Advanced topics in control systems theory II

Nonlinear Adaptive Stabilization via System Immersion: Control Design and ApplicationsD.. 1.1 Introduction The problem of adaptive stabilization of nonlinear systems has been anactive ar

in Control and Information Sciences 311

Editors: M Thoma · M Morari

E Panteley (Eds.)

Advanced Topics

in Control Systems Theory

Lecture Notes from FAP 2004

With 12 Figures

A.B Kurzhanski · H Kwakernaak · J.N Tsitsiklis

Editors

Dr Fran¸coise Lamnabhi-Lagarrigue

Dr Antonio Lor´ıa

Dr Elena Panteley

Laboratoire des Signaux et Syst`emes

Centre National de la Recherche Scientifique

British Library Cataloguing in Publication Data

Advanced topics in control systems theory : lecture notes

from FAP 2004 - (Lecture notes in control and information sciences ; 311)

1 Automatic control 2 Automatic control - Mathematical models

3 Control theory 4 Systems engineering

I Lamnabhi-Lagarrigue, F (Francoise), 1953- II Loria, Antonio

III Panteley, Elena

629.8’312

ISBN 1852339233

Library of Congress Control Number: 2004117782

Apart from any fair dealing for the purposes of research or private study, or criticism or review,

as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing

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Lecture Notes in Control and Information Sciences ISSN 0170-8643

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Advanced topics in control systems theory is a byproduct of the Europeanschool “Formation d’Automatique de Paris” (Paris Graduate School on Au-tomatic Control) which took place in Paris through February and March 2004.The school benefited of the valuable participation of 17 European renownedcontrol researchers and about 70 European PhD students While the programconsisted of the modules listed below, the contents of the present monographcollects selected notes provided by the lecturers and is by no means exhaustive.Program of FAP 2004:

P1 Nonlinear control of electrical and electromechanical systems

P4 Modeling and control of chemical and biotechnological processes

Jan van Impe, D Dochain,

P5 Modeling and boundary control of infinite dimensional systems

B Maschke, A.J van der Schaft, H Zwart

P6 Linear systems, algebraic theory of modules, structural properties

H Bourles, M Fliess

P7 Lyapunov-based control: state and output feedback

L Praly, A Astolfi, A Lor´ıa

P8 Nonlinear control and mechanical systems

B Bonnard

P9 Tools for analysis and control of time-varying systems

a one-term general advanced course on non linear control theory, thereby voting a few lectures to each topic, or it may be used in support to morefocused intensive courses at graduate level The academic requirement for theclass student or the reader in general is a basic knowledge on control theory(linear and non linear)

de-Advanced topics in control systems theory also constitutes an ideal startfor researchers in control theory who wish to broaden their general culture or

to get involved in fields different to their expertise, while avoiding a thoroughbook-keeping Indeed, the monograph presents in a concise but pedagogicalmanner diverse aspects of modern control theory

This book is the first of a series of yearly volumes, which shall prevail yond the lectures taught in class during each FAP season Further information

be-on FAP, in particular, be-on the scientific program for the subsequent years isupdated in due time on our URL http://www.supelec.lss/cts/fap.FAP is organized within the context of the European teaching network

“Control Training Site” sponsored by the European Community through theMarie Curie program The editors of the present text greatefully acknowledgesuch sponsorship We also take this oportunity to acknowledge the Frenchnational center for scientific research (C.N.R.S.) which provides us with aworking environment and ressources probably unparalleled in the world

