following assumption.

Assumption 9.4.

for all given by

where are eigenvalues of the matrix (0).


Proof. For a given /, equations (9.36) actually take the same form as the linear regulatorequations (1.108). Thus, by Theorem 1.9, equations (9.36) have a solution for any E1 andFI if and only if equality (9.48) holds for all A. in the spectrum of

We now show that the eigenvalues of A[/1 are precisely those described by (9.49). To thisend, again define Pl as the vector space of all homogeneous polynomials in vi,..., vq ofdegree /; then the components of u[/1 give a basis of Pl. Also define a linear mappingLAiv '• Pl -» Pl such that, for each <f> 6 P't

Note that

Thus (At/J)r is the matrix of the linear mapping L&lV : Pl -*• Pl under the ordered basis

Thus, the spectrum of A[/1 is the same as that of the linear mapping (9.51).Now let the Jordan canonical form of A i be

where

is an Hi x «/ Jordan block with eigenvalue A.,-. Suppose the generalized row eigenvectors ofAI are

which satisfy

Clearly,


also constitutes a basis for Pl. Furthermore, for j = ni,

and for j < nt,

Now define an order on (9.57) in the following "lexicographic" way:

if and only if there exist positive integers i0 and 70 (< n,i0) such that

and

if i < I0, j < «,- or i = /o» j < jo- Then (9.57) constitutes an ordered basis of Pl. Using(9.58) and (9.60) gives

Thus, the matrix of the linear mapping LA}V on P1 is upper triangular, with the diagonalelements being

Therefore, the eigenvalues of LAIV on Pl are exactly given by equation (9.49).

Remark 9.12. If the solution of equations (9.36) is such that (9.34) has a positive convergentradius, (9.34) is an exact solution of equations (9.16) in power series form. In particular, ifthe solution of equations (9.16) is a polynomial in v[l], then Lemma 9.10 gives an approachto exactly solve equations (9.16). Note that equation (9.48) represents the constraints onthe transmission zeros of the Jacobian linearization of the plant which can be viewed as thediscrete-time counterpart of the transmission zeros condition for the continuous-time outputregulation problem, as studied in Chapter 4. •

Assume that the transmission zeros condition in equation (9.48) holds up to somepositive integer k. Let

9.3. Robust Output Regulation for Discrete-Time Uncertain Nonlinear Systems 279

Then, it is not difficult to see from the proof of Lemma 9.10 that there exist degree kpolynomials x(k)(v) and u(k)(u) such that equations (9.29) are satisfied. By Theorem 9.9,we immediately obtain the following sufficient conditions for the solvability of the fcth-ordernonlinear output regulation problem.

Theorem 9.13.

(i) Under Assumptions 9.1, 9.2, and 9.4, for any integer k, the kth-order nonlinear outputregulation problem is solvable by the state feedback control law of the form (9.30).

(ii) Under additional Assumption 9.3, the kth-order nonlinear output regulation problemis solvable by the measurement output feedback control law (9.31).

9.3 Robust Output Regulation for Discrete-TimeUncertain Nonlinear Systems

Consider a discrete-time nonlinear system described by

where x(t) Rn is the plant state, u(t) Rm the plant input, e(t) Rp the plant outputrepresenting the tracking error, w Rnw the plant uncertain parameters, and v(t) Rqthe exogenous signal representing the disturbance and/or the reference input. Again, it isassumed that v(t) is generated by the autonomous system (9.2).

The class of control laws is described by

where z(t) is the compensator state vector of dimension nz to be specified later. The abovecontroller encompasses three cases.

3. Dynamic State Feedback: When v(t) does not appear in (9.63), that is,

4. Dynamic Output Feedback: When x(t) and v(t) do not appear in (9.63), that is,

5. Dynamic Output Feedback with Feedforward: When x(t) does not appear in (9.63),that is,


Letting xc = col(x, z), the resulting closed-loop system can be written as

where

For simplicity, all the functions involved in this setup are assumed to be sufficiently smoothand defined globally on the appropriate Euclidean spaces, with the value zero at the respectiveorigins. Throughout this chapter, we use V and W to denote some open neighborhoods ofthe origins of Rq and Rnw, respectively. For convenience of presentation, we allow V andW to be made arbitrarily small.

The discrete-time kth-order robust output regulation problem and the discrete-timerobust output regulation problem are formulated as follows.

Discrete-Time kth-Order Robust Nonlinear Output Regulation Problem (DKRNORP).Find a controller of the form (9.63) such that the closed-loop system (9.67) satisfies thefollowing properties.

Property 9.5. The matrix (0, 0, 0) is Schur.

Property 9.6. For all sufficiently small xco, VQ, and w, the trajectory xc(t) of the closed-loopsystem (9.67) satisfies

where k is some given positive integer and ok(v) is some sufficiently smooth function of vzero up to kth order.

Discrete-Time Robust Nonlinear Output Regulation Problem (DRNORP). Find a con-troller of the form (9.63) such that the closed-loop system (9.67) satisfies Property 9.5 andthe following:

Property 9.7. For all sufficiently small XCQ, VQ, and w, the trajectory xc(t) of the closed-loopsystem (9.67) satisfies

The two problems defined above are discrete-time counterparts of the fcth-order robustoutput regulation problem and the robust output regulation problem for continuous-timesystems described in Chapter 5. They can also be viewed as extensions of the fcth-order


discrete-time output regulation problem and the discrete-time output regulation problemstudied in the last two sections by further taking into account the model uncertainty. Viewingw as being generated by an exosystem of the form w(t + 1) = w(t), a solvability conditioncan be obtained by slightly modifying Lemma 9.4, as follows.

