Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari
273
Springer Berlin Heidelberg NewYor...

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

273

Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo

Fritz Colonius, Lars Grüne (Eds)

Dynamics, Bifurcations, and Control With 85 Figures

13

Series Advisory Board

A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Editors Professor Fritz Colonius Universität Augsburg Institut für Mathematik Universitätsstraße 86150 Augsburg Germany

Dr. Lars Grüne J.W. Goethe-Universität Fachbereich Mathematik Postfach 11 19 32 60054 Frankfurt am Main Germany

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Dynamics, Bifurcations, and Control / Fritz Colonius, Lars Grüne (eds) Berlin; Heidelberg; NewYork; Barcelona; Hong Kong; London; Milano; Paris; Tokyo: Springer, 2002 (Lecture Notes in control and information sciences; 273) ISBN 3-540-42890-9

ISBN 3-540-42890-9

Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by author. Data-conversion by PTP-Berlin, Stefan Sossna Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10791459 62/3020Rw - 5 4 3 2 1 0

Preface

This volume originates from the Third Nonlinear Control Workshop "Dynamics, Bifurcations and Control", held in Kloster Irsee, April 1-3 2001. As the preceding workshops held in Paris (2000) and in Ghent (1999), it was organized within the framework of Nonlinear Control Network funded by the European Union (http://www.supelec.fr/lss/NCN). The papers in this volume center around those control problems where phenomena and methods from dynamical systems theory play a dominant role. Despite the large variety of techniques and methods present in the contributions, a rough subdivision can be given into three areas: Bifurcation problems, stabilization and robustness, and global dynamics of control systems. A large part of the fascination in nonlinear control stems from the fact that is deeply rooted in engineering and mathematics alike. The contributions to this volume reflect this double nature of nonlinear control. We would like to take this opportunity to thank all the contributors and the referees for their careful work. Furthermore, it is our pleasure to thank Franchise Lamnabhi-Lagarrigue, the coordinator of our network, for her support in organizing the workshop and the proceedings and for the tremendous efforts she puts into this network bringing the cooperation between the different groups to a new level. In particular, the exchange and the active participation of young scientists, also reflected in the Pedagogical Schools within the Network, is an asset for the field of nonlinear control. We, as all participants, enjoyed the pleasant atmosphere created by the Schwabisches Bildungszentrum Kloster Irsee and its staff during the workshop. Last but not least, we appreciate the financial support from the European Union which made it all possible.

Augsburg, Frankfurt a.M., September 2001

Fritz Colonius Lars Griine

Contents

I

Bifurcation Problems

Controlling a n Inverted P e n d u l u m with B o u n d e d Controls . . .

Diego M. Alonso, Eduardo E. Paolini, Jorge L. Moiola 1 Introduction 2 Description of the system 3 Bounded control law 4 Local nonlinear analysis 5 Numerical analysis of the global dynamical behavior 6 Desired operating behaviour 7 Conclusions References Bifurcations of Neural Networks with Almost Symmetric Interconnection Matrices Mauro Di Marco, Mauro Forti, Alberto Test 1 Introduction 2 Neural network model and preliminaries 3 Limit cycles in a competitive neural network 4 Hopf bifurcations in sigmoidal neural networks 5 Period-doubling bifurcations in a third-order neural network 6 Conclusion References Bifurcations in Systems with a Rate Limiter Francisco Gordillo, Ismael Alcald, Javier Aracil 1 Introduction 2 Behaviour of rate limiters 3 Describing function of rate limiters 4 Limit cycle analysis of systems with rate limiters 5 Bifurcations in systems with a rate limiter 6 Conclusions References

1 3

3 4 5 7 8 14 15 16 17 17 19 23 26 30 32 32 37 37 38 41 42 43 49 50

Monitoring and Control of Bifurcations Using Probe Signals.. 51 Munther A. Hassouneh, Hassan Yaghoobi, Eyad H. Abed 1 Introduction 51 2 Hopf bifurcation 52 3 Analysis of the effects of near-resonant forcing 54 4 Numerical example 57 5 Combined Stability Monitoring and Control 58

Vm

Table of Contents

6 Detection of Impending Bifurcation in a Power System Model 7 Conclusions References Normal Form, Invariants, and Bifurcations of Nonlinear Control Systems in the Particle Deflection Plane Wei Kang 1 Introduction 2 Problem formulation 3 Normal form and invariants 4 Bifurcation of control systems 5 Bifurcation control using state feedback 6 The cusp bifurcation and hysteresis 7 Other related issues 8 Conclusions References Bifurcations of Reachable Sets Near an Abnormal Direction and Consequences Emmanuel Trelat 1 Setup and definitions 2 Asymptotics of the reachable sets 3 Applications References

II

Stabilization and Robustness

Oscillation Control in Delayed Feedback Systems Fatihcan M. Atay 1 Introduction 2 Perturbations of linear retarded equations 3 The harmonic oscillator under delayed feedback 4 Controlling the amplitude and frequency of oscillations 5 Conclusion References Nonlinear Problems in Friction Compensation Antonio Barreiro, Alfonso Banos, Francisco Gordillo, Javier Aracil 1 Introduction 2 Conic analysis of uncertain friction 3 Harmonic balance 4 Frequencial synthesis using QFT 5 Discussion References

60 64 64 67 67 68 70 75 77 81 83 84 85 89 89 91 94 98

101 103 103 105 106 Ill 115 115 117 117 121 124 127 128 129

Table of Contents

IX

Time-Optimal Stabilization for a Third-Order Integrator: a Robust State-Feedback Implementation 131 Giorgio Bartolini, Siro Pillosu, Alessandro Pisano, Elio Usai 1 Introduction 131 2 Closed loop time-optimal stabilization for a third-order integrator .. 133 3 Sliding-mode implementation of the time-optimal controller 137 4 Simulation results 141 5 Conclusions 143 References 144 Stability Analysis of Periodic Solutions via Integral Quadratic Constraints 145 Michele Basso, Lorenzo Giovanardi, Roberto Genesio 1 Introduction 145 2 A motivating example 146 3 Problem formulation and preliminary results 148 4 Sufficient conditions for stability of periodic solutions 151 5 Application example 154 6 Conclusions 156 References 156 Port Controller Hamiltonian Synthesis Using Evolution Strategies Jose Cesdreo Raimundez Alvarez 1 Introduction 2 Port controlled Hamiltonian systems 3 Controller design 4 Preliminaries on evolution strategies 5 Evolutionary formulation 6 Case study - ball k beam system 7 Conclusions References Feedback Stabilization and l-L^ Control of Nonlinear Systems Affected by Disturbances: the Differential Games Approach . . Pierpaolo Soravia 1 Introduction 2 Differential games approach to nonlinear %oo control 3 Other stability questions 4 Building a feedback solution for nonlinear Hex, control References A Linearization Principle for Robustness with Respect to Time-Varying Perturbations Fabian Wirth 1 Introduction

