Communications and Control Engineering

Published titles include:

Stability and Stabilization ofInfinite Dimensional Systems with Applications
Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul
Nonsmooth Mechanics (Second edition)
Bernard Brogliato
Nonlinear Control Systems II
Alberto Isidori
LrGain and Passivity Techniques in nonlinear Control
Arjan van der Schaft
Control ofLinear Systems with Regulation and Input Constraints
Ali Saberi, Anton A. Stoorvogel and PeddapuUaiab Sannuti
Robust and H,., Control
BenM.Chen
Computer Controlled Systems
Efim N. Rosenwasser and Bernhard P. Lampe
Dissipative Systems Analysis and Control
Rogeüo Lozano, Bemard Brogliato, Olav Egeland and Bernhard Maschke
Control ofComplex and Uncertain Systems
Stanislav V. Emelyanov and Sergey K. Korovin
Robust Control Design Using H,.,Methods
!an R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin
Model Reduction for Control System Design
Goro Obinata and Brian D.O. Anderson
Control Theory for Linear Systems
Harry L. Trentelman, Anton Stoorvogel and Malo Hautus
Functional Adaptive Control
Sirnon G. Fabri and Visakan Kadirkamanathan
Positive lD and 2D Systems
Tadeusz Kaczorek

Identification and Control Using Volterra Models

F.J. Doyle III, R.K. Pearson and B.A. Ogunnaike
Non-linear Control for Underactuated Mechanical Systems
IsaheUe Fantoni and Rogelio Lozsno
Robust Control (Second edition)
Jürgen Ackermann
Flow Control by Feedback
Oie Morten Aamo and Miroslav Krstic
Leaming and Generalization (Second edition)
Mathukumalli Vidyasagar
Constrained Control and Estimation
Graham C. Goodwin, Marfa M. Seron and Josc! A. Oe Don&
Randomized Algorithms for Analysis and Control of Uncertain Systems
Roberto Tempo, Giuaeppe Calafiore and Fabrizio Dabbene
Switched Linear Systems
Zhendong Sun and Shuzhi S. Ge
Subspace Methods for System Identification
Tohru Katayama
Digital Control Systems
Ioan D. Landau and Gianluca Zito
Alberto Isidori

Nonlinear Control
Systems
Third Edition

With 47 Figures

~Springer
Professor Alberto Isidori
Dipartimento di Informatica e Sistemistica
"Antonio Ruberti"
Via Eudossiana 18
00184Roma
Italy

Series Editors
E.D. Sontag • M. Thoma • A. lsidori • J.H. van Schuppen

British Library Cataloguing in Publication Data

Isidori, Alberto
Nonlinear Control Systems. - 3Rev.ed -
(Communications & Control Engineering Series)
I. Title II. Series
629.836
ISBN 978-1-4471-3909-6

Library of Congress Cataloging-in-Publication Data

lsidori, Alberto
Nonlinear control systems/Alberto Isidori.- 3rd ed
p. cm. - (Communications and control engineering series)
Includes bibliographical references and index.
ISBN 978-1-4471-3909-6 ISBN 978-1-84628-615-5 (eBook)
DOI 10.1007/978-1-84628-615-5
1. Feedback control systems. 2. Nonlinear control theory.
3. Geometry, Differential. I. Title II. Series
QA402.3.174 1995 95-14976
629.8'36-dc20

Communications and Control Engineering Series ISSN 0178-5354

ISBN 978-1-4471-3909-6 Printedon acid-free paper

@ Springer-Verlag London 1995

Originally published by Springer-Verlag London Limited in 2000
Softcover reprint of the bardeover 3rd edition 2000

Frrstpublished1985
Second edition 1989

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted
under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or
transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case
of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing
Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.

The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a
specific Statement, that such names are exempt from the relevant laws and regulations and therefore free for
general use.

The publisher makes no representation, express or implied, with regard to the accuracy of the information
contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that
maybemade.

9876
For Maria Adelaide
Preface to the second edition

The purpose of this book is to present a self-contained description of the fun-

damentals of the theory of nonlinear control systems, with special emphasis
on the differential geometric approach. The book is intended as a graduate
text as weil as a reference to scientists and engineers involved in the analysis
and design of feedback systems.
The first version of this book was written in 1983, while I was teach-
ing at the Department of Systems Science and Mathematics at Washington
University in St. Louis. This new edition integrates my subsequent teaching
experience gained at the University of Illinois in Urbana-Champaign in 1987,
at the Carl-Cranz Gesellschaft in Oberpfaffenhofen in 1987, at the University
of California in Berkeley in 1988. In addition to a major rearrangement of
the last two Chapters of the first version, this new edition incorporates two
additional Chapters at a more elementary level and an exposition of some
relevant research findings which have occurred since 1985.
In the past few years differential geometry has proved to be an effective
means of analysis and design of nonlinear control systems as it was in the
past for the Laplace transform, complex variable theory and linear algebra
in relation to linear systems. Synthesis problems of longstanding interest like
disturbance decoupling, noninteracting control, output regulation, and the
shaping of the input-output response, can be dealt with relative ease, on
the basis of mathematical concepts that can be easily acquired by a control
scientist. The objective of this text is to render the reader familiar with
major methods and results, and enable him to follow the new significant
developments in the constantly expanding literature.
The book is organized as follows. Chapter 1 introduces invariant dis-
tributions, a fundamental tool in the analysis of the internal structure of
nonlinear systems. With the aid of this concept, it is shown that a non-
linear system locally exhibits decompositions into "reachable/unreachable"
parts and/or "observablejunobservable" parts, similar to those introduced
by Kaiman for linear systems. Chapter 2 explains to what extent global
decompositions may exist, corresponding to a partition of the whole state
space into "lower dimensional" reachability and/or indistinguishability sub-
sets. Chapter 3 describes various "formats" in which the input-output map
of a nonlinear system may be represented, and provides a short description
viii

of the fundamentals of realization theory. Chapter 4 illustrates how a series

of relevant design problems can be solved for a single-input single-output
nonlinear system. It explains how a system can be transformed into a linear
and controllable one by means of feedback and coordinates transformation,
discusses how the nonlinear analogue of the notion of "zero" plays an im-
portant role in the problern of achieving local asymptotic stability, describes
the problems of asymptotic tracking, model matehing and disturbance decou-
pling. The approach is somehow "elementary", in that requires only standard
mathematical tools. Chapter 5 covers similar subjects for a special dass of
multivariable nonlinear systems, namely those systems which can be rendered
noninteractive by means of static state feedback. For this dass of systems,
the analysis is a rather Straightforward extension of the one illustrated in
Chapter 4. Finally, the last two Chapters are devoted to the solution of the
problems of output regulation, disturbance decoupling, noninteracting con-
trol with stability via static state feedback, and noninteracting control via
dynamic feedback, for a broader dass of multivariable nonlinear systems. The
analysis in these Chapters is mostly based on a number of key differential
geometric concepts that, for convenience, are treated separately in Chapter
6.
It has not been possible to include all the most recent developments in this
area. Significant omissions are, for instance: the theory of globallinearization
and global controlled invariance, the notions of left- and right-invertibility
and their control theoretic consequences. The bibliography, which is by no
means complete, indudes those publications which were actually used and
several works of interest for additional investigation.
The reader should be familiar with the basic concepts of linear system
theory. Although the emphasis of the book is on the application of differential
geometric concepts to control theory, most of Chapters 1, 4 and 5 do not
require a specific background in this field. The reader who is not familiar with
the fundamentals of differential geometry may skip Chapters 2 and 3 in a first
approach to the book, and then come back to these after having acquired the
necessary skill. In order to make the volume as self-contained as possible,
the most important concepts of differential geometry used throughout the
book are described-without proof-in Appendix A. In the exposition of each
design problem, the issue of local asymptotic stability is also discussed. This
also presupposes a basic knowledge of stability theory, for which the reader is
referred to well-known standard reference books. Some specific results which
arenot frequently found in these references are induded in Appendix B.
I wish to express my sincerest gratitude to Professor A. Ruberti, for his
constant encouragement, to Professors J. Zaborszky, P. Kokotovic, J. Acker-
mann, C.A. Desoer who offered me the opportunity to teach the subject of
this book in their academic institutions, and to Professor M. Thoma for his
continuing interest in the preparation of this book. I am indebted to Pro-
fessor A.J. Krener from whom-in the course of a joint research venture-I
 ix

learned many of the methodologies which have been applied in the book. I
wish to thank Professor C.l. Byrnes, with whom I recently shared intensive
research activity and Professors T.J. Tarn, J.W. Grizzle and S.S. Sastry with
whom I had the opportunity to cooperate on a number of relevant research
issues. I also would like to thank Professors M. Fliess, S. Monaco and M.D.
Di Benedetto for their valuable advice.

Rome, March 1989 Alberto Isidori

Preface to the third edition

In the last six years, feedback design for nonlinear systems has experienced
a growing popularity and many issues of major interest, which at the time
of the preparation of the second edition of this book were still open, have
been successfully addressed. The purpose of this third edition is to describe
a few significant new findings as well as to streamline and improve some of
the earlier passages.
Chapters from 1 to 4 are unchanged. Chapter 5 now includes also the
discussion of the problern of achieving relative degree via dynamic extension,
which in the second edition was presented in Chapter 7 (former sections
7.5 and 7.6). The presentation is now based on a new "canonical" dynamic
extension algorithm, which proves itself very convenient from a number of
different viewpoints. Chapter 6 is also unchanged, with the only exception of
the proof of the main result of section 6.2, namely the construction of feedback
laws rendering invariant a given distribution, which has been substantially
simplified due to a valuable suggestion of C.Scherer. Chapter 7 no Ionger
includes the subject of tracking and regulation (former section 7.2) which
has been expanded and moved to a separate new Chapter and, as explained
before, the discussion of how to obtain relative degree via dynamic extension.
It includes, on the other hand, a rather detailed exposition of the subject of
noninteracting control with stability via dynamic feedback, which was not
covered in the second edition.
Chapters 8 and 9 are new. The first one of these covers the subject of
tracking and regulation, in a improved exposition which very easily leads to
the solution of the problern of how to obtain a "structurally stable" design.
The last Chapter deals with the design of feedback laws to the purpose of
achieving global or "semiglobal" stability as well as global disturbance atten-
uation. This particular area has been the subject of major research efforts in
the last years. Among the several and indeed outstanding progresses in this
domain, Chapter 9 concentrates only on those contributions whose develop-
ment seems to have been particularly influenced by concepts and methods
presented in the earlier Chapters of the book. The bibliography of the sec-
ond edition has been updated only with those references which were actually
used in the preparation of the new material, namely sections 5.4, 7.4, 7.5 and
Chapters 8 and 9.
xii