Gif-sur-Yvette, France Fran¸coise Lamnabhi-Lagarrigue,

Elena Panteley

1 Nonlinear Adaptive Stabilization via System Immersion:

Control Design and Applications

D Karagiannis, R Ortega, A Astolfi 1

1.1 Introduction 1

1.2 Nonlinear Stabilization via System Immersion 3

1.3 Adaptive Control via System Immersion 4

1.3.1 Systems Linear in the Unknown Parameters 5

1.3.2 Systems in Feedback Form 7

1.4 Output Feedback Stabilization 11

1.4.1 Linearly Parameterized Systems 12

1.4.2 Control Design Using a Separation Principle 14

1.5 Applications 16

1.5.1 Aircraft Wing Rock Suppression 16

1.5.2 Output Voltage Regulation of Boost Converters 17

1.6 Conclusions 21

References 21

2 Cascaded Nonlinear Time-Varying Systems: Analysis and Design Antonio Lor´ıa, Elena Panteley 23

2.1 Preliminaries on Time-Varying Systems 24

2.1.1 Stability Definitions 25

2.1.2 Why Uniform Stability? 27

2.2 Cascaded Systems 29

2.2.1 Introduction 29

2.2.2 Peaking: A Technical Obstacle to Analysis 31

2.2.3 Control Design from a Cascades Point of View 33

2.3 Stability of Cascades 36

2.3.1 Brief Literature Review 36

2.3.2 Nonautonomous Cascades: Problem Statement 38

2.3.3 Basic Assumptions and Results 39

2.3.4 An Integrability Criterion 43

2.3.5 Growth Rate Theorems 44

2.4 Applications in Control Design 48

2.4.1 Output Feedback Dynamic Positioning of a Ship 49

2.4.2 Pressure Stabilization of a Turbo-Diesel Engine 51

2.4.3 Nonholonomic Systems 54

2.5 Conclusions 60

References 61

3 Control of Mechanical Systems from Aerospace Engineering Bernard Bonnard, Mohamed Jabeur, Gabriel Janin 65

3.1 Introduction 65

3.2 Mathematical Models 67

3.2.1 The Attitude Control Problem 68

3.2.2 Orbital Transfer 69

3.2.3 Shuttle Re-entry 71

3.3 Controllability and Poisson Stability 73

3.3.1 Poisson Stability 73

3.3.2 General Results About Controllability 74

3.3.3 Controllability and Enlargement Technique (Jurdjevi´c-Kupka) 76

3.3.4 Application to the Attitude Problem 77

3.3.5 Application to the Orbital Transfer 77

3.4 Constructive Methods 78

3.4.1 Stabilization Techniques 78

3.4.2 Path Planning 82

3.5 Optimal Control 84

3.5.1 Geometric Framework 84

3.5.2 Weak Maximum Principle 84

3.5.3 Maximum Principle 86

3.5.4 Extremals in SR-Geometry 87

3.5.5 SR-Systems with Drift 88

3.5.6 Extremals for Single-Input Affine Systems 93

3.5.7 Second-Order Conditions 95

3.5.8 Optimal Controls with State Constraints 101

3.6 Indirect Numerical Methods in Optimal Control 109

3.6.1 Shooting Techniques 109

3.6.2 Second-Order Algorithms in Orbital Transfer 112

References 113

4 Compositional Modelling of Distributed-Parameter Systems Bernhard Maschke, Arjan van der Schaft 115

4.1 Introduction 115

4.2 Systems of Two Physical Domains in Canonical Interaction 117

4.2.1 Conservation Laws, Interdomain Coupling and Boundary Energy Flows: Motivational Examples 118

4.2.2 Systems of Two Conservation Laws in Canonical Interaction 123

4.3 Stokes-Dirac Structures 129

4.3.1 Dirac Structures 129

4.3.2 Stokes-Dirac Structures 130

4.3.3 Poisson Brackets Associated to Stokes-Dirac Structures 132

4.4 Hamiltonian Formulation of Distributed-Parameter Systems with Boundary Energy Flow 134

4.4.1 Boundary Port-Hamiltonian Systems 134

4.4.2 Boundary Port-Hamiltonian Systems with Distributed Ports and Dissipation 136

4.5 Examples 138

4.5.1 Maxwell’s Equations 138

4.5.2 Telegraph Equations 140

4.5.3 Vibrating String 141

4.6 Extension of Port-Hamiltonian Systems Defined on Stokes-Dirac Structures 143

4.6.1 Burger’s Equations 143

4.6.2 Ideal Isentropic Fluid 143

4.7 Conserved Quantities 148

4.8 Conclusions and Final Remarks 151

References 152

5 Algebraic Analysis of Control Systems Defined by Partial Differential Equations Jean-Fran¸cois Pommaret 155

5.1 Introduction 155

5.2 Motivating Examples 161

5.3 Algebraic Analysis 168

5.3.1 Module Theory 168

5.3.2 Homological Algebra 179

5.3.3 System Theory 183

5.4 Problem Formulation 197

5.5 Problem Solution 200

5.6 Poles and Zeros 211

5.7 Conclusion 220

5.8 Exercises 220

References 222

6 Structural Properties of Discrete and Continuous Linear Time-Varying Systems: A Unified Approach Henri Bourl`es 225

6.1 Introduction 225

6.2 Differential Polynomials 227

6.2.1 Differential Fields 227

6.2.2 Rings of Differential Polynomials 229

6.2.3 Properties of General Rings 230

6.3 Modules and Systems of Linear Differential Equations 236

6.3.1 Modules 236

6.3.2 Autonomous Linear Differential Equations 244

6.3.3 Systems of Linear Differential Equations 249

6.4 Linear Time-Varying Systems: A Module-Theoretic Setting 256

6.4.1 Basic Structural Properties 256

6.4.2 Finite Poles and Zeros 263

6.5 Duality and Behaviors 265

6.5.1 The Functor Hom 265

6.5.2 Behaviors 274

6.6 Concluding Remarks 277

References 278

1.1 State-space trajectory of the wing rock system Dashed

line: Full-information controller Dotted line: Adaptive

backstepping controller Solid line: Proposed controller 17

1.2 Diagram of the DC–DC boost converter 18

1.3 State and control histories of the boost converter 20

2.1 Block-diagram of a cascaded system 23

2.2 The peaking phenomenon 32

2.3 Synchronisation of two pendula 33

2.4 Turbo charged VGT-EGR diesel engine 52

2.5 Mobile robot: tracking problem 56

3.1 Weierstrass variations 102

3.2 Time optimal synthesis 106

3.3 Geodesics on a Flat torus 110

5.1 Exact commutative diagrams 193

A Astolfi

Department of Electrical and

Electronic Engineering, Imperial

SATIE, ENS de Cachan et CNAM,

61 Ave du Pr´esident Wilson,

B.P 47870, 21078 Dijon Cedex,France

gabriel.janin@ensta.org

D KaragiannisDepartment of Electrical andElectronic Engineering, ImperialCollege,

Exhibition Road, London SW72BT, UK

d.karagiannis@imperial.ac.uk

A Lor´ıaLaboratoire des Signaux etSyst`emes, Sup´elec,

3, Rue Joliot Curie, 91192Gif-sur-Yvette, France

loria@lss.supelec.fr

B MaschkeLaboratoire d’Automatique et deG´enie des Proc´ed´es,

Universit´e Claude Bernard Lyon-1,CPE Lyon - Bˆatiment 308 G, 43,

Bd du 11 Novembre 1918, F-69622Villeurbanne cedex, France.maschke@lagep.univ-lyon1.fr

P.O Box 217, 7500 AE Enschede,The Netherlands.

twarjan@math.utwente.nl

Nonlinear Adaptive Stabilization via System Immersion: Control Design and Applications