Lemma 9.14. Assume the exosystem satisfies Assumption 9.1', and the closed-loop system(9.67) has Property 9.5. Then

(i) The closed-loop system (9.67) has Property 9.6 if and only if

Property 9.8. There exists a sufficiently smooth function x^(u, u;) with x£fc) (0,0) = 0that satisfies, for v e V and w € W, the following algebraic equations:

(ii) The closed-loop system (9.67) has Property 9.7 if and only if

Property 9.9. There exists a sufficiently smooth function Xc(v, w) with Xc(0,0) = 0that satisfies, for v V and w € W, the following algebraic equations:

Various assumptions needed for the solvability of the above two problems are listedas follows.

Assumption 9.5. There exist sufficiently smooth functions x(v, w) and u(v, w) withx(0,0) = 0 and u(0,0) = 0 such that, for v V, w W,

where V c Rq, W RW are some open neighborhoods of the origin of *R,q and Tinw,respectively.

Assumption 9.6. The pair ( (0,0,0,0), (0,0, 0, 0)) is stabilizable.

Assumption 9.7. The pair ( (0, 0,0,0), (0,0,0,0)) is detectable.

Assumption 9.8. For / = 1, 2 , . . . ,

for all X given by

where are eigenvalues of the matrix (0).


We will employ a discrete-time version of the internal model principle to solve theabove two problems. Like continuous-time systems, we can also convert the discrete-timefcth-order robust output regulation problem of the given nonlinear plant with the givenexosystem into a discrete-time robust output regulation problem of a linearized plant witha fc-fold exosystem. For this purpose, let

where A(w), B(w), E(w), and so forth are given by

For convenience, in what follows, we will use the shorthand notation A, B, E, and so forthto denote A(0), fi(0), E(0), and so forth.

Now, assume a control law of the form (9.63) with g(z, e) = G1z + G2e that makesthe closed-loop system (9.67) satisfy Property 9.5. Then, by Theorem 2.31, there exists alocally defined sufficiently smooth function Xc(u, w) with Xc(0, 0) = 0 such that, for v V,weW,

By partitioning Xc(v, w;) = col(x(v, w), z(v, w)), (9.78) becomes

where

Express x(v, w), z(v, w), and e(y, u;) uniquely as


where (Xiw, Ziw) are constant matrices of appropriate dimensions depending perhaps on w.In analogy of the derivation of equation (9.36), substituting (9.81) into (9.79) and (9.80),expanding (9.79) and (9.80) into power series in vl/1, and identifying the coefficients of y[1]

yield, for/ = 1,2,...,*,

and

where, for/ = 1,...,*, Af /] = M/Afty, (Elw, Fiw) = (E(u>), F(w)), and, for/ =2, 3, . . . ,& , (£/„,, Ftw) depend only on Xlw, • • • , X(/-i)u, and Zliy, • • • , Z(/_i)w.

Since, for the given /, equations (9.82) and (9.83) take the same form as (1.118) and(1.119), the fact that the closed-loop system has Property 9.5 means that the matrix

is Schur. Thus, by Lemma 1.38, YIW = 0 for all w e W if the pair (g1 G2) incorporates ap-copy internal model of the matrix Af/]. Moreover, let

If the pair (gi , g2) incorporates a p-copy internal model of the matrix Akf, it also incor-porates a p-copy internal model of all the matrices A1'1 for / = 1 ,...,k. Therefore, thecontrol law renders Ylw = 0 for all l = 1, . . . ,£, thereby solving the discrete-time kth-orderrobust output regulation problem. As a result, we have the following result.

Lemma 9.15. Under Assumption 9.1, assume that a control law of the form (9.63) withg(z, e) = Giz + G2e renders the closed-loop system (9.67) into Property 9.5. Then,

(i) for any l > 1, YIW = 0for all w e W if the pair (C/i, £2) incorporates a p-copyinternal model of the matrix Af/1;

(ii) the kth-order robust output regulation problem is solved if the pair (G1, G2) incorpo-rates a p-copy internal model of the matrix Akf-

Now consider the linear approximation of discrete-time nonlinear system (9.62) asfollows:


Let the pair (Gi, G2) be a minimal p-copy internal model of the matrix Akf. Since theeigenvalues of the matrix At/] are given by

where A.I, . . . , kq are eigenvalues of AI, under Assumption 9.8, G\ satisfies the followingtransmission zeros condition: for all A. (Gi),

By Lemma 1.37, the pair

is stabilizable. Thus, there exist feedback gains K\ and K2 such that the matrix

is Schur; that is, there exists a static state feedback control law u(t) = K\x(t) + K2z(t) thatexponentially stabilizes the following system:

That is, the following dynamic state feedback control law:

solves the ^th-order robust output regulation problem of the discrete-time nonlinear system(9.62).

Next, assume that (9.90) solves the fcth-order robust output regulation problem of theoriginal plant. Under Assumption 9.7, there exists an L such that A — LC is Schur. LetK = [Ki, K2],

Then, by exactly the same argument as in the proof of part (ii) of Theorem 5.7 for thecontinuous-time case, the dynamic output feedback control law of the form

solves the fcth-order robust output regulation problem for the discrete-time nonlinearsystem (9.62).


In summary, we have the following discrete-time counterpart of Theorem 5.7.

Theorem 9.16.

(i) Under Assumptions 9.1, 9.6, and 9.8, for any positive integer k, the discrete-timekth-order robust output regulation problem is solvable by a linear state feedbackcontroller of the form (9.90), where (Gi, 62) is a minimal p-copy internal model ofthe matrix Akf-

(ii) Under Assumptions 9.1 and 9.6 to 9.8, for any positive integer k, the discrete-timekth-order robust output regulation problem is solvable by a linear output feedbackcontroller of the form (9.92), where (G1, G2) w given by (9.91).