159 159 160 160 162 165 167 169 170 173 173 175 181 182 188 191 191

X

Table of Contents

2 Preliminaries 3 The discrete time case 4 Continuous time 5 Conclusion References

192 195 197 199 200

III

201

Global Dynamics of Control Systems

On Constrained Dynamical Systems and Algebroids Jesus Clemente-Gallardo, Bernhard M. Maschke, Arjan J. van der Schaft 1 Introduction: Constrained Hamiltonian systems 2 What is a Lie algebroid? 3 Dirac structures and Port Controlled Hamiltonian systems 4 Constrained mechanical systems and algebroids 5 Control of constrained mechanical systems References On the Classification of Control Sets Fritz Colonius, Marco Spadini 1 Introduction 2 Basic definitions 3 Strong inner pairs 4 The dynamic index 5 The index of a control set near a periodic orbit References On the Frequency Theorem for Nonperiodic Systems Roberta Fabbri, Russell Johnson, Carmen Nunez 1 Introduction 2 Nonautonomous Hamiltonian systems 3 Generalization of Yakubovich's theorem References Longtime Dynamics in Adaptive Gain Control Systems Gennady A. Leonov, Klaus R. Schneider 1 Introduction 2 Assumptions and preliminaries 3 Localization of the global attractor 4 Longtime behavior and estimates of the Hausdorff dimension of the global attractor References

203

203 205 208 213 214 216 217 217 218 219 221 224 230 233 233 235 238 240 241 241 242 245 248 253

Table of Contents

XI

Model Reduction for Systems with Low-Dimensional Chaos . . 255 Carlo Piccardi, Sergio Rinaldi 1 Introduction 255 2 Peak-to-peak dynamics 256 3 The control problem 260 4 Examples of application 261 5 Delay-differential systems 263 6 Concluding remarks 265 References 267 Feedback Equivalence to Feedforward Forms for Nonlinear Single-Input Control Systems 269 Issa Amadou Tall, Witold Respondek 1 Introduction 269 2 Definitions and notations 271 3 Feedforward normal form 274 4 m-invariants 275 5 Main results 276 6 Examples 281 7 Feedforward systems in E 4 283 References 285 Conservation Laws in Optimal Control Delfim F. M. Torres 1 Introduction 2 Preliminaries 3 Main results 4 Examples References

287 287 289 291 294 295

List of Participants

297

Bifurcations in Systems with a Rate Limiter Francisco Gordillo, Ismael Alcala , and Javier Aracil Dept. Ingeniera de Sistemas y Automatica, Universidad de Sevilla. Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n. 41092 Sevilla, Spain

Abstract. Limit cycles analysis of feedback systems with rate limiters in the ac-

tuator can be implemented by a classical method in the frequency domain, the harmonic balance method. In this paper, the rate limiter describing function is obtained and applied to the search for limit cycles in such control systems. Three examples with three dierent bifurcations (saddle-node bifurcation of limit cycles, subcritical Hopf bifurcation at in nity and supercritical Hopf-like bifurcation) are included. The method is approximate but its main advantage is that intuition is gained into a dicult problem.

1

Introduction

Nonlinear control of nonlinear plants leads to nonlinear dynamical systems. The last class of systems can display complex dynamical behaviours, and, what is more important, after (even small) changes in parameters or in their system structure, one can observe qualitative changes in their behaviour modes. This is the realm of bifurcation theory [7,9], which supplies tools to study the points where these changes are produced and the archetypical forms of the state portrait changes in these points. These archetypes are of a great value for the control systems designer as they supply a uni ed and global perspective on the behaviour modes of the system (he can see what is expected to happen for all the parameter values involved). Bifurcation theory is a valuable tool for understanding the behaviour richness of nonlinear systems. Roughly speaking, when by moving system parameters one observes a qualitative change in the system response (to be deduced from the state portrait, for instance) it is said that the system undergoes a bifurcation phenomenon. These phenomena can lead to a dierent number of stationary solutions (equilibrium points), to the appearance of oscillations, or even more complex behaviours (chaos, for instance). After a bifurcation analysis it is possible to split the parameter space into several regions with dierent asymptotic dynamics [10]. Furthermore, from a certain point of view, bifurcations are related to robustness issues, since only far from the bifurcation points the system displays behaviours that are structurally stable. In this paper, the presence of a concrete nonlinearity, the rate limiter, in the actuator of control systems is considered. The study of the rate limiter is of a great importance due to fact that this nonlinearity exists in a large type of actuators in which the speed of response is bounded. Furthermore, F. Colonius, L. Grüne (Eds.): Dynamics, Bifurcations, and Control, LNCIS 273, pp. 37−50, 2002. Springer-Verlag Berlin Heidelberg 2002

38

F. Gordillo, I. Alcalá, and J. Aracil

the presence of rate limiters in plants that are controlled with PID gives rise to dicult problems, mainly with unstable plants [16,17]. Anti-windup techniques give good solutions for systems with saturation, but they do not work well in systems with rate limiters. In this paper, the rate limiter is studied with the describing function method [3,5,6,8] which helps to analyze the stability of the system, the appearance of limit cycles and other phenomena such as bifurcations. The method is approximate, but the fact that it allows to analyze not only the local stability but also the global one [1,2,11,15] together with the simplicity and facility of the use of the method, justi es the current study. Furthermore, the analysis leads to detect the occurrence of stable and unstable limit cycles that are organized by the archetypes supplied by several types of bifurcations. The describing function method has been previously applied to bifurcation analysis [4,11,14]. In this paper, three examples of systems with rate limiters are presented. These examples show three dierent kinds of bifurcations: saddle-node bifurcation of limit cycles, subcritical Hopf bifurcation at in nity and supercritical Hopf-like bifurcation. In this way an overall perspective of the behaviour modes displayed by the systems can be reached. This perspective is of a great interest for the control system designer. In Section 2, the dynamical response of rate limiters is described and the dierent behaviour modes when the input is sinusoidal are characterized. This analysis is used in Section 3 to obtain the analytical expression of the describing function of rate limiters. In Section 4 this describing function is used to analyze the existence of limit cycles, while in Section 5 the three examples of bifurcations are presented. Finally, some conclusions are drawn in Section 6.