D Karagiannis1, R Ortega2, and A Astolfi1

1 Department of Electrical and Electronic Engineering, Imperial College,

Exhibition Road, London SW7 2AT, UK Email:

in applications where a controller for a reduced-order model is known and

we would like to robustify it with respect to higher-order dynamics This isachieved by immersing the dynamics of the controlled plant into the desireddynamics of the reduced-order model The method is illustrated with severalpractical and academic examples

1.1 Introduction

The problem of (adaptive) stabilization of nonlinear systems has been anactive area of research in the past years with several constructive method-ologies being introduced, such as feedback linearization [3, 7], sliding-modecontrol [10], backstepping [6] and passivity-based control [8], see also themonograph [4] Most of the existing methods rely on the use of (control)Lyapunov functions, i.e the control law and/or the adaptive law are designed

so that a candidate Lyapunov function—typically a quadratic function of thestates and the (parameter) estimation error—is rendered negative definite.For systems with certain “triangular” structures, this approach has provenparticularly successful [6, 7]

More recently in [1] the concepts used in the theory of output regulation [2]have been exploited to develop a novel framework for solving nonlinear sta-bilization and adaptive control problems This new approach makes use oftwo classical tools from nonlinear regulator theory and geometric nonlinear

F Lamnabhi-Lagarrigue et al (Eds.): Adv Top in Cntrl Sys Theory, LNCIS 311, pp 1–21, 2005

© Springer-Verlag London Limited 2005

control: (system) immersion and (manifold) invariance For this reason themethod is referred to as immersion and invariance (I&I).

The basic idea in this methodology is to immerse the plant dynamics into

a stable (lower-order) target system To illustrate this, consider the system

˙

is the image through the mapping π(·) of a trajectory ξ(t) of the target tem (1.2) From (1.4), this implies that x(t) converges to the origin Thusthe stabilization problem for the system (1.1) can be recasted as a problem

sys-of solving the partial differential equation (1.5) with the boundary tions (1.3)-(1.4)

condi-A geometric interpretation of (1.3)–(1.5) is the following Consider theclosed-loop system (1.6) and a manifold in the n-dimensional state-space de-fined by

M = {x ∈ Rn | x = π(ξ), ξ ∈ Rp}

From (1.5), the manifold M is invariant with internal dynamics (1.2), henceall trajectories x(t) that start on the manifold remain there and asymptoticallyconverge to the point x = π(0), which is the origin by (1.4) Moreover, thecondition (1.3) guarantees that the initial state of (1.6) lies on the manifoldM

The above formulation is impractical for two reasons First, from (1.3), themapping π(·) will, in general, depend on the initial conditions Second, even if

3 Note that, since the dimension of ξ is strictly lower than the dimension of x, themapping x = π(ξ) is an immersion

with x ∈ Rn and u ∈ Rm and the problem of finding, whenever possible, astate feedback control law u = υ(x) such that the closed-loop system is (glob-ally) asymptotically stable This is equivalent to finding a target dynamicalsystem

with ξ ∈ Rpand p < n, which is (globally) asymptotically stable, a mapping3

x = π(ξ) and a control law υ(x) such that

the target system (1.2) is globally asymptotically stable, we cannot concludethat the trajectories of (1.6) are bounded, without additional conditions onthe mapping π(·).

These obstacles can be removed by modifying the control law u = υ(x)

so that, for all initial conditions, the trajectories of the system (1.1) remainbounded and asymptotically converge to the manifold M, i.e M is renderedattractive The attractivity of the manifold can be expressed in terms of thedistance

z = dist(x, M),which should be driven to zero Notice that the variable z, which is referred

to as the off-the-manifold coordinate, is not uniquely defined This provides

an additional degree of freedom in the control design

1.2 Nonlinear Stabilization via System Immersion

The present section reviews the basic theoretical results of [1], namely a set ofsufficient conditions for the construction of globally asymptotically stabilizingstate feedback control laws for general nonlinear systems

Theorem 1.1 [1] Consider the system (1.1) with an equilibrium point

x∗∈ Rn to be stabilized and assume we can find mappings α(·), π(·), c(·),φ(·) and ψ(·) such that the following hold

(A1) (Target system) The system (1.2) has a globally asymptotically stableequilibrium at ξ∗∈ Rp and x∗= π(ξ∗)

(A2) (Immersion) For all ξ ∈ Rp

f(π(ξ), c(π(ξ))) = ∂π∂ξα(ξ) (1.7)(A3) (Implicit manifold) The following set identity holds

M = {x ∈ Rn | φ(x) = 0} = {x ∈ Rn | x = π(ξ), ξ ∈ Rp} (1.8)(A4) (Manifold attractivity and trajectory boundedness) All trajectories of thesystem

Then x∗ is a globally asymptotically stable equilibrium of the closed-loop tem

sys-˙x = f(x, ψ(x, φ(x)))

Remark 1.1 The result in Theorem 1.1 implies that the stabilization problemfor the system (1.1) can be divided into two subproblems First, given thetarget system (1.2) find, if possible, a manifold M described implicitly by{x ∈ Rn | φ(x) = 0} and in parameterized form by {x ∈ Rn | x = π(ξ),for some ξ ∈ Rp}, which can be rendered invariant with internal dynamics acopy of the target dynamics Second, design a control law u = ψ(x, z) thatdrives to zero the off-the-manifold coordinate z = φ(x) and keeps the systemtrajectories bounded

Remark 1.2 The convergence condition (1.11) can be relaxed, i.e to proveasymptotic stability of the equilibrium x∗ it suffices to require

lim

t→∞[f(x(t), ψ(x(t), z(t))) − f(x(t), ψ(x(t), 0))] = 0 (1.12)