Remark 9.17. Similar to the continuous-time case, if v(t) satisfies v(t + 1) = AI v(t), thenwe have vll](t + 1) = Amv[V](t). Let

Then the matrix Akf is such that

System (9.94) can be considered as a generalized exosystem which generates not only theexogenous signal v (when a(v) = A\v), but also the higher order terms of the exogenoussignal v up to order k. We call system (9.94) a discrete-time &-fold exosystem. Now considerthe following linear system:

Lemma 9.15 effectively asserts that designing a discrete-time Ath-order robust servoregulatorfor a discrete-time nonlinear system (9.62) is equivalent to designing a linear discrete-timerobust servoregulator for the linear system (9.95). Theorem 9.16 further gives the conditionsunder which the above linear discrete-time robust output regulation problem is solvable. I

Next, we will further show that, under some additional assumptions on the solutionof the discrete regulator equations, a control law solving the discrete-time fcth-order robustoutput regulation problem for the given plant (9.62) with the exosystem (9.2) also solvesthe discrete-time robust output regulation problem for the same plant and the exosystem.

Lemma 9.18. Under Assumption 9.1, suppose a control law of the form (9.63) is such thatthe closed-loop system satisfies Property 9.5. Then the control law solves the robust outputregulation problem if there exist sufficiently smooth junctions (x(u, u;), u(u, w), z(u, u;))


locally defined in v E V, w E W with (x(0, 0), u(0,0), z(0,0)) = (0,0,0) such thatx(u, w) and u(v, w) are the solution of the discrete-time nonlinear regulator equations(9.75) and z(u, w) satisfies

Proof. By Lemma 9.14, we only need to show that there exists a sufficiently smoothfunction x c(v, w) with Xc(0, 0) = 0 that satisfies (9.73) and (9.74). To this end, defineX C ( V , w) = col(x(u, w), z(v, w)). Using (9.68) yields


Using the regulator equations (9.75) and equation (9.97) in (9.100) and (9.101) gives

To solve the discrete-time robust output regulation problem, we need to impose anadditional assumption on the exosystem (9.2).

Assumption 9.9. a(v) = A\v for some matrix A\, and all the eigenvalues of A1 are simpleand lie on the unit circle.

Theorem 9.19.

(i) Under Assumptions 9.5, 9.6, 9.8, and 9.9, assume the solution x(v, w) and u(v, w)of the discrete-time regulator equations (9.75) are degree k polynomials in v. Thenif the state feedback controller (9.90) solves the discrete-time kth-order robust outputregulation problem, it also solves the discrete-time robust output regulation problem.

(ii) Under Assumptions 9.5 to 9.9, assume the solution u(v, w) of the discrete-time reg-ulator equations (9.75) is a degree k polynomial in v. Then if the output feedbackcontroller (9.92) solves the discrete-time kth-order robust output regulation problem,it also solves the discrete-time robust output regulation problem.


Proof. Part (i). Assume that the controller (9.90) solves the discrete-time fcth-order robustoutput regulation problem. By Lemma 9.18, it suffices to show that there exists a sufficientlysmooth function z(v, w) such that

To this end, let \(v, w) and z(v, w) be sufficiently smooth functions satisfying (9.79)with a(v) = AIV, and let e(v, w) be as defined in (9.80). Again, express x(v, w), z(v, w),and e(v, w) as in (9.81). Since the controller (9.90) solves the discrete-time kth-orderrobust output regulation problem, for / = 1,...,k, k1w and Ziw satisfy (9.82) and (9.83)with YIW = 0, where

Let Uiw = KiXiw + K2Ztw. Then (9.82) and (9.83) imply, for / = 1,..., k,

By Lemma 9.10, there exist sufficiently smooth functions xk(u, w) = ok(u) and uk(v, w) —ok(u) such that

However, by the assumption of this theorem, x(u, w) and u(v, w) are degree k polynomialsin v, and thus

Let

Clearly, (9.102) is satisfied. Now using (9.82) and (9.83) yields


Multiplying (9.104) from the right by v[l] and then summarizing from l = 1 to k gives

Thus,

Part (ii). The proof of part (ii) is almost the same as that of part (i). Assume thata controller of the form (9.92) solves the discrete-time fcth-order robust output regulationproblem. By Lemma 9.18, we need to show the existence of a sufficiently smooth functionz(v, w) with z(0,0) = 0, which satisfies

Let x(v, w) and z(v, w) be sufficiently smooth functions satisfying (9.79), and e(v, w) beas defined in (9.80). Again, express x(v, w), z(v, w), and e(v, w) as in (9.81). Since thecontroller (9.92) solves the discrete-time fcth-order robust output regulation problem, forl = 1, . . . , k, Xlw and Zlw, satisfy (9.82) and (9.83) with Ylw = 0, where

Let Uiw = KZiw. Then (9.82) and (9.83) imply, for / = 1, . . . ,k,

Again, by Lemma 9.10, there exist sufficiently smooth functions xk(v, w) = ok(v) andUK(u, w) = ok(v) such that


However, by the assumption of this theorem, u(u, w) is a degree k polynomial in u, andthus

Let

Clearly, (9.106) is satisfied. The proof of satisfaction of (9.107) is the same as that of (9.103)in part (i), and thus is omitted.

If the exogenous signal v is available for control, it is possible to somehow relax therestriction on x(v, w) and u(v, w) as shown by the following theorem.

Theorem 9.20.

(i) Under Assumptions 9.5, 9.6, 9.8, and 9.9, suppose that there exists some integer k > 0such that x(u, w) and u(v, w) take the following form:

where x[k](v, w) and u[k](v, w) are degree k polynomials of v with coefficients de-pending on w, and x h k(v) and uhk(v) are some sufficiently smooth Junctions of v,independent of w, vanishing at the origin together with their derivatives up to or-der k. If the state feedback controller (9.90) solves the discrete-time kth-order robustoutput regulation problem, then the following controller:

solves the discrete-time robust output regulation problem.(ii) Under Assumptions 9.5 to 9.9, suppose that there exists some integer k > 0 such that

u(v, w) takes the form of

If the output feedback controller (9.92) solves the discrete-time kth-order robust outputregulation problem, then the following controller:

solves the discrete-time robust output regulation problem.