2 Behaviour of rate limiters In this study the rate-limiter nonlinearity is considered as the block of Fig. 1 with one input and one output. The output attempts to follow its input but with the constraint that the slope of the output is bounded in m. 1

Input

Fig. 1.

1

Rate Limiter (slope m)

Output

Simulink block for a rate limiter.

In this way a saturation is applied not to the input of the nonlinearity but to the derivative of this input. Furthermore, if the output, , is less than the

Bifurcations in Systems with a Rate Limiter

39

input, u, the output will grow at the maximum speed, _ = m, and if > u then _ = m. A sinusoidal input is considered in the describing function method. In this case,

y(t) = a sin !t =) dydt(t) = a ! cos !t =) a! dydt(t) a!: For small values of a and !, the maximum slope of the input does not violate the bound m and, therefore, the output of the rate limiter y(t) follows its input y(t). This behaviour continues while a! m. When this bound is violated, the output of the nonlinearity y(t) is not able to follow exactly the input y(t) during some periods in which the output evolves at the maximum speed (m). Depending on the values of a and ! three qualitatively dierent behaviour modes of the output may exist:

mode (a) The slope of the input y(t) never violates the constraint m. The

input y(t) and the output y(t) are overlapped during the whole period of the input. This behaviour mode is not longer valid when

m !a = 1:

max dydt(t) = a! = m ()

(1)

mode (b) The output in this mode has time periods in which it follows the sinusoidal input and periods where it follows a straight line of slope m (see Fig. 2).

100

80

60

Input and Output

40

20

0

−20

−40

0

−60

−80

−100

0

10.5

11

11.5

12

12.5

13

Time

Fig. 2. Response of the rate limiter (solid line) to sinusoidal inputs (dashed) in mode (b).

40

F. Gordillo, I. Alcalá, and J. Aracil

Assume that the steady state has been reached. Let and be the values of !t when y(t) reaches y(t) (in Fig. 2). Notice that = + . Analogously, let and be the values of !t when y(t) leaves y(t), at these points. 0

0

0

d y(t) dy(t) dt = dt =

(

m for ! t = m for ! t = :

(2)

0

Notice that = + . It is evident that y(t) = y(t) in [; ] [ [ ; ]. During [ ; ] y(t) is a straight line slope equal to m and during [ ; + 2] y(t) is a straight line with slope equal to m. mode (c) In this case the output y(t) is too slow and is not able to follow the input y(t). Therefore , y(t) evolves at the maximum rate in both directions m. Figure 3 shows how, each time that the output reaches the input, the slope of the output changes. In the frontier between modes (b) and (c) = and = . Furthermore, into this critical case, the slope of the input is equal to m in !t = and equal to m in !t = . Thus, 0

0

0

0

0

0

0

0

m ( + ) sin = sin + !a a! cos = m

m = cos = p 2 : = arctan 2 =) !a 4 + 2

100

80

60

Input and Output

40

20

0

−20

−40

0

−60

−80

−100

5.2

5.4

5.6

5.8

6

6.2

6.4

Fig. 3. Response of the rate limiter (solid line) to sinusoidal inputs (dashed) in mode (c). Time

Bifurcations in Systems with a Rate Limiter

41

Notice the delay of y(t) with respect to y(t) in modes (b) and (c). This delay has an important eect in the global behaviour of the system. As it can be seen, the value of the adimensional parameter m=(!a) de nes the qualitative behaviour mode of the nonlinearity, as appears in Table 1. Modes (a) and (c) are particular cases of mode (b). Indeed, in mode (a) = 0 and = while in mode (c) = . This fact will be used in the next section to obtain the describing function of rate limiters. Modes

m Range of !a m (x) @@H x yR G>R (x) @@H x

#

(3)

with uR = RyR for some positive semi-de nite symmetric matrix R. Incorporating in (1) leads to models with the structure (

R(x)] @@Hx (x) + G(x)u : x_ = [J >(x) @ H y = G (x) @ x (x)

(4)

3 Controller design Considering the closed loop dynamics for the pair plant-controller given by x_ = [Jd(x) Rd (x)]

@Hd @x

(5)

The problem is to nd a static feed-back control u = (x) in (4) responsible for the new closed loop energy shaping. De ning

Jd (x) = J (x) + Ja (x) Rd (x) = R(x) + Ra (x)

Port Controller Hamiltonian Synthesis Using Evolution Strategies

Hd (x) = H(x) + Ha (x)

161

(6)

and considering x_ = [Jd (x) Rd (x)] @@Hxd @H = [J + Ja (R + Ra )] @@H x + @ xa @H = [J R] @@H x + [Ja Ra ] @ x + [Jd (x) @ H = [J R] @ x + G(x) (x) we conclude

Rd

(x)] @@Hxa

(7)

= G(x) (x) (8) )] Hxa + [ a a ] H x which is the basic relationship involving (x) and Ha . Considering K (x) = @@Hxa , the controller synthesis problem can be stated as: Given (x) (x) H G(x) and the desired equilibrium to be stabilized x 2 (x) + a (x) = [ (x) + a (x)]> 0 2. Integrability K (x) is the gradient of a scalar function: J

J

R

R

J

R

J

R

> @K K ( x) = ( x) @x @x 3. Equilibrium Assignment K (x) at x veri es @

K (x ) =

@

H (x )

x 4. Lyapunov Stability The Jacobian of K (x) at x satis es the bound K (x ) @x

@

@

@

2

@

H (x )

x2

(9)

162

J.C. Raimúndez Álvarez

Under these conditions, the closed-loop system u = (x) will be a Port Controlled Hamiltonian (PCH) system with dissipation of the form (5), where Hd (x) is given by (6) and @ Ha (x) = K (x) @x

(10)

Supposing that such a K (x) can be found, the control can be calculated using the formula h

i 1

(x) = G> (x)G(x) G> (x) n [Jd (x) Rd (x)] K (x) + [Ja (x)

o

Ra (x)] @@H x (x)

(11)

4 Preliminaries on evolution strategies 4.1

Evolutionary basics

Evolution Strategies (ES) belongs to the class of stocastic optimization techniques, commonly described as evolutionary algorithms. Simulated Evolution is based on the collective learning processes within a population of individuals, in the quest for survival [1]. Each individual represents a search point in the space of potential solutions to a given problem. There are currently three main lines of research strongly related but independently developed in simulated evolution : Genetic Algorithms (GA), Evolution Strategies (ES), and Evolutionary Programming (EP). In each of these methods, the population of individuals is arbitrarily initialized and evolves towards better regions of the search space by means of a stochastic process of selection, mutation, and recombination if appropriate. These methods dier in the speci c representation, mutation operators and selection procedures. While genetic algorithms emphasize chromosomal operators based on observed genetic mechanisms (e.g., cross-over and bit mutation), evolution strategies and evolutionary programming emphasize the adaptation and diversity of behavior from parent to ospring over successive generations. Evolution is the result of interplay between the creation of new genetic information and its evaluation and selection. A single individual of a population is aected by other individuals of the population as well as by the environment. The better an individual performs under these conditions the greater is the chance for the individual to survive for a longer while and generate ospring, which inherit the parental genetic information. The main contributions in the evolutionary computation approach are:

Model regularity independence (Applicable to nonsmooth problems).