I wish to express my sincere gratitude to all colleagues who have kindly

expressed comments and advice on the earlier versions of the book. In par-
ticular, I wish to thank Prof. Ying-Keh Wu, Prof. M.Zeitz and Dr. U. Knöpp
for their valuable suggestions and careful help.

St.Louis, December 1994 Alberto Isidori

Table of Contents

1. Local Decompositions of Control Systems . . . . . . . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •. 1
1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 The Differential Geometrie Point of View . . . . . . . . . . . . . . . . . . 33
1.6 Invariant Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1. 7 Local Decompositions of Control Systems . . . . . . . . . . . . . . . . . . 49
1.8 Local Reachability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.9 Local Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2. Global Decompositions of Control Systems . . . . . . . . . . . . . . . 77

2.1 Sussmann's Theorem and Global Decompositions........... 77
2.2 The Control Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.3 The Observation Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4 Linear Systems and Bilinear Systems . . . . . . . . . . . . . . . . . . . . . 91
2.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3. Input-Output Maps and Realization Theory .............. 105

3.1 Fliess Functional Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2 Valterra Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.3 Output Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.4 Realization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.5 Uniqueness of Minimal Realizations ....................... 132

4. Elementary Theory of Nonlinear Feedback for Single-Input

Single-Output Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.1 Local Coordinates Transformations ....................... 137
4.2 Exact Linearization Via Feedback ........................ 147
4.3 The Zero Dynamics ..................................... 162
4.4 Local Asymptotic Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.5 Asymptotic Output Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.6 Disturbance Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4. 7 High Gain Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
xiv Table of Contents

4.8 Additional Results on Exact Linearization ................. 194

4.9 Observers with Linear Error Dynamics .................... 203
4.10 Examples .............................................. 211

5. Elementary Theory of Nonlinear Feedback for Multi-Input

Multi-Output Systems .................................... 219
5.1 Local Coordinates Transformations ....................... 219
5.2 Exact Linearization via Feedback ......................... 227
5.3 Noninteracting Control .................................. 241
5.4 Achieving Relative Degree via Dynamic Extension .......... 249
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
5.6 Exact Linearization of the Input-Output Response .......... 277

6. Geometrie Theory of State Feedback: Tools . . . . . . . . . . . . . . . 293

6.1 The Zero Dynamics ..................................... 293
6.2 Contralied Invariant Distributions ........................ 312
6.3 The Maximal Contralied Invariant Distribution in ker(dh) ... 317
6.4 Cantrollability Distributions ............................. 333

7. Geometrie Theory of Nonlinear Systems: Applications .... 339

7.1 Asymptotic Stabilization via State Feedback ............... 339
7.2 Disturbance Decoupling ................................. 342
7.3 Noninteracting Control with Stability via Static Feedback ... 344
7.4 Noninteracting Control with Stability: Necessary Conditions . 364
7.5 Noninteracting Control with Stability: Sufficient Conditions .. 373

8. Tracking and Regulation .................................. 387

8.1 The Steady State Response in a Nonlinear System .......... 387
8.2 The Problem of Output Regulation ....................... 391
8.3 Output Regulation in the Case of Full Information. . . . . . . . . . 396
8.4 Output Regulation in the Case of Error Feedback .......... 403
8.5 Structurally Stahle Regulation ........................... 416

9. Global Feedback Design for Single-Input Single-Output Sys-

tems ...................................................... 427
9.1 Global Normal Forms ................................... 427
9.2 Examples of Global Asymptotic Stabilization .............. 432
9.3 Examples of Semiglobal Stabilization ...................... 439
9.4 Artstein-Sontag's Theorem .............................. 448
9.5 Examples of Global Disturbance Attenuation .............. 450
9.6 Semiglobal Stabilization by Output Feedback .............. 460
 Table of Contents xv

A. Appendix A .............................................. 471

A.1 Some Facts from Advanced Calculus ...................... 471
A.2 Some Elementary Notions of Topology .................... 473
A.3 Smooth Manifolds ...................................... 474
A.4 Submanifolds .......................................... 479
A.5 Tangent Vectors ........................................ 483
A.6 Vector Fields .......................................... 493

B. Appendix B .............................................. 503

B.1 Center Manifold Theory ................................. 503
B.2 Some Useful Properties .................................. 511
B.3 Local Geometrie Theory of Singular Perturbations .......... 517

Bibliographical N otes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

Index .................................................. .' ...... 545