nonlinear systems

Control Engineering Series 84

Nonlinear and Adaptive Control

SystemsN

onlinear and Adaptive C

ontrol Systems

Ding Zhengtao Ding

The Institution of Engineering and Technologywww.theiet.org 978-1-84919-574-4

Nonlinear and Adaptive Control Systems

Zhengtao Ding is a Senior Lecturer in Control Engineering and Director for MSc in Advanced Control and Systems Engineering at the Control Systems Centre, School of Electrical and Electronic Engineering, The University of Manchester, UK. His research interests focus on nonlinear and adaptive control design. He pioneered research in asymptotic rejection of general periodic disturbances in nonlinear systems and produced a series of results to systematically solve this problem in various situations. He also made significant contributions in output regulation and adaptive control of nonlinear systems with some more recent results on observer design and output feedback control as well. Dr Ding has been teaching ‘Nonlinear and Adaptive Control Systems’ to MSc students for 9 years, and he has accumulated tremendous experiences in explaining difficult control concepts to students.

An adaptive system for linear systems with unknown parameters is a nonlinear system. The analysis of such adaptive systems requires similar techniques to analyse nonlinear systems. Therefore it is natural to treat adaptive control as a part of nonlinear control systems.

Nonlinear and Adaptive Control Systems treats nonlinear control and adaptive control in a unified framework, presenting the major results at a moderate mathematical level, suitable for MSc students and engineers with undergraduate degrees. Topics covered include introduction to nonlinear systems; state space models; describing functions for common nonlinear components; stability theory; feedback linearization; adaptive control; nonlinear observer design; backstepping design; disturbance rejection and output regulation; and control applications, including harmonic estimation and rejection in power distribution systems, observer and control design for circadian rhythms, and discrete-time implementation of continuous-time nonlinear control laws.

Nonlinear and Adaptive Control.indd 1 04/03/2013 09:32:20

IET CONTROL ENGINEERING SERIES 84

Nonlinear andAdaptive Control

Systems

Other volumes in this series:

Volume 8 A history of control engineering, 1800–1930 S. BennettVolume 18 Applied control theory, 2nd Edition J.R. LeighVolume 20 Design of modern control systems D.J. Bell, P.A. Cook and N. Munro (Editors)Volume 28 Robots and automated manufacture J. Billingsley (Editor)Volume 33 Temperature measurement and control J.R. LeighVolume 34 Singular perturbation methodology in control systems D.S. NaiduVolume 35 Implementation of self-tuning controllers K. Warwick (Editor)Volume 37 Industrial digital control systems, 2nd Edition K. Warwick and D. Rees (Editors)Volume 39 Continuous time controller design R. BalasubramanianVolume 40 Deterministic control of uncertain systems A.S.I. Zinober (Editor)Volume 41 Computer control of real-time processes S. Bennett and G.S. Virk (Editors)Volume 42 Digital signal Processing: principles, devices and applications N.B. Jones and

J.D.McK. Watson (Editors)Volume 44 Knowledge-based systems for industrial control J. McGhee, M.J. Grimble and A. Mowforth

(Editors)Volume 47 A History of control engineering, 1930–1956 S. BennettVolume 49 Polynomial methods in optimal control and filtering K.J. Hunt (Editor)Volume 50 Programming industrial control systems using IEC 1131-3 R.W. LewisVolume 51 Advanced robotics and intelligent machines J.O. Gray and D.G. Caldwell (Editors)Volume 52 Adaptive prediction and predictive control P.P. KanjilalVolume 53 Neural network applications in control G.W. Irwin, K. Warwick and K.J. Hunt (Editors)Volume 54 Control engineering solutions: a practical approach P. Albertos, R. Strietzel and N. Mort

(Editors)Volume 55 Genetic algorithms in engineering systems A.M.S. Zalzala and P.J. Fleming (Editors)Volume 56 Symbolic methods in control system analysis and design N. Munro (Editor)Volume 57 Flight control systems R.W. Pratt (Editor)Volume 58 Power-plant control and instrumentation: the control of boilers and HRSG systems

D. LindsleyVolume 59 Modelling control systems using IEC 61499 R. LewisVolume 60 People in control: human factors in control room design J. Noyes and M. Bransby (Editors)Volume 61 Nonlinear predictive control: theory and practice B. Kouvaritakis and M. Cannon (Editors)Volume 62 Active sound and vibration control M.O. Tokhi and S.M. VeresVolume 63 Stepping motors, 4th edition P.P. AcarnleyVolume 64 Control theory, 2nd Edition J.R. LeighVolume 65 Modelling and parameter estimation of dynamic systems J.R. Raol, G. Girija and J. SinghVolume 66 Variable structure systems: from principles to implementation A. Sabanovic, L. Fridman

and S. Spurgeon (Editors)Volume 67 Motion vision: design of compact motion sensing solution for autonomous systems

J. Kolodko and L. VlacicVolume 68 Flexible robot manipulators: modelling, simulation and control M.O. Tokhi and

A.K.M. Azad (Editors)Volume 69 Advances in unmanned marine vehicles G. Roberts and R. Sutton

(Editors)Volume 70 Intelligent control systems using computational intelligence techniques A. Ruano

(Editor)Volume 71 Advances in cognitive systems S. Nefti and J. Gray (Editors)Volume 72 Control theory: a guided tour, 3rd Edition J.R. LeighVolume 73 Adaptive sampling with Mobile WSN K. Sreenath, M.F. Mysorewala, D.O. Popa

and F.L. LewisVolume 74 Eigenstructure control algorithms: applications to aircraft/rotorcraft handling

qualities design S. SrinathkumarVolume 75 Advanced control for constrained processes and systems F. Garelli, R.J. Mantz and

H. De BattistaVolume 76 Developments in control theory towards glocal control L. Qiu, J. Chen, T. Iwasaki and

H. FujiokaVolume 77 Further advances in unmanned marine vehicles G.N. Roberts and R. Sutton (Editors)Volume 78 Frequency-domain control design for high-performance systems J. O’Brien

Nonlinear andAdaptive Control

Systems

Zhengtao Ding

The Institution of Engineering and Technology

Published by The Institution of Engineering and Technology, London, United Kingdom

The Institution of Engineering and Technology is registered as a Charity in England &Wales (no. 211014) and Scotland (no. SC038698).

© 2013 The Institution of Engineering and Technology

First published 2013

This publication is copyright under the Berne Convention and the Universal CopyrightConvention. All rights reserved. Apart from any fair dealing for the purposes of research orprivate study, or criticism or review, as permitted under the Copyright, Designs and Patents Act1988, this publication may be reproduced, stored or transmitted, in any form or by any means,only with the prior permission in writing of the publishers, or in the case of reprographicreproduction in accordance with the terms of licences issued by the Copyright LicensingAgency. Enquiries concerning reproduction outside those terms should be sent to thepublisher at the undermentioned address:

The Institution of Engineering and TechnologyMichael Faraday HouseSix Hills Way, StevenageHerts, SG1 2AY, United Kingdom

www.theiet.org

While the author and publisher believe that the information and guidance given inthis work are correct, all parties must rely upon their own skill and judgement whenmaking use of them. Neither the author nor the publisher assumes any liability toanyone for any loss or damage caused by any error or omission in the work, whethersuch an error or omission is the result of negligence or any other cause. Any and allsuch liability is disclaimed.

The moral rights of the author to be identified as author of this work have beenasserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication DataA catalogue record for this product is available from the British Library

ISBN 978-1-84919-574-4 (hardback)ISBN 978-1-84919-575-1 (PDF)

Typeset in India by MPS LimitedPrinted in the UK by CPI Group (UK) Ltd, Croydon

To my family, Yinghong, Guangyue and Xiang

Contents

Preface ix

1 Introduction to nonlinear and adaptive systems 11.1 Nonlinear functions and nonlinearities 11.2 Common nonlinear systems behaviours 41.3 Stability and control of nonlinear systems 5

2 State space models 92.1 Nonlinear systems and linearisation around an operating point 92.2 Autonomous systems 112.3 Second-order nonlinear system behaviours 122.4 Limit cycles and strange attractors 18

3 Describing functions 253.1 Fundamentals 263.2 Describing functions for common nonlinear components 293.3 Describing function analysis of nonlinear systems 34

4 Stability theory 414.1 Basic definitions 414.2 Linearisation and local stability 454.3 Lyapunov’s direct method 464.4 Lyapunov analysis of linear time-invariant systems 51

5 Advanced stability theory 555.1 Positive real systems 555.2 Absolute stability and circle criterion 595.3 Input-to-state stability and small gain theorem 655.4 Differential stability 71

6 Feedback linearisation 756.1 Input–output linearisation 756.2 Full-state feedback linearisation 83

7 Adaptive control of linear systems 897.1 MRAC of first-order systems 907.2 Model reference control 947.3 MRAC of linear systems with relative degree 1 997.4 MRAC of linear systems with high relatives 1027.5 Robust adaptive control 103

viii Nonlinear and adaptive control systems

8 Nonlinear observer design 1098.1 Observer design for linear systems 1098.2 Linear observer error dynamics with output injection 1118.3 Linear observer error dynamics via direct state transformation 1208.4 Observer design for Lipschitz nonlinear systems 1228.5 Reduced-order observer design 1278.6 Adaptive observer design 136

9 Backstepping design 1419.1 Integrator backstepping 1419.2 Iterative backstepping 1449.3 Observer backstepping 1479.4 Backstepping with filtered transformation 1529.5 Adaptive backstepping 1599.6 Adaptive observer backstepping 167

10 Disturbance rejection and output regulation 17510.1 Asymptotic rejection of sinusoidal disturbances 17510.2 Adaptive output regulation 18610.3 Output regulation with nonlinear exosystems 19410.4 Asymptotic rejection of general periodic disturbances 204

11 Control applications 21911.1 Harmonics estimation and rejection in power distribution

systems 21911.1.1 System model 22011.1.2 Iterative observer design for estimating frequency

modes in input 22411.1.3 Estimation of specific frequency modes in input 23211.1.4 Rejection of frequency modes 23411.1.5 Example 235

11.2 Observer and control design for circadian rhythms 23811.2.1 Circadian model 23911.2.2 Lipschitz observer design 24111.2.3 Phase control of circadian rhythms 243

11.3 Sampled-data control of nonlinear systems 24711.3.1 System model and sampled-data control 24911.3.2 Stability analysis of sampled-data systems 25111.3.3 Simulation 260

Bibliographical Notes 263

References 268

Index 275

Preface

This book is intended for the use as a textbook at MSc and senior undergraduatelevel in control engineering and related disciplines such as electrical, mechanical,chemical and aerospace engineering and applied mathematics. It can also be used asa reference book by control engineers in industry and research students in automationand control. It is largely, although not entirely, based on the course unit bearing thesame name as the book title that I have been teaching for several years for the MSccourse at Control Systems Centre, School of Electrical and Electronic Engineering,The University of Manchester. The beginning chapters cover fundamental conceptsin nonlinear control at moderate mathematical level suitable for students with a firstdegree in engineering disciplines. Simple examples are used to illustrate importantconcepts, such as the difference between exponential stability and asymptotic stability.Some advanced and recent stability concepts such as input-to-state stability are alsoincluded, mainly as an introduction at a less-demanding mathematical level comparedwith their normal descriptions in the existing books, to research students who mayencounter those concepts in literature. Most of the theorems in the beginning chaptersare introduced with the proofs, and some of the theorems are simplified with lessgeneral scopes, but without loss of rigour. The later chapters cover several topicswhich are closely related to my own research activities, such as nonlinear observerdesign and asymptotic disturbance rejection of nonlinear systems. They are includedto demonstrate the applications of fundamental concepts in nonlinear and adaptivecontrol to MSc and research students, and to bridge the gap between a normal textbooktreatment of control concepts and that of research articles published in academicjournals. They can also be used as references for the students who are working onthe related topics. At the end of the book, applications to less traditional areas suchas control of circadian rhythms are also shown, to encourage readers to explore newapplied areas of nonlinear and adaptive control.

This book aims at a unified treatment of adaptive and nonlinear control.It is well known that the dynamics of an adaptive control system for a linear dynamicsystem with unknown parameters are nonlinear. The analysis of such adaptive sys-tems requires similar techniques to the analysis for nonlinear systems. Some morerecent control design techniques such as backstepping relies on Lyapunov functionsto establish the stability, and they can be directly extended to adaptive control of non-linear systems. These techniques further reduce the traditional gap between adaptivecontrol and nonlinear control. Therefore, it is now natural to treat adaptive controlas a part of nonlinear control systems. The foundation for linear adaptive controland nonlinear adaptive control is the positive real lemma, which is related to passivesystems in nonlinear control and Lyapunov analysis. It is decided to use the positive

x Nonlinear and adaptive control systems

real lemma and related results in adaptive control and nonlinear control as themain theme of the book, together with Lyapunov analysis. Other important resultssuch as circle criterion and backstepping are introduced as extensions and furtherdevelopments from this main theme.

For a course unit of 15 credits on nonlinear and adaptive control at the ControlSystems Centre, I normally cover Chapters 1–4, 6 and 7, and most of the contents ofChapter 5, and about half of the materials in Chapter 9. Most of the topics coveredin Chapters 8, 10 and 11 have been used as MSc dissertation projects and some ofthem as PhD projects. The contents may also be used for an introductory courseon nonlinear control systems, by including Chapters 1–5, 8 and the first half ofChapter 9, and possibly Chapter 6. For a course on adaptive control of nonlinearsystems, an instructor may include Chapters 1, 2, 4, 5, 7 and 9. Chapter 8 may beused alone as a brief introduction course to nonlinear observer design. Some resultsshown in Chapters 8, 10 and 11 are recently published, and can be used as referencesfor the latest developments in related areas.

Nonlinear and adaptive control is still a very active research area in automationand control, with many new theoretic results and applications continuing to merge.I hope that the publication of this work will have a good impact, however small,on students’ interests to the subject. I have been benefited from my students, bothundergraduate and MSc students, through my teaching and other interactions withthem, in particular, their questions to ask me to explain many of the topics coveredin this book with simple languages and examples. My research collaborators andPhD students have contributed to several topics covered in the book through jointjournal publications, whose names may be found in the references cited at the end ofthe book. I would like to thank all the researchers in the area who contributed to thetopics covered in the book, who are the very people that make this subject fascinating.

Chapter 1

Introduction to nonlinear and adaptive systems

Nonlinearity is ubiquitous, and almost all the systems are nonlinear systems. Manyof them can be approximated by linear dynamic systems, and significant amount ofanalysis and control design tools can then be applied. However, there are intrinsicnonlinear behaviours which cannot be described using linear systems, and analysisand control are necessarily based on nonlinear systems. Even for a linear system,if there are uncertainties, nonlinear control strategies such as adaptive control mayhave to be used. In the last two decades, there have been significant developments innonlinear system analysis and control design. Some of them are covered in this book.In this chapter, we will discuss typical nonlinearities and nonlinear behaviours, andintroduce some basic concepts for nonlinear system analysis and control.

1.1 Nonlinear functions and nonlinearities

A dynamic system has its origin from dynamics in classic mechanics. The behaviour ofa dynamic system is often specified by differential equations. Variables in a dynamicsystem are referred to as states, and they can be used to determine the status of asystem. Without external influences, the state variables are sufficient to determine thefuture status for a dynamic system. For a dynamic system described by continuousdifferential equations, state variables cannot be changed instantly, and this reflects thephysical reality. Many physical and engineering systems can be modelled as dynamicsystems, using ordinary differential equations. Application areas of dynamic systemshave expanded rapidly to other areas such as biological systems, financial systems,etc. Analysis of the behaviours of dynamic systems is essential to the understandingof various applications in many science and engineering disciplines. The behaviourof a dynamic system may be altered by exerting external influences, and quite oftenthis kind of influences is based on knowledge of the current state. In this sense, thedynamic system is controlled to achieve certain behaviours.

State variables are denoted by a vector in an appropriate dimension for conve-nience, and the dynamic systems are described by first-order differential equations ofthe state vector. A linear dynamic system is described by

x = Ax + Bu

y = Cx + Du,(1.1)

2 Nonlinear and adaptive control systems

where x ∈ Rn is the state, u ∈ Rm is the external influence, which is referred to as

the input, and y ∈ Rs is a vector that contains the variables for measurement, which

is referred to as the output, and A ∈ Rn×n, B ∈ R

n×m, C ∈ Rs×n and D ∈ R

s×m arematrices that may depend on time. If the system matrices A, B, C and D are constantmatrices, the system (1.1) is a linear time-invariant system. For a given input, the stateand the output can be computed using the system equation (1.1). More importantly,the superposition principle holds for linear dynamic systems.

Nonlinear dynamic systems are the dynamic systems that contain at least onenonlinear component, or in other words, the functions in the differential equationscontain nonlinear functions. For example we consider a Single-Input-Single-Output(SISO) system with input saturation

x = Ax + Bσ (u)

y = Cx + Du,(1.2)

where σ : R → R is a saturation function defined as

σ (u) =⎧⎨

⎩

−1 for u < −1,u otherwise,1 for u > 1.

(1.3)

The only difference between the systems (1.2) and (1.1) is the saturation function σ .It is clear that the saturation function σ is a nonlinear function, and therefore thissystem is a nonlinear system. Indeed, it can be seen that the superposition principledoes not apply, because after the input saturation, any increase in the input amplitudedoes not change the system response at all.

A general nonlinear system is often described by

x = f (x, u, t)

y = h(x, u, t),(1.4)

where x ∈ Rn, u ∈ R

m and y ∈ Rs are the system state, input and output respectively,

and f : Rn × R

m × R → Rn and h : R

n × Rm × R → R

s are nonlinear functions.Nonlinearities of a dynamic system are described by nonlinear functions. We may

roughly classify nonlinear functions in nonlinear dynamic systems into two types.The first type of nonlinear functions are analytical functions such as polynomials,sinusoidal functions and exponential functions, or composition of these functions.The derivatives of these functions exist, and their Taylor series can be used to obtaingood approximations at any points. These nonlinearities may arise from physicalmodelling of actual systems, such as nonlinear springs and nonlinear resistors, or dueto nonlinear control design, such as nonlinear damping and parameter adaptation lawfor adaptive control. There are nonlinear control methods such as backstepping whichrequires the existence of derivatives up to certain orders.

Introduction to nonlinear and adaptive systems 3

Other nonlinearities may be described by piecewise linear functions, but with afinite number of points where the derivatives do not exist or the functions are not evencontinuous. The saturation function mentioned earlier is a piecewise linear functionthat is continuous, but not smooth at the two joint points. A switch, or an idea relay,can be modelled using a signum function which is not continuous. There may also benonlinearities which are multi-valued, such as relay with hysteresis, which is describedin Chapter 3 in detail. A nonlinear element with multi-valued nonlinearity returns asingle value at any instance, depending on the history of the input. In this sense,the multi-valued nonlinearities have memory. Other single-valued nonlinearities arememoryless. The nonlinearities described by piecewise linear functions are also calledhard nonlinearities, and the common hard nonlinearities include saturation, relay, deadzone, relay with hysteresis, backlash, etc. One useful way to study hard nonlinearitiesis by describing functions. Hard nonlinearities are also used in control design, forexample signum function is used in siding mode control, and saturation functionsare often introduced for control inputs to curb the peaking phenomenon for the semi-global stability. In the following example, we show a nonlinearity that arises fromadaptive control of a linear system.

Example 1.1. Consider a first-order linear system

x = ax + u,

where a is an unknown parameter. How to design a control system to ensure thestability of the system? If a range a− < a < a+ is known, we can design a controllaw as

u = −cx − a+x

with c > 0, which results in the closed-loop system

x = −cx + (a − a+)x.

Adaptive control can be used in the case of completely unknown a,

u = −cx − ax

˙a = x2.

If we let a = a − a, the closed-loop system is described by

x = −cx + ax

˙a = −x2.

This adaptive system is nonlinear, even though the original uncertain system is linear.This adaptive system is stable, and it does need the stability theory introduced laterin Chapter 7 of the book.


1.2 Common nonlinear systems behaviours

Many nonlinear systems may be approximated by linearised systems around operatingpoints, and their behaviours in close neighbourhoods can be predicted from the lineardynamics. This is the justification of applying linear system theory to control study,as almost all the practical systems are nonlinear systems.

There are many nonlinear features that do not exist in linear dynamic systems,and therefore linearised dynamic models cannot describe the behaviours associatedwith these nonlinear features. We will discuss some of those nonlinear features in thefollowing text.

Multiple equilibrium points are common for nonlinear systems, unlike the linearsystem. For example

x = −x + x2.

This system has two equilibria at x = 0 and x = 1. The behaviours around these equi-librium points are very different, and they cannot be described by a single linearisedmodel.

Limit cycles are a phenomenon that periodic solutions exist and attract nearbytrajectories in positive or negative time. Closed curve solutions may exist for linearsystems, such as solution to harmonic oscillators. But they are not attractive to nearbytrajectories and not robust to any disturbances. Heart beats of human body can bemodelled as limit cycles of nonlinear systems.

High-order harmonics and subharmonics occur in the system output when subjectto a harmonic input. For linear systems, if the input is a harmonic function, theoutput is a harmonic function, with the same frequency, but different amplitudeand phase. For nonlinear systems, the output may even have harmonic functions withfractional frequency of the input or multiples of the input frequency. This phenomenonis common in power distribution networks.

Finite time escape can happen in a nonlinear system, i.e., the system state tendsto infinity at a finite time. This will never happen for linear systems. Even for anunstable linear system, the system state can only grow at an exponential rate. Thefinite time escape can cause a problem in nonlinear system design, as a trajectorymay not exist.

Finite time convergence to an equilibrium point can happen to nonlinear systems.Indeed, we can design nonlinear systems in this way to achieve fast convergence.This, again, cannot happen for linear systems, as the convergence rate can only beexponential, i.e., a linear system can only converge to its equilibrium asymptotically.

Chaos can only happen in nonlinear dynamic systems. For some class of non-linear systems, the trajectories are bounded, but not converge to any equilibriumor limit cycles. They may have quasi-periodic solutions, and the behaviour is verydifficult to predict.

There are other nonlinear behaviours such as bifurcation, etc., which cannothappen in linear systems. Some of the nonlinear behaviours are covered in detail inthis book, such as limit cycles and high-order harmonics. Limit cycles and chaos are


discussed in Chapter 2, and limit cycles also appear in other problems consideredin this book. High-order harmonics are discussed in disturbance rejection. When thedisturbance is a harmonic signal, the internal model for disturbance rejection has toconsider the high-order harmonics generated due to nonlinearities.

1.3 Stability and control of nonlinear systems

Nonlinear system behaviours are much more complex than those of linear systems.The analytical tools for linear systems often cannot be applied to nonlinear systems.For linear systems, the stability of a system can be decided by eigenvalues of systemmatrix A, and obviously for nonlinear systems, this is not the case. For some nonlinearsystems, a linearised model cannot be used to determine the stability even in a verysmall neighbourhood of the operating point. For example for a first-order system

x = x3,

linearised model around the origin is x = 0. This linearised model is critically stable,but the system is unstable. In fact, if we consider another system

x = −x3,

which is stable, but the linearised model around the origin is still x = 0. Frequencydomain methods also cannot be directly applied to analysing input–output relationshipfor nonlinear systems, as we cannot define a transfer function for a general nonlinearsystem.

Of course, some basic concept may still be applicable to nonlinear systems,such as controllability, but these systems are often in different formulation anduse different mathematical tools. Frequency response method can be applied toanalysing a nonlinear system with one nonlinear component based on approximationin frequency domain using describing function method. High gain control and zerodynamics for nonlinear systems originate from their counterparts in linear systems.

One important concept that we need to address is stability. As eigenvalues areno longer a suitable method for nonlinear systems, stability concepts and methodsto check stability for nonlinear systems are necessary. Among various definition,Lyapunov stability is perhaps the most fundamental one. It can be checked by using aLyapunov function for stability analysis. Some of the other stability concepts such asinput-to-state stability may also be interpreted using Lyapunov functions. Lyapunovfunctions can also provide valid information in control design. Lyapunov stabilitywill be the main stability concept used in this book.

Compared with the stability concepts of nonlinear systems, control design meth-ods for nonlinear systems are even more diversified. Unlike control design for linearsystems, there is a lack of systematic design methods for nonlinear systems. Most ofthe design methods can only apply to specific classes of nonlinear systems. Becauseof this, nonlinear control is often more challenging and interesting. It is impossiblefor the author to cover all the major areas of nonlinear control design and analysis,partially to author’s knowledge base, and partially due to the space constraint.


People often start with linearisation of a nonlinear system. If the control designbased on a linearised model works, then there is no need to worry about nonlinearcontrol design. Linearised models depend on operating points, and a switching strat-egy might be needed to move from one operating point to the others. Gain schedulingand linear parameter variation (LPV) methods are also closely related to linearisationaround operating points.

Linearisation can also be achieved for certain class of nonlinear systems througha nonlinear state transformation and feedback. This linearisation is very much differ-ent from linearisation around operating points. As shown later in Chapter 6, a numberof geometric conditions must be satisfied for the existence of such a nonlinear trans-formation. The linearisation obtained in this way works globally in the state space,not just at one operating point. Once the linearised model is obtained, further controldesign can be carried out using design methods for linear systems.

Nonlinear functions can be approximated using artificial neuron networks, andfuzzy systems and control methods have been developed using these approximationmethods. The stability analysis of such systems is often similar to Lyapunov function-based design method and adaptive control. We will not cover them in this book. Othernonlinear control design methods such as band–band control and sliding mode controlare also not covered.

In the last two decades, there were developments for some more systematiccontrol design methods, such as backstepping and forwarding. They require the systemto have certain structures so that these iterative control designs can be carried out.Among them, backstepping method is perhaps the most popular one. As shown inChapter 9 in the book, it requires the system state space function in a sort of lower-triangular form so that at each step a virtual control input can be designed. Significantamount of coverage of this topic can be found in this book. Forwarding control designcan be interpreted as a counterpart of backstepping in principle, but it is not coveredin this book.

When there are parametric uncertainties, adaptive control can be introduced totackle the uncertainty. As shown in a simple example earlier, an adaptive control sys-tem is nonlinear, even for a linear system. Adaptive technique can also be introducedtogether with other nonlinear control design methods, such as backstepping method.In such a case, people often give it a name, adaptive backstepping. Adaptive controlfor linear systems and adaptive backstepping for nonlinear systems are covered indetails in Chapter 7 and Chapter 9 in this book.

Similar to linear control system design, nonlinear control design methods canalso be grouped as state-feedback control design and output-feedback control design.The difference is that the separation principle is not valid for nonlinear control designin general, that is if we replace the state in the control input by its estimate, we wouldnot be able to guarantee the stability of the closed-loop system using state estimate.Often state estimation must be integrated in the control design, such as observerbackstepping method.

State estimation is an important topic for nonlinear systems on its own. Over thelast three decades, various observer design methods have been introduced. Some ofthem may have their counterparts in control design. Design methods are developed


for different nonlinearities. One of them is for systems with Lipschitz nonlinearity,as shown in Chapter 8. A very neat nonlinear observer design is the observer designwith output injection, which can be applied to a class of nonlinear systems whosenonlinearities are only of the system output.

In recent years, the concept of semi-global stability is getting more popular.Semi-global stability is not as good as global stability, but the domain of attractioncan be as big as you can specify. The relaxation in the global domain of attraction doesgive control design more freedom in choosing control laws. One common strategy isto use high gain control together with saturation. We will not cover it in this book,but the design methods in semi-global stability can be easily followed once a readeris familiar with the control design and analysis methods introduced in this book.

Chapter 2

State space models

The nonlinear systems under consideration in this book are described by differentialequations. In the same way as for linear systems, we have system state variables, inputsand outputs. In this chapter, we will provide basic definitions for state space models ofnonlinear systems, and tools for preliminary analysis, including linearisation aroundoperating points. Typical nonlinear behaviours such as limit cycles and chaos willalso be discussed with examples.

2.1 Nonlinear systems and linearisation aroundan operating point

A system is called a dynamic system if its behaviours depend on its history. Statesof a system are the variables that represent the information of history. At any timeinstance, the current state value decides the system’s future behaviours. In this sense,the directives of the state variables decide the system’s behaviours, and hence wedescribe a dynamic system by a set of first-order differential equations in vectorform as

x = f (x, u, t), x(0) = x0, (2.1)

where x ∈ Rn is the state of the system, f : R

n → Rn is a continuous function and

u ∈ Rm denotes the external influence, which is usually referred to as the input to the

system.For the differential equation (2.1) to exist as a unique solution for a given initial

condition, we need to impose a restriction on the nonlinear function f that f must beLipschitz with respect to the variable x. The definition of Lipschitz condition is givenbelow.

Definition 2.1. A function f : Rn × R

m × R → Rn is Lipschitz with a Lipschitz

constant γ if for any vectors x, x ∈ Dx ⊂ Rn, and u ∈ Du ⊂ R

m and t ∈ It ⊂ R, withDx, Du being the regions of interest and It being an time interval,

‖f (x, u, t) − f (x, u, t)‖ ≤ γ ‖x − x‖, (2.2)

with γ > 0.

Note that Lipschitz condition implies continuity with respect to x. The existenceand uniqueness of a solution for (2.1) are guaranteed by the function f being Lipschitzand being continuous with respect to t.


Remark 2.1. The continuity of f with respect to t and state variable x might bestronger than we have in real applications. For example, a step function is not con-tinuous in time. In the case that there are finite number of discontinuities in a giveninterval, we can solve the equation of a solution in each of the continuous region, andjoin them together. There are situations of discontinuity with state variable, such asan ideal relay. In such a case, the uniqueness of the solution can be an issue. Furtherdiscussion on this is beyond the scope of this book. In the systems considered inthe book, we would assume that there would be no problem with the uniqueness ofa solution. �

The system state contains the whole information of the behaviour. However,for a particular application, only a subset of the state variables or a function of statevariables is of interest, which can be denoted as y = h(x, u, t) with h : R

n × Rm ×

R → Rs, normally with s < n. We often refer to y as the output of the system. To

write them together with the system dynamics, we have

x = f (x, u, t), x(0) = x0

y = h(x, u, t).

In this book, we mainly deal with time-invariant systems. Hence, we can drop thevariable t in f and h and write the system as

x = f (x, u), x(0) = x0

y = h(x, u),(2.3)

where x ∈ Rn is the state of the system, y ∈ R

s and u ∈ Rm are the output and the

input of the system respectively, and f : Rn × R

m → Rn and h : R

n × Rm → R

s arecontinuous functions.

Nonlinear system dynamics are much more complex than linear systems ingeneral. However, when the state variables are subject to small variations, we wouldexpect the behaviours for small variations to be similar to linear systems, based onthe fact that

f (x + δx, u + δu) ≈ f (x, u) + ∂f

∂x(x, u)δx + ∂f

∂u(x, u)δu,

when δx and δu are very small.An operating point at (xe, ue) is taken with x = xe and u = ue being constants

such that f (xe, ue) = 0. A linearised model around the operation point can then beobtained. Let

x = x − xe,

u = u − ue,

y = h(x, u) − h(xe, ue),

then the linearised model is given by

State space models 11

˙x = Ax + Bu

y = Cx + Du,(2.4)

where A ∈ Rn×n, B ∈ R

n×m, C ∈ Rs×n, D ∈ R

s×m matrices with elements ai,j , bi,j , ci,j

and di,j respectively shown by, assuming that f and h are differentiable,

ai,j = ∂fi

∂xj(xe, ue),

bi,j = ∂fi

∂uj(xe, ue),

ci,j = ∂hi

∂xj(xe, ue),

di,j = ∂hi

∂uj(xe, ue).

Remark 2.2. For a practical system, a control input can keep the state in an equilib-rium point, i.e., at a point such that x = 0, and therefore it is natural to look at thelinearisation around this point. However, we can obtain linearised model at points thatare not at equilibrium. If (xe, ue) is not an equilibrium point, we have f (xe, ue) = 0.We can carry out the linearisation in the same way, but the resultant linearised systemis given by

˙x = Ax + Bu + d

y = Cx + Du,

where d = f (xe, ue) is a constant vector. �

2.2 Autonomous systems

For a system in (2.3), the external influence can only be exerted through the input u.If a system does not take any input, its future state only depends on the initial state.It means that the system is not influenced by external factors, and the behaviours arecompletely determined by the system state. Such a system is often referred to as anautonomous system. A definition is given below.

Definition 2.2. An autonomous system is a dynamic system whose behaviour doesnot explicitly depend on time.

In terms of differential equations, an autonomous system can be expressed as

x = f (x), (2.5)

where x ∈ Rn is the state of the system, and f : R

n → Rn.


Remark 2.3. It is easy to see that for a system in (2.3), if the control input remainsconstant, then it is an autonomous system. We only need to re-define the function f asfa(x) := f (x, uc) where uc is a constant input. Even if the inputs are polynomials andsinusoidal functions of time, we can convert the system to the autonomous system bymodelling the sinusoidal and polynomial functions as the state variables of a lineardynamic system, and integrate this system into the original system. The augmentedsystem is then an autonomous system. �Definition 2.3. For an autonomous system (2.5), a point xe ∈ R

n is a singular pointif f (xe) = 0.

It is easy to see that singular points are equilibrium points. Singular points aremore preferred for autonomous systems, especially for second-order systems.

Since autonomous system do not have external input, the set of all the tra-jectories provides a complete geometrical representation of the dynamic behaviour.This is often referred to as the phase portrait, especially for second-order systemsin the format

x1 = x2

x2 = φ(x1, x2).

In the above system, if we interpret x1 as the displacement, then x2 is the velocity. Thestate variables often have clear physical meanings. Phase portraits can be obtained bya number of methods, including analysing the behaviours near the singular points. Infact, singular points might get the name from their positions in the phase portrait. In aphase portrait, the lines usually do not intercept each other due to the uniqueness of thesolutions. However, they meet at the points where f (x) = 0, seemingly interceptingeach other. Those points are singular in this sense.

2.3 Second-order nonlinear system behaviours

For a second-order system, we can write the system equation as

x1 = f1(x1, x2)

x2 = f2(x1, x2).

For an equilibrium point (x1e, x2e), the linearised model is given by[ ˙x1˙x2

]

= Ae

[x1

x2

]

, (2.6)

where

Ae =

⎡

⎢⎢⎣

∂f1

∂x1(x1e, x2e)

∂f1

∂x2(x1e, x2e)

∂f2

∂x1(x1e, x2e)

∂f2

∂x2(x1e, x2e)

⎤

⎥⎥⎦ .


Therefore, the behaviour about this equilibrium or singular point is determined bythe properties of matrix Ae. Based on the eigenvalues of Ae, we can classify thesingular points in the following six different cases. We use λ1 and λ2 to denote thetwo eigenvalues of Ae.

Stable node, for λ1 < 0, λ2 < 0. This is a case when both eigenvalues are negativereal numbers. The linearised model is stable, and a typical phase portrait around thissingular point is shown below.

x1

x2

Stable node

Unstable node, for λ1 > 0, λ2 > 0. This singular point is unstable, and the trajectoriesdiverge from the point, but not spiral around it.

x1

x2

Unstable node

Saddle point, for λ1 < 0, λ2 > 0. With one positive and one negative eigenvalues, thehyperplane in three dimensions may look like a saddle. Some trajectories convergeto the singular point, and others diverge, depending on the directions of approachingthe point.

x1

x2

Saddle point

Stable focus, for λ1,2 = μ± jν, (μ < 0). With a negative real part for a pair of con-jugate poles, the singular point is stable. Trajectories converge to the singular point,


spiralling around. In time domain, the solutions are similar to decayed sinusoidalfunctions.

x1

x2

Stable focus

Unstable focus, for λ1,2 = μ± jν, (μ > 0). The real part is positive, and thereforethe singular point is unstable, with the trajectories spiralling out from the singularpoint.

x1

x2

Unstable focus

Centre, for λ1,2 = ±jν, (ν > 0). For the linearised model, when the real part is zero,the norm of the state is constant. In phase portrait, there are closed orbits around thesingular point.

x1

x2

Centre

To draw a phase portrait, the analysis of singular points is the first step. Basedon the classification of the singular points, the behaviours in neighbourhoods of thesepoints are more or less determined. For other regions, we can calculate the directionsof the movement from the directives.

At any point, the slope of trajectory can be computed by

dx2

dx1= f2(x1, x2)

f1(x1, x2).


With enough points in the plane, we should be able to sketch phase portraits connect-ing the points in the directions determined by the slopes.

Indeed, we can even obtain curves with constant slopes, which are named asisoclines. An isocline is a curve on which (f2(x1, x2)/f1(x1, x2)) is constant. This, again,can be useful in sketching a phase portrait for a second-order nonlinear system.

It should be noted that modern computer simulation can provide accurate solu-tions to many nonlinear differential equations. For this reason, we will not go tofurther details of drawing phase portraits based on calculating slope of trajectoriesand isoclines.

Example 2.1. Consider a second-order nonlinear system

x1 = x2

x2 = −x2 − 2x1 + x21.

Setting

0 = x2,

0 = −x2 − 2x1 + x21,

we obtain two singular points (0, 0) and (−2, 0).Linearised system matrix for the singular point (0, 0) is obtained as

A =[

0 1−1 −2

]

,

and the eigenvalues are obtained as λ1 = λ2 = −1. Hence, this singular point is astable node.

For the singular point (−2, 0), the linearised system matrix is obtained as

A =[

0 12 −1

]

,

and the eigenvalues are λ1 = −2 and λ2 = 1. Hence, this singular point is a saddlepoint. It is useful to obtain the corresponding eigenvectors to determine which direc-tion is converging and which is diverging. The eigenvalues v1 and v2, for λ1 = −2and λ2 = 1, are obtained as

v1 =[

1−2

]

, v2 =[

11

]

This suggests that along the direction of v1, relative to the singular point, the stateconverges to the singular point, while along v2, the state diverges. We can clearlysee from Figure 2.1 that there is a stable region near the singular point (0, 0).However, in the neighbourhood of (−2, 1) one part of it is stable, and the other partis unstable.

�


−4 −3 −2 −1 0 1 2 3−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

x1

x 2

Figure 2.1 Phase portrait of Example 2.1

Example 2.2. Consider the swing equation of a synchronous machine

H δ = Pm − Pe sin (δ), (2.7)

where H is the inertia, δ is the rotor angle, Pm is the mechanical power and Pe is themaximum electrical power generated. We may view Pm as the input and Pe sin δ asthe output. For the convenience of presentation, we take H = 1, Pm = 1 and Pe = 2.

The state space model is obtained by letting x1 = δ and x2 = δ as

x1 = x2

x2 = 1 − 2 sin (x1).

For the singular points, we obtain

x1e = 1

6π or

5

6π ,

x2e = 0.

Note that there are an infinite number of singular points, as x1e = 2kπ + 16π and

x1e = 2kπ + 56π are also solutions for any integer value of k .


Let us concentrate on the analysis of the two singular points ( 16π , 0) and ( 5

6π , 0).The linearised system matrix is obtained as

A =[

0 1−2 cos (x1e) 0

]

.

For ( 16π , 0), the eigenvalues are λ1,2 = ±31/4j, and therefore this singular point it a

centre.For ( 5

6π , 0), the eigenvalues are λ1 = −31/4 and λ2 = 31/4. Hence, this singularpoint it a saddle point. The eigenvalues v1 and v2, for λ1 = −31/4 and λ2 = 31/4, areobtained as

v1 =[

1−31/4

]

, v2 =[

131/4

]

.

−2 0 2 4 6 8−3

−2

−1

0

1

2

3

x1

x 2

Figure 2.2 Phase portrait of Example 2.2

Figure 2.2 shows a phase portrait obtained from computer simulation. The centre at( 1

6π , 0) and the saddle point at ( 56π , 0) are clearly shown in the figure. The directions

of the flow can be determined from the eigenvectors of the saddle point. For examplethe trajectories start from the points around (5, −3) and move upwards and to the left,along the direction pointed by the eigenvector v1 towards the saddle point. Along thedirection pointed by v2, the trajectories depart from the saddle point. �


2.4 Limit cycles and strange attractors

Some trajectories appear as closed curves in phase portrait. For autonomous systems,they represent periodic solutions, as the solutions only depend on the states, and theyare referred to as cycles. For linear systems, periodic solutions appear in harmonicoscillators. For example a harmonic oscillator described by

x1 = x2

x2 = −x1

has solutions as sinusoidal functions, and the amplitude is determined by the initialvalues. It is easy to see that the function V = x2

1 + x22 remains a constant, as we have

V = 2x1(−x2) + 2x2x1 = 0.

Therefore, the solutions of this oscillator are circles around the origin. For a giveninitial state, the radius does not change. When two initial points are very close, theirsolutions will be very close, but they will not converge to one cycle, no matter howclose the initial values. Also the solution for this harmonic oscillator is not robust, asany small disturbance will destroy the cycle.

There are cycles in the phase portrait shown in Figure 2.2. Even though they aresolutions to a nonlinear dynamic system, they are similar to the cycles obtained fromthe harmonic oscillator in the sense that cycles depend on the initial values and theydo not converge or attract to each other, no matter how close the two cycles are.

The cycles discussed above are not limit cycles. For limit cycles we have thefollowing definition.

Definition 2.4. A closed curve solution, or in other word, a cycle, of an autonomoussystem is a limit cycle, if some non-periodic solutions converge to the cycle as t → ∞or t → −∞.

A limit cycle is stable if nearby trajectories converge to it asymptotically, unstableif move away. One property of a limit cycle is that amplitude of the oscillation maynot depend on the initial values. A limit cycle may be attractive to the nearby region.One of the most famous ones is van der Pol oscillator. This oscillator does have aphysical meaning. It can be viewed as a mathematical model of an RLC circuit, withthe resistor being possible to take negative values in certain regions.

Example 2.3. One form of van der Pol oscillator is described in the followingdifferential equation:

y − ε(1 − y2)y + y = 0, (2.8)

where ε is a positive real constant.If we take x1 = y and x2 = y, we obtain the state space equation

x1 = x2

x2 = −x1 + ε(1 − x21)x2.


From this state space realisation, it can be seen that when ε = 0, van der Pol oscillatoris the same as a harmonic oscillator. For ε with small values, one would expect thatit behaves like a harmonic oscillator.

A more revealing state transformation for ε with big values is given by

x1 = y

x2 = 1

εy + f (y),

where f (y) = y3/3 − y. Under the above transformation, we have the system as

x1 = ε(x2 − f (x1))

x2 = − 1εx1.

(2.9)

It can be obtained that

dx2

dx1(x2 − f (x1)) = x1

ε2(2.10)

This equation suggests that as ε → ∞, we have dx2dx1

= 0 or x2 = f (x1). This can beseen from the phase portrait for very big values of ε in Figure 2.4.

Let us stick with the state space model (2.9). The only singular point is at theorigin (0, 0). The linearised system matrix at the origin is obtained as

A =[ε 1

− 1ε

0

]

.

From the eigenvalues of A, we can see that this singular point is either an unstablenode or an unstable focus, depending on the value of ε. Phase portrait of van derPol oscillator with ε = 1 is shown in Figure 2.3 for two trajectories, one with initialcondition outside the limit cycle and one from inside. The broken line shows x2 =f (x1). Figure 2.4 shows the phase portrait with ε = 10. It is clear from Figure 2.4 thatthe trajectory sticks with the line x2 = f (x1) along the outside and then moves almosthorizontally to the other side, as predicted in the analysis earlier. �

Limit cycles also exist in high-order nonlinear systems. As seen later in Chap-ter 11, circadian rhythms can also be modelled as limit cycles of nonlinear dynamicsystems. For second-order autonomous systems, limit cycles are very typical trajec-tories. The following theorem, Poincare–Bendixson theorem, describes the featuresof trajectories of the second-order systems, from which a condition on the existenceof a limit cycle can be drawn.

Theorem 2.1. If a trajectory of the second-order autonomous system remains in afinite region, then one of the following is true:


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−3

−2

−1

0

1

2

3

x1

x 2

Figure 2.3 Phase portrait of van der Pol oscillator with ε = 1

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−3

−2

−1

0

1

2

3

x 2

x1

Figure 2.4 Phase portrait of van der Pol oscillator with ε = 10


● The trajectory goes to an equilibrium point.● The trajectory tends to an asymptotically stable limit cycle.● The trajectory is itself a cycle.

For high-order nonlinear systems, there are more complicated features if thetrajectories remain in a bounded region. For the asymptotic behaviours of dynamicsystems, we define positive limit sets.

Definition 2.5. Positive limit set of a trajectory is the set of all the points for whichthe trajectory converges to, as t → ∞.

Positive limit sets are also referred to as ω-limit sets, as ω is the last letter ofGreek letters. Similarly, we can define negative limit sets, and they are called α-limitsets accordingly. Stable limit cycles are positive limit sets, so do stable equilibriumpoints. The dimension for ω-limit sets is zero or one, depending on singular points orlimit cycles.

Strange limit sets are those limit sets which may or may not be asymptoticallyattractive to the neighbouring trajectories. The trajectories they contain may be locallydivergent from each other, within the attracting set. Their dimensions might be frac-tional. Such structures are associated with the quasi-random behaviour of solutionscalled chaos.

Example 2.4. The Lorenz attractor. This is one of the most widely studied examplesof strange behaviour in ordinary differential equations, which is originated fromstudies of turbulent convection by Lorenz. The equation is in the form

x1 = σ (x2 − x1)

x2 = (1 + λ− x3)x1 − x2

x3 = x1x2 − bx3,

(2.11)

where σ , λ and b are positive constants. There are three equilibrium points (0, 0, 0),(√

bλ,√

bλ, λ) and (−√bλ, −√

bλ, λ). The linearised system matrix around the originis obtained as

A =⎡

⎣−σ σ 0λ+ 1 −1 0

0 0 −b

⎤

⎦ ,

and its eigenvalues are obtained as λ1,2 = −(σ − 1) ±√(σ − 1)2 + 4σλ/2 and

λ3 = −b. Since the first eigenvalue is positive, this equilibrium is unstable. It canbe shown that the other equilibrium points are unstable when the parameters satisfies

σ > b + 1,

λ >(σ + 1)(σ + b + 1)

σ − b − 1.


−15−10

−50

510

15

−20

−10

0

10

200

5

10

15

20

25

30

35

x1x2

x 3

Figure 2.5 A typical trajectory of Lorenz attractor

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

x1

x 2

Figure 2.6 Projection to (x1, x2)


−15 −10 −5 0 5 10 150

5

10

15

20

25

30

35

x1

x 3

Figure 2.7 Projection to (x1, x3)

0 20 40 60 80 100−15

−10

−5

0

5

10

15

x 1

Time (s)

Figure 2.8 Time trajectory of x1


It can be established that the trajectories converge to a bounded region specified by

(λ+ 1)x21 + σx2

2 + σ (x3 − 2(λ+ 1))2 ≤ C

for a positive constant C. When all the three equilibria are unstable, the behaviour ofLorenz system is chaotic. A trajectory is plotted in Figure 2.5 for b = 2, σ = 10 andλ = 20. Projections to (x1, x2) and (x1, x3) are shown in Figures 2.6 and 2.7. The timetrajectory of x1 is shown in Figure 2.8. �

Chapter 3

Describing functions

In classical control, frequency response is a powerful tool for analysis and controldesign of linear dynamic systems. It provides graphical presentation of system dynam-ics and often can reflect certain physical features of engineering systems. The basicconcept of frequency response is that for a linear system, if the input is a sinusoidalfunction, the steady-state response will still be a sinusoidal function, but with adifferent amplitude and a different phase. The ratio of the input and output ampli-tudes and the difference in the phase angles are determined by the system dynamics.When there is a nonlinear element in a control loop, frequency response methodscannot be directly applied. When a nonlinear element is a static component, i.e., theinput and output relationship can be described by an algebraic function, its outputto any periodic function will be a periodic function, with the same period as theinput signal. Hence, the output of a static nonlinear element is a periodic functionwhen the input is a sinusoidal function. It is well known that any periodic func-tion with piece-wise continuity has its Fourier series which consists of sinusoidalfunctions with the same period or frequency as the input with a constant bias, andother sinusoidal functions with high multiple frequencies. If we take the term withthe fundamental frequency, i.e., the same frequency as the input, as an approxima-tion, the performance of the entire dynamic system may be analysed using frequencyresponse techniques. Describing functions are the frequency response functions ofnonlinear components with their fundamental frequency terms as their approximateoutputs. In this sense, describing functions are first-order approximation in frequencydomain. It can also be viewed as a linearisation method in frequency domain fornonlinear components.

Describing function analysis remains as an important tool for analysis of non-linear systems with static components despite several more recent developments innonlinear control and design. It is relatively easy to use, and closely related to fre-quency response analysis of linear systems. It is often used to predict the existenceof limit cycles in a nonlinear system, and it can also be used for prediction of subhar-monics and jump phenomena of nonlinear systems. In this chapter, we will presentbasic concept of describing functions, calculation of describing functions of commonnonlinear elements and how to use describing functions to predict the existence oflimit cycles.


3.1 FundamentalsFor a nonlinear component described by a nonlinear function f : R → R, its

f (x)A sin(wt) w(t)

output

w(t) = f (A sin(ωt))

to a sinusoidal input A sin (ωt) is a periodical function, although it may not be sinu-soidal in general. Assuming that the function f is piecewise-continuous, w(t) is apiecewise-continuous periodic function with the same period as the input signal. Apiecewise periodical function can be expanded in Fourier series

w(t) = a0

2+

∞∑

n=1

(an cos(nωt) + bn sin(nωt)), (3.1)

where

a0 = 1

π

∫ π

−πw(t)d(ωt)

an = 1

π

∫ π

−πw(t) cos(nωt)d(ωt)

bn = 1

π

∫ π

−πw(t) sin(nωt)d(ωt).

Remark 3.1. For a piecewise-continuous function w(t), the Fourier series on theright-hand side of (3.1) converges to w(t) at any continuous point, and to the averageof two values obtained by taking limits from both sides at a dis-continuous point. Ifwe truncate the series up to order k ,

wk (t) = a0

2+

k∑

n=1

(an cos(nωt) + bn sin(nωt)),

where wk is the best approximation in least squares, i.e., in L2. �

Taking the approximation to the first order, we have

w1 = a0

2+ a1 cos(ωt) + b1 sin(ωt).

If a0 = 0, which can be guaranteed by setting the nonlinear function f to an oddfunction, we have the approximation

w1 = a1 cos(ωt) + b1 sin(ωt) (3.2)

which is an approximation at the fundamental frequency. The above discussion showsthat for a nonlinear component described by the nonlinear function f , the approxima-tion at the fundamental frequency, i.e., the frequency of the input signal, to an input

Describing functions 27

signal A sin(ωt), is a sinusoidal function in (3.2) with the Fourier coefficients a1 andb1 shown in (3.1). Hence, we can analyse the frequency response of this nonlinearcomponent.

We can rewrite w1 in (3.2) as

w1 = M sin(ωt + φ), (3.3)

where

M (A,ω) =√

a21 + b2

1,

φ(A,ω) = arctan(a1/b1).

In complex expression, we have

w1 = Mej(ωt+φ) = (b1 + ja1)ejωt .

The describing function is defined, similar to frequency response, as the complex ratioof the fundamental component of the nonlinear element against the input by

N (A,ω) = Mejωt+φ

Aejωt= b1 + ja1

A. (3.4)

Remark 3.2. A clear difference between the describing function of a nonlinear ele-ment and the frequency response of a linear system is that the describing functiondepends on the input amplitude. This reflects the nonlinear nature of the describingfunction. �

Remark 3.3. If f is a single-valued odd function, i.e., f (−x) = f (x), we have

a1 = 1

π

∫ π

−πf (A sin(ωt)) cos(ωt)d(ωt)

= 1

π

∫ 0

−πf (A sin(ωt) cos(ωt)d(ωt) + 1

π

∫ π

0f (A sin(ωt) cos(ωt)d(ωt)

= 1

π

∫ π

0f (A sin(−ωt) cos(−ωt)d(ωt) + 1

π

∫ π

0f (A sin(ωt) cos(ωt)d(ωt)

= 0.

If a1 = 0, the describing function is a real value. �Example 3.1. The characteristics of a hardening spring are given by

f (x) = x + x3

2.

Given the input A sin(ωt), the output is

w(t) = f (A sin(ωt))

= A sin(ωt) + A3

2sin3(ωt).


Since f is an odd function, we have a1 = 0. The coefficient b1 is given by

b1 = 1

π

∫ π

−π

[

A sin(ωt) + A3

2sin3(ωt)

]

sin(ωt)d(ωt)

= 4

π

∫ π/2

0

[

A sin2(ωt) + A3

2sin4(ωt)

]

d(ωt).

Using the integral identity

∫ π/2

0sinn(ωt)d(ωt) = n − 1

n

∫ π/2

0sinn−2(ωt)d(ωt) for n > 2,

we have

b1 = A + 3

8A3.

Therefore, the describing function is

N (A,ω) = N (A) = b1

A= 1 + 3

8A2.

Alternatively, we can also use the identity

sin(3ωt) = 3 sin(ωt) − 4 sin3(ωt)

to obtain

w(t) = A sin(ωt) + A3

2sin3(ωt)

= A sin(ωt) + A3

2

(3

4sin(ωt) − 1

4sin(3ωt)

)

=(

A + 3

8A3

)

sin(ωt) − 1

8A3 sin(3ωt).

Hence, we obtain b1 = A + 38 A3 from the first term. �

Through the above discussion, describing functions are well defined for nonlinearcomponents whose input–output relationship can be well defined by piecewise-continuous functions. These functions are time-invariant, i.e., the properties ofnonlinear elements do not vary with time. This is in line with the assumption for fre-quency response analysis, which can only be applied to time-invariant linear systems.We treat describing functions as the approximations at the fundamental frequencies,and therefore in our analysis, we require a0 = 0 which is guaranteed by odd functionsfor the nonlinear components. With the describing function of a nonlinear compo-nent, we can then apply analysis in frequency responses for the entire system. For theconvenience of this kind of analysis, we often assume that the nonlinear componentfor which the describing function is used to approximate its behaviours is the only


r x w yf (x) G(s)Σ

–

+

Figure 3.1 Block diagram for describing function analysis

nonlinear component in the system, as shown in Figure 3.1. Hence, in the remain-ing part of this chapter, we use the following assumptions for describing functionanalysis:

● There is only a single nonlinear component in the entire system.● The nonlinear component is time-invariant.● The nonlinearity is odd.

3.2 Describing functions for common nonlinear components

In this section, we will calculate the describing functions of common nonlinearelements in a number of examples.

Example 3.2. Saturation. A saturation function shown in Figure 3.2 is described by

f (x) ={

kx, for |x| < a,

sign(x)ka, otherwise.(3.5)

The output to the input A sin(ωt), for A > a, is symmetric over quarters of a period,and in the first quarter,

f(x)

a–a

x

Figure 3.2 Saturation

w(t) ={

kA sin(ωt), 0 ≤ ωt ≤ γ ,

ka, γ < ωt ≤ π/2,(3.6)

where γ = sin−1(a/A). The function is odd, hence we have a1 = 0, and the symmetryof w1(t) implies that


b1 = 4

π

∫ π/2

0w1 sin(ωt)d(ωt)

= 4

π

∫ γ

0kA sin2(ωt)d(ωt) + 4

π

∫ π/2

γ

ka sin(ωt)d(ωt)

= 2kA

π

(

γ − 1

2sin(2γ )

)

+ 4ka

πcos(γ )

= 2kA

π

(γ − a

Acos(γ )

)+ 4ka

πcos(γ )

= 2kA

π

(γ + a

Acos(γ )

)

= 2kA

π

(

γ + a

A

√

1 − a2

A2

)

.

Note that we have used sin γ = aA and cos γ =

√

1 − a2

A2 . Therefore, thedescribing function is given by

N (A) = b1

A= 2k

π

(

sin−1 a

A+ a

A

√

1 − a2

A2

)

. (3.7)

�

Example 3.3. Ideal relay. The output from the ideal relay shown in Figure 3.3(signum function) is described by, with M > 0,

w(t) ={−M , −π ≤ ωt < 0,

M , 0 ≤ ωt < π.(3.8)

It is again an odd function, hence we have a1 = 0. The coefficient b1 is given by

b1 = 2

π

∫ π

0M sin(ωt)d(ωt) = 4M

π

f(x)

x

M

–M

Figure 3.3 Ideal relay


f(x)

a–a

x

Figure 3.4 Dead zone

and therefore the describing function is given by

N (A) = 4M

πA. (3.9)

�

Example 3.4. Dead zone. A dead zone is a complement to saturation. A dead zoneshown in Figure 3.4 can be described by a nonlinear function

f (x) =

⎧⎪⎨

⎪⎩

k(x − a), for x > a,

0, for |x| < a,

k(x + a), for x < −a.

(3.10)

The output to the input A sin(ωt), for A > a, is symmetric over quarters of aperiod, and in the first quarter,

w(x) ={

0, 0 ≤ ωt ≤ γ ,

k(A sin(ωt) − a), γ < ωt ≤ π/2,(3.11)

where γ = sin−1(a/A). The function is odd, hence we have a1 = 0, and the symmetryof w(t) implies that

b1 = 4

π

∫ π/2

0w(t) sin(ωt)d(ωt)

= 4

π

∫ π/2

γ

k(A sin(ωt) − a)d(ωt)

= 2kA

π

((π

2− γ

)+ 1

2sin(2γ )

)

− 4ka

πcos(γ )

= kA − 2kA

π

(γ + a

Acos(γ )

)

= kA − 2kA

π

(

γ + a

A

√

1 − a2

A2

)

.


Similar to the calculation of the describing function for saturation, we have used

sin γ = aA and cos γ =

√

1 − a2

A2 . The describing function for a dead zone is given by

N (A) = b1

A= k − 2k

π

(

sin−1 a

A+ a

A

√

1 − a2

A2

)

. (3.12)

�

Remark 3.4. The dead-zone function shown in (3.10) complements the saturationfunction shown in (3.5) in the sense that if we use fs and fd to denote the saturationfunction and dead-zone function, we have fs + fd = k for the describing functionsshown in (3.7) and (3.12), the same relationship holds. �

Example 3.5. Relay with hysteresis. Consider a case when there is a delay in theideal relay as shown in Figure 3.5. The nonlinear function for relay with hysteresiscan be described by

f (x) =

⎧⎪⎪⎨

⎪⎪⎩

M , for x ≥ a,−M , for |x| < a, x > 0,M , for |x| < a, x < 0,−M , for x ≤ −a.

(3.13)

When this nonlinear component takes A sin (ωt) as the input with A > a, theoutput w(t) is given by

w(t) =⎧⎨

⎩

M , for −π ≤ ωt < (π − γ ),−M , for −(π − γ ) ≤ ωt < γ ,M , for γ ≤ ωt < π ,

(3.14)

where γ = sin−1( aA ). In this case, we still have a0 = 0, but not a1. For a1 we have

f(x)

x

M

–M

Figure 3.5 Relay with hysteresis


a1 = 1

π

∫ −(π−γ )

−πM cos(ωt)d(ωt) + 1

π

∫ γ

−(π−γ )− M cos(ωt)d(ωt)

+ 1

π

∫ π

γ

M cos(ωt)d(ωt)

= − 4M

πsin(γ )

= − 4M

π

a

A.

Similarly, we have

b1 = 1

π

∫ −(π−γ )

−πM sin(ωt)d(ωt) + 1

π

∫ γ

−(π−γ )− M sin(ωt)d(ωt)

+ 1

π

∫ π

γ

M sin(ωt)d(ωt)

= 4M

πcos(γ )

= 4M

π

√

1 − a2

A2.

From

N (A,ω) = b1 + ja1

A,

we have

N (A) = 4M

πA

(√

1 − a2

A2− j

a

A

)

. (3.15)

Using the identity cos(γ ) + j sin(γ ) = ejr , we can rewrite the describing function as

N (A) = 4M

πAe−j arcsin(a/A).