Port Controller Hamiltonian Synthesis Using Evolution Strategies

163

Parallelization to cope with intensive cost tness computation. Population search versus individual search (classical). General meta-heuristics. Good convergence properties.

Evolutionary algorithms mimic the process of neo-Darwinian organic evolution and involves concepts such as: t Time or epoch. ; Individual. Exogenous parameters. (Search Space). Endogenous parameters. (Adaptation). P Population. P = 1 ; 1 ; : : : ; n ; n + (P ) Fitness. (P ) = (( 1 ); : : : ; (n )) ( ) : operators (Mutation, Selection, Variation, etc.) where ni is the number of individuals in the population. A simple evolutionary algorithm follows: t 0 initialize P evaluate (P ) f

g

ff

g

f

i gg

i

j

ospring and the best are selected as the next population [Q = ]. ( + ). Selects the survivors from the union of parents and ospring, such that a monotonic course of evolution is guaranteed [Q = P ] i

k

k

;

4.2

Fitness evaluation

Each individual is characterized by a set of exogenous and endogenous parameter values and respectively. The exogenous parameters are inherited from the response function. The endogenous parameters also called strategic parameters, they do not in uence the tness measure. Each individual represents a set of independent paths over the phase space beginning at dierent initial conditions, under the in uence of the same controller. This set of ns orbits should cover conveniently the phase space and can be randomly generated at the very beginning, being common to all the individuals of the population. Care must be taken in the process of initial conditions generation. The set of initial conditions must spread over the expected attraction basin and to avoid over tting [9] a minimum must be imposed over ns and to avoid prohibitive computational costs a maximum must be imposed over ns . Given an open-loop system represented as x_ = F (x) + G(x)u

(15)

and a stabilizing controller u = (; x) in which represents a set of parameters to be xed, the controller stabilizing behaviour can be measured taking the set of ns orbits described by the closed-loop system, beginning at a

Port Controller Hamiltonian Synthesis Using Evolution Strategies

165

set of initial conditions spread over a region of interest. Under the stabilizing controller action, the resultant set of orbits should approach the origin considered as an equilibrium point. The tness should detect and measure this performance to serve as learning factor. Thus being xk (0); k = 1; ; ns initial conditions, each orbit starting at xk (0) under the in uence of the controller parameters can be represented as: f

X k (; t) = X (; xk (0); t)

g

(16)

Settling time performance is measured through a function (X ) > 0 which normally has one of the following structures:

max X t (X ) = R tmax8t k

0

(17)

k

X k t dt

k

with > 1. A typical tness measure can be obtained as: f () = k1

X k

(X k ()) + k2

X k

b(X k ()) + k3 g()

(18)

k1 ; k2 and k3 being positive scale factors and g() is a measure of closeness from the parameters to the origin, as a means to guarantee regularity [9] in the approximator. Usually g() = and b(xk ()) is a barrier function [2] which penalizes unwanted states or control eorts. k

k

5 Evolutionary formulation Our individual will be represented by a set of exogenous parameters which are the controller parameters. (The endogenous parameters are related to the search process). As can be seen in (11), the controller depends on a , Ja and Ra . In this paper the Ja and Ra values will be heuristically chosen according to Remark 1 later. The evolutionist process will act only in a de ned according to (19). Under the controller action, a set of independent orbits started at previously de ned points xk (0); k = 1; ; ns (initial conditions), will reach the equilibrium point x and will remain there, assuring asymptotic stability. The asymptotic convergence task is performed by the evolutionist process unsupervised learning capabilities, through a behaviour measure ( tness) minimization. The controller derives from an energy function a whose gradient eld is modulated according (10),(11). Our purpose is then to nd a vector eld K (x) which is the gradient of an energy function a (x) with structure H

H

f

g

H

H

166

J.C. Raimúndez Álvarez

Ha (x) = > ( 1 ; 2 ; x) 0 ( 1 ; 2 ; x)

(19)

where ( 1 ; 2 ; x) is a neural net with n inputs, n outputs where n = dimfxg, with the structure ( 1 ; 2 ; x) = ( 1 ( 2 x)) and (x) = 2=(1 + exp( 2x)) 1. 0 is a positive de nite matrix of weights. ( k = fijk g; k = 0; 1; 2) are square matrices, so the vector of design parameters is obtained by putting inside a vector all the independent coecients of the square matrices 0 ; 1 ; 2 of size n n each, giving n = 3n(n +1)=2 Adopting for Ha (x) the structure (19) implies to assume x = 0. If rank( k ) = n; k = 1; 2 then x = x is the only point which obeys ( 1 ( 2x)) = x In order to characterize the controller performance, a set of initial conditions xk (0); fk = 1; ; ns g spread over the desired attraction basin are given, remaining constant during the calculations. The feasible controllers are those which obey the conditions stated in items 1 to 4. For the feasible controllers, the minimization process involves a measure over the plant-controller behaviour. The better the controller, the smaller the measure ( tness). This measure is achieved through the steps:

For a given calculate (x) (; x) according to (11) with K (x) K (x) = 2> ( 1 ; 2 ; x) 0

@ ( 1; 2; x) @x

Integrate for t 2 [0; tmax] the dierential equation @H x_ = [J R] + G(x) (; x)

(20)

@x

Being x(0) = fx1 (0); : : : ; xns (0)ig the set of ns initial conditions de ning the needed attraction basin, X (; t) the closed loop path with initial conditions xi (0) and 1, calculate the tness index

X i ( ; t) t i () = (X i ()) = t2[0max ;tmax ]

(21)

with a suitable norm kk so the tness is calculated as

=

ns X i

i())

(22)

Incorporating the feasibility search in the tness calculation gives the following procedure: function fitness n f

f = min eigen

@ > 0 @ @2H @xi @xj + @x @x (x ); eigen

0 ;

Port Controller Hamiltonian Synthesis Using Evolution Strategies

if

167

(f < 0) f = k1 + k2 jj;

else;

f =

end return

g

P

ns i

i ();

f;

This procedure applies a penalization to () in the case that the desired positive-de niteness conditions fail. k1 and k2 are large positive numbers.