In other words, it is not necessary to reach the manifold M, in order tostabilize the equilibrium x∗

Remark 1.3 If we can find a partition of x = col(x1, x2) with x1 ∈ Rp and

x2∈ Rn−p and a corresponding partition of π(·) = col(π1(·), π2(·)) such that

x1= π1(ξ) is a global change of coordinates, then (A3) is satisfied with z =φ(x) = x2−π2(π−1

1 (x1)) As a result, instead of considering the trajectories ofthe extended system (1.9)-(1.10) in (A4), it suffices to study the trajectories

of the system with state (x1, z)

We conclude this section by recalling a definition introduced in [1], whichwill be used in the rest of the chapter

Definition 1.1 The system (1.1) is said to be I&I-stabilizable with targetdynamics (1.2), if the assumptions (A1)–(A4) of Theorem 1.1 are satisfied.1.3 Adaptive Control via System Immersion

In this section we show how the general theory of Section 1.2 can be used todesign adaptive stabilizing controllers for nonlinear systems with parametricuncertainties The first step is to assume that a “full-information” control law(i.e a control law that depends on the unknown parameters) is available.4

4 Obviously, if it is not available, we could design one using Theorem 1.1

Then the problem is reduced to finding an adaptive law such that the loop system is immersed into the system that would result if we applied thefull-information controller.

closed-To illustrate this approach, consider again the system (1.1), where thevector field f(·) may depend on an unknown parameter vector θ ∈ Rq Assumethat there exists a parameterized control law u = υ(x, θ) such that the closed-loop system

as nonlinear PI adaptation

1.3.1 Systems Linear in the Unknown Parameters

In this section, to demonstrate the application of the foregoing theory and forsimplicity, we will consider linearly parameterized, control affine systems ofthe form

˙x = f0(x) + f1(x)θ + g(x)u (1.15)with state x ∈ Rn and input u ∈ Rm, where θ ∈ Rq is an unknown constantvector

Proposition 1.1 Consider the system (1.15) with an equilibrium point x∗to

be stabilized and assume the following hold

(B1) There exists a full-information control law u = υ(x, θ) such that theclosed-loop system

˙x = f∗(x) := f0(x) + f1(x)θ + g(x)υ(x, θ) (1.16)has a globally asymptotically stable equilibrium at x∗

(B2) There exists a mapping β1(·) such that all trajectories of the system

˙z = −∂β∂x1f1(x)z

˙x = f∗(x) + g(x) (υ(x, θ + z) − υ(x, θ)) (1.17)are bounded and satisfy

limt→∞[υ(x(t), θ + z(t)) − υ(x(t), θ)] = 0

Then the system (1.15) with assumptions (B1)-(B2) is adaptively I&I lizable

stabi-Proof We will verify that for the extended system

˙x = f0(x) + f1(x)θ + g(x)υ(x, ˆθ + β1(x))

˙ˆθ = β2(x, ˆθ)the conditions of Theorem 1.1 hold with (1.12) replacing (1.11) as pointed out

in Remark 1.2 First, (A1) is automatically satisfied from (B1) for ˙ξ = f∗(ξ).Second, for the immersion condition (A2), we are looking for mappings π1(·),

π2(·), c1(·) and c2(·) with

x = π1(ξ), ˆθ = π2(ξ)that solve the equations

φ(x, ˆθ) = ˆθ − θ + β1(x) = 0

It remains to prove that (A4) holds To this end, note that the dynamics ofthe off-the-manifold coordinate z = φ(x, ˆθ) are given by the equation

˙z = β2(x, ˆθ) + ∂β1

∂x f0(x) + f1(x) ˆθ + β1(x) − z + g(x)υ(x, ˆθ + β1(x)) Selecting the adaptation law

β2(x, ˆθ) = −∂β1

∂x f0(x) + f1(x) ˆθ + β1(x) + g(x)υ(x, ˆθ + β1(x))yields the first equation in (1.17), while the second equation is obtaineddirectly from (1.15) using (B1) and the definition of z Hence, by assump-tion (B2), the condition (A4) with (1.11) replaced by (1.12) holds

Example 1.1 Consider the stabilization to zero of the unstable first-order ear system

where θ > 0 is an unknown constant and note that (B1) is satisfied with

u = −kx − θx, for any k > 0, hence the I&I adaptive control law is given by

u = −kx − ˆθ + β1(x) x, ˙ˆθ = β2(x, ˆθ) (1.19)Defining

limt→∞[x(t)z(t)] = 0,i.e the condition (B2) holds As a result, the closed-loop system (1.18)-(1.19)has a globally asymptotically stable equilibrium at x = 0

1.3.2 Systems in Feedback Form

The assumption (B2), on which the result in Proposition 1.1 relies, is quiterestrictive in the sense that there is no systematic way of treating the cascade

system (1.17), unless the function υ(x, θ) is linear in θ or satisfies a chitz condition.5 In this section we depart from this approach and do notassume that a control law u = υ(x, θ) is known Instead, we design a dynamiccontroller directly using Theorem 1.1.

Lips-Consider a class of systems described by equations of the form

˙x1= x2

˙xp−1= xp

˙xp= xp+1+ φT(x1, , xp)θ

˙xp+1= xp+2

˙xn= xn+1= u

(1.21)

with states xi ∈ R, i = 1, , n and input u ∈ R, where θ ∈ Rq is a vector ofunknown parameters We will show that an adaptive, globally asymptoticallystabilizing control law for the system (1.21) can be obtained by applyingTheorem 1.1 In particular, we have the following result

Proposition 1.2 Consider the system (1.21) and the adaptation law

with ˆθ ∈ Rq The system (1.21)-(1.22) with inputs u and β2is I&I stabilizablewith target dynamics

˙ξ1= ξ2

˙ξp−1= ξp

˙ξp= −KTξ,

(1.23)

where ξ = [ξ1, , ξp]T and K is a constant vector

Proof To begin with, note that the vector K can be selected so that the targetsystem (1.23) is globally asymptotically stable, hence (A1) holds Considernow the immersion condition (A2) and the mappings

xi= πi(ξ), i = 1, , n

ˆθ = πn+1(ξ)

Setting πi(ξ) = ξi, for i = 1, , p, the equations (1.7) reduce to

5 See [1, Section IV-C] for more detail

c1(·) and c2(·) Note now that, from (A3), the off-the-manifold coordinates

z = x − π(ξ)are given by

∂β1

∂xjxj+1+∂β∂x1

pφT(x1, , xp) ˆθ + β1(x1, , xp) − zn−p+1

6 Notice that the last n − p + 1 equations are redundant, since they are globallydiffeomorphic to the first n − p + 1 equations (see Remark 1.3)

∂xjxj+1−∂β∂x1

pφT(x1, , xp) ˆθ + β1(x1, , xp)u(x, ˆθ) = −ΛT[z1, , zn−p]T +∂πn

∂ ˆθ β2(x, ˆθ) +

p j=1

∂πn

∂xjxj+1+∂π∂xn

pφT(x1, , xp) ˆθ + β1(x1, , xp)

−21 ∂π∂xn

p

2[z1, , zn−p]TP [0, , 1]T,where > 0 is an arbitrary constant, the vector Λ is chosen so that the matrix

1.4 Output Feedback Stabilization

In this section we extend the results of Section 1.3 to the output feedbackstabilization problem, where we seek a dynamic stabilizing control law thatdoes not depend on the unmeasured states

Consider the system

˙η = f(η, y, u)

with state (η, y) ∈ Rn × Rr and input u ∈ Rm and suppose that only thestate y is available for measurement Note that the system (1.25) may alsoinclude unknown parameters, i.e equations of the form ˙ηi= 0 Following theideas of Section 1.3 we assume that there exists a full-information control law

u = υ(η, y) such that all trajectories of the closed-loop system

Note that we only require that the output y converges to a set-point and that

η remains bounded This is because it may not be possible to drive the wholestate (η, y) to a desired equilibrium This is the case, for instance, when ηcontains unknown parameters Consider now the system

1.4.1 Linearly Parameterized Systems

As in Section 1.3.1, we will first consider linearly parameterized, control affinesystems of the form

˙η = f0(y) + f1(y)η + g1(y)u

˙y = h0(y) + h1(y)η + g2(y)u (1.29)with state (η, y) ∈ Rn× Rr, output y and input u ∈ Rm

Proposition 1.3 Consider the system (1.29) and assume the following hold.(C1) There exists a full-information control law u = υ(η, y) such that alltrajectories of the closed-loop system

˙η = f∗(η, y) := f0(y) + f1(y)η + g1(y)υ(η, y)

˙y = h∗(η, y) := h0(y) + h1(y)η + g2(y)υ(η, y) (1.30)are bounded and satisfy (1.27)

(C2) There exists a mapping β1(·) such that all trajectories of the system

Then there exists a dynamic output feedback control law described by equations

of the form

u = υ(ˆη + β1(y), y), ˙ˆη = β2(y, ˆη) (1.32)such that all trajectories of the closed-loop system (1.25)-(1.32) are boundedand satisfy (1.27)

Proof As in the proof of Proposition 1.1, we will construct a function β2(y, ˆη)

so that the closed-loop system (1.29)-(1.32) is transformed into (1.31) To thisend, let

z = ˆη − η + β1(y),which represents the distance of the system trajectory from the manifoldφ(η, y, ˆη) = ˆη − η + β1(y) = 0, and note that the dynamics of the variable zare given by

˙z = β2(y, ˆη) − f0(y) − f1(y) (ˆη + β1(y) − z) − g1(y)υ(ˆη + β1(y), y)+∂β∂y1[h0(y) + h1(y) (ˆη + β1(y) − z) + g2(y)υ(ˆη + β1(y), y)] Selecting the adaptation law

β2(y, ˆη) = f0(y) + f1(y) (ˆη + β1(y)) + g1(y)υ(ˆη + β1(y), y)

−∂β∂y1[h0(y) + h1(y) (ˆη + β1(y)) + g2(y)υ(ˆη + β1(y), y)]yields the first equation of (1.31), while the other two equations are obtainedfrom (1.29) with u = υ(ˆη + β1(y), y) = υ(η + z, y) by adding and subtractingthe term υ(η, y) Hence, by assumptions (C1)-(C2), all trajectories of theclosed-loop system (1.25)-(1.32) are bounded and satisfy (1.27)

Example 1.2 Consider the stabilization to zero of the second-order nonlinearsystem

˙η = η + y

with input u and output y, where η is an unknown state It is interesting tonote that the zero dynamics are described by the equation ˙η = η, hence thesystem (1.33) is not minimum-phase To begin with, note that (C1) is satisfiedwith the function

υ(η, y) = −η(y2+ 1) − (1 + k1+ k2)(η + y) − k1k2η

with k1 > 0, k2 > 0 Consider now the dynamic control law (1.32) and thechange of coordinates ˜y = y + (1 + k1)η Selecting

β2(y) = ∂β∂y1[(1 + k1+ k2) (ˆη + β1+ y) + k1k2(ˆη + β1)] + ˆη + β1+ y

yields the error system

The result in Proposition (1.3) relies on our ability to stabilize the cascadesystem (1.31) by assigning the function β1(·) Note that, by construction, when

z = 0, the (η, y)-subsystem in (1.31) is globally stable with y converging tothe desired equilibrium Therefore, it is natural to ask whether we can neglectthe (η, y)-subsystem and concentrate on the stabilization of the z-subsystemalone In the case of Proposition (1.3) this is not possible because, even if zconverges to zero exponentially, the term υ(η + z, y) − υ(η, y) may destabilizethe (η, y)-subsystem This clearly poses a restriction on the function υ(η, y).The present section provides an answer to this problem by establishing

a condition under which the design of the full-information control law u =υ(η, y) can be decoupled from the design of the function β1(·) and so thestabilization of the z-subsystem can be considered independently In this sense

it can be considered as a nonlinear counterpart of the well-known separationprinciple used in linear systems Interestingly, this result can be applied tosystems that are not affine in the input Moreover, the result can be extended

to systems that are nonlinear in the unmeasured states.7

We consider systems described by equations of the form

˙η = f0(y, u) + f1(y, u)η

˙y = h0(y, u) + h1(y, u)η (1.35)with state (η, y) ∈ Rn× Rr, output y and input u ∈ Rm As before, we areseeking a dynamic output feedback control law

7 See [5] for more detail

u = υ(ˆη + β1(y), y), ˙ˆη = β2(y, ˆη) (1.36)such that all trajectories of the closed-loop system (1.35)-(1.36) are boundedand

Then there exists a dynamic output feedback control law described by equations

of the form (1.36) such that all trajectories of the closed-loop system (1.36) are bounded and satisfy (1.37)

(1.35)-The proof is similar to the one of Proposition 1.3, hence it is omitted.Remark 1.4 Assumption (D2) can be replaced by the following (stronger)condition

(D3) There exists a mapping β1(·) such that the system (1.39) is uniformlyglobally stable for any y and u and z(t) is such that, for any y and u,

limt→∞z(t) = 0

If (D3) holds, then ˆη can be used to construct an asymptotic estimate of theunmeasured states η, which is given by ˆη + β1(y) However, to achieve thedesired control goal, only the reconstruction of the full-information controllaw υ(η, y) is necessary

1.5 Applications

1.5.1 Aircraft Wing Rock Suppression

In this section we apply the result in Section 1.3.2 to the problem of wing rockelimination in high-performance aircrafts This example has been adoptedfrom [6, Section 4.6], where a classical controller, based on the adaptive back-stepping method, has been proposed

Consider the system

˙x1 = x2

˙x2 = x3+ φ(x1, x2)Tθ (1.40)

˙x3 = 1τu −τ1x3,where the states x1, x2 and x3 represent the roll angle, roll rate and ailerondeflection angle respectively, τ is the aileron time constant, u is the controlinput, θ ∈ R5 is an unknown constant vector and

φ(x1, x2)T = 1, x1, x2, |x1|x2, |x2|x2 The control objective is to regulate x1to zero Note that, despite the presence

of extra terms in the dynamics of x3, the result in Section 1.3.2 still applies.The target dynamics are defined as

β2(x, ˆθ) = −∂β∂x1

1x2−∂β∂x1

2 x3+ φT(x1, x2) ˆθ + β1(x1, x2)1

Figure 1.1 shows the trajectory of the closed-loop system from the initialstate x(0) = [0.4, 0, 0] for the data provided in [6, Section 4.6], namely

θ = [0, −26.67, 0.76485, −2.9225, 0]

and τ = 1/15 The design parameters are k1 = 25, k2 = 10, λ = 5, γ =

100 and = 5000 We see that the proposed adaptive scheme recovers theperformance of the full-information controller The speed of response can befurther increased (or reduced) by tuning the parameter γ

Full-1.5.2 Output Voltage Regulation of Boost Converters

In this section we consider the problem of regulating the output voltage of aDC–DC boost converter with the circuit topology shown in Figure 1.2 Theaveraged model of the system can be described by equations of the form (1.35),namely

˙η1= 0

˙η2= −L1uy + L1η1

˙y = −RC1 y +C1uη2,

(1.42)

where the states η1, η2and y represent the input voltage, inductor current andoutput voltage respectively, L, C and R are positive constants and u ∈ [ , 1],with 0 < < 1, is the modulating signal of the PWM circuit controlling theswitch and acts as the control input The control objective is to regulate theoutput voltage y to a set-point y using output feedback only Note that, due

to the constraint ≤ u ≤ 1, the set-point y∗ must be such that

+ y

−

Fig 1.2 Diagram of the DC–DC boost converter

and the mapping β1(y) = λ1 λ2 T y, with λ1 > 0, λ2 > 0, yields the errordynamics

λ2> 0), in closed loop with a time-varying gain (¯u − u)/¯u, which is such that

|¯u − u¯u | ≤1 −1 + Note now that the H∞norm of the system (1.45) with output r is given by

1 + 2µ2 1 + 2/µ − 2µ2− 2µ, µ =

λ1

λ2 CL¯u.

Hence, a simple application of the small-gain theorem8 shows that the tem (1.44) is asymptotically stable provided

sys-γ(µ)1 −1 + < 1

Since γ(µ) ≥ 1 for any µ ≥ 0, this condition is equivalent to

>γ(µ) − 1γ(µ) + 1 (1.46)Note that the function on the right-hand side of (1.46) is zero for µ = 0 and

it is monotonically increasing and tends to one for µ → ∞, hence, for any, it is always possible to select µ > 0, thus λ1 > 0 and λ2 > 0, such that(1.46) holds As a result, the system (1.44) is uniformly globally asymptoticallystable Moreover, since the condition (D3) in Remark 1.4 holds, an asymptoticestimate of the unmeasured state η is given by

8 See [9, Proposition 3.4.7]

ˆη + β1(y) = ˆηˆη1+ λ1y

2+ λ2y = ηη12+ z+ z12 .The closed-loop system (1.42)-(1.43) has been simulated using the param-eters L = 20mH, C = 20µF and R = 30Ω The desired output voltage is

y∗ = 30V and the design parameters are = 0.1 and λ1 = λ2= 0.01, whichsatisfy the condition (1.46) It is assumed that the input voltage is η1= 15V

at t = 0s and it changes to η1= 10V at t = 0.02s In Figure 1.3 the input age η1 and the inductor current η2 together with their estimates, the outputvoltage y and the control signal u are displayed We see that the output volt-age y tracks the desired value y∗, despite partial state measurement and thechange in the input voltage In addition, the input voltage and the inductorcurrent estimates converge to the true values

1 1.5 2 2.5 3 3.5

1.6 Conclusions

The problem of designing globally stabilizing adaptive control laws for eral nonlinear systems has been addressed from a new perspective using thenotion of system immersion The proposed methodology is applicable to bothstate and output feedback control problems and treats unmeasured parame-ters and unknown states in a unified way The method has been illustratedwith several academic and practical examples, highlighting the potential ofthe proposed approach in solving nonlinear control problems with partial stateand/or parameter information

gen-References

1 A Astolfi and R Ortega (2003) Immersion and invariance: a new tool forstabilization and adaptive control of nonlinear systems IEEE Trans AutomaticControl, 48(4):590–606

2 C.I Byrnes, F Delli Priscoli, and A Isidori (1997) Output regulation of tain nonlinear systems Birkh¨auser

uncer-3 A Isidori (1995) Nonlinear Control Systems Springer-Verlag, 3rd edition

4 A Isidori (1999) Nonlinear Control Systems II Springer-Verlag

5 D Karagiannis, A Astolfi, and R Ortega (2003) Two results for adaptiveoutput feedback stabilization of nonlinear systems Automatica, 39(5):857–866

6 M Krsti´c, I Kanellakopoulos, and P Kokotovi´c (1995) Nonlinear and AdaptiveControl Design John Wiley and Sons

7 R Marino and P Tomei (1995) Nonlinear Control Design: Geometric, Adaptiveand Robust Prentice Hall

8 R Ortega, A Lor´ıa, P.J Nicklasson, and H Sira-Ram´ırez (1998) based Control of Euler-Lagrange Systems Springer-Verlag

Passivity-9 A van der Schaft (2000) L2-Gain and Passivity Techniques in Nonlinear trol Springer-Verlag, 2nd edition

Con-10 V.I Utkin (1992) Sliding Modes in Control and Optimization Springer-Verlag

Cascaded Nonlinear Time-Varying Systems: Analysis and Design

Antonio Lor´ıa and Elena Panteley

CNRS–LSS, Sup´elec, 3 rue Joliot Curie, 91192 Gif s/Yvette, France

loria@lss.supelec.fr, panteley@lss.supelec.fr

The general topic of study is Lyapunov stability of nonlinear time-varyingcascaded systems Roughly speaking these are systems in “open loop” as il-lustrated in the figure below

Fig 2.1 Block-diagram of a cascaded system

The results that we present here are not original, they have been published

in different scientific papers so the adequate references are provided in theBibliography The chapter gathers the material of lectures given by the firstauthor within the “Formation d’Automatique de Paris” 2004

The material is organised in three main sections In the first, we introducethe reader to several definitions and state our motivations to study time-varying systems We also state the problems of design and analysis of cascadedcontrol systems The second part contains the main stability results All thetheorems and propositions in this section are on conditions to guarantee Lya-punov stability of cascades No attention is paid to the control design problem.The third section contains some selected practical applications where controldesign aiming at obtaining a cascaded system in closed loop reveals to bebetter than classical Lyapunov-based designs such as Backstepping

The technical proofs of the main stability results are omitted here but theinterested readers are invited to see the references cited in the Bibliography

at the end of the chapter

Throughout this chapter we use the following nomenclature

F Lamnabhi-Lagarrigue et al (Eds.): Adv Top in Cntrl Sys Theory, LNCIS 311, pp 23–64, 2005

© Springer-Verlag London Limited 2005

Notation The solution of a differential equation ˙x = f(t, x), where f :

R≥0×Rn→ Rn, with initial conditions (t◦, x◦) ∈ R≥0×Rnand x◦= x(t◦), isdenoted by x(· ; t◦, x◦) or simply by x(·) We say that the system ˙x = f(t, x)

is uniformly globally stable (UGS) if the trivial solution x(· ; t◦, x◦) ≡ 0 isUGS (cf Definition 2.4) Respectively for uniform global asymptotic stability(UGAS, cf Definition 2.5) These properties will be precisely defined later |·|stands for the Euclidean norm of vectors and induced norm of matrices, and

· p, where p ∈ [1, ∞], denotes the Lp norm of time signals In particular, for

a measurable function φ : R≥t ◦ → Rn, φ pdenotes ( t∞◦ |φ(t)|pdt)1/p for p ∈[1, ∞) and φ ∞denotes the quantity ess supt≥t◦|φ(t)| We denote by Brtheopen ball Br:= {x ∈ Rn : |x| < r} We denote by ˙V(#)(t, x) := ∂V

∂t+∂V

∂xf(t, x)the time derivative of the (differentiable) Lyapunov function V (t, x) along thesolutions of the differential equation ˙x = f(t, x) labelled by (#) When clearfrom the context we use the compact notation V (t, x(t)) = V (t) We also use