�

Remark 3.5. Comparing the describing function of the relay with hysteresis withthat of ideal relay in (3.9), the describing functions indicate that there is a delay in therelay with hysteresis by arcsin(a/A) in terms of phase angle. There is indeed a delayof γ = arcsin(a/A) in the time response w(t) shown in (3.14) with that of the idealrelay. In fact, we could use this fact to obtain the describing function for the relaywith hysteresis. �


3.3 Describing function analysis of nonlinear systems

One of the most important applications of describing functions is to predict the exis-tence of a limit cycle in a closed-loop system that contains a nonlinear componentwith a linear transfer function, as shown in Figure 3.1. Consider a system with a lin-ear transfer function G(s) and a nonlinear element with describing function N (A,ω)in the forward path, under unit feedback. The input–output relations of the systemcomponent by setting r = 0 can be described by

w = N (A,ω)x

y = G(jω)w

x = −y,

with y as the output and x as the input to the nonlinear component. From the aboveequations, it can be obtained that

y = G(jω)N (A,ω)(−y),

and it can be arranged as

(G(jω)N (A,ω) + 1)y = 0.

If there exists a limit cycle, then y �= 0, which implies that

G(jω)N (A,ω) + 1 = 0, (3.16)

or

G(jω) = − 1

N (A,ω). (3.17)

Therefore, the amplitude A and frequency ω of the limit cycle must satisfy the aboveequation. Equation (3.17) is difficult to solve in general. Graphic solutions can befound by plotting G(jω) and −1/N (A,ω) on the same graph to see if they intersecteach other. The intersection points are the solutions, from which the amplitude andfrequency of the oscillation can be obtained.

Remark 3.6. The above discussion is based on the assumption that the oscillation,or limit cycle, can be well approximated by a sinusoidal function, and the nonlinearcomponent is well approximated by its describing function. The describing functionanalysis is an approximate method in nature. �

Only a stable limit cycle may exist in real applications. When we say stable limitcycle, we mean that if the state deviates a little from the limit cycle, it should comeback. With the amplitude as an example, if A is perturbed from its steady condition,say with a very small increase in the amplitude, for a stable limit cycle, the systemwill decay to its steady condition.


As describing functions are first-order approximations in the frequency domain,stability criteria in the frequency domain may be used for the stability analysis oflimit cycles. Nyquist criterion can be extended to give the conditions for stability oflimit cycles.

Recall the case for a linear system with the forward transfer function G(s) withunit feedback. The characteristic equation is given by

G(s) + 1 = 0, or G(s) = −1.

The Nyquist criterion determines stability of the closed-loop system from the numberof encirclements of the Nyquist plot around point −1, or (−1, 0) in the complex plain.

In the case that there is a control gain K in the forward transfer function, thecharacteristic equation is given by

KG(s) + 1 = 0, or G(s) = −1

K.

In this case, the Nyquist criterion can be extended to determine the stability of theclosed loop by counting the encirclements of the Nyquist plot around (−1/K , 0) in thecomplex plain in the same way as around (−1, 0). The Nyquist criterion for non-unityforward path gain K is also referred to as the extended Nyquist criterion. The sameargument holds when k is a complex number.

We can apply the extended Nyquist criterion to determine the stability of a limitcycle. When the condition specified in (3.17) is satisfied for some (A0,ω0), A0 andω0 are the amplitude and frequency of the limit cycle respectively, and N (A0,ω0)is a complex number. We can use the extended Nyquist criterion to determine thestability of the limit cycle with the amplitude A0 and frequency ω0 by considering aperturbation of A around A0.

To simplify our discussion, let us assume that G(s) is stable and minimum phase.It is known from the Nyquist criterion that the closed-loop system with constantgain K is stable if the Nyquist plot does not encircle (−1/K , 0). Let us consider aperturbation in A to A+ with A+ > A0. In such a case, −1/N (A+,ω0) is a complexnumber in general. If the Nyquist plot does not encircle the point −1/N (A+,ω0), weconclude that the closed-loop system is stable with the complex gain −1/N (A+,ω0).Therefore, in a stable closed-loop system, the oscillation amplitude decays, whichmakes A+ return to A0. This implies that the limit cycle (A0,ω0) is stable. Alternatively,if the Nyquist plot encircles the point −1/N (A+,ω0), we conclude that the closed-loop system is unstable with the complex gain −1/N (A+,ω0). In such a case, theoscillation amplitude may grow even further, and does not return to A0. Therefore,the limit cycle is unstable.

Similar arguments can be made for the perturbation to a smaller amplitude. Foran A− < A0, if the Nyquist plot does encircle the point −1/N (A−,ω0), the limit cycleis stable. If the Nyquist plot does not encircle the point −1/N (A−,ω0), the limit cycleis unstable.

When we plot −1/N (A,ω0) in the complex plane with A as a variable, we obtaina line with direction of the increment of A. Based on the discussion above, the way


Im

Re

Stable

1–

A–

A+

N(A,w0)

G( jw )

Figure 3.6 Digram for stable limit cycle

Im

Re

Unstable1–

A–A+

N(A,w0)

G( jw )

Figure 3.7 Digram for unstable limit cycle

of the line for −1/N (A,ω0) intersects with the Nyquist plot determines the stabilityof the limit cycle. Typical Nyquist plots of stable minimum-phase systems are shownin Figures 3.6 and 3.7 for stable and unstable limit cycles with nonlinear elementsrespectively.

We can summarise the above discussion for the stability criterion of limit cyclesusing describing function.

Theorem 3.1. Consider a unity-feedback system with the forward path with stableminimum phase transfer function G(s) and a nonlinear component with the describingfunction N (A,ω), and suppose that the plots, −1/N and G(jω) intersect at the pointwith A = A0 andω = ω0. The limit cycle at (A0,ω0) is stable if the plot of −1/N (A,ω0)crosses the Nyquist plot from the inside of the encirclement to the outside of theencirclement as A increases. The limit cycle at (A0,ω0) is unstable if the plot of−1/N (A,ω0) crosses the Nyquist plot from the outside of the encirclement to theinside of the encirclement as A increases.

Remark 3.7. Theorem 3.1 requires the transfer function to be stable and minimumphase, for the simplicity of the presentation. This theorem can be easily extended to


the case when G(s) is unstable or has unstable zeros by using corresponding stabilityconditions based on the Nyquist criterion. For example if G(s) is stable and has oneunstable zero, then the stability criterion for the limit cycle will be opposite to thecondition stated in the theorem, i.e., the limit cycle is stable if the plot of −1/N (A,ω0)crosses the Nyquist plot from the outside of the encirclement to the inside of theencirclement as A increases. �

Example 3.6. Consider a linear transfer function G(s) = K

s(s + 1)(s + 2)with K a

positive constant and an ideal relay in a closed loop, as shown in Figure 3.8. We willdetermine if there exists a limit cycle and analyse the stability of the limit cycle.

r x w yΣ

–

+ Ks(s + 1)(s + 2)

Figure 3.8 Closed-loop system for Example 3.6

For the ideal relay, we have N = 4M

πA. For the transfer function, we can

obtain that

G(jω) = K

jω(jω + 1)(jω + 2)

= K−3ω2 − jω(2 − ω2)

(−3ω2)2 + ω2(2 − ω2)2.

From

G(jω) = − 1

N,

we obtain two equations for real and imaginary parts respectively as

�(G(jω)) = 0,

(G(jω)) = − πA

4M.

From the equation of the imaginary part, we have

K−ω(2 − ω2)

(−3ω2)2 + ω2(2 − ω2)2= 0,

which gives ω = √2. From the equation of the real part, we have

K−3ω2

(−3ω2)2 + ω2(2 − ω2)2= − πA

4M,

which gives A = 2KM/3π .


Hence, we have shown that there exists a limit cycle with amplitude and frequencyat (A, ω) = (2KM/3π ,

√2).

The plot of −(1/N (A)) = −(πA/4M ) overlaps with the negative side of the realaxis. As A increases from 0, −(1/N (A)) moves from the origin towards left. Therefore,as A increases, −(1/N (A)) moves from inside of the encirclement of the Nyquist plotto outside of the encirclement, and the limit cycle is stable, based on Theorem 3.1.

A simulation result for K = M = 1 is shown in Figure 3.9 with the amplitudeA = 0.22 and period T = 4.5 s, not far from the values A = 0.2212 and T = 4.4429,predicted from the describing function analysis. �Example 3.7. In this example, we consider a van der Pol oscillator described by

y + ε(3y2 − 1)y + y = 0. (3.18)

We will use describing function analysis to predict the existence of a limit cycle, andcompare the predicted amplitudes and periods for different ε values with the simulatedones.

To use the describing analysis, we need to formulate the system in the format ofone linear transfer function and a nonlinear element. Rearranging (3.18), we have

y − εy + y = −ε d

dty3.

34 36 38 40 42 44 46−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time (s)

y

Figure 3.9 The simulated output for Example 3.6


Hence, the system (3.18) can be described by a closed-loop system with a nonlinearcomponent

f (x) = x3

and a linear transfer function

G(s) = εs

s2 − εs + 1.

Using the identity

sin(3ωt) = 3 sin(ωt) − 4 sin3(ωt)

in a similar way as in Example 3.1, we obtain the describing function forf (x) = x3 as

N (A) = 3

4A2.

Setting

�G(jω) = −�(

1

N

)

= 0,

we have

εω(1 − ω2)

(1 − ω2)2 + ε2ω2= 0,

which gives ω = 1. From the equation for the real part, we obtain

−ε2ω2

(1 − ω2)2 + ε2ω2= − 4

3A2

which gives A = 2√

3/3.The linear part of the transfer function has one unstable pole. We need to take

this into consideration for the stability of the limit cycle. As A increases, −1/N (A)moves from the left to the right along the negative part of the real axis, basicallyfrom the outside of the encirclement of the Nyquist plot to the inside of the encir-clement. This suggests that the limit cycle is stable, as there is an unstable pole inthe linear transfer function. The simulation results for ε = 1 and ε = 30 are shown inFigures 3.10 and 3.11. In both cases, the amplitudes are very close to the predictedone from the describing function analysis. For the period, the simulation result forε = 1 in Figure 3.10 is very close to 2π , but the period for ε = 30 is much better than2π . This suggests that the describing function analysis gives a better approximationfor the case of ε = 1 than ε = 30. In fact, for a small value of ε the oscillation isvery similar to a sinusoidal function. With a big value of ε, the wave form is verydifferent from a sinusoidal function, and therefore the describing function methodcannot provide a good approximation. �


10 15 20 25 30 35

−1

−0.5

0

0.5

1

Time (s)

y

Figure 3.10 The simulated output for Example 3.7 with ε = 1

30 40 50 60 70 80 90 100 110 120

−1

−0.5

0

0.5

1

y

Figure 3.11 The simulated output for Example 3.7 with ε = 30

Chapter 4

Stability theory

For control systems, design, one important objective is to ensure the stability of theclosed-loop system. For a linear system, the stability can be evaluated in time domainor frequency domain, by checking the eigenvalues of the system matrix or the polesof the transfer function. For nonlinear systems, the dynamics of the system cannot bedescribed by equations in linear state space or transfer functions in general. We needmore general definitions about the stability of nonlinear systems. In this chapter, wewill introduce basic concepts of stability theorems based on Lyapunov functions.

4.1 Basic definitions

Consider a nonlinear system

x = f (x), (4.1)

where x ∈ D ⊂ Rn is the state of the system, and f : D ⊂ R

n −→ Rn is a continuous

function, with x = 0 as an equilibrium point, that is f (0) = 0, and with x = 0 as aninterior point of D. Here we use D to denote a domain around the equilibrium x = 0.This domain can be interpreted as a set with 0 as its interior point, or it can alsobe simplified as D = {x|‖x‖ < r} for some positive r. In the remaining part of thischapter, we will use D in this way.

When we say the stability of the system, we refer to the behaviour of the systemaround the equilibrium point. Here we assume that x = 0 is an equilibrium pointwithout loss of generality. In case that the system has an equilibrium point at x0, wecan always define a state transformation with x − x0 as the new state, to shift theequilibrium point to the origin.

The system in (4.1) is referred to as an autonomous system, as it does not dependon the signals other than the system state. For nonlinear control systems, we can write

x = f (x, u), (4.2)

where u ∈ Rm is the control input. With u as an external signal, the system (4.2) is

not autonomous. However, for such a system, if we design a feedback control lawu = g(x) with g : R

n −→ Rm as a continuous function, the closed-loop system

x = f (x, g(x))

becomes an autonomous system.


In this chapter, we will present basic definitions and results for stability ofautonomous systems. As discussed above, control systems can be converted toautonomous systems by state feedback control laws.

There are many different definitions of stability for dynamics systems. Oftendifferent definitions are needed for different purposes, and many of them are actuallythe same when the system is linear. Among different definitions, the most fundamentalone is the Lyapunov stability.

Definition 4.1 (Lyapunov stability). For the system (4.1), the equilibrium point x = 0is said to be Lyapunov stable if for any given positive real number R, there exists apositive real number r to ensure that ‖x(t)‖ < R for all t ≥ 0 if ‖x(0)‖ < r. Otherwisethe equilibrium point is unstable.

x(t)

x(0)

r R

Figure 4.1 Lyapunov stability

The definition of Lyapunov stability concerns with the behaviours of a dynamicsystem with respect to the initial state. If a system is Lyapunov stable, we can imposea restriction on the initial state of the system to make sure that the state variables stayin a certain region. For the two positive numbers in the definition, R and r, the defi-nition did not explicitly require R ≥ r. However, if we set r > R, from the continuityof the solution, we cannot ensure ‖x(t)‖ < R for t close to 0, because of ‖x(0)‖ > R.Therefore, when using this definition, we need r ≤ R.

Example 4.1. Consider a linear system

x = Ax, x(0) = x0,

where

A =[

0 ω

−ω 0

]

with ω > 0. For this linear system, we can explicitly solve the differential equationto obtain

x(t) =[

cosωt sinωt−sinωt cosωt

]

x0.

Stability theory 43

It is easy to check that we have ‖x(t)‖ = ‖x0‖. Hence, to ensure that ‖x(t)‖ ≤ R, weonly need to set r = R, i.e., if ‖x0‖ ≤ R, we have ‖x(t)‖ ≤ R for all t > 0. �

Note that for the system in Example 4.1, the system matrix has two eigenvalueson the imaginary axis, and this kind of systems is referred to as critically stable inmany undergraduate texts. As shown in the example, this system is Lyapunov stable.It can also be shown that for a linear system, if all the eigenvalues of the systemmatrix A are in the closed left half of the complex plane, and the eigenvalues on theimaginary axis are simple, the system is Lyapunov stable. However, if the systemmatrix has multiple poles on the imaginary axis, the system is not Lyapunov stable.For example let x1 = x2, and x2 = 0 with x1(0) = x1,0, x2(0) = x2,0. It is easy to obtainthat x1(t) = x1,0 + x2,0t and x2(t) = x2,0. If we want ‖x(t)‖ ≤ R, there does not exista positive r for ‖x(0)‖ ≤ r to guarantee ‖x(t)‖ ≤ R. Therefore, this system is notLyapunov stable.

For linear systems, when a system is stable, the solution will converge to the equi-librium point. This is not required by Lyapunov stability. For more general dynamicsystems, we have the following definition concerning with the convergence to theequilibrium.

Definition 4.2 (Asymptotic stability). For the system (4.1), the equilibrium pointx = 0 is asymptotically stable if it is stable (Lyapunov) and furthermore limt→∞x(t) = 0.

x(t)

x(0)

r R

Figure 4.2 Asymptotic stability

Linear systems with poles in the open left half of the complex plane are asymp-totically stable. The asymptotic stability only requires that a solution converges theequilibrium point, but it does not specify the rate of convergence. In the followingdefinition, we specify a stability property with an exponential rate of convergence.

Definition 4.3 (Exponential stability). For the system (4.1), the equilibrium pointx = 0 is exponentially stable if there exist two positive real numbers a and λ such thatthe following inequality holds:

‖x(t)‖ < a‖x(0)‖e−λt (4.3)

for t > 0 in some neighbourhood D ⊂ Rn containing the equilibrium point.


For linear systems, the stability properties are relatively simple. If a linear systemis asymptotically stable, it can be shown that it is exponentially stable. Of course,for nonlinear systems, we may have a system that is asymptotically stable, but notexponentially stable.

Example 4.2. Consider a nonlinear system

x = −x3, x(0) = x0 > 0,

where x ∈ R. Let us solve this differential equation. From the system equation wehave

−dx

x3= dt,

which gives

1

x2(t)− 1

x20

= 2t

and

x(t) = x0√

1 + 2x20t.

It is easy to see that x(t) decreases as t increases, and also limt→∞ x(t) = 0. Therefore,this system is asymptotically stable. However, this system is not exponentially stable,as there does not exist a pair of a and γ to satisfy

x0√

1 + 2x20t

≤ ax0e−γ t .

Indeed, if there exist such constants a and γ , we have

√

1 + 2x20te−γ t ≥ 1

a,

which is not satisfied for any choices of a and γ , because the left-hand side convergesto zero. Hence, the system considered in this example is asymptotically stable, butnot exponentially stable. �

Lyapunov, asymptotic and exponential, stabilities are defined around equilibriumpoints. If the properties hold for any initial points in the entire state space, they are

Stability theory 45

referred to as global stability properties. In the following definitions, we give theirglobal versions.

Definition 4.4 (Globally asymptotic stability). If the asymptotic stability defined inDefinition 4.2 holds for any initial state in R

n, the equilibrium point is said to beglobally asymptotically stable.

Definition 4.5 (Globally exponential stability). If the exponential stability definedin Definition 4.3 holds for any initial state in R

n, the equilibrium point is said to beglobally exponentially stable.

The stability property discussed in Example 4.2 is globally and asymptoticallystable. In the two examples shown in this section, the stability properties are checkedbased on the actual solutions of the systems. In general, explicit solutions of nonlinearsystems are difficult to obtain, and it is expected to check the stability properties ofnonlinear systems without knowing the solutions. In the later part of this chapter, wewill show a number of results to establish stability properties without their solutions.

4.2 Linearisation and local stability

In this section, we introduce a result for checking the stability of nonlinear systemsbased on its linearised model.

Theorem 4.1 (Lyapunov’s linearisation method). For a linearised model, there arethree cases:

● If the linearised system has all the system’s poles in the open left half of the complexplane, the equilibrium point is asymptotically stable for the actual nonlinearsystem.

● If the linearised system has poles in the open right half of the complex plane, thenthe equilibrium point is unstable.

● If the linearised system has poles on the imaginary axis, then the stability of theoriginal system cannot be concluded using the linearised model.

We do not show a proof of this theorem here. It is clear that this theorem canbe applied to check local stabilities of nonlinear systems around equilibrium points.For the case that the linearised model has poles on the imaginary axis, this theoremcannot give conclusive result about the stability. This is not a surprise, because stableand unstable systems can have the same linearised model. For example the systemsx = −x3 and x = x3 have the same linearised model at x = 0, that is x = 0, which ismarginally stable. However, as we have seen in Example 4.2, the system x = −x3 isasymptotically stable, and it is not difficult to see that x = x3 is unstable. For both thestable and unstable cases of linearised models, the linearised model approximates theoriginal system better when the domain around the equilibrium point gets smaller.Hence, the linearised model is expected to reflect on the stability behaviours aroundthe equilibrium point.



x1 = x2 + x1 − x31,

x2 = −x1.

It can be seen that x = (0, 0) is an equilibrium point of the system. The linearisedmodel around x = (0, 0) is given by

x = Ax

where

A =[

1 1−1 0

]

The linearised system is unstable as λ(A) = 1±√3j

2 . Indeed, this nonlinear system is avan der Pols system, and the origin is unstable. Any trajectories that start from initialpoint close to the origin and within the limit cycle will spiral out, and converge to thelimit cycle. �

4.3 Lyapunov’s direct method

Lyapunov’s linearisation method can only be used to check local stabilities, and alsothere is a limitation in the case of marginal stability. Fortunately, there is a directmethod to check the stability of dynamic systems. This method is based on Lyapunovfunctions. We need a few definitions before we can show some of the results onstability based on Lyapunov functions.

Definition 4.6 (Positive definite function). A function V (x) : D ⊂ Rn → R is said to

be locally positive definite if V (x) > 0 for x ∈ D except at x = 0 where V (0) = 0. IfD = R

n, i.e., the above property holds for the entire state space, V (x) is said to beglobally positive definite.

There are many examples of positive definite functions, such as xT Px for P beinga positive definite matrix, or even ‖x‖.

Definition 4.7 (Lyapunov function). If in D ⊂ Rn containing the equilibrium point

x = 0, the function V (x) is positive definite and has continuous partial derivatives,and if its time derivative along any state trajectory of system (4.1) is non-positive,i.e.,

V (x) ≤ 0 (4.4)

then V (x) is a Lyapunov function.

Stability theory 47

Stability analysis based on a Lyapunov function is probably the most commonlyused method to establish the stability of nonlinear dynamic systems. A fundamentaltheorem on Lyapunov function is given below.

Theorem 4.2 (Lyapunov theorem for local stability). Consider the system (4.1). Ifin D ⊂ R

n containing the equilibrium point x = 0, there exists a function V (x) :D ⊂ R

n → R with continuous first-order derivatives such that

● V (x) is positive definite in D● V (x) is non-positive definite in D

then the equilibrium point x = 0 is stable. Furthermore, if V (x) is negative definite,i.e., −V (x) is positive definite in D, then the stability is asymptotic.

Proof. We need to find a value for r such that when ‖x(0)‖ < r, we have ‖x(t)‖ < R.Define

BR := {x|‖x‖ ≤ R} ⊂ D,

and let

a = min‖x‖=R

V (x).

Since V (x) is positive definite, we have a > 0. We then define the level set within BR

�c := {x ∈ BR|V (x) < c},where c is a positive real constant and c < a. The existence of such a positive realconstant c is guaranteed by the continuity and positive definiteness of V . From thedefinition of �c, x ∈ �c implies that ‖x‖ < R. Since V ≤ 0, we have V (x(t)) ≤V (x(0)). Hence, for any x(0) ∈ �c, we have

V (x(t)) ≤ V (x(0)) < c,

which implies

‖x(t)‖ < R.

Since �c contains 0 as an interior point, and V is a continuous function, theremust exist a positive real r such that

Br := {x|‖x‖ < r} ⊂ �c.

Hence, we have

Br ⊂ �c ⊂ BR.


Therefore, for any x(0) ∈ Br , we have

V (x(t)) ≤ V (x(0)) < c,

and ‖x(t)‖ < R. We have established that if V is non-positive, the system is Lyapunovstable.

Next, we will establish the asymptotic stability from the negative definiteness ofV . For any initial point in D, V (x(t)) monotonically decreases with time t. Therefore,there must be a lower limit such that

limt→∞ V (x(t)) = β ≥ 0.

The asymptotic stability can be established if we can show that β = 0. We canprove it by seeking a contradiction. Suppose β > 0. Let

α = minx∈D−�β

(−V (x)),

where �β := {x ∈ D|V (x) < β}. Since V is negative definite, we have α > 0. Fromthe definition of α, we have

V (x(t)) ≤ V (x(0)) − αt.

The right-hand side turns to negative when t is big enough, which is a contradiction.Therefore, we can conclude that limt→∞ V (x(t)) = 0, which implies limt→∞ x(t) = 0.

�

Example 4.4. A pendulum can be described by

θ + θ + sin θ = 0,

where θ is the angle. If we let x1 = θ and x2 = θ , we re-write the dynamic system as

x1 = x2

x2 = −sin x1 − x2.

Consider the scalar function

V (x) = (1 − cos x1) + x22

2. (4.5)

The first term (1 − cos x1) in (4.5) can be viewed as the potential energy and the

second termx2

22 as the kinetic energy. This function is positive definite in the domain

D = {|x1| ≤ π , x2 ∈ R}. A direct evaluation gives

Stability theory 49

V (x) = −sin x1x1 + x2x2

= −x22.

Hence, the system is stable at x = 0. However, we cannot conclude the asymptoticstability of the system from Theorem 4.2. This system is in fact asymptotically stableby using more advanced stability theorem such as invariant set theorem, which is notcovered in this book. �

When establishing global stability using Lyapunov functions, we need the func-tion V (x) to be unbounded as x tends to infinity. This may sound strange. The reasonbehind this point is that we need the property that if V (x) is bounded, then x isbounded, in order to conclude the boundedness of x from the boundedness of V (x).This property is defined in the following function as the radial unboundedness of V .

Definition 4.8 (Radially unbounded function). A positive definite function V (x) :R

n → R is said to be radially unbounded if V (x) → ∞ as ‖x‖ → ∞.

Theorem 4.3 (Lyapunov theorem for global stability). For the system (4.1) with D =R

n, if there exists a function V (x) : Rn → R with continuous first order derivatives

such that

● V (x) is positive definite● V (x) is negative definite● V (x) is radially unbounded

then the equilibrium point x = 0 is globally asymptotically stable.

Proof. The proof is similar to the proof of Theorem 4.2, except that for any givenpoint in R

n, we need to show that there is a level set defined by

�c = {x ∈ Rn|V (x) < c}

to contain it.Indeed, since the function V is radially unbounded, for any point in Br with any

positive real r, there exists a positive real constant c such that Br ⊂ �c. It is clearthat the level set �c is invariant for any c, that is, for any trajectory that starts in �c

remains in �c. The rest of the proof follows the same argument as in the proof ofTheorem 4.2. �

Example 4.5. Consider the nonlinear system

x = −x3, x(0) = x0 > 0,


where x ∈ R. In Example 4.2, we have shown that the equilibrium point x = 0 isasymptotically stable by checking the solution of the differential equation. In thisexample, we use a Lyapunov function.

Let

V = 1

2x2

and it is easy to see that this function is globally positive definite. Its derivative isgiven by

V = −x4

which is negative definite. Hence, from Theorem 4.3, we conclude x = 0 isasymptotically stable. �

To conclude this section, we have another result for exponential stability.

Theorem 4.4 (Exponential stability). For the system (4.1), if there exists a functionV (x) : D ⊂ R

n → R with continuous first-order derivatives such that

a1‖x‖b ≤ V (x) ≤ a2‖x‖b, (4.6)

∂V

∂xf (x) ≤ −a3‖x‖b, (4.7)

where a1, a2, a3 and b are positive real constants, the equilibrium point x = 0 isexponentially stable. Furthermore, the conditions hold for the entire state space, thenthe equilibrium point x = 0 is globally exponentially stable.

The proof of this theorem is relatively simple, and we are going to show it here.We need a technical lemma, which is also needed later for stability analysis of robustadaptive control systems.

Lemma 4.5 (Comparison lemma). Let g, V : [0, ∞) → R. Then

V (t) ≤ −aV (t) + g(t), ∀t ≥ 0 (4.8)

implies that

V (t) ≤ e−atV (0) +∫ t

0e−α(t−τ )g(τ )dτ , ∀t ≥ 0 (4.9)

for any finite constant a.

Stability theory 51

Proof. From the derivative of Veat , we have

d

dt(Veat) = V eat + aVeat .

Substituting V from (4.8) in the above equation, we have

d

dt(Veat) ≤ eatg(t). (4.10)

Integrating (4.10), we have

V (t)eat ≤ V (0) +∫ t

0eaτg(τ )dτ. (4.11)

Multiplying both sides of (4.11) by e−aτ gives (4.8). This completes the proof. �

Now we are ready to prove Theorem 4.4.

Proof. From (4.6) and (4.7), we have

V ≤ −a3

a2V .

Applying the comparison lemma (Lemma 4.5), we have

V (t) ≤ V (0)e−(a3/a2)t . (4.12)

Then from (4.6) and (4.12), we have

‖x(t)‖ ≤(

1

a1V (t)

)1/b

≤(

1

a1V (0)

)1/b

e−(a3/a2b)t

≤(

a2

a1

)1/b

‖x(0)‖e−(a3/a2b)t .

Hence, (4.3) is satisfied with a =(

a2a1

)1/band λ = a3

a2b , and the equilibrium point is

exponentially stable. �

4.4 Lyapunov analysis of linear time-invariant systems

In this section, we will apply Lyapunov stability analysis to linear time-invariant (LTI)systems. Consider an LTI system

x = Ax (4.13)


where x ∈ Rn and A ∈ R

n×n, x is the state variable, and A is a constant matrix. Fromlinear system theory, we know that this system is stable if all the eigenvalues of A arein the open left half of the complex plane. Such a matrix is referred to as a Hurwitzmatrix. Here, we would like to carry out the stability analysis using a Lyapunovfunction. We can state the stability in the following theorem.

Theorem 4.6. For the linear system shown in (4.13), the equilibrium x = 0 is globallyand exponentially stable if and only if there exist positive definite matrices P and Qsuch that

AT P + PA = −Q (4.14)

holds.

Proof. For sufficiency, let

V (x) = xT Px, (4.15)

and then the direct evaluation gives

V = xT AT Px + xT PAx = −xT Qx. (4.16)

Let us use λmax(·) and λmin(·) to denote maximum and minimum eigenvalues of apositive definite matrix. From (4.15), we have

λmin(P)‖x‖2 ≤ V (x) ≤ λmax(P)‖x‖2. (4.17)

From (4.17) and (4.16), we obtain

V ≤ −λmin(Q)‖x‖2

≤ −λmin(Q)

λmax(P)‖x‖2. (4.18)

Now we can applyTheorem 4.4 with (4.17) and (4.18) to conclude that the equilibriumpoint is globally and exponentially stable. Furthermore, we can identify a1 = λmin(P),a2 = λmax(P), a3 = λmin(Q) and b = 2. Following the proof of Theorem 4.4, we have

‖x(t)‖ ≤√λmax(P)

λmin(P)‖x(0)‖e− λmin(Q)

2λmax (P) t. (4.19)

For the necessary part, we have

‖x(t)‖ ≤ a‖x(0)‖e−λt

Stability theory 53

for some positive real constants a and λ, which implies limt→∞ x(t) = 0. Since

x(t) = eAtx(0),

we can conclude limt→∞ eAt = 0. In such a case, for a positive definite matrix Q, wecan write

∫ ∞

0d[exp (AT t)Q exp (At)] = −Q. (4.20)

For the left-hand side, we can obtain

∫ ∞

0d[exp (AT t)Q exp (At)]

= AT

∫ ∞

0exp (AT t)Q exp (At)t +

∫ ∞

0exp (AT t)Q exp (At)tA

Let

P =∫ ∞

0exp (AT t)Q exp (At)t

and if we can show that P is positive definite, then we obtain (4.14), and hencecomplete the proof. Indeed, for any z ∈ R

n �= 0, we have

zT Pz =∫ ∞

0zT exp (AT t)Q exp (At)zdt.

Since Q is positive definite, and eAt is non-singular for any t, we have zT Pz > 0, andtherefore P is positive definite. �

Chapter 5

Advanced stability theory

Lyapunov direct method provides a tool to check the stability of a nonlinear systemif a Lyapunov function can be found. For linear systems, a Lyapunov function canalways be constructed if the system is asymptotically stable. In many nonlinear sys-tems, a part of the system may be linear, such as linear systems with memorylessnonlinear components and linear systems with adaptive control laws. For such a sys-tem, a Lyapunov function for the linear part may be very useful in the constructionfor the Lyapunov function for the entire nonlinear system. In this chapter, we willintroduce one specific class of linear systems, strict positive real systems, for which,an important result, Kalman–Yakubovich lemma, is often used to guarantee a choiceof the Lyapunov function for stability analysis of several types of nonlinear systems.The application of Kalman–Yakubovich lemma to analysis of adaptive control sys-tems will be shown in later chapters, while in this chapter, this lemma is used forstability analysis of systems containing memoryless nonlinear components and therelated circle criterion. In Section 5.3 of this chapter, input-to-state stability (ISS) isbriefly introduced.

5.1 Positive real systems

Consider a first-order system

y = −ay + u,

where a > 0 is a constant, and y and u ∈ R are the output and input respectively. Thisis perhaps the simplest dynamic system we could possibly have. The performanceof such a system is desirable in control design. One of the characteristics for such asystem is that its transfer function 1

s+a is with positive real part if s = σ + jω withσ > 0. If such a property holds for other rational transfer functions, they are referredto as positive real transfer functions. For this, we have the following definition.

Definition 5.1. A rational transfer function G(s) is positive real if

● G(s) is real for real s● �(G(s)) ≥ 0 for s = σ + jω with σ ≥ 0

For analysis of adaptive control systems, strictly positive real systems are morewidely used than the positive real systems.


Definition 5.2. A proper rational transfer function G(s) is strictly positive real ifthere exists a positive real constant ε such that G(s − ε) is positive real.

Example 5.1. For the transfer function G(s) = 1s+a , with a > 0, we have, for

s = σ + jω,

G(s) = 1

a + σ + jω= a + σ − jω

(a + σ )2 + ω2

and

�(G(s)) = a + σ

(a + σ )2 + ω2> 0.

Hence, G(s) = 1s+a is positive real. Furthermore, for any ε ∈ (0, a), we have

�(G(s − ε)) = a − ε + σ

(a − ε + σ )2 + ω2> 0

and therefore G(s) = 1s+a is also strictly positive real. �

Definition 5.1 shows that a positive real transfer function maps the closed righthalf of the complex plane to itself. Based on complex analysis, we can obtain thefollowing result.

Proposition 5.1. A proper rational transfer function G(s) is positive real if

● all the poles of G(s) are in the closed left half of the complex plane● any poles on the imaginary axis are simple and their residues are non-negative● for all ω ∈ R, �(G(jω)) ≥ 0 when jω is not a pole of G(s)

It can be seen that G(s) = 1s+a , with a < 0, is not positive real. If G(s) is posi-

tive real, we must have �(G(jω)) ≥ 0. Similarly, other necessary conditions can beobtained for a transfer function G(s) to be positive real. We can state those conditionsin an opposite way.

Proposition 5.2. A transfer function G(s) cannot be positive real if one of thefollowing conditions is satisfied:

● The relative degree of G(s) is greater than 1.● G(s) is unstable.● G(s) is non-minimum phase (i.e., with unstable zero).● The Nyquist plot of G(jω) enters the left half of the complex plane.

Advanced stability theory 57

Based on this proposition, the transfer functions G1 = s − 1

s2 + as + b, G2 =

s + 1

s2 − s + 1and G3 = 1

s2 + as + bare not positive real, for any real numbers a

and b, because they are non-minimum phase, unstable, and with relative degree 2

respectively. It can also be shown that G(s) = s + 4

s2 + 3s + 2is not positive real as

G(jω) < 0 for ω > 2√

2.One difference between strictly positive real transfer functions and positive real

transfer functions arises due to the poles on imaginary axis.

Example 5.2. Consider G(s) = 1

s. For s = σ + jω, we have?

�(G(s)) = �(

1

σ + jω

)

= σ

σ 2 + ω2.

Therefore, G(s) = 1s is positive real. However, G(s) = 1

s is not strictly positive real.�

For the stability analysis later in the book, we only need the result on strictlypositive real transfer functions.

Lemma 5.3. A proper rational transfer function G(s) is strictly positive real if andonly if

● G(s) is Hurwitz, i.e., all the poles of G(s) are in the open left half of the complexplane.

● The real part of G(s) is strictly positive along the jω axis, i.e.,

∀ω ≥ 0, �(G(jω)) > 0,

● lims→∞ G(s) > 0, or in case of lims→∞ G(s) = 0, limω→∞ ω2�(G(jω)) > 0.

Proof. We show the proof for sufficiency here, and omit the necessity, as it is moreinvolved. For sufficiency, we only need to show that there exists a positive real constantε such that G(s − ε) is positive real.

Since G(s) is Hurwitz, there must exist a positive real constant δ such that forδ ∈ (0, δ ], G(s − δ) is Hurwitz. Suppose (A, b, cT , d) is a minimum state spacerealisation for G(s), i.e.,

G(s) = cT (sI − A)−1b + d.

We have

G(s − δ) = cT (sI − δI − A)−1b + d

= cT (sI − A)−1((sI − δI − A) + δI )(sI − δI − A)−1b + d

= G(s) + δE(s), (5.1)


where

E(s) = cT (sI − A)−1(sI − δI − A)−1b.

E(s) is Hurwitz, and strictly proper. Therefore, we have

�(E(jω)) < r1, ∀ω ∈ R, δ ∈ (0, δ ] (5.2)

for some positive real r1 and the existence of limω→∞ ω2�(E(jω)), which implies

ω2�(E(jω)) < r2, for |ω| > ω1, δ ∈ (0, δ ] (5.3)

for some ω1 > 0.If limω→∞ �(G(jω)) > 0, we have

�(G(jω)) > r3, ∀ω ∈ R (5.4)

for some r3 > 0. Hence, combining (5.2) and (5.4), we obtain, from (5.1), that

�(G(jω − δ)) > r3 − δr1, ∀ω ∈ R. (5.5)

Then we have �(G(jω − δ)) > 0, by setting δ < r3r1

.In the case that limω→∞ �(G(jω)) = 0, the condition

limω→∞ω

2�(G(jω)) > 0

implies that

ω2�(G(jω)) > r4, for |ω| > ω2 (5.6)

for some positive reals r4 and ω2. From (5.1), (5.3) and (5.6), we obtain that

ω2�(G(jω − δ)) > r4 − δr2, for |ω| > ω3 (5.7)

whereω3 = max{ω1,ω2}. From the second condition of the lemma, we have, for somepositive real constant r5,

�(G(jω)) > r5, for |ω| ≤ ω3. (5.8)

Then from (5.1), (5.2) and (5.8), we obtain that

�(G(jω − δ)) > r5 − δr1, for |ω| ≤ ω3. (5.9)

Combining the results in (5.7) and (5.9), we obtain that �(G(jω − δ)) > 0 by settingδ = min{ r4

r2, r5

r1}. Therefore, we have shown that there exists a positive real δ such that

G(s − δ) is positive real. �


The main purpose of introducing strictly positive real systems is for the followingresult, which characterises the systems using matrices in time domain.

Lemma 5.4 (Kalman–Yakubovich lemma). Consider a dynamic system

x = Ax + bu

y = cT x,(5.10)

where x ∈ Rn is the state variable; y and u ∈ R are the output and input respectively;

and A, b and c are constant matrices with proper dimensions, and (A, b, cT ) iscontrollable and observable. Its transfer function G(s) = cT (sI − A)−1b is strictlypositive real if and only if there exist positive definite matrices P and Q such that

AT P + PA = −Q,

Pb = c.(5.11)

Remark 5.1. We do not provide a proof here, because the technical details in theproof such as finding the positive definite P and the format of Q are beyond the scopeof this book. In the subsequent applications for stability analysis, we only need toknow the existence of P and Q, not their actual values for a given system. For examplein the stability analysis for adaptive control systems in Chapter 7, we only need tomake sure that the reference model is strictly positive real, which then implies theexistence of P and Q to satisfy (5.11). �

5.2 Absolute stability and circle criterion

In this section, we will consider a dynamic system which consists of a linear partand a memoryless nonlinear component. Surely some engineering systems can bemodelled in this format, such as linear systems with nonlinearity in sensors. Let usconsider a closed-loop system

x = Ax + bu

y = cT x

u = −F(y)y,

(5.12)

where x ∈ Rn is the state variable; y and u ∈ R are the output and input respectively;

and A, b and c are constant matrices with proper dimensions. The nonlinear componentis in the feedback law. Similar systems have been considered earlier using describingfunctions for approximation to predict the existence of limit cycles. Nonlinear ele-ments considered in this section are sector-bounded, i.e., the nonlinear feedback gaincan be expressed as

α < F(y) < β (5.13)

for some constants α and β, as shown in Figure 5.2.


F(y)

u .x = Ax + buy = cTx

r = 0Σ

+

–

y

Figure 5.1 Block digram of system (5.12)

F(y)yb

a

y

Figure 5.2 Sector-bounded nonlinear feedback gain

The absolute stability refers to the globally asymptotic stability of the equilib-rium point at the origin for the system shown in (5.12) for a class of sector-boundednonlinearities shown in (5.13). We will use Kalman–Yakubovich lemma for the sta-bility analysis. If the transfer function for the linear part is strictly positive real, wecan establish the stability by imposing a restriction on the nonlinear element.

Lemma 5.5. For the system shown in (5.12), if the transfer function cT (sI − A)−1bis strictly positive real, the system is absolutely stable for F(y) > 0.

Proof. The proof is straightforward by invoking Kalman–Yakubovich lemma. Sincethe linear part of the system is strictly positive real, then there exist positive definitematrices P and Q such that (5.11) is satisfied.

Consider a Lyapunov function candidate

V = xT Px.


Its derivative is given by

V = xT (AT P + PA)x + 2xT PBbu

= −xT Qx − 2xT PbF(y)y

= −xT Qx − 2xT cF(y)y

= −xT Qx − 2F(y)y2,

where in obtaining the third line of equation, we used Pb = c from the Kalman–Yakubovich lemma. Therefore, if F(y) > 0, we have

V ≤ −xT Qx,

and then system is exponentially stable by Theorem 4.4. �

Note that the conditions specified in Lemma 5.4 are the sufficient conditions.With the result shown in Lemma 5.5, we are ready to consider the general case

for α < F(y) < β. Consider the function defined by

F = F − α

β − F(5.14)

and obviously we have F > 0. How to use this transformation for analysis of systemsstability?

With G(s) = cT (sI − A)−1b, the characteristic equation of (5.12) can bewritten as

G(s)F + 1 = 0. (5.15)

Manipulating (5.15) by adding and subtracting suitable terms, we have

G(s)(F − α) = −αG − 1, (5.16)

G(s)(β − F) = βG + 1. (5.17)

With (5.16) being divided by (5.17), we can obtain that

1 + βG

1 + αG· F − α

β − F+ 1 = 0. (5.18)

Let

G := 1 + βG

1 + αG(5.19)

and we can write (5.18) as

GF + 1 = 0 (5.20)


which implies that the stability of the system (5.12) with the nonlinear gain shown in(5.13) is equivalent to the stability of the system with the forward transfer function Gand the feedback gain F . Based on Lemma 5.5 and (5.20), we can see that the system(5.12) is stable if G is strictly positive real.

The expressions of F in (5.14) and G in (5.19) cannot deal with the case β = ∞.In such a case, we re-define

F = F − α (5.21)

which ensures that F > 0. With this F , we can obtain the manipulated characteristicequation as

G

1 + αG· (F − α) + 1 = 0

which enables us to re-define

G := G

1 + αG. (5.22)

We summarise the results in the following theorem.

Theorem 5.6 (Circle criterion). For the system (5.12) with the feedback gain satisfyingthe condition in (5.13), if the transfer function G defined by

G(s) = 1 + βG(s)

1 + αG(s)

or in case of β = ∞, by

G(s) = G(s)

1 + αG(s)

is strictly positive real, with G(s) = cT (sI − A)−1b, the system is absolutely stable.

What is the condition of G if G is strictly positive real? Let us assume thatβ > α > 0. Other cases can be analysed similarly. From Lemma 5.3, we know thatfor G to be strictly positive real, we need G to be Hurwitz, and �(G(jω)) > 0, that is

�(

1 + βG(jω)

1 + αG(jω)

)

> 0, ∀ω ∈ R,

which is equivalent to

�(

1/β + G(jω)

1/α + G(jω)

)

> 0, ∀ω ∈ R. (5.23)


If 1β

+ G(jω) = r1ejθ1 and 1α

+ G(jω) = r2ejθ2 , the condition in (5.23) is satisfied by

−π2< θ1 − θ2 <

π

2,

which is equivalent to the point G(jω) that lies outside the circle centered at(− 1

2 (1/α + 1/β), 0) with radius of 12 (1/α − 1/β) in the complex plane. This cir-

cle intersects the real axis at (− 1α

, 0) and (− 1β

, 0). Indeed, 1β

+ G(jω) is represented

as a vector from the point (− 1β

, 0) to G(jω), and 1α

+ G(jω) as a vector from the

point (− 1α

, 0) to G(jω). The angle between the two vectors will be less than π

2 whenG(jω) is outside the circle, as shown in Figure 5.3. Since the condition must holdfor all ω ∈ R, the condition �(G(jω)) > 0 is equivalent to the Nyquist plot of G(s)that lies outside the circle. The condition that G is strictly positive real requires thatthe Nyquist plot of G(s) does not intersect with the circle and encircles the circlecounterclockwise the same number of times as the number of unstable poles of G(s),as illustrated in Figure 5.4.

Alternatively, the circle can also be interpreted from complex mapping. From(5.19), it can be obtained that

G = G − 1

β − αG. (5.24)

The mapping shown in (5.24) is a bilinear transformation, and it maps a line to a lineor circle. For the case of β > α > 0, we have

G = − 1

α−(

1

α− 1

β

)β/α

G − β/α. (5.25)

The function

β/α

G − β/α

– 1

Im

Re

G( jw)

a b– 1

q2 q1

Figure 5.3 Diagram of |θ1 − θ2| < π

2


– 1

Im

Re

G( jw)

a – 1b

Figure 5.4 Circle criterion

maps the imaginary axis to a circle centred as (−1/2, 0) with the radius 1/2, i.e., theline from (−1, 0) to (0, 0) on the complex plane is the diameter of the circle. Thenthe function

−(

1

α− 1

β

)β/α

G − β/α

maps the imaginary axis to a circle with the diameter on the line from (0, 0) to( 1α

− 1β

, 0) on the complex plane. Finally, it can be seen from (5.25) that this map of G

to G maps the imaginary axis to the circle with the diameter on the line from (− 1α

, 0)to (− 1

β, 0), or in other words, the circle centered as (− 1

2 (1/α + 1/β), 0) with radius

of 12 (1/α − 1/β). It can also be shown that the function maps the open left-hand

complex plane to the domain inside the circle.Indeed, we can evaluate the circle directly from (5.24). Let u and v denote the

real and imaginary parts of the mapping of the imaginary axis, and we have

u = � jω − 1

β − αjω= − α + βω2

α2 + β2ω2,

v = � jω − 1

β − αjω= (α − β)ω

α2 + β2ω2.

Denoting μ = β

αω, we obtain

u = −1/α + (1/β)μ2

1 + μ2= −1

2

(1

α+ 1

β

)

+ 1

2

(1

β− 1

α

)1 − μ2

1 + μ2,

v = (1/β − 1/α)μ

1 + μ2= 1

2

(1

β− 1

α

)2μ

1 + μ2.


It is now easy to see that

(

u + 1

2

(1

α+ 1

β

))2

+ v2 =(

1

2

(1

α− 1

β

))2

.

which describes the circle that is discussed before.

5.3 Input-to-state stability and small gain theorem

In this section, we continue to consider the stability properties for systems with inputs.For linear systems, if a system is asymptotically stable, the state will remain boundedwhen the input is bounded. Does this property still hold for nonlinear systems?


x = f (x, u) (5.26)

where x ∈ Rn is the state of the system, and u ∈ R

m is a bounded input, and f :R

n × Rm −→ R

n is a continuous and locally Lipschitz function. If the autonomoussystem

x = f (x, 0)

is asymptotically stable, will the state remain bounded for a bounded input signal u?


x = −x + (1 + 2x)u, x(0) = 0

where x ∈ R. When u = 0, we have

x = −x

which is asymptotically (exponentially) stable. However, the state of this system maynot remain bounded for a bounded input. For example if we let u = 1, we have

x = x + 1

of which the state variable x is unbounded. �

From the above example, it can be seen that even the corresponding autonomoussystem is asymptotically stable, the state may not remain bounded subject to a boundedinput. We introduce a definition for systems with the property of bounded state withbounded input.

To show a definition of input-to-state stable (ISS), we need to use comparisonfunctions, which are defined below.


Definition 5.3. A function γ : [0, a) → [0, ∞) is a class K function if γ is contin-uous, and strictly increasing with γ (0) = 0. If a = ∞ and limr→∞ γ (r) = ∞, thefunction is a class K∞ function.

Definition 5.4. A function β : [0, a) × [0, ∞) → [0, ∞) is a class KL function if itis continuous, for a fixed t = t0, β(·, t0) is a class K function, and for a fixed x(0),limt→∞ β(x(0), t) = 0.

Definition 5.5. The system (5.26) is ISS if there exist a class KL function β and aclass K function γ such that

‖x(t)‖ ≤ β(‖x(0)‖, t) + γ (‖u‖∞), ∀t > 0. (5.27)

There is an alternative definition, which may be convenient to use in somesituations. Here, we state it as a proposition.

Proposition 5.7. The system shown in (5.26) is ISS if and only if there exist a classKL function β and a class K function γ such that

‖x(t)‖ ≤ max{β(‖x(0)‖, t), γ (‖u‖∞)}, ∀t > 0. (5.28)

Proof. From (5.28), we have

‖x(t)‖ ≤ β(‖x(0)‖, t) + γ (‖u‖∞)

and hence the system is ISS. From (5.27), there exist a class KL function β1 and aclass K function γ1

‖x(t)‖ ≤ β1(‖x(0)‖, t) + γ1(‖u‖∞)

≤ max{2β1(‖x(0)‖, t), 2γ1(‖u‖∞)}

and therefore the system satisfies (5.28) with β = 2β1 and γ = 2γ1. �

Example 5.4. Consider a linear system

x = Ax + Bu,

where x ∈ Rn and u ∈ R

m are the state and input respectively, and A and B are matriceswith appropriate dimensions and A is Hurwitz. For this system, when u = 0, thesystem is asymptotically stable, as A is Hurwitz. From its solution

x(t) = eAtx(0) +∫ t

0eA(t−τ )Bu(τ )dτ ,


we have

‖x(t)‖ ≤ ‖eAt‖‖x(0)‖ +∫ t

0‖eA(t−τ )‖dτ‖B‖‖u‖∞.

Since A is Hurwitz, there exist positive real constants a and λ such that ‖eA(t)‖ ≤ae−λt . Hence, we can obtain

‖x(t)‖ ≤ ae−λt‖x(0)‖ +∫ t

0ae−λ(t−τ )‖dτ‖B‖‖u‖∞

≤ ae−λt‖x(0)‖ + a

λ‖B‖‖u‖∞

It is easy to see that the first term in the above expression is a KL function of t and‖x(0)‖ and the second term is a K function of ‖u‖∞. Therefore, the linear system withHurwitz system matrix A is ISS. �

Next, we show a result on establishing ISS property from a Lyapunov function.

Theorem 5.8. For the system (5.26), if there exists a function V (x) : Rn → R with

continuous first-order derivatives such that

a1‖x‖b ≤ V (x) ≤ a2‖x‖b, (5.29)

∂V

∂xf (x, u) ≤ −a3‖x‖b, ∀‖x‖ ≥ ρ(‖u‖) (5.30)

where a1, a2, a3 and b are positive real constants and ρ is a class K function, thesystem (5.26) is ISS.

Proof. From Theorem 4.4, we can see that the system is exponentially stable whenu = 0, and V (x) is a Lyapunov function for the autonomous system x = f (x, 0). Toconsider the case for non-zero input, let us find a level set based on V , defined by

�c := {x|V (x) ≤ c}and determine the constant c such that for x /∈ �c, we have ‖x‖ > ρ(‖u‖∞). Indeed,let

c = a2(ρ(‖u‖∞))b.

In this case, from V (x) > c, we have

V (x) > a2(ρ(‖u‖∞))b,

which implies that

a2‖x‖b > a2(ρ(‖u‖∞))b

and hence ‖x‖ > ρ(‖u‖∞).


Therefore, for any x outside�c, we can obtain from (5.29) and (5.30), in a similarway to the proof of Theorem 4.4,

V ≤ −a3

a2V

and then

‖x(t)‖ ≤(

a2

a1

)1/b

‖x(0)‖e− a3a2b t.

There will be a time, say t1, at which the trajectory enters �c.We can show that �c is invariant, i.e., the trajectory starts from �c will remain

in �c, and this is shown by the fact that on the boundary of �c, V ≤ 0. Therefore,after t1, the trajectory will remain in �c. Furthermore, for x ∈ �c, we have,

V (x) < a2(ρ(‖u‖∞))b

which implies that

‖x‖ ≤(

a2

a1

)1/b

ρ(‖u‖∞).

Combining the cases for x outside and in �c, we conclude that

‖x‖ ≤ max

{(a2

a1

)1/b

‖x(0)‖e− a3a2b t ,

(a2

a1

)1/b

ρ(‖u‖∞)

}

.

Hence, the system is ISS with the gain function γ (·) = ( a2a1

)1/bρ(·).There is a more general result than Theorem 5.8 that requires x = f (x, 0) to be

asymptotically stable, not necessarily exponential stable. The proof of that theoremis beyond the level of this text, and we include it here for completeness.

Theorem 5.9. For the system (5.26), if there exists a function V (x) : Rn → R with


α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), (5.31)

∂V

∂xf (x, u) ≤ −α3(‖x‖), ∀‖x‖ ≥ ρ(‖u‖), (5.32)

where α1, α2, α3 are K∞ functions and ρ is a class K function, the system (5.26) isISS with the gain function γ (·) = α−1

1 (α2(ρ(·))).Note that class K∞ functions are class K functions that satisfy the property

limr→∞α(r) = ∞.


There is a slightly different version of Theorem 5.9 which is shown below.

Corollary 5.10. For the system (5.26), if there exists a function V (x) : Rn → R with


α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), (5.33)

∂V

∂xf (x) ≤ −α(‖x‖) + σ (‖u‖), (5.34)

where α1, α2, α are K∞ functions and σ is a class K function, the system (5.26)is ISS.

The function which satisfies (5.31) and (5.32) or (5.33) and (5.34) is referred to asISS-Lyapunov function. In fact, it can be shown that the existence of an ISS-Lyapunovfunction is also a necessary condition for the system to be ISS. Referring to (5.34),the gain functions α and σ characterise the ISS property of the system, and they arealso referred to as an ISS pair. In other words, if we say a system is ISS with ISS pair(α, σ ), we mean that there exists an ISS-Lyapunov function that satisfies (5.34).


x = −x3 + u,

where x ∈ R is the state, and u is input. The autonomous part x = −x3 is consideredin Example 4.2, and it is asymptotically stable, but not exponentially stable. Consideran ISS-Lyapunov function candidate

V = 1

2x2.


V = −x4 − xu

≤ −1

2x4 − 1

2|x|(|x|3 − 2|u|)

≤ −1

2x4, for|x| ≥ (2|u|)1/3.

Hence the system is ISS with the gain function ρ(|u|) = (2|u|)1/3, based onTheorem 5.9. Alternatively, using Young’s inequality, we have

|x||u| ≤ 1

4|x|4 + 3

4|u|4/3


which gives

V ≤ −3

4x4 + 3

4|u|4/3.

Therefore, the system is ISS based on Corollary 5.10 with α(·) = 34 (·)4 and

σ (·) = 34 (·)4/3. �

ISS property is useful in establishing the stability of interconnected systems. Weinclude two results here to end this section.

Theorem 5.11. If for the cascade connected system

x1 = f1(x1, x2), (5.35)

x2 = f1(x2, u), (5.36)

the subsystem (5.35) is ISS with x2 as the input, and the subsystem (5.36) is ISS withu as input, the overall system with state x = [xT

1 , xT2 ]T and input u is ISS.

Theorem 5.12 (ISS small gain theorem). If for the interconnected system

x1 = f1(x1, x2), (5.37)

x2 = f1(x2, x2, u), (5.38)

the subsystem (5.37) is ISS with x2 as the input with γ1 as the ISS gain for x2, andthe subsystem (5.38) is ISS by viewing x1 and u as the inputs, with the ISS input gainfunction γ2 for x1, the overall system with state x = [xT

1 , xT2 ]T and input u is ISS if

γ1(γ2(r)) < r, ∀ r > 0. (5.39)

From Theorem 5.11, it can be seen that if the subsystem x2 is globally asymptoti-cally stable when u = 0, the overall system is globally asymptotically stable. Similarly,Theorem 5.12 can be used to establish the stability of the following system:

x1 = f1(x1, x2)

x2 = f1(x2, x2),

and the global and asymptotic stability of the entire system can be concluded if thegain condition shown in Theorem 5.12 is satisfied.

Example 5.6. Consider the second-order nonlinear system

x1 = −x31 + x2,

x2 = x1x2/32 − 3x2.


From Example 5.5, we know that x1-subsystem is ISS with the gain functionγ1(·) = (2·)1/3. For the x2-subsystem, we choose

V2 = 1

2x2

2

and we have

V2 = −3x22 + x1x5/3

2

≤ −x22 − |x2|5/3(2|x2|1/3 − |x1|)

≤ −x22, for |x2| >

( |x1|2

)3

Hence, the x2-subsystem is ISS with the gain function γ2(·) = ( ·2 )3. Now we have,

for r > 0,

γ1(γ2(r)) =(

2( r

2

)3)1/3

=(

1

4

)1/3

r < r.

Therefore, the system is globally asymptotically stable. �

5.4 Differential stability


x = f (x, u), (5.40)

where x ∈ Rn is the state vector, u ∈ R

s is the input and f : Rn × R

s → Rn is a

nonlinear smooth vector field with f (0, u) = 0.

Definition 5.6. This system (5.40) has differential stability if there exists a Lyapunovfunction V (x) such that V : R

n → R for all x, x ∈ Rn, u ∈ R

s, satisfies

γ1(‖x‖) ≤ V (x) ≤ γ2(‖x‖),

∂V (x − x)

∂x(f (x, u) − f (x, u)) ≤ −γ3(‖x − x‖),

c1

∥∥∥∥∂V (x)

∂x

∥∥∥∥

c2

≤ γ3(‖x‖),

(5.41)

where γi, i = 1, 2, 3, are K∞ functions and ci, i = 1, 2, are positive real constants withc2 > 1.


Remark 5.2. The conditions specified in (5.41) are useful for observer design, inparticular, for the stability analysis of the reduced-order observers in Chapter 8.A similar definition to differential stability is incremental stability. However, theconditions specified in (5.41) are not always satisfied by the systems with incrementalstability. When x = 0, i.e., in the case for one system only, the conditions specifiedin (5.41) are then similar to the properties of the nonlinear systems with exponentialstability. The last condition in (5.41) is specified for interactions with other systems.This condition is similar to the conditions for the existence of changing the supplyfunctions for inter-connection of ISS systems. �

We include two illustrative examples below for the properties of differentialstability.

Example 5.7. A linear system is differentially stable if the system is asymptoticallystable. Consider

x = Ax,

where A ∈ Rn×n. If the system is asymptotically stable, A must be Hurwitz. Therefore

there exist positive definite matrices P and Q such that

AT P + PA = −Q.

Let V (x) = xT Px. In this case, the conditions (5.41) are satisfied with

γ1(‖x‖) = λmin(P)‖x‖2,

γ2(‖x‖) = λmax(P)‖x‖2,

γ3(‖x‖) = λmin(Q)‖x‖2,

c1 = λmin(Q)

4(λmax(P))2, c2 = 2,

where λmin(·) and λmax(·) denote the minimum and maximum eigenvalues of apositive definite matrix. �

The differential stability is closely related to observer design. Consider a systemwith input u ∈ R

s

x = Ax + Bu. (5.42)

If x = Ax is differentially stable, an observer for (5.42) can be designed as

˙x = Ax + Bu, x(0) = 0. (5.43)

It is easy to see that x − x converges to 0 exponentially.However, for nonlinear systems, differential stability is not guaranteed by the

asymptotic or even exponential stability of a system.


Example 5.8. We consider a first-order nonlinear system

x = −x − 2 sin x.

Take V = 12 x2, and we have

V = −x2 − 2x sin x.

For |x| ≤ π , we have x sin x ≥ 0, and therefore

V ≤ −x2.

For |x| > π , we have

V ≤ −x2 − 2x sin x

≤ −x2 + 2|x|= −

(

1 − 2

π

)

x2 − 2|x|( |x|π

− 1)

≤ −(

1 − 2

π

)

x2.

Combining both the cases, we have

V ≤ −(

1 − 2

π

)

x2.

Hence, the system is exponentially stable. But this system is not differentially stable.Indeed, let e = x − x. We have

e = −e − 2( sin (x) − sin (x + e)).