6 Case study - ball & beam system r x Jb

Jp

Fig. 1.

Ball & Beam diagram

Consider the Ball & Beam plant as can be depicted in Figure 1. The beam is made to rotate in a vertical plane by applying a torque at the center of rotation, and the ball is free to roll along the beam which is one-dimensional. The ball must remain in contact with the beam and the rolling must occur without slipping, which imposes a constraint on the rotational acceleration of the beam as well as in the friction coecient. This plant is a well known example of a nonlinear system which is neither feedback linearizable nor minimum phase. Controllers for tracking purposes can be found in [6],[5]. Our goal is to drive the ball to the rest position over a set of initial conditions with values taken on a neighborhood of the origin given by 0:5 xi 0:5; i = 1; : : : ; 4 on the phase space, which characterize the needed attraction basin. Let the moment of inertia of the beam be Jp , the mass and moment of inertia of the ball be mb and Jb respectively, the radius of the ball be r and the

168

J.C. Raimúndez Álvarez

acceleration of gravity be g. Choosing the beam angle and the ball position over the beam x as generalized coordinates for the system and according with the above diagram, the kinetic energy is given by

T (; x; ;_ x_ ) = 12 Jp + mb x2 _2 + Jr2b + mb x_ 2 and the potential by

V (; x) = mb gx sin transforming to the hamiltonian formalism in which

x1 = x x2 = x3 = px x4 = p

(23)

the total system energy is given by

H(x) = 12 J =rx + m + J +xm x + mb gx sin x b b p b 2 3

2 4

2

2 1

1

2

(24)

and the movement has the description:

0 x_ BB x_ @ x_

1 2 3

x_ 4

1 0 CC = BB A @

0 0 1 0

0 @H 1 0 1 @x C 0 10 B C B BB @@xH CC + G(x)u = BB 0 0 1C C B@ 0 0 0AB @H C C B @x 100 @ A 1 2 3

@H @x4

x3 Jb =r2 +mb x4 J2p +mb x21 mb x1 x4 mb g sin x2 2 (Jp +mb x2 1) mb gx1 cos x2 +u

1 CC CA (25)

calculating with the values

r = 0:04 Jb = 0:001 Jp = 2 mb g = 4:9

Remark 1 The choice of Ja and Ra should accomplish the conditions previ-

ously stated within section 3. There are no general rules for their determination. The considered Ball & Beam plant has no dissipation so it is convenient

Port Controller Hamiltonian Synthesis Using Evolution Strategies

169

to de ne Ra 0 in order to inject dissipation through the controller action. Concerning Ja , their contribution should be considered for interconnection enhancement in plants which have poor interconnection between states. An example of interconnection enhancement between mechanical and electrical parts using the participation of Ja can be seen in [3]. For our plant we choose

0 B Ja = B @

1

0 101 1 0 0 0C C 0 0 0 0A; 1000

04 0 0 01 B 0 4 0 0 CC Ra = B @0 0 0 0A

(26)

0000

Processing in a Pentium II at 600Mz after 20 minutes we obtained: PI0 = [ 6.966073 -5.319661 -2.699448 -4.489864 PI1 = [ 0.379863 -0.460202 0.004198 0.172915

-0.011644 -1.817432 -0.208203 -0.018196

-1.027616 -0.597628 -0.054679 -0.085693

-5.319661 7.976571 3.093717 1.910575

-0.185061 -0.615594 0.459153 -0.274629 ]';

-2.699448 3.093717 1.860503 0.797894

-4.489864 1.910575 0.797894 7.964292 ]';

PI2 = [ -0.882976 0.259639 -0.547722 -1.013208

-2.350738 -0.259648 0.3397 -0.782236

-0.035688 0.056688 -0.572971 -0.46785

0.296927 1.268853 -1.985099 2.375783 ]';

In Figures 2,3 the local potential and kinetic energy modi cations introduced by Ha can be observed. In Figure 4 a set of four orbits with initial conditions over the needed attraction basin are shown characterizing the controller eectiveness. 7

Conclusions

The main contribution of this technique is associated to the vector eld

K (x) determination, avoiding the cumbersome resolution of a partial dierential equation. The saturated nature of neural nets also facilitates the bounding of the control action as practically desired. The vector eld K (x) maximum strength is xed previously, depending on the desired attraction basin, after a judicious eort consideration. The maximum strength is controlled as a barrier on the maximum eigenvalue of 0 and limiting a suitable norm over 1 and 2 (18). The controller performance depends on the choice of Ja ,Ra which models state interconnections and dissipation and which models closed-loop settling time. The plant model can be described with class C 0 or even piecewise C 0 functions and state space restrictions are also allowed (anti slippage for instance)

170

J.C. Raimúndez Álvarez

H

H

1

5

0.8

4

0.6 3

0.4

2

0.2

1 0

0

−1

−0.2 −2

−0.4

−3 −4

−0.6 1 0.5

−5 −1

−0.5

−0.8

0 0

0.5

−0.5 1

−1

H + Ha

−1 −1

−0.8

−0.6

−0.4

−0.2

−0.8

−0.6

−0.4

−0.2

1

14

0.8

12

0.6

10

0.4

H + Ha 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.2

8

0 6

−0.2 4

−0.4 2

−0.6 1 0.5

0 −1

−0.5

0 0

0.5

−0.8

−0.5 1

−1

−1 −1

Comparison of H and Hd = H + Ha for f 1 = 0 4 = 0. (Potential).

Fig. 2.

1g

; x3

x1

1g f 1 ;

x2

; x

As future work we are also introducing the automatic con guration of and Ra using an additional neural net that will evolve jointly with the controller.

Ja

Acknowledgements This work is supported by CICYT, under project TAP99-0926-C04-03.