LψV = ∂V

∂xψ for a vector field ψ : R≥0× Rq → Rn

2.1 Preliminaries on Time-Varying Systems

The first subject of study in this report is to establish sufficient (and for somecases, necessary) conditions to guarantee uniform global asymptotic stability(UGAS) (cf Definition 2.5) of the origin, for nonlinear ordinary differentialequations (ODE)

Most of the literature for nonlinear systems in the last decades has beendevoted to time-invariant systems nonetheless, the importance of nonau-tonomous systems should not be underestimated; these arise for instance asclosed-loop systems in nonlinear trajectory tracking control problems, that is,where the goal is to design a control input u(t, x) for the system

such that the output y follows asymptotically a desired time-varying ence yd(t) For a “feasible” trajectory yd(t) = h(xd(t)) some “desired” statetrajectory xd(t), satisfying ˙xd= f(xd, u), system (2.2) in closed loop with thecontrol input u = u(x, xd(t), yd(t)) may be written as

refer-˙˜x = ˜f(t, ˜x) , ˜x(t◦) = ˜x◦, (2.3a)

where ˜x = x−xd, ˜y = y−ydand ˜f(t, x) := f(˜x+xd(t), u(˜x+xd(t), xd(t), yd(t)).The so stated tracking control problem, applies to many physical systems, e.g

mechanical and electromechanical, for which there is a large body of literature(see [39] and references therein).

Another typical situation in which closed-loop systems of the form (2.3)arise, is in regulation problems (that is, when the desired set-point (xd, yd) isconstant) such that the open-loop plant is not stabilisable by continuous time-invariant feedbacks u = u(x) This is the case, for instance, of some driftless(e.g nonholonomic) systems, ˙x = g(x)u See e.g [4, 22]

A classical approach to analyse the stability of the nonautonomous system(2.1) is to search a so-called Lyapunov function with certain properties (seee.g [66, 20]) Consequently, for the tracking control design problem previouslydescribed one looks for so-called Control Lyapunov Function (CLF) for system(2.2) so that the control law u is derived from the CLF (see e.g [23, 47, 57])

In general, finding an adequate LF or CLF is very hard and one has to focus

on systems with specific structural properties This gave rise to approachessuch as the so-called integrator backstepping [23] and feedforwarding [30, 57].The second subject of study in this chapter, is a specific structure of sys-tems, which are wide enough to cover a large number of applications whilesimple enough to allow criteria for stability which are easier to verify thanfinding an LF for the closed-loop system These are cascaded systems We dis-tinguish between two problems: stability analysis and control design For thedesign problem, we do not offer a general methodology as in [23, 57] however,

we show through different applications, that simple (from a mathematicalviewpoint) controllers can be obtained by aiming at giving the closed-loopsystem a cascaded structure

2.1.1 Stability Definitions

There are various types of asymptotic stability that can be pursued for varying nonlinear systems As we shall see in this section, from a robustnessviewpoint, the most useful are uniform (global) asymptotic stability and uni-form (local) exponential stability (ULES) For clarity of exposition and self-containedness let us recall a few definitions (cf e.g [19, 66, 10]) as we usethem throughout the chapter

time-Definition 2.1 A continuous function α : R≥0 → R≥0 is said to belong toclass K if it is strictly increasing and α(0) = 0 It is said to be of class K∞ ifmoreover α(s) → ∞ as s → ∞

Definition 2.2 A continuous function β : R≥0×R≥0→ R≥0is said to belong

to class KL if, for each fixed s β(·, s) is of class K and for each fixed r, β(r, ·)

is strictly decreasing and β(r, s) → 0 as s → ∞

For system (2.1), we define the following

Definition 2.3 (Uniform boundedness) We say that the solutions of(2.1) are uniformly (resp globally) bounded (UB, resp., UGB) if there ex-ist a class K (resp K∞) function α and a number c > 0 such that

|x(t; t◦, x◦)| ≤ α(|x◦|) + c ∀ t ≥ t◦ (2.4)Definition 2.4 (Uniform stability) The origin of system (2.1) is said to

be uniformly stable (US) if there exist a constant r > 0 and γ ∈ K such that,for each (t◦, x◦) ∈ R≥0 × Br,

clas-2 It is clear from the above that uniform global boundedness is a necessarycondition for uniform global stability, that is, in the case that γ ∈ K∞wehave that (2.5) implies (2.4) However, a system may be (locally) uniformlystable and yet, have unbounded solutions

3 Another common characterization of UGS and which we use in someproofs is the following (see e.g [23]): “the system is UGS if it is USand uniformly globally bounded (UGB)” Indeed, observe that US im-plies that there exists γ ∈ K such that (2.5) holds Then, using (2.4) andletting b ≤ r, one can construct ¯γ ∈ K∞ such that ¯γ(s) ≥ α(s) + c for all

s ≥ b > 0 and ¯γ(s) ≥ α(s) for all s ≤ b ≤ r Hence (2.5) holds with ¯γ.Definition 2.5 (Uniform asymptotic stability) The origin of system(2.1) is said to be uniformly asymptotically stable (UAS) if it is uniformlystable and uniformly attractive, i.e., there exists r > 0 and for each σ > 0there exists T > 0, such that

|x◦| ≤ r =⇒ |x(t; t◦, x◦)| ≤ σ ∀ t ≥ t◦+ T (2.6)

If moreover the origin of system is UGS and the bound (2.6) holds for each

r > 0 then the origin is uniformly globally asymptotically stable (UGAS)

1 “Sufficiently small” here means that s is taken to belong to the domain of γ−1.Recall that in general, class K functions are not invertible on R≥0 For instance,tanh(|·|) ∈ K but tanh−1: (−1, 1) → R≥0