Proof. The proof of this theorem is almost the same as that of Theorem 5.14 and is thusomitted.


9.4 The Inverted Pendulum on a Cart ExampleIn this section, we will consider the asymptotic tracking problem for the discretized modelof the inverted pendulum on a cart system. The continuous-time model is given in equation(2.110). Discretizing the continuous-time model (2.110) via Euler's method with T as thesampling period gives the discrete-time model as follows:

Again, consider the asymptotic tracking of the output y(t) to a sinusoidal function yd(t) =Am sin (wt). Thus the exosystem is given by

with

It is clear that t»i(t) = Am sin(wt). Thus, we can define the error equation as follows:

It can be verified that the matrix AI has two distinct eigenvalues, cos w j sin a), which areclearly located on the unit circle. Thus, the exosystem satisfies Assumption 9.1.

If we consider the coefficient of viscous friction b as an uncertain parameter andassume that b = bo + b with bo = 12.98 kg/sec, then the Jacobian linearization of thediscrete-time inverted pendulum on a cart system (9.110) can be calculated as follows:

9.4. The Inverted Pendulum on a Cart Example 291

It is now possible to verify that the pair (A, B) is controllable, and none of thetransmission zeros of the linearized plant are on the unit circle. Thus the plant also satisfiesAssumptions 9.6 and 9.8. By Theorem 9.16, for any k > 0, the discrete-time fcth-orderrobust output regulation problem is solvable by dynamic state feedback control. Of course,the nominal plant also satisfies Assumptions 9.1,9.2, and 9.4, and thus the kth-order outputregulation problem for this system is also solvable for any integer k assuming b = b0. Inwhat follows, we will design both a third-order state feedback servoregulator and a third-order state feedback robust servoregulator for this system.

Third-Order State Feedback Servoregulator: The discrete regulator equations associatedwith the inverted pendulum on a cart system are

By an inspection, equations (9.112) can be partially solved as follows:

with x3(v) and X4(v) satisfying the following equations:

The above two equations can be viewed as center manifold equations associated with thefollowing nonlinear difference equations:


When v is set to zero, the Jacobian linearization of (9.117) at the origin is

It can be easily verified that the two eigenvalues of (9.118) are not on the unit circle forall 0, and the two eigenvalues of the matrix A1 are on the unit circle. Therefore, byTheorem 2.31 (the Center Manifold Theorem for Maps), equations (9.116) admit a solution.However, the complex nonlinearity of (9.116) precludes an attempt to obtain an analyticsolution. Therefore, let us find an approximate solution of (9.116) as follows.

Eliminating x 4 (v ) from (9.116) gives

Therefore, as long as we can obtain the function x3(V) by solving (9.119), we can thenobtain xi(V), X2(V), and u(v) through (9.113) to (9.115) and X4(v) through

A third-order polynomial approximation for x3 (v) denoted by (v) can be obtainedby solving (9.119) and is given as follows:

where

and


where

For example, when w = 0.05 rad/sec, g = 9.8 m/sec2, / = 0.325 m, and T = 0.1 sec,

and when w = 0.1 rad/sec, g = 9.8 m/sec2, / = 0.325 m, and T = 0.1 sec,

With at hand, we can obtain the third-order approximations of x(v) and u(v),denoted by x(3)(u) and u(3)(v), by using (9.113), (9.114), (9.120), and (9.115). Thus athird-order state feedback controller is given as follows:

where the feedback gain Kx is selected such that the eigenvalues of the matrix A + BKX are

which are obtained by bilinear transformation from the ITAE prototype design for thecontinuous-time systems with the cutoff frequency equal to 4.0 rad/sec.

Third-Order Robust State Feedback Servoregulator: To design a third-order robust statefeedback controller, we need to find a pair of matrices (G1 , G2) that incorporates a one-copyinternal model of A3 f . Since the solution of the discrete-time regulator equations does notcontain the second-order term, the output equation of the closed-loop system under any statefeedback control law of the form (9.90) will not contain the second-order term either. Thus,it suffices to find a pair of matrices (G1, G2) that incorporates a one-copy internal model ofA[1] and A[13]. The minimal polynomials of A[1J and A[3J are computed as follows:

Thus, the minimal polynomial of the matrix block diag (A[1], A[3]) is


Figure 9.1. Tracking performance: Nominal case Am = 1.25 and w = 0.05

Therefore, following the discussion in Section 5.5, we can specify GI and G2 as follows:

The compensator, together with the plant, forms an eight-dimensional system. The feedbackgain (K1 , K2) is chosen such that the eigenvalues of the linearized closed-loop system are

which, again, are obtained by bilinear transformation from the ITAE prototype design forthe continuous-time systems with the cutoff frequency equal to 4.0 rad/sec.

Both controllers are designed based on the nominal values of the system parameters,which are given as follows: feo = 12.98 kg/sec, M = 1.378 kg, l = 0.325 m, g =9.8 m/sec2, m = 0.051 kg.

Let us first compare the performance of the linear controller, the third-order controller,and the third-order robust controller for the nominal case, that is, = 0. The frequency ofthe reference input is fixed at w = 0.05 rad/sec while the amplitude Am of the reference in-put takes Am = 0.75,1.0,1.25, 1.5, respectively. Table 9.1 shows the maximal steady-state


Figure 9.2. Tracking performance: Perturbed system with Am = 1.25, w0.057 , and = 1.0.

tracking errors of the closed-loop systems under various control laws for w = 0.05 rad/secand Am = 0.75, 1.0, 1.25, 1.5. It is seen that the tracking performance of all controllersis quite good. The steady-state tracking error of the third-order robust controller is muchsmaller than that of the other two controllers, while the third-order controller is better thanthe linear controller. Figure 9.1 shows the tracking performance of the nominal closed-loopsystem resulting from the third-order controller and the third-order robust controller withAm = 1.25 and w = 0.05 rad/sec.

Next, we compare the performance of the various controllers in the presence of theparameter uncertainty with Am = 1.25 and w = 0.05 . Assume that the parameter b isperturbed to b = 12.98 + Aft with = -1.0, -0.5,0.5, 1.0,1.5. Table 9.2 shows thesteady-state tracking error of the perturbed closed-loop systems. As shown in Table 9.2,the third-order robust controller maintains small maximal steady-state tracking errors whenthe value of ft varies. In contrast, the tracking performance of both the linear and thethird-order controller greatly deteriorates when the parametric uncertainties are present. Itis interesting to note that, while the third-order controller performs much better than thelinear controller in the nominal case, it shows no advantage over the linear controller whenthe parameter uncertainties are present. Figure 9.2 shows the tracking performance of theperturbed closed-loop system resulting from the third-order controller and the third-orderrobust controller with Am = 1.25, w = 0.05 , and Aft = 1.0.


Amplitude0.751.001.251.50

CO

0.05

0.050.05

0.05

Linear0.00950.02260.04460.0788

Third order0.00020.00070.00210.0053

Third-order robust0.00000.00000.00020.0008

Table 9.1. The maximal steady-state tracking errors of the nominal system.

0.00-1.00-0.500.501.001.50

Linear0.0446

Unstable0.15020.17870.41250.7367

Third order0.0021

Unstable0.14080.17920.41500.7399

Third-order robust0.00020.00140.00060.00010.00010.0009

Table 9.2. The maximal steady-state tracking errors of the perturbed system withAm = l,25 and w = O.Q5n.

Appendix A

Kronecker Productand SylvesterEquation

Let A = [ a i j ] E Rmxq and B = [bij] E Kpxn. Then the Kronecker product of A and B,denoted by A B, is defined by

Let vec : Rnxm Rnmxl be a vector-valued function of a matrix such that, for any

where, for i = 1, . . . , m, X1 is the ith column of X.

Proposition A.I.

(i) For any matrices A, B,C, D of conformable dimensions,

Proof, (i) follows directly from the definition of Kronecker product.

297

298 Appendix A. Kronecker Product and Sylvester Equation

and

Thus

More detailed discussion on the properties of the Kronecker product can be found in [1].Consider the linear matrix equation of the following form:

where M, B E Rp x n , A,N E Rmxq and Q E RPxq are known matrices, and X e Knxm

is an unknown matrix. Using property (ii) of Proposition A.I, (A.3) can be converted intothe following standard form:

When m = q, n = p, and M and N are identity matrices, (A.3) becomes

and is called the Sylvester equation. Correspondingly, (A.4) becomes the following:

The Sylvester equation has the following properties.

Proposition A.2.

(i) The Sylvester equation (A.5), where A € Rmxm and B & RnXn, has a unique solutionif and only if A and B have no eigenvalues in common.

(ii) Let A e Um xm and B E nnxn. A linear mapping S : Rnxm Rnxm such that

Appendix A. Kronecker Product and Sylvester Equation 299

is called a Sylvester map. Let 1C be the kernel ofS, that is,

Let [ ,i = 1,..., n1} and {e/, 7 = 1,..., n2) be the lists of invariant factors of Band A, respectively. Let , i = 1,..., n1 j = 1,. . . , «2, be the greatest commondivisor of 8i and €j. Then

(iii) Consider the Sylvester equation (A.5) with m = n. Assume A and B have no commoneigenvalues and there exist N E Rnxl and € R lxn such that Q = N with (B, N)controllable and ( , A) observable. Then the Sylvester equation (A.5) has a uniquesolution X e Rnxn which is nonsingular.

Proof. For simplicity, assume that A and B have distinct eigenvalues denoted by { , . . . , }and {u1 , . . . , un}, respectively. Suppose ai- and Bj are eigenvectors of AT and B corre-sponding to the eigenvalues , and uj, respectively. By property (i) of Proposition A.1,a, Bj is the eigenvector of (AT I — I B) corresponding to the eigenvalue ,- — u j . Thus,the eigenvalues of (AT I — / B) are given by { — uj, i = 1,..., m, j = 1,..., n}.That is, the matrix (AT I — / B) is nonsingular if and only if the matrices A and Bhave no common eigenvalues.

Proof of property (ii) is suggested on page 25 of [112] and is outlined here. Firstshow that (A.9) holds when A and B are in Jordan form. Then letting A = JATA andB = T , where JA and JB are the Jordan form of A and B, respectively, gives

where Y = . LetK: = [X E Rnxm \ YJA-JBY = 0}. Clearlydim(K) = dim(K).Thus (A.9) holds for any A and B.

Property (iii) is a special case of Theorem 7-10 of [10]. The proof is outlined below.Let the characteristic polynomial of A be

Then it can be shown that

Clearly, the right-hand side of (A. 10) is invertible since (B, N) is controllable and ( , A) isobservable. Moreover, A(A) = 0 by the Cayley-Hamilton theorem, and (B) is invertiblesince the eigenvalues of A(B) are ( (u1),..., (un)} and A and B have no commoneigenvalues. Thus X is invertible.

Appendix B

ITAE PrototypeDesign

A convenient way to select the desirable pole locations for a closed-loop system is to makea member of a set of the so-called prototype polynomials as the characteristic polynomialof the closed-loop system. There are several sets of prototype polynomials, one of which isshown in Table B.I.

k12345678

Pole locations for wo = 1 rad/sec5 + 1

s +0.7071 0.7071;(s + 0.7081)(5 + 0.5210 ± 1.0687)(5 + 0.4240 ± 1.2630/)(f + 0.6260 ± 0.4141.;)(s + 0.8955)(s + 0.3764 ± 1.2920y)(s + 0.5758 ± 0.5339»(s + 0.3099 ± 1.2634/)(j + 0.5805 ± 0.78287)0? + 0.7346 ± 0.28737(5 + 0.6816)0* + 1.2123 ± 1.00707)0? + 0.2492 ± \.VJV7j)(s + 0.4214 ± 0.5579;)(s + 2.0782)0$ + 0.6675)(s + 0.2031 ± 1.1 7747 )(j +0.3945 ±0.74797)0* + 0.62% ±0.55677)

Table B.I. Pole locations of ITAEprototype design.

This table was worked out by Graham and Lathrop [30] based on the criterion ofminimizing the integral of the time multiplied by the absolute value of the error (ITAE),that is,

In Table B.I, the nominal cutoff frequency is CDQ = 1 rad/sec. Pole locations for other valuesof O)Q can be obtained by substituting S/Q)Q for s everywhere [27].

301

Notes and References

Chapter 1. Various versions of the linear output regulation problem have been thoroughlystudied since the early 1970s. The problem was first treated for the special case whereboth the reference input and disturbance are step functions by Johnson [71] and Smith andDavison [99]. Extension to the general case with various versions can be found in Chengand Pearson [18], Davison [21], [22], [23], Francis [28], Francis and Wonham [29], andWonham and Pearson [113], to name just a few. A self-contained treatment on this topic wasgiven by Desoer and Wang [26]. Extensive exposition on this topic can be found in severaltextbooks, such as Chen [10], Knobloch, Isidori, and Flockerzi [77], Saberi, Stoorvogel, andSannuti [94], and Wonham [112]. The main references for this chapter are Davison [23],Desoer and Wang [26], and Wonham [112]. Most results in Section 1.3 can be found in [28],in which the solvability of the regulator equations is tied to the solvability of the regulationproblem. Most results in Section 1.4 can be found in [23] and [26], but the exposition ismore close, in spirit, to Huang [41]. The exposition in Section 5 is based on the work of[28] and [29]. Frequency domain synthesis of linear regulators can be found in [18]. Outputregulation of linear systems with input saturation was studied by Lin and Saberi [85].

Chapter 2. The materials in Section 2.2 are quite standard. The notion of input-to-state stability summarized in Section 2.3 was first proposed by Sontag [100] and [101].Properties of input-to-state stability were further elaborated by Krichman, Sontag, and Wang[79], Sontag [102], Sontag and Wang [103], [104], and [105]. A discrete-time version of theconcept of input-to-state stability was treated by Jiang and Wang [70]. A concise yet quiteself-contained introduction to input-to-state stability concepts was given by Isidori [64] forautonomous systems and by Khalil for nonautonomous systems [74]. The exposition andnotation of Section 2.3 is quite close to Section 10.4 of the book [64] with a major differencethat nonautonomous systems are treated here. Various versions of the Small Gain Theoremcan be found in [36], [37], [64], [66], and [69]. Theorem 2.18 as well as Corollaries 2.19and 2.20 are taken from [16], and it can be viewed as a special case of Theorem 1 of[66]. Sections 2.4 and 2.5 are mainly based on Carr [7]. Normal form and zero dynamicsas summarized in Sections 2.6 and 2.7 have now become a standard topic in nonlinearcontrol textbooks after the trendsetting book of Isidori [63]. Other main references for thesetwo sections are Khalil [74], Nijmeijer and van der Schaft [88], and Slotine and Li [98].The models of the three typical nonlinear systems, that is, the RTAC system, the invertedpendulum on a cart, and the ball and beam system of Section 2.8 are taken from [2], [31 ],and [32], respectively.

303

304 Notes and References

Chapter 3. The output regulation problem for nonlinear systems was first treated forthe special case in which the exogenous signals are constant by Francis and Wonham [29],Further elaboration of this case was given by Hepburn and Wonham [33] to [35], Desoerand Lin [25], and Huang and Rugh [58]. In particular, Huang and Rugh tied the solvabilityof the nonlinear robust output regulation with constant exogenous signals to the solvabilityof a set of nonlinear algebraic equations, which are a special case of the nonlinear regulatorequations. The nonlinear output regulation problem with time-varying exogenous signalswas first studied in 1990 by Isidori and Byrnes without considering parameter uncertainty[65]. They fundamentally established the solvability of the nonlinear output regulationproblem in terms of the solvability of the nonlinear regulator equations. The formulationof the nonlinear output regulation given in Section 3.2 is slightly more general than whatwas given in [65]. Results in Section 3.3 were basically covered in [65]. Solvability ofthe nonlinear regulator equations was investigated in several papers by Cheng, Tarn, andSpurgeon [17], Huang [47], Huang and Lin [56], and Isidori and Byrnes [65]. Section 3.4 isessentially taken from the work of Huang [47]. The output regulation of nonlinear systemswith nonhyperbolic zero dynamics was studied by Huang in [38] and [45]. Section 3.5 isbased on the work of [38]. The output regulation of nonlinear systems was studied in Wangand Huang [111], and Section 3.6 is a refinement of the work in [111]. The estimate of theconvergence region of output regulation, an important issue but not touched on in this book,was addressed by very recent work of Pavlov, van de Wouw, and H. Nijmeijer in [90].

Chapter 4. This chapter is mainly based on two papers by Huang and Rugh [59],[60]. Similar work on the formal Taylor series solution of the regulator equations can befound in Krener [78]. The proof of Lemma 4.8 in Section 4.2 is from [72]. Section 4.3is an expansion of Theorem 1 of [45]. Section 4.4 on the approximation solution of theasymptotic tracking of the inverted pendulum on a cart system is based on the work ofHuang [44]. Approximation approaches based on neural networks were studied in [109]and [110].