References 1. Schwefel H.-P., Rudolph, G. (1995), Contemporary Evolution Strategies. In: Moran F., Moreno A., Merelo J.J., Chacon, P. (eds.) Advances in Arti cial Life. Third International Conference on Arti cial Life, vol. 929 of Lecture Notes in Computer Science, 893{907, Springer, Berlin 2. Fiacco A.V., McCormick, G.P. (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley and Sons, Inc. 3. Ortega R., van der Schaft A.J., Mareels I., Maschke B. (2001), Putting Energy Back in Control. Control Systems Magazine, 21, 18{33

Port Controller Hamiltonian Synthesis Using Evolution Strategies

H

H

1

0.7

0.8

0.6

0.6

171

0.4

0.5

0.2 0.4

0 0.3

−0.2 0.2

−0.4 0.1

−0.6 1 0.5

0 −1

−0.8

0

−0.5

0

0.5

−0.5 1

−1

H + Ha

−1 −1

−0.8

−0.6

−0.4

−0.2

−0.8

−0.6

−0.4

−0.2

1

12

H + Ha 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.8

0.6

10

0.4 8

0.2 6

0

−0.2

4

−0.4 2

−0.6 1 0.5

0 −1

0

−0.5

0

0.5

−0.8

−0.5 1

−1

−1 −1

Fig. 3. Comparison of H and Hd = H + Ha for 1g f 1 4 1g. (Kinetic). ;

x1

=0

; x2

=0 f 1 ;

x3

x

4. Ortega R., Spong M.W. (2000), Stabilization of Underactuated Mechanical Systems Via Interconnection and Damping Assignment. CNRS-SUPELEC, University of Illinois, Proceedings of Lagrangian and Hamiltonian Methods for Nonlinear Control, 1, 69{74 5. Liu P., Zinober A.S.I (1994), Recursive Interlacing Regulation of Flat and NonFlat Systems. School of Mathematics and Statistics, University of Sheeld. 6. Hauser J., Sastry S., Kokotovich P. (1992), Nonlinear Control Via Approximate Input-Output Linearization: The Ball and Beam Example. School of Mathematics and Statistics, University of Sheeld, Centre for Systems & Control, University of Glasgow | Internal Report, 37, 392{398 7. van der Schaft A.J. (1996), L2 Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Science, Vol. 218, Springer Verlag, London 8. Back T., Hammel U., Schwefel H.P. (1997), Evolutionary Computation: Comments on the History and Current State. In: IEEE Trans. on Evolutionary Computation, vol. 1 n 1 9. Sjoberg J., Ljung L. (1992), Overtraining, regularization and searching for minimum in neural networks. In: 4th IFAC Symposium on Adaptive Systems in Control and Signal Processing, Grenoble, France, 669{674

172

J.C. Raimúndez Álvarez

0.5

0.4

0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5 −1.5

−1

−0.5

0

0.5

1

1.5

6

4

2

0

−2

−4

−6

0

Fig. 4. x1

1

2

3

4

5

6

x2 phase space projection (upper) and u(t) (lower)

7

8

A Linearization Principle for Robustness with Respect to Time-Varying Perturbations ? Fabian Wirth Zentrum fur Technomathematik, Universitat Bremen, 28334 Bremen, Germany, fabian@math.uni-bremen.de

Abstract. We study nonlinear systems with an asymptotically stable xed point

subject to time-varying perturbations that do not perturb the xed point. Based on linearization theory we show that in discrete time the linearization completely determines the local robustness properties at exponentially stable xed points of nonlinear systems. In the continuous time case we present a counterexample for the corresponding statement. Sucient conditions for the equality of the stability radii of nonlinear respective linear systems are given. We conjecture that they hold on an open and dense set.

1

Introduction

A natural question in perturbation or robustness theory of nonlinear systems concerns the information that the linearization of a nonlinear system at a singular point contains with respect to local robustness properties. This question has been treated for time-invariant perturbations in [8] for continuous time, (see the references therein for the discrete time case). The result obtained in these papers was that generically the linearization determines the local robustness of the nonlinear system, where genericity is to be understood in the sense of semi-algebraic geometry (on the set of linearizations). Speci cally, the objects under consideration are the local stability radius of the nonlinear system and the stability radius of the linear system, where as usual the stability radius of a system is the in mum of the norms of destabilizing perturbations in a prescribed class. The question is then, whether these two quantities are equal or, more precisely, when this is case, see also [4, Chapter 11]. In this paper we treat this problem for nonlinear systems subject to timevarying perturbations. Our analysis is based on recent results on the generalized spectral radius of linear inclusions. In particular, we see a surprising dierence between continuous and discrete time. While the linearization always determines the robustness of the nonlinear system if the nominal system is exponentially stable this fails to be true for continuous time. On the other hand we are able to give a sucient condition which guarantees equality between linear and nonlinear stability radius on an open set of systems. As it is known from [9] that the Lebesgue measure of those linearizations for which ?

Research supported by the European Nonlinear Control Network.

F. Colonius, L. Grüne (Eds.): Dynamics, Bifurcations, and Control, LNCIS 273, pp. 191−200, 2002. Springer-Verlag Berlin Heidelberg 2002

192

F. Wirth

it is possible that the nonlinear stability radius is dierent from the linear is zero it seems therefore natural to conjecture that the set of systems where these two quantities coincide is open and dense. We proceed as follows. In Section 2 we recall the de nition of the stability radius for nonlinear systems with time varying perturbations and state some relevant results from the theory of linear inclusions. In particular, we recall upper and lower bounds of the stability radius of the nonlinear system in terms of the stability radius and the strong stability radius of the linearization. In Section 3 we develop a local robustness theory based on the linearization of the system for the discrete time case. It is shown that the two linear stability radii coincide under weak conditions, demonstrating that one need only consider the linearization in order to determine the local nonlinear robustness properties of a system. The continuous time case is treated in Section 4. We rst present a counterexample showing that analogous statements to the discrete time case cannot be expected in continuous time. We then present a sucient condition for the equality of the two linear stability radii on an open set. Concluding remarks are found in Section 5.