By linearising the system at x = π and x = π , and denoting the error at this point byel , we have

el = −el − 2 cos (x)|x=π (x − π ) + 2 cos (x)|x=π (x − π )

= el

and the system is unstable in a neighbourhood of this point. �

Chapter 6

Feedback linearisation

Nonlinear systems can be linearised around operating points and the behaviours inthe neighbourhoods of the operating points are then approximated by their linearisedmodels. The domain for a locally linearised model can be fairly small, and this mayresult in that a number of linearised models are needed to cover an operating rangeof a system. In this chapter, we will introduce another method to obtain a linearmodel for nonlinear systems via feedback control design. The aim is to convert anonlinear system to a linear one by state transformation and redefining the controlinput. The resultant linear model describes the system dynamics globally. Of course,there are certain conditions for the nonlinear systems to satisfy so that this feedbacklinearisation method can be applied.

6.1 Input–output linearisation

The basic idea for input–output linearisation is fairly straightforward. We use anexample to demonstrate this.


x1 = x2 + x31

x2 = x21 + u

y = x1.

(6.1)

For this system, taking the derivatives of y, we have

y = x2 + x31,

y = 3x21(x2 + x3

1) + x21 + u.

Now, let us define

v = 3x21(x2 + x3

1) + x21 + u,

and we obtain

y = v.


Viewing v as the new control input, we see that the system is linearised. Indeed, letus introduce a state transformation

ξ1 := y = x1,

ξ2 := y = x2 + x31.

We then obtain a linear system

ξ1 = ξ2

ξ2 = v.

We can design a state feedback law as

v = −a1ξ1 − a2ξ2

to stabilise the system with a1 > 0 and a2 > 0. The control input of the original systemis given by

u = −3x21(x2 − x3

1) − x21 + v. (6.2)

We say that the system (6.1) is linearised by the feedback control law (6.2). Noticethat this linearisation works for the entire state space. �

As shown in the previous example, we can keep taking the derivatives of theoutput y until the input u appears in the derivative, and then a feedback linearisationlaw can be introduced. The derivatives of the output also introduce a natural statetransformation.


x = f (x) + g(x)u

y = h(x),(6.3)

where x ∈ D ⊂ Rn is the state of the system; y and u ∈ R are output and input respec-

tively; and f and g : D ⊂ Rn → R

n are smooth functions and h : D ⊂ Rn → R is a

smooth function.

Remark 6.1. The functions f (x) and g(x) are vectors for a given point x in the statespace, and they are often referred to as vector fields. All the functions in (6.3) arerequired to be smooth in the sense that they have continuous derivatives up to certainorders when required. We use the smoothness of functions in the remaining part ofthe chapter in this way. �

The input–output feedback linearisation problem is to design a feedback controllaw

u = α(x) + β(x)v (6.4)

with β(x) �= 0 for x ∈ D such that the input–output dynamics of the system

Feedback linearisation 77

x = f (x) + g(x)α(x) + g(x)β(x)v

y = h(x)(6.5)

are described by

y(ρ) = v (6.6)

for 1 ≤ ρ ≤ n.For the system (6.3), the first-order derivative of the output y is given by

y = ∂h(x)

∂x(f (x) + g(x)u)

:= Lf h(x) + Lgh(x)u,

where the notations Lf h and Lgh(x) are Lie derivatives.For any smooth function f : D ⊂ R

n → Rn and a smooth function

h : D ⊂ Rn → R, the Lie derivative Lf h, referred to as the derivative of h along

f , is defined by

Lf h(x) = ∂h(x)

∂xf (x).

This notation can be used iteratively, that is

Lf (Lf h(x)) = L2f h(x),

Lkf (h(x)) = Lf (Lk−1

f (h(x))),

where k ≥ 0 is an integer.The solution to this problem depends on the appearance of the control input in

the derivatives of the output, which is described by the relative degree of the dynamicsystem.

Definition 6.1. The dynamic system (6.3) has relative degree ρ at a point x if thefollowing conditions are satisfied:

LgLkf h(x) = 0, for k = 0, . . . , ρ − 2,

LgLρ−1f h(x) �= 0.

Example 6.2. Consider the system (6.1). Comparing it with the format shown in(6.3), we have

f (x) =[

x31 + x2

x21

]

, g(x) =[

01

]

, h(x) = x1.


Direct evaluation gives

Lgh(x) = 0,

Lf Lgh(x) = 1.

Therefore, the relative degree of the system (6.1) is 2. �

For SISO linear systems, the relative degree is the difference between the ordersof the polynomials in the numerator and denominator of the transfer function. Withthe definition of the relative degree, we can present the input–output feedbacklinearisation using Lie derivatives.

Example 6.3. Consider the system (6.1) again, and continue from Example 6.2. WithLgh(x) = 0, we have

y = Lf h(x)

where

Lf h(x) = x31 + x2.

Taking the derivative of Lf h(x), we have

y = L2f h(x) + LgLf h(x)u

where

L2f h(x) = 3x2

1(x2 + x31) + x2

1,

LgLf h(x) = 1.

Therefore, we have

v = L2f h(x) + LgLf h(x)u

or

u = − L2f h(x)

LgLf h(x)+ 1

LgLf h(x)v

which gives the same result as in Example 6.1. �

The procedure shown in Example 6.3 works for systems with any relative degrees.Suppose that the relative degree for (6.3) is ρ, which implies that LgLk

f h(x) = 0 fork = 0, . . . , ρ − 2. Therefore, we have the derivatives of y expressed by

y(k) = Lkf h(x), for k = 0, . . . , ρ − 1, (6.7)

y(ρ) = Lρf h(x) + LgLρ−1f h(x)u. (6.8)


If we define the input as

u = 1

LgLρ−1f h(x)

(−Lρf h(x) + v)

it results in

y(ρ) = v.

We can consider to use ξi := y(i−1) = Li−1f h as coordinates for the linearised

input–output dynamics. The only remaining issue is to establish that ∂ξ

∂x has fullrank. To do that, we need to introduce a few notations.

For any smooth functions f , g : D ⊂ Rn → R

n, the Lie bracket [f , g] isdefined by

[f , g](x) = ∂g(x)

∂xf (x) − ∂g(x)

∂xf (x)

and we can introduce a notation which is more convenient for high-order Liebrackets as

ad0f g(x) = g(x),

ad1f g(x) = [f , g](x),

adkf g(x) = [f , adk−1

f g](x).

For the convenient of presentation, let us denote

dh = ∂h

∂x

which is a row vector. With this notation, we can write

Lf h = < dh, f >.

Based on the definition of Lie Bracket, it can be obtained by a direct evaluationthat, for a smooth function h : R

n → R,

L[f , g]h = Lf Lgh − LgLf h = Lf < dh, g > − < dLf h, g >,

that is

< dh, [f , g] >= Lf < dh, g > − < dLf h, g > .

Similarly it can be obtained that, for any non-negative integers k and l,

< dLkf h, adl+1

f g >= Lf < dLkf h, adl

f g > − < dLk+1f h, adl

f g > (6.9)


By now, we have enough tools and notations to show that ∂ξ∂x has full rank. From

the definition of the relative degree,

< dLkf h, g >= 0, for k = 0, . . . , ρ − 2

and by using (6.9) iteratively, we can show that

< dLkf h, adl

f g >= 0, for k + l ≤ ρ − 2, (6.10)

and

< dLkf h, adl

f g >= (−1)l < dLρ−1f h, g >, for k + l = ρ − 1. (6.11)

From (6.10) and (6.11), we have

⎡

⎢⎢⎢⎢⎣

dh(x)dLf h(x)...

dLρ−1f h

⎤

⎥⎥⎥⎥⎦

[g(x) adf g(x) . . . adρ−1

f g(x)]

=

⎡

⎢⎢⎢⎣

0 . . . 0 (−1)ρ−1r(x)0 . . . (−1)ρ−2r(x) ∗...

......

...

r(x) . . . ∗ ∗

⎤

⎥⎥⎥⎦

(6.12)

where r(x) =< dLρ−1f h, g >. Therefore, we conclude that

∂ξ

∂x=

⎡

⎢⎢⎢⎣

dh(x)dLf h(x)...

dLρ−1f h

⎤

⎥⎥⎥⎦

has full rank. We summarise the result about input–output feedback linearisation inthe following theorem.

Theorem 6.1. If the system in (6.3) has a well-defined relative degree ρ in D, theinput–output dynamics of the system can be linearised by the feedback control law

u = 1

LgLρ−1f h(x)

(−Lρf h(x) + v) (6.13)

and the linearised input–output dynamics are described by


ξ1 = ξ2

...

ξρ−1 = ξρ

ξρ = v

(6.14)

with a partial state transformation

ξi = Li−1f h(x) for i = 1, . . . , ρ. (6.15)

When ρ = n, the whole nonlinear system dynamics are fully linearised.

The results shown in (6.10) and (6.11) can also be used to conclude the followingresult which is needed in the next section.

Lemma 6.2. For any functions f , g : D ⊂ Rn → R

n, and h : D ⊂ Rn → R, all

differentiable to certain orders, the following two statements are equivalent for r > 0:

● Lgh(x) = LgLf h(x) = · · · = LgLrf h(x) = 0,

● Lgh(x) = L[f , g]h(x) = · · · = Ladrf gh(x) = 0.

Remark 6.2. The input–output dynamics can be linearised based on Theorem 6.1. Inthe case of ρ < n, the system for ρ < n can be transformed under certain conditionsto the normal form

z = f0(z, ξ ),

ξ1 = ξ2,

...

ξρ−1 = ξρ ,

ξρ = Lρf h + uLgLρ−1f h,

y = ξ1

where z ∈ Rn−ρ is the part of the state variables which are not in the input–output

dynamics of the system, and f0 : Rn → R

n−ρ is a smooth function. It is clear thatwhen ρ < n, the input–output linearisation does not linearise the dynamicsz = f0(z, ξ ). Also note that the dynamics z = f0(z, 0) are referred to as the zerodynamics of the system. �

To conclude this section, we use an example to demonstrate the input–outputlinearisation for the case ρ < n.



x1 = x31 + x2

x1 = x21 + x3 + u

x2 = x21 + u

y = x1.

For this system we have

f (x) =⎡

⎣x3

1 + x2

x21 + x3

x21

⎤

⎦ , g(x) =⎡

⎣011

⎤

⎦ , h(x) = x1.

It is easy to check that

Lf h = x31 + x2,

Lgh = 0,

LgLf h = 1,

L2f h = 3x2

1(x31 + x2) + x2

1 + x3

and therefore the relative degree ρ = 2. For input–output linearisation, we set

u = 1

LgLf h(−L2

f h + v)

= −3x21(x3

1 + x2) − x21 − x3 + v.

Introduce the partial state transformation

ξ1 = h = x1,

ξ2 = Lf h = x31 + x2,

and it is easy to see that dh and dLf h are linearly independent. The linearisedinput–output dynamics are described by

ξ1 = ξ2,

ξ2 = v.

If we like to transform the system to the normal form shown in Remark 6.2, we needto introduce another state, in addition to ξ1 and ξ2. For this, we have the additionalstate

z = x3 − x2.

The inverse transformation is given by

x1 = ξ1,

x2 = ξ2 − ξ 31 ,

x3 = z + ξ2 − ξ 31 .


With the coordinates z, ξ1 and ξ2, we have the system in the normal form

z = −z − ξ2 + ξ 31

ξ1 = ξ2

ξ2 = z + ξ2 + ξ 21 − ξ 3

1 + 3ξ 21 ξ2 + u.

It is clear that the input–output linearisation does not linearise the dynamics of z. Alsonote that the zero dynamics for this system are described by

z = −z.

�

6.2 Full-state feedback linearisation


x = f (x) + g(x)u, (6.16)

where x ∈ D ⊂ Rn is the state of the system; u ∈ R is the input; and f , g :

D ⊂ Rn → R

n are smooth functions. The full-state linearisation problem is to find afeedback control design

u = α(x) + β(x)v (6.17)

with β(x) �= 0 for x ∈ D such that the entire system dynamics are linearised by a statetransformation with v as the new control input.

It is clear from the results shown in the input–output linearisation that the com-plete linearisation can only be achieved when the relative degree of the system equalsthe order of the system. If we can find an output function h(x) for the system (6.16),the input–output linearisation result shown in the previous section can be appliedto solve the full-state feedback linearisation problem. Therefore, we need to find anoutput function h(x) such that

LgLkf h(x) = 0 for k = 0, . . . , n − 2 (6.18)

LgLn−1f h(x) �= 0 (6.19)

Based on Lemma 6.2, the condition specified in (6.18) is equivalent to the condition

Ladkf gh(x) = 0 for k = 0, . . . , n − 2 (6.20)

and furthermore, the condition in (6.19) is equivalent to

Ladn−1f gh(x) �= 0. (6.21)

The output function h(x) that satisfies the condition shown in (6.20) is a solution ofthe partial differential equation

[g, adf g, . . . , adn−2f g]

∂h

∂x= 0. (6.22)


To discuss the solution of this partial differential equation, we need a few notationsand results. We refer to a collection of vector fields as a distribution. For example iff1(x), . . . , fk (x) are vector fields, with k a positive integer,

� = span{f1(x), . . . , fk (x)}

is a distribution. The dimension of distribution is defined as

dim (�(x)) = rank[f1(x), . . . , fn(x)].

A distribution � is said to be involutive, if for any two vector fields f1, f2 ∈ �, wehave [f1, f2] ∈ �. Note that not all the distributions are involutive, as shown in thefollowing example.

Example 6.5. Consider the distribution

� = span{f1(x), f2(x)}

where

f1(x) =⎡

⎣2x2

10

⎤

⎦ , f2(x) =⎡

⎣10x2

⎤

⎦ .

A direct evaluation gives

[f1, f2] = ∂f2

∂xf1 − ∂f1

∂xf2

=⎡

⎣0 0 00 0 00 1 0

⎤

⎦

⎡

⎣2x2

10

⎤

⎦−⎡

⎣0 2 00 0 00 0 0

⎤

⎦

⎡

⎣10x2

⎤

⎦

=⎡

⎣001

⎤

⎦ .

It can be shown that [f1, f2] /∈ �, and therefore � is not involutive. Indeed, the rankof the matrix [f1, f2, [f1, f2]] is 3, which means that [f1, f2] is linearly independent of f1

and f2, and it cannot be a vector field in �. The rank of [f1, f2, [f1, f2]] can be verifiedby its non-zero determinant as

|[f1, f2, [f1, f2]]| =∣∣∣∣∣∣

⎡

⎣2x2 1 01 0 00 x2 1

⎤

⎦

∣∣∣∣∣∣= −1.

�


We say that a distribution � is integrable if there exists a non-trivial h(x) suchthat for any vector field f ∈ �,< dh, f >= 0. The relationship between an involutivedistribution and its integrability is stated in the following theorem.

Theorem 6.3 (Frobenius theorem). A distribution is integrable if and only if it isinvolutive.

Now we are ready to state the main result for full-state feedback linearisation.

Theorem 6.4. The system (6.16) is full-state feedback linearisable if and only if∀x ∈ D● the matrix G = [g(x), adf g(x), . . . , adn−1

f g(x)] has full rank● the distribution Gn−1 = span{g(x), adf g(x), . . . , adn−2

f g(x)} is involutive

Proof. For the sufficiency, we only need to show that there exists a function h(x) suchthat the relative degree of the system by viewing h(x) as the output is n, and the restfollows from Theorem 6.1. From the second condition that Gn−1 is involutive, andFrobenius theorem, there exists a function h(x) such that

∂h

∂x[g, adf g, . . . , adn−2

f g] = 0

which is equivalent to, from Lemma 6.2,

LgLkf h(x) = 0, for k = 0, . . . , n − 2.

From the condition that G is full rank, we can establish

LgLn−1f h(x) �= 0.

In fact, if LgLn−1f h(x) = 0, then from Lemma 6.2, we have

∂h

∂xG = 0

which is a contradiction as G has full rank.For necessity, we show that if the system is full-state feedback linearisable, then

the two conditions hold. From the problem formulation, we know that the full-statelinearisability is equivalent to the existence of an output function h(x) with relativedegree n. From the definition of the relative degree and Lemma 6.2, we have

Ladkf gh(x) = 0, for k = 0, . . . , n − 2,

which implies

∂h

∂x[g, adf g, . . . , adn−2

f g] = 0.


From Frobenius theorem, we conclude that Gn−1 is involutive. Furthermore, from thefact that the system with h(x) as the output has a relative degree n, we can show, inthe same way as the discussion that leading to Lemma 6.2, that

⎡

⎢⎢⎢⎣

dh(x)dLf h(x)...

dLn−1f h

⎤

⎥⎥⎥⎦

[g(x) adf g(x) · · · adn−1

f g(x)]

=

⎡

⎢⎢⎢⎣

0 . . . 0 (−1)n−1r(x)0 . . . (−1)n−2r(x) ∗...

......

...

r(x) . . . ∗ ∗

⎤

⎥⎥⎥⎦

where r(x) = LgLn−1f h(x). This implies that G has rank n. This concludes the

proof. �

Remark 6.3. Let us see the conditions in Theorem 6.4 for linear systems with

f (x) = Ax, g(x) = b.

where A is a constant matrix and b is a constant vector. A direct evaluation gives

[f , g] = −Ab,

and

adkf g = (−1)kAkb

for k > 0. Therefore, we have

G = [b, −Ab, . . . , (−1)n−1An−1b].

It can be seen that the full rank condition of G is equivalent to the full controllabilityof the linear system. �

In the next example, we consider the dynamics of the system that was consid-ered in Example 6.4 for input–output linearisation for the input h(x) = x1. We willshow that the full-state linearisation can be achieved by finding a suitable outputfunction h(x).



x1 = x31 + x2

x1 = x21 + x3 + u

x2 = x21 + u,

for full-state feedback linearisation. With

f (x) =⎡

⎣x3

1 + x2

x21 + x3

x21

⎤

⎦ , g(x) =⎡

⎣011

⎤

⎦ ,

we have

[f , g] =⎡

⎣−1−10

⎤

⎦ ,

and

ad2f g =

⎡

⎣3x2

1 + 12x1

2x1

⎤

⎦ .

Hence, we have

G2 = span

⎧⎨

⎩

⎡

⎣011

⎤

⎦ ,

⎡

⎣−1−10

⎤

⎦

⎫⎬

⎭

and

G =⎡

⎣0 −1 3x2

1 + 11 −1 2x1

1 0 2x1

⎤

⎦ .

The distribution G2 is involutive, as it is spanned by constant vectors. The matrix Ghas full rank, as shown by its determinant

|G| = 3x21 + 1 �= 0.

Hence, the conditions in Theorem 6.4 are all satisfied, and the system is full-statelinearisable. Indeed, we can find

h(x) = x1 − x2 + x3

by solving

∂h

∂x

⎡

⎣0 −11 −11 0

⎤

⎦ = 0.

For this h(x), it is easy to check that


Lgh = 0,

Lf h = x2 + x31 − x3,

LgLf h = 0,

L2f h = 3x2

1(x31 + x2) + x3,

LgL2f h = 3x2

1 + 1,

L3f h = (15x4

1 + 6x1x2)(x31 + x2) + 3x2

1(x3 + x21) + x2

1.

It can be seen that the relative degree, indeed, equals 3 as

LgL2f h = 3x2

1 + 1 �= 0.

For the full-state linearisation, we have the state transformation

ξ1 = x1 − x2 + x3,

ξ2 = x2 + x31 − x3,

ξ2 = 3x21(x3

1 + x2) + x3,

and the feedback law

u = 1

3x21 + 1

(v − (15x4

1 + 6x1x2)(x31 + x2) − 3x2

1(x3 + x21) − x2

1

).

�

Chapter 7

Adaptive control of linear systems

The principle of feedback control is to maintain a consistent performance when thereare uncertainties in the system or changes in the setpoints through a feedback con-troller using the measurements of the system performance, mainly the outputs. Manycontrollers are with fixed controller parameters, such as the controllers designed bynormal state feedback control, and H∞ control methods. The basic aim of adaptivecontrol also is to maintain a consistent performance of a system in the presence ofuncertainty or unknown variation in plant parameters, but with changes in the con-troller parameters, adapting to the changes in the performance of the control system.Hence, there is an adaptation in the controller setting subject to the performance of theclosed-loop system. How the controller parameters change is decided by the adaptivelaws, which are often designed based on the stability analysis of the adaptive controlsystem.

A number of design methods have been developed for adaptive control. ModelReference Adaptive Control (MRAC) consists of a reference model which pro-duces the desired output, and the difference between the plant output and thereference output is then used to adjust the control parameters and the controlinput directly. MRAC is often in continuous-time domain, and for deterministicplants. Self-Tuning Control (STC) estimates system parameters and then computesthe control input from the estimated parameters. STC is often in discrete-time and forstochastic plants. Furthermore, STC often has a separate identification procedure forestimation of the system parameters, and is referred to as indirect adaptive control,while MRAC adapts to the changes in the controller parameters, and is referred toas direct adaptive control. In general, the stability analysis of direct adaptive con-trol is less involved than that of indirect adaptive control, and can often be carriedout using Lyapunov functions. In this chapter, we focus on the basic design methodof MRAC.

Compared with the conventional control design, adaptive control is moreinvolved, with the need to design the adaptation law. MRAC design usually involvesthe following three steps:

● Choose a control law containing variable parameters.● Design an adaptation law for adjusting those parameters.● Analyse the stability properties of the resulting control system.


7.1 MRAC of first-order systems

The basic design idea can be clearly demonstrated by first-order systems. Consider afirst-order system

y + apy = bpu, (7.1)

where y and u ∈ R are the system output and input respectively, and ap and bp areunknown constant parameters with sgn(bp) known. The output y is to follow the outputof the reference model

ym + amym = bmr. (7.2)

The reference model is stable, i.e., am > 0. The signal r is the reference input. Thedesign objective is to make the tracking error e = y − ym converge to 0.

Let us first design a Model Reference Control (MRC), that is, the control designassuming all the parameters are known, to ensure that the output y follows ym.Rearrange the system model as

y + amy = bp

(

u − ap − am

bpy

)

and therefore we obtain

e + ame = bp

(

u − ap − am

bpy − bm

bpr

)

:= bp (u − auy − arr) ,

where

ay = ap − am

bp,

ar = bm

bp.

If all the parameters are known, the control law is designed as

u = arr + ayy (7.3)

and the resultant closed-loop system is given by

e + ame = 0.

The tracking error converges to zero exponentially.One important design principle in adaptive control is the so-called the certainty

equivalence principle, which suggests that the unknown parameters in the controldesign are replaced by their estimates. Hence, when the parameters are unknown,let ar and ay denote their estimates of ar and ay, and the control law, based on thecertainty equivalence principle, is given by

u = arr + ayy. (7.4)

Adaptive control of linear systems 91

Note that the parameters ar and ay are the parameters of the controllers, and theyare related to the original system parameters ap and bp, but not the original systemparameters themselves.

The certainty equivalence principle only suggests a way to design the adaptivecontrol input, not how to update the parameter estimates. Stability issues must beconsidered when deciding the adaptive laws, i.e., the way how estimated parametersare updated. For first-order systems, the adaptive laws can be decided from Lyapunovfunction analysis.

With the proposed adaptive control input (7.4), the closed-loop system dynamicsare described by

e + ame = bp(−ayy − arr), (7.5)

where ar = ar − ar and ay = ay − ay. Consider the Lyapunov function candidate

V = 1

2e2 + |bp|

2γra2

r + |bp|2γy

a2y , (7.6)

where γr and γy are constant positive real design parameters. Its derivative along thetrajectory (7.5) is given by

V = −ame2 + ar

(

|bp|˙ar

γr− ebpr

)

+ ay

(

|bp|˙ay

γy− ebpy

)

.

If we can set

|bp|˙ar

γr− ebpr = 0, (7.7)

|bp|˙ay

γy− ebpy = 0, (7.8)

we have

V = −ame2. (7.9)

Noting that ˙ar = −˙ar and ˙ay = −˙ay, the conditions in (7.7) and (7.8) can be satisfiedby setting the adaptive laws as

˙ar = −sgn(bp)γrer, (7.10)

˙ay = −sgn(bp)γyey. (7.11)

The positive real design parameters γr and γy are often referred to as adaptive gains,as they can affect the speed of parameter adaptation.

From (7.9) and Theorem 4.2, we conclude that the system is Lyapunov stablewith all the variables e, ar and ay bounded, and hence the boundedness of ar and ay.

However, based on the stability theorems introduced in Chapter 4, we cannotconclude anything about the tracking error e other than its boundedness. In orderto do it, we need to introduce an important lemma for stability analysis of adaptivecontrol systems.


Lemma 7.1 (Barbalat’s lemma). If a function f (t) : R −→ R is uniformly continuousfor t ∈ [0, ∞), and

∫∞0 f (t)dt exists, then limt→∞ f (t) = 0.

From (7.9), we can show that∫ ∞

0e2(t)dt = V (0) − V (∞)

am< ∞. (7.12)

Therefore, we have established that e ∈ L2 ∩ L∞ and e ∈ L∞. Since e and e arebounded, e2 is uniformly continuous. Therefore, we can conclude from Barbalat’slemma that limt→∞ e2(t) = 0, and hence limt→∞ e(t) = 0.

We summarise the stability result in the following lemma.

Lemma 7.2. For the first-order system (7.1) and the reference model (7.2), the adap-tive control input (7.4) together with the adaptive laws (7.10) and (7.11) ensures theboundedness of all the variables in the closed-loop system, and the convergence tozero of the tracking error.

Remark 7.1. The stability analysis ensures the convergence to zero of the trackingerror, but nothing can be told about the convergence of the estimated parameters. Theestimated parameters are assured to be bounded from the stability analysis. In general,the convergence of the tracking error to zero and the boundedness of the adaptiveparameters are stability results that we can establish for MRAC. The convergenceof the estimated parameters may be achieved by imposing certain conditions of thereference signal to ensure the system is excited enough. This is similar to the conceptof persistent excitation for system identification. �

Example 7.1. Consider a first-order system

Gp = b

s + a,

where b = 1 and a is an unknown constant parameter. We will design an adaptivecontroller such that the output of the system follows the output of the reference model

Gm = 1

s + 2.

We can directly use the result presented in Lemma 7.2, i.e., we use the adaptivelaws (7.10) and (7.11) and the control input (7.4). Since b is known, we only have oneunknown parameter, and it is possible to design a simpler control based on the samedesign principle.

From the system model, we have

y + ay = u,

which can be changed to

y + 2y = u − (a − 2)y.


Subtracting the reference model

ym + 2ym = r,

we obtain that

e + 2e = u − ayy − r.

where ay = a − 2. We then design the adaptive law and control input as

˙ay = −γyey,

u = ayy + r.

The stability analysis follows the same discussion that leads to Lemma 7.2. Simulationstudy has been carried out with a = −1, γ = 10 and r = 1. The simulation results areshown in Figure 7.1. The figure shows that the estimated parameter converges to thetrue value ay = −3. The convergence of the estimated parameters is not guaranteedby Lemma 7.2. Indeed, some strong conditions on the input or reference signal areneeded to generate enough excitation for the parameter estimation to achieve theconvergence of the estimated parameters in general. �

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

t (s)

y an

d y m

yym

0 2 4 6 8 10−4

−3

−2

−1

0

t (s)

Estim

ated

par

amet

er

Figure 7.1 Simulation results of Example 7.1


7.2 Model reference control

It is clear from MRAC design for first-order systems that an MRC input is designedfirst which contains unknown parameters, and the adaptive control input is thenobtained based on the certainty equivalence principle. Hence, MRC design is thefirst step for MRAC. Furthermore, MRC itself deserves a brief introduction, as it isdifferent from the classical control design methods shown in standard undergraduatetexts. In this section, we will start with MRC for systems with relative degree 1, andthen move on to MRC of systems with high-order relative degrees.

Consider an nth-order system with the transfer function

y(s) = kpZp(s)

Rp(s)u(s), (7.13)

where y(s) and u(s) denote the system output and input in frequency domain; kp is thehigh frequency gain; and Zp and Rp are monic polynomials with orders of n − ρ andn respectively with ρ as the relative degree. The reference model is chosen to havethe same relative degree of the system, and is described by

ym(s) = kmZm(s)

Rm(s)r(s), (7.14)

where ym(s) is the reference output for y(s) to follow; r(s) is a reference input; andkm > 0 and Zm and Rm are monic Hurwitz polynomials.

Remark 7.2. A monic polynomial is a polynomial whose leading coefficient, thecoefficient of the highest power, is 1. A polynomial is said to be Hurwitz if all its rootsare with negative real parts, i.e., its roots locate in the open left half of the complexplane. The high-frequency gain is the leading coefficient of the numerator of a transferfunction. �

The objective of MRC is to design a control input u such that the output of thesystem asymptotically follows the output of the reference model, i.e., limt→∞ (y(t) −ym(t)) = 0.

Note that in this chapter, we abuse the notations of y, u and r by using samenotations for the functions in time domain and their Laplace transformed functionsin the frequency domain. It should be clear from the notations that y(s) is the Laplacetransform of y(t) and similarly for u and r.

To design MRC for systems with ρ = 1, we follow a similar manipulation to thefirst-order system by manipulating the transfer functions. We start with

y(s)Rp(s) = kpZp(s)u(s)

and then

y(s)Rm(s) = kpZp(s)u(s) − (Rp(s) − Rm(s))y(s).


Note that Rp(s) − Rm(s) is a polynomial with order n − 1, and Rm(s) − Rp(s)Zm(s) is a proper

transfer function, as Rp(s) and Rm(s) are monic polynomials. Hence, we can write

y(s)Rm(s) = kpZm(s)(

Zp(s)

Zm(s)u(s) + Rm(s) − Rp(s)

Zm(s)y(s)

)

.

If we parameterise the transfer functions as

Zp(s)

Zm(s)= 1 − θT

1 α(s)

Zm(s),

Rm(s) − Rp(s)

Zm(s)y(s) = −θ

T2 α(s)

Zm(s)y(s) − θ3,

where θ1 ∈ Rn−1, θ2 ∈ R

n−1 and θ3 ∈ R are constants and

α(s) = [sn−2, . . . , 1]T ,

we obtain that

y(s) = kpZm(s)

Rm(s)

(

u(s) − θT1 α(s)

Zm(s)u(s) − θT

2 α(s)

Zm(s)y(s) − θ3y(s)

)

. (7.15)

Hence, we have the dynamics of tracking error given by

e1(s) = kpZm(s)

Rm(s)

(

u(s) − θT1 α(s)

Zm(s)u(s) − θT

2 α(s)

Zm(s)y(s) − θ3y(s) − θ4r

)

, (7.16)

where e1 = y − ym and θ4 = kmkp

.The control input for MRC is given by

u(s) = θT1 α(s)

Zm(s)u(s) + θT

2 α(s)

Zm(s)y + θ3y + θ4r(s)

:= θTω, (7.17)

where

θT = [θT1 , θT

4 , θ3, θ4],

ω = [ωT1 ,ωT

2 , y, r]T ,

with

ω1 = α(s)

Zm(s)u,

ω2 = α(s)

Zm(s)y.


Remark 7.3. The control design shown in (7.17) is a dynamic feedback con-

troller. Each element in the transfer matrixα(s)

Zm(s)is strictly proper, i.e.,

with relative degree greater than or equal to 1. The total number of parameters inθ equals 2n. �

Lemma 7.3. For the system (7.13) with relative degree 1, the control input (7.17)solves MRC problem with the reference model (7.14) and limt→∞ (y(t) − ym(t)) = 0.

Proof. With the control input (7.17), the closed-loop dynamics are given by

e1(s) = kpZm(s)

Rm(s)ε(s),

where ε(s) denotes exponentially convergent signals due to non-zero initial values.The reference model is stable, and then the track error e1(t) converges to zeroexponentially. �

Example 7.2. Design MRC for the system

y(s) = s + 1

s2 − 2s + 1u(s)

with the reference model

ym(s) = s + 3

s2 + 2s + 3r(s).

We follow the procedures shown early to obtain the MRC control. From the transferfunction of the system, we have

y(s)(s2 + 2s + 3) = (s + 1)u(s) + (4s + 2)y(s),

which leads to

y(s) = s + 3

s2 + 2s + 3

(s + 1

s + 3u(s) + 4s + 2

s + 3y(s)

)

= s + 3

s2 + 2s + 3

(

u(s) − 2

s + 3u(s) − 10

s + 3y(s) + 4y(s)

)

.

Subtracting it by the reference model, we have

e1(s) = s + 3

s2 + 2s + 3

(

u(s) − 2

s + 3u(s) − 10

s + 3y(s) + 4y(s) − r(s)

)

,

which leads to the MRC control input

u(s) = 2

s + 3u(s) + 10

s + 3y(s) − 4y(s) + r(s)

= [2 10 − 4 1][ω1(s) ω2(s) y(s) r(s)]T ,


where

ω1(s) = 1

s + 3u(s),

ω2(s) = 1

s + 3y(s).

Note that the control input in the time domain is given by

u(s) = [2 10 − 4 1][ω1(t) ω2(t) y(t) r(t)]T ,

whereω1 = −3ω1 + u,

ω2 = −3ω2 + y.

�For a system with ρ > 1, the input in the same format as (7.17) can be obtained.

The only difference is that Zm is of order n − ρ < n − 1. In this case, we let P(s) be amonic and Hurwitz polynomial with order ρ − 1 so that Zm(s)P(s) is of order n − 1.We adopt a slightly different approach from the case of ρ = 1.

Consider the identity

y(s) = Zm(s)

Rm(s)

(Rm(s)P(s)

Zm(s)P(s)y(s)

)

= Zm(s)

Rm(s)

(Q(s)Rp(s) +�(s)

Zm(s)P(s)y(s)

)

. (7.18)

Note that the second equation in (7.18) follows from the identity

Rm(s)P(s) = Q(s)Rp(s) +�(s),

where Q(s) is a monic polynomial with order n − ρ − 1, and �(s) is a polynomialwith order n − 1. In fact Q(s) can be obtained by dividing Rm(s)P(s) by Rp(s) usinglong division, and�(s) is the remainder of the polynomial division. From the transferfunction of the system, we have

Rp(s)y(s) = kpZp(s)u(s).

Substituting it into (7.18), we have

y(s) = kpZm(s)

Rm(s)

(Q(s)Zp(s)

Zm(s)P(s)u + k−1

p �(s)

Zm(s)P(s)y(s)

)

.

Similar to the case for ρ = 1, if we parameterise the transfer functions as

Q(s)Zp(s)

Zm(s)P(s)= 1 − θT

1 α(s)

Zm(s)P(s),

k−1p �

Zm(s)P(s)= − θT

2 α(s)

Zm(s)P(s)− θ3


where θ1 ∈ Rn−1 and θ2 ∈ R

n−1 and θ3 ∈ R are constants and

α(s) = [sn−2, . . . , 1]T ,

we obtain that

y(s) = kpZm(s)

Rm(s)

(

u(s) − θT1 α(s)

Zm(s)P(s)u(s) − θT

2 α(s)

Zm(s)P(s)y(s) − θ3y(s)

)

.

Hence, we have the dynamics of tracking error given by

e1(s) = kpZm(s)

Rm(s)

(

u(s) − θT1 α(s)

Zm(s)P(s)u(s) − θT

2 α(s)

Zm(s)P(s)y(s) − θ3y(s) − θ4r

)

,

where e1 = y − ym and θ4 = kmkp

. The control input is designed as

u = θT1 α(s)

Zm(s)P(s)u + θT

2 α(s)

Zm(s)P(s)y + θ3y + θ4r

:= θTω (7.19)

with the same format as (7.17) except

ω1 = α(s)

Zm(s)P(s)u,

ω2 = α(s)

Zm(s)P(s)y.

Remark 7.4. The final control input is in the same format as shown for the caseρ = 1. The filters for w1 and w2 are in the same order as in the case for ρ = 1, as theorder of Zm(s)P(s) is still n − 1. �

Lemma 7.4. For the system (7.13) with relative degreeρ > 1, the control input (7.19)solves MRC problem with the reference model (7.14) and limt→∞ (y(t) − ym(t)) = 0.

The proof is the same as the proof for Lemma 7.3.

Example 7.3. Design MRC for the system

y(s) = 1

s2 − 2s + 1u

with the reference model

ym(s) = 1

s2 + 2s + 3r.

The relative degree of the system is 2. We set P = s + 1. Note that

(s2 + 2s + 3)(s + 1) = (s + 5)(s2 − 2s + 1) + (14s − 2).


From the reference model, we have

y(s) = 1

s2 + 2s + 3

((s2 + 2s + 3)(s + 1)

s + 1y(s)

)

= 1

s2 + 2s + 3

((s + 5)(s2 − 2s + 1)y(s) + (14s − 2)y(s)

s + 1

)

= 1

s2 + 2s + 3

((s + 5)u(s) + (14s − 2)y(s)

s + 1

)

= 1

s2 + 2s + 3

(

u(s) + 4

s + 1u(s) − 16

1

s + 1y(s) + 14y(s)

)

.

The dynamics of the tacking error are given by

e1(s) = 1

s2 + 2s + 3

(

u(s) + 4

s + 1u(s) − 16

1

s + 1y(s) + 14y(s) − r(s)

)

.

We can then design the control input as

u = [−1, −16, 14, 1][ω1, ω2, y, r]T

with

ω1 = 1

s + 1u,

ω2 = 1

s + 1y.

�

7.3 MRAC of linear systems with relative degree 1

Adaptive control deals with uncertainties in terms of unknown constant parameters.It may be used to tackle some changes or variations in model parameters in adaptivecontrol application, but the stability analysis will be carried under the assumption theparameters are constants. There are other common assumptions for adaptive controlwhich are listed below:

● the known system order n● the known relative degree ρ● the minimum phase of the plant● the known sign of the high frequency gain sgn(kp)

In this section, we present MRAC design for linear systems with relativedegree 1.


y(s) = kpZp(s)

Rp(s)u(s), (7.20)


where y(s) and u(s) denote the system output and input in frequency domain; kp isthe high frequency gain; and Zp and Rp are monic polynomials with orders of n − 1and n respectively. This system is assumed to be minimum phase, i.e., Zp(s) is aHurwitz polynomial, and the sign of the high-frequency gain, sgn(kp), is known. Thecoefficients of the polynomials and the value of kp are constants and unknown. Thereference model is chosen to have the relative degree 1 and strictly positive real, andis described by

ym(s) = kmZm(s)

Rm(s)r(s), (7.21)

where ym(s) is the reference output for y(s) to follow, r(s) is a reference input, andZm(s) and Rm(s) are monic polynomials and km > 0. Since the reference model isstrictly positive real, Zm and Rm are Hurwitz polynomials.

MRC shown in the previous section gives the control design in (7.17). Based onthe certainty equivalence principle, we design the adaptive control input as

u(s) = θTω, (7.22)

where θ is an estimate of the unknown vector θ ∈ R2n, and ω is given by

ω = [ωT1 ,ωT

2 , y, r]T

with

ω1 = α(s)

Zm(s)u,

ω2 = α(s)

Zm(s)y.

With the designed adaptive control input, it can be obtained, from the trackingerror dynamics shown in (7.16), that

e1(s) = kpZm(s)

Rm(s)(θTω − θTω)

= kmZm(s)

Rm(s)

(

− kp

kmθTω

)

(7.23)

where θ = θ − θ .To analyse the stability using a Lyapunov function, we put the error dynamics in

the state space form as

e = Ame + bm

(− kp

kmθTω

)

e1 = cTme

(7.24)

where (Am, bm, cm) is a minimum state space realisation of kmZm(s)Rm(s) , i.e.,

cTm(sI − Am)−1bm = km

Zm(s)

Rm(s).


Since (Am, bm, cm) is a strictly positive real system, from Kalman–Yakubovich lemma(Lemma 5.4), there exist positive definite matrices P and Q such that

ATmPm + PmAm = −Qm, (7.25)

Pmbm = cm. (7.26)

Define a Lyapunov function candidate as

V = 1

2eT Pme + 1

2

∣∣∣∣

kp

km

∣∣∣∣ θ

T−1θ ,

where ∈ R2n is a positive definite matrix. Its derivative is given by

V = 1

2eT (AT

mPm + PmAm)e + eT Pmbm

(

− kp

kmθTω

)

+∣∣∣∣

kp

km

∣∣∣∣ θ

T−1 ˙θ

Using the results from (7.25) and (7.26), we have

V = −1

2eT Qme + e1

(

− kp

kmθTω

)

+∣∣∣∣

kp

km

∣∣∣∣ θ

T−1 ˙θ

= −1

2eT Qme +

∣∣∣∣

kp

km

∣∣∣∣ θ

T(−1 ˙

θ − sgn(kp)e1ω).

Hence, the adaptive law is designed as

˙θ = −sgn(kp)e1ω, (7.27)

which results in

V = −1

2eT Qme.

We can now conclude the boundedness of e and θ . Furthermore it can be shown thate ∈ L2 and e1 ∈ L∞. Therefore, from Barbalat’s lemma we have limt→∞ e1(t) = 0.The boundedness of other system state variables can be established from the minimum-phase property of the system.

We summarise the stability analysis for MRAC of linear systems with relativedegree 1 in the following theorem.

Theorem 7.5. For the first-order system (7.20) and the reference model (7.21), theadaptive control input (7.22) together with the adaptive law (7.27) ensures the bound-edness of all the variables in the closed-loop system, and the convergence to zero ofthe tracking error.

Remark 7.5. The stability result shown in Theorem 7.5 only guarantees the conver-gence of the tracking error to zero, not the convergence of the estimated parameters.In the stability analysis, we use Kalman–Yakubovich lemma for the definition ofLyapunov function and the stability proof. That is why we choose the reference modelto be strictly positive real. From the control design point of view, we do not needto know the actual values of Pm and Qm, as long as they exist, which is guaranteed


by the selection of a strictly positive real model. Also it is clear from the stabilityanalysis, that the unknown parameters must be constant. Otherwise, we would not

have ˙θ = − ˙

θ . �

7.4 MRAC of linear systems with high relatives

In this section, we will introduce adaptive control design for linear systems with theirrelative degrees higher than 1. Similar to the case for relative degree 1, the certaintyequivalence principle can be applied to the control design, but the designs of theadaptive laws and the stability analysis are much more involved, due to the higherrelative degrees. One difficulty is that there is not a clear choice of Lyapunov functioncandidate as in the case of ρ = 1.


y(s) = kpZp(s)

Rp(s)u(s), (7.28)

where y(s) and u(s) denote the system output and input in frequency domain, kp isthe high frequency gain, Zp and Rp are monic polynomials with orders of n − ρ andn respectively, with ρ > 1 being the relative degree of the system. This system isassumed to be minimum phase, i.e., Zp(s) is Hurwitz polynomial, and the sign of thehigh-frequency gain, sgn(kp), is known. The coefficients of the polynomials and thevalue of kp are constants and unknown. The reference model is chosen as

ym(s) = kmZm(s)

Rm(s)r(s) (7.29)

where ym(s) is the reference output for y(s) to follow; r(s) is a reference input; andZm(s) and Rm(s) are monic polynomials with orders n − ρ and n respectively andkm > 0. The reference model (7.29) is required to satisfy an additional condition thatthere exists a monic and Hurwitz polynomial P(s) of order n − ρ − 1 such that

ym(s) = kmZm(s)P(s)

Rm(s)r(s) (7.30)

is strictly positive real. This condition also implies that Zm and Rm are Hurwitzpolynomials.

MRC shown in the previous section gives the control design in (7.19). We designthe adaptive control input, again using the certainty equivalence principle, as

u = θTω, (7.31)

where θ is an estimate of the unknown vector θ ∈ R2n, and ω is given by

ω = [ωT1 ,ωT

2 , y, r]T


with

ω1 = α(s)

Zm(s)P(s)u,

ω2 = α(s)

Zm(s)P(s)y.

The design of adaptive law is more involved, and we need to examine the dynamicsof the tacking error, which are given by

e1 = kpZm

Rm(u − θTφ)

= kmZmP(s)

Rm

(k(uf − θTφ)

), (7.32)

where

k = kp

km, uf = 1

P(s)u and φ = 1

P(s)ω.

An auxiliary error is constructed as

ε = e1 − kmZmP(s)

Rm

(k(uf − θTφ)

)− km

ZmP(s)

Rm

(εn2

s

), (7.33)

where k is an estimate of k , n2s = φTφ + u2

f . The adaptive laws are designed as

˙θ = −sgn(bp)εφ, (7.34)

˙k = γ ε(uf − θTφ). (7.35)

With these adaptive laws, a stability result can be obtained for the boundedness ofparameter estimates and the convergence of the tracking error. For the completeness,we state the theorem below without giving the proof.

Theorem 7.6. For the system (7.28) and the reference model (7.29), the adaptivecontrol input (7.31) together with the adaptive laws (7.34) and (7.35) ensures theboundedness of all the variables in the closed-loop system, and the convergence tozero of the tracking error.

7.5 Robust adaptive control

Adaptive control design and its stability analysis have been carried out under thecondition that there is only parametric uncertainty in the system. However, manytypes of non-parametric uncertainties do exist in practice. These include

● high-frequency unmodelled dynamics, such as actuator dynamics or structuralvibrations

● low-frequency unmodelled dynamics, such as Coulomb frictions


● measurement noise● computation roundoff error and sampling delay

Such non-parametric uncertainties will affect the performance of adaptive controlsystems when they are applied to practical systems. They may cause instability. Thedifference between adaptive control of linear systems and other design methods isparameter estimation. Without specific requirement for input signals, such as persis-tent excitation, we can only establish the boundedness of estimated parameters, andthe asymptotic convergence to zero of the output error. Therefore, the closed-loopadaptive system is not even asymptotically stable. For other design methods for linearsystems without parameter adaptation, the closed-loop systems are normally expo-nentially stable. For a linear system with exponential stability, the state inherentlyremain bounded under any bounded input. This is not the case for a linear systemunder adaptive control, due to the difference in the stability properties.

Many non-parametric uncertainties can be represented by a bounded disturbanceto the nominal system. Even a bounded disturbance can cause serious problem inparameter adaptation. Let us consider a simple example.

Consider the system output is described by

y = θω. (7.36)

The adaptive law

˙θ = γ εω, (7.37)

where

ε = y − θω

will render the convergence of the estimate θ by taking

V = 12γθ 2

as a Lyapunov function candidate and the analysis

V = −θ (y − θω)ω

= −θ2ω2. (7.38)

The boundedness of θ can then be concluded, no matter what the signal ω is.Now, if the signal is corrupted by some unknown bounded disturbance d(t),

i.e.,

y = θω + d(t).


the same adaptive will have a problem. In this case,

V = −θ (y − θω)ω

= −θ (θω + d − θω)ω

= −θ 2ω2 − θdω

= − θ2ω2

2− 1

2(θω + d)2 + d2

2.

From the above analysis, we cannot conclude the boundedness of θ even though ω isbounded. In fact, if we take θ = 2, γ = 1 and ω = (1 + t)−1/2 ∈ L∞ and let

d(t) = (1 + t)−1/4

(5

4− 2(1 + t)−1/4

)

,

it can then be obtained that

y(t) = 5

4(1 + t)−1/4, → 0 as t → ∞,

˙θ = 5

4(1 + t)−3/4 − θ (1 + t)−1

which has a solution

θ = (1 + t)1/4 → ∞ as t → ∞. (7.39)

In this example, we have observed that adaptive law designed for the disturbance-free system fails to remain bounded even though the disturbance is bounded andconverges to zero as t tends to infinity.

Remark 7.6. If ω is a constant, then from (7.38) we can show that the estimateexponentially converges to the true value. In this case, there is only one unknownparameter. If θ is a vector, the requirement for the convergence is much stronger. Inthe above example, ω is bounded, but not in a persistent way. It does demonstrate thateven a bounded disturbance can cause the estimated parameter divergent. �

Robust adaptive control issue is often addressed by modifying parameter adaptivelaws to ensure the boundedness of estimated parameters. It is clear from the exampleshown above that bounded disturbance can cause estimated parameters unbounded.Various robust adaptive laws have been introduced to keep estimated parametersbounded in the presence of bounded disturbances. We will show two strategies usingthe following simple model:

y = θω + d(t) (7.40)

with d as a bounded disturbance. In the following we keep using ε = y − θω andV = θ2

2γ . Once the basic ideas are introduced, it is not difficult to extend the robustadaptive laws to adaptive control of dynamic systems.

Dead-zone modification is a modification to the parameter adaptive law to stopparameter adaptation when the error is very close to zero. The adaptive law ismodified as


˙θ =

{γ εω |ε| > g0 |ε| ≤ g

(7.41)

where g is a constant satisfying g > |d(t)| for all t. For ε > g, we have

V = −θ εω= −(θω − θω)ε

= −(y − d(t) − θω)ε

= −(ε − d(t))ε

< 0.

Therefore, we have

V

{< 0, |ε| > g= 0, |ε| ≤ g

and we can conclude that V is bounded. Intuitively, when the error ε is small, thebounded disturbance can be more dominant, and therefore, the correct adaptationdirection is corrupted by the disturbance. In such a case, a simple strategy would bejust to stop parameter adaptation. The parameter adaptation stops in the range |ε| ≤ g,and for this reason, this modification takes the name ‘dead-zone’ modification. Thesize of the dead zone depends on the size of the bounded disturbances. One problemwith the dead-zone modification is that the adaptive law is discontinuous, and thismay not be desirable in some applications.

σ -Modification is another strategy to ensure the boundedness of estimated param-eters. The adaptive law is modified by adding an additional term −γ σ θ to the normaladaptive law as

˙θ = γ εω − γ σ θ (7.42)

where σ is a positive real constant. In this case, we have

V = −(ε − d(t))ε + σ θ θ

= −ε2 + d(t)ε − σ θ 2 + σ θθ

≤ −ε2

2+ d2

0

2− σ

θ 2

2+ σ

θ 2

2

≤ −σγV + d20

2+ σ

θ 2

2, (7.43)

where d0 ≥ |d(t)|, ∀t ≥ 0. Applying Lemma 4.5 (comparison lemma) to (7.43),we have

V (t) ≤ e−σγ t + V (0)∫ t

0e−σγ (t−τ )

(d2

0

2+ σ

θ 2

2

)

dτ ,


and therefore we can conclude that V ∈ L∞, which implies the boundedness of theestimated parameter. A bound can be obtained for the bounded parameter as

V (∞) ≤ 1

σγ

(d2

0

2+ σ

θ 2

2

)

. (7.44)

Note that this modification does not need a bound for the bounded disturbances, andalso it provides a continuous adaptive law. For these reasons, σ -modification is oneof the most widely used modifications for parameter adaptation.

Remark 7.7. We re-arrange the adaptive law (7.42) as

˙θ + γ σ θ = γ εω.

Since (γ σ ) is a positive constant, the adaptive law can be viewed as a stable first-orderdynamic system with (εω) as the input and θ as the output. With a bounded input,obviously θ remains bounded. �

The robust adaptive laws introduced here can be applied to various adaptivecontrol schemes. We demonstrate the application of a robust adaptive law to MRACwith ρ = 1. We start directly from the error model (7.24) with an additional boundeddisturbance

e = Ame + bm(−k θTω + d(t))

e1 = cTme, (7.45)

where k = kp/km and d(t) are a bounded disturbance with the bound d0, which rep-resents the non-parametric uncertainty in the system. As discussed earlier, we need arobust adaptive law to deal with the bounded disturbances. If we take σ -modification,then the robust adaptive law is

˙θ = −sgn(kp)e1ω − σθ. (7.46)

We will show that this adaptive law will ensure the boundedness of the variables.Let

V = 1

2eT Pe + 1

2|k|θT−1θ .

Similar to the analysis leading to Theorem 7.5, the derivative of V is obtained as

V = −1

2eT Qe + e1( − k θTω + d) + |k|θT−1 ˙

θ

≤ −1

2λmin(Q)‖e‖2 + e1d + |k|σ θT θ

≤ −1

2λmin(Q)‖e‖2 + |e1d| − |k|σ‖θ‖2 + |k|σ θT θ.


Note that

|e1d| ≤ 1

4λmin(Q)‖e‖2 + d2

0

λmin(Q),

|θT θ | ≤ 1

2‖θ‖2 + 1

2‖θ‖2.

Hence, we have

V ≤ −1

4λmin(Q)‖e‖2 − |k|σ

2‖θ‖2 + d2

0

λmin(Q)+ |k|σ

2‖θ‖2

≤ −αV + d20

λmin(Q)+ |k|σ

2‖θ‖2,

where α is a positive real and

α = min{(1/2)λmin(Q), |k|σ }max{λmax(P), |k|/λmin()} .

Therefore, we can conclude the boundedness of V from Lemma 4.5 (comparisonlemma), which further implies the boundedness of the tracking error e1 and theestimate θ .

From the above analysis, it is clear that the adaptive law with σ -modificationensures the boundedness of all the variables in the closed-loop adaptive control system.It is worth noting that the output tracking error e1 will not asymptotically convergeto zero, even though the bounded disturbance d(t) becomes zero. That is the price topay for the robust adaptive scheme.

Chapter 8

Nonlinear observer design

Observers are needed to estimate unmeasured state variables of dynamic systems.They are often used for output feedback control design when only the outputs areavailable for the control design. Observers can also be used for other estimation pur-poses such as fault detection and diagnostics. There are many results on nonlinearobserver design in literature, and in this chapter, we can introduce only a number ofresults. Observer design for linear systems is briefly reviewed before the introductionof observers with linear error dynamics. We then introduce another observer designmethod based on Lyapunov’s auxiliary theorem, before the observer design for sys-tems with Lipschitz nonlinearities. At the end of this chapter, adaptive observers arebriefly described.

8.1 Observer design for linear systems

We briefly review the results for linear systems. Consider

x = Ax

y = Cx,(8.1)

where x ∈ Rn is the state; y ∈ R

m is the system output with m < n; and A ∈ Rn×n

and C ∈ Rm×n are constant matrices. From linear system theory, we know that this

system, or the pair (A, C), is observable if the matrix

Po =

⎡

⎢⎢⎢⎣

CCA...

CAn−1

⎤

⎥⎥⎥⎦

has rank n. The observability condition is equivalent to that the matrix[λI − A

C

]

has rank n for any value of λ ∈ C.When the system is observable, an observer can be designed as

˙x = Ax + L(y − Cx), (8.2)


where x ∈ Rn is the estimate of the state x, and L ∈ R

n×m is the observer gain suchthat (A − LC) is Hurwitz.

For the observer (8.2), it is easy to see the estimate x converges to x asymptoti-cally. Let x = x − x, and we can obtain

˙x = (A − LC)x.

Remark 8.1. In the observer design, we do not consider control input terms in thesystem in (8.1), as they do not affect the observer design for linear systems. In fact, ifBu term is added to the right-hand side of the system (8.1), for the observer design,we can simply add it to the right-hand side of the observer in (8.2), and the observererror will still converge to zero exponentially. �

Remark 8.2. The observability condition for the observer design of (8.1) can berelaxed to the detectability of the system, or the condition that (A, C) is detectable,for the existence of an observer gain L such that (A − LC) is Hurwitz. Detectability isweaker than observability, and it basically requires the unstable modes of the systemobservable. The pair (A, C) is detectable if the matrix

[λI − A

C

]

has rank n for any λ in the closed right half of the complex plan. Some other designmethods shown in this chapter also need only the condition of detectability, although,for simplicity, we state the requirement for the observability. �

There is another approach to full-state observer design for linear systems (8.1).Consider a dynamic system

z = Fz + Gy, (8.3)

where z ∈ Rn is the state; F ∈ R

n×n is Hurwitz; and G ∈ Rn×m. If there exists an

invertible matrix T ∈ Rn×n such that Z converges to Tx, then (8.3) is an observer with

the state estimate given by x = T −1z. Let

e = Tx − z.

A direction evaluation gives

e = TAx − (Fz + GCx)

= F(Tx − z) + (TA − FT − GC)x

= Fe + (TA − FT − GC)x.

If we have

TA − FT − GC = 0,

the system (8.3) is an observer with an exponentially convergent estimation error. Wehave the following lemma to summarise this observer design.

Nonlinear observer design 111

Lemma 8.1. The dynamic system (8.3) is an observer for the system (8.1) if and onlyif F is Hurwitz and there exists an invertible matrix T such that

TA − FT = GC. (8.4)

Proof. The sufficiency has been shown in the above analysis. For necessity, we onlyneed to observe that if any of the conditions is not satisfied, we cannot guaran-tee the convergence of e to zero for a general linear system (8.1). Indeed, if (8.4)is not satisfied, then e will be a state variable with a non-zero input, and we can setup a case such that e does not converge to zero. So does for the condition that F isHurwitz. �

How to find matrices F and G such that the condition (8.4) is satisfied? We listthe result in the following lemma without the proof.

Lemma 8.2. Suppose that F and A have exclusively different eigenvalues. The nec-essary condition for the existence of a non-singular solution T to the matrix equation(8.4) is that the pair (A, C) is observable and the pair (F , G) is controllable. Thiscondition is also sufficient when the system (8.1) is single output, i.e., m = 1.

This lemma suggests that we can choose a controllable pair (F , G) and make surethat the eigenvalues of F are different from those of A. An observer can be designed ifthere is a solution of T from (8.4). For single output system, the solution is guaranteed.

8.2 Linear observer error dynamics with output injection

Now consider

x = Ax + φ(y, u)

y = Cx,(8.5)

where x ∈ Rn is the state, y ∈ R

m is the system output with m < n, u ∈ Rs is the

control input, or other known variables, A ∈ Rn×n and C ∈ R

m×n are constant matricesand φ : R

m × Rs → R

n is a continuous function. This system is a nonlinear system.However, comparing with the system (8.1), the only difference is the additional termφ(y, u). The system (8.5) can be viewed as the linear system (8.1) perturbed by thenonlinear term φ(y, u).

If the pair (A, C) is observable, we can design an observer as

˙x = Ax + L(y − Cx) + φ(y, u), (8.6)

where x ∈ Rn is the estimate of the state x and L ∈ R

n×m is the observer gain suchthat (A − LC) is Hurwitz. The only difference between this observer and the one in(8.2) is the nonlinear term φ(y, u). It can be seen that the observer error still satisfies

˙x = (A − LC)x.


Note that even the system (8.5) is nonlinear, the observer error dynamics arelinear. The system in the format of (8.5) is referred to as the system with linearobserver errors. More specifically, if we drop the control input u in the function φ,the system is referred to as the output injection form for observer design.

Let us summarise the result in the following proposition.

Proposition 8.3. For the nonlinear system (8.5), a full-state observer can be designedas in (8.6) if (A, C) is observable. Furthermore, the observer error dynamics are linearand exponentially stable.

For a nonlinear system in a more general form, there may exist a state transfor-mation to put the system in the format of (8.5), and then the proposed observer canbe applied.

We will introduce the conditions for the existence of a nonlinear state transfor-mation to put the system in the format shown in (8.6). Here, we only consider singleoutput case, and for the simplicity, we do not consider the system with a control input.

The system under consideration is described by

x = f (x)

y = h(x),(8.7)

where x ∈ Rn is the state vector, y ∈ R is the output, f : R

n → Rn and h : R

n → R

are continuous nonlinear functions with f (0) = 0, and h(0) = 0. We will show thatunder what conditions there exists a state transformation

z = �(x), (8.8)

where � : Rn → R

n, such that the transformed system is in format

z = Az + φ(Cz)

y = Cz,(8.9)

where A ∈ Rn×n, C ∈ R

1×n, (A, C) is observable and φ : R → Rn. Without loss of

generality, we take the transformed system (8.9) as

z1 = z2 + φ1(y)z2 = z3 + φ2(y). . .

zn−1 = zn + φn−1(y)zn = φn(y)y = z1.

(8.10)

This system is in the output injection form for nonlinear observer design.

Remark 8.3. Assume that the system (8.9) is different from (8.10) with (A, C) asa general observable pair, instead of having the special format implied by (8.10).


In this case, we use a different variable z to denote the state for (8.10), and it can bewritten as

˙z = Az + φ(y)

y = Cz.(8.11)

It is clear that there exists a linear transformation

z = Tz,

which transforms the system (8.9) to the system (8.11), because (A, C) is observable.The transformation from (8.7) to (8.11) is given by

z = Tφ(x) := �(x).

Therefore, if there exists a nonlinear transformation from (8.7) to (8.9), there mustexist a nonlinear transformation from (8.7) to (8.11), and vise versa. That is whywe can consider the transformed system in the format of (8.10) without loss ofgenerality. �

If the state transformation transforms the system (8.7) to (8.9), we must have

[∂�(x)

∂xf (x)

]

x=�(z)

= Az + φ(Cz)

h(�(z)) = Cz,(8.12)

where � = �−1. For notational convenience, let us denote

f (z) = Az + φ(Cz)

h(z) = Cz.

From the structure shown in (8.10), we can obtain

h(z) = z1,

Lf h(z) = z2 + φ1(z1),

L2fh(z) = z3 + ∂φ1

∂z1(z2 + φ1(z1))

:= z3 + φ2(z1, z2),

. . .

Ln−1f

h(z) = zn +n−2∑

k=1

∂φn−2

∂zk(zk+1 + φk (z1))

:= zn + φn−1(z1, . . . , zn−1).


From the above expression, it is clear that

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂ h

∂z∂Lf h(z)

∂z...

∂Ln−1f

h(z)

∂z

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎣

1 0 . . . 0∗ 1 . . . 0...

.... . .

...

∗ ∗ . . . 1

⎤

⎥⎥⎥⎦

This implies that

dh, dLf h, . . . , dLn−1f

h

are linearly independent. This property is invariant under state transformation, andtherefore, we need the condition under the coordinate x, that is

dh, dLf h, . . . , dLn−1f h

are linearly independent. Indeed, we have

Ln−1f h(x) = (Ln−1

fh(z))z=�(x), for k = 0, 1, . . . , n − 1

and therefore

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂h(x)

∂x∂Lf h(x)

∂x...

∂Ln−1f h(x)

∂x

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂ h(z)

∂z∂Lf h(z)

∂z...

∂Ln−1f

h(z)

∂z

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z=�(x)

∂�(x)

∂x.

The linear independence of dh, dLf h, . . . , dLn−1f h is a consequence of the observ-

ability of (A, C) in (8.9). In some literature, this linear independence condition isdefined as the observability condition for nonlinear system (8.7). Unlike linear sys-tems, this condition is not enough to design a nonlinear observer. We state necessaryand sufficient conditions for the transformation to the output injection form in thefollowing theorem.

Theorem 8.4. The nonlinear system (8.7) can be transformed to the output injectionform in (8.10) if and only if

● the differentials dh, dLf h, . . . , dLn−1f h are linearly independent

● there exists a map � : Rn → R

n such that


∂�(z)

∂z=[adn−1

−f r, . . . , ad−f r, r]

x=�(z), (8.13)

where r is a vector field solved from

⎡

⎢⎢⎢⎣

dhdLf h...

dLn−1f h

⎤

⎥⎥⎥⎦

r =

⎡

⎢⎢⎢⎣

0...

01

⎤

⎥⎥⎥⎦. (8.14)

Proof. Sufficiency. From the first condition and (8.14), we can show, in a similar wayas for (6.12), that

⎡

⎢⎢⎢⎣

dh(x)dLf h(x)...

dLρ−1f h

⎤

⎥⎥⎥⎦

[adn−1

−f r(x) . . . ad−f r(x) r(x)]

=

⎡

⎢⎢⎢⎣

1 0 . . . 0∗ 1 . . . 0...

.... . .

...

∗ ∗ . . . 1

⎤

⎥⎥⎥⎦. (8.15)

Therefore,[adn−1

−f r(x) . . . ad−f r(x) r(x)]

has full rank. This implies that there exists

an inverse mapping for �. Let us denote it as � = �−1, and hence we have

∂�(x)

∂x

[adn−1

−f r(x) . . . ad−f r(x) r(x)] = I . (8.16)

Let us define the state transformation as z = �(x) and denote the functions after thistransformation as

f (z) =[∂�(x)

∂xf (x)

]

x=�(z)

,

h(z) = h(�(z)).

We need to show that the functions f and h are in the format of the output injectionform as in (8.10). From (8.16), we have

∂�(x)

∂xadn−k

−f r(x) = ek , for k = 1, . . . , n,

where ek denotes the kth column of the identity matrix. Hence, we have, fork = 1, . . . , n − 1,


[∂�(x)

∂xadn−k

−f r(x)]

x=�(z)

=[∂�(x)

∂x[− f (x), adn−(k+1)

−f r(x)]]

x=�(z)

=[

−∂�(x)

∂xf (x),

∂�(x)

∂xadn−(k+1)

−f r(x)]

x=�(z)

= [−f (z), ek+1

]

= ∂ f (z)

∂zk+1.

This implies that

∂ f (z)

∂zk+1= ek , for k = 1, . . . , n − 1,

i.e.,

∂ f (z)

∂z=

⎡

⎢⎢⎢⎢⎢⎣

∗ 1 0 . . . 0∗ 0 1 . . . 0...

......

. . ....

∗ 0 0 . . . 1∗ 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

.

Therefore, we have shown that f is in the output injection form.From the second condition, we have

∂�(z)

∂zk= [adn−k

−f (x)r(x)]x=�(z) for k = 1, . . . , n.

Hence, we obtain that, for k = 1, . . . , n,

∂ h(z)

∂zk=[∂h(x)

∂x

]

x=�(z)

∂�(z)

∂zk

=[∂h(x)

∂x

]

x=�(z)

[adn−k−f (x)r(x)]x=�(z)

= [Ladn−k−f (x)r(x)h(x)]x=�(z).

Furthermore from (8.15), we have

Ladn−1−f (x)r(x)h(x) = 1,

Ladn−k−f (x)r(x)h(x) = 0, for k = 2, . . . , n.

Therefore, we have

∂ h(z)

∂z= [1, 0, . . . , 0].

This concludes the proof for sufficiency.


Necessity. The discussion prior to this theorem shows that the first condition isnecessary. Assume that there exists a state transformation z = �(x) to put the systemin the output injection form, and once again, we denote

f (z) =[∂�(x)

∂xf (x)

]

x=�(z)

h(z) = h(�(z)),

where � = �−1. We need to show that when the functions f and h are in the formatof the output injection form, the second condition must hold. Let

g(x) =[∂�(z)

∂zn

]

z=�(x)

.

From the state transformation, we have

f (x) =[∂�(z)

∂zf (z)

]

z=�(x)

.

Therefore, we can obtain

[−f (x), g(x)] =[

−[∂�(z)

∂zf (z)

]

z=�(x)

,[∂�(z)

∂zn

]

z=�(x)

]

=[∂�(z)

∂z

]

z=�(x)

[−f (z), en

]

z=�(x)

=[∂�(z)

∂z

]

z=�(x)

∂ f (z)

∂zn

=[∂�(z)

∂z

]

z=�(x)

en−1

=[∂�(z)

∂zn−1

]

z=�(x)

.

Similarly, we can show that

adn−k−f g =

[∂�(z)

∂zn−k

]

z=�(x)

, for k = n − 2, . . . , 1.

Hence, we have established that

∂�(z)

∂z= [

adn−1−f g, . . . , ad−f g, g

]

x=�(z). (8.17)


The remaining part of the proof is to show that g(x) coincides with r(x) in (8.14).From (8.17), we have

∂ h(z)

∂z= ∂h(x)

∂x

∂�(z)

∂z

= ∂h(x)

∂x

[adn−1

−f g, . . . , ad−f g, g]

x=�(z)

= [Ladn−1−f gh(x), . . . , Lad−f gh(x), Lgh(x)]x=�(z).

Since h(z) is in the output feedback form, the above expression implies that

Ladn−1−f gh(x) = 1,

Ladn−k−f gh(x) = 0, for k = 2, . . . , n,

which further imply that

LgLn−1h(x) = 1,

LgLn−kh(x) = 0, for k = 2, . . . , n,

i.e.,⎡

⎢⎢⎢⎣

dhdLf h...

dLn−1f h

⎤

⎥⎥⎥⎦

g =

⎡

⎢⎢⎢⎣

0...

01

⎤

⎥⎥⎥⎦.

This completes the proof of necessity. �

Remark 8.4. It would be interesting to revisit the transformation for linear single-output systems to the observer canonical form, to reveal the similarities between thelinear case and the conditions stated in Theorem 8.4. For a single output system

x = Ax

y = Cx,(8.18)

where x ∈ Rn is the state; y ∈ R is the system output; and A ∈ R

n×n and C ∈ R1×n

are constant matrices. When the system is observable, we have Po full rank.Solving r from Por = e1, i.e.,

⎡

⎢⎢⎢⎣

CCA...

CAn−1

⎤

⎥⎥⎥⎦

r =

⎡

⎢⎢⎢⎣

0...

01

⎤

⎥⎥⎥⎦

,

and the state transformation matrix T is then given by

T −1 = [An−1r, . . . , Ar, r]. (8.19)


We can show that the transformation z = Tx which transforms the system to theobserver canonical form

z = TAT −1z := Az

y = CT −1z := Cz

where

A =

⎡

⎢⎢⎢⎢⎢⎣

−a1 1 0 . . . 0−a2 0 1 . . . 0...

......

. . ....

−an−1 0 0 . . . 1−an 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

,

C = [1 0 . . . 0 0],

with constants ai, i = 1, . . . , n, being the coefficients of the characteristic polynomial

|sI − A| = sn + a1sn−1 + · · · + an−1s + an.

Indeed, from (8.19), we have

AT −1 = [Anr, . . . , A2r, Ar] (8.20)

Then from Cayley–Hamilton theorem, we have

An = −a1An−1 − · · · − an−1A − anI .

Substituting this into the previous equation, we obtain that

AT −1 = [An−1r, . . . , Ar, r]

⎡

⎢⎢⎢⎢⎢⎣

−a1 1 0 . . . 0−a2 0 1 . . . 0...

......

. . ....

−an−1 0 0 . . . 1−an 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

= T −1A,

which gives

TAT −1 = A.

Again from (8.19), we have

CT −1 = [CAn−1r, . . . , CAr, Cr] = [1, 0, . . . , 0] = C.

If we identify f (x) = Ax and h(x) = Cx, the first condition of Theorem 8.4 is that Po

has full rank, and the condition in (8.13) is identical as (8.19). �

When the conditions for Theorem 8.4 are satisfied, the transformation z = �(x)exists. In such a case, an observer can be designed for the nonlinear system (8.7) as

˙z = Az + L(y − Cz) + φ(y)

x = �(z),(8.21)


where x ∈ Rn is the estimate of the state x of (8.7); z ∈ R

n is an estimate of z in (8.10);and L ∈ R

n is the observer gain such that (A − LC) is Hurwitz.

Corollary 8.5. For nonlinear system (8.7), if the conditions for Theorem 8.4 aresatisfied, the observer in (8.21) provides an asymptotic state estimate of the system.

Proof. From Theorem 8.4, we conclude that (A, C) is observable, and there exists anL such that (A − LC) is Hurwitz. From (8.21) and (8.10), we can easily obtain that

˙z = (A − LC)z

where z = z − z. We conclude that x asymptotically converges to x as limt→∞z(t) = 0. �

8.3 Linear observer error dynamics via direct statetransformation

Observers presented in the last section achieve the linear observer error dynamics bytransforming the nonlinear system to the output injection form, for which an observerwith linear observer error dynamics can be designed to estimate the transformed state,and the inverse transformation is used to transform the estimate for the original state. Inthis section, we take a different approach by introducing a state transformation whichdirectly leads to a nonlinear observer design with linear observer error dynamics, andshow that this observer can be directly implemented in the original state space.

We consider the same system (8.7) as in the previous section, and describe it hereunder a different equation number for the convenience of presentation

x = f (x)

y = h(x),(8.22)

where x ∈ Rn is the state vector, y ∈ R is the output, f : R

n → Rn and h : R

n → R

are continuous nonlinear functions with f (0) = 0, and h(0) = 0. We now consider ifthere exists a state transformation

z = �(x), (8.23)

where � : Rn → R

n, such that the transformed system is in format

z = Fz + Gy, (8.24)

for a chosen controllable pair (F , G) with F ∈ Rn×n Hurwitz, and G ∈ R

n. Comparing(8.24) with (8.9), the matrices F and G in (8.24) are chosen for the observer design,while in (8.9), A and C are any observable pair which depends on the original system.Therefore, the transformation in (8.24) is more specific. There is an extra benefitgained from this restriction in the observer design as shown later.


From (8.22) and (8.24), the nonlinear transformation must satisfy the followingpartial differential equation

∂�(x)

∂xf (x) = F�(x) + Gh(x). (8.25)

Our discussion will be based on a neighbourhood around the origin.

Definition 8.1. Let λi(A) for i = 1, . . . , n are the eigenvalues of a matrix A ∈ Rn. For

another matrix F ∈ Rn, an eigenvalue of F is resonant with the eigenvalues of A if

there exists an integer q = ∑ni=1 qi > 0 with qi being non-negative integers such that

for some j with 1 ≤ j ≤ n

λj(F) =n∑

i=1

qiλi(A).

The following theorem states a result concerning with the existence of a nonlinearstate transformation around the origin for (8.22).

Theorem 8.6. For the nonlinear system (8.22), there exists a state transformation,i.e., a locally invertible solution to the partial differential equation (8.25) if

● the linearised model of (8.22) around the origin is observable● the eigenvalues of F are not resonant with the eigenvalues of ∂f

∂x (0)

● the convex hall of{λ1

(∂f∂x (0)

), . . . , λn

(∂f∂x (0)

)}does not contain the origin

Proof. From the non-resonant condition and the exclusion of the origin of the convexhall of the eigenvalues of ∂f

∂x (0), we can establish the existence of a solution to thepartial differential equation (8.25) by invoking Lyapunov Auxiliary Theorem. Thatthe function� is invertible around the origin is guaranteed by the observability of thelinearised model and the the controllability of (F , G). Indeed, with the observability of(∂f∂x (0), ∂h

∂x (0))

and the controllability of (F , G), we can apply Lemma 8.2 to establish

that ∂�∂x (0) is invertible. �

With the existence of the nonlinear transformation to put the system in the formof (8.24), an observer can be designed as

˙z = Fz + Gy

x = �−1(z),(8.26)

where � is the inverse transformation of �. It is easy to see that the observer errordynamics are linear as

˙z = Fz.


Note that once the transformation is obtained, the observer is directly given withoutdesigning observer gain, unlike the observer design based on the output injectionform.

The observer can also be implemented directly in the original state as

˙x = f (x) +(∂�

∂ x(x))−1

G(y − h(x)), (8.27)

which is in the same structure as the standard Luenberger observer for linear systemsby viewing

(∂�

∂ x (x))−1

G as the observer gain. In the following theorem, we show thatthis observer also provides an asymptotic estimate of the system state.

Theorem 8.7. For the nonlinear system (8.22), if the state transformation in (8.25)exists, the observer (8.27) provides an asymptotic estimate. Furthermore, thedynamics of the transformed observer error (�(x) −�(x)) are linear.

Proof. Let e = �(x) −�(x). Direct evaluation gives

e = ∂�(x)

∂xf (x) − ∂�(x)

∂ x

(

f (x) +(∂�

∂ x(x))−1

G(y − h(x))

)

= ∂�(x)

∂xf (x) − ∂�(x)

∂ xf (x) − G(y − h(x))

= F�(x) + Gy − (F�(x) + Gh(x)) − G(y − h(x))

= F(�(x) −�(x))

= Fe.

Therefore, the dynamics of the transformed observer error are linear, and the trans-formed observer error converges to zero exponentially, which implies the asymptoticconvergence of x to x. �

Remark 8.5. For linear systems, we have briefly introduced two ways to designobservers. For the observer shown in (8.2), we have introduced the observer shownin (8.6) to deal with nonlinear systems. The nonlinear observer in (8.27) can beviewed as a nonlinear version of (8.3). For both cases, the observer error dynamicsare linear. �

8.4 Observer design for Lipschitz nonlinear systems

In this section, we will deal with observer design for nonlinear systems with Lipschitznonlinearity. We introduced this definition for a time-varying function in Chapter 2 forthe existence of a unique solution for a nonlinear system. For a time-invariant function,the definition is similar, and we list below for the convenience of presentation.


Definition 8.2. A function φ : Rn × R

s → Rn is Lipschitz with a Lipschitz constant

γ if for any vectors x, x ∈ Rn and u ∈ R

s

‖φ(x, u) − φ(x, u)‖ ≤ γ ‖x − x‖, (8.28)

with γ > 0.

Once again we consider linear systems perturbed by nonlinear terms as

x = Ax + φ(x, u)

y = Cx,(8.29)


m is the system output with m < n; u ∈ Rs is the control

input, or other known variables, A ∈ Rn×n and C ∈ R

m×n are constant matrices with(A, C) observable; and φ : R

n × Rs → R

n is a continuous function with Lipschitzconstant γ with respect to the state variable x. Comparing with the system (8.5), theonly difference is the nonlinear term φ(x, u). Here, it is a function of state variable,not only the output, as in (8.5), and therefore the observer design by output injectiondoes not work.

We can still design an observer based on the linear part of the system, and replacethe unknown state in the nonlinear function by its estimate, that is,

˙x = Ax + L(y − Cx) + φ(x, u), (8.30)

where x ∈ Rn is the estimate of the state x and L ∈ R

n×m is the observer gain. However,the condition that (A − LC) is Hurwitz is not enough to guarantee the convergenceof the observer error to zero. Indeed, a stronger condition is needed, as shown in thefollowing theorem.

Theorem 8.8. The observer (8.30) provides an exponentially convergent state esti-mate if for the observer gain L, there exists a positive definite matrix P ∈ R

n×n suchthat

(A − LC)T P + P(A − LC) + γ 2PP + I + εI = 0, (8.31)

where γ is the Lipschitz constant of φ and ε is any positive real constant.

Proof. Let x = x − x. From (8.29) and (8.31), we have

˙x = (A − LC)x + φ(x, u) − φ(x, u).

Let

V = xT Px.

Its derivative along the observer error dynamics is obtained as

V = xT ((A − LC)T P + P(A − LC))x + 2xT P(φ(x, u) − φ(x, u)).


For the term involved with the nonlinear function φ, we have

2xT P(φ(x, u) − φ(x, u))

≤ γ 2xT PPx + 1

γ 2(φ(x, u) − φ(x, u))T (φ(x, u) − φ(x, u))

= γ 2xT PPx + 1

γ 2‖φ(x, u) − φ(x, u)‖2

≤ γ 2xT PPx + ‖x‖2

= xT (γ 2PP + I )x.

Applying the condition shown in (8.31), we have the derivative of V satisfying

V ≤ xT ((A − LC)T P + P(A − LC) + γ 2PP + I )x

= −εxT x.

This implies that the observer error converges to zero exponentially. �

In the condition (8.31), ε is an arbitrary positive real number. In this case, wecan use inequality to replace the equality, that is

(A − LC)T P + P(A − LC) + γ 2PP + I < 0. (8.32)

Using the same Lyapunov function candidate as in the proof of Theorem 8.8, we canshow that

V < 0

which implies that the observer error converges to zero asymptotically. We summarisethis result below.

Corollary 8.9. The observer (8.30) provides an asymptotically convergent state esti-mate if for the observer gain L, there exists a positive definite matrix P ∈ R

n×n thatsatisfies the inequality (8.32).

Remark 8.6. By using the inequality (8.32) instead of the equality (8.31), we onlyestablish the asymptotic convergence to zero of the observer error, not the exponentialconvergence that is established inTheorem 8.8 using (8.31). Furthermore, establishingthe asymptotic convergence of the observer error from V < 0 requires the stabilitytheorems based on invariant sets, which are not covered in this book. �

The condition shown in (8.32) can be relaxed if one-side Lipschitz constant isused instead of the Lipschitz constant.

Definition 8.3. A function φ : Rn × R

s → Rn is one-sided Lipschitz with respect to

P and one-sided Lipschitz constant ν if for any vectors x, x ∈ Rn and u ∈ R

s

(x − x)T P(φ(x, u) − φ(x, u)) ≤ ν‖x − x‖2 (8.33)

where P ∈ Rn×n is a positive real matrix, and ν is a real number.


Note that the one-sided Lipschitz constant ν can be negative. It is easy to see fromthe definition of the one-sided Lipschitz condition that the term (x − x)T P(φ(x, u) −φ(x, u)) is exactly the cross-term in the proof Theorem 8.8 which causes the termγ 2PP + I in (8.32). Hence, with the Lipschitz constant ν with respect to P, thecondition shown in (8.32) can be replaced by

(A − LC)T P + P(A − LC) + 2νI < 0. (8.34)

This condition can be further manipulated to obtain the result shown in the followingtheorem.

Theorem 8.10. The observer (8.30) provides an asymptotically convergent stateestimate if the following conditions hold:

L = σP−1CT , (8.35)

AT P + PA + 2νI − 2σCT C < 0, (8.36)

where P ∈ Rn×n is a positive real matrix, σ is a positive real constant and ν is the

one-sided Lipschitz constant of φ with respect to x and P.

Proof. From (8.35), we have(√

σCP−1 − LT

√σ

)T (√σCP−1 − LT

√σ

)

= 0,

which gives

P−1CT LT + LCP−1 = σP−1CT CP−1 + LLT

σ.

Using (8.35) and multiplying the above equation by P on both sides, we obtain theidentity

CT LT P + PLC = 2σCT C.

From this identity and (8.36), we can easily obtain the inequality (8.34). Similar tothe proof of Theorem 8.8, we let

V = xT Px,

and obtain, using the one-sided Lipschitz condition of φ,

V = xT ((A − LC)T P + P(A − LC))x + 2xT P(φ(x, u) − φ(x, u))

≤ xT ((A − LC)T P + P(A − LC))x.+ 2νxT x.

Applying the inequality (8.34) to the above expression, we have

V < 0,

which implies that the observer error asymptotically converges to zero. �


To end this section, we consider a class of systems with nonlinear Lipschitz outputfunction

x = Ax

y = h(x),(8.37)

where x ∈ Rn is the state vector, y ∈ R

m is the output, A ∈ Rn×n is a constant matrix

and h : Rn → R

m is a continuous function. We can write the nonlinear function h ash = Hx + h1(x) with Hx denoting a linear part of the output, and the nonlinear parth1 with Lipschitz constant γ .

An observer can be designed as

˙x = Ax + L(y − h(x)), (8.38)

where the observer gain L ∈ Rn×m is a constant matrix.

Theorem 8.11. The observer (8.38) provides an exponentially convergent stateestimate of (8.37) if the observer gain L can be chosen to satisfy the followingconditions:

L = 1

γ 2P−1H T , (8.39)

PA + AT P − H T H

γ 2+ (1 + ε)I = 0, (8.40)

where P ∈ Rn×n is a positive definite matrix and ε is a positive real constant.

Proof. Let x = x − x. From (8.37) and (8.38), we have

˙x = (A − LH )x + L(h1(x) − h1(x)).

Let

V = xT Px,

where x = x − x. It can be obtained that

V = xT ((A − LH )T P + P(A − LH ))x + 2xT PL(h1(x) − h1(x))

≤ xT ((A − LH )T P + P(A − LH ))x + xT (I + γ 2PLLT P)x

= xT

(

AT P + PA − H T H

γ 2+ I

)

x + xT

(H

γ− γLT P

)T (H

γ− γLT P

)

x

= −εxT x.

Therefore, we can conclude that x converges to zero exponentially. �

Remark 8.7. The nonlinearity in the output function with linear dynamics mayoccur in some special cases such as modelling a periodic signal as the output of asecond-order linear system. This kind of formulation is useful for internal modeldesign to asymptotically reject some general periodic disturbances, as shown inChapter 10. �


8.5 Reduced-order observer design

The observers introduced in the previous sections all have the same order as theoriginal systems. One might have noticed that an observer provides estimate of theentire state variables, which include the system output. Only the variables whichare not contained in the output are needed for state estimation from an observer.This was noticed at the very early stage of the development of observer design oflinear systems, and leads to the design of observers with less order than the originalsystems. The observers with less order than the original system are referred to asreduced-order observers. This section devotes to reduced-order observer design fornonlinear systems.


x = f (x, u)

y = h(x),(8.41)

where x ∈ Rn is the state vector, u ∈ R

s is the known input, y ∈ Rm is the output

and f : Rn × R

s → Rn is a nonlinear smooth vector field. To design a reduced-order

observer, we need to study the dynamics of other state variables other than the output.We can define a partial-state transformation.

Definition 8.4. A function g : Rn → R

n−m is an output-complement transformationif the function T given by

T (x) :=[

h(x)g(x)

]

is a diffeomorphism, where h(x) is the output function. The transformed states arereferred to as output-complement states.

Clearly an output-complement transformation defines a state transformationtogether with the output function. If the output-complement states are known, thenthe state variables can be fully determined. In fact, if we can design an observer forthe output-complement state, this observer is a reduced-order observer.

Definition 8.5. The dynamic model

z = p(z, y) + q(y, u) (8.42)

is referred to as reduced-order observer form for the system (8.41) if the z = g(x) isoutput-complement transformation, and the dynamic system

z = p(z, y)

is differentially stable.


For a system which can be transformed to a reduced-order observer form, we canpropose a reduced-order observer design as

˙z = p(z, y) + q(y, u)

x = T −1

([yz

])

.(8.43)

Theorem 8.12. If the system (8.41) can be transformed to the reduced-order observerform (8.42), the state estimate x provided by the reduced-order observer in (8.43)asymptotically converges to the state variable of (8.41).

Proof. First, let us establish the boundedness of z. Since z = p(z, y) is differentiallystable, from the definition of the differential stability in Chapter 5, there exists aLyapunov function, V (z), such that the conditions (5.41) are satisfied. Take the samefunction V (z) here as the Lyapunov function candidate with z = 0. From (5.41) and(8.42), we have

V ≤ −γ3(‖z‖) + ∂V

∂zq(y, u)

≤ −γ3(‖z‖) + ‖∂V

∂z‖‖q(y, u)‖. (8.44)

Let us recallYoung’s inequality in a simplified form that for any a ∈ R and b ∈ R,and a pair of constants p > 1 and q > 1 with 1

p + 1q = 1, we have

|ab| ≤ εp

p|a|p + 1

qεq|b|q

for any positive real constant ε.Applying Young’s inequality to the second term on the right-hand side of (8.44)

gives∥∥∥∥∂V (z)

∂z

∥∥∥∥

∥∥q(y, u)

∥∥ ≤ cc2

4

c2

∥∥∥∥∂V (z)

∂z

∥∥∥∥

c2

+ 1

c3cc34

‖q(y, u)‖c3 , (8.45)

where c3 = c2c2−1 , and c4 is an arbitrary positive real constant. We set c4 = ( c1c2

2 )1/c2 ,which results in

∥∥∥∥∂V (z)

∂x

∥∥∥∥ ‖q(y, u)‖ ≤ c1

2

∥∥∥∥∂V (z)

∂z

∥∥∥∥

c2

+ 1

2c5‖q(y, u)‖c3 , (8.46)

where c5 = 1c3

( 12 c1c2)−

c3c2 . Substituting (8.46) into (8.44), we have

V ≤ −1

2γ3(‖z‖) + 1

2c5‖q(y, u)‖c3 . (8.47)

Since q(y, u) is continuous, there exists a class K function g such that, for all y ∈ Rm

and u ∈ Rs, we have

‖q(y, u)‖ ≤ g(‖y‖ + ‖u‖).


We then choose χ (·) = γ −13 (2c5(g(·))c3 ). For ‖z‖ ≥ χ (‖y‖ + ‖u‖), we have

γ3(‖z‖) ≥ 2c5(g(‖y‖ + ‖u‖))c3

≥ 2c5‖q(y, u)‖c3 ,

which further implies that

V ≤ −1

4γ3(‖z‖). (8.48)

Hence, V (z) is an ISS-Lyapunov function, and therefore z is bounded when y isbounded.

The dynamics of e are given by

e = p(z(t), y) − p((z(t) − e), y).

Taking V (e) as the Lyapunov function candidate, we have

V = ∂V

∂e(p(z(t), y) − p((z(t) − e), y))

≤ −γ3(‖e‖).

Therefore, we can conclude that the estimation error asymptotically converges to zero.With z as a convergent estimate of z, we can further conclude that x is an asymptoticestimate of x. �

To demonstrate the proposed reduced-order observer design, let us consider anexample.

Example 8.1. Consider a second-order system

x1 = x21 − 3x2

1x2 − x31

x2 = −x2 + x21 − 6x2x2

1 + 3x22x1 − x3

2

y = x1.

Let us check if the system can be transformed to the reduced-order observerform. For this, we need to find g(x). Take

z = g(x) = x2 − x1.

We have

z = −x2 − (x2 − x1)3

= −(1 + z2)z + y.

Comparing with the format shown in (8.42), we have p(z, y) = −(1 + z2)z + y. Notethat for the system without input, we always have z = p(z, y). Let V = 1

2 z2. It is easyto see the first and the third conditions in (5.41) are satisfied. For the second condition,we have


∂V (z − z)

∂z(p(z) − p(z))

= −(z − z)(z − z + z3 − z3)

= −(z − z)2(1 + z2 − zz + z2)

= −(z − z)2(1 + 1

2(z2 + z2 + (z − z)2))

≤ −(z − z)2.

Therefore, the system satisfies the conditions specified in (5.41). We design thereduced-order observer as

˙z = −(1 + z2)z + y

x2 = z + y.

Simulation study has been carried out, and the simulation results are shown inFigures 8.1 and 8.2. �

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

x 1 a

nd x

2

x1x2

Figure 8.1 State variables


0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (s)

x 2 a

nd e

stim

ate

x2Estimate of x2

Figure 8.2 Unmeasured state and its estimate

There are lack of systematic design methods for nonlinear observers with globalconvergence when there are general nonlinear terms of unmeasured state variablesin the systems. With the introduction of the reduced-order observer form, we like tofurther explore the class of nonlinear systems which can be transformed to the formatshown in (8.42), and therefore a nonlinear observer can then be designed accordingly.

We consider a multi-output (MO) nonlinear system

x = Ax + φ(y, u) + Eϕ(x, u)

y = Cx,(8.49)


m is the output; u ∈ Rs is the control; φ is a known

nonlinear smooth vector field; ϕ : Rn × R

s → Rm is a smooth nonlinear function;

C ∈ Rm×n, E ∈ R

n×m and A ∈ Rn×n are constant matrices, with (A, C) observable.

When ϕ = 0, the system (8.49) degenerates to the well-known form of the non-linear systems with the linear observer error dynamics, and nonlinear observer can beeasily designed by using nonlinear output injection. With this additional term ϕ, thenonlinear output injection term can no longer be used to generate a linear observererror dynamics. We will convert the system to the reduced-order observer formconsidered, and then apply the reduced-order observer design for state observation.


Without loss of generality, we can assume that C has full row rank. There existsa nonsingular state transformation M such that

CM −1 = [Im, 0m×(n−m)].

If span{E} is a complement subspace of ker{C} in Rn, we have (CM −1)(ME) = CE

invertible. If we partition the matrix ME as

ME :=[

E1

E2

]

with E1 ∈ Rm×m, then we have CE = E1. Furthermore, if we partition Mx as

Mx :=[χ1

χ2

]

with χ1 ∈ Rm, we have

z = g(x) = χ2 − E2E−11 χ1. (8.50)

Note that we have χ1 = y. With the partition of

MAM −1 :=[

A1,1 A1,2

A2,1 A2,2

]

and

Mφ :=[φ1

φ2

]

,

we can write the dynamics χ1 and χ2 as

χ1 = A1,1χ1 + A1,2χ2 + φ1 + E1ϕ

χ2 = A2,1χ1 + A2,2χ2 + φ2 + E2ϕ.

Then we can obtain the dynamics of z as

z = A2,1χ1 + A2,2χ2 + φ2 − E2E−11 (A1,1χ1 + A1,2χ2 + φ1)

= (A2,2 − E2E−11 A1,2)χ2 + (A2,1 − E2E−1

1 A1,1)χ1 + φ2 − E2E−11 φ1

= (A2,2 − E2E−11 A1,2)z + q(y, u), (8.51)

where

q(y, u) = (A2,2 − E2E−11 A1,2)E2E−1

1 y + (A2,1 − E2E−11 A1,1)y

+φ2(y, u) − E2E−11 φ1(y, u).

Note that the nonlinear function ϕ(x, u) does not appear in the dynamics of zin (8.51) due to the particular choice of z in (8.50).

Remark 8.8. After the state transformation, (8.51) is in the same format as (8.42).Therefore, we can design a reduced-order observer if z = (A2,2 − E2E−1

1 A1,2)z is dif-ferentially stable. Notice that it is a linear system. Hence, it is differentially stable if it


is asymptotically stable, which means the eigenvalues of (A2,2 − E2E−11 A1,2) are with

negative real parts. �

Following the format shown in (8.43), a reduced-order observer can then bedesigned as

˙z = (A2,2 − E2E−11 A1,2)z + q(y, u) (8.52)

and the estimate of x is given by

x = M −1

[y

z + E2E−11 y

]

. (8.53)

Theorem 8.13. For a system (8.49), if

● C has full row rank, and span{E} is a complement subspace of ker{C} in Rn

● all the invariant zeros of (A, E, C) are with negative real parts

it can be transformed to the reduced-order observer form. The estimates z given in(8.52) and x (8.53) converge to the respective state variables z and x exponentially.

Proof. We only need to show that z = p(z, y) is differentially stable, and then we canapply Theorem 8.12 to conclude the asymptotic convergence of the reduced-orderobserver error. The exponential convergence comes as a consequence of the linearityin the reduced-order observer error dynamics. For (8.52), we have

p(z, y) = (A2,2 − E2E−11 A1,2)z,

which is linear in z. Therefore, the proof can be completed by proving that thematrix (A2,2 − E2E−1

1 A1,2) is Hurwitz. It can be shown that the eigenvalues of(A2,2 − E2E−1

1 A1,2) are the invariant zeros of (A, E, C). Indeed, we have[

M 00 Im

] [sI − A E

C 0

] [M −1 0

0 Im

]

=[

sI − MAM −1 MECM −1 0

]

=⎡

⎣sIm − A1,1 −A1,2 E1

−A2,1 sIn−m − A2,2 E2

Im 0 0

⎤

⎦ . (8.54)

Let us multiply the above matrix in the left by the following matrix to perform a rowoperation:

⎡

⎣Im 0 0

−E2E−11 In−m 0

0 0 Im

⎤

⎦ ,


we result in the following matrix:⎡

⎣sIm − A1,1 −A1,2 E1

� sIn−m − (A2,2 − E2E−11 A1,2) 0

Im 0 0

⎤

⎦

with

� = −E2E−11 (sIm − A1,1) − A2,1.

Since E1 is invertible, any values of s which make the matrix[

sI − A EC 0

]

rank deficient must be the eigenvalues of (A2,2 − E2E−11 A1,2). From the second condi-

tion that all the invariant zeros of (A, E, C) are with negative real part, we can concludethat (A2,2 − E2E−1

1 A1,2) is Hurwitz. �

Example 8.2. Consider a third-order system

x1 = −x1 + x2 − y1 + u + x2x3 − x1x3

x2 = −x1 + x2 + x3 − 2y1 + u + y1y2 + x2x3

x3 = −y21 + x1x3 + x2x3

y1 = x1

y2 = −x1 + x2.

We can identify

φ =⎡

⎣−y1 + u

−2y1 + y1y2 + u−y2

1

⎤

⎦, ϕ =[

x2x3 − x1x3

x1x3

]

, (8.55)

and we have

A =⎡

⎣−1 1 0−1 1 10 0 0

⎤

⎦ , E =⎡

⎣1 01 11 2

⎤

⎦ , C =[

1 0 0−1 1 0

]

. (8.56)

It can be easily checked that (A, C) is observable and the invariant zero of(A, E, C) is at −2. Therefore, the conditions in Theorem 8.13 are satisfied. It isalso easy to see that x2 = y2 + y1. Therefore, the only unmeasured state variable isx3. Indeed, following the procedures introduced earlier, we have

M =⎡

⎣1 0 0

−1 1 00 0 1

⎤

⎦ , E1 =[

1 00 1

]

, E2

[1 2

],

χ1 =[

x1

x2 − x1

]

, χ2 = x3,


and

z = x3 − y1 − 2y2 = x3 + x1 − 2x2.

With

A1,1 =[

0 10 0

]

, A1,2 =[

01

]

, A2,1

[0 0

], A2,2 = 0,

φ1 =[ −y1 + u

−y1 + y1y2

]

, φ2 = −y21,

we have the dynamics of z as

z = −2z + q(y, u),

where

q(y, u) = y1 − 5y2 − y21 − 2y1y2 − u.

A reduced-order observer can then be designed as

˙z = −2z + q(y, u)

x3 = z + y1 + 2y2.

0 2 4 6 8 10 12 14 16 18 20−6

−5

−4

−3

−2

−1

0

1

Time (s)

x

x1x2x3



0 2 4 6 8 10 12 14 16 18 20−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time (s)

x 3 a

nd e

stim

ate

x3

Estimate of x3

Figure 8.4 Unmeasured state and its estimate

Simulation study has been carried with x(0) = [1, 0, 0.5]T and u = sin t. Theplots of the state variables are shown in Figure 8.3 and the estimated state is shownin Figure 8.4 . �

8.6 Adaptive observer design

When there are unknown parameters in dynamic systems, observers can still bedesigned under certain conditions to provide estimates of unknown states by usingadaptive parameters. These observers are referred to as adaptive observers. One wouldexpect more stringent conditions imposed on nonlinear systems for which adaptiveobservers can be designed. In this section, we will consider two classes of nonlinearsystems. The first one is based on the nonlinear systems in the output injection formor output feedback form with unknown parameters and the other class of the systemsis nonlinear Lipschitz systems with unknown parameters.

Consider nonlinear systems which can be transformed to a single-outputsystem

x = Ax + φ0(y, u) + bφT (y, u)θ

y = Cx,(8.57)


where x ∈ Rn is the state, y ∈ R is the system output, u ∈ R

s is the control input, orother known variables, A ∈ R

n×n, b ∈ Rn×1 and C ∈ R

1×n are constant matrices, with(A, C) observable, θ ∈ R

r is an unknown constant vector, φ0 : Rm × R

s → Rn and

φ : Rm × R

s → Rr are continuous functions. If the parameter vector θ is known, the

system is in the output-injection form.One would expect that additional conditions are required for the design of

an adaptive observer. If the linear system characterised by (A, b, C) is of relativedegree 1 and minimum phase, we can design an observer gain L ∈ R

n such that thelinear system characterised by (A − LC, b, C) is a positive real system. In such a case,there exist positive real matrices P and Q such that

(A − LC)T P + P(A − LC) = −Q

Pb = CT .(8.58)

Let us consider the observer designed as

˙x = Ax + φ0(y, u) + bφT (y, u)θ + L(y − Cx), (8.59)

where x is the estimate of x and θ is the estimate of θ . We need to design an adaptivelaw for θ . Let x = x − x. The observer error dynamics are obtained as

˙x = (A − LC)x + bφT (y, u)θ , (8.60)

where θ = θ − θ .Consider a Lyapunov function candidate

V = xT Px + θT�−1θ ,

where � ∈ Rn×n is a positive definite matrix. From (8.58) and (8.60), we have

V = xT ((A − LC)T P + P(A − LC))x + 2xT Pbφ(y, u)T θ + 2 ˙θT�−1θ

= −xT Qx + 2xT CTφ(y, u)T θ − 2 ˙θT�−1θ

= −xT Qx − 2( ˙θ − (y − Cx)�φ(y, u))T�−1θ

If we set the adaptive law as

˙θ = (y − Cx)�φ(y, u),

we obtain

V = −xT Qx.

Then similar to stability analysis of adaptive control systems, we can conclude thatlimt→∞ x(t) = 0, and θ is bounded. The above analysis leads to the following theorem.

Theorem 8.14. For the observable single-output system (8.57), if the linear systemcharacterised by (A, b, C) is minimum phase and has relative degree 1, there exists an


observer gain L ∈ Rn that satisfies the conditions in (8.58) and an adaptive observer

designed as

˙x = Ax + φ0(y, u) + bφT (y, u)θ + L(y − Cx)

˙θ = (y − Cx)�φ(y, u),

(8.61)

where � ∈ Rn×n is a positive definite matrix and provides an asymptotically

convergent state estimate with adaptive parameter vector remaining bounded.

Remark 8.9. In this remark, we will show a particular choice of the observergain for the adaptive control observer for a system that satisfies the conditions inTheorem 8.14. Without loss of generality, we assume

A =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, b =

⎡

⎢⎢⎢⎢⎢⎣

b1

b2...

bn−1

bn

⎤

⎥⎥⎥⎥⎥⎦

,

C = [1 0 . . . 0 0],

where b1 �= 0, since the relative degree is 1. Indeed, for an observer system {A, b, C}with relative degree 1, there exists a state transformation, as shown in Remark 8.4, totransform the system to the observable canonical form. Furthermore, if we move thefirst column of A in the canonical form, and combine it with φ0(y, u), we have A as inthe format shown above. Since the system is minimum phase, b is Hurwitz, i.e.,

B(s) := b1sn−1 + b2sn−2 + · · · + bn−1s + bn = 0

has all the solutions in the left half of the complex plane. In this case, we can designthe observer gain to cancel all the zeros of the system. If we denote L = [l1, l2, . . . , ln],we can choose L to satisfy

sn + l1sn−1 + l2sn−2 + · · · + ln−1s + ln = B(s)(s + λ),

where λ is a positive real constant. The above polynomial equation implies

L = (λI + A)b.

This observer gain ensures

C (sI − (A − LC))−1 b = 1

s + λ,

which is a strict positive real transfer function. �

Now we will consider adaptive observer design for a class of Lipschitz nonlinearsystems. Consider

x = Ax + φ0(x, u) + bφT (x, u)θ

y = Cx,(8.62)


where x ∈ Rn is the state, y ∈ R is the system output, u ∈ R

s is the control input,or other known variables, A ∈ R

n×n, b ∈ Rn×1 and C ∈ R

1×n are constant matrices,with (A, C) observable, θ ∈ R

r is an unknown constant vector, φ0 : Rn × R

s → Rn

and φ : Rm × R

s → Rr are Lipschitz nonlinear functions with Lipschitz constants γ1

and γ2 with respect to x.This system is only different in nonlinear functions from the system (8.57) consid-

ered earlier for adaptive observer design, where the nonlinear functions are restrictedto the functions of the system output only, but the functions do not have to be globallyLipschitz for a globally convergent observer.

We propose an adaptive observer for (8.62) as

˙x = Ax + φ0(x, u) + bφT (x, u)θ + L(y − Cx)

˙θ = (y − Cx)�φ(x, u),

(8.63)

where L ∈ Rn is an observer gain and � ∈ R

n×n is a positive definite matrix.When state variables other than the system output are involved in the nonlinear

functions, we often impose the conditions that the nonlinear function are Lipschitz,and the conditions are more stringent for designing an observer than the nonlinearsystems with only nonlinear functions of the system output. We have consideredadaptive observer design for a class of nonlinear systems (8.57) with only the nonlinearfunctions of the system output. For the adaptive observer (8.63) to work for the system(8.62), we will expect stronger conditions for adaptive observer design. For this, westate the following result.

Theorem 8.15. The adaptive observer proposed in (8.63) provides an asymptoticallyconvergent state estimate for the system (8.62) if there exists a positive definite matrixP such that

(A − LC)T P + P(A − LC) + (γ1 + γ2γ3‖b‖)(PP + I ) + εI ≤ 0

Pb = CT ,(8.64)

where γ3 ≥ ‖θ‖, and ε is a positive real constant.

Proof. Let x = x − x. The observer error dynamics are obtained as

˙x = (A − LC)x + φ0(x, u) − φ0(x, u) + b(φT (y, u)θ − φT (x, u)T θ )

= (A − LC)x + φ0(x, u) − φ0(x, u) + b(φ(x, u) − φ(x, u))T θ

+ bφT (x, u)θ , (8.65)

where θ = θ − θ . Consider a Lyapunov function candidate

V = xT Px + θT�−1θ .

Its derivative along the dynamics in (8.65) is obtained as

V = xT ((A − LC)T P + P(A − LC))x + 2xT P(φ0(x, u) − φ0(x, u))

+ 2xT Pb(φ(x, u) − φ(x, u))T θ + 2xT PbφT (x, u)θ + 2 ˙θT�−1θ .


From the Lipschitz constants of φ0 and φ, we have

2xT P(φ0(x, u) − φ0(x, u)) ≤ γ1xT (PP + I )x,

2xT Pb(φ(x, u) − φ(x, u))T θ ≤ γ2γ3‖b‖xT (PP + I )x.

Then from the adaptive law and (8.64), we have

V ≤ xT ((A − LC)T P + P(A − LC) + (γ1 + γ2γ3‖b‖)(PP + I ))xT

+ 2xT CTφT (x, u)θ − 2 ˙θT�−1θ

≤ − εxT x.

This implies that the variables x and θ are bounded, and furthermore x ∈ L2 ∩ L∞.Since all the variables are bounded, ˙x is bounded, from x ∈ L2 ∩ L∞ and theboundedness of ˙x, we conclude limt→∞ x(t) = 0 by invoking Babalat’s Lemma. �

Chapter 9

Backstepping design

For a nonlinear system, the stability around an equilibrium point can be establishedif one can find a Lyapunov function. Nonlinear control design can be carried out byexploring the possibility of making a Lyapunov function candidate as a Lyapunovfunction through control design. In Chapter 7, parameter adaptive laws are designedin this way by setting the parameter adaptive laws to make the derivative of a Lyapunovfunction candidate negative semi-definite. Backstepping is a nonlinear control designmethod based on Lyapunov functions. It enables a designed control to be extended toan augmented system, provided that the system is augmented in some specific way.One scheme is so-called adding an integrator in the sense that if a control input isdesigned for a nonlinear system, then one can design a control input for the augmentedsystem of which an integrator is added between the original system input and theinput to be designed. This design strategy can be applied iteratively. There are a fewsystematic control design methods for nonlinear systems, and backstepping is oneof them. In this chapter, we start with the fundamental form of adding an integrator,and then introduce the method for iterative backstepping with state feedback. We alsointroduce backstepping using output feedback, and adaptive backstepping for certainnonlinear systems with unknown parameters.

9.1 Integrator backstepping

Consider

x = f (x) + g(x)ξξ = u,

(9.1)

where x ∈ Rn and ξ ∈ R are the state variables; u ∈ R is the control input; and f :

Rn → R

n with f (0) = 0 and g : Rn → R

n are continuous functions. Viewing thefirst equation of (9.1) as the original system with x as the state and ξ as the input,the integration of u gives ξ , which means that an integrator is added to the originalsystem.

We consider the control design problem under the assumption that a knowncontrol input exists for the original system. Furthermore, we assume that the Lyapunovfunction is also known, associated with the known control for x-subsystem. Supposethat control input for the x-subsystem is α(x) with α differentiable and α(0) = 0, andthe associated Lyapunov function is V (x). We assume that


∂V

∂x(f (x) + g(x)α(x)) ≤ −W (x), (9.2)

where W (x) is positive definite.Condition (9.2) implies that the system x = f (x) + g(x)α(x) is asymptotically

stable. Consider

x = f (x) + g(x)α(x) + g(x)(ξ − α(x)).

Intuitively, if we can design a control input u = u(x, ξ ) to force ξ to converge to α(x),we have a good chance to ensure the stability of the entire system. Let us define

z = ξ − α(x).

It is easy to obtain the dynamics under the coordinates (x, z) as

x = f (x) + g(x)α(x) + g(x)z

z = u − α = u − ∂α

∂x(f (x) + g(x)ξ ).

Consider a Lyapunov function candidate

Vc(x, z) = V (x) + 1

2z2. (9.3)


Vc = ∂V

∂x(f (x) + g(x)α(x))+ ∂V

∂xg(x)z

+ z

(

u − ∂α

∂x(f (x) + g(x)ξ )

)

= −W (x) + z

[

u + ∂V

∂xg(x) − ∂α

∂x(f (x) + g(x)ξ )

]

.

Let

u = −cz − ∂V

∂xg(x) + ∂α

∂x(f (x) + g(x)ξ ) (9.4)

with c > 0 which results in

Vc = −W (x) − cz2. (9.5)

It is clear that −W (x) − cz2 is negative definite with respect to variables (x, z). Hence,we can conclude that Vc(x, z) is a Lyapunov function, and that (0, 0) in the coordinates(x, z) is a globally asymptotic equilibrium. From α(0) = 0, we can conclude that (0, 0)in the coordinates (x, ξ ) is also a globally asymptotic equilibrium, which means thatthe system (9.1) is globally asymptotically stable under the control input (9.4). Wesummarise the above result in the following lemma.

Lemma 9.1. For a system described in (9.1), if there exist differentiable function α(x)and a positive-definite function V (x) such that (9.2) holds, the control design givenin (9.4) ensures the global asymptotic stability of the closed-loop system.

Backstepping design 143

Remark 9.1. Considering the structure of (9.1), if ξ is viewed as the control input forthe x-subsystem, ξ = α(x) is the desired control, ignoring the ξ -system. This is whyξ can be referred to as a virtual control input for the x-subsystem. The control inputu for the overall system is designed with the consideration of the dynamics back tothe control design for the x-subsystem, and it may suggest the name of this particulardesign method as backstepping. �

Example 9.1. Consider

x1 = x21 + x2

x2 = u.

We design a control input using backstepping. Comparing with (9.1), we can identify

x ⇒ x1, ξ ⇒ x2, f (x) ⇒ x21, g(x) ⇒ 1.

First, we need to design α(x1) to stabilise

x1 = x21 + α(x1).

An obvious choice is

α(x1) = −c1x1 − x21

with c1 > 0 a constant. This design leads to

x1 = x21 + α(x1) = −c1x1.

Hence, we take

V (x1) = 1

2x2

1

with

V (x1) = −c1x21.

Therefore, the condition specified in Lemma 9.1 is satisfied with α(x1) = −c1x1 −x2

1, V (x1) = 12 x2

1 and W (x1) = −c1x21. The control input u can then be obtained

from (9.4) by substituting proper functions in the equation. Alternatively, we canobtain the control input by directly following the backstepping method. Indeed, letz = x2 − α(x1). The dynamics of the system in coordinate (x1, z) are obtained as

x1 = −c1x1 + z

z = u − ∂α(x1)

∂x1(x2

1 + x2),

where∂α(x1)

∂x1= −c1 − 2x1.

Let

Vc(x1, z) = 1

2x2

1 + 1

2z2.


Its derivative along the system dynamics is obtained as

Vc(x1, z) = −c1x21 + x1z + z

(

u − ∂α(x1)

∂x1(x2

1 + x2))

.

Designing the control u as

u = −x1 − c2z + ∂α(x1)

∂x1(x2

1 + x2)

results in

Vc(x1, z) = −c1x21 − c2z2.

Hence, the system is asymptotically stable with (x1, z). As α(0) = 0, we con-clude limt→∞ x2(t) = limt→∞ (z(t) + α(x1(t))) = 0, which implies that the systemis asymptotically stable in the equilibrium (0, 0) in (x1, x2). Note that the closed-loopsystem in (x1, z) is written as

[x1

z

]

=[−c1 1

−1 −c2

][x1

z

]

.

�

In Example 9.1, backstepping design has been used to design a control input for anonlinear system with unmatched nonlinearities. When a nonlinear function appearsin the same line as the control input, it is referred as a matched nonlinear function,and it can be cancelled by adding the same term in u with an opposite sign. From thesystem considered in Example 9.1, the nonlinear function x2

1 does not appear in thesame line as the control input u, and therefore it is unmatched. However, it is in thesame line as x2, which is viewed as a virtual control. As a consequence, α(x1), whichis often referred to as a stabilising function, can be designed to cancel the nonlinearitywhich matches with the virtual control, and backstepping method enables the controlin the next line to be designed to stabilise the entire system. This process can berepeated by identifying a virtual control, designing a stabilising function and usingbackstepping to design control input for more complicated nonlinear systems.

9.2 Iterative backstepping


x1 = x2 + φ1(x1)

x2 = x3 + φ2(x1, x2)

. . . (9.6)

xn−1 = xn + φn−1(x1, x2, . . . , xn−1)

xn = u + φn(x1, x2, . . . , xn)


where xi ∈ R for i = 1, . . . , n are state variables; φi :

i︷︸︸︷R × · · · × R → R for

i = 1, . . . , n are differentiable functions up to the order n − i with φi(0, . . . , 0) = 0;and u ∈ R is the control input.

When xi+1 in the ith equation of (9.6) is identified as the virtual control, the non-linear function φi is then matched with respect to the virtual control, and backsteppingmethod can be applied to move down the control design to (i + 1)th equation withxi+2 as the next virtual control. This process starts from i = 1 and can be repeateduntil i = n − 1 when u is reached. We will present this iterative backstepping designfor the system (9.6) in n steps. In each step, we could show the Lyapunov functionand other details for which Lemma 9.1 can be applied to. Although, for the simplicityof the control design, we leave the stability analysis to the end, stability is consideredin designing the stabilising function at each step. The control design will be shownin n steps.

Let

z1 = x1,

zi = xi − αi−1(x1, . . . , xi−1), for i = 2, . . . , n,

where αi−1 for i = 2, . . . , n are stabilising functions obtained in the iterativebackstepping design.

Step 1. For the design of the stabilising function, we arrange the dynamics of z1 as

z1 = (x2 − α1) + α1 + φ1(x1)

= z2 + α1 + φ1(x1).

Let

α1 = −c1z1 − φ1(x1). (9.7)

The resultant dynamics of z1 are

z1 = −c1z1 + z2. (9.8)

Step 2. The dynamics of z2 are obtained as

z2 = x2 − α1

= x3 + φ2(x1, x2) − ∂α1

∂x1(x2 + φ1(x1))

= z3 + α2 + φ2(x1, x2) − ∂α1

∂x1(x2 + φ1(x1)).

Design α2 as

α2 = −z1 − c2z2 − φ2(x1, x2) + ∂α1

∂x1(x2 + φ1(x1)). (9.9)

The resultant dynamics of z2 are given by

z2 = −z1 − c2z2 + z3. (9.10)


Note that the term −z1 in α2 is used to tackle a cross-term caused by z2 in the dynamicsof z1 in the stability analysis. Other terms in α2 are taken to cancel the nonlinear termsand stabilise the dynamics of z2.

Step i. For 2 < i < n, the dynamics of zi are given by

zi = xi − αi−1(x1, . . . , xi−1)

= xi+1 + φi(x1, . . . , xi)

−i−1∑

j=1

∂αi−1

∂xj(xj+1 + φj(x1, . . . , xj))

= zi+1 + αi + φi(x1, . . . , xi)

−i−1∑

j=1

∂αi−1

∂xj(xj+1 + φj(x1, . . . , xj)).

Design αi as

αi = −zi−1 − cizi − φi(x1, . . . , xi)

+i−1∑

j=1

∂αi−1

∂xj(xj+1 + φj(x1, . . . , xj)). (9.11)

The resultant dynamics of zi are given by

zi = −zi−1 − cizi + zi+1. (9.12)

Note that similar to the design of α2, the term −zi−1 is used to tackle a cross-termcaused by zi in the dynamics of zi−1 in the stability analysis, and the other terms in αi

are used to stabilise the dynamics of zi.

Step n. At the final step, we have

zn = xn − αn−1(x1, . . . , xn−1)

= u + φn(x1, . . . , xn)

−n−1∑

j=1

∂αn−1

∂xj(xj+1 + φj(x1, . . . , xj)).

Design the control input as

u = −zn−1 − cnzn − φn(x1, . . . , xn)

+n−1∑

j=1

∂αn−1

∂xj(xj+1 + φj(x1, . . . , xj)). (9.13)

The resultant dynamics of zn are given by

zn = −zn−1 − cnzn. (9.14)


Note that we can write u = αn by setting i = n in the expression of αi in (9.11).Let us establish the stability of the closed-loop system under the proposed

control. The closed-loop dynamics of the system in coordinate z = [z1, . . . , zn]T canbe written as

z =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

−c1 1 0 . . . 0

−1 −c2 1. . . 0

0 −1 −c3. . . 0

.... . .

. . .. . . 1

0 0 0. . . −cn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

z := Azz.

Notice that all the off-diagonal Az are skew symmetric. Let

V = 1

2zT z. (9.15)

Its derivative along the dynamics of z is obtained as

V = −n∑

i=1

ciz2i ≤ −2

nmini=1

ciV . (9.16)

Therefore, we can conclude that the system is exponentially stable in z-coordinate.From the property that φi(0, . . . , 0) = 0 for i = 1, . . . , n, we can establish thatαi(0, . . . , 0) = 0 for i = 1, . . . , n − 1 and u(0, . . . , 0) = 0, which implies thatlimt→∞ xi(t) = 0 for i = 1, . . . , n. Hence, we have established the following result.

Theorem 9.2. For a system in the form of (9.6), the control input (9.13) renders theclosed-loop system asymptotically stable.

9.3 Observer backstepping

We have presented integrator backstepping and iterative backstepping based on statefeedback. In this section, we present a control design for nonlinear systems usingoutput feedback. An observer is designed, and the observer state, or estimate of thestate variables of the original system, is then used for backstepping design.

Consider a nonlinear system which can be transformed to

x = Acx + bu + φ(y)

y = Cx(9.17)

with


Ac =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

T

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0bρ...

bn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where x ∈ Rn is the state vector; u ∈ R is the control; φ : R → R

n with φ(0) = 0 isa nonlinear function with element φi being differentiable up to the (n − i)th order;and b ∈ R

n is a known constant Hurwitz vector with bρ �= 0, which implies that therelative degree of the system is ρ. This form of the system is often referred to as theoutput feedback form. Since b is Hurwitz, the linear system characterised by (Ac, b, C)is minimum phase. Note that a vector is said Hurwitz if its corresponding polynomialis Hurwitz.

Remark 9.2. For a system in the output feedback form, if the input is zero, thesystem is in exactly the same form as (8.10). We have shown the geometric conditionsin Chapter 8 for systems to be transformed to the output injection form (8.10), andsimilar geometric conditions can be specified for nonlinear systems to be transformedto the output feedback form. Clearly we can see that for the system (9.17) with anyobservable pair (A, C), there exists a linear transformation to put the system in theform of (9.17) with the specific (Ac, C). �

Since the system (9.17) is in the output injection form, we design an observer as

˙x = Acx + bu + φ(y) + L(y − Cx), (9.18)

where x ∈ Rn is the state estimate and L ∈ R

n is an observer gain designed such that(A − LC) is Hurwitz. Let x = x − x, and it is easy to see

˙x = (Ac − LC)x. (9.19)

The backstepping design can be carried out with the state estimate x in ρ steps.From the structure of the system (9.17) we have y = x1. The backstepping design willstart with the dynamics of y. In the following design, we assume ρ > 1. In the caseof ρ = 1, control input can be designed directly without using backstepping.

To apply the observer backstepping through x in (9.18), we define

z1 = y,

zi = xi − αi−1, i = 2, . . . , ρ, (9.20)

zρ+1 = xρ+1 + bρu − αρ ,

where αi, i = 1, . . . , ρ, are stabilising functions decided in the control design.Consider the dynamics of z1

z1 = x2 + φ1(y). (9.21)


We use x2 to replace the unmeasurable x2 in (9.21), resulting at

z1 = x2 + x2 + φ1(y)

= z2 + α1 + x2 + φ1(y). (9.22)

We design α1 as

α1 = −c1z1 − k1z1 − φ1(y), (9.23)

where ci and ki for i = 1, . . . , ρ are positive real design parameters. Comparing thebackstepping design using the output feedback with the one using state feedback, wehave one additional term, −k1z1, which is used to tackle the observer error x2 in theclosed-loop system dynamics. Then from (9.22) and (9.23), we have

z1 = z2 − c1z1 − k1z1 + x2. (9.24)

Note that α1 is a function of y, i.e., α1 = α1(y).For the dynamics of z2, we have

z2 = ˙x2 − α1

= x3 + φ2(y) + l2(y − x1) − ∂α1

∂y(x2 + φ1(y))

= z3 + α2 + φ2(y) + l2(y − x1) − ∂α1

∂y(x2 + x2 + φ1(y)).

where l2 is the second element of the observer gain L, and in the subsequent design,li is the ith element of L. We design α2 as

α2 = −z1 − c2z2 − k2

(∂α1

∂y

)2

z2 − φ2(y) − l2(y − x1)

+∂α1(y)

∂y(x2 + φ1(y)). (9.25)

Note that α2 = α2(y, x1, x2). The resultant dynamics of z2 are given by

z2 = −z1 − c2z2 − k2

(∂α1

∂y

)2

z2 + z3 − ∂α1

∂yx2.

For the dynamics of zi, 2 < i ≤ ρ, we have

zi = ˙xi − αi−1

= zi+1 + αi + φi(y) + li(y − x1)

− ∂αi−1

∂y(x2 + x2 + φ1(y)) −

i−1∑

j=1

∂αi−1

∂ xj

˙xj.


We design αi, 2 < i ≤ ρ, as

αi = −zi−1 − cizi − ki

(∂αi−1

∂y

)2

zi − φi(y) − li(y − x1)

+ ∂αi−1

∂y(x2 + φ1(y)) +

i−1∑

j=1

∂αi−1

∂ xj

˙xj. (9.26)

Note that αi = αi(y, x1, . . . , xi). The resultant dynamics of zi, 2 < i ≤ ρ, are given by

zi = −zi−1 − cizi − ki

(∂αi−1

∂y

)2

zi + zi+1 − ∂αi−1

∂yx2. (9.27)

When i = ρ, the control input appears in the dynamics of zi, and it is included in zρ+1,as shown in the definition (9.20). We design the control input by setting zρ+1 = 0,which gives

u = αρ(y, x1, . . . , xρ) − xρ+1

bρ. (9.28)

The stability result of the above control design is given in the following theorem.

Theorem 9.3. For a system in the form of (9.17), the dynamic output feedback controlwith the input (9.28) and the observer (9.18) asymptotically stabilise the system.

Proof. From the observer error dynamics, we know that the error exponentially con-verges to zero. Since (Ac − LC) is Hurwitz, there exists a positive definite matrixP ∈ R

n×n such that

(Ac − LC)T P + P(Ac − LC) = −I .

This implies that for

Ve = xT Px,

we have

Ve = −‖x‖2. (9.29)

Let

Vz =ρ∑

i=1

z2i .

From the dynamics of z1, z2 and zi, we can obtain that

Vz =ρ∑

i=1

(

−ciz2i − ki

(∂αi−1

∂y

)2

z2i − ∂αi−1

∂yzix2

)

,

where we define α0 = y for notational convenience. Note that if we ignore the twoterms concerning with ki and x2 in the dynamics of zi, the evaluation of the derivative


of Vz will be exactly the same as the stability analysis that leads to Theorem 9.2. Forthe cross-term concerning with x2, we have, from Young’s inequality,

∣∣∣∣∂αi−1

∂yzix2

∣∣∣∣ ≤ ki

(∂αi−1

∂y

)2

z2i + 1

4kix2

2.

Hence, we obtain that

Vz ≤ρ∑

i=1

(

−ciz2i + 1

4kix2

2

)

. (9.30)

Let

V = Vz +(

1 + 1

4d

)

Ve,

where d = minρi=1 ki. From (9.29) and (9.30), we have

V ≤ρ∑

i=1

(

−ciz2i + 1

4kix2

2

)

−(

1 + 1

4d

)

‖x‖2

≤ −ρ∑

i=1

ciz2i − ‖x‖2.

Therefore, we can conclude that zi for i = 1, . . . , ρ and x exponentially converge tozero. Noticing that y = z1, and α1(0) = 0, we can conclude that limt→∞ x2(t) = 0,which further implies that limt→∞ x2(t) = 0. Following the same process, we canshow that limt→∞ xi(t) = 0 for i = 1, . . . , ρ.

We still need to establish the stability property for xi, with i = ρ + 1, . . . , n. Let

ξ =⎡

⎢⎣

xρ+1...

xn

⎤

⎥⎦−

⎡

⎢⎣

bρ+1...

bn

⎤

⎥⎦

xρbρ

,

and from the system dynamics (9.17), it can be shown that

ξ = Bξ +⎡

⎢⎣

φρ+1(y)...

φn(y)

⎤

⎥⎦−

⎡

⎢⎣

bρ+1...

bn

⎤

⎥⎦φρ(y)

bρ+ B

⎡

⎢⎣

bρ+1...

bn

⎤

⎥⎦

xρbρ

, (9.31)

where

B =

⎡

⎢⎢⎢⎢⎢⎢⎣

−bρ+1/bρ 1 . . . 0

−bρ+2/bρ 0. . . 0

......

......

−bn−1/bρ 0 . . . 1−bn/bρ 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎥⎦


is a companion matrix associated with vector b. Since b is Hurwitz, the matrix B isHurwitz. All the terms other than Bξ in the right-hand side of (9.31) converge to zero,which implies that limt→∞ ξ (t) = 0. Therefore, we can conclude limt→∞ xi(t) = 0 fori = ρ + 1, . . . , n. This concludes the proof. �

9.4 Backstepping with filtered transformation

We have presented backstepping design for a class of dynamic systems using outputfeedback in the previous section. The control design starts from the system outputand the subsequent steps in the backstepping design are carried out with estimates ofstate variables provided by an observer. The backstepping design, often referred toas observer backstepping, completes in ρ steps, with ρ being the relative degree ofthe system, and other state estimates of xi for i > ρ are not used in the design. Thoseestimates are redundant, and make the dynamic order of the controller higher. Thereis an alternative design method to observer backstepping for nonlinear systems in theoutput feedback form, of which the resultant order of the controller is exactly ρ − 1.In this section, we present backstepping design with filtered transformation for thesystem (9.17), of which the main equations are shown here again for the convenienceof presentation,


y = Cx.(9.32)

Define an input filter

ξ1 = −λ1ξ1 + ξ2

... (9.33)

ξρ−1 = −λρ−1ξρ−1 + u,

where λi > 0 for i = 1, . . . , ρ − 1 are the design parameters. Define the filteredtransformation

ζ = x −ρ−1∑

i=1

diξi, (9.34)

where di ∈ Rn for i = 1, . . . , ρ − 1 and they are generated recursively by

dρ−1 = b,

di = (Ac + λi+1I )di+1 for i = ρ − 2, . . . , 1.

We also denote

d = (Ac + λ1I )d1.


From the filtered transformation, we have

˙ζ = Acx + bu + φ(y) −ρ−2∑

i=1

di(−λiξi + ξi+1) − dρ−1(−λiξρ−1 + u)

= Acζ +ρ−1∑

i=1

Adiξi + φ(y) +ρ−1∑

i=1

diλiξi −ρ−2∑

i=1

diξi+1

= Acζ +ρ−1∑

i=1

(A + λiI )diξi + φ(y) −ρ−2∑

i=1

diξi+1

= Acζ + φ(y) + dξ1.

The output under the coordinate ζ is given by

y = C ζ +ρ−1∑

i=1

Cdiξi

= C ζ

because Cdi = 0 for i = 1, . . . , ρ − 1, from the fact that di,j = 0 for i = 1, . . . ,ρ − 1, 1 ≤ j ≤ i. Hence, under the filtered transformation, the system (9.32) is thentransformed to

˙ζ = Acζ + φ(y) + dξ1,

y = C ζ .(9.35)

Let us find a bit more information of d. From the definition

dρ−2 = (Ac + λρ−1I )b

we have

n∑

i=ρ−1

dρ−2,isn−i = (s + λρ−1)

n∑

ρ

bisn−i.

Repeating the process iteratively, we can obtain

n∑

i=1

disn−i =

ρ−1∏

i=1

(s + λi)n∑

ρ

bisn−i (9.36)

which implies that d1 = bρ and that d is Hurwitz if b Hurwitz. In the special formof Ac and C used here, b and d decide the zeros of the linear systems characterisedby (Ac, b, C) and (Ac, d, C) respectively as the solutions to the following polynomialequations:


n∑

ρ

bisn−i = 0,

n∑

i=1

disn−i = 0.

Hence, the invariant zeros of (Ac, d, C) are the invariant zeros of (Ac, b, C) plus λi fori = 1, . . . , ρ − 1. For the transformed system, ξ1 can be viewed as the new input. Inthis case, the relative degree with ξ1 as the input is 1. The filtered transformation liftsthe relative degree from ρ to 1.

As the filtered transformation may have its use independent of backsteppingdesign shown here, we summarise the property of the filtered transformation in thefollowing lemma.

Lemma 9.4. For a system in the form of (9.32) with relative degree ρ, the filteredtransformation defined in (9.34) transforms the system to (9.35) of relative degree 1,with the same high frequency gain. Furthermore, the zeros of (9.35) consist of thezeros of the original system (9.32) and λi for i = 1, . . . , ρ − 1.

We introduce another state transform to extract the internal dynamics of (9.35)with ζ ∈ R

n−1 given by

ζ = ζ2:n − d2:n

d1y, (9.37)

where ζ ∈ Rn−1 forms the state variable of the transformed system together with y,

the notation (·)2:n refers to the vector or matrix formed by the 2nd row to the nth row.With the coordinates (ζ , y), (9.35) is rewritten as

ζ = Dζ + ψ(y)

y = ζ1 + ψy(y) + bρξ1,(9.38)

where D is the companion matrix of d given by

D =

⎡

⎢⎢⎢⎢⎢⎢⎣

−d2/d1 1 . . . 0

−d3/d1 0. . . 0

......

......

−dn−1/d1 0 . . . 1−dn/d1 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎥⎦

,

and

ψ(y) = Dd2:n

d1y + φ2:n(y) − d2:n

d1φ1(y),

ψy(y) = d2

d1y + φ1(y).


If we view ξ1 as the input, the system (9.35) is of relative degree 1 with the stablezero dynamics. For such a system, there exists an output feedback law to globally andexponentially stabilise the system.

For this, we have the following lemma, stating in a more stand alone manner.

Lemma 9.5. For a nonlinear system (9.32), if the relative degree is 1, there exista continuous function ϕ : R → R with ϕ(0) = 0 and a positive real constant c suchthat the control input in the form of

u = −cy − ϕ(y) (9.39)

globally and asymptotically stabilises the system.

Proof. Introducing the same transformation as (9.37), i.e.,

ζ = x2:n − b2:n

b1y,

we can obtain exactly the same transformed system as in (9.38) with d being replacedby b and ξ1 by u. Since D is Hurwitz, there exists a positive definite matrix P suchthat

DT P + PD = −3I . (9.40)

We set

ϕ(y) = ψy(y) + ‖P‖2‖ψ(y)‖2

y. (9.41)

Note thatψ(0) = 0, and therefore ‖ψ(y)‖2/y is well defined. The closed-loop systemis then obtained as

ζ = Dζ + ψ(y),

y = ζ1 − cy − ‖P‖2‖ψ(y)‖2

y.

Let

V = ζ T Pζ + 1

2y2.

We have

V = −cy2 − 3‖ζ‖2 + 2ζ T Pψ(y) + yζ1 − ‖P‖2‖ψ(y)‖2.

With the inequalities of the cross terms

|2ζ T Pψ(y)| ≤ ‖ζ‖2 + ‖P‖2‖ψ(y)‖2,

|yζ1| ≤ 1

4y2 + ‖ζ‖2,

we have

V ≤ −(

c − 1

4

)

y2 − ‖ζ‖2.


Therefore, the proposed control design with c > 14 and (9.41) exponentially stabilises

the system in the coordinate (ζ , y), which implies the exponential stability of theclosed-loop system in x coordinate, because the transformation from (ζ , y) to x islinear. �

From Lemma 9.5, we know the desired value of ξ1. But we cannot directly assigna function to ξ1 as it is not the actual control input. Here backstepping can be appliedto design control input based on the desired function of ξ1. Together with the filteredtransformation, the overall system is given by

ζ = Dζ + ψ(y)

y = ζ1 + ψy(y) + bρξ1

ξ1 = −λ1ξ1 + ξ2 (9.42)

. . .

ξρ−1 = −λρ−1ξρ−1 + u,

to which the backstepping design is then applied. Indeed, in the backstepping design,ξi for i = 1, . . . , ρ − 1 can be viewed as virtual controls.

Let

z1 = y,

zi = ξi−1 − αi−1, for i = 2, . . . , ρ

zρ+1 = u − αρ ,

where αi for i = 2, . . . , ρ are stabilising functions to be designed. We also use thepositive real design parameters ci and ki for i = 1, . . . , ρ and γ > 0.

Based on the result shown in Lemma 9.5, we have

α1 = −c1z1 − k1z1 + ψy(y) − γ ‖P‖2‖ψ(y)‖2

y(9.43)

and

z1 = z2 − c1z1 − k1z1 − γ ‖P‖2‖ψ(y)‖2. (9.44)

For the dynamics of z2, we have

z2 = −λ1ξ1 + ξ2 − ∂α1

∂yy

= z3 + α2 − λ1ξ1 − ∂α1

∂y(ζ1 + ψ(y)).

The design of α2 is then given by

α2 = −z1 − c2z2 − k2

(∂α1

∂y

)2

z2 + ∂α1

∂yψ(y) + λ1ξ1. (9.45)


The resultant dynamics of z2 is obtained as

z2 = −z1 − c2z2 − k2

(∂α1

∂y

)2

z2 + z3 − ∂α1

∂yζ1. (9.46)

Note that α2 = α2(y, ξ1).For the subsequent steps for i = 3, . . . , ρ, we have

zi = −λi−1ξi−1 + ξi − ∂αi−1

∂yy −

i−2∑

j=1

∂αi−1

∂ξjξj

= zi+1 + αi − λi−1ξi−1 − ∂αi−1

∂y(ζ1 + ψ(y))

−i−2∑

j=1

∂αi−1

∂ξj(−λjξi−1 + ξj+1).

The design of αi is given by


(∂αi−1

∂y

)2

zi + ∂αi−1

∂yψ(y)

+i−2∑

j=1

∂αi−1

∂ξj(−λjξj + ξj+1) + λi−1ξi−1. (9.47)

The resultant dynamics of zi is obtained as


(∂αi−1

∂y

)2

zi + zi+1 − ∂αi−1

∂yζ1. (9.48)

Note that αi = αi(y, ξ1, . . . , ξi−1).When i = ρ, the control input appears in the dynamics of zi, through the term

zρ+1. We design the control input by setting zρ+1 = 0, which gives u = αρ , that is

u = −zρ−1 − cρzρ − kρ

(∂αρ−1

∂y

)2

zρ + ∂αρ−1

∂yψ(y)

+i−2∑

j=1

∂αρ−1

∂ξj(−λjξj + ξj+1) + λρ−1ξρ−1. (9.49)

For the control design parameters, ci, i = 1, . . . , ρ and γ can be any positive, and fordi, the following condition must be satisfied:

ρ∑

1=1

1

4ki≤ γ.

The stability result of the above control design is given in the followingtheorem.


Theorem 9.6. For a system in the form of (9.32), the dynamic output feedbackcontrol (9.49) obtained through backstepping with the input filtered transformationasymptotically stabilises the system.

Proof. Let

Vz =ρ∑

i=1

z2i .

From the dynamics for zi shown in (9.44), (9.46) and (9.48) we can obtain that

Vz =ρ∑

i=1

(

−ciz2i − ki

(∂αi−1

∂y

)2

z2i − ∂αi−1

∂yziζ1

)

− γ ‖P‖2‖ψ(y)‖2,

where we define α0 = y for notational convenience. For the cross-term concerningwith ζ1, we have

∣∣∣∣∂αi−1

∂yziζ1

∣∣∣∣ ≤ ki

(∂αi−1

∂y

)2

z2i + 1

4kiζ 2

1 .

Hence, we obtain that

Vz ≤ρ∑

i=1

(

−ciz2i + 1

4kiζ 2

1

)

− γ ‖P‖2‖ψ(y)‖2. (9.50)

Let

Vζ = ζ T Pζ

and we can obtain, similar to the proof of Lemma 9.5,

Vζ ≤ −2‖ζ‖2 + ‖P‖2‖ψ(y)‖2.

Let

V = Vz + γVζ .

and we have

V ≤ρ∑

i=1

(

−ciz2i + 1

4kiζ 2

1

)

− 2γ ‖ζ‖2 (9.51)

≤ −ρ∑

i=1

ciz2i − γ ‖ζ‖2. (9.52)

Therefore, we have shown that the system (9.42) is exponentially stable under thecoordinate (ζ , z1, . . . , zρ−1). With y = z1, we can conclude limt→∞ ζ (t) = 0. Fromy = z1, and α1(0) = 0, we can conclude that limt→∞ ξ1(t) = 0. Following the sameprocess, we can show that limt→∞ ξ (t) = 0 for i = 1, . . . , ρ − 1. Finally from thefiltered transformation (9.34), we can establish limt→∞ x(t) = 0. �


9.5 Adaptive backstepping

We have shown backstepping design for two classes of systems with state feedbackand output feedback respectively. In this section, we will show the nonlinear adaptivecontrol design for a class of system of which there are unknown parameters.

Consider a first-order nonlinear system described by

y = u + φT (y)θ (9.53)

where φ : R → Rp is a smooth nonlinear function, and θ ∈ R

p is an unknown vectorof constant parameters. For this system, adaptive control law can be designed as

u = −cy − φT (y)θ (9.54)

˙θ = yφ(y) (9.55)

where c is a positive real constant, and ∈ Rp×p is a positive definite gain matrix.

The closed-loop dynamics is given by

y = −cy + φT (y)θ

with the usual notation θ = θ − θ .For stability analysis, let

V = 1

2y2 + 1

2θT −1θ

and its derivative is obtained as

V = −cy2,

which ensures the boundedness of y and θ . We can show limt→∞ y(t) = 0 in the sameway by invoking Babalat’s Lemma as in the stability analysis of adaptive controlsystems shown in Chapter 7.

Remark 9.3. The system considered above is nonlinear with unknown parameters.However, the unknown parameters are linearly parameterised, i.e., the terms relatingto the unknown parameters, φT (y)θ are linear with the unknown parameters, instead ofsome nonlinear functions φ(y, θ ), which are referred to as nonlinearly parameterised.Obviously, nonlinear parameterised unknown parameters are much more difficult todeal with in adaptive control. In this book, we only consider linearly parameterisedunknown parameters. �

For the first-order system, the control input is matched with the uncertainty andthe nonlinear function. Backstepping can also be used with adaptive control to dealwith nonlinear and unknown parameters which are not in line with the input, or,unmatched.



x1 = x2 + φ1(x1)T θ

x1 = x3 + φ2(x1, x2)T θ

. . . (9.56)

xn−1 = xn + φn−1(x1, x1, . . . , xn−1)T θ

xn = u + φn(x1, x2, . . . , xn)T θ ,

where xi ∈ R for i = 1, . . . , n are state variables; θ ∈ Rp is an unknown vector of

constant parameters; φi :

i︷︸︸︷R × · · · × R → R

p for i = 1, . . . , n are differentiablefunctions up to the order n − i with φi(0, . . . , 0) = 0; and u ∈ R is the control input.

Note that this system is exactly the same as (9.6) if the parameter vector θ isknown.

The backstepping design method will be applied iteratively in a similar way tothe control design for (9.6), with only difference of including adaptive parameters inthe control design.

Let

z1 = x1,

zi = xi − αi−1(xi, . . . , xi−1, θ ), for i = 2, . . . , n,

zn+1 = u − αn(xi, . . . , xn, θ ),

ϕ1 = φ1,

ϕi = φi −i−1∑

j=1

∂αi−1

∂xjφj , for i = 2, . . . , n,

where αi−1, for i = 2, . . . , n, are stabilising functions obtained in the adaptivebeackstepping design, and θ denotes an estimate of θ .

We start the adaptive backstepping from the dynamics of z1

z1 = z2 + α1 + ϕT1 θ.

Design α1 as

α1 = −c1z1 − ϕT1 θ , (9.57)

where ci, for i = 1, . . . , n, are set of positive design parameters. The closed-loopdynamics are obtained as

z1 = −c1z1 + z2 + ϕT1 θ ,

where θ = θ − θ .



z2 = x2 − α1

= x3 + φT2 θ − ∂α1

∂x1x1 − ∂α1

∂θ

˙θ

= z3 + α2 + ϕT2 θ − ∂α1

∂x1x2 − ∂α1

∂θ

˙θ.

The dynamics of z2 involve the adaptive law ˙θ . Even though the adaptive law has

not been designed, it is surely known to the control design, and can be used in thecontrol input. However, the dynamics of zi with i > 2 will also affect the design of theadaptive law. For this reason, we would like to leave the design of the adaptive law tothe end. Inevitably, the adaptive law will include zi for i > 2. This causes a problem,

if we use α2 to cancel ˙θ at this step, because z3 depends on α2. Instead, we only deal

with the part of the adaptive law that depends on z1 and z2 at this step, and we denotethat as τ2, which is a function of z1, z2 and θ . In the subsequent steps, we use notationsτi = τi(z1, . . . , zi, θ ) for i = 3, . . . , n, which are often referred to as tuning functions.

Based on the above discussion, we design α2 as

α2 = −z1 − c2z2 − ϕT2 θ + ∂α1

∂x1x2 + ∂α1

∂θτ2. (9.58)

The closed-loop dynamics are obtained as

z2 = −z1 − c2z2 + z3 + ϕT2 θ − ∂α1

∂θ( ˙θ − τ2).

Then the adaptive backstepping can be carried on for zi with 2 < i ≤ n. Thedynamics of zi can be written as

zi = xi − αi−1

= xi+1 + φTi θ −

i−1∑

j=1

∂αi−1

∂xjxj − ∂αi−1

∂θ

˙θ

= zi+1 + αi + ϕTi θ −

i−1∑

j=1

∂αi−1

∂xjxj+1 − ∂αi−1

∂θ

˙θ.

The stabilising function αi, for 2 < i ≤ n, are designed as

αi = −zi−1 − cizi − ϕTi θ +

i−1∑

j=1

∂αi−1

∂xjxj+1 + ∂αi−1

∂θτi + βi, (9.59)

where βi = βi(z1, . . . , zi, θ ), for i = 3, . . . , n, are functions to be designed later to

tackle the terms ( ˙θ − τi) in stability analysis. The closed-loop dynamics are obtained

as, for 2 < i ≤ n,

zi = −zi−1 − cizi + zi+1 + ϕTi θ − ∂αi−1

∂θ( ˙θ − τi) + βi.


The control input u appears in the dynamics zn, in the term zn+1. When i = n,

we have τi = ˙θ by definition. We obtain the control input by setting zn+1 = 0, which

gives

u = αn

= −zn−1 − cnzn − ϕTn θ

+n−1∑

j=1

∂αn−1

∂xjxj+1 + ∂αn−1

∂θ

˙θ + βn. (9.60)

We need to design the adaptive law, tuning functions and βi to complete thecontrol design. We will do it based on Lyapunov analysis. For notational convenience,we set β1 = β2 = 0.

Let

V = 1

2

n∑

i=1

z2i + 1

2θT −1θ , (9.61)

where ∈ Rp×p is a positive definite matrix. From the closed-loop dynamics of zi,

for i = 1, . . . , n, we obtain

V =n∑

i=1

(−ciz2i + ziϕ

Ti θ + ziβi) −

n∑

i=2

zi∂αi−1

∂θ( ˙θ − τi) + ˙

θT −1θ

= −n∑

i=1

ciz2i +

n∑

i=2

(

ziβi − zi∂αi−1

∂θ( ˙θ − τi)

)

+(

n∑

i=1

ziϕi − −1 ˙θ

)T

θ .

We set the adaptive law as

˙θ =

n∑

i=1

ziϕi. (9.62)

We can conclude the design by setting the tuning functions to satisfyn∑

i=2

(

ziβi − zi∂αi−1

∂θ( ˙θ − τi)

)

= 0.

Substituting the adaptive law in the above equation, and with some manipulation ofindex, we obtain that

0 =n∑

i=2

⎛

⎝ziβi − zi∂αi−1

∂θ

⎛

⎝

n∑

j=1

zjϕj − τi

⎞

⎠

⎞

⎠

=n∑

i=2

ziβi −n∑

i=2

n∑

j=2

zizj∂αi−1

∂θ ϕj +

n∑

i=2

zi∂αi−1

∂θ(τi − z1 ϕ1)


=n∑

i=2

ziβi −n∑

i=2

n∑

j=i+1

zizj∂αi−1

∂θ ϕj

−n∑

i=2

i∑

j=2

zizj∂αi−1

∂θ ϕj +

n∑

i=2

zi∂αi−1

∂θ(τi − z1 ϕ1)

=n∑

i=2

ziβi −n∑

j=3

j−1∑

i=2

zizj∂αi−1

∂θ ϕj

+n∑

i=2

zi∂αi−1

∂θ

⎛

⎝τi − z1 ϕ1 −i∑

j=2

zj ϕj

⎞

⎠

=n∑

i=3

zi

⎛

⎝βi −i−1∑

j=2

zj∂αj−1

∂θ ϕi

⎞

⎠

+n∑

i=2

zi∂αi−1

∂θ

⎛

⎝τi − z1 ϕ1 −i∑

j=2

zj ϕj

⎞

⎠ .

Hence, we obtain

βi =i−1∑

j=2

zj∂αj−1

∂θ ϕi, for i = 3, . . . , n, (9.63)

τi =i∑

j=1

zj ϕj , for i = 2, . . . , n. (9.64)

With the complete design of u and ˙θ , we finally obtain that

V = −n∑

i=1

ciz2i .

from which we can deduce that zi ∈ L2 ∩ L∞, i = 1, . . . , n, and θ is bounded. Sinceall the variables are bounded, we have zi bounded. From Barbalat’s Lemma, we havelimt→∞ zi(t) = 0, i = 1, . . . , n. Noticing that x1 = z1 and φ1(0) = 0, we can concludethat α1 converges to zero, which further implies that limt→∞ x2(t) = 0. Repeating thesame process, we can show that limt→∞ xi(t) = 0 for i = 1, . . . , n.

Theorem 9.7. For a system in the form of (9.57), the control input (9.60) and adaptivelaw (9.62) designed by adaptive backstepping ensure the boundedness of all thevariables and limt→∞ xi(t) = 0 for i = 1, . . . , n.


Example 9.2. Consider a second-order system

x1 = x2 + (ex1 − 1)θ

x2 = u,

where θ ∈ R is the only unknown parameter. We will follow the presented designprocedure to design an adaptive control input to stabilise the system.

Let z1 = x1 and z2 = x2 − α1. We design the stabilising function α1 as

α1 = −c1z1 − (ex1 − 1)θ .

The resultant dynamics of z1 are given by

z1 = −c1z1 + z2 + (ex1 − 1)θ .

The dynamics of z2 are obtained as

z2 = u − ∂α1

∂x1(x2 + (ex1 − 1)θ ) − ∂α1

∂θ

˙θ ,

where

∂α1

∂x1= −c1 − ex1 θ ,

∂α1

∂θ= −(ex1 − 1).

Therefore, we design the control input u as

u = −z1 − c2z2 + ∂α1

∂x1(x2 + (ex1 − 1)θ ) + ∂α1

∂θ

˙θ.

Note that the control input u contains ˙θ , which is to be designed later, as a function

of the state variables and θ . The resultant dynamics of z2 are obtained as

z2 = −z1 − c2z2 − ∂α1

∂x1(ex1 − 1)θ .

Let

V = 1

2

(

z21 + z2

2 + θ 2

γ

)

.

We obtain that

V = −c1z21 − c2z2

2 +(

z1(ex1 − 1) − z2∂α1

∂x1(ex1 − 1)

)

θ − 1

γθ

˙θ.

Therefore, an obvious choice of the adaptive law is

˙θ = γ z1(ex1 − 1) − γ z2

∂α1

∂x1(ex1 − 1)

which gives

V = −c1z21 − c2z2

2 .

The rest part of the stability analysis follows Theorem 9.7. �


Example 9.3. In Example 9.2, the nonlinear system is in the standard format asshown in (9.56). In this example, we show adaptive control design for a system whichis slightly different from the standard form (9.56), but the same design procedure canbe applied with some modifications.

Consider a second-order nonlinear system

x1 = x2 + x31θ + x2

1

x2 = (1 + x21)u + x2

1θ ,

where θ ∈ R is the only unknown parameter.Let z1 = x1 and z2 = x2 − α1. The stabilising function α1 is designed as

α1 = −c1z1 − x31 θ − x2

1,

which results in the dynamics of z1 as

z1 = −c1z1 + z2 + x31 θ .

The dynamics of z2 are obtained as

z2 = (1 + x21)u + x2

1θ − ∂α1

∂x1(x2 + x3

1θ + x21) − ∂α1

∂θ

˙θ ,

where∂α1

∂x1= −c1 − 3x2

1 θ − 2x1,∂α1

∂θ= −x3

1.

Therefore, we design the control input u as

u = 1

1 + x21

(

−z1 − c2z2 − x21 θ + ∂α1

∂x1(x2 + x3

1 θ + x21) + ∂α1

∂θ

˙θ

)

.

The resultant dynamics of z2 are obtained as

z2 = −z1 − c2z2 + x21 θ − ∂α1

∂x1x3

1 θ .

Let

V = 1

2

(

z21 + z2

2 + θ 2

γ

)

.

We obtain that

V = −c1z21 − c2z2

2 +(

z1x13 + z2

(

x21 − ∂α1

∂x1x3

1

))

θ − 1

γθ

˙θ.

We can set the adaptive law as

˙θ = γ z1x1

3 + γ z2

(

x21 − ∂α1

∂x1x3

1

)

to obtain

V = −c1z21 − c2z2

2 .


0 2 4 6 8 10−6

−5

−4

−3

−2

−1

0

1

2

t (s)

Stat

e va

riabl

es

x1

x2


0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

t (s)

Estim

ated

par

amet

er

Figure 9.2 Estimated parameter θ


The rest part of the stability analysis follows Theorem 9.7. Simulation results areshown in Figures 9.1 and 9.2 with x(0) = [1, 1]T , c1 = c2 = γ = θ = 1. The statevariables converge to zero as expected, and the estimated parameter converges toa constant, but not to the correct value θ = 1. In general, estimated parameters inadaptive control are not guaranteed to converge to their actual values. �

9.6 Adaptive observer backstepping

Backstepping can also be used to design control input for a class of nonlinear systemswith unknown parameters using output feedback. We consider a system which can betransformed to the output feedback form with unknown parameters

x = Acx + bσ (y)u + φ0(y) +p∑

i=1

φi(y)ai

:= Acx + bσ (y)u + φ0(y) +�(y)a (9.65)

y = Cx

with

Ac =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

T

,

a =

⎡

⎢⎢⎢⎣

a1

a2...

ap

⎤

⎥⎥⎥⎦

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0bρ...

bn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

:=[

0(ρ−1)×1

b

]

,

where x ∈ Rn is the state vector, u ∈ R is the control and b ∈ R

n is an unknownconstant Hurwitz vector with bρ �= 0, which implies that the relative degree of thesystem is ρ, a denotes unknown system parameters, φi : R → R

n for 0 ≤ i ≤ pare nonlinear functions with their elements being differentiable up to the ρth order,σ : R → R is a continuous function and σ (y) �= 0, ∀y ∈ R.

For the system (9.65), we assume that only the system output is available forcontrol design. Both observer backstepping and backstepping with filtered transfor-mation design methods presented earlier can be applied for the control design of(9.65) if there are no unknown parameters. Even with the unknown parameters, thesame filtered transformation introduced earlier in (9.34) can still be applied to thesystem (9.65), and adaptive backstepping can then be applied to design the controlinput. Here we take an approach similar to observer backstepping. Because there


are unknown parameters, we will not be able to design an observer in the same wayas in the observer backstepping. Instead, we can design filters similar to observers,and obtain an expression of state estimation which contains unknown parameters. Theunknown parameters in the state estimation are then tackled by adaptive backstepping.

For the state estimation, we re-arrange the system as

x = Acx + φ0(y) + FT (y, u)θ , (9.66)

where the vector θ ∈ Rq, with q = n − ρ + 1 + p, is defined by

θ =[

ba

]

and

F(y, u)T =[[

0(ρ−1)×(n−ρ+1)

In−ρ+1

]

σ (y)u, �(y)]

.

Similar to observer design, we design the following filters:

ξ = A0ξ + Ly + φ0(y), (9.67)

�T = A0�T + F(y, u)T , (9.68)

where ξ ∈ Rn, �T ∈ R

n×q and

L = [l1, . . . , ln]T , A0 = Ac − LC

with L being chosen so that A0 is Hurwitz. An estimate of the state is then given by

x = ξ +�T θ. (9.69)

Let

ε = x − x

and from direct evaluation, we have

ε = A0ε. (9.70)

Therefore, the state estimate shown in (9.69) is an exponentially convergent one.Notice that this estimate contains unknown parameter vector θ , and therefore x cannotbe directly used in control design. The relationship between an convergent estimatex and the unknown parameter vector can be used in adaptive backstepping controldesign.

We can reduce the order of the filters. Let us partition �T as �T = [v,�] withv ∈ R

n×(n−ρ+1) and � ∈ Rn×p. We then obtain, from (9.68), that

� = A0�+�(y),

vj = A0vj + ejσ (y)u, for j = ρ, . . . , n,

where ej denotes jth column of identity matrix I in Rn. For 1 < j < n, we have

A0ej = (Ac − LC)ej = Acej = ej+1.


This implies that

vj = An−j0 vn.

Finally, we summarise the filters for �T as

�T = [vρ , . . . , vn,�],� = A0�+�(y),λ = A0λ+ enσ (y)u,vj = An−j

0 λ, for j = ρ, . . . , n.

(9.71)

With the filters being designed, control design can be carried out using adaptivebackstepping in a similar way as the observer backstepping, by combining the designof tuning functions in parameter adaptation. During the control design, the statevariable x will be replaced by

x = ξ +�T θ + ε,

and in particular, whenever we encounter x2, we will replace it by

x2 = ξ2 +�T(2)θ + ε2,

where the subscript (i) denotes the ith row of a matrix. In the following control design,we will consider the tracking control instead of stabilisation. The output y is designedto track a trajectory yr with its derivatives available for the control design.

Let us define a number of notations:

z1 = y − yr ,

zi = vρ,i − �y(i−1)r − αi−1, i = 2, . . . , ρ,

zρ+1 = σ (y)u + vρ,ρ+1 − �y(ρ)r − αρ ,

ϕ0 = ξ2 + φ0,1,

ϕ = [vρ,2, . . . , vn,2,�(1) +�(2)]T ,

ϕ = [0, vρ+1,2, . . . , vn,2,�(1) +�(2)]T ,

λi = [λ1, . . . , λi]T ,

yi = [yr , yr , . . . , y(i)r ]T ,

Xi = [ξT , vec(�)T , �, λTi , yT

i ]T ,

σj,i = ∂αj−1

∂θ ∂αi−1

∂yϕ,

where �, an estimate of � = 1/bρ , and αi, i = 1, . . . , ρ, are the stabilising functionsto be designed.

Consider the dynamics of z1

z1 = x2 + φ0,1(y) +�(1)θ − yr

= ξ2 +�T(2)θ + ε2 + φ0,1(y) +�(1)θ − yr

= bρvρ,2 + ϕ0 + ϕT θ + ε2 − yr. (9.72)


Note that bρ is unknown. To deal with unknown control coefficient, we often estimateits reciprocal, instead of itself, to avoid using the reciprocal of an estimate. This is thereason why we define

z2 = vρ,2 − α1 − �yr

:= vρ,2 − �α1 − �yr.

From bρ� = 1, we have

bρ� = 1 − bρ�,

where � = � − �. Then from (9.72), we have

z1 = bρ(z2 + �α1 + �yr) + ϕ0 + ϕT θ + ε2 − yr

= bρz2 − bρ�(α1 + yr) + α1 + ϕ0 + ϕT θ + ε2. (9.73)

Hence, we design

α1 = −c1z1 − k1z1 − ϕ0 − ϕT θ , (9.74)

where ci and ki for i = 1, . . . , ρ are positive real design parameters. Note that withα = �α1, we have α1 = α1(y, X1, θ ).

The resultant closed-loop dynamics are obtained as

z1 = −c1z1 − k1z1 + bρz2 − bρ�(α1 + yr) + ϕT θ + ε2

= −c1z1 − k1z1 + bρz2 + bρz2 − bρ�(α1 + yr) + ϕT θ + ε2

= −c1z1 − k1z1 + bρz2 − bρ�(α1 + yr)

+ (ϕ − �(α1 + yr)e1)T θ + ε2, (9.75)

where θ = θ − θ . Note that bρ = θT e1.As shown in the observer backstepping, the term −k1z1 is used to tackle the

error term ε2. In the subsequent steps, the terms headed by ki are used to deal withthe terms caused by ε2 in stability analysis. Indeed, the method of tackling observererrors is exactly the same as in the observer backstepping when all the parameters areknown. The adaptive law for � can be designed in this step, as it will not appear inthe subsequent steps,

˙� = −γ sgn(bρ)(α1 + yr))z1, (9.76)

where γ is a positive real constant.Similar to adaptive backstepping shown in the previous section, the unknown

parameter vector θ will appear in the subsequent steps, and tuning functions can beintroduced in a similar way. We define the tuning functions τi as

τ1 = (ϕ − �(yr + α1)e1)z1,

τi = τi−1 − ∂αi−1∂y ϕzi, i = 2, . . . , ρ,

(9.77)

where αi are the stabilising functions to be designed, and ∈ Rq×q is a positive

definite matrix, as the adaptive gain.



z2 = vρ,2 − α1 − ˙�yr − �yr

= vρ,3 − l2vρ,1 − ∂α1

∂X1X1 − ∂α1

∂yy − ∂α1

∂θ

˙θ − ˙�yr − �yr.

With

y = ϕ0 + ϕT θ + ε2,

z3 = vρ,3 − α2 − �yr ,

we obtain that

z2 = z3 + α2 − l2vρ,1 − ∂α1

∂X1X1 − ∂α1

∂y(ϕ0 + ϕT θ + ε2) − ∂α1

∂θ

˙θ − ˙�yr ,

from which the stabilising function α2 is defined as

α2 = −bρz1 − c2z2 − k2

(∂α1

∂y

)2

z2 + l2vρ,1 + ˙�yr

+ ∂α1

∂X1X1 + ∂α1

∂y(ϕ0 + ϕT θ ) + ∂α1

∂θτ2. (9.78)

The resultant dynamics of z2 are obtained as

z2 = −bρz1 − c2z2 − k2

(∂α1

∂y

)2

z2 + z3

−∂α1

∂yϕT θ − ∂α1

∂yε2 − ∂α1

∂θ( ˙θ − τ2). (9.79)

For the dynamics of zi, 2 < i ≤ ρ, we have

zi = zi+1 + αi − livρ,1 − ∂αi−1

∂Xi−1Xi−1 − ∂αi−1

∂y(ϕ0 + ϕT θ + ε2)

−∂αi−1

∂θ

˙θ − ˙�y(i−1)

r .

We design αi, 2 < i ≤ ρ, as


(∂αi−1

∂y

)2

zi + livρ,1

+ ˙�y(i−1)r + ∂αi−1

∂Xi−1Xi−1 + ∂αi−1

∂y(ϕ0 + ϕT θ )

+ ∂αi−1

∂θτi −

i−1∑

j=2

σj,izj , i = 3, . . . , ρ, (9.80)

where the last term −∑i−1j=2 σj,izj is similar to the term βi in the adaptive backstepping

with tuning functions. The resultant dynamics of zi are obtained as



(∂αi−1

∂y

)2

zi + zi+1

− ∂αi−1

∂yϕT θ − ∂αi−1

∂yε2 −

i−1∑

j=2

σj,izj

− ∂αi−1

∂θ( ˙θ − τi) i = 3, . . . , ρ. (9.81)

Now we design the adaptive law for θ as

˙θ = τρ. (9.82)

The control input is obtained by setting zρ+1 = 0 as

u = 1

σ (y)(αρ − vρ,ρ+1 + �y(ρ)

r ). (9.83)

For the adaptive observer backstepping, we have the following stability result.

Theorem 9.8. For a system in the form of (9.65), the control input (9.83) and adap-tive laws (9.76) and (9.82) designed by adaptive observer backstepping ensure theboundedness of all the variables and limt→∞ (y(t) − yr(t)) = 0.

Proof. With the control design and adaptive laws presented earlier for the case ρ > 1,the dynamics of zi, for i = 1, . . . , ρ can be written as

z1 = −c1z1 − k1z1 + ε2 + (ϕ − �(yr + α1)e1)T θ

− bρ(yr + α1)� + bρz2, (9.84)

z2 = −bρz1 − c2z2 − k2

(∂α1

∂y

)2

z2 + z3

− ∂α1

∂yϕT θ − ∂α1

∂yε2 +

ρ∑

j=3

σ2,jzj , (9.85)


(∂αi−1

∂y

)2

zi + zi+1

− ∂αi−1

∂yϕT θ − ∂αi−1

∂yε2 +

ρ∑

j=i+1

σi,jzj

−i−1∑

j=2

σj,izj i = 3, . . . , ρ. (9.86)


Let

Vρ = 1

2

ρ∑

i=1

z2i + 1

2θT −1θ + |bρ |

2γ�2 +

ρ∑

i=1

1

4kiεT Pε, (9.87)

where P is a positive definite matrix that satisfies

AT0 P + PA0 = −I .

From (9.84)–(9.86), (9.82) and (9.76), it can be shown that

Vρ = −ρ∑

i=1

(

ci + ki

(∂αi−1

∂y

)2)

z2i −

ρ∑

i=1

zi∂αi−1

∂yε2 −

ρ∑

i=1

1

4ki‖ε‖2, (9.88)

where we set ∂α0∂y = −1. Noting

∣∣∣∣zi∂αi−1

∂yε2

∣∣∣∣ ≤ 1

4ki‖ε2‖2 + ki

(∂αi−1

∂y

)2

z2i , (9.89)

we have

Vρ ≤ −ρ∑

i=1

ciz2i . (9.90)

This implies that zi, i = 1, . . . , ρ, θ , � and ε are bounded. The boundedness of θfurther implies that θ is bounded. The boundedness of � follows from the fact that� = 1/bρ is a constant. Since y = z1 + yr , the boundedness of ξ and � follows theboundedness of y. The boundedness of λ can be established from the minimum phaseproperty of the system. Therefore, u is bounded and we can conclude that all thevariables of the feedback control system are bounded. From the above analysis, wecan deduce that zi ∈ L2 ∩ L∞, i = 1, . . . , ρ and θ are bounded. Since all the variablesare bounded, we have zi bounded. From Barbalat’s Lemma, we have limt→∞ zi(t) =0, i = 1, . . . , ρ, of which the result for z1 means the asymptotic output tracking,limt→∞ (y(t) − yr(t)) = 0. �

Chapter 10

Disturbance rejection and output regulation

Disturbances are often inevitable in control system design. A well-performed con-troller is expected to suppress undesirable effects of the disturbances in the system.There are various types of disturbances in physical systems, from random distur-bances, wide-band, narrow-band disturbances, in terms of disturbance power spectra,to deterministic disturbances that include harmonic disturbances, i.e., sinusoidal func-tions, general periodic disturbances and other deterministic signals generated fromnonlinear dynamic systems such as limit cycles. The spectral information of randomdisturbances may be considered in loop-shaping and other conventional design meth-ods. In this chapter, we concentrate on suppression and rejection of deterministicperiodic disturbances. One control design objective is to track a specific signal. If wetake the tracking error as the state variable, the tracking problem could be convertedto a stabilisation problem. Indeed, we can formulate both the disturbance rejectionand output tracing problems in terms of output regulation. This will be discussed laterin this chapter.

Disturbance rejection and output regulation are big topics in control systems,design. We have to be selective to limit the contents in one chapter. In this chapter, wewill concentrate on rejection of deterministic disturbances in a class of nonlinear out-put feedback systems. As for disturbances, we will start from sinusoidal disturbanceswith unknown frequencies, then disturbances generated from nonlinear exosystemsand then general periodical disturbances, etc. Adaptive control techniques are usedto deal with the unknown disturbance frequencies and unknown parameters in thesystem. The presented design concepts can be applied to other classes of nonlinearsystems.

10.1 Asymptotic rejection of sinusoidal disturbances

In this section, we consider asymptotic rejection of sinusoidal disturbances withunknown frequencies for dynamic systems in the output feedback form.

We consider a SISO nonlinear system which can be transformed into the outputfeedback form

ζ = Acζ + bu + φ(y) + Ew

y = Cζ ,(10.1)


with

Ac =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0.......... . .

...

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

T

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0bρ...

bn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where ζ ∈ Rn is the state vector; u ∈ R is the control; φ : R → R

n with φ(0) = 0 isa nonlinear function with element φi being differentiable up to the (n − i)th order;b ∈ R

n is a known constant Hurwitz vector with bρ �= 0, which implies the relativedegree of the system isρ; E ∈ R

n×m is a constant matrix; and w ∈ Rm are disturbances,

and they are generated from an unknown exosystem

w = Sw

of which, S is a constant matrix with distinct eigenvalues of zero real parts.

Remark 10.1. For the system (10.1), if the disturbance w = 0, it is exactly sameas (9.17), of which observer backstepping can be used to design a control input.Due to the unknown disturbance, although the observer backstepping presented inChapter 9 cannot be applied directly for control design, a similar technique can bedeveloped using an observer and an adaptive internal model. Also note that thelinear system characterised by (Ac, b, C) is minimum phase. �

Remark 10.2. The dynamic model w = Sw is referred to as an exosystem, becausew is a disturbance, not a part of the system state. This is a convention adopted fordisturbance rejection and output regulation. Of course, one could argue that w couldbe considered as a part of the system state, or at least, of an augmented system. Insuch a case, w is not controllable. �

Remark 10.3. With the assumption that S has distinct eigenvalues with zero realparts, w is restrict to sinusoidal signals (sinusoidal disturbances) with a possibleconstant bias. This is a common assumption for disturbance rejection. Roughlyspeaking, all the periodic signal can be approximated by finite number of sinusoidalfunctions. �

The disturbance rejection problem to be solved here is to design a control inputthat ensures the boundedness of the variables in the closed-loop system, and theconvergence to zero of the system output.

To solve the problem, we start from state transformation, based on an invariantmanifold. The basic idea for disturbance suppression or output regulation is the inter-nal model principle. A controller for disturbance rejection or output regulation shouldgenerate a feedforward input term to cancel the influence caused by the disturbance orto track a desired trajectory. This feedforward term is also referred to as the equivalent

Disturbance rejection and output regulation 177

input disturbance. For nonlinear systems, asymptotic disturbance rejection dependson the existence of an invariant manifold. We shall show that there exists an invariantmanifold for any exosystem specified in (10.1). This invariant manifold is then usedin the state transformation. The following lemma summarises the results.

Lemma 10.1. For the system (10.1) with an exosystem whose eigenvalues are distinctand with zero real parts, there exist π (w) ∈ R

n with π1(w) = 0 and α(w) ∈ R suchthat

∂π (w)

∂wSw = Acπ (w) + Ew + bα(w). (10.2)

Proof. Asymptotic disturbance rejection aims at y = 0, which implies that π1 = 0.The manifold π is invariant and it should satisfy the system equation (10.1) withy ≡ 0. From the first equation of (10.1), we have

π2 = −E1w (10.3)

where E1 denotes the first row of E. Furthermore, we have, for 2 ≤ i ≤ ρ,

πi = d

dtπi−1 − Ei−1w. (10.4)

From equations ρ to n of (10.1), we obtainn∑

i=ρ

dn−i

dtn−ibiα(w) = dn−ρ+1

dtn−ρ+1πρ −

n∑

i=ρ

dn−i

dtn−iEiw. (10.5)

A solution of α(w) can always be found from (10.5). With α(w), we can write, forρ < i ≤ n,

πi = d

dtπi−1 − Ei−1w − bi−1α(w). (10.6)

�

With the invariant manifold π (w), we define a transformation of state as

x = ζ − π (w). (10.7)

It can be easily shown from (10.1) and (10.2) that

x = Acx + b(u − α) + φ(y)

y = Cx.(10.8)

The stabilisation and disturbance suppression problem of (10.1) degenerates to thestabilisation problem of (10.8).

For dynamic output feedback control, state variables need to be estimated forthe control design directly or indirectly. The difficulty for designing a state estimatorfor (10.8) is that α(w), the feedforward control input for disturbance suppression, isunknown. Let us consider the observers

p = (Ac − LC)p + φ(y) + bu + Ly (10.9)

q = (Ac − LC)q + bα(w), (10.10)


where L ∈ Rn is chosen so that Ac − LC is Hurwitz, and q denotes the steady-state

contribution of α(w) to the state variable. Notice that the observer (10.10) cannot beimplemented becauseα(w) is unknown, due to the unknown exosystem. Nevertheless,if we define

x = p − q (10.11)

then the error of the observer defined by ε = x − x is an exponentially decaying signalwith the dynamics

ε = (Ac − LC)ε, (10.12)

which can be obtained by a direct evaluation from (10.8) to (10.10).Observe from (10.5) that α(w) is a linear combination of w. Since the filter

(10.10) is just a stable linear system, the steady-state solution of state variables islinear combinations of w as well. Therefore, there exists an l ∈ R

m for q2, and we canwrite

w = Sw

q2 = lT w.(10.13)

We re-parameterise (10.13) for state estimation here for q2. For any known controllablepair (F , G) with F ∈ R

m×m being Hurwitz and G ∈ Rm, there exists a ψ ∈ R

m so that

η = (F + GψT )η

q2 = ψTη,(10.14)

with the initial value η(0) dependent on exogenerous variables.

Remark 10.4. The importance of (10.14) compared with (10.13) is the re-formulationof the uncertainty caused by the unknown exosystem. The uncertainty in (10.13)parameterised by unknown S and l is represented by a single vector ψ in (10.14). Therelation between the two parameterisations is discussed here. Suppose that M ∈ R

m×m

is the unique solution of

MS − FM = GlT . (10.15)

The existence of a non-singular M is ensured by the fact that S and F have exclu-sively different eigenvalues, and (S, l) and (F , G) are observable and controllablerespectively. From (10.15), we have

MSM−1 = F + GlT M −1, (10.16)

which implies η = Mw and ψT = lT M −1

To achieve asymptotic rejection with global stability using output feedback, anasymptotic state estimator, probably depending on unknown parameters, is instru-mental. In the control design shown later for the problem considered, the estimate ofstate variable x2 is crucial, and therefore we introduce the internal model (10.14) todescribe the influence of disturbance w on this state. �

The following lemma summarises the results on state observation.


Lemma 10.2. The state variable x can be expressed as

x = p − q + ε, (10.17)

where p is generated from (10.9) with q and ε satisfying (10.10) and (10.12)respectively. In particular,

x2 = p2 − ψTη + ε2, (10.18)

where η satisfies (10.14).

Remark 10.5. The expression (10.18) cannot be directly implemented, because ψand η are not available. The relation shown in (10.18) is very useful in the controldesign, and it allows adaptive control technique to be introduced to deal with theunknown parameter ψ later. �

Based on the parameterisation (10.14) of the internal model, we design anestimator of η as

ξ = (F + GψT )ξ + ι(y)

q2 = ψT ξ ,(10.19)

where ι(y) is an interlace function to be designed later.With the state estimation introduced in previous chapter, in particular, the filter

(10.9), the relation (10.18) and estimator (10.19), the control design can be carried outusing adaptive observer backstepping technique described in previous chapter. In thebackstepping design, ci, for i = 1, . . . , ρ, denote constant design parameters whichcan be set in the design, while ki, for i = 1, . . . , ρ, denote the unknown parametersdepending on an upper bound of ‖ψ‖. The estimates, ki, for i = 1, . . . , ρ, are thenused. To apply the observer backstepping through p in (10.9), we define

z1 = y, (10.20)

zi = pi − αi−1, i = 2, . . . , ρ, (10.21)

zρ+1 = bρu + pρ+1 − αρ , (10.22)

where αi, for i = 1, . . . , ρ, are stabilising functions decided in the control design.Consider the dynamics of z1

z1 = x2 + φ1(y). (10.23)

We use (10.18) to replace the unmeasurable x2 in (10.23), resulting at

z1 = p2 − ψTη + ε2 + φ1(y)

= z2 + α1 − ψTη + ε2 + φ1(y). (10.24)

We design α1 as

α1 = −c1z1 − k1z1 − φ1(y) + ψT ξ. (10.25)


Then from (10.25) and (10.24), we have

z1 = z2 − c1z1 − k1z1 + ε2 + (ψT ξ − ψTη). (10.26)

After the design of α1, the remaining stabilising functions can be designed in a similarway to the standard adaptive backstepping with tuning functions shown in the previouschapter. Omitting the deriving procedures, we briefly describe the final results


(∂αi−1

∂y

)2

zi − li(y − p1) − φi(y)

+∂αi−1

∂y(p2 − ψT ξ + φ1) +

i−1∑

j=1

∂αi−1

∂pjpj +

i−1∑

j=1

∂αi−1

∂ kj

˙kj + ∂αi−1

∂ξξ

+∂αi−1

∂ψτi +

i−1∑

j=2

∂αj−1

∂ψ�∂αi−1

∂yξzj , i = 2, . . . , ρ, (10.27)

where li are the ith element of the observer gain L in (10.9), � is a positive definitematrix, τi, i = 2, . . . , ρ, are the tuning functions defined by

τi =i∑

j=1

�∂αj−1

∂yξzj , i = 2, . . . , ρ, (10.28)

where we set ∂α0∂y = −1. The adaptive law for ψ is set as

˙ψ = τρ. (10.29)

The adaptive laws for ki, i = 1, . . . , ρ, are given by

˙ki = γi

(∂αi−1

∂y

)2

z2i , (10.30)

where γi is a positive real design parameter. The control input is obtained by settingzρ+1 = 0 as

u = 1

bρ(αρ − pρ+1). (10.31)

Considering the stability, we set a restriction on c2 by

c2 > ‖PG‖2, (10.32)

where P is a positive definite matrix, satisfying

FT P + PF = −2I . (10.33)

To end the control design, we set the interlace function in (10.19) by

ι(y) = −(FG + c1G + k1G)y. (10.34)

Remark 10.6. The adaptive coefficients ki, i = 1, . . . , ρ, are introduced to allow theexosystem to be truly unknown. It implies that the proposed control design can reject


disturbances completely at any frequency. If an upper bound of ‖ψ‖ is known, wecan replace ki, i = 1, . . . , ρ, by constant positive real parameters. �

Remark 10.7. The final control u in (10.31) does not explicitly contain α(w), thefeedforward control term for disturbance rejection. Instead, the proposed controldesign considers q2, the contribution of the influence of α(w) to x2, from the first stepin α1 throughout to the final step in αρ . �

Theorem 10.3. For the system (10.1), the control input (10.31) ensures the bound-edness of all the variables, and asymptotically rejects the unknown disturbances inthe sense that limt→∞ y(t) = 0. Furthermore, if w(0) ∈ R

m is such that w(t) containsthe components at m/2 distinct frequencies, then limt→∞ ψ(t) = ψ .

Proof. We start from the analysis of the internal model. Define

e = ξ − η − Gy. (10.35)

From (10.14), (10.19), (10.34) and (10.26), it can be obtained that

e = Fe − G(z2 + ε2). (10.36)

Based on the stabilising functions shown in (10.27), the dynamics of zi, fori = 2, . . . , ρ, can be written as


(∂αi−1

∂y

)2

zi + zi+1 − ∂αi−1

∂y(ψT ξ − ψTη)

−∂αi−1

∂yε2 −

ρ∑

j=i+1

∂αi−1

∂ψ�∂αj

∂yξzj

+i−1∑

j=2

∂αj−1

∂ψ�∂αi−1

∂yξzj , i = 2, . . . , ρ, (10.37)

where the term ψT ξ − ψTη can be rephrased by

ψT ξ − ψTη = ψT e − ψT ξ + ψT Gz1 (10.38)

with ψ = ψ − ψ .In the following analysis, we denote κ0, κi,j , i = 1, . . . , 3, and j = 1, . . . , ρ, as

constant positive reals, which satisfy

κ0 +ρ∑

j=1

κ3,j <1

2. (10.39)


Furthermore, we set constant positive reals ki, i = 1, . . . , ρ, satisfying the followingconditions:

k1 > |ψT G| + κ1,1 +ρ∑

j=2

κ2,i|ψT G|2 + ‖ψ‖2

4κ3,1, (10.40)

ki > κ1,i + 1

4κ2,i+ ‖ψ‖2

4κ3,ii = 2, . . . , ρ. (10.41)

Define a Lyapunov function candidate

V = 1

2

(

eT Pe +ρ∑

i=1

z2i +

ρ∑

i=1

γ −1i k2

i + ψT�−1ψ + βεT Pεε

)

, (10.42)

where ki = ki − ki, i = 1, . . . , ρ, Pε is a positive definite matrix satisfying

(Ac − LC)T Pε + Pε(Ac − LC) = −2I ,

and β is a constant positive real satisfying

β >‖PG‖2

4κ0+

ρ∑

j=1

1

4κ1,i. (10.43)

Evaluating the derivative of V along the dynamics in (10.12), (10.26), (10.36)and (10.37) together with adaptive laws in (10.29) and (10.30), we have

V = −ρ∑

i=1

(

ci + ki

(∂αi−1

∂y

)2)

z2i −

ρ∑

i=1

zi∂αi−1

∂yε2

−ρ∑

i=1

zi∂αi−1

∂y(ψT e + ψT Gz1)

−eT e − eT PG(z2 + ε2) − βεT ε. (10.44)

For the cross-terms in (10.44), we have

∣∣eT PGε2

∣∣ < κ0eT e + 1

4κ0‖PG‖2ε2

2 ,

∣∣∣∣zi∂αi−1

∂yε2

∣∣∣∣ < κ1,i

(∂αi−1

∂y

)2

z2i + 1

4κ1,iε2

2 , i = 1, . . . , ρ,

∣∣∣∣∂αi−1

∂yψT Gz1zi

∣∣∣∣ < κ2,i|ψT G|2z2

1 + 1

4κ2,i

(∂αi−1

∂y

)2

z2i , i = 2, . . . , ρ,

∣∣∣∣∂αi−1

∂yψT ezi

∣∣∣∣ < κ3,ie

T e + ‖ψ‖2

4κ3,i

(∂αi−1

∂y

)2

z2i , i = 1, . . . , ρ,

|eT PGz2| < eT e

2+ ‖PG‖2z2

2/2.


Then based on the conditions specified in (10.32), (10.39), (10.40), (10.41) and(10.43), we can conclude that there exist positive real constants δi, for i = 1, 2, 3,such that

V ≤ −δ1

ρ∑

i=1

z2i − δ2eT e − δ3ε

T ε. (10.45)

We conclude zi ∈ L2 ∩ L∞, i = 1, . . . , ρ, and ‖e‖ ∈ L2 ∩ L∞ and the boundednessof the variables ψ , and ki, i = 1, . . . , ρ. With y = z1 ∈ L∞, the boundedness of pcan be established from the minimum phase property of the system and the neutralstability of w. Therefore, we can conclude that all the variables in the proposedclosed-loop system are bounded. Since the derivatives of zi and e are bounded, fromzi ∈ L2 ∩ L∞, i = 1, . . . , ρ, ‖e‖ ∈ L2 ∩ L∞ and Babalat’s lemma, we further concludethat limt→∞ zi = 0, i = 1, . . . , ρ, and limt→∞ ‖e‖ = 0.

From (10.35), we have limt→∞ ‖ξ (t) − η(t)‖ = 0, which means that ξ asymptot-ically converges to η. From the boundedness of all the variables and limt→∞ zi = 0,

i = 1, . . . , ρ, we can conclude that limt→∞˙ψ = 0 and limt→∞

˙ki = 0, i = 1, . . . , ρ,which implies that the adaptive controller will converge to a fixed-parameter type.

We now establish the convergence of ψ . Since limt→∞˙ψ = 0, there exists a

ψ∞ = limt→∞ ψ(t). From (10.14), (10.19) and (10.35), we obtain

d

dt(η − ξ ) = F(η − ξ ) + G(ψ − ψ∞)Tη + ε(t), (10.46)

where

ε(t) = G(ψ∞ − ψ)T ξ + GψT∞(e + Gy) − ι.

From limt→∞ (η − ξ ) = 0 and limt→∞ ε(t) = 0, we can conclude that

(ψ − ψ∞)Tη = 0, (10.47)

because, if otherwise, (ψ − ψ∞)Tη would be a persistently excited signal, and wecould never have limt→∞ (η − ξ ) = 0. If the disturbance w(t) contains componentsat m/2 distinct frequencies, the signal η(t) is persistently excited, and from (10.47)we have ψ − ψ∞ = 0, i.e., limt→∞ ψ(t) = ψ . �

Remark 10.8. The condition imposed on w(0) does not really affect the convergenceof ψ . It is added to avoid the situation that w(t) degenerates to have less independentfrequency components. In that situation, we can reform the exosystem and E in (10.1)with a smaller dimension m such that the reformed w(t) is persistently excited inreduced space R

m and accordingly we have η,ψ ∈ Rm. With the estimate ψ , together

with (F , G), the disturbance frequencies can be estimated. �


ζ1 = ζ2 + (ey − 1) + w1

ζ2 = u + w1

y = ζ1,


where w1 ∈ R is a disturbance with two sinusoidal components at unknown frequen-cies ω1 and ω2. An augmented disturbance vector w = [w1, w2, w3, w4]T satisfiesw = Sw with eigenvalues of S at {±jω1, ±jω2}. It is easy to obtain that π = [0, −w1]T

and α = −w1 − w1. Using the state transform x = ζ − π , we have

x1 = x2 + (ey − 1)

x2 = u − α (10.48)

y = x1.

A transform w = T w can also be introduced to the disturbance model so that thedisturbance is generated by

˙w =

⎡

⎢⎢⎣

[0 ω1

−ω1 0

]

02×2

02×2

[0 ω2

−ω2 0

]

⎤

⎥⎥⎦ w := Sw

w1 = [t11, t12, t13, t14]w,

where T , ω1 and ω2 are unknown. With the coordinate w, we have

α(w) = [−t11 + ω1t12, −t12 − ω1t11, −t13 + ω2t14, −t14 − ω2t13, ]w.

The steady-state contribution of w to x2 is given by q2 = Q2w, i.e., lT = Q2T −1, whereQ2 is the second row of Q which satisfies

QS = (Ac + kC)Q + [0, 1]T [−t11 + ω1t12, −t12 − ω1t11, −t13 + ω2t14, −t14 − ω2t13].

The system (10.48) is of relative degree 2, and the control input and parameterestimator can be designed by following the steps presented earlier.

In the simulation study, we chose the pair

F =

⎡

⎢⎢⎣

0 1 0 00 0 1 00 0 0 1

−1 −4 −6 −4

⎤

⎥⎥⎦ , G =

⎡

⎢⎢⎣

000

1/10

⎤

⎥⎥⎦ ,

which is controllable with the eigenvalues of F at {−1, −1, −1, −1} and ‖PG‖ < 1.We set c1 = c2 = 10, with the condition (10.32) being satisfied. Other parametersin the control design were set as γ1 = γ2 = 1, � = 1000I , L1 = 3 and L2 = 2. Thedisturbance was set as

w1 = 4 sinω1t + 4 sinω2t

with ω1 = 1 and

ω2 ={

2, 250 > t ≥ 0,1.5, t ≥ 250.

A set of simulation results are presented here. Figure 10.1 shows the system outputtogether with the control input. Figure 10.2 shows the estimates ofψ . The ideal valuesofψ are [−30, 40, 10, 40]T and [−12.5, 40, 27.5, 40]T for ω1 = 1,ω2 = 2 and ω1 = 1,


0 50 100 150 200 250 300 350 400 450 500−0.2

0

0.2

0.4

0.6

Time (s)

y

0 50 100 150 200 250 300 350 400 450 500−20

−10

0

10

20

30

Time (s)

u

Figure 10.1 System output and control input

0 50 100 150 200 250 300 350 400 450 500−80

−60

−40

−20

0

20

40

60

80

Time (s)

Estim

ated

par

amet

ers

Figure 10.2 Estimated parameters ψ1 (dashed), ψ2 (dotted), ψ3 (dashdot) andψ4 (solid)


ω2 = 1.5 respectively. Under both sets of the frequencies, ψ converged to the idealvalues. It can be seen through this example that the disturbance of two unknownfrequencies has been rejected completely. �

10.2 Adaptive output regulation

In the previous section, we presented a control design for asymptotic rejection ofharmonic disturbances in nonlinear systems when the disturbance frequencies areunknown. When the measurement, or the output, does not explicitly contain thedisturbances, the control design is referred to as asymptotic rejection problem forthe boundedness of the signals and the asymptotic convergence to zero. When themeasurement contains the disturbance or exogenous signals, the control design toensure the measurement converge to zero asymptotically is often referred to as theoutput regulation problem. When the measurement contains the exogenous signal, theoutput is normally different from the measurement, and in such a case, the output canbe viewed as to track an exogenous signal. In this section, we will present a controldesign for output regulation when the measurement is different from the controlledoutput.

We consider a SISO nonlinear system which can be transformed into the outputfeedback form

x = Acx + φ(y, w, a) + bu

y = Cx

e = y − q(w),

(10.49)

with

Ac =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0.......... . .

...

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

T

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0bρ...

bn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where x ∈ Rn is the state vector; u ∈ R is the control; y ∈ R is the output; e is

the measurement output; a ∈ Rq and b ∈ R

n are vectors of unknown parameters,with b being a Hurwitz vector with bρ �= 0, which implies the relative degree of thesystem is ρ; φ : R × R

m × Rq → R

n is a smooth vector field with each element beingpolynomials with a known upper order of its variables and satisfying φ(0, w, a) = 0;q is an unknown polynomial of w; and w ∈ R

m are disturbances, and they are generatedfrom an unknown exosystem

w = S(σ )w


with unknown σ ∈ Rs, of which, S ∈ R

m×m is a constant matrix with distincteigenvalues of zero real parts.

The control design problem considered in this section is to design a control inputusing the measurement feedback to ensure that the regulated measurement convergesto zero asymptotically while keeping all other variables bounded.

The system (10.1) has unknown parameters in the system and in the exosystem,and therefore adaptive control techniques are used to tackle the uncertainties. For thisreason, the problem under consideration is referred to as adaptive output regulation.

Remark 10.9. Different from (10.1), the system (10.49) has a measurement e that isperturbed by an unknown polynomial of the unknown disturbance. This is the reasonwhy the problem to be solved is an output regulation problem, rather than a disturbancerejection problem. �

Remark 10.10. The nonlinear functions in (10.49) are restricted to be polynomialsof its variables, to guarantee the existence of invariant manifold for the solutionof output regulation problem. Comparing with (10.1), the nonlinear term φ(y, w, a)has a more complicated structure, containing the output, disturbance, and unknownparameters all together. Assuming the nonlinear functions to be polynomials alsoallows that the nonlinear terms in the control design can be bounded by polynomialsof the measurement. �

Remark 10.11. In the system, all the parameters are assumed unknown, includingthe sign of the high-frequency gain, bρ , and the parameters of the exosystem. ANussbaum gain is used to deal with the unknown sign of the high-frequency gain.The adaptive control techniques presented in this control design can be easily appliedto other adaptive control schemes introduced in this book. The nonlinear functionsφ(y, w, a) and q(w) are only assumed to have known upper orders. This class ofnonlinear systems perhaps remains as the largest class of uncertain nonlinear systemsof which global output regulation problem can be solved with unknown disturbancefrequencies. �

When we set the unknown disturbance w and unknown parameter a to zero,the system (10.49) is in exactly the same format as (9.32), to which backsteppingwith filtered transformation has been applied. With unknown parameters, adaptivebackstepping with filtered transformation can be applied to solve the problem here.One reason for us to use backstepping with filtered transformation is due to theuncertainty in the nonlinear function φ(y, w, a), which prevents the application ofadaptive observer backstepping. In the following design, we only consider the casefor relative degree greater than 1.

For the system (10.49) with relative degree ρ > 1, we introduce the same filteredtransformation as in Section 9.3 with the filter

ξ1 = −λ1ξ1 + ξ2

. . .

ξρ−1 = −λρ−1ξρ−1 + u,(10.50)


where λi > 0, for i = 1, . . . , ρ − 1, are the design parameters, and the filteredtransformation

z = x − [d1, . . . , dρ−1]ξ , (10.51)

where ξ = [ξ1, . . . , ξρ−1]T and di ∈ Rn for i = 1, . . . , ρ − 1, and they are generated

recursively by dρ−1 = b and di = (Ac + λi+1I )di+1 for i = ρ − 2, . . . , 1. The system(10.49) is then transformed to

˙z = Acz + φ(y, w, a) + dξ1

y = Cz,(10.52)

where d = (Ac + λ1I )d1. It has been shown in (9.36) that d1 = bρ and

n∑

i=1

disn−i =

ρ−1∏

i=1

(s + λi)n∑

ρ

bisn−i. (10.53)

With ξ1 as the input, the system (10.52) is with relative degree 1 and minimum phase.We introduce another state transform to extract the internal dynamics of (10.52) withz ∈ R

n−1 given by

z = z2:n − d2:n

d1y, (10.54)

where (·)2:n refers to the vector or matrix formed by the 2nd row to the nth row. Withthe coordinates (z, y), (10.52) is rewritten as

z = Dz + ψ(y, w, θ )

y = z1 + ψy(y, w, θ ) + bρξ1,(10.55)

where the unknown parameter vector θ = [aT , bT ]T , and D is the left companionmatrix of d given by

D =

⎡

⎢⎢⎢⎢⎢⎢⎣

−d2/d1 1 . . . 0

−d3/d1 0. . . 0

......

......

−dn−1/d1 0 . . . 1−dn/d1 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎥⎦

, (10.56)

and

ψ(y, w, θ ) = Dd2:n

d1y + φ2:n(y, w, a) − d2:n

d1φ1(y, w, a),

ψy(y, w, θ ) = d2

d1y + d2:n

d1φ1(y, w, a).

Notice that D is Hurwitz, from (9.36), and that the dependence of d on b is reflected inthe parameter θ inψ(y, w, θ ) andψy(y, w, θ ), and it is easy to check thatψ(0, w, θ ) = 0and ψy(0, w, θ ) = 0.


The solution of the output regulation problem depends on the existence of certaininvariant manifold and feedforward input. For this problem, we have the followingresult.

Proposition 10.4. Suppose that an invariant manifold π (w) ∈ Rn−1 satisfies

∂π (w)

∂wS(σ )w = Dπ (w) + ψ(q(w), w, θ ). (10.57)

Then there exists an immersion for the feedforward control input

∂τ (w, θ , σ )

∂wS(σ )w = �(σ )τ (w, θ , σ )

α(w, θ , σ ) = �τ (w, θ , σ ),

where

α(w, θ , σ ) = b−1ρ

(∂q(w)

∂wS(σ )w − π1(w) − ψy(q(w), w, θ )

)

.

Furthermore, this immersion can be re-parameterised as

η = (F + GlT )η

α = lTη,(10.58)

where (F , G) is a controllable pair with compatible dimensions, η = Mτ andl = �M−1 with M satisfying

M (σ )�(σ ) − FM (σ ) = G�. (10.59)

Proof. With ξ1 being viewed as the input, α is the feedforward term used for outputregulation to tackle the disturbances, and from the second equation of (10.55), wehave

α(w, θ , σ ) = b−1ρ

(∂q(w)

∂wS(σ )w − π1(w) − ψy(q(w), w, θ )

)

.

From the structure of the exosystem, the disturbances are sinusoidal functions.Polynomials of sinusoidal functions are still sinusoidal functions, but with some high-frequency terms. Since all the nonlinear functions involved in the system (10.49) arepolynomials of their variables, the immersion in (10.58) always exists. For a control-lable pair (F , G), M is an invertible solution of (10.59) if (�,�) is observable, whichis guaranteed by the immersion. �

We now introduce the last transformation based on the invariant manifold with

z = z − π (10.60)


Finally we have the model for the control design

˙z = Dz + ψe = z1 + ψy + bρ(ξ − lTη)

ξ1 = −λ1ξ1 + ξ2

. . .

ξρ−1 = −λρ−1ξρ−1 + u,

(10.61)

where

ψ = ψ(y, w, θ ) − ψ(q(w), w, θ )

and

ψy = ψy(y, w, θ ) − ψy(q(w), w, θ ).

Since the state in the internal model η is unknown, we design the adaptive internalmodel

˙η = F η + Gξ1. (10.62)

If we define the auxiliary error

η = η − η + b−1ρ Ge, (10.63)

it can be shown that

˙η = F η − FGb−1ρ e + b−1

ρ Gz1 + b−1ρ Gψy. (10.64)

If the system (10.49) is of relative degree 1, then ξ1 in (10.61) is the controlinput. For the systems with higher relative degrees, adaptive backstepping will beused to find the final control input u from the desirable value of ξ1. Supposing that ξ1

is desirable value for ξ1, we introduce a Nussbaum gain N (κ) such that

ξ1 = N (κ)ξ1

κ = eξ1,(10.65)

where the Nussbaum gain N is a function (e.g. N (κ) = κ2 cos κ) which satisfies thetwo-sided Nussbaum properties

limκ→±∞ sup

1

κ

∫ κ

0N (s)ds = +∞, (10.66)

limκ→±∞ inf

1

κ

∫ κ

0N (s)ds = −∞, (10.67)

where κ → ±∞ denotes κ → +∞ and κ → −∞ respectively. From (10.61)and the definition of the Nussbaum gain, we have

e = z1 + (bρN − 1)ξ1 + ξ1 + bρ ξ1 + bρ ξ1 − lTb η + ψy,


where lb = bρ l, bρ is an estimate of bρ and bρ = bρ − bρ , and ξ1 = ξ1 − ξ1. Sine thenonlinear functions involved in ψ and ψy are polynomials with ψ(0, w, θ , σ ) = 0 andψy(0, w, θ , σ ) = 0, w is bounded, and the unknown parameters are constants, it canbe shown that

|ψ | < rz(|e| + |e|p),

|ψy| < ry(|e| + |e|p),

where p is a known positive integer, depending on the polynomials in ψ and ψy, andrz and ry are unknown positive real constants. We now design the virtual control ξ1

as, with c0 > 0,

ξ1 = −c0e − k0(e + e2p−1) + lTb η.

Using (10.63), we have the resultant error dynamics

e = −c0e − k0(e + e2p−1) + z1 + (bρN − 1)ξ1 + bρ ξ1 + bρ ξ1

− lTb η − lT

b η + lT Ge + ψy. (10.68)

The adaptive laws are given by

˙k0 = e2 + e2p,

τb,0 = ξ1e, (10.69)

τl,0 = −ηe,

where τb,0 and τl,0 denote the first tuning functions in adaptive backstepping designfor the final adaptive laws for bρ and lb. If the relative degree ρ = 1, we set u = ξ1.For ρ > 1, adaptive backstepping can be used to obtain the following results:

ξ2 = −bρe − c1ξ1 − k1

(∂ξ1

∂e

)2

ξ1

+ ∂ξ1

∂e(bρξ1 − lT

b η) + ∂ξ1

∂η˙η

+ ∂ξ1

∂ k0

˙k0 + ∂ξ1

∂ lb

τl,1, (10.70)


ξi = −ξi−2 − ci−1ξi−1 − ki−1

(∂ξi−1

∂e

)2

ξi−1

+ ∂ξi−1

∂e(bρξ1 − lT

b η) + ∂ξi−1

∂η˙η

+ ∂ξi−1

∂ k0

˙k0 + ∂ξi−1

∂ bρτb,i−1 + ∂ξi−1

∂ lb

τl,1

−i∑

j=4

∂ξi−1

∂e

∂ξj−2

∂ bρξ1ξj−2

+i∑

j=3

∂ξi−1

∂e

∂ξj−2

∂ lb

ηξj−2 for i = 2, . . . , ρ, (10.71)

where ξi = ξi − ξi for i = 1, . . . , ρ − 2, ci and ki, i = 2, . . . , ρ − 1, are positive realdesign parameters, and τb,i and τl,i, for i = 1, . . . , ρ − 2, are tuning functions. Theadaptive law and tuning functions are given by

τb,i = τb,i−1 − ∂ξi

∂eξ1ξi, for i = 1, . . . , ρ − 1,

τl,i = τl,i−1 + ∂ξi

∂eηξi, for i = 1, . . . , ρ − 1,

˙bρ = τb,ρ−1, (10.72)

˙lb = τl,ρ−1. (10.73)

Finally we design the control input as

u = ξρ . (10.74)

For the proposed control design, we have the following result for stability.

Theorem 10.5. For a system (10.49) satisfying the invariant manifold condition(10.57), the adaptive output regulation problem is globally solved by the feedbackcontrol system consisting the ξ -filters (10.50), the adaptive internal model (10.62),Nussbaum gain parameter (10.65), the parameter adaptive laws (10.69), (10.72),(10.73) and the feedback control (10.74), which ensures the convergence to zero ofthe regulated measurement, and the boundedness of all the variables in the closed-loopsystem.

Proof. Define a Lyapunov function candidate

V = β1ηT Pηη + β2zT Pzz

+ 1

2

(

e2 +ρ−1∑

i=1

ξ 2i + (k0 − k0)2 + b2

ρ + lTb lb

)

,


where β1 and β2 are two positive reals and Pz and Pη are positive definite matricessatisfying

PzD + DT Pz = −I ,

PηF + FT Pη = −I .

With the design of ξi, for i = 1, . . . , ρ, the dynamics of ξi can be easily evaluated.From the dynamics of z in (10.61) and the dynamics of η in (10.64), virtual controlsand adaptive laws designed earlier, we have the derivative of V as

V = β1(−ηT η − 2ηT PηFb−1ρ Ge + 2ηT Pηb

−1ρ Gz1 + 2ηT Pηb

−1ρ Gψy)

+ β2(−zT z + 2zT Pzψ) + (k0 − k0)(e2 + e2p)

− c0e2 − k0(e2 + e2p) + (bρN − 1)eξ1

+ ez1 + eψy − elTb η + lT Ge2

+ρ−1∑

i=1

⎛

⎝−ci ξ2i − ki

(∂Aξ i

∂e

)2

ξ 2i − ξi

∂ξi

∂ez1

−ξi∂ξi

∂eψy + ξi

∂ξi

∂elTb η − ξi

∂ξi

∂elT Ge

)

.

The stability analysis can be proceeded by using the inequalities 2xy < rx2 + y2/ror xy < rx2 + y2/(4r) for x > 0, y > 0 and r being any positive real, to tackle thecross-terms between the variables z, η, e, ξi, for i = 1, . . . , ρ − 1. It can be shownthat there exist sufficiently big positive real β1, and then sufficiently big positive realβ2, and finally the sufficient big k0 such that the following result holds:

V ≤ (bρN (κ) − 1)κ − 1

3β1η

T η − 1

4β2zT z − c0e2 −

ρ−1∑

i=1

ci ξ2i . (10.75)

The boundedness of V can be established based on the Nussbaum gain properties(10.66) and (10.67) via an argument of contradiction. In fact, integrating (10.75) gives

V (t) +∫ t

0

(1

3β1η

T η + 1

4β2zT z + c0e2 +

ρ−1∑

i=1

ci ξ2i

)

dt

≤ bρ

∫ κ(t)

0N (s)ds − κ(t) + V (0). (10.76)

If κ(t), ∀t ∈ R+, is not bounded from above or below, then from (10.66) and (10.67)

it can be shown that the right-hand side of (10.76) will be negative at some instancesof time, which is a contradiction, since the left-hand side of (10.76) is non-negative.


Therefore, κ is bounded, which implies the boundedness of V . The boundedness ofV further implies η, z, e, ξi ∈ L2 ∩ L∞ for i = 1, . . . , ρ − 1, and the boundednessof k0, bρ and lb. Since the disturbance w is bounded, e, z, η ∈ L∞ implies theboundedness of y, z and η, which further implies the boundedness of ξ1 and thenthe boundedness of ξ1. The boundedness of ξ1 and ξ1, together with the boundednessof e, η, k0, bρ and lb, implies the boundedness of ξ2, and then the boundedness ofξ2 follows the boundedness of ξ2. Applying the above reasoning recursively, we canestablish the boundedness of ξi for i > 2 to i = ρ − 1. We then conclude that all thevariables are bounded.

The boundedness of all the variables implies the boundedness of ˙η, ˙z, e and˙ξi, which further implies, together with η, z, e, ξi ∈ L2 ∩ L∞ and Barbalat’slemma, limt→∞ η = 0, limt→∞ z = 0, limt→∞ e(t) = 0 and limt→∞ ξi = 0 fori = 1, . . . , ρ − 1. �

10.3 Output regulation with nonlinear exosystems

In the previous sections, the disturbances are sinusoidal functions which are generatedfrom linear exosystems with the restriction that the eigenvalues of the exosystemmatrix are distinct and with zero real parts. Sinusoidal disturbances are important asdisturbances in practical systems can often be approximated by a finite number ofsinusoidal functions. However, there are situations where disturbances are generatedfrom nonlinear exosystems, such as nonlinear vibration, etc. Such disturbances canstill be approximated by a finite number of sinusoidal functions, possibly with a bignumber of sinusoidal functions for a good accuracy. The more sinusoidal functionsinvolved in the approximation of a periodic signal, the higher order will be thecorresponding system matrix for the linear exosystem. If an internal model canbe designed directly based on the nonlinear exosystem, it is possible to achieveasymptotic rejection of the disturbances, which cannot be achieved by approximationusing sinusoidal functions, and the order of the internal model can also remain muchlower. Of course, it is expected to be a very difficult problem to directly designan internal model to deal with nonlinear exosystems, even though there exists aninvariant manifold for output regulation. In this section, we will show nonlinearinternal model design for a class of nonlinear exosystem to achieve output regulation.For the internal model design, we exploit a technique for nonlinear observer designbased on conditions similar to circle criteria. The dynamic model considered foroutput regulation is still the class of output feedback form.

We consider a SISO nonlinear system

x = Acx + φ(y)a + E(w) + bu

y = Cx (10.77)

e = y − q(w),


with

Ac =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0.......... . .

...

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

T

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0bρ...

bn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where x ∈ Rn is the state vector; u ∈ R is the control; y ∈ R is the output; e is the

measurement output; a ∈ Rq and b ∈ R

n are vectors of unknown parameters, with bbeing a Hurwitz vector with bρ �= 0, which implies the relative degree of the systemis ρ, and with known sign of bρ , E : R

m → Rn,φ : R → R

n×q with φ(0) = 0 and|φ(y1) − φ(y2)| ≤ �1(|y1|)δ1(|y1 − y2|) and δ1(·) ∈ K and �1(·) is non-decreasingand the function δ1(·) is a known smooth function; and w ∈ R

m are disturbances, andthey are generated from a nonlinear exosystem

w = s(w) (10.78)

of which the flows are bounded and converge to periodic solutions.

Remark 10.12. The assumption about the function φ is satisfied for many kinds offunctions, for example polynomial functions. �

Remark 10.13. The nonlinear exosystem (10.78) includes nonlinear systems thathave limit cycles. �

Remark 10.14. The system (10.78) is very similar to the system (10.49) consideredin the previous section. The main difference is that the exosystem is nonlinear. �

The system (10.78) has the same structure of Ac, bu and C as in (10.49), andtherefore the same filtered transformation as in the previous section can be used here.We can use the same filtered transformation, and the transformation for extractingthe zero dynamics of the system as in the previous section. Using the transformations(10.51) and (10.54), the system (10.78) is put in the coordinate (z, y) as

zi = −di+1

d1z1 + zi+1 +

(di+2

d1− di+1d2

d21

)

y + (φi+1(y) − di+1

d1φ1(y))a

+ Ei+1(w) − di+1

d1E1(w), i = 1, . . . , n − 2, (10.79)

zn−1 = −dn

d1z1 − dnd2

d21

y +(

φn(y) − dn

d1φ1(y)

)

a + En(w) − dn

d1E1(w),

y = z1 + d2

d1y + φ1(y)a + E1(w) + bρξ1

where di are defined in (10.53).


It is necessary to have the existence of certain invariant manifolds for a solutionto the output regulation problem. When the exosystem is nonlinear, it is even morechallenging to have the necessary conditions for invariant manifold to exist.

Proposition 10.6. Suppose that there exist� (w) ∈ Rn and ι(w) with�1(w) = q(w)

for each a, b such that

∂�

∂ws(w) = Ac� + φ(q(w))a + E(w) + bι(w). (10.80)

Then there exists π (w) ∈ Rn−1 along the trajectories of exosystem satisfying

∂πi(w)

∂ws(w) = −di+1

d1π1(w) + πi+1(w) + q(w)

(di+2

d1− di+1d2

d21

)

+ Ei+1(w) − E1(w)di+1

d1+ (φi+1(q(w))

−di+1

d1φ1(q(w)))a, i = 1, . . . , n − 2,

∂πn−1(w)

∂ws(w) = −dn

d1π1(w) − dnd2

d21

q(w) + (φn(q(w))

−dn

d1φ1(q(w)))a + En(w) − dn

d1E1(w).

Proof. Since the last equation of input filter (10.50) used for the filtered transformationis an asymptotically stable linear system, there is a static response for every externalinput u(w), i.e., there exists a function χρ−1(w) such that

∂χρ−1(w)

∂ws(w) = −λρ−1χρ−1(w) + ι(w).

Recursively, if there exists χi(w) such that

∂χi(w)

∂ws(w) = −λiχi(w) + χi+1(w),

then there exists χi−1(w) such that

∂χi−1(w)

∂ws(w) = −λi−1χi−1(w) + χi(w).

Define[π (w)q(w)

]

= Da(� (w) − [d1, . . . , dρ−1]χ ),

where χ = [χ1, . . . ,χρ−1]T and

Da =

⎡

⎢⎢⎢⎣

−d2/d1 1 . . . 0...

.... . .

...

−dn/d1 0 . . . 11 0 . . . 0

⎤

⎥⎥⎥⎦.


It can be seen that π (w) satisfies the dynamics of z along the trajectories of (10.78)as shown in (10.80), and hence the proposition is proved. �

Based on the above lemma, we have

∂q(w)

∂ws(w) = π1(w) + d2

d1q(w) + φ1(q(w))a + E1(w) + bρα(w),

where α(w) = χ1(w). With ξ1 being viewed as the input, α(w) is the feedforward termused for output regulation to tackle the disturbances, and it is given by

α = b−1ρ

(∂q(w)

∂ws(w) − π1(w) − d2

d1q(w) − φ1(q(w))a − E1(w)

)

.

We now introduce the last transformation based on the invariant manifold with

z = z − π (w(t)).

Finally we have the model for the control design

˙zi = −di+1

d1z1 + zi+1 +

(di+2

d1− di+1d2

d21

)

e

+ (φi+1(y) − φi+1(q(w)))a

− di+1

d1(φ1(y) − φ1(q(w)))a, i = 1, . . . , n − 2

˙zn−1 = −dn

d1z1 − dnd2

d21

e + (φn(y) − φn(q(w)))a

− dn

d1(φ1(y) − φ1(q(w)))a

e = z1 + d2

d1e + (φ1(y) − φ1(q(w)))a + bρ(ξ1 − α(w)),

i.e., the system can be represented as

˙z = Dz +�e +�(y, w, d)a

e = z1 + d2

d1e + (φ1(y) − φ1(q(w)))a + bρ(ξ1 − α(w)),

(10.81)

where D is a companion matrix of d shown in (10.56), and

� =(

d3

d1− d2

2

d21

, . . . ,dn

d1− dn−1d2

d21

, −dnd2

d21

)T

,

�(y, w, d) =

⎛

⎜⎜⎜⎜⎝

φ2(y) − φ2(q(w)) − d2

d1(φ1(y) − φ1(q(w)))

...

φn(y) − φn(q(w)) − dn

d1(φ1(y) − φ1(q(w)))

⎞

⎟⎟⎟⎟⎠.


Lemma 10.7. There exists a known function ζ (·) which is non-decreasing and anunknown constant�, which is dependent on the initial state w0 of exosystem, such that

|�(y, w, d)| ≤ �|e|ζ (|e|),|φ1(y) − φ1(q(w))| ≤ �|e|ζ (|e|).

Proof. From the assumption of φ we can see that

|φ(y) − φ(q(w))| ≤ �1(|q(w)|)δ1(|e|).Since the trajectories of exosystem are bounded and δ1(·) is smooth there exist smoothnondecreasing known function ζ (·) and a nondecreasing known function �2(|w0|),such that

δ1(|e|) ≤ |e|ζ (|e|),�1(|q(w)|) ≤ �2(|w0|).

From previous discussion the result of the lemma is obtained. �

Let

Vz = zT Pd z,

where

PdD + DT Pd = −I .

Then using 2ab ≤ ca2 + c−1b2 and ζ 2(|e|) ≤ ζ 2(1 + e2), there exist unknown positivereal constants �1 and �2 such that

Vz = −zT z + 2zT Pd(�e +�(y, w, d)a)

≤ −3

4zT z +�1e2 +�2e2ζ 2(1 + e2), (10.82)

noting that

2zT Pd�e ≤ 1

8zT z + 8eT�T P2

d�e

≤ 1

8zT z +�1e2,

and

2zT Pd�(y, w, d)a ≤ 1

8zT z + 8aT�T P2

d�a

≤ 1

8zT z +�1

2|�|2

≤ 1

8zT z +�1

2�2|e|2ζ 2(|e|)

≤ 1

8zT z +�2e2ζ 2(1 + e2),

where �12 is an unknown positive real constant.


Now let us consider the internal model design. We need an internal model toproduce a feedforward input that converges to the ideal feedforward control termα(w), which can be viewed as the output of the exosystem as

w = s(w)

α = α(w).

Suppose that there exists an immersion of the exosystem

η = Fη + Gγ (Jη)

α = Hη,(10.83)

where η ∈ Rr , H = [1, 0, . . . , 0], (H , F) is observable

(v1 − v2)T (γ (v1) − γ (v2)) ≥ 0,

and G and J are some appropriate dimensional matrices. We then design an internalmodel as

˙η = (F − KH )(η − b−1ρ Ke) + Gγ (J (η − b−1

ρ Ke)) + Kξ1, (10.84)

where K ∈ Rr is chosen such that F0 = F − KH is Hurwitz and there exist a positive

definite matrix Pf and a semi-positive definite matrix Q satisfying⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

Pf F0 + FT0 Pf = −Q

Pf G + J T = 0

ηT Qη ≥ γ0|η1|2, γ0 > 0, η ∈ Rr

span(PF K) ⊆ span(Q).

(10.85)

Remark 10.15. It reminds us a challenging problem to design internal models foroutput regulation with nonlinear exosystems. It is not clear at the moment whatgeneral conditions are needed to guarantee the existence of an internal model foroutput regulation of nonlinear systems with nonlinear internal models. Here we usethe condition of the existence of an immersion (10.83) for the internal model design.Also note that even when the exosystem is linear, an internal model can be nonlinearfor a nonlinear dynamic system. �

Remark 10.16. Note the condition specified in (10.85) is weaker than the conditionthat there exist Pf > 0 and Q > 0 satisfying

{PF F0 + FT

0 PF = −Q

PF G + J T = 0,(10.86)

which can be checked by LMI. This will be seen in the example later in this section. Inparticular, if G and J T are two column vectors, (F0, G) controllable, (J , F0) observableand Re[−J (jωI − F0)−1G] > 0, ∀ω ∈ R, then there exists a solution of (10.86) fromKalman–Yacubovich lemma. �


If we define the auxiliary error

η = η − η + b−1ρ Ke,

it can be shown that

˙η = F0η + G(γ (Jη) − γ (J (η − b−1ρ Ke)))

+ b−1ρ K

(

z1 + d2

d1e + (φ1(y) − φ1(q(w)))a

)

.

Let

Vη = ηPF η.

Then following the spirit of (10.82), there exist unknown positive real constants �1

and �2 such that

Vη = −ηT Qη + 2ηT PF b−1ρ K

(

z1 + d2

d1e

)

+ 2ηT PF b−1ρ K(φ1(y) − φ1(q(w)))a

+ 2ηT PF G(γ (Jη) − γ (J (η − b−1ρ Ke))

≤ −3

4γ0|η1|2 + 12

γ0b−2ρ z2

1 +�1e2 +�2e2ζ 2(1 + e2). (10.87)

Let us proceed with the control design. From (10.81) and

α = η1 = η1 + η1 − b−1ρ K1e,

we have

e = z1 + d2

d1e + (φ1(y) − φ1(q(w)))a + ξ1 + bρ(ξ1 − η1 − η1 + b−1

ρ K1e),

where ξ1 = ξ1 − ξ1 and

ξ1 = b−1ρ ξ1. (10.88)

For the virtual control ξ1, we design ξ1 as, with c0 > 0,

ξ1 = −c0e + bρη1 − K1e − le(1 + ζ 2(1 + e2)), (10.89)

where l is an adaptive coefficient. Then we have the resultant error dynamics

e = z1 − c0e + d2

d1e − le(1 + ζ 2(1 + e2)) + (φ1(y) − φ1(q(w)))a + bρ(ξ1 − η1).

Then for

Ve = 1

2e2,


there exist unknown positive real constants 1 and 2, and a sufficiently largeunknown positive constant β such that

Ve = −c0e2 + ez1 + d2

d1e2 + ebρ(ξ1 − η1)

+ e(φ1(y) − φ1(q(w)))a − le2(1 + ζ 2(1 + e2))

≤ −c0e2 + 1

8β z2

1 + 1

4γ0η

21 + 1e2 + 2e2ζ (1 + e2)

−le2(1 + ζ 2(1 + e2)) + bρeξ1. (10.90)

Let

V0 = βVz + Vη + Ve + 1

2γ −1(l − l)2,

where β ≥ 96γ0

b−2ρ is chosen and l = 1 + 2 +�1 +�2 + β(�1 +�2) is an

unknown constant. Let

˙l = γ e2(1 + ζ 2(1 + e2)).

Then, it can be obtained that

V0 ≤ −1

2β zT z − 1

2γ0|η1|2 − c0e2 + bρeξ1.

If the system (10.78) has relative degree 1, the virtual control ξ1 shown in (10.88)together with ξ1 in (10.88) gives the input, i.e., u = ξ1. For the system with higherrelative degrees, the control design can be proceeded with backstepping using (10.50)in the same way as the adaptive backstepping with filtered transformation shown in theprevious section for adaptive output regulation with linear exosystems. We summarisethe stability result in the following theorem.

Theorem 10.8. For the system (10.78) with the nonlinear exosystem (10.78), if thereexists an invariant manifold (10.80) and an immersion (10.83), then there existsK ∈ R

r such that F0 = F − KH is Hurwitz and there exist a positive definite matrix PF

and a semi-positive definite matrix Q satisfying (10.85), and there exists a controllerto solve the output regulation in the sense the regulated measurement converges tozero asymptotically while other variables remain bounded.

We use an example to illustrate the proposed control design, concentrating onthe design of nonlinear internal model.

Example 10.2. Consider a first-order system

y = 2y + θ sin y − y3 − θ sin w1 + w2 + u

e = y − w1,


where θ is an unknown parameter, and the disturbance w is generated by

w1 = w1 + w2 − w31

w2 = −w1 − w32.

It is easy to see that V (w) = 12 w2

1 + 12 w2

2 satisfies

dV

dt= w2

1 − w41 − w4

2 ≤ 0, when |w1| ≥ 1,

and that

q(w) = w1,

π = w1,

α(w) = −w1.

From the exosystem and the desired feedforward input α, it can be seen that thecondition specified in (10.85) is satisfied with η = −w and

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

F =(

1 1

−1 0

)

, G =(

−1 0

0 −1

)

γ1(s) = γ2(s) = s3, J =(

1 0

0 1

)

.

Let K = [2, 0]T . Then with

F0 =(−1 1

−1 0

)

, PF = I , Q = diag(2, 0),

the internal model is designed as the following:

˙η1 = −(η1 − 2e) + η2 − (η1 − 2e)3 + 2u

˙η2 = −(η1 − 2e) − η32.

The control input and the adaptive law are given by

u = −ce + η1 − le(1 + (e2 + 1)2),

˙l = γ e2(1 + (e2 + 1)2).

For simulation study, we set c = 1, θ = 1, γ = 1, and the initial states arey(0) = 1, w1(0) = 2 and w2(0) = 2. The initial state of dynamic controller is zero.The system output and input are shown in Figure 10.3, while the feedforward termand its estimation are shown in Figure 10.4 and the portrait of the exosystem isshown in Figure 10.5. As shown in the figures, the internal model successfully repro-duces the feedforward control needed after a transient period, and the system outputmeasurement is regulated to zero, as required. �


0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Time t (s)

Trac

king

err

or e

0 5 10 15−2

−1

0

1

2

3

4

5

Time t (s)

Con

trol u

Figure 10.3 The system’s output e and input u

0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1

Time t (s)

h 1 a

nd h

1

Figure 10.4 The systems’s feedforward control η1 and its estimation η1


−1 −0.5 0 0.5 1 1.5 2−1

−0.5

0

0.5

1

1.5

2

w1

w2

Figure 10.5 The portrait of exosystems

10.4 Asymptotic rejection of general periodic disturbances

We have presented design methods for asymptotic rejection and output regulationof disturbances generated from linear exosystems, i.e., sinusoidal functions and dis-turbances from a specific class of nonlinear exosystems, which generally producenon-harmonic but still periodic disturbances. For disturbances from linear exosys-tem, internal models can normally be designed under some mild assumptions, whilefor nonlinear exosystems, the conditions are more restrictive for designing an inter-nal model for asymptotic rejection and output regulation. The difficulty lies in theguaranteed existence of the invariant manifold, and then the nonlinear internal modeldesign, which is often more involved even than nonlinear observers for nonlinearsystems.

We consider some more general periodic disturbances than harmonic distur-bances in this section. These general periodic disturbances can be modelled as outputsof nonlinear systems, and in particular, as the outputs of linear dynamic systems withnonlinear output functions. For the systems with Lipschitz nonlinearities, nonlinearobservers can be designed as shown in Section 8.4 and other results in literature. Ofcourse, the problem addressed in this chapter cannot be directly solved by nonlinearobserver design, not even the state estimation of the disturbances system, as the dis-turbance is not measured. However, there is an intrinsic relationship between observerdesign and internal model design, as evidenced in the previous results of disturbancerejection and output regulation in the earlier sections in this chapter.


With the formulation of general periodic disturbances as the nonlinear outputsof a linear dynamic system, the information of the phase and amplitude of a generalperiodic disturbance is then embedded in the state variables, and the information ofthe wave profile in the nonlinear output function. The nonlinear output functions areassumed to be Lipschitz. By formulating the general periodic disturbances in this way,we are able to explore nonlinear observer design of nonlinear systems with outputLipschitz nonlinearities shown in Section 8.4. We will show that general periodicdisturbances can be modelled as nonlinear outputs of a second-order linear systemwith a pair of pure imaginary poles which depend on the frequencies. For this specificsystem with Lipschitz nonlinear output, a refined condition on the Lipschitz constantwill be given by applying the proposed method in this section, and observer gain willbe explicitly expressed in terms of the Lipschitz constant and the period or frequencyof the disturbance.

An internal model design is then introduced based on the proposed Lipschitzoutput observer for a class of nonlinear systems. Conditions are identified for thenonlinear system, and control design is carried out using the proposed internal model.Two examples are included to demonstrate the proposed internal model and controldesign procedures. These examples also demonstrate that some other problems canbe converted to the problem addressed in this section.

We consider a nonlinear system

y = a(z) + ψ0(y) + ψ(y, v) + b(u − μ(v))

z = f (z, v, y),(10.91)

where y ∈ R is the output; a and ψ0 : Rn → R are continuous functions; v ∈ R

m

denotes general periodic disturbances; μ : Rm → R is a continuous function; ψ :

R × Rm → R is a continuous function and satisfies the condition that |ψ(y, v)|2 ≤

yψ(y) with ψ being a continuous function; b is a known constant; u ∈ R is the input;z ∈ R

n is the internal state variable; and f : Rn × R

m × R → Rn is a continuous

function.

Remark 10.17. For the convenience of presentation, we only consider the systemwith relative degree 1 as in (10.91). The systems with higher relative degreescan be dealt with similarly by invoking backstepping. The second equation in(10.91) describes the internal dynamics of the system states, and if we set v = 0and y = 0, z = f (z, 0, 0) denotes the zero dynamics of this system. �

Remark 10.18. The system in (10.91) specifies a kind of standard form forasymptotic rejection of general periodic disturbances. For example, consider

x = Ax + φ(y, v) + bu

y = cT x,


with b, c ∈ Rn and

A =

⎡

⎢⎢⎢⎢⎣

−a1 1 . . . 0

−a2 0. . . 0

......

. . ....

−an . . . . . . 0

⎤

⎥⎥⎥⎥⎦

, b =

⎡

⎢⎢⎢⎣

b1

b2...

bn

⎤

⎥⎥⎥⎦

, c =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

,

where x ∈ Rn is the state vector; y and u ∈ R are the output and input respectively

of the system; v ∈ Rm denotes general periodic disturbances; and φ : R × R

m → Rn

is a nonlinear smooth vector field in Rn with φ(0, 0) = 0. This system is similar to

the system (10.49) with q(w) = 0. For this class of nonlinear systems, the asymptoticdisturbance rejection depends on the existence of state transform to put the systemsin the form shown in (10.91), and it has been shown in Section 10.2 that such atransformation exists under some mild assumptions. �

The wave profile information of a general periodic disturbance is used to constructa nonlinear function as the output function for a linear exosystem, for the generationof the desired feedforward input. By doing that, an observer with nonlinear outputfunction can be designed, viewing the feedforward input as the output, and an internalmodel can then be designed based on the nonlinear observer. The problem to solve inthis section is to design a control scheme, using a nonlinear observer-based internalmodel, to asymptotically reject a class of general periodic disturbances for the systemin (10.91).

We start with modelling general periodic disturbances as the outputs of a linearoscillator with nonlinear output functions, and then propose nonlinear observer designfor such a system, for the preparation of internal model design.

Many periodic functions with period T can be modelled as outputs of a second-order system

w = Aw, with A =[

0 ω

−ω 0

]

μ(v) = h(w),

where ω = 2πT . Here, the desired feedforward input μ(v) is modelled as the nonlinear

output h(w) of the second-order system. With

eAt =[

cosωt sinωt−sinωt cosωt

]

,

the linear part of the output h(w), Hw, is always in the form of a sin (ωt + φ) wherea and φ denote the amplitude and phase respectively. Hence, we can set H = [1 0]without loss of generality, as the amplitude and the phase can be decided by the initialvalue with

w(0) = [a sin (φ) a cos (φ)]T .


Based on the above discussion, the dynamic model for general periodic disturbanceis described by

w1 = ωw2

w2 = −ωw1 (10.92)

μ = w1 + h1(w1, w2),

where h1(w1, w2) is a Lipschitz nonlinear function with Lipschitz constant γ .

Remark 10.19. General periodic disturbances can be modelled as af (t + φ) with aandφ for the amplitude and phase of a disturbance, and the wave profile is specified bya periodic function f . In the model shown in (10.93), the amplitude and phase of thedisturbance are determined by the system state variables w1 and w2, and the profile isdetermined by the nonlinear output function. In some results shown in literature, thephase and amplitude are obtained by delay and half-period integral operations. Here,we use nonlinear observers for the estimation of phases and amplitudes of generalperiodic disturbances. �

For the model shown in (10.93), the dynamics are linear, but the output functionis nonlinear. Many results in literature on observer design for nonlinear Lipschitzsystems are for the system with nonlinearities in the system dynamics while the outputfunctions are linear. Here we need the results for observer design with nonlinear outputfunctions. Similar techniques to the observer design of nonlinearities in dynamics canbe applied to the case when the output functions are nonlinear.

We have shown the observer design for a linear dynamic system with a nonlinearLipschitz output function in Section 8.4 with the observer format in (8.38) and gainin Theorem 8.11. Now we can apply this result to observer design for the model ofgeneral periodic disturbances. For the model shown in (10.93), the observer shown

in (8.38) can be applied with A =[

0 ω

−ω 0

]

and H = [1 0]. We have the following

lemma for the stability of this observer.

Lemma 10.9. An observer in the form of (8.38) can be designed to provide an expo-nentially convergent state estimate for the general periodic disturbance model (10.93)if the Lipschitz constant γ for h1 satisfies γ < 1√

2.

Proof. Our proof is constructive. Let

P =

⎡

⎢⎢⎣

p − 1

4γ 2ω

− 1

4γ 2ωp

⎤

⎥⎥⎦ ,

where p > 14γ 2ω

. It is easy to see that P is positive definite. A direct evaluation gives

PA + AT P − H T H

γ 2= − 1

2γ 2I .


Therefore, the second condition in (8.40) is satisfied. Following the first conditionspecified in (8.39), we set

L =

⎡

⎢⎢⎣

4ω(4pγ 2ω)

(4pγ 2ω)2 − 14ω

(4pγ 2ω)2 − 1

⎤

⎥⎥⎦ . (10.93)

The rest part of the proof can be completed by invoking Theorem 8.11. �

Hence from the above lemma, we design the observer for the general disturbancemodel as

˙x = Ax + L(y − h(x)), (10.94)

where A =[

0 ω

−ω 0

]

and H = [1 0] with the observer gain L as shown in (10.93).

Before introducing the control design, we need to examine the stability issues ofthe z−subsystem, and hence introduce a number of functions that are needed laterfor the control design and stability analysis of the entire system.

Lemma 10.10. Assuming that the subsystem

z = f (z, v, y)

is ISS with state z and input y, characterised by an ISS pair (α, σ ), and furthermore,α(s) = O(a2(s)) as s → 0, there exist a differentiable positive definite function V (z)and a K∞ function β satisfying β(‖z‖) ≥ a2(z) such that

˙V (z) ≤ −β(‖z‖) + σ (y). (10.95)

where σ is a continuous function.

Proof. From Corollary 5.10, there exists a Lyapunov function Vz(z) that satisfies that

α1(‖z‖) ≤ Vz(z) ≤ α2(‖z‖)

Vz(z) ≤ −α(‖z‖) + σ (|y|), (10.96)

where α, α1 and α2 are class K∞ functions, and σ is a class K function. Let βbe a K∞ function such that β(‖z‖) ≥ a2(z) and β(s) = O(a2(s)) as s → 0. Sinceβ(s) = O(a2(s)) = O(α(s)) as s → 0, there exists a smooth nondecreasing (SN )function q such that, ∀r ∈ R

+

1

2q(r)α(r) ≥ β(r).

Let us define two functions

q(r) := q(α−11 (r)),

ρ(r) :=∫ r

0q(t)dt.


Define

V (z) := ρ(V (z)),

and it can be obtained that˙V (z) ≤ −q(V (z))α(z) + q(V (z))σ (|y|)

≤ −1

2q(V (z))α(z) + q(θ (|y|))σ (|y|)

≤ −1

2q(α1(‖z‖))α(z) + q(θ (|y|))σ (|y|)

= −1

2q(‖z‖)α(z) + q(θ (|y|))σ (|y|),

where θ is defined as

θ (r) := α2(α−1(2σ (r))

for r ∈ R+. Let us define a smooth function σ such that

σ (r) ≥ q(θ (|r|))σ (|r|)for r ∈ R and σ (0) = 0, and then we have established (10.95). �

Based on observer design presented in (10.94), we design the following internalmodel:

η = Aη + b−1Lψ0(y) + Lu − b−1ALy − Lh(η − b−1Ly), (10.97)

where L is designed as in (10.93).The control input is then designed as

u = −b−1

(

ψ0(y) + k0y + k1y + k2σ (y)

y+ k3ψ(y)

)

+ h(η − b−1Ly), (10.98)

where k0 is a positive real constant, and

k1 = κ−1b2(γ + ‖H‖)2 + 3

4,

k2 = 4κ−1||b−1PL‖2 + 2,

k3 = 4κ−1||b−1PL‖2 + 1

2.

For the stability of the closed-loop system, we have the following theorem.

Theorem 10.11. For a system in the form shown in (10.91), if

● feedforward term μ(v) can be modelled as the output of a system in the formatshown in (10.93) and the Lipschitz constant of the output nonlinear function γsatisfies γ < 1√

2


● the subsystem z = f (z, v, y) is ISS with state z and input y, characterized by ISSpair (α, σ ), and furthermore, α(s) = O(a2(s)) as s → 0

the output feedback control design with the internal model (10.97) and the controlinput (10.98) ensures the boundedness of all the variables of the closed-loop systemand the asymptotic convergence to zero of the state variables z and y and the estimationerror (w − η + b−1Ly).

Proof. Let

ξ = w − η + b−1Ly.

It can be obtained from (10.97) that

ξ = (A − LH )ξ + b−1L(h1(w) − h1(w − ξ )) + b−1La(z) + b−1Lψ(y, v).

Let Vw = ξT Pξ . It can be obtained that

Vw(ξ ) ≤ −κ‖ξ‖2 + 2|ξT b−1PLa(z)| + 2|ξT b−1PLψ(y, v)|≤ −1

2κ‖ξ‖2 + 2κ−1||b−1PL‖2(a2(z) + ψ(y, v)|2)

≤ −1

2κ‖ξ‖2 + (k2 − 2)β(‖z‖) +

(

k3 − 1

2

)

yψ(y) (10.99)

where κ = 12γ 2 − 1.

Based on the control input (10.98), we have

y = −k0y − k1y − k2σ (y)

y− k3ψ(y) + a(z) + ψ(y, v) + b(h(w − ξ ) − h(w)).

Let Vy = 12 y2. It follows from the previous equation that

Vy = − (k0 + k1)y2 − k2σ (y) − k3yψ(y) + ya(z) + yψ(y, v) + yb(h(w − ξ ) − h(w))

≤ −k0y2 − k2σ (y) −(

k3 − 1

2

)

yψ(y) + β(‖z‖) + 1

4κ‖ξ‖2. (10.100)

Let us define a Lyapunov function candidate for the entire closed-loop system as

V = Vy + Vw + k2Vz.

Following the results shown in (10.96), (10.99) and (10.100), we have

V ≤ −k0y2 − 1

4κ‖ξ‖2 − β(‖z‖).

Therefore, we can conclude that closed-loop system is asymptotically stable withrespect to the state variables y, z and the estimation error ξ .

Several types of disturbance rejection and output regulation problems can beconverted to the form (10.91). In this section, we show two examples. The firstexample deals with rejection of general periodic disturbances, and the second exampledemonstrates how the proposed method can be used for output regulation.


Example 10.3. Consider

x1 = x2 + φ1(x1) + b1u

x2 = φ2(x1) + ν(w) + b2u

w = Aw

y = x1,

(10.101)

where y ∈ R is the measurement output; φi : R → R, for i = 1, 2, are continuousnonlinear functions; ν : R

2 → R is a nonlinear function which produces a periodicdisturbance from the exosystem state w; and b1 and b2 are known constants with thesame sign, which ensures that stability of the zero dynamics. The control objectiveis to design an output feedback control input to ensure the overall stability of theentire system, and the asymptotic convergence to zero of the measurement output.The system shown in (10.101) is not in the form of (10.91) and the disturbance is notmatched. We will show that the problem can be transformed to the problem consideredin the previous section.

Let

z = x2 − b2

b1x1.

In the coordinates (y, z), we have

y = z + b2

b1y + φ1(y) + b1u

˙z = −b2

b1z + φ2(y) − b2

b1φ1(y) −

(b2

b1

)2

y + ν(w).

Consider

πz = −b2

b1πz + ν(w).

It can be shown that there exists a steady-state solution, and furthermore, we canexpress the solution as a nonlinear function of w, denoted by πz(w). Let us introduceanother state transformation with z = z − πz(w). We then have

y = z + b2

b1y + φ1(y) + b1(u + b−1

1 πz(w))

z = −b2

b1z + φ2(y) − b2

b1φ1(y) −

(b2

b1

)2

y. (10.102)


Comparing (10.102) with (10.91), we have

a(z) = z,

ψ(y) = b2

b1y + φ1(y),

b = b1,

h(w) = −b−11 πz(w),

f (z, v, y) = φ2(y) − b2

b1φ1(y) −

(b2

b1

)2

y.

From a(z) = z, we can set β(‖z‖) = ‖z‖2 = z2.It can be shown that the second condition of Theorem 10.11 is satisfied by

(10.102). Indeed, let V (z) = 12 z2, and we have

Vz = −b2

b1z2 + z

(

φ2(y) − b2

b1φ1(y) −

(b2

b1

)2

y

)

≤ −1

2

b2

b1z2 + 1

2

b1

b2

(

φ2(y) − b2

b1φ1(y) −

(b2

b1

)2

y

)2

.

Let

Vz = 2b1

b2Vz

and finally we have

˙Vz ≤ −β(|z|) +(

b1

b2

)2(

φ2(y) − b2

b1φ1(y) −

(b2

b1

)2

y

)2

. (10.103)

It can be seen that there exists a class K function σ (|y|) to dominate the second termon the right-hand side of (10.103), and the z-subsystem is ISS. For the control design,we can take

σ (y) =(

b1

b2

)2(

φ2(y) − b2

b1φ1(y) −

(b2

b1

)2

y

)2

.

The rest part of the control design follows the steps shown earlier.For the simulation study, we set the periodic disturbance as a square wave. For

convenience, we abuse the notations of ν(w(t)) and h(w(t)) as ν(t) and h(t). For νwith t in one period, we have

ν =

⎧⎪⎨

⎪⎩

d, 0 ≤ t <T

2,

−d,T

2≤ t < T ,

(10.104)


where d is an unknown positive constant, denoting the amplitude. It can be obtainedthat

h = dh(t),

where

h(t) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

− 1

b2(1 − e− b2

b1t) + 1

b2e− b2

b1t tanh

(T

4

b2

b1

)

, 0 ≤ t <T

2,

1

b2(1 + e− b2

b1t − 2e

b2b1

( T2 −t)) + 1

b2e− b2

b1t tanh

(T

4

b2

b1

)

,T

2≤ t < T.

(10.105)

Eventually we have the matched periodic disturbance h(w) given by

h(w) =√

w21 + w2

2 h

(

arctan(

w2

w1

))

.

Note that√

w21 + w2

2 decides the amplitude, which can be determined by the initialstate of w.

In the simulation study, we set T = 1, d = 10, φ1 = y3, φ2 = y2 and b1 = b2 = 1.The simulation results are shown in Figures 10.6–10.9. It can be seen from Figure 10.6that the measurement output converges to zero and the control input converges to aperiodic function. In fact, the control input converges to h(w) as shown in Figure 10.7.

0 5 10 15−0.05

0

0.05

0.1

0.15

0.2

Time (s)

y

0 5 10 15−20

−15

−10

−5

0

5

Time (s)

u

Figure 10.6 The system input and output


0 5 10 15−20

−15

−10

−5

0

5

Time (s)

u an

d h

uh

Figure 10.7 Control input and the equivalent input disturbance

0 5 10 15−4

−2

0

2

4

6

8

Time (s)

h an

d es

timat

e

hEstimate of h

Figure 10.8 The equivalent input disturbance and its estimate


0 5 10 15−20

−10

0

10

20

30

40

Time (s)

w a

nd e

stim

ates

w1w2h1h2

Figure 10.9 The exosystem states and the internal model states

As for the internal model and state estimation, it is clear from Figure 10.8 that theestimated equivalent input disturbance converges to h(w), and η converges to w.

�

Example 10.4. In this example, we briefly show that an output regulation problemcan also be converted to the form in (10.91). Consider

x1 = x2 + (ey − 1) + u

x2 = (ey − 1) + 2w1 + u

w = Aw

y = x1 − w1,

(10.106)

where y ∈ R is the measurement output and w1 = [1 0]w. In this example, the mea-sured output contains the unknown disturbance, unlike Example 10.3. The controlobjective remains the same, to design an output feedback control law to ensure theoverall stability of the system and the convergence to zero of the measured output.The key step in the control design is to show that the system shown in (10.106) canbe converted to the form as shown in (10.91).

Let

πz = 1

1 + ω2[1 − ω]w,


and it is easy to check that πz satisfies

πz = −πz + [1 0]w.

Let z = x2 − πz − x1. It can be obtained that

y = z + y + ew1 (ey − 1) + (u − h(w))

z = −z − y + 2w1,

where

h(w) = 2 + ω2

1 + ω2[−1ω]w − (ew1 − 1).

It can be seen that we have transformed the system to the format as shown in(10.91) with ψ(y, v) = ew1 (ey − 1).

To make H = [1 0], we introduce a state transform for the disturbance model as

ζ = 2 + ω2

1 + ω2

[ −1 ω

−ω −1

]

w.

It can be easily checked that ζ = Aζ . The inverse transformation is given as

w = 1

2 + ω2

[−1 −ωω −1

]

ζ.

With ζ as the disturbance state, we can write the transformed system as

y = z + y + e1/(2+ω2)[−1−ω]ζ (ey − 1) + (u − h(ζ ))

z = −z − y + 2

2 + ω2[−1 −ω]ζ (10.107)

ζ = Aζ ,

where

h(ζ ) = ζ1 − (e1/(2+ω2)[−1−ω]ζ − 1).

Note that ey−1y is a continuous function, and we can take ψ(y) = d0y( ey−1

y )2 whered0 is a positive real constant depending on the frequency and the knowledge of anupper limit of the disturbance amplitude. The control design presented in the previoussection can then be applied to (10.108). Simulation studies were carried out with theresults shown in Figures 10.10–10.13. �


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.05

0

0.05

0.1

0.15

0.2

Time (s)

y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−300

−250

−200

−150

−100

−50

0

50

Time (s)

u

Figure 10.10 The system input and output

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−300

−250

−200

−150

−100

−50

0

50

Time (s)

u an

d h

uh

Figure 10.11 Control input and the equivalent input disturbance


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−25

−20

−15

−10

−5

0

5

10

15

20

Time (s)

h an

d es

timat

ehEstimate of h

Figure 10.12 The equivalent input disturbance and its estimate

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−30

−20

−10

0

10

20

30

40

Time (s)

m an

d es

timat

es

m1m2h1h2

Figure 10.13 The exosystem states and the internal model states

Chapter 11

Control applications

In this chapter, we will address a few issues about control applications. Severalmethods of disturbance rejection are presented in Chapter 10, including rejectionof general periodic disturbances. A potential application can be the estimation andrejection of undesirable harmonics in power systems. Harmonics, often referred tohigh-order harmonics in power systems, are caused by nonlinearities in power systems,and the successful rejection depends on accurate estimation of amplitudes and phaseof harmonics. We will show an iterative estimation method based on a new observerdesign method.

There are tremendous nonlinearities in biological systems, and there have beensome significant applications of nonlinear system analysis and control methods insystem biology. We will show a case that nonlinear observer and control are appliedto circadian rhythms. A Lipschitz observer is used to estimate unknown states, andbackstepping control design is then applied to restore circadian rhythms.

Most of the control systems are implemented in computers or other digital deviceswhich are in discrete-time in nature. Control implementation using digital devicesinevitably ends with sample-data control. For linear systems, the sampled systems arestill linear, and the stability of the sampled-date system can be resolved in stabilityanalysis using standard tools of linear systems in discrete-time. However, when anonlinear system is sampled, the system description may not have a closed form, andthe structure cannot be preserved. The stability cannot be assumed for a sampled-dataimplementation of nonlinear control strategy. We will show that for certain nonlinearcontrol schemes, the stability can be preserved by fast sampling in the last section.

11.1 Harmonics estimation and rejection in powerdistribution systems

There is a steady increase in nonlinear loading in power distribution networks due tothe increase in the use of electrical cars, solar panels for electricity generation, etc.Nonlinear loading distorts the sinusoidal waveforms of voltage and current in net-works. The distorted waveforms are normally still periodic, and they can be viewed asgeneral periodic disturbances. Based on Fourier series, a general periodic signal canbe decomposed into sinusoidal functions with multiple frequencies of the base fre-quency. The components with high frequencies are referred to as harmonics. In otherwords, they are individual frequency modes at frequencies which are multiples of thebase frequency.


Harmonics are undesirable in power distribution networks for various reasons.They occupy the limited power capacity, and can be harmful to electrical and elec-tronic devices. Often in power distribution networks, active and passive power filtersare used to reduce the undesired harmonics. Effective rejection of harmonics dependson accurate phase and amplitude estimation of individual frequency modes. In a powerdistribution network, harmonics with certain frequencies are more critical for rejectionthan others. For example the double frequency harmonics normally disappear in thesystem due to a particular connection in power distribution networks, and third-orderharmonics would be most important to reject, perhaps due to high-attenuation high fre-quencies of distribution networks. Based on this discussion, estimation and rejectionof specific frequency modes will be of interest to power distribution networks.

As explained above, harmonics appear as general periodic signals. Rejection ofgeneral periodic disturbances is discussed in Chapter 10, for matched cases, i.e., theinput is in exactly the same location as the disturbances, which is a restriction tocertain applications, even though we may be able to convert an unmatched case to amatched one for a certain class of nonlinear systems. The method in Chapter 10 doesnot apply to rejection of individual frequency modes. We will show the conversionof an unmatched disturbance to a matched one, and show how individual frequencymodes can be estimated and rejected.

We will consider a class of nonlinear systems that has a similar structure tothe systems considered in the earlier chapters, but with unmatched general periodicdisturbances. We will show how an equivalent input, also a periodic disturbance, canbe obtained for this system, and then propose an estimation and rejection methodfor the system. The presentation at this stage is not exactly based on power systems,but the proposed methods can be directly applied to power distribution systems. Wewill show an example of estimation of harmonics using the proposed method forindividual frequency modes.

11.1.1 System model

Consider a SISO nonlinear system which can be transformed into the output feedbackform

ζ = Acζ + φ(y) + bu + dw

y = Cζ ,(11.1)

with

Ac =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0bρ...

bn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

T

, d =⎡

⎢⎣

d1...

dn

⎤

⎥⎦ ,

where ζ ∈ Rn is the state vector; u ∈ R is the control input; φ : R → R

n withφ(0) = 0is a nonlinear function with element φi being differentiable up to the (n − i)th order;

Control applications 221

b ∈ Rn is a known constant Hurwitz vector, with bρ �= 0, which implies the relative

of the system is ρ; d is an unknown constant vector; and w ∈ R is a bounded periodicdisturbance, which has continuous derivative up to the order of max{ρ − ι, 0} with ιbeing the index of the first non-zero element of vector d.

Remark 11.1. The disturbance-free system of (11.1) is in the output feedback formdiscussed in the previous chapters. With the disturbance, it is similar to the system(10.1), with the only difference that w is a general periodic disturbance. It is alsodifferent from the system (10.91) of which the disturbance is in the matched form. �

Remark 11.2. The continuity requirement specified for the general periodic distur-bance w in (11.1) is for the existence of a continuous input equivalent disturbanceand a continuous invariant manifold in the state space. For the case of ρ < ι, we mayallow disturbance to have finite discontinuous points within each period, and for eachof the discontinuous points, the left and right derivatives exist. �

Remark 11.3. The minimum phase assumption is needed for the convenience ofpresentation of the equivalent input disturbance and the control design based onbackstepping. It is not essential for control design, disturbance rejection or disturbanceestimation. We could allow the system to be non-minimum phase, provided that thereexists a control design for the disturbance-free system which renders the closed-loopsystem exponentially stable. �

The zero dynamics of (11.1) is linear. To obtain the equivalent input disturbance,we need a result for steady-state response for stable linear systems.

Lemma 11.1. For a linear system

x = Ax + bw, (11.2)

where x ∈ Rn is the system state, A is Hurwitz, b ∈ R

n is a constant vector and w is theperiodic disturbance with period T , the steady state under the input of the periodicdisturbance w is given by

xs(t) =∫ t

0eA(t−τ )bw(τ )dτ + eAt(I − eAT )−1eAT WT , (11.3)

where WT is a constant vector in Rn given by

WT =∫ T

0e−Aτbw(τ )dτ.

Proof. The state response to the input w is given by

x(t) = eAtx(0) +∫ t

0eA(t−τ )bw(τ )dτ.


Considering the response after a number of periods, we have

x(NT + t) = eA(NT+t)x(0) + eA(NT+t)

∫ NT+t

0e−Aτbw(τ )dτ

= eA(NT+t)x(0) + eA(NT+t)N−1∑

i=0

∫ (i+1)T

iTe−Aτbw(τ )dτ

+eA(NT+t)

∫ NT+t

NTe−Aτbw(τ )dτ ,

where N is a positive integer. Since w(t) is a periodic function, we have

∫ (i+1)T

iTe−Aτbw(τ )dτ =

∫ (i+1)T

iTe−Aτbw(τ − iT )dτ

=∫ T

0e−A(iT+τ )bw(τ )dτ

= e−iAT

∫ T

0e−Aτbw(τ )dτ.

Therefore, we have

x(NT + t) = eA(NT+t)x(0) + eAtN−1∑

i=0

eA(N−i)T

∫ T

0e−Aτbw(τ )dτ

+∫ t

0eA(t−τ )bw(τ )dτ. (11.4)

The steady-state response in (11.3) is obtained by taking the limit of (11.4) fort → ∞. �

To obtain the equivalent input disturbance, we need to introduce state trans-formation to (11.1). To extract the zero dynamics, we introduce a partial statetransformation for system (11.1) as

z =⎡

⎢⎣

ζρ+1...

ζn

⎤

⎥⎦ −

ρ∑

i=1

Bρ−ibζi,

where

B =

⎡

⎢⎢⎢⎢⎣

−bρ+1/bρ 1 . . . 0...

.... . .

...

−bn−1/bρ 0 . . . 1

−bn/bρ 0 . . . 0

⎤

⎥⎥⎥⎥⎦

, b =⎡

⎢⎣

bρ+1/bρ...

bn/bρ

⎤

⎥⎦ .


The dynamics with the coordinates (ζ1, . . . , ζρ , z) can be obtained as

ζi = ζi+1 + φi(y) + diw, i = 1, . . . , ρ − 1

ζρ = z1 +ρ∑

i=1

riζi + φρ(y) + dρw + bρu (11.5)

z = Bz + φz(y) + dzw,

where

ri = (Bρ−ib)i, for i = 1, . . . , ρ,

φz(y) =⎡

⎢⎣

φρ+1...

φn

⎤

⎥⎦ −

ρ∑

i=1

Bρ−ibφi + Bρ by,

and

dz =⎡

⎢⎣

dρ+1...

dn

⎤

⎥⎦ −

ρ∑

i=1

Bρ−ibdi.

The periodic trajectory and the equivalent input disturbance can be found usingthe system in the coordinate (ζ1, . . . , ζρ , z). Since the system output y does notcontain the periodic disturbance, we have the invariant manifold for π1 = 0. FromLemma 11.1 which is used for the result of the steady-state response of linear systemsto the periodic input, we have, for 0 ≤ t < T ,

πz(t) =∫ t

0eB(t−τ )dzw(τ )dτ + eBt(I − eBT )−1eBT WT

with WT = ∫ T0 e−Bτdzw(τ )dτ . From the first equation of (11.6), we have, for

i = 1, . . . , ρ − 1

πi+1(t) = dπi(t)

dt− diw.

Based on the state transformation introduced earlier, we can use its inversetransformation to obtain

⎡

⎢⎣

πρ+1...

πn

⎤

⎥⎦ = πz +

ρ∑

i=1

Bρ−ibπi.

Therefore, the periodic trajectory in the state space is obtained as

π = [π , . . . ,πn]T . (11.6)


Finally, the equivalent input disturbance μ is given by

μ = 1

bρ

(dπρ(t)

dt− πz,1 −

ρ∑

i=1

riπz,i − dρw

)

. (11.7)

Let x = ζ − π denote the difference between the state variable ζ and the periodictrajectory.

The periodic trajectory, π , plays a similar role as the invariant manifold in theset-up for the rejection of disturbances generated from linear exosystems. For this,we have the following result.

Theorem 11.2. For the general periodic disturbance w in (11.1), the periodic tra-jectory given in (11.6) and the equivalent input disturbance given in (11.7) are welldefined and continuous, and the difference between the state variable (11.1) and theperiodic trajectory, denoted by x = ζ − π , satisfies the following equation:

x = Acx + φ(y) + b(u − μ)

y = Cx.(11.8)

The control design and disturbance rejection will be based on (11.8) insteadof (11.1).

Remark 11.4. The control design and disturbance rejection only use the output y,with no reference to any of other state of the system. Therefore, there is no differencewhether we refer to (11.1) or (11.8) for the system, because they have the same output.The format in (11.8) shows that there exist an invariant manifold and an equivalentinput disturbance. However, the proposed control design does not depend on anyinformation of μ, other than its period, which is the same as the period of w. In otherwords, control design only relies on the form shown in (11.1). The form shown in(11.8) is useful for the analysis of the performance of the proposed control design,including the stability. In this section, we start our presentation from (11.1) ratherthan (11.8) in order to clearly indicate the class of the systems to which the proposedcontrol design can be applied, without the restriction to the rejection of matcheddisturbances. �

11.1.2 Iterative observer design for estimating frequencymodes in input

We will propose an iterative observer design method to estimate specific frequencymodes in the input to a linear system from its output. For the convenience of discussion,we have the following definitions.


Definition 11.1. A T -periodic function f is said to be orthogonal to a frequency modewith frequency ωk if

∫ T

0f (τ ) sinωkτdτ = 0, (11.9)

∫ T

0f (τ ) cosωkτdτ = 0. (11.10)

Definition 11.2. A function f is said to be asymptotically orthogonal to a frequencymode with frequency ωk if

limt→∞

∫ t+T

tf (τ ) sinωkτdτ = 0, (11.11)

limt→∞

∫ t+T

tf (τ ) cosωkτdτ = 0. (11.12)

We consider a stable linear system

x = Ax + bμ

y = Cx,(11.13)

where x ∈ Rn is the state variable; and y and u ∈ R are output and input, the matrices

A, b and C are with proper dimensions; and the system is stable and observable withthe transfer function Q(s) = C(sI − A)−1b. The problem considered in this subsectionis to create a signal such that specific frequency modes can then be removed from theinput μ, which is a general periodic disturbance described by

μ(t) =∞∑

k=1

ak sin (ωk t + φk ),

where ωk = 2πkT and ak and φk are the amplitude and phase angle of the mode for

frequency ωk . The dynamics for a single frequency mode can be described as

wk = Skwk , (11.14)

where

Sk =[

0 ωk

−ωk 0

]

.

For a single frequency mode with the frequency ωk as the input denoted byμk = ak sin (ωk t + φk ), its output in y, denoted by yk , is a sinusoidal function withthe same frequency, if we only consider the steady-state response. In fact, based onthe frequency response, we have the following result.

Lemma 11.3. Consider a stable linear system (A, b, C) with no zero at jωk for anyinteger k. For the output yk of a single frequency mode with the frequency ωk , thereexists an initial state wk (0) such that

yk = gT wk , (11.15)


where g = [1 0]T and wk is the state of (11.14). Furthermore, the input μk for thisfrequency mode can be expressed by

μk = gTk wk , (11.16)

where

gk = 1

mk

[cos θk sin θk

−sin θk cos θk

]

g := Qkg (11.17)

with θk = ∠Q(jωk ) and mk = |Q(jωk )|.

Proof. For the state model (11.14), we have

eSk t =[

1 1j −j

] [ejωk t

ejωk t

] [1 1j −j

]−1

=[

cos (ωk t) sin (ωk t)−sin (ωk t) cos (ωk t)

]

.

For the single frequency mode ωk , the output is given by

yk = mkak sin (ωk t + φk + θk ).

With g = [1 0]T , we have

wk (0) =[

mkak sin (φk + θk )mkak cos (φk + θk )

]

and

wk (t) = eSk twk (0) =[

mkak sin (ωk t + φk + θk )mkak cos (ωk t + φk + θk )

]

such that

yk = gT eSk twk (0) = gT wk (t).

Considering the gain and phase shift of Q(s), we have

μk = 1

mkgT eSk (t−(θk /ωk ))wk (0)

= 1

mkgT e−(θk /ωk )Sk eSk twk (0).

Hence, we have

gk = 1

mke−(θk /ωk )ST

k g

and therefore

Qk = 1

mke−(θk /ωk )ST

k = 1

mke(θk /ωk )Sk = 1

mk

[cos θk sin θk

−sin θk cos θk

]

.


For a stable single-input linear system, if the input is a T -periodic signal that isorthogonal to a frequency ωk , the steady state, as shown earlier, is also T -periodic.Furthermore, we have the following results.

Lemma 11.4. If the input to a stable single-input linear system (A, b) is T -periodicsignal that is orthogonal to a frequency modeωk , for any positive integer k, the steadystate is orthogonal to the frequency mode and the state variable is asymptoticallyorthogonal to the frequency mode. Furthermore, if the linear system (A, b, C) hasno zero at jωk , the steady-state output is orthogonal to the frequency mode ωk if andonly if the input to the system is orthogonal to the frequency mode.

Proof. We denote the input as μ, and the state variable x satisfies

x = Ax + bμ.

Since μ is T -periodic, the steady-state solution of the above state equation, denotedby xs, is also T -periodic and

xs = Axs + bμ.

Let

Jk =∫ T

0xs(τ ) sinωkτdτ.

Using integration by part, we have

Jk = −ω−1k

∫ T

0xsd cosωkτ

= ω−1k

∫ T

0xs cosωkdτ

= ω−1k

∫ T

0(Axs + bμ(t)) cosωkτdτ

= ω−1k A

∫ T

0xs cosωkτdτ

= ω−2k A

∫ T

0xsd sinωkτ

= −ω−2k A

∫ T

0(Axs + bμ(t)) sinωkτdτ

= −ω−2k A2

kJk .

Hence, we have

(ω2k I + A2)Jk = 0.


Since A is a Hurwitz matrix which cannot have ±ωk j as its eigenvalues, we concludeJk = 0. Similarly we can establish

∫ T

0xs(τ ) cosωkτdτ = 0

and therefore xs is orthogonal to the frequency mode ωk .If we denote ex = x − xs, we have ex = Aex. It is clear that ex exponentially

converges to zero, and therefore we can conclude that x is asymptotically orthogonalto the frequency mode ωk . This completes the proof of the first part.

For the second part of the lemma, the ‘if ’ part follows directly from the first partof the lemma, and we now establish the ‘only if’ part by seeking a contradiction.Suppose that the output y is orthogonal to the frequency mode ωk , and the input μ isnot. In this case, it can be shown in a similar way as in the proof Lemma 11.3, thatthere exists a proper initial condition for wk (0) such that μ− gT

k wk is orthogonal toωk . Since the system is linear, we can write

y = y⊥ + yw,

where y⊥ denotes the steady-state output generated by μ− gTk wk and yw generated

by gTk wk . We have y⊥ orthogonal to the frequency mode ωk . However, yw would be

a sinusoidal function with frequency ωk and it is definitely not orthogonal. Thus weconclude y is not orthogonal to ωk , which is a contradiction. Therefore, μ must beorthogonal to ωk if y is. This completes the proof. �

To remove frequency modes in the inputμ is to find an estimate μ such thatμ− μ

does not contain those frequency modes asymptotically. For a single frequency modewith frequency ωk , the task to obtain a μ is accomplished by

˙wk = Skwk + lk (y(t) − gT wk )

μ = (Qkg)T wk = gTk wk ,

(11.18)

where lk is chosen such that Sk − lkgT is Hurwitz. For this observer, we have a usefulresult stated in the following lemma.

Lemma 11.5. For any positive integer k, with the observer as designed in (11.18),(μ(τ ) − gT

k wk ) is asymptotically orthogonal to the frequency mode ωk , i.e.,

limt→∞

∫ t+T

t(μ(τ ) − gT

k wk ) sinωkτdτ = 0, (11.19)

limt→∞

∫ t+T

t(μ(τ ) − gT

k wk ) cosωkτdτ = 0. (11.20)


Proof. Consider the steady-state output y of input μ. From Lemma 11.3, there existsan initial state wk (0) such that

∫ T

0(y(τ ) − gT wk (τ )) sinωkτdτ = 0,

∫ T

0(y(τ ) − gT wk (τ )) cosωkτdτ = 0,

which implies that μ− gTk wk is orthogonal to the frequency mode ωk , again based on

Lemma 11.3.Let wk = wk − wk . The dynamics of wk can be obtained from (11.14) and

(11.18) as

˙wk = Sk wk − lk (y − gT wk ), (11.21)

where Sk = Sk − lkgT . Note that Sk is a Hurwitz matrix and (y − gT wk ) is aT -periodic signal. There exists a periodic steady-state solution of (11.21) such that

πk = Skπk − lk (y − gT wk ).

From Lemma 11.4, πk is orthogonal to the frequency mode ωk because (y − gT wk )is. Let ek = wk − πk . We have

ek = Skek ,

which implies that ek exponentially converges to zero. The observer state wk can beexpressed as

wk = wk − πk − ek .

Therefore, (11.19) and (11.20) can be established, and this completes the proof. �

The result in Lemma 11.5 shows how an individual frequency mode can beremoved with the observer designed in the way as if the output would not containother frequency modes. From the proof of Lemma 11.5, it can be seen that thereis an asymptotic error, πk , between the observer state and the actual state variablesassociated with the frequency mode ωk . Although πk is orthogonal to the frequencymodeωk , it does in general contain components generated from all the other frequencymodes. Because of this, a set of observers of the same form as shown in (11.18) wouldnot be able to extract multiple frequency modes simultaneously. To remove multiplefrequency modes, it is essential to find an estimate which is asymptotically orthogonalto the multiple frequency modes. For this, the interactions between the observers mustbe dealt with.

Suppose that we need to remove a number of frequency modes ωk for all the kin a finite set of positive integers K = {ki}, for = 1, . . . , m. To estimate the frequencymodes for ωk ,i, i = 1, . . . , m, we propose a sequence of observers,

˙wk ,1 = Sk ,1wk ,1 + lk ,1(y − gT wk ,1) (11.22)


and, for i = 2, . . . , m,

ηk ,i−1 = Aηk ,i−1 + bgTk ,i−1wk ,i−1 (11.23)

˙wk ,i = Sk ,iwk ,i + lk ,i

⎛

⎝y −i−1∑

j=1

Cηk ,j − gT wk ,i

⎞

⎠ , (11.24)

where lk ,i, for i = 1, . . . , m, are designed such that Sk ,i := Sk ,i − lk ,igT are Hurwitz,and

gk ,i = 1

mk ,i

[cosφk ,i sin φk ,i

−sin φk ,i cosφk ,i

]

g := Qk ,ig

with mk ,i = |C(jwk ,i − A)−1b| and φk ,i = ∠C(jwk ,i − A)−1b.The estimate for the input disturbance which contains the required frequency

modes for asymptotic rejection is given by

μm =m∑

i=1

gTk ,iwk ,i. (11.25)

The estimate μm contains all the frequency modes ωk ,i, for i = 1, . . . , m. The usefulproperty of the estimate is given in the following theorem.

Theorem 11.6. For the estimate μm given in (11.25), μ− μm is asymptoticallyorthogonal to the frequency modes ωk ,i for i = 1, . . . , m.

Proof. In the proof, we will show how to establish the asymptotic orthogonality indetail by induction.

We introduce the notations wk ,i = wk ,i − wk ,i. We use πk ,i to denote the steady-state solutions of wk ,i and ek ,i = wk ,i − πk ,i, for i = 1, . . . , m.

Lemma 11.5 shows that the results hold for m = 1. Let

μ1 = gTk ,1(wk ,1 − πk ,1)

and μ− μ1 is orthogonal to the frequency mode ωk ,1.We now establish the result for m = 2. From Lemma 11.3, there exists an initial

state variable wk ,2(0) for the dynamic system

wk ,2 = Sk ,2wk ,2 (11.26)

such that y − Cqk ,1 − gT wk ,2 is orthogonal to the frequency mode ωk ,2 where qk ,i, fori = 1, . . . , m − 1, denote the steady-state solution of

qk ,i = Aqk ,i + bgTk ,i(wk ,i − πk ,i).

Note that gT wk ,2 can be viewed as the output for the input gTk ,2wk ,2 to the system

(A, b, C), based on Lemma 11.3. Hence, y − Cqk ,1 − gT wk ,2 is the output for the inputμ− gT

k ,1(wk ,1 − πk ,1) − gTk ,2wk ,2 to the system (A, b, C). Therefore, from Lemma 11.4,

μ− gTk ,1(wk ,1 − πk ,1) − gT

k ,2wk ,2 is orthogonal to ωk ,2. Furthermore, from the previous


step πk ,1 is orthogonal to ωk ,1. Hence, μ− gTk ,1(wk ,1 − πk ,1) − gT

k ,2wk ,2 is orthogonalto the frequency modes ωk ,j for j = 1, 2.

The dynamics of wk ,2 and πk ,2 are obtained as

˙wk ,2 = Sk ,2wk ,2 − lk ,2(y − Cηk ,1 − gT wk ,2), (11.27)

πk ,2 = Sk ,2πk ,2 − lk ,2(y − Cqk ,1 − gT wk ,2), (11.28)

and the error ek ,2 = wk ,2 − πk ,2 satisfies

ek ,2 = Sk ,2ek ,2 − lk ,2C(qk ,1 − ηk ,1).

Since qk ,1 − ηk ,1 exponentially converges to zero, so does ek ,2. From (11.28) andLemma 11.4, πk ,2 is orthogonal to the frequency modes ωk ,j for j = 1, 2. Therefore,by letting

μ2 = gTk ,1(wk ,1 − πk ,1) + gT

k ,2(wk ,2 − πk ,2),

μ− μ2 is orthogonal to the frequency modes ωk ,j for j = 1, 2.Notice that

μ2 − μ2 =2∑

j=1

gTk ,j(wk ,j − πk ,j − wk ,j)

=2∑

j=1

gTk ,1ek ,j.

Hence μ2 − μ2 converges to zero exponentially. Therefore, we conclude that μ− μ2

is asymptotically orthogonal to the frequency modes ωk ,j for j = 1, 2.Now suppose that the result holds for m = i and therefore μ− μi is orthogonal

to the frequency modes ωk ,j for j = 1, . . . , i, with

μi =i∑

j=1

gTk ,j(wk ,j − πk ,j).

We need to establish that the result holds for m = i + 1.For the frequency mode ωk ,i+1, there exists an initial state variable wk ,i(0) for the

dynamic system

wk ,i+1 = Sk ,iwk ,i+1

such that y − C∑i

j=1 qk ,j − gT wk ,i+1 is orthogonal to the frequency modeωk ,i+1. Note

that y − C∑i

j=1 qk ,j − gT wk ,i+1 is the output for the inputμ− μi − gTk ,i+1wk ,i+1 to the

system (A, b, C). Therefore from Lemma 11.4, μ− μi − gTk ,i+1wk ,i+1 is orthogonal

to ωk ,i+1. Since μ− μi is orthogonal to the frequency modes ωk ,j for j = 1, . . . , i,μ− μi − gT

k ,i+1wk ,i+1 is orthogonal to the frequency modes ωk ,j for j = 1, . . . , i + 1

and so is (y − C∑i

j=1 qk ,j − gT wk ,i+1).


The dynamics of wk ,i+1 and πk ,i+1 are obtained as

˙wk ,i+1 = Sk ,i+1wk ,i+1

− lk ,i+1

⎛

⎝y − Ci∑

j=1

ηk ,j − gT wk ,i+1

⎞

⎠ , (11.29)

πk ,i+1 = Sk ,i+1πk ,i+1

− lk ,i+1

⎛

⎝y − Ci∑

j=1

qk ,j − gT wk ,i+1

⎞

⎠ . (11.30)

Since the input is orthogonal to the frequency modes ωk ,j for j = 1, . . . , i + 1, weconclude from Lemma 11.4 that πk+i+1 is orthogonal to ωk ,j for j = 1, . . . , i + 1, andthereforeμ− μi+1 is orthogonal to the frequency modesωk ,j for j = 1, . . . , i + 1 with

μi+1 = μi + gTk ,i+1(wk ,i+1 − πk+i+1).

The error ek ,i+1 = wk ,i+1 − πk ,i+1 satisfies

ek ,i+1 = Sk ,i+1ek ,i+1 − lk ,i+1Ci∑

j=1

(qk ,j − ηk ,j).

Since qk ,j − ηk ,j , for j = 1, . . . , i, exponentially converge to zero, so does ek ,i+1. With

μi+1 − μi+1 =i+1∑

j=1

gTk ,j(wk ,j − πk ,j − wk ,j)

=i∑

j=1

gTk ,jek ,j ,

μi+1 − μi+1 converges to zero exponentially. Therefore, we conclude that μ− μi+1

is asymptotically orthogonal to the frequency modes ωk ,j for j = 1, . . . , i + 1.Since i is an arbitrary integer between 1 and m, the result holds for i = m − 1,

and this completes the proof. �

11.1.3 Estimation of specific frequency modes in input

From the previous analysis on the equivalent input disturbance, we can convert theunmatched disturbance case to the matched one. Therefore, we can now carry on thedisturbance rejection based on the matched case:

x = Acx + φ(y) + b(u − μ)

y = Cx.(11.31)


If the disturbance does not exist in (11.31), the system (11.31) is in the linearobserver error with output injection that is shown in Chapter 8. In that case, we candesign a state observer as

p = (Ac − LC)p + φ(y) + bu + Ly, (11.32)

where p ∈ Rn, L ∈ R

n is chosen so that Ac − LC is Hurwitz. The difficulty in the stateestimation is due to the unknown disturbance μ. Consider

q = (Ac − LC)q + bμ, (11.33)

where q denotes the steady-state solution. Such a solution exists, and an explicitsolution is given earlier. Each element of q is a periodic function, as μ is periodic.We have an important property for p and q stated in the following lemma.

Lemma 11.7. The state variable x can be expressed as

x = p − q + ε, (11.34)

where p is generated from (11.32) with q satisfying (11.33) and ε satisfying

ε = (Ac − LC)ε. (11.35)

From (11.33), we have Cq as the steady-state output of the system ((Ac −kC), b, C) for the equivalent input disturbanceμ. If Cq is available, we are ready to usethe results presented in the previous section for disturbance estimation of the specifiedfrequency modes. The result shown in Lemma 11.7 indicates that Cq = Cp − y − Cε,and hence Cp − y exponentially converges to Cq. Therefore, we now propose anobserver for frequency modes in the input disturbances using Cp − y. To estimate thefrequency modes for ωk ,i i = 1, . . . , m, we have

˙wk ,1 = Sk ,1wk ,1 + lk ,1(Cp − y − gT wk ,1) (11.36)

and, for i = 2, . . . , m,

ηk ,i−1 = (Ac − kC)ηk ,i−1 + bgTk ,i−1wk ,i−1, (11.37)


⎛

⎝Cp − y −i−1∑

j=1

Cηk ,j − gT wk ,i

⎞

⎠ , (11.38)

where lk ,i, for i = 1, . . . , m, are designed such that Sk ,i := Sk ,i − lk ,igT are Hurwitz,and

gk ,i = 1

mk ,i

[cosφk ,i sin φk ,i

−sin φk ,i cosφk ,i

]

g := Qk ,ig (11.39)

with mk ,i = |C(jwk ,i − (A − kC))−1b| and φk ,i = ∠C(jwk ,i − (A − kC))−1b.


The estimate for the input disturbance which contains the required frequencymodes for asymptotic rejection is given by

μm =m∑

i=1

gTk ,iwk ,i. (11.40)

The estimate μm contains all the frequency modes ωk ,i, for i = 1, . . . , m. The usefulproperty of the estimate is given in the following theorem.

Theorem 11.8. The estimate μm given in (11.40) is bounded and satisfies thefollowing:

limt→∞

∫ t+T

t[μ(τ ) − μm(τ )] sinωk ,iτdτ = 0, (11.41)

limt→∞

∫ t+T

t[μ(τ ) − μm(τ )] cosωk ,iτdτ = 0, (11.42)

for i = 1, . . . , m.

Proof. From Theorem 11.6, the results in (11.41) and (11.42) hold if the inputCp − y in (11.36), (11.37) and (11.38) equals Cq. Since Cp − y − Cq convergesexponentially to 0, and (11.36)–(11.38) are stable linear systems, the errors inthe estimation of μ caused by replacing Cq by Cp − y in (11.36)–(11.38) are alsoexponentially convergent to zero. Therefore, (11.41) and (11.42) can be establishedfrom Theorem 11.6. �

11.1.4 Rejection of frequency modes

With the estimated frequency modes μm, we set the control input as

u = v + μm,

where v is designed based on backstepping design which is shown in Chapter 9. Dueto the involvements of disturbance in the system, we need certain details of stabilityanalysis with explicit expression of Lyapunov functions. For the convenience of sta-bility analysis, we adopt an approach using backstepping with filtered transformationshown in Section 9.4. The control input v can be designed as a function

v = v(y, ξ ),

where ξ is the state variable of the input filter for the filtered transformation. The finalcontrol input is designed as

u = v(y, ξ ) + μ. (11.43)

For the stability of the closed-loop system with disturbance rejection, we have thefollowing theorem.


Theorem 11.9. The closed-loop system of (11.1) under the control input (11.43)ensures the boundedness of all the state variables, and the disturbance modes ofspecified frequencies ωk ,i, for i = 1, . . . , m, are asymptotically rejected from thesystem in the sense that all the variables of the closed-loop system is bounded andthe system is asymptotically driven by the frequency modes other than the specifiedmodes.

Proof. For the standard backstepping design for the nonlinear systems, we establishthe stability by considering the Lyapunov function

V = βzT Pz + 1

2y2 + 1

2

ρ−1∑

i=1

ξ 2i , (11.44)

where β is a constant and ξi = ξi − ξi and P is a positive definite matrix satisfy-ing DT P + PD = −I . The standard backstepping design ensures that there exists apositive real constant γ1 such that

V ≤ −γ1V − [2βzT Pbz + yby]μ

≤ −γ1V − γ2

√V μ

≤ −γ3V + γ4μ2 (11.45)

for some positive real constants γi, i = 2, 3, 4. Hence, the boundedness of μ ensures

the boundedness of all the state variables. It is clear from Theorem 11.8 that all thespecified frequency modes are asymptotically removed from μ. This completes theproof.

Remark 11.5. In the control design, we have used the control input obtained with thebackstepping design for the proof of Theorem 11.9. In case that there exists a knowncontrol design for the disturbance-free system which ensures exponential stability ofthe disturbance-free case, we can pursue the proof of Theorem 11.9 in a similar wayto obtain the result shown in (11.45). This is very useful in particular for the case ofnon-minimum phase systems, to which the standard backstepping presented earlierdoes not apply. �

11.1.5 Example

In this section, we use a simple example to demonstrate estimation of harmonics usingthe method proposed earlier in this section.

We consider a voltage signal with diados as shown in Figure 11.1. We aregoing to use iterative observers to estimate harmonics in this signal. To simplifythe presentation, we assume that the signal is directly measurable. In this case,there are no dynamics between the measurement and the point of estimation.Hence, we can simplify the algorithms presented in Subsection 11.1.2 ((11.22),


0.7 0.72 0.74 0.76 0.78 0.8–1

–0.5

0

0.5

1

1.5

t (s)

Vol

tage

Figure 11.1 Voltage signal of a nonlinear load

(11.23) and (11.24)). In this case, there is no need to use the observer model in(11.23). The simplified algorithm for direct estimation of harmonics is presentedbelow.

To estimate the frequency modes for ωk ,i, i = 1, . . . , m, the iterative observersare given by

˙wk ,1 = Sk ,1wk ,1 + lk ,1(y − gT wk ,1) (11.46)

and, for i = 2, . . . , m,


⎛

⎝y −i∑

j=1

gT wk ,j

⎞

⎠ , (11.47)

where lk ,i, for i = 1, . . . , m, are designed such that Sk ,i := Sk ,i − lk ,igT are Hurwitz.For the electricity, the base frequency is 50 Hz. We will show the estimates

of harmonics from the base frequency, that is, we set ωk ,1 = 100π , ωk ,2 = 200π ,ωk ,3 = 300π , etc. For these frequency modes, we have

Sk ,i =[

0 i100π−i100π 0

]

.


The observer gain lk ,i can be easily designed to place the close-loop poles at anyspecified positions. For the simulation study, we place the poles at {−200, −200}. Forsuch pole positions, we have

lk ,i =[

40040000i100π − 200π

]

.

The harmonics of second and third orders are plotted with the original signal andthe estimated base frequency component in Figure 11.2. The approximations ofaccumulative harmonics to the original signal are shown in Figure 11.3.

In this example, we have shown the estimation of harmonics from directmeasurements. The advantage of this estimation is that the estimation is on-lineand can be implemented in real time. Also, the estimation using the proposedmethod can provide the phase information, which is very important for harmonicrejection. We did not show the rejection of harmonics in this example. However,it is not difficult to see that rejection can be simulated easily with the proposedmethod.

0.74 0.745 0.75 0.755 0.76 0.765 0.77 0.775 0.78–1

–0.5

0

0.5

1

1.5

t (s)

Sign

al &

har

mon

ics

Original1st order2nd order3rd order

Figure 11.2 Estimated harmonics


Original1st orderUp to 2nd orderUp to 3rd order

0.74 0.745 0.75 0.755 0.76 0.765 0.77 0.775 0.78–1

–0.5

0

0.5

1

1.5

t (s)

Sign

al &

app

roxi

mat

ion

Figure 11.3 Approximation using harmonic components

11.2 Observer and control design for circadian rhythms

Circadian rhythms play an important role in daily biological activities of livingspecies. Circadian rhythms, or biological clocks, normally have a period of 24 h, andchanges to the rhythms can be referred to as changes in phases. Circadian disorder isa phenomenon of circadian rhythms which occurs when internal rhythms cannot keepup with the changes of external environmental rhythms. Changes of environmentalrhythms, presented by the change of light/dark cycles or by irregular rhythms, resultin phase shifts between internal and external rhythms. For example, jet lags arecaused when people travel from one time zone to another. The existence of thesephase shifts in longer term has negative effect to health. Therefore, in biologicalstudy of circadian rhythms, it is important to find methods to recover the shiftedphases, such as jet lags, to their normal rhythms. These methods could then lead totreatments of circadian disorder, like sleep disorder and jet lag. Circadian rhythms canbe accurately modelled as limit cycles of nonlinear dynamic systems. We will applythe observer and control design shown in earlier chapters for analysis and controlof circadian rhythms. The phase restoration is carried out by the synchronisation oftrajectories generated from a controlled model with the trajectories of a referencesystem via nonlinear control design. Both reference and controlled systems are based


on a given third-order model of Neurospora circadian rhythms. An observer is alsodesigned to estimate two unknown states from the measurement of a single state.

11.2.1 Circadian model

For almost every living organism on the Earth, their daily biological activities aregoverned by rhythms. These biological rhythms are called circadian rhythms. Thecircadian rhythms exist as self-sustained and periodic oscillations, and they are alsoknown with their entrainment to 24-h day/night cycle. The entrainment is based onthe activity of a circadian clock gene. For mammals, the circadian clock gene, alsoknown as the circadian pacemaker, is found in the suprachiasmatic nuclei (SCN) ofthe anterior hypothalamus. The circadian pacemaker captures information sent froman external environment cue such as light, and then coordinates the timing of otherslave clocks or slave oscillators in other parts of the body. Any changes of environmentcues which cause the mismatch between external and internal rhythms can lead todisruption of circadian rhythms. This phenomenon is known as circadian disorders.Jet lags due to trans-continent flights and sleeping disorders due to irregular sleep–wake cycles are two typical examples of circadian disorders. In practice, one of theknown medical treatments for circadian disorders is the application of light. Light ismajor external environmental cue, and with light input, the circadian phase can beadjusted to the light/dark rhythms at destination.

In this section, we aim for the restoration of circadian phase using nonlinearcontrol. In order to achieve this objective, we propose an alternative control designmethod which synchronises trajectories generated from a controlled model with thetrajectories generated from a reference model via backstepping approach. Both ref-erence system and controlled system are based on a third-order mathematical modelof Neurospora circadian rhythms. The trajectories generated by controlled systemrepresent the altered rhythms. Meanwhile, the reference trajectories represent thedesired rhythms which the trajectories of controlled system are adjusted to match. Wealso present an observer design for this circadian rhythm based on observer designpresented in Chapter 8 for nonlinear systems with Lipschitz nonlinearities.

A third-order mathematical model is developed to describe molecular mechanismof circadian rhythms in Neurospora. Its molecular mechanism is based on the negativefeedback exerted by FRQ proteins on the expression of frq gene. Transcription of frqgene yields messenger RNA (mRNA), and the translation of which synthesises FRQprotein. These synthesised FRQ proteins are then transferred back into nucleus wherethey inhibit the transcription of frq gene. A new activation of frq gene transcriptionwill restart the cycle. Dynamics of these variables, frq mRNA, FRQ protein andnuclear FRQ protein, are by the following dynamic model:

x1 = vsKn

i

Kni + xn

3

− vmx1

KM + x1

x2 = ksx1 − vdx2

Kd + x2− k1x2 + k2x3 (11.48)

x3 = k1x2 − k2x3,


where x1, x2 and x3 denote concentration of frq mRNA, concentration of FRQ proteinoutside nucleus and concentration of nucleus FRQ protein respectively. Values ofthree state variables x1, x2 and x3 are assumed to be positive values. In system(11.49), the parameter vs denotes the transcription rate of frq gene. The otherparameters involved in system (11.49) are Ki, n, vm, KM , ks, k1, k2, vd and Kd . Theparameters Ki, n, vm, KM represent the threshold constant beyond which nuclear FRQprotein inhibits the transcription of frq, the Hill coefficient showing the degree ofco-operativity of the inhibition process, the maximum rate of frq mRNA degradationand the Michaelis constant related to the latter process respectively. The parametersks, k1 and k2 denote the rate constant measuring the rate of FRQ synthesis, the rate con-stants of the transport of FRQ into and out of the nucleus respectively. The parametervd denotes maximum rate of FRQ degradation and Kd is the Michaelis constant relatedto this process. For the third-order Neurospora model, the parameters have their typ-ical values as vs = 1.6 nM h−1, Ki = 1 nM, n = 4, vm = 0.7 nM h−1, KM = 0.4 nM,ks = 1 h−1, vd = 4 nMh−1, Kd = 1.4 nM, k1 = 0.3 h−1, k2 = 0.15 h−1.

Circadian rhythms are self-sustained and periodic oscillations. With the setparameters as above, dynamics of state variables of (11.48) can sustain periodic oscil-lations, and in fact, a limit cycle. Figure 11.4 shows the plots of three state variablesobtained in simulation with the model (11.48). For simulation study, we choose theinitial values x(0) = [

5 1 1]T

. This particular initial value is actual a point on thelimit cycle. If the initial value starts from a point outside the limit cycle, then theretrajectory will converge to the limit cycle. There can be other selections for values ofinitial conditions.

0 10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

Time (h)

Stat

e va

riabl

es o

f Neu

rosp

ora

mod

el

x1x2x3

Figure 11.4 Circadian rhythm of Neurospora


The model (11.48) is not in the right format for control and observer design. Weintroduce a state transformation as

z = Tx (11.49)

with

T =⎡

⎣0 0 10 k1 0

k1ks 0 0

⎤

⎦ ,

and the transformed system is described by

z1 = z2 − k2z1

z2 = z3 − k1z2 + k1k2z1 − vdk1z2

k1Kd + z2(11.50)

z3 = vsk1ksKn

i

Kni + zn

1

− vmk1ksz3

KM k1ks + z3.

Now the transformed model (11.50) is in the lower-triangular format.

11.2.2 Lipschitz observer design

We have presented observer design for nonlinear dynamic systems with Lipschitznonlinearity in Section 8.4. Basically, for a nonlinear system described by (8.29)

x = Ax + φ(x, u)

y = Cx,

a nonlinear observer can be designed as in (8.30)

˙x = Ax + L(y − Cx) + φ(x, u), (11.51)

provided that the conditions specified in Theorem 8.10 are satisfied. That suggeststhat we need find the Lipschitz constant ν together with a positive definite matrix Pfor nonlinearities in the circadian model (11.48) such that the inequality (8.36)

AT P + PA + 2νI − 2σCT C < 0

is satisfied, and then we obtain the observer gain as

L = σP−1CT .

The one-sided Lipschitz condition shown in (8.36) can be checked using theLipschitz constant for individual elements in the vector φ. Let us denote the Lipschitzconstants of φi by γi. The condition shown in (8.36) can be guaranteed by

AT P + PA + 2nn∑

i=1

γiλiI − 2σCT C < 0, (11.52)


where λi are positive real constants, and P ∈ ∩ni=1Pn(i, λi). The definition of

P ∈ ∩ni=1Pn(i, λi) is given by

Pn(i, λi) = {P : |pji| < λi, for j = 1, 2, . . . , n}.For the system (11.50), we set

C1 = [1 0 0

]

and therefore state variables z2 and z3 are unknown. Comparing with the structure in(8.29), we have

φ(z, u) =⎡

⎣0φ2

φ3a + φ3b

⎤

⎦ ,

where clearly φ1 = 0, and

φ2 = −vdk1z2

k1Kd + z2,

φ3a = vsk1ksKn

i

Kni + zn

1

,

φ3b = −vmk1ksz3

KM k1ks + z3.

From the definition of Lipschitz constant in Definition 8.2, we have∥∥∥∥

−vdk1z2

k1Kd + z2+ vdk1z2

k1Kd + z2

∥∥∥∥ ≤ vd

Kd

∥∥z2 − z2

∥∥ ,

∥∥∥∥

−vmk1ksz3

KM k1ks + z3+ vmk1ksz3

KM k1ks + z3

∥∥∥∥ ≤ vm

KM

∥∥z3 − z3

∥∥ ,

and therefore we obtain the Lipschitz constants as γ2 = vdKd

=10.7962 for ϕ2(z2), andγ3b = vm

KM=1.01 for ϕ3b(z3). For nonlinear function ϕ3a(z1), its Lipschitz constant can

be computed by using mean value theorem which is described by

∣∣f ′(ζ )

∣∣ =

∣∣∣∣f (x) − f (x)

x − x

∣∣∣∣ ,

where ζ ∈ [x, x]. Setting f = φ3a, we have

∣∣f ′(ζ )

∣∣ =

∣∣∣∣∣−nvsk1ksKn

i ζn−1

(Kn

i + ζ n)2

∣∣∣∣∣=

∣∣∣∣φ3a(z1) − φ3a(z1)

z1 − z1

∣∣∣∣ ,

where ζ ∈ [min (z1, z1), max (z1, z1)]. We find maximum value of |f ′(ζ )| by solving|f ′′(ζ )| = 0, and obtain the result as 0.325. Since the value of Lipschitz constant isequivalent to maximum value of f ′(ζ ), the Lipschitz constant is given by γ3a = 0.325.The Lipschitz constant λ3 is given by λ3 = λ3a + λ3b = 1.326.


After the Lipschitz constants are found, we solve (11.52) to obtain

σ = 67.8032, L =⎡

⎣3.98872.60671.3469

⎤

⎦

for the observer (11.51). Simulation study for the observer has been carried out withthe results shown in Figures 11.5 and 11.6.

0 10 20 30 40 50 60 70 80 90 100 110 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (h)

Tran

sfor

med

stat

e z 2

and

its e

stim

ate

z2Estimate of z2

Figure 11.5 Observer state for circadian model

11.2.3 Phase control of circadian rhythms

The objective of phase control is to reset a distorted phase. Our strategy is to controlthe distorted circadian rhythm to follow a target rhythm. In order to design a controlstrategy, we need to define a sensible control input for the circadian model. Amongthe parameters appeared in (11.48), parameter vs, which denotes the rate of frq mRNAtranscription, is sensitive to light input. Therefore, for Neurospora circadian rhythms,this parameter is usually used as control input in many results which have beenpresented in literature. If vs is taken as the control input, the circadian model (11.48)is rewritten as

x1 = (vs + u)Kn

i

Kni + xn

3

− vmx1

KM + x1

x2 = ksx1 − vdx2

Kd + x2− k1x2 + k2x3

x3 = k1x2 − k2x3,


z3Estimate of z3

0 10 20 30 40 50 60 70 80 90 100 110 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (h)

Tran

sfor

med

stat

e z 3

and

its e

stim

ate

Figure 11.6 Observer state for circadian model

where u is the control input. The system model after the state transformation (11.49)is then obtained as

z1 = z2 − k2z1

z2 = z3 − k1z2 + k1k2z1 − vdk1z2

k1Kd + z2

z3 = vsk1ksKn

i

Kni + zn

1

− vmk1ksz3

KM k1ks + z3+ u

k1ksKni

Kni + zn

1

. (11.53)

This model (11.53) is then used for phase control design. We use q to denote the statevariable for the target circadian model

q1 = q2 − k2q1

q2 = q3 − k1q2 + k1k2q1 − vdk1q2

k1Kd + q2

q3 = vsk1ksKn

i

Kni + qn

1

− vmk1ksq3

KM k1ks + q3, (11.54)

to which the variable z is controlled to follow.The transformed dynamic models (11.53) and (11.54) are in the triangular form,

and therefore the iterative backstepping method shown in Section 9.2 can be applied.There is a slight difference in the control objective here from the convergence to zero


in Section 9.2. For this, we define

e1 = z1 − q1,

e2 = z2 − q2, (11.55)

e3 = z3 − q3,

where ei, for i = 1, 2 and 3, are the tracking errors. The error dynamics are obtained as

e1 = e2 − k2e1

e2 = e3 − k1e2 + k1k2e1 − vdk21 Kde2

(k1Kd + z2) (k1Kd + q2)

e3 = vsk1ksKn

i

Kni + zn

1

− vsk1ksKn

i

Kni + qn

1

− vmKM (k1ks)2 e3

(KM k1ks + z3) (KM k1ks + q3)+ u

k1ksKni

Kni + zn

1

.

(11.56)

It is easy to see that the model (11.56) is still in the triangular form. The phase resettingis achieved if we can ensure that the errors ei, i = 1, 2, 3, converge to zero. To themodel (11.56), iterative backstepping can be applied. Following the procedures shownin Section 9.2, we define

w1 = e1,

w2 = e2 − α1, (11.57)

w3 = e3 − α2,

where α1 and α2 are stabilising functions to be designed.From the first equation of (11.56), we obtain dynamics of w1 as

w1 = e2 − k2e1

= w2 + α1 − k2e1.

The stabilising function α1 is then designed as

α1 = −c1w1 + k2e1, (11.58)

where c1 is a positive real constant. The resultant dynamics of w1 are given by

w1 = −c1w1 + w2. (11.59)

The dynamics of w2 are obtained as

w2 = e3 − k1e2 + k1k2e1 − vdk21 Kde2

(k1Kd + z2) (k1Kd + q2)− ∂α1

∂e1e1

= w3 + α2 − k1e2 + k1k2e1 − vdk21 Kde2

(k1Kd + z2) (k1Kd + q2)− ∂α1

∂e1e1.

The stabilising function α2 can be designed as

α2 = −w1 − c2w2 + k1e2 − k1k2e1 + vdk1k1Kde2

(k1Kd + z2) (k1Kd + q2)+ ∂α1

∂e1e1


with c2 being a positive real constant, which results in the dynamics of w2

w2 = −w1 − c2w2 + w3. (11.60)

From the dynamics of w3

w3 = vsk1ksKn

i

Kni + zn

1

− vsk1ksKn

i

Kni + qn

1

+ uk1ksKn

i

Kni + zn

1

− vmKM (k1ks)2 e3

(KM k1ks + z3) (KM k1ks + q3)− ∂α2

∂e1e1 − ∂α2

∂e2e2,

we design the control input u as

u = Kni + zn

1

k1ksKni

[

−w2 − c3w3 − vsk1ksKn

i

Kni + zn

1

+ vsk1ksKn

i

Kni + qn

1

+ ∂α2

∂e1(e2 − k2e1)

+ ∂α2

∂e2(e3 − k1e2 + k1k2e1) − ∂α2

∂e2

vdk21 Kde2

(k1Kd + z2) (k1Kd + q2)

]

, (11.61)

with c3 as a positive real constant. The resultant dynamics of w3 are then obtained as

w3 = −w2 − c3w3. (11.62)

The stability analysis of the proposed control design can be established in thesame way as the proof shown in Section 9.2 for iterative backstepping control design.Indeed, consider a Lyapunov function candidate

V = 1

2

(w2

1 + w22 + w2

3

). (11.63)

Using the dynamics of wi, i = 1, 2, 3, in (11.59), (11.60) and (11.62), we obtain

V = w1w1 + w2w2 + w3w3

= w1 (−c1w1 + w2)+ w2 (−w1 − c2w2 + w3)+ w3 (−w2 − c3w3)

= −c1w21 − c2w2

2 − c3w23.

Therefore, we can conclude that the closed-loop system under the control of u in(11.61) is exponentially stable with respect to the variables wi, i = 1, 2, 3, whichsuggests that the controlled circadian rhythm asymptotically tracks the targeted one.

Simulation study of the control design proposed above has been carried with thecontrol parameters c1 = c2 = c3 = 0.1. The states z2 and z3 are shown in Figures 11.7and 11.8 when the control input is applied at t = 50 h, and the control input is shownin Figure 11.9. The plot of z1 is very similar to the plot of z2, and it is omitted. It canbe seen from the figures that there is a phase difference before the control input isapplied, and the control input resets the phase of the control circadian rhythm to thetargeted one.


q2z2

0 10 20 30 40 50 60 70 80 90 100 110 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (h)

q 2 a

nd z 2

Control input applied

Figure 11.7 Circadian phase reset for state 2

q3z3

q 3 a

nd z 3

0 10 20 30 40 50 60 70 80 90 100 110 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (h)

Control input applied

Figure 11.8 Circadian phase reset for state 3

11.3 Sampled-data control of nonlinear systems

A practical issue of applying nonlinear control strategies is to implement them incomputers or microprocessors which are discrete-time in nature. For a continuous-time system with a discrete-time controller, inevitably we end with a sampled-data


0 10 20 30 40 50 60 70 80 90 100 110 120–30

–25

–20

–15

–10

–5

0

5

Time (h)

Con

trol i

nput

Control input u

Figure 11.9 Control input for circadian phase reset

control system. It is well known for linear systems, we can use emulation method,that is, to design a controller in continuous-time and then implement its discrete-timeversion. Alternative to emulation method is the direct design in discrete-time. Stabilityanalysis for sampled-data control of linear dynamic systems can then be carried out inthe framework of linear systems in discrete-time. Sampled-data control of nonlinearsystems is a much more challenging task.

For a nonlinear system, it is difficult or even impossible to obtain a nonlineardiscrete-time model after sampling. Even with a sampled-data model in discrete-time, the nonlinear model structure cannot be preserved in general. It is well knownthat nonlinear control design methods do require certain structures. For examplebackstepping can be applied to lower-triangular systems, but a lower-triangular systemin continuous-time will not have its discrete-time model in the triangular structure ingeneral after sampling. Hence, it is difficult to carry out directly design in discrete-timefor continuous-time nonlinear systems.

Foracontrol inputdesignedbasedon thecontinuous-timemodel, it canbesampledand implemented in discrete-time. However, the stability cannot be automaticallyguaranteed for the sampled-data system resulted from emulation method. The stabilityanalysis is challenging because there is no corresponding method in discrete-time fornonlinear systems after sampling, unlike linear systems. One would expect that if thesampling is fast enough, the stability of the sampled-data system might be guaranteedby the stability of the continuous-time system. In fact, there is a counter example tothis claim. Therefore, it is important to analyse the stability of sampled-data systemin addition to the stability of the continuous-time control, for which the stability isguaranteed in control design.

In this section, we will show the stability analysis of sampled-data control for aclass of nonlinear systems in the output feedback form. A link between the sampling


period and initial condition is established, which suggests that for a given domainof initial conditions, there always exists a sampling time such that if the sampling isfaster than that time, the stability of the sampled-data system is guaranteed.

11.3.1 System model and sampled-data control

In this section, we consider sampled-data control for a class of nonlinear systems inthe output feedback form. This class of nonlinear systems (9.17) has been consideredfor control design in continuous-time in Chapter 9, using observer backstepping inSection 9.3 and using backstepping with filtered transformation in Section 9.4. We willpresent sampled-data control based on the control input in continuous-time obtainedusing backstepping with filtered transformation. For the convenience of presentation,we will show the system again and outline the key steps in the continuous-time here.The system, as shown in (9.17) and (9.32), is described as


y = Cx,(11.64)

with

Ac =

⎡

⎢⎢⎢⎢⎢⎣

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 10 0 0 . . . 0

⎤

⎥⎥⎥⎥⎥⎦

, C =

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

T

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0bρ...

bn

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where x ∈ Rn is the state vector; u ∈ R is the control input; φ : R → R

n withφ(0) = 0is a nonlinear function with element φi being differentiable up to the (n − i)th order;and b ∈ R

n is a known constant Hurwitz vector with bρ �= 0, which implies the relativeof the system is ρ.

We design an output feedback control for the system in (11.64). As shown inChapter 9, the continuous-time control input can be designed using the backstep-ping with filtered transformation in Section 9.4. In this section, we carry out thestability analysis based on the control input obtained using backstepping with filteredtransformation.

In this section, we need to deal with control in continuous-time and discrete-time.For notation, we use uc for the control input obtained in continuous-time and ud forthe resultant sampled-data control input. For notational convenience, we re-write thefilter (9.33) as

ξ = �ξ + bf u (11.65)

u = uc(y, ξ ), (11.66)


where

� =

⎡

⎢⎢⎢⎣

−λ1 1 0 . . . 00 −λ2 1 . . . 0...

......

......

0 0 0 . . . −λρ−1

⎤

⎥⎥⎥⎦

, bf =

⎡

⎢⎢⎢⎣

0...

01

⎤

⎥⎥⎥⎦.

For a given uc, the sampled-data controller based on emulation method is given as

ud(t) = uc(y(mT ), ξ (mT )), ∀t ∈ [mT , mT + T ), (11.67)

ξ (mT ) = e�T ξ ((m − 1)T ) + bf uc(y((m − 1)T ), ξ ((m − 1)T ))∫ T

0e�τdτ , (11.68)

where y(mT ) is obtained by sampling y(t) at each sampling instant; ξ (mT ) is thediscrete-time implementation of the filter shown in (11.65); T is the fixed samplingperiod; and m is the discrete-time index, starting from 0.

Before the analysis of the sampled-data control, we briefly review the controldesign in continuous-time.

As shown in Section 9.4, through the state transformations, we obtain

ζ = Dζ + ψ(y)

y = ζ1 + ψy(y) + bρξ1.(11.69)

For the system with relative degree ρ = 1, as shown in Lemma 9.5 and its proof,the continuous-time control uc1 can be designed as

uc1 = −(

c0 + 1

4

)

y − ψy(y) − ‖P‖2‖ψ(y)‖2

y, (11.70)

where c0 is a positive real constant and P is a positive real constant that satisfies

DT P + PD = −3I .

For the control (11.70), the stability of the continuous-time system can be establishedwith the Lyapunov function

V = ζ T Pζ + 1

2y2

and its derivative

V ≤ −c0y2 − ‖ζ‖2.

For the case of ρ > 1, backstepping is used to obtain the final control input uc2.We introduce the same notations zi, for i = 1, . . . , ρ, as in Section 9.4 for backstepping

z1 = y, (11.71)

zi = ξi−1 − αi−1, for i = 2, . . . , ρ, (11.72)

zρ+1 = u − αρ , (11.73)

where αi for i = 2, . . . , ρ are stabilising functions to be designed. We also use thepositive real design parameters ci and ki for i = 1, . . . , ρ and γ > 0.


The control design has been shown in Section 9.4. We list a few key steps herefor the convenience of the stability analysis. The stabilising functions are designed as

α1 = −c1z1 − k1z1 + ψy(y) − γ ‖P‖2‖ψ(y)‖2

y,

α2 = −z1 − c2z2 − k2

(∂α1

∂y

)2

z2 + ∂α1

∂yψ(y) + λ1ξ1


(∂αi−1

∂y

)2

zi + ∂αi−1

∂yψ(y) + λi−1ξi−1

+ ∑i−2j=1

∂αi−1

∂ξj(−λjξj + ξj+1) for i = 3, . . . , ρ.

(11.74)

The continuous-time control input is given by uc2 = αρ , that is

uc2 = −zρ−1 − cρzρ − kρ

(∂αρ−1

∂y

)2

zρ + ∂αρ−1

∂yψ(y)

+i−2∑

j=1

∂αρ−1

∂ξj(−λjξj + ξj+1) + λρ−1ξρ−1. (11.75)

For the control input (11.75) in the continuous-time, the stability result has beenshown in Theorem 9.6. In the stability analysis, the Lyapunov function candidate ischosen as

V =ρ∑

i=1

z2i + γ ζ T Pζ

and its derivative is shown to satisfy

V ≤ −ρ∑

i=1

ciz2i − γ ‖ζ‖2. (11.76)

11.3.2 Stability analysis of sampled-data systems

The following lemma is needed for stability analysis in sampled-data case.

Lemma 11.10. Let V : Rn → R

+ be a continuously differentiable, radiallyunbounded, positive definite function. Define D := {χ ∈ R

n|V (χ ) ≤ r} with r > 0.Suppose

V ≤ −μV + βVm, ∀t ∈ (mT , (m + 1)T ], (11.77)


hold for allχ (mT ) ∈ D, whereμ, β are any given positive reals withμ > β, T > 0 thefixed sampling period and Vm := V (χ (mT )). If χ (0) ∈ D, then the following holds:

limt→∞χ (t) = 0.

Proof. Since χ (0) ∈ D, then (11.77) holds for t ∈ (0, T ] with the following form:

V ≤ −μV + βV (χ (0)).

Using the comparison lemma (Lemma 4.5), it is easy to obtain from the above thatfor t ∈ (0, T ]

V (χ (t)) ≤ e−μtV0 + 1 − e−μt

μβV0 = q(t)V0, (11.78)

where q(t) := (e−μt + β

μ(1 − e−μt)). Since μ > β > 0, then q(t) ∈ (0, 1), ∀t ∈

(0, T ]. Then we have

V (χ (t)) < V0, ∀t ∈ (0, T ]. (11.79)

Particularly, setting t = T in (11.78) leads to

V1 ≤ q(T )V0, (11.80)

which means that χ (T ) ∈ D. Therefore, (11.77) holds for t ∈ (T , 2T ]. By induction,we have

V (χ (t)) < Vm, ∀t ∈ (mT , (m + 1)T ]. (11.81)

which states inter-sample behaviour of the sampled-data system concerned, and inparticular

Vm+1 ≤ q(T )Vm (11.82)

indicating that V decreases at two consecutive sampling points with a fixed ratio.From (11.82),

Vm ≤ q(T )Vm−1 ≤ qm(T )V0, (11.83)

which implies that limm→∞ Vm = 0. The conclusion then follows from (11.81), whichcompletes the proof. �

Remark 11.6. Lemma 11.10 plays an important role in stability analysis for thesampled-data system considered in this section. When sampled-data systems areanalysed in discrete-time in literature, only the performances at sampling instances areconsidered. In this section, Lemma 11.10 provides an integrated analysis frameworkwhere the behaviour of the sampled-data system both at and between sampling instantscan be characterised. In particular, the system’s behaviour at sampling instants isportrayed in (11.82), and (11.81) shows that the inter-sample behaviour of the systemis well bounded by a compact set defined by Vm. �


For the case of relative degree 1, the sampled-data system takes the followingform:

ζ = Dζ + ψ(y)y = ζ1 + ψy(y) + ud1,

(11.84)

where ud1 is the sampled-data controller and can be simply implemented via a zero-order hold device as the following:

ud1(t) = uc1(y(m)), ∀t ∈ [mT , mT + T ). (11.85)

Define χ := [ζ T , yT ]T and we have the following result.

Theorem 11.11. For system (11.84) and the sampled-data controller ud1 shown in(11.85), and a given neighbourhood of the origin Br := {χ ∈ R

n| ‖χ‖ ≤ r} with rany given positive real, there exists a constant T1 > 0 such that, for all 0 < T < T1

and for all χ (0) ∈ Br, the system is asymptotically stable.

Proof. We choose

V (χ ) = ζ T Pζ + 1

2y2

as the Lyapunov function candidate for the sampled-data system. We start with somesets used throughout the proof. Define

c := maxχ∈Br

V (χ )

and the level set

�c := {x ∈ Rn|V (χ ) ≤ c}.

There exist two class K functions �1 and �2 such that

�1(‖χ‖) ≤ V (χ ) ≤ �2(‖χ‖).

Let l > �−11 (c), and define Bl := {χ ∈ R

n| ‖χ‖ ≤ l}. Then we have

Br ⊂ �c ⊂ Bl.

The constants Lu1, L1 and L2 are Lipschitz constants of the functions uc1 and ψy andψ with respect to Bl .

These local Lipschitz conditions establish that for the overall sampled-data systemwith χ (0) ∈ �c, there exists a unique solution χ (t) over some interval [0, t1). Noticethat t1 might be finite. However, later analysis shows that the solution can be extendedone sampling interval after another, and thus exists for all t ≥ 0 with the propertythat limt→∞ χ (t) = 0. Particularly, we intend to formulate the time derivative of theLyapunov function V into the form shown in (11.77) or (11.93), which is shown below.

Consider the case when t = 0, χ (0) ∈ Br ⊂ �c. First, choose a sufficiently largel such that there exists a T ∗

1 > 0, and for all T ∈ (0, T ∗1 ), the following holds:

χ (t) ∈ Bl , ∀t ∈ [0, T ], χ (0) ∈ �c. (11.86)


The existence of T ∗1 is ensured by continuous dependency of the solution χ (t) on the

initial conditions.Next, calculate the estimate of |y(t) − y(0)| forced by the sampled-data control

ud1 during the interval [0, T ], provided that χ (0) ∈ Br ⊂ �c and T ∈ (0, T ∗1 ). From

the second equation of (11.84), the dynamics of y are given by

y = ζ1 + ψy(y) + ud1.

It follows that

y(t) = y(0) +∫ t

0ζ1(τ )dτ +

∫ t

0ud1(τ )dτ

+∫ t

0

(ψy(y) − ψy (y(0))

)dτ +

∫ t

0ψy (y(0)) dτ.

Then we have

|y(t) − y(0)| ≤∫ t

0‖ζ (τ )‖dτ

︸︷︷︸�1

+∫ t

0Lu1|y(0)|dτ

+∫ t

0L1|y(τ ) − y(0)|dτ +

∫ t

0L1|y(0)|dτ. (11.87)

We first calculate the integral�1. From the first equation of system (11.69), we obtain

ζ (t) = eDtζ (0) +∫ t

0eDtψ(y(τ ))dτ. (11.88)

Since D is a Hurwitz matrix, there exist positive reals κ1 and σ1 such that ‖eDt‖ ≤κ1e−σ1t . Thus, from (11.88)

‖ζ (t)‖ ≤ κ1e−σ1t‖ζ (0)‖ +∫ t

0κ1e−σ1(t−τ )‖ψ(y(τ ))− ψ(y(0))‖dτ

+∫ t

0κ1e−σ1(t−τ )‖ψ(y(0))‖dτ

≤ κ1e−σ1t‖ζ (0)‖ + L2

∫ t

0κ1e−σ1(t−τ )|y(τ ) − y(0)|dτ

+ L2

∫ t

0κ1e−σ1(t−τ )|y(0)|dτ.

Then the following inequality holds:

�1 ≤ κ1‖ζ (0)‖σ1

(1 − e−σ1t)+ κ1L2

σ1|y(0)|t

+ κ1L2

σ1

∫ t

0|y(τ ) − y(0)|dτ. (11.89)


Now we are ready to compute |y(t) − y(0)|. In fact, we have from (11.87) and (11.89)

|y(t) − y(0)| ≤ A1(1 − e−σ1t)+ B1t + H∫ t

0|y(τ ) − y(0)|dτ , (11.90)

where

A1 = σ−11 κ1‖ζ (0)‖,

B1 = Lu1|y(0)| + L1|y(0)| + σ−11 κ1L2|y(0)|,

H = σ−11 κ1L2 + L1.

Applying Gronwall–Bellman inequality to (11.90) produces

|y(t) − y(0)| ≤ A1(1 − e−σ1t)+ B1

H(eHt − 1)

+ A1(σ1eHt + He−σ1t − (H + σ1))(H + σ1)−1 (11.91)

Setting t = T on the right side of (11.91) leads to

|y(t) − y(0)| ≤ δ1(T )|y(0)| + δ2(T )‖ζ (0)‖,

where

δ1(T ) = H−1(Lu1 + L1 + σ−11 κ1L2)(eHT − 1)

δ2(T ) = σ−11 κ1(σ1eHT + He−σ1T − (H + σ1))(H + σ1)−1

+ σ−11 κ1(1 − e−σ1T ). (11.92)

Note that δ1(T ) and δ2(T ) only depend on the sampling period T once the Lipschitzconstants and the control parameters are chosen.

Next we shall study the behaviour of the sampled-data system during each inter-val using a Lyapunov function candidate V (y, ζ ) = ζ TPζ + 1

2 y2. Consider χ (0) =[ζ (0), y(0)]T ∈ Br . When t ∈ (0, T ], its time derivative satisfies

V = −3‖ζ‖2 + 2ζ TPψ + y(ζ1 + ψy + ud1)

≤ −c0y2 − ‖ζ‖2 + |y||ud1(y(0)) − uc1|≤ −c0y2 − ‖ζ‖2 + Lu1|y − y(0)||y|≤ −

(

c0 − Lu1

2(δ1(T ) + δ2(T ))

)

y2 − ‖ζ‖2

+ Lu1

2δ1(T )|y(0)|2 + Lu1

2δ2(T )‖ζ (0)‖2

≤ −μ1(T )V + β1(T )V (ζ (0), y(0)), (11.93)


where

μ1 = min{

2c0 − Lu1δ1 − Lu1δ2(T ),1

λmax(P)

}

,

β1 = max{

Lu1δ1(T ),Lu1δ2(T )

2λmin(P)

}

,

(11.94)

withλmax( · ) andλmin( · ) denoting the maximum and minimum eigenvalues of a matrixrespectively.

Next we shall show that there exists a constant T ∗2 > 0 such that the condi-

tion μ1(T ) > β1(T ) > 0 is satisfied for all 0 < T < T ∗2 . Note from (11.92) that both

δ1(T ) and δ2(T ) are actually the continuous functions of T with δ1(0) = δ2(0). Definee1(T ) := μ1(T ) − β1(T ) and we have e1(0) > 0. It can also be established from(11.94) that e1(T ) is a decreasing and continuous function of T , which asserts bythe continuity of e1(T ) the existence of T ∗

2 so that for 0 < T < T ∗2 , e1(T ) > 0, that is

0 < β1(T ) < μ1(T ).Finally, set T1 = min (T ∗

1 , T ∗2 ), and from Lemma 11.10, it is known that V1 ≤ c,

which means χ (T ) ∈ �c, and subsequently, all the above analysis can be repeated forevery interval [mT , mT + T ]. Applying Lemma 11.10 completes the proof. �

For systems with relative degree ρ > 1, the implementation of the sampled-datacontroller ud2 is given in (11.67) and (11.68). It is easy to see from (11.68) that ξ (mT )is the exact, discrete-time model of the filter

ξ = −�ξ + bf ud2 (11.95)

due to the fact that ud remains constant during each interval and the dynamics of ξshown in (11.95) is linear. Then (11.68) and (11.95) are virtually equivalent at eachsampling instant. This indicates that we can use (11.95) instead of (11.68) for stabilityanalysis of the sampled-data system.

Let χ := [ζ T , zT ]T and we have the following result.

Theorem 11.12. For the extended system consisting (11.65), (11.69) and thesampled-data controller ud2 shown in (11.67) and (11.68), and a given neighbour-hood of the origin Br := {χ ∈ Rn| ‖χ‖ ≤ r} with r any given positive real, there existsa constant T2 > 0 such that, for all 0 < T < T2 and for all χ (0) ∈ Br, the system isasymptotically stable.

Proof. The proof can be carried out in a similar way to that for the case ρ = 1,except that the effect of the dynamic filter of ξ has to be dealt with.

For the overall sampled-data system, a Lyapunov function candidate is chosenthe same as for continuous-time case, that is

V = γ ζ T Pζ + 1

2

ρ∑

i=1

z2i .


Similar to the case of ρ = 1, the sets Br , �c and Bl can also be defined such thatBr ⊂ �c ⊂ Bl , and there exists a T ∗

3 > 0 such that for all T ∈ (0, T ∗3 ), the following

holds:

χ (t) ∈ Bl , ∀t ∈ (0, T ], χ (0) ∈ �c.

As in the proof for Theorem 11.11, we also aim to formulate the time derivativeof V (χ ) into form (11.77). Next we shall derive the bounds for ‖ξ (t) − ξ (0)‖ and|y(t) − y(0)| during t ∈ [0, T ] with 0 < T < T ∗

3 . Consider the case where χ (0) ∈ Br .We have from (11.95)

ξ (t) = e�tξ (0) +∫ t

0e�(t−τ )bf ud2dτ. (11.96)

Since � is a Hurwitz matrix, there exist positive reals κ2, κ3 and σ2 such, that

‖e�t‖ ≤ κ2e−σ2t ,

‖e�t − I‖ ≤ κ3(1 − e−σ2t),

where I is the identity matrix. Then, using the Lipschitz property of uc2 with respectto the set Bl and the fact that ud2(0, 0) = 0, it can be obtained from (11.96) that

∫ t

0‖ξ (τ )‖dτ ≤ κ2‖ξ (0)‖

σ2

(1 − e−σ2t

) + κ2Lu2

σ2(|y(0)| + ‖ξ (0)‖)t (11.97)

and

‖ξ (t) − ξ (0)‖ ≤ κ3‖ξ (0)‖(1 − e−σ2t) + ‖ud2 (y(0), ξ (0)) ‖∫ t

0κ2e−σ2(t−τ )dτ

≤ δ3(T )|y(0)| + δ4(T )‖ξ (0)‖, (11.98)

where

δ3(T ) = σ−12 κ2Lu2(1 − e−σ2T ),

δ4(T ) = (κ3 + σ−12 κ2Lu2)(1 − e−σ2T ),

and Lu2 is a Lipschitz constant of uc2. As for |y − y(0)|, we have from (11.69)

y(t) = y(0) +∫ t

0ζ1(τ )dτ +

∫ t

0ξ (τ )dτ

+∫ t

0

(ψy(y) − ψy(y(0))

)dτ +

∫ t

0ψy(y(0))dτ.

It can then be shown that

|y(t) − y(0)| ≤∫ t

0‖z(τ )‖dτ

︸︷︷︸�1

+∫ t

0‖ξ (τ )‖dτ

︸︷︷︸�2

+∫ t

0L1|y(τ ) − y(0)|dτ +

∫ t

0L1|y(0)|dτ , (11.99)


where �1 is already shown in (11.89) and �2 in (11.97). With (11.89), (11.97) and(11.99), it follows that

|y(t) − y(0)| ≤ A1(1 − e−σ1t) + A2(1 − e−σ2t) + B2t

+ H∫ t

0|y(τ ) − y(0)|dτ (11.100)

where A1 and H are defined in (11.90)

A2 = σ−12 κ2‖ξ (0)‖,

B2 = L1|y(0)| + σ−11 κ1L2|y(0)| + σ−1

2 κ2Lu2|y(0)| + σ−12 κ2Lu2‖ξ (0)‖.

Defining A3 := A1 + A2 and σ0 := max (σ1, σ2), we have

|y(t) − y(0)| ≤ A3(1 − e−σ0t) + B2t + H∫ t

0|y(τ ) − y(0)|dτ.

Applying the Gronwall–Bellman lemma produces

|y(t) − y(0)| ≤ δ5(T )|y(0)| + δ6(T )‖z(0)‖ + δ7(T )‖ξ (0)‖, (11.101)

where

δ5(T ) = H−1(L1 + σ−11 κ1L2 + σ2κ

−12 Lu2)(eHT − 1),

δ6(T ) = σ−11 κ1(σ0eHT + He−σ0T − (H + σ0))(H + σ0)−1 + σ−1

1 κ1(1 − e−σ0T ),

δ7(T ) = σ−12 κ2(1 − e−σ0T ) + σ−1

2 κ2Lu2(eHT − 1)

+ σ−12 κ2Lu2(σ0eHT + He−σ0T − (H + σ0))(H + σ0)−1.

Note that ‖ξ (0)‖ appears in (11.98) and (11.101) while for the analysis using Lyapunovfunction to carry on, ‖z(0)‖ is needed. Therefore, it is necessary to find out anexpression of ‖ξ (0)‖ that exclusively involves ‖z(0)‖, which is shown below.

Notice that due to the special structure of the filter (11.65) and the backsteppingtechnique, each stabilising function has the property that α1 = α1(y), α1(0) = 0, andαi = αi(y, ξ1, . . . , ξi−1) and αi(0, . . . , 0) = 0, i = 2, . . . , ρ − 1. From (11.74) we have

|ξ1(0)| ≤ |z2(0)| + |α1(0)| ≤ z2(0)| + L1|y(0)||ξ2(0)| ≤ |z3(0)| + |α2(0)| ≤ |z3(0)| + L2|y(0)| + L2|ξ1(0)|

...

|ξρ−1(0)| ≤ |zρ(0)| + |αρ−1(0)| ≤ |zρ(0)| + Lρ−1|y(0)| + Lρ−1

ρ−2∑

i=1

|ξi(0)|

where with a bit abuse of notation, Li is the Lipschitz constant of ξi with respect tothe set Bl . Thus, a constant L0 can be found such that the following holds:

‖ξ (0)‖ ≤ L0(‖z(0)‖ + |y(0)|). (11.102)


with z = [z2, . . . , zρ]T , which implies that if [ζ (0)T , y(0)T , z(0)T ]T ∈ Bl , then[ζ (0)T , y(0)T , ξ (0)T ]T will be confined in a bounded set, denoted by B′

l .Then the time derivative of

Vd2 = γ ζ T Pζ + 1

2y2 + 1

2

ρ∑

i=2

z2i

during the interval (0, T ] satisfies

Vd2 ≤ −γ ‖ζ‖2 −ρ∑

i=1

ciz2i + zρ(ud2 − uc2)

≤ −c1y2 − γ ‖ζ‖2 − λ0

ρ∑

i=2

z2i + ‖z‖|ud2 − uc2|, (11.103)

where λ0 = min{c2, . . . , cρ}. In addition, we have

‖ξ‖|ud2 − uc2| ≤ Lu2‖ξ‖(|y − y(0)| + ‖ξ − ξ (0)‖)

≤ Lu2‖ξ‖(δ3(T ) + δ5(T ))|y(0)| + Lu2‖ξ‖δ6(T )‖ζ (0)‖

+ Lu2‖ξ‖(δ4(T ) + δ7(T ))‖ξ (0)‖

≤ ε1(T )|y(0)|2 + ε2(T )‖ζ (0)‖2 + ε3(T )‖ξ (0)‖2

+ ε4(T )‖z‖2, (11.104)

where ε1(T ) = Lu22 (δ3(T ) + δ5(T )), ε2 = Lu2

2 δ6(T ), ε3 = Lu22 (δ4(T ) + δ7(T )), ε4 =

Lu22

∑7i=3 δi(T ) and Lu2 is a Lipschitz constant of uc2 with respect to the set B′

ldependent on Bl .

From (11.102) to (11.103), we then have

Vd2 ≤ −c1y2 − γ ‖ζ‖2 − (λ0 − ε4)z2

+ (ε1(T ) + 2L20ε3(T ))|y(0)|2 + ε2(T )‖ζ (0)‖2 + 2L2

0ε3(T )‖z(0)‖2

= −α2(T )Vd2 + β2(T )Vd2(ζ (0), y(0), z(0)),

where

α2(T ) = min{

2c1,γ

λmax(P), 2(λ0 − ε4(T ))

}

,

β2(T ) = max{

2(ε1(T ) + 2L20ε3(T )),

ε2(T )

λmin(P), 4L2

0ε3(T )}

.


Note from (11.98), (11.101) and (11.104) that each εi (1 ≤ i ≤ 4) is a continuousfunction of T with εi(0) = 0. Then, as shown in the analysis for the case ρ = 1, itcan be claimed that given arbitrary positive reals c1, γ and λ, there exists a constantT ∗

4 > 0 such that for all 0 < T < T ∗4 , α2(T ) > β2(T ) > 0.

Finally, letting T2 := min (T ∗3 , T ∗

4 ) proves the theorem. �

Remark 11.7. In the results shown in Theorems 11.11 and 11.12, the radius r of Br

can be any positive value. It means that we can set the stability region as large as welike, and the stability can still be guaranteed by determining a fast enough samplingtime. Therefore, Theorems 11.11 and 11.12 establish the semi-global stability of thesampled-data control of nonlinear systems in the output feedback form. �

11.3.3 Simulation

The following example is a simplified model of a jet engine without stall:

φm = ψ + 3

2φ2

m − 1

2φ3

m

ψ = u,

where φm is the mass flow and ψ the pressure rise. Take φm as the output and then theabove system is in the output feedback form. The filter ξ = −λξ + u is introduced sothat the filtered transformation y = x1 and ζ = x2 − ξ , and the state transformationζ = ζ − λy can render the system into the following form:

ζ = −λζ − λ

(3

2y2 − 1

2y3

)

− λ2y

y = ζ + λy +(

3

2y2 − 1

2y3

)

+ ξ.

Finally, the stabilising function

α1 = −y −(

3

2y2 − 1

2y3

)

− λ2y

(3

2y − 1

2y2 + λ

)2

with P = 1, and the control uc can be obtained using α2 as shown in (11.74). Forsimulation, we choose λ = 0.5.

Simulations are carried out with the initial values set as x1(0) = 1 andx2(0) = 1, which are for simulation purpose only and have no physical meaning.Results shown in Figures 11.10 and 11.11 indicate that the sampled-data systemis asymptotically stable when T = 0.01 s, which is confirmed by a closer lookat the convergence of V shown in Figure 11.12. Further simulations show thatthe overall system is unstable if T = 0.5 s. In summary, the example illustratesthat for a range of sampling period T , the sampled-data control design presentedearlier in this section can asymptotically stabilise the sampled-data system.


0 1 2 3 4 5 6 7 8 9 10–0.2

0

0.2

0.4

0.6

0.8

1

1.2

t (s)

x 1

Figure 11.10 The time response of x1 for T = 0.01 s

–0.2 0 0.2 0.4 0.6 0.8 1 1.2–2

–1.5

–1

–0.5

0

0.5

1

x1

x 2

Figure 11.11 Phase portrait of the system for T = 0.01 s


–0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t (s)

V

Figure 11.12 The convergence of V for T = 0.01 s

Bibliographical Notes

Chapter 1. The basic concepts of systems and states of nonlinear systems dis-cussed in this chapter can be found in many books on nonlinear systems andnonlinear control systems, for example Cook [1], Slotine and Li [2], Vidyasagar[3], Khalil [4] and Verhulst [5]. For the background knowledge of linear systems,readers can consult Ogata [6], Kailth [7]. Antsaklis and Michel [8], Zheng [9],and Chen, Lin and Shamash [10]. The existence of unique solutions of differen-tial equations is discussed in the references such as Arnold [11] and Borrelli andColeman [12]. Many concepts such as stability and backstepping control design arecovered in detail in later chapters of the book. For system with saturation, read-ers may consult Hu and Lin [13]. Sliding mode control is covered in detail inEdwards and Spurgeon [14] and feedforward control in Isidori [15]. Limit cyclesand chaos are further discussed in Chapter 2. Semi-global stability is often relatedto systems with saturation, as discussed in Hu and Lin [13], and Isidori [15].

Chapter 2. Lipschitz conditions and the existence of solutions for differentialequations are covered in detail in Arnold [11] and Borrelli and Coleman [12]. Classi-fications of singular points of second-order nonlinear systems are covered in almostall the introductory texts on nonlinear control systems, Cook [1], Slotine and Li[2], Vidyasagar [3] and Khalil [4]. More phase portraits of systems similar to theone in Example 2.1 can be found in Slotine and Li [2], Khalil [4] and Verhulst [5].The swing machine equation in Example 2.2 was a simplified model of the systemshown in Elgerd [16] and a similar treatment is shown in Cook [1]. More examplesof similar systems can also be found in Verhulst [5]. The van der Pol oscillator wasfirst discussed in his publication van der Pol [17], and perhaps it is one of the bestknown systems with limit cycles. Detailed coverage of van der Pol oscillator can befound in several books, including Verhulst [5] and Borrelli and Coleman [12]. Lorenzattractor was first shown in Lorenz [18] for his research on heat transfer. The bound-edness of the trajectory shown in the book was taken from Borrelli and Coleman [12].

Chapter 3. Describing function analysis is covered in many classic texts on non-linear control, such as Cook [1], Slotine and Li [2], Vidyasagar [3], and Khalil [4].The stability criterion of limit cycles with describing functions is influenced by thetreatment in Slotine and Li [2]. The describing function analysis for van der Pol sys-tems is adapted from Cook [1]. More results on describing functions are to be foundin Atherton [19] and Gelb and Velde [20].


Chapter 4. Lyapunov’s work on stability was published in Russian in 1892 andwas translated in French in 1907 [21]. Lyapunov stability is covered in almost all thebooks on nonlinear systems and control. One of the early books on motion stability ispublished by Hahn [22]. Example 4.4 is adapted from Slotine and Li [2], and a moregeneral form is shown in Reference [5]. More discussions on radially unboundedfunctions and global stability can be found in Khalil [4] and in Slotine and Li [2].Further results on comparison lemma and its applications are to be found in Khalil[4] and Vidyasagar [3].

Chapter 5. The definition of positive real systems are common in many bookssuch as Slotine and Li [2], Khalil [4] and Vidyasagar [3]. For strictly positive realsystems, there may have some variations in definitions. A discussion on the varia-tions can be found in Narendra and Annaswamy [23]. The definition in this bookfollows Slotine and Li [2], Khalil [4]. Lemma 5.3 was adapted from Khalil [4],where a poof of necessity can also be found. Kalman–Yakubovich Lemma can appearin different variations. A proof of it can be found in Khalil [4] and Marino andTomei [24]. A book on absolute stability was published by Narendra and Taylor [25].Circle criterion is covered in a number of books such as Cook [1], Slotine and Li[2], Vidyasagar [3] and Khalil [4]. The treatment of loop-transformation to obtainthe condition of circle criterion is similar to Cook [1]. The interpretation based onFigure 5.3 is adapted from Khalil [4]. Complex bilinear mapping is also used tojustify the circle interpretation. Basic concepts of complex bilinear mapping canbe found from text books on complex analysis, such as Brown and Churchill [26].Input-to-state stability was first introduced by Sontag [27]. For the use of comparisonfunctions of classes K and K∞, more are to be found in Khalil [4]. The defini-tions for ISS are adapted from Isidori [15]. Theorem 5.8 is a simplified version, byusing powers of the state norm, rather than K∞ to characterise the stability for theconvenience of presentation. The original result of Theorem 5.8 was shown by Son-tag and Wang [28]. Using ISS pair (α, σ ) for interconnected systems is based ona technique, changing supply functions, which is shown in Sontag and Teel [29].The small gain theorem for ISS was shown in Jiang, Teel and Praly [30]. A sys-tematic presentation of ISS stability issues can be found in Isidori [15]. Differentialstability describes the stability issue of nonlinear systems for observer design. It isintroduced in Ding [31], and the presentation in Section 5.4 is adapted from thatpaper.

Chapter 6. Fundamentals of Lie derivatives and differential manifolds can befound in Boothby [32]. A brief description of these concepts, sufficient for theconcepts used in this book, is presented in Isidori [33]. Early results on exact lineari-sation in state space started from Brockett [34], and the presentation in this chapterwas greatly influenced by the work of Isidori [33] and Marino and Tomei [24]. Aproof of Frobenius Theorem can be found in Isidori [33].

Chapter 7. For self-tuning control, readers can refer to the textbooks such asAstrom and Wittenmark [35] and Wellstead and Zarrop [36]. The approach to obtain


the model reference control is based on Ding [37]. Early work of using Lyapunovfunction for stability analysis can be found in Parks [38]. A complete treatment ofMRAC of linear systems with high relative degrees can be found in Narendra andAnnaswamy [23] and Ioannou and Sun [39]. The example to show the divergenceof parameter estimation under a bounded disturbance is adapted from Ioannou andSun [39], where more robust adaptive laws for linear systems are to be found. Resultson robust adaptive control and on relaxing assumptions of the sign of high-frequencygain, minimum-phase, etc., for nonlinear systems can be found in Ding [37, 40–45].

Chapter 8. Linear observer design is covered in many books on linear systems.A proof of Lemma 8.2 can be found in Zheng [9]. Early results on nonlinearobserver design can be traced back to 1970s, for example in Thau [46], and Kou,Ellitt and Tarn [47]. The result on the output injection form was shown in Krenerand Isidori [48]. The geometric conditions for the existence of a state transforma-tion to the output injection form can also be found in Isidori [33] and Marino andTomei [24]. Basic knowledge of differential manifolds can be found in Boothby[32]. Linear observer error dynamics via direct state transformation was initiallyshown by Kazantzis and Kravaris [49]. Further results on this topic can be foundin Xiao [50] and Ding [51]. In the latter one, polynomial approximation to non-linear functions for a solution to a nonlinear state transformation is discussed andan explicit region of convergence of such an approximation is given. The basicidea of observer design for systems with Lipschitz nonlinearities is to use the lin-ear part of the observer dynamics to dominate the nonlinear terms. This idea wasshown in some of the early results on nonlinear observers, but the systematic intro-duction to the topic was due to Rajamani [52], in which the condition (8.31) wasshown. Some later results on this topic are shown in Zhu and Han [53]. One-sided Lipschtiz condition for nonlinear observer design was shown in Zhao, Taoand Shi [54]. The result on observer design for systems with Lipschitz output non-linearities is adapted from Ding [55]. The result on reduced-order observer of linearsystems can be traced back to the very early stage of observer design in Luenberger[56]. Early results on this topic were discussed in Kailath [7]. The reduced-orderobserver design is closely related to observer design for systems with unknowninputs, of which some results may be found in Hou and Muller [57]. For nonlin-ear systems, a reduced-order observer was used for estimation of unknown statesfor control design of a class of nonlinear systems with nonlinearities of unmea-sured states by Ding [58]. The result shown here is based on Ding [31]. Anearly result on adaptive observer for nonlinear systems is reported in Bastin andGevers [59]. Results on adaptive observers with output nonlinearities are coveredin Marino and Tomei [24]. The results shown in Section 8.6 is mainly based on Choand Rajamani [60]. The method to construct a first-order positive real system inRemark 8.9 is based on Ding [40].

Chapter 9. Early ideas of backstepping appeared in a number of papers, suchas Tsinias [61]. Iterative backstepping for state feedback is shown in Kanella-kopoulos, Kokotovic and Morse [62] with adaptive laws for unknown parameters.


Backstepping designs without adaptive laws shown in this chapter are basically sim-plified versions of their adaptive counterparts. Filtered transformations were used forbackstepping in Marino and Tomei [63], and the presentation in this chapter is dif-ferent from the forms used in [63]. Initially, multiple estimation parameters are usedfor adaptive control with backstepping, for example the adaptive laws in [63]. Tun-ing function method was first introduced in Krstic, Kanellakopoulos and Kokotovic[64] to remove multiple adaptive parameters for one unknown parameter vector withbackstepping. Adaptive backstepping with filtered transformation was based on Ding[65], without the disturbance rejection part. Adaptive observer backstepping is largelybased on the presentation in Krstic, Kanellakopoulos and Kokotovic [66]. Nonlinearadaptive control with backstepping is also discussed in details in Marino and Tomei[24]. Some further developments of nonlinear adaptive control using backstepping canbe found in Ding [40] for robust adaptive control with dead-zone modification, robustadaptive control with σ -modification in Ding [67], with unknown control directionsand unknown high-frequency gains in Ding [41, 42]. Adaptive backstepping was alsoapplied to nonlinear systems with nonlinear parameterisation in Ding [68].

Chapter 10. Early results on rejection of sinusoidal disturbances can be foundin Bodson, Sacks and Khosla [69] and Bodson and Douglas [70]. Asymptoticrejection of sinusoidal disturbances for nonlinear systems in the output feedbackform is reported in Ding [71] when the frequencies are known. The result shownin Section 10.1 is adapted mainly from Ding [72]. Asymptotic rejection of peri-odic disturbances can be formulated as an output regulation problem when theperiodic disturbances are generated from a dynamic system, which is known asan exosystem. Output regulation for linear systems is shown in Davison [73]and Francis [74], and for nonlinear systems with local stability in Huang andRugh [75], and Isidori and Byrnes [76]. More results on output regulation withlocal stability can be found in the books by Isidori [33] and Huang [77]. Theresult shown in Section 10.2 is based on Ding [65]. The format of the exosys-tem for internal model design was inspired by Nikiforov [78]. The Nussbaumgain was first used by Nussbaum [79], and the treatment here is similar to Yeand Ding [80] used in the control design to tackle the unknown sign of high fre-quency gain. Early results on output regulation with nonlinear exosystems canbe found in Ding [81, 82], and Chen and Huang [83]. The result shown inSection 10.3 is based on Xi and Ding [84]. Asymptotic rejection of general dis-turbances was studied as an extension of output regulation of nonlinear exosystems.Waveform profiles of the general periodic disturbances were used to estimate theequivalent input disturbance in some early results in Ding [85–87]. The approachfor asymptotic rejection with an observer-like internal model is adapted fromDing [55, 88].

Chapter 11. Harmonic estimation is treated as a special case of state estimation.The recursive estimation and rejection method in Section 11.1 is based on Ding[89]. Practical issues on harmonics estimation and rejections are covered in Arrillaga,Watson and Chen [90] and Wakileh [91].


Clock gene was reported in 1994 by Takahashi et al. [92]. Reports on key genesin circadian oscillators are in Albrecht et al. [93], van der Horst et al. [94] and Bungeret al. [95]. The effects of jet lags and sleep disorders to circadian rhythms are reportedin Sack et al. [96] and Sack et al. [97]. The effects of light and light treatmentsto disorders of circadian rhythms are shown by Boulos et al. [98], Kurosawa andGoldbeter [99] and Geier et al. [100]. The circadian model (11.48) is adapted fromGonze, Leloup and Goldbeter [101]. The condition (11.52) is shown by Zhao, Taoand Shi [54]. The observer design is adapted from That and Ding [102], and the con-trol design is different from the version shown in that paper. More results on observerdesign for a seventh-order circadian model are published by That and Ding [103].

There are many results in literature on sampled-data control of linear systems,for example see Chen and Francis [104] and the references therein. The difficultyof obtaining exact discrete-time model for nonlinear systems is shown by Nesic andTeel [105]. Structures in nonlinear systems cannot be preserved as shown in Grizzleand Kokotovic [106]. Approximation to nonlinear discrete-time models is shownby several results, for example Nesic, Teel and Kokotovic [107] and Nesic and Laila[108]. The effect on fast-sampling of nonlinear static controllers is reported in Owens,Zheng and Billings [109]. Several other results on emulation method for nonlinearsystems are shown in Laila, Nesic and Teel [110], Shim and Teel [111] and Bianand French [112]. The results presented in Section 10.3 are mainly based on Wu andDing [113]. Sampled-data control for disturbance rejection of nonlinear systems isshown by Wu and Ding [114]. A result on sampled-data adaptive control of a class ofnonlinear systems is to be found in Wu and Ding [115].

References

[1] Cook, P. A. Nonlinear Dynamic Systems. London: Prentice-Hall, 1986.[2] Slotine, J.-J. E. and Li, W. Applied Nonlinear Control. London: Prentice-

Hall, 1991.[3] Vidyasagar, M. Nonlinear Systems Analysis, 2nd ed. Englewood Cliffs,

New Jersey: Prentice-Hall International, 1993.[4] Khalil, H. K. Nonlinear Systems, 3rd ed. Upper Saddle River, New Jersey:

Prentice Hall, 2002.[5] Verhulst, F. Nonlinear Differential Equations and Dynamic Systems. Berlin:

Springer-Verlag, 1990.[6] Ogata, K. Modern Control Engineering, 3rd ed. London: Prentice-Hall,

1997.[7] Kailath, T. Linear Systems. London: Prentice-Hall, 1980.[8] Antsaklis, P. J. and Michel, A. N. Linear Systems. New York: McGraw-Hill,

1997.[9] Zheng, D. Z. Linear Systems Theory (in Chinese). Beijing: Tsinghua

University Press, 1990.[10] Chen, B. M., Lin, Z. and Shamash, Y. Linear Systems Theory. Boston:

Birkhauser, 2004.[11] Arnold, V. I. Ordinary Differential Equations. Berlin: Springer International,

1994.[12] Borrelli, R. L. and Coleman, C. S. Differential Equations. New York: John

Wiley & Sons, 1998.[13] Hu, T. and Lin, Z. Control Systems With Actuator Saturation: Analysis And

Design. Boston, MA: Birhäuser, 2001.[14] Edwards, C. and Spurgeon, S. Sliding Mode Control. London: Taylor &

Francis, 1998.[15] Isidori, A. Nonlinear Control Systems II. London: Springer-Verlag, 1999.[16] Elgerd, O. I. Electric Energy Systems Theory. New York: McGraw-Hill,

1971.[17] van der Pol, B. ‘On relaxation oscillations,’ Phil. Mag., vol. 2, pp. 978–992,

1926.[18] Lorenz, E. N. ‘Deterministic nonperiodic flow,’ J. Atmos. Sci., vol. 20,

pp. 130–141, 1963.[19] Atherton, D. P. Nonlinear Control Engineering. Workingham: Van Nostraind

Reinhold, 1975.

References 269

[20] Gelb,A. andVelde, W. E.V. Multi-Input Describing Functions and NonlinearSystem Design. New York: McGraw-Hill, 1963.

[21] Lyapunov, A. M. ‘The general problem of motion stability (translated infrench),’ Ann. Fac. Sci. Toulouse, vol. 9, pp. 203–474, 1907.

[22] Hahn, W. Stability of Motion. Berlin: Springer, 1967.[23] Narendra, K. S. andAnnaswamy, A. M. StableAdaptive Systems. Englewood

Cliffs, New Jersey: Prentice-Hall, 1989.[24] Marino, R. and Tomei, P. Nonlinear Control Design: Geometric, Adaptive,

and Robust. London: Prentice-Hall, 1995.[25] Narendra, K. S. and Taylor, J. H. Frequency Domain Stability for Absolute

Stability. New York: Academic Press, 1973.[26] Brown, J.W. and Churchill, R.V. ComplexVariables andApplications, 7th ed.

New York: McGraw-Hill, 2004.[27] Sontag, E. D. ‘Smooth stabilization implies coprime factorization,’ IEEE

Transaction on Automatic Control, vol. 34, pp. 435–443, 1989.[28] Sontag, E.D. and Wang,Y. ‘On characterization of the input-to-state stability

properties,’ Syst. Contr. Lett., vol. 24, pp. 351–359, 1995.[29] Sontag, E. and Teel, A. ‘Changing supply functions input/state stable

systems,’ IEEE Trans. Automat. Control, vol. 40, no. 8, pp. 1476–1478,1995.

[30] Jiang, Z. -P., Teel, A. R. and Praly, L. ‘Small-gain theorem for iss systemsand applications,’ Math. Contr. Sign. Syst., vol. 7, pp. 95–120, 1994.

[31] Ding, Z. ‘Differential stability and design of reduced-order observers fornonlinear systems,’ IET Control Theory and Applications, vol. 5, no. 2,pp. 315–322, 2011.

[32] Boothby, W. A. An Introduction to Differential Manifolds and RiemaniannGeometry. London: Academic Press, 1975.

[33] Isidori, A. Nonlinear Control Systems, 3rd ed. Berlin: Springer-Verlag,1995.

[34] Brockett, R. W. ‘Feedback invariants for non-linear systems,’ inThe Proceedings of 7th IFAC Congress, vol. 6, Helsinki, Finland.

[35] Astrom, K. J. and Wittenmark, B. Adaptive Control, 2nd ed. Reading, MA:Addison-Wesley, 1995.

[36] Wellstead, P. E. and Zarrop, M. B. Self-Tuning Systems: Control and SignalProcessing. New York: John Wiley & Sons, 1991.

[37] Ding, Z. ‘Model reference adaptive control of dynamic feedback linearis-able systems with unknown high frequency gain,’ IEE Proceedings ControlTheory and Applications, vol. 144, pp. 427–434, 1997.

[38] Parks, P. C. ‘Lyapunov redesign of model reference adaptive controlsystems,’ IEEE Trans. Automat. Contr., vol. 11, pp. 362–367, 1966.

[39] Ioannou, P. A. and Sun, J. Robust Adaptive Control. Upper Saddle River,New Jersey: Prentice Hall, 1996.

[40] Ding, Z. ‘Robust adaptive control of nonlinear output-feedback systemsunder bounded disturbances,’ IEE Proc. Control Theory Appl., vol. 145,pp. 323–329, 1998.


[41] Ding, Z. ‘Global adaptive output feedback stabilization of nonlinear systemsof any relative degree with unknown high frequency gain,’ IEEE Trans.Automat. Control, vol. 43, pp. 1442–1446, 1998.

[42] Ding, Z. ‘Adaptive control of nonlinear systems with unknown virtual con-trol coefficients,’ Int. J. Adaptive Control Signal Proc., vol. 14, no. 5,pp. 505–517, 2000.

[43] Ding, Z. ‘Analysis and design of robust adaptive control for nonlinear outputfeedback systems under disturbances with unknown bounds,’ IEE Proc.Control Theory Appl., vol. 147, no. 6, pp. 655–663, 2000.

[44] Ding, Z. and Ye, X. ‘A flat-zone modification for robust adaptive controlof nonlinear output feedback systems with unknown high frequency gains,’IEEE Trans. Automat. Control, vol. 47, no. 2, pp. 358–363, 2002.

[45] Ding, Z. ‘Adaptive stabilization of a class of nonlinear systems with unsta-ble internal dynamics,’ IEEE Trans. Automat. Control, vol. 48, no. 10,pp. 1788–1792, 2003.

[46] Thau, F. ‘Observing the states of nonlinear dynamical systems,’ Int. J.Control, vol. 18, pp. 471–479, 1973.

[47] Kou, S., Ellitt, D. and Tarn, T. ‘Exponential observers for nonlineardynamical systems,’ Inf. Control, vol. 29, pp. 204–216, 1975.

[48] Krener, A. J. and Isidori, A. ‘Linearization by output injection and nonlinearobservers,’ Syst. Control Lett., vol. 3, pp. 47–52, 1983.

[49] Kazantzis, M. and Kravaris, C. ‘Nonlinear observer design using lyapunov’sauxiliary theorem,’ Syst. Control Lett., vol. 34, pp. 241–247, 1998.

[50] Xiao, M. ‘The global existence of nonlinear observers with linear errordynamics: A topological point of view,’ Syst. Control Lett., vol. 55, no. 10,pp. 849–858, 2006.

[51] Ding, Z. ‘Observer design in convergent series for a class of nonlinearsystems,’ IEEE Trans. Automat. Control, vol. 57, no. 7, pp. 1849–1854,2012.

[52] Rajamani, R. ‘Observers for Lipschitz nonlinear systems,’ IEEE Trans.Automat. Control, vol. 43, no. 3, pp. 397–401, 1998.

[53] Zhu, F. and Han, Z. ‘A note on observer design for Lipschitz nonlinearsystems,’ IEEE Trans. Automat. Control, vol. 47, no. 10, pp. 1751–1754,2002.

[54] Zhao, Y., Tao, J. and Shi, N. ‘A note on observer design for one-sidedLipschitz nonlinear systems,’ Syst. Control Lett., vol. 59, pp. 66–71, 2010.

[55] Ding, Z. ‘Asymptotic rejection of unmatched general periodic disturbanceswith nonlinear Lipschitz internal model,’ Int. J. Control, vol. 86, no. 2,pp. 210–221, 2013.

[56] Luenberger, D. G. ‘Observing the state of a linear system,’ IEEE Trans. Mil.Electron., vol. 8, pp. 74–80, 1964.

[57] Hou, M. and Muller, P. ‘Design of observers for linear systems with unknowninputs,’ IEEE Trans. Automat. Control, vol. 37, no. 6, pp. 871–875, 1992.

[58] Ding, Z. ‘Global output feedback stabilization of nonlinear systems withnonlinearity of unmeasured states,’ IEEE Trans. Automat. Control, vol. 54,no. 5, pp. 1117–1122, 2009.

References 271

[59] Bastin, G. and Gevers, M. ‘Stable adaptive observers for nonlinear timevarying systems,’ IEEETrans. Automat. Control, vol. 33, no. 7, pp. 650–658,1988.

[60] Cho, Y. M. and Rajamani, R. ‘A systematic approach to adaptive observersynthesis for nonlinear systems,’ IEEE Trans. Automat. Control, vol. 42,no. 4, pp. 534–537, 1997.

[61] Tsinias, J., ‘Sufficient lyapunov-like conditions for stabilization,’ Math.Control Signals Systems, vol. 2, pp. 343–357, 1989.

[62] Kanellakopoulos, I., Kokotovic, P. V. and Morse, A. S. ‘Systematic designof adaptive controllers for feedback linearizable systems,’ IEEE Trans.Automat. Control, vol. 36, pp. 1241–1253, 1991.

[63] Marino, R. andTomei, P. ‘Global adaptive output feedback control of nonlin-ear systems, part i: Linear parameterization,’ IEEE Trans. Automat. Control,vol. 38, pp. 17–32, 1993.

[64] Krstic, M., Kanellakopoulos, I. and Kokotovic, P. V. ‘Adaptive nonlinearcontrol without overparametrization,’ Syst. Control Lett., vol. 19, pp. 177–185, 1992.

[65] Ding, Z. ‘Adaptive output regulation of class of nonlinear systems withcompletely unknown parameters,’ in Proceedings of 2003American ControlConference, Denver, CO, 2003, pp. 1566–1571.

[66] Krstic, M., Kanellakopoulos, I. and Kokotovic, P. V. Nonlinear and AdaptiveControl Design. New York: John Wiley & Sons, 1995.

[67] Ding, Z. ‘Almost disturbance decoupling of uncertain output feedbacksystems,’ IEE Proc. Control Theory Appl., vol. 146, pp. 220–226, 1999.

[68] Ding, Z. ‘Adaptive control of triangular systems with nonlinear parameter-ization,’ IEEE Trans. Automat. Control, vol. 46, no. 12, pp. 1963–1968,2001.

[69] Bodson, M., Sacks, A. and Khosla, P. ‘Harmonic generation in adaptivefeedforward cancellation schemes,’ IEEE Trans. Automat. Control, vol. 39,no. 9, pp. 1939–1944, 1994.

[70] Bodson, M. and Douglas, S. C. ‘Adaptive algorithms for the rejection ofsinusoidal disturbances with unknown frequencies,’ Automatica, vol. 33,no. 10, pp. 2213–2221, 1997.

[71] Ding, Z. ‘Global output regulation of uncertain nonlinear systems withexogenous signals,’Automatica, vol. 37, pp. 113–119, 2001.

[72] Ding, Z. ‘Global stabilization and disturbance suppression of a class ofnonlinear systems with uncertain internal model,’Automatica, vol. 39, no. 3,pp. 471–479, 2003.

[73] Davison, E. J. ‘The robust control of a servomechanism problem for lin-ear time-invariant multivariable systems,’ IEEE Trans. Automat. Control,vol. 21, no. 1, pp. 25–34, 1976.

[74] Francis, B. A. ‘The linear multivariable regulator problem,’ SIAM J. ControlOptimiz., vol. 15, pp. 486–505, 1977.

[75] Huang, J. and Rugh, W. J. ‘On a nonlinear multivariable servomechanismproblem,’Automatica, vol. 26, no. 6, pp. 963–972, 1990.


[76] Isidori, A. and Byrnes, C. I. ‘Output regulation of nonlinear systems,’ IEEETrans. Automat. Control, vol. 35, no. 2, pp. 131–140, 1990.

[77] Huang, J. Nonlinear Output Regulation Theory and Applications. Philadel-phia, PA: SIAM, 2004.

[78] Nikiforov, V. O. ‘Adaptive non-linear tracking with complete compensationof unknown disturbances,’ Eur. J. Control, vol. 4, pp. 132–139, 1998.

[79] Nussbaum, R. D. ‘Some remarks on a conjecture in parameter adaptivecontrol,’ Syst. Control Lett., vol. 3, pp. 243–246, 1983.

[80] Ye, X. and Ding, Z. ‘Robust tracking control of uncertain nonlinear systemswith unknown control directions,’ Syst. Control Lett., vol. 42, pp. 1–10,2001.

[81] Ding, Z. ‘Output regulation of uncertain nonlinear systems with nonlin-ear exosystems,’ in Proceeding of the 2004 American Control Conference,Boston, MA, 2004, pp. 3677–3682.

[82] Ding, Z. ‘Output regulation of uncertain nonlinear systems with nonlinearexosystems,’ IEEE Trans. Automat. Control, vol. 51, no. 3, pp. 498–503,2006.

[83] Chen, Z. and Huang, J. ‘Robust output regulation with nonlinear exosys-tems,’Automatica, vol. 41, pp. 1447–1454, 2005.

[84] Xi, Z. and Ding, Z. ‘Global adaptive output regulation of a class ofnonlinear systems with nonlinear exosystems,’ Automatica, vol. 43, no. 1,pp. 143–149, 2007.

[85] Ding, Z. ‘Asymptotic rejection of general periodic disturbances in output-feedback nonlinear systems,’ IEEE Trans. Automat. Control, vol. 51, no. 2,pp. 303–308, 2006.

[86] Ding, Z. ‘Asymptotic rejection of asymmetric periodic disturbancesin output-feedback nonlinear systems,’ Automatica, vol. 43, no. 3,pp. 555–561, 2007.

[87] Ding, Z. ‘Asymptotic rejection of unmatched general periodic disturbancesin a class of nonlinear systems,’ IET Control Theory Appl., vol. 2, no. 4,pp. 269–276, 2008.

[88] Ding, Z. ‘Observer design of Lipschitz output nonlinearity with applicationto asymptotic rejection of general periodic disturbances,’ inThe Proceedingsof The 18th IFAC Congress, vol. 8, Milan, Italy, 2011.

[89] Ding, Z. ‘Asymptotic rejection of finite frequency modes of general periodicdisturbances in output-feedback nonlinear systems,’ Automatica, vol. 44,no. 9, pp. 2317–2325, 2008.

[90] Arrillaga, J., Watson, N. R. and Chen, S. Power System Quality Assessment.Chichester: John Wiley & Sons, 2000.

[91] Wakileh, G. J. Power Systems Harmonics. Berlin: Springer-Verlag,2010.

[92] Takahashi, J., Vitaterna, M., King, D., Chang, A., Kornhauser, J., Lowrey, P.,McDonald, J., Dove, W., Pinto, L., and Turek, F. ‘Mutagenesis and mappingof a mouse gene, clock, essential for circadian behaviour,’ Science, vol. 264,pp. 719–725, 1994.

References 273

[93] Albrecht, U., Sun, Z., Eichele, G. and Lee, C. ‘A differential response of twoputative mammalian circadian regulators, mper1 and mper2 to light,’ CellPress, vol. 91, pp. 1055–1064, 1997.

[94] van der Horst, G., Muijtjens, M., Kobayashi, K., Takano, R., Kanno,S., Takao, M., de Wit, J., Verkerk, A., Eker, A., van Leenen, D., Buijs,R., Bootsma, D., Hoeijmakers, J., and Yasui, A. ‘Mammalian cry1 andcry2 are essential for maintenance of circadian rhythms,’ Nature, vol. 398,pp. 627–630, 1999.

[95] Bunger, M., Wilsbacher, L., Moran, S., Clendenin, C., Radcliffe, L.,Hogenesch, J., Simon, M., Takahashi, J., and Bradfield, C., ‘Mop3 is anessential component of the master circadian pacemaker in mammals,’ Cell,vol. 103, pp. 1009–1017, 2000.

[96] Sack, R., Auckley, D., Auger, R., Carskadon, M., Wright, K., Vitiello, M.,and Zhdanova, I. ‘Circadian rhythm sleep disorders: part i, basic principles,shift work and jet lag disorders,’ Sleep, vol. 30, pp. 1460–1483, 2007.

[97] Sack, R., Auckley, D., Auger, R., Carskadon, M., Wright, K., Vitiello, M.,and Zhdanova, I. ‘Circadian rhythm sleep disorders: Part ii, advanced sleepphase disorder, delayed sleep phase disorder, free-running disorder, andirregular sleep-wake rhythm,’ Sleep, vol. 30, pp. 1484–1501, 2007.

[98] Boulos, Z., Macchi, M., Sturchler, M., Stewart, K., Brainard, G., Suhner,A.,Wallace, G., and Steffen, R., ‘Light visor treatment for jet lag after westward travel across six time zones,’ Aviat. Space Environ. Med., vol. 73,pp. 953–963, 2002.

[99] Kurosawa, G. and Goldbeter, A. ‘Amplitude of circadian oscillationsentrained by 24-h light-dark cycles,’ J. Theor. Biol., vol. 242, pp. 478–488,2006.

[100] Geier, F., Becker-Weimann, S., Kramer, K., and Herzel, H. ‘Entrainment ina model of the mammalian circadian oscillator,’ J. Biol. Rhythms, vol. 20,pp. 83–93, 2005.

[101] Gonze, D., Leloup, J., and Goldbeter, A., ‘Theoretical models for circadianrhythms in neurospora and drosophila,’ Comptes Rendus de l’Academie desSciences -Series III -Sciences de la Vie, vol. 323, pp. 57–67, 2000.

[102] That, L. T. and Ding, Z. ‘Circadian phase re-setting using nonlinear output-feedback control,’ J. Biol. Syst., vol. 20, no. 1, pp. 1–19, 2012.

[103] That, L. T. and Ding, Z. ‘Reduced-order observer design of multi-outputnonlinear systems with application to a circadian model,’Trans. Inst. Meas.Control, to appear, 2013.

[104] Chen, T. and Francis, B. Optimal Sampled-Data Control Systems. London:Springer-Verlag, 1995.

[105] Nesic, D. and Teel, A. ‘A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models,’ IEEE Trans.Automat. Control, vol. 49, pp. 1103–1122, 2004.

[106] Grizzle, J. W. and Kokotovic, P. V. ‘Feedback linearization of sampled-datasystems,’ IEEE Trans. Automat. Control, vol. 33, pp. 857–859, 1988.


[107] Nesic, D., Teel, A., and Kokotovic, P.V. ‘Sufficient conditions for stabiliza-tion of sampled-data nonlinear systems via discrete-time approximations,’Syst. Control Lett., vol. 38, pp. 259–270, 1999.

[108] Nesic, D. and Laila, D. ‘A note on input-to-state stabilization for nonlinearsampled-data systems,’ IEEE Trans. Automat. Control, vol. 47, pp. 1153–1158, 2002.

[109] Owens, D. H., Zheng, Y. and Billings, S. A. ‘Fast sampling and stability ofnonlinear sampled-data systems: Part 1. Existing theorems,’ IMA J. Math.Control Inf., vol. 7, pp. 1–11, 1990.

[110] Laila, D. S., Nesic, D. and Teel, A. R. ‘Open-and closed-loop dissipa-tion inequalities under sampling and controller emulation,’ Eur. J. Control,vol. 18, pp. 109–125, 2002.

[111] Shim, H. and Teel, A. R. ‘Asymptotic controllability and observabilityimply semiglobal practical asymptotic stabilizability by sampled-data outputfeedback,’Automatica, vol. 39, pp. 441–454, 2003.

[112] Bian, W. and French, M. ‘General fast sampling theorems for nonlinearsystems,’ Syst. Control Lett., vol. 54, pp. 1037–1050, 2005.

[113] Wu, B. and Ding, Z. ‘Asymptotic stablilisation of a class of nonlinear systemsvia sampled-data output feedback control,’ Int. J. Control, vol. 82, no. 9,pp. 1738–1746, 2009.

[114] Wu, B. and Ding, Z. ‘Practical disturbance rejection of a class of nonlin-ear systems via sampled output,’ J. Control Theory Appl., vol. 8, no. 3,pp. 382–389, 2010.

[115] Wu, B. and Ding, Z. ‘Sampled-data adaptive control of a class of nonlinearsystems,’ Int. J. Adapt. Control Signal Process., vol. 25, pp. 1050–1060,2011.

Index

σ -modification, 106absolute stability, 60adaptive control

direct, 89indirect, 89model reference, 89nonlinear, 159

adaptive lawlinear system, 101nonlinear, 162, 172robust, 107

asymptotic stability, 43autonomous system, 11

backsteppingadaptive, 159adaptive observer, 167filtered transformation, 152integrator, 141iterative, 144observer, 147

Barbalat’s lemma, 92

centre, 14certainty equivalence principle, 90chaos, 4, 21circadian disorders, 238circadian rhythm, 239

Neurospora control input, 246Neurospora model, 239

circle criterion, 62comparison functions

class K 66class KL 66class K∞, 66

comparison lemma, 50

dead zone, 31dead-zone modification, 105describing function, 27

dead zone, 32ideal relay, 30relay with hysteresis, 32saturation, 30stability of limit cycle, 35

detectabilitylinear system, 110

differential stability, 71distribution, 84

involutive, 84disturbance

bounded, 104, 107equivalent input, 176–77general periodic, 204, 205

dynamic model, 207generated from nonlinear

exosystem, 194sinusoidal, 175

disturbance rejectiongeneral periodic, 204sinusoidal, 175

eigenvalueresonant, 121

equivalent input disturbance, 176–77exosystem, 175, 176

linear, 176, 186nonlinear, 195

exponential stability, 43

feedback linearisation, 76feedforward


control, 181, 189, 199input, 176, 189, 206term, 189, 197, 209

finite time convergence, 4finite time escape, 4frequency mode, 219, 224

asymptotically orthogonal, 225asymptotically rejected, 235multiple, 229

Frobenius theorem, 85full state feedback linearisable, 85

global stability properties, 45globally asymptotic stability, 45globally exponential stability, 45

harmonics, 4estimation, 235power system, 219

high-frequency gain, 94, 100, 102, 187Hurwitz

matrix, 52polynomial, 94transfer function, 57vector, 148

ideal relay, 30immersion, 189, 199inter-sample behaviour, 252internal model, 199

adaptive, 179, 190nonlinear, 199, 201observer based, 209

internal model principle, 176invariant manifold, 177, 189, 196invariant zero, 133, 134ISS pair (α, σ ), 69, 208ISS stability, 66ISS-Lyapunov function, 69

jet engine model, 260

Kalman Yakubovich lemma, 59

level set, 47, 49, 253Lie bracket, 79

Lie derivative, 77limit cycle, 4, 18, 34, 175, 195, 238, 240linear observer error dynamics, 112,121linearisation

around operation point, 10feedback, 75, 76, 80full state, 83, 85linearised model, 10

Lipschitz constant, 9, 123, 207, 241one-sided, 124, 241

Lorenz attractor, 21Lyapunov function, 46

adaptive control, 101adaptive observer, 139ISS, 67–70linear systems, 52

Lyapunov stability, 42Lyapunov theorem

global stability, 49local stability, 47

Lyapunov’s direct method, 46Lyapunov’s linearisation method, 45

model reference controlrelative degree ρ > 1, 97–98relative degree one, 94–96

monic polynomial, 94MRAC, 89multiple equilibrium points, 4

nonlinearity, 1, 3dead zone, 31hard, 3ideal relay, 30Lipschitz, 7, 122multi-valued, 3relay with hysteresis, 32saturation, 29sector-bounded, 59

normal form, 81Nussbaum gain, 190

observabilitylinear system, 109

Index 277

observergeneral periodic disturbance, 207

observer canonical formlinear system, 118

observer designadaptive, 137direct state transformation, 121Lipschitz nonlinearity, 122output injection, 112output nonlinearity, 126reduced-order, 127

output feedback form, 148, 152with disturbance, 175, 186, 194with unknown parameter, 167

output feedback stabilisation, 158output injection form, 112

state transformation, 114output regulation, 186

adaptive, 186nonlinear exosystem, 194

output-complement transformation, 127

pendulum model, 48phase portrait, 12Poincare-Bendixson theorem, 19positive definite function, 46positive limit set, 21positive real transfer function, 55

strictly, 56

radially unbounded function, 49relative degree, 77relay with hysteresis, 32robust adaptive law, 107σ -modification, 106dead-zone modification, 105

saddle point, 13sampled-data

emulation method, 248sampled-data controller, 250saturation, 2, 29sector-bounded nonlinearity, 59self-tuning control, 89semi-global stability, 3, 260singular point, 12small gain theorem

ISS, 70stability

absolute, 60asymptotic, 43differential, 71exponential, 43globally asymptotic, 45globally exponential, 45ISS, 66Lyapunov, 42semi-global, 3, 260

stable focus, 13–14stable node, 13STC, 89

transformationfiltered, 152, 187output-complement, 127

tuning function, 161unstable focus, 14unstable node, 13

van der Pol oscillator, 18, 38

Young’s inequality, 128

zero dynamics, 81, 155

Control Engineering Series 84

Nonlinear and Adaptive Control

Systems

Nonlinear and Adaptive

Control System

sD

ing Zhengtao Ding

The Institution of Engineering and Technologywww.theiet.org 978-1-84919-574-4

Nonlinear and Adaptive Control Systems

Zhengtao Ding is a Senior Lecturer in Control Engineering and Director for MSc in Advanced Control and Systems Engineering at the Control Systems Centre, School of Electrical and Electronic Engineering, The University of Manchester, UK. His research interests focus on nonlinear and adaptive control design. He pioneered research in asymptotic rejection of general periodic disturbances in nonlinear systems and produced a series of results to systematically solve this problem in various situations. He also made significant contributions in output regulation and adaptive control of nonlinear systems with some more recent results on observer design and output feedback control as well. Dr Ding has been teaching ‘Nonlinear and Adaptive Control Systems’ to MSc students for 9 years, and he has accumulated tremendous experiences in explaining difficult control concepts to students.

An adaptive system for linear systems with unknown parameters is a nonlinear system. The analysis of such adaptive systems requires similar techniques to analyse nonlinear systems. Therefore it is natural to treat adaptive control as a part of nonlinear control systems.

Nonlinear and Adaptive Control Systems treats nonlinear control and adaptive control in a unified framework, presenting the major results at a moderate mathematical level, suitable for MSc students and engineers with undergraduate degrees. Topics covered include introduction to nonlinear systems; state space models; describing functions for common nonlinear components; stability theory; feedback linearization; adaptive control; nonlinear observer design; backstepping design; disturbance rejection and output regulation; and control applications, including harmonic estimation and rejection in power distribution systems, observer and control design for circadian rhythms, and discrete-time implementation of continuous-time nonlinear control laws.

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