Chapter 5. Francis and Wonham discovered as early as 1976 that, for the special casewhere the exogenous signals are constant, the linear internal model that works for linearsystems also works for nonlinear systems. However, this technique does not work for thegeneral case where the exogenous signals are time-varying, as shown by a counterexampleby Byrnes and Isidori [6], Huang and Lin first revealed in 1991 that the linear internalmodel principle fails because, unlike for linear systems, the steady-state tracking error of anonlinear system is a nonlinear function of the exogenous signals [51]. They also introducedthe notion of fan-order robust output regulation in [51], [54]. Huang and Lin further showedthat when the solution of the regulator equations is polynomial, the kth-order robust regulatoralso solves the robust output regulation problem [39], [53]. Other aspects of robust outputregulation were studied in Byrnes et al. [4], [5], Delli Prescoli [24], Huang [40], [43], andKhalil [75]. Sections 5.1 to 5.3 are essentially based on the work of Huang [39], [43].Section 5.4 is taken from the work of [40]. /rth-order robust control of the ball and beamsystem was studied in Huang and Lin [57]. A frequency approach can be found in [42].

Chapter 6. This chapter is mainly based on the papers by Huang [46], Huang andChen [49], and Chen and Huang [14]. The new design framework presented in Section 6.1was first proposed in Huang and Chen [48]. The notion of the steady-state generator isclosely related to the concept of system immersion suggested by Byrnes et al. [5]. Using

Notes and References 305

the system immersion concept, Byrnes et al. gave an alternative sufficient condition forsolvability of the robust output regulation problem, which requires that the solution of theregulator equations satisfy some partial differential equation [5]. This result leads directlyto Proposition 6.12. Proposition 6.14 is based on the work of Huang [46]. Lemma 6.17 ofSection 6.2 and most parts of Section 6.3 are based on the work of Chen and Huang [14].The example on the RTAC system is based on the work of Huang and Hu [50].

Chapter 7. The formulation of the global robust output regulation problem for generalnonlinear systems given in Section 7.1 is taken from Huang and Chen [49]. The mainreferences for Section 7.2 are [19], [64], [66], [68], and [84]. In particular, the paper byJiang and Mareels [66] studied the robust stabilization of lower triangular continuous systemswith dynamic uncertainties. Theorem 7.6 can be viewed as a refinement of the results givenin [68]. The robust stabilization problem of lower triangular continuous systems withoutdynamic uncertainties was also treated in Section 11.4 of the book by Isidori [64]. Furtherextensions of the results in Section 11.4 of the book [64] can be found in [12] and [84]. Useof the inequality given in Lemma 7.8 and its variations has been made in several papers, suchas [84] and [91]. The proof of Lemma 7.8 was also suggested in [84] and [91]. The robuststabilization of the systems in output feedback form was studied by Marino and Tomei in[86]. A somewhat alternative treatment is also given in Section 11.3 of the book by Isidori[64]. The global robust regulation of systems in output feedback form for the special casewhere the system admits a linear internal model was studied by Serrani and Isidori [95],and the more general case was studied by Chen and Huang [15]. The result in Section 7.4is mainly taken from [49]. Examples 7.26 and 7.32 are worked out by my Ph.D. studentZhiyong Chen. The semiglobal robust output regulation problem for various nonlinearsystems was studied by Isidori in [62], Serrani, Isidori, and Marconi in [96], and Khalil in[75] and [76]. The adaptive output regulation for systems with uncertain exosystems wasstudied in Chen and Huang [13], Nikiforov [89], Serrani, Isidori, and Marconi [97], and Yeand Huang [107]. A more extensive exposition of global robust stabilization of nonlinearsystems can be found in books by Kristic, Kanellakopoulos, and Kokotovic [80], Marinoand Tomei [87], and Qu [92], and in the papers [73], [67], and [106].

Chapter 8. Output regulation of nonlinear singular systems was first studied in Huangand Zhang in [61]. A comprehensive treatment for singular linear systems was given byDai [20], which is also the main reference for Section 8.2. Section 8.3 is based on the workof Huang and Zhang in [61]. A major portion of Section 8.4 is taken from [11]. Outputregulation of linear singular systems with input saturation is studied by Lan and Huangin [81].

Chapter 9. The output regulation for discrete-time nonlinear systems was studied byCastillo et al. [8], [9] and Huang and Lin [52], [55]. The approximate output regulationproblem for discrete-time nonlinear systems is treated by Wang and Huang [108]. Robustoutput regulation for discrete-time nonlinear systems was given by Lan and Huang [82],The output regulation and the robust output regulation of the inverted pendulum on a cartexample were studied in [82] and [108], respectively.

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Index

Asymptotic regulation, 2Asymptotic tracking, 2,78, 127

Ball and Beam system, 153Bilinear transformation, 293, 294

Cay ley-Hamilton Theorem, 170Center Manifold Theorem, 46, 102

for Maps, 49Compact set, bound of, 189Companion matrix, 171Coordinate and input transformation, 163,

181,208,218Coordinate transformation, 232, 247Critical case, 39

Decoupling matrix, 60Detectable, 5, 77, 137,234

strongly, see Singular system, de-tectable, strongly

Diffeomorphismglobal, 53local, 53

Disturbance rejection, 2,78,106robust asymptotic, 179

DKNORP, see Output regulation prob-lem, nonlinear, discrete-time kth-order

DKRNORP, see Output regulation prob-lem, nonlinear, discrete-time fcth-order robust

DLORP, see Output regulation problem,linear, discrete-time

DLRORP, see Output regulation problem,linear, discrete-time robust

DNORP, see Output regulation problem,nonlinear, discrete-time

DRNORP, see Output regulation problem,nonlinear, discrete-time robust

E-vector, 60,93Equilibrium point, 37Exosystem, 3,4,187, 266

Mold, 144

Feedbackdynamic measurement output, 4,29,

75, 266dynamic output, 16, 32, 134, 188,

230,279dynamic output with feedforward, 134,

279dynamic state, 16, 32,134,188,279normal output, 246,249singular output, 240static state, 4, 29, 75, 81, 230, 240,

266Feedback gain, 7Feedforward gain, 7Function

class kL, 41class K, 40class K , 40Lyapunov, see Lyapunov function

Gain function, 41,192Generator, 160,166

global, 160linearly observable, 161steady-state, see Steady-state gener-

atorGradient, 50GRORP, see Output regulation problem,

nonlinear, global robust

315

316 Index

GRSP, see Stabilization problem, globalrobust

H-vector, 53, 61, 92Hypersurface, 45

Implicit Function Theorem, 83, 99, 241,248, 258

Input-output linearizationcontrol of, 52

Input-to-state stable, 41robust, 44, 192, 193, 197

Internal modelcharacterization of, 162existence of, 166-175nonlinear, 162, 165, 174, 175, 189p-copy, 20-22, 24, 33, 34, 141, 162,

283minimal, 21

Internal model principle, 27Invariant manifold equations, solvability

of, 126Inverted pendulum on a cart system, 68,

78,127,290ISS, see Input-to-state stableITAE, 110, 130,293

Jacobian linearization, 39, 179Jordan block, 124Jordan form, 10, 124, 277

KNORP, see Output regulation problem,nonlinear, fcth-order

Kronecker product, 9, 118, 119, 274KRORP, see Output regulation problem,

nonlinear, Kth-order robust

Lie derivative, 50Lipschitz, locally, 37LORP, see Output regulation problem, lin-

ear, 18LRORP, see Output regulation problem,

linear, robustLuenburger observer, 13Lyapunov function, 39

global, 39, 40

ISS-, 41RISS-, 45

Manifold, 45center, 46, 138

stable, 81,83, 242zero-error, 81

invariant, 45control, 55, 83equation of, 46

output zeroing, 55, 63, 83locally maximal, 56maximal, 89

Minimal polynomial, see Polynomial, min-imal

Minimum phase system, see System, min-imum phase

Nonlinear systemsin low triangular form, 216

strictly feedback, 192in output feedback form, 201, 202

Normal form, 54, 62

Output regulation problem, 3linear, 5

discrete-time, 30discrete-time robust, 33robust, 3,18

nonlinear, 76discrete-time, 267discrete-time kth-order, 272discrete-time kth-order robust, 280discrete-time robust, 280global robust, 188, 190, 201, 221kth-order, 114fcth-order robust, 135,140-144robust, 75, 135,145-151, 175with exponential stability, 77, 87,

267singular, 240, 244, 246, 247

robust, 257Output regulation property, 5

robust, 18

Pairwise coprime, 171

L 41

Index 317

PBH test, 172, 176Poisson stable, 77, 78, 81, 268Polynomial

characteristic, 21Hurwitz, 85minimal, 21,149, 156,170

roots of, 151trigonometric, 170,171zeroing, see Zeroing polynomial

Polynomial assumption, 162Power series, 118,126,141,169, 253

Radially unbounded, 40Reduction Theorem, 47, 49Regulator equations, 8, 31, 189

discrete-time, 31nonlinear, 82

discrete-time, 270of the uncertain systems, 137

singular, 246, 253solvability of, 89-101, 125

Relative degree, 50,59, 94vector, 59,92

RISS, see Input-to-state stable, robustRORP, see Output regulation problem, non-

linear, robustRotational/Translational Actuator system,

see RTACRTAC,66,78,106,179

Servomechanism problem, see Output reg-ulation problem

Servoregulator, 5, 267kth-order, 114measurement output feedback, 77,

267dynamic, 5Mi-order, 114

nonlinear, 77output feedback

dynamic, 18robust, 18, 136

kth-order, 136output feedback, 136state feedback, 136,293

state feedback, 5, 77, 267, 291

dynamic, 18kth-order, 114

Singular systemdetectable, 231

strongly, 231,233, 234impulse free, 231normalizable, 231stabilizable, 230

strongly, 231, 233, 234stable, 230

strongly, 231, 233standard, 231, 233,234, 237, 238

Small Gain Theorem, 42, 194Spectrum, 9Stabilizable, 5, 22,77, 137, 234, 281

strongly, see Singular system, stabi-lizable, strongly

Stabilization problemglobal robust, 190

of systems in low triangular form,192

solvability of, 193Stable

asymptotically, 37, 39,48globally, 37,40,48locally, 38uniformly, 38,40uniformly globally, 38,40

exponentially, 1, 5,17input-to-state, see Input-to-state sta-

bleLyapunov, 37, 39,48of singular system, see Singular sys-

tem, stablePoisson, see Poisson stableuniformly, 37,40

Steady-state generator, 161,165,167,180existence of, 161,166-175global, 189linearly observable, 207

Steady-state state, 9Sylvester equation, 6,138,174,179,180Sylvester's inequality, 22

318

Systemautonomous, 36

nonlinear, 74composite, 74minimum phase, 11, 57, 65nonautonomous, 36

nonlinear, 74nonlinear control, 36

affine, 36autonomous, 36nonautonomous, 36

nonminimum phase, 57, 65, 178singular, 229

Taylor series, 118, 141, 155Transmission zeros, 11, 125, 179

condition, 11,127, 255, 278, 284

Index

Uncertaintyof plant, 15, 188,279

dynamic, 192static, 192

Unstable, 37, 38, 48

Zero dynamics, 57, 64, 93, 192hyperbolic, 58nonhyperbolic, 58,101

Zero up to kth order, 113Zero-error constrained

control, 9equilibrium, 9input, 83, 270state, 9, 83, 270

Zeroing polynomial, 170minimal, 170

nonlinear output regulation

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