2 Preliminaries Consider nominal discrete and continuous time nonlinear systems of the form x(t + 1) = f0 (x(t)) ; t 2 N ; (1a) x_ (t) = f0 (x(t)) ; t 2 R+ ; (1b) which are exponentially stable at a xed point which we take to be 0. By this we mean that there exists a neighborhood U of 0 and constants c > 1; < 0 such that the solutions '(t; x; 0) of (1a),(1b) satisfy k'(t; x; 0)k ce tkxk for all x 2 U . As the concepts we will discuss do not dier in continuous and discrete time we will summarize our notation by writing T = N ; R + for the time-scale and x+ (t) := x_ (t); x(t + 1) according to the time-scale we are working on. Assume that (1a),(1b) are subject to perturbations of the form

x+ (t) = f0 (x(t)) +

Xm di(t)fi(x(t)) =: F (x(t); d(t)) ; i=1

(2)

where the perturbation functions fi leave the xed point invariant, i.e. fi (0) = 0; i = 0; 1; : : : ; m. We assume that the fi are continuously dierentiable in 0 (and locally Lipschitz in the case T = R+ ). The unknown perturbation function d is assumed to take values in D Rm ,

d : T ! D ;

A Linearization Principle for Robustness

193

where in the case T = R+ we impose that d is measurable. Here > 0 describes the perturbation intensity, which we intend to vary in the sequel, while the perturbation set D is xed. Thus structural information about the perturbations one wants to consider can be included in the functions fi ; i = 1; : : : ; m and in the set D. For the perturbation set D Rm we assume that it is compact, convex, with nonempty interior, and 0 2 int D. Solutions to the initial value problem (2) with x(0) = x0 for a particular time-varying perturbation d will be denoted '(t; x0 ; d). The question we are interested in concerns the critical perturbation intensity at which the system (2) becomes unstable. The stability radius is thus de ned as

rnl (f0 ; (fi )) := inf f > 0 j 9d : T ! D : x+ (t) = F (x(t); d (t)) is not asymptotically stable at 0g :

(3)

By linearizing the perturbed system in (2) we are led to the system

x+ (t) = A0 +

m

X

i=1

!

di (t)Ai x(t) ; t 2 T :

(4)

This is a (discrete or dierential) linear inclusion, which is in principle determined by the set (

M(A0 ; : : : ; Am ; ) := A0 +

m

X

i=1

)

di Ai kdk :

If the matrices Ai are xed we will denote this set by M() for the sake of succinctness. The inclusion (4) is called exponentially stable, if there are constants M 1; < 0 such that

k (t)k Me tk (0)k ; 8t 2 T for all solutions of (4). Exponential stability is characterized by the number

(M(A0 ; : : : ; Am ; )) := sup lim sup k (t)k1=t ; t!1

where the supremum is taken over all solutions of (4). Namely, (4) is exponentially stable i (M(A0 ; : : : ; Am ; )) < 1. Again we will write () if there is no fear of confusion. In the discrete time case the number is known as the joint or the generalized spectral radius. We refer to [2,10] for further characterizations of this number and for further references. In the continuous time case it is more

194

F. Wirth

customary to consider the quantity () := log (), which is known under the name of maximal Lyapunov exponent, see [4] and references therein. As in the nonlinear case we now de ne stability radii by rLy (A0 ; (Ai )) := inf f 0 j () 1g ; rLy (A0 ; (Ai )) := inf f 0 j () > 1g : The relation between the linear and the nonlinear stability radii is indicated by the following result which is contained in [3] for the continuous and in [7] for the discrete time case. Lemma 1. Let T = N; R + and consider system (2) and its linearization (4), then T (A0 ; (Ai )) rT (f0 ; (fi )) rT (A0 ; (Ai )) : rLy Ly nl

It is the aim of this paper to obtain further results on the information the linear stability radii contain for the nonlinear system. The following set of matrix sets will play a vital role in our analysis. Recall that a set of matrices M is called irreducible if only the trivial subspaces of Rn are invariant under all A 2 M. We de ne

I (Rnn ) := fM Rnn j M compact and irreducibleg : Note that this set is open and dense in the set of compact subsets of Rnn endowed with the usual Hausdor metric. The proof of the following statements can be found in [10]. They are the foundation for our analysis of linearization principles. Theorem 1. (i) The generalized spectral radius is locally Lipschitz continuous on I (Rnn ). (ii) The maximal Lyapunov exponent is locally Lipschitz continuous on I (Rnn ). Furthermore in the discrete time case a strict monotonicity property can be shown to hold, under the assumption that the following condition can be satis ed. Given A 2 Rnn we denote by PA the reducing projection corresponding to the eigenvalues 2 (A) with jj = r(A). Property 1. The set M K(Rnn ) is said to have Property 1 if n = 1; 2 or if there exists an A 2 conv M such that

r(A) < (M) ; or rank PA 6= 2 ; or ((I PA )A) 6= f0g : It is easy to construct a set M that does not possess Property 1 (just take a set of matrices with entries A11 = A22 = 1 and A12 = c and zero

elsewhere). The interesting question, however, is whether it is possible to do

A Linearization Principle for Robustness

195

this in a way so that M is irreducible. We would assume that this is not the case, but this matter remains unresolved for the moment. In any case, the negation of Property 1 is highly nongeneric, as it requires that the spectrum of all A 2 M are contained in f0g [ fz 2 C j jz j = (M)g. Modulo this point we now state a monotonicity property of the generalized spectral radius valid in the discrete time case. In the following we denote the ane subspace generated by a set M Rnn by a M while int a M Y denotes the interior of Y with respect to this ane subspace. Proposition 1. Let M1; M2 2 I (Rnn ) satisfy M1 6= M2 and M1 int a M2 conv M2 : (5) Assume that M1 has Property 1 then (M1 ) < (M2 ) :

3 The discrete time case In discrete time the situation turns out to be particularly simple. In fact, if Property 1 holds then we can immediately conclude the following linearization principle. Theorem 2. Let T = N and consider the discrete-time system (1a) and the perturbed system (2) along with its linearization (4). If for some < rLy (A0 ; (Ai )) the set M( ) is irreducible and satis es Property 1, then rLy (A0 ; (Ai )) = rnl (f0 ; (fi )) = rLy (A0 ; (Ai )) : Proof. The assumptions guarantee that the map 7! () is strictly increasing on [ ; 1) by Proposition 1. This implies rLy (A0 ; (Ai )) = rLy (A0 ; (Ai )). The assertion now follows from Lemma 1. The situation simpli es even more if we assume that the unperturbed system (1a) is exponentially stable. We can use this natural assumption to replace the somewhat awkward condition concerning Property 1. The reason for this is simple. Exponential stability implies that r(A0 ) < 1. The stability radius rLy (A0 ; (Ai )) equals only if (M(A0 ; A1 ; : : : ; Am ; )) = 1 > r(A0 ). These two things enforce that M(A0 ; A1 ; : : : ; Am ; ) has Property 1. Corollary 1. Let T = N and consider the discrete-time system (1a) and the perturbed system (2) along with its linearization (4). If the point x = 0 is exponentially stable for the unperturbed system x(t + 1) = f0 (x(t)) then rLy (A0 ; (Ai )) = rnl (f0 ; (fi )) = rLy (A0 ; (Ai )) :

196

F. Wirth

Proof. There exists a similarity transformation T such that all Ai , i = 0, : : : ,

m are similar to matrices of the form 2 i i 3 A11 A12 : : : : : : Ai1d 66 0 Ai22 Ai23 : : : Ai2d 7 66 .. 777 i 0 0 A . 7; 33 TA T 1 = 6 i

66 .. . . . . . . .. 77 . 7 64 . 5 i 0 : : : 0 Add where each of the sets Mj := fAijj j i = 0; : : : ; mg; j = 1; : : : ; d is irreducible. It holds that () = maxj=1;::: ;d (Mj ()). Thus it is sucient to consider the blocks individually to determine rLy , resp. rLy . Under the assumption of exponential stability we have r(A0 ) < 1. Hence for each j we have r(A0jj ) < 1 and the set Mj () has Property 1 for all > 0 such that (Mj ()) > r(A0 ). Now the result follows from Theorem 2. Corollary 2. Let T = N . The stability radius of linear systems with respect to time-varying perturbations rLy is continuous on the set f(A0 ; : : : ; Am ) 2 (Rnn )m+1 j r(A0 ) 6= 1g : Furthermore, the set f(A0 ; : : : ; Am ) 2 (Rnn )m+1 j rLy (A0 ; : : : ; Am ) 6= rLy (A0 ; : : : ; Am )g is contained in a lower dimensional algebraic set. Proof. It was shown in [7] that rLy ; rLy are upper respectively lower semicontinuous on (Rnn )m+1 . The preceding Corollary 1 shows that these two functions coincide if r(A0 ) < 1, which shows continuity in this case. If r(A0 ) > 1 the statement is obvious as both functions are equal to 0. The second statement now follows because a necessary condition for the condition rLy (A0 ; : : : ; Am ) 6= rLy (A0 ; : : : ; Am ) is r(A0 ) = 1. The latter condition de nes a lower dimensional algebraic set. The result for the linear stability radii extends to the case of nonlinear systems as follows. First, denote by C 1 (Rn ; Rn ; 0) the set of continuously dierentiable maps from Rn to itself satisfying f (0) = 0. This space may be endowed with the C 1 topology inherited from the topologies on the space C 1 (Rn ; Rn ), (see [6, Chapter 17]). Corollary 3. Given n; m 2 N , the set W of functions (f0 ; f1 ; : : : ; fm ) 2 C 1 (Rn ; Rn ; 0)m+1 for which rnl (f0 ; (fi )) = rLy (A0 ; (Ai )) (6) contains an open and dense subset of C 1 (Rn ; Rn ; 0)m+1 with respect to both the coarse and the ne C 1 topology. Proof. This is immediate from the de nition of the C 1 topology.

A Linearization Principle for Robustness

197

4 Continuous time A natural question is if statements similar to those of Theorem 2 and Corollary 1 hold in continuous time. The fundamental tool for this results is the monotonicity property given by Proposition 1. This statement is unfortunately in general false in continuous time, as any subset M1 of the skewsymmetric matrices generates a linear inclusion whose system semigroup is a subset of the orthogonal group and for which the maximal Lyapunov exponent is therefore equal to 0. Taking a set M2 which contains M1 in its interior (with respect to the skew-symmetric matrices) does not yield a Lyapunov exponent larger than zero, so that the strict monotonicity property fails to hold. This example leaves still some hope that maybe a statement corresponding to Corollary 1 remains true in continuous time. The following example shows that even such expectations are unfounded. Example 1. Consider the matrices A(d) := 0d 2d+ d : It is easy to see that A (d) + A(d) 0 for all d 2 ( 1; 2). Hence for D ( 1; 2) it is immediate that (D) 0 as the Euclidean unit ball is forward invariant under the associated time-varying linear system. On the other hand while for the spectral abscissa (A(0)) = 0, we have (A(d)) < 0 for all d 2 (0; 2), see Figure 1.

spectral abscissa

2 1 0 −1 −2 −3 Fig. 1.

−2

−1

0 d

1

2

3

The spectral abscissa of A(d) in dependence of d.

The consequence of this is the following. If we de ne A0 = A(1=2) and A1 := 01 11 ; then 0 < rLy (A0 ; A1 ) 21 < 32 = rLy (A0 ; A1 ) ;

198

F. Wirth

because at least A0 1=2A1 = A(0) is not asymptotically stable. While on the other hand for

273

Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo

Fritz Colonius, Lars Grüne (Eds)

Dynamics, Bifurcations, and Control With 85 Figures

13

Series Advisory Board

A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Editors Professor Fritz Colonius Universität Augsburg Institut für Mathematik Universitätsstraße 86150 Augsburg Germany

Dr. Lars Grüne J.W. Goethe-Universität Fachbereich Mathematik Postfach 11 19 32 60054 Frankfurt am Main Germany

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Dynamics, Bifurcations, and Control / Fritz Colonius, Lars Grüne (eds) Berlin; Heidelberg; NewYork; Barcelona; Hong Kong; London; Milano; Paris; Tokyo: Springer, 2002 (Lecture Notes in control and information sciences; 273) ISBN 3-540-42890-9

ISBN 3-540-42890-9

Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by author. Data-conversion by PTP-Berlin, Stefan Sossna Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10791459 62/3020Rw - 5 4 3 2 1 0

Preface

This volume originates from the Third Nonlinear Control Workshop "Dynamics, Bifurcations and Control", held in Kloster Irsee, April 1-3 2001. As the preceding workshops held in Paris (2000) and in Ghent (1999), it was organized within the framework of Nonlinear Control Network funded by the European Union (http://www.supelec.fr/lss/NCN). The papers in this volume center around those control problems where phenomena and methods from dynamical systems theory play a dominant role. Despite the large variety of techniques and methods present in the contributions, a rough subdivision can be given into three areas: Bifurcation problems, stabilization and robustness, and global dynamics of control systems. A large part of the fascination in nonlinear control stems from the fact that is deeply rooted in engineering and mathematics alike. The contributions to this volume reflect this double nature of nonlinear control. We would like to take this opportunity to thank all the contributors and the referees for their careful work. Furthermore, it is our pleasure to thank Franchise Lamnabhi-Lagarrigue, the coordinator of our network, for her support in organizing the workshop and the proceedings and for the tremendous efforts she puts into this network bringing the cooperation between the different groups to a new level. In particular, the exchange and the active participation of young scientists, also reflected in the Pedagogical Schools within the Network, is an asset for the field of nonlinear control. We, as all participants, enjoyed the pleasant atmosphere created by the Schwabisches Bildungszentrum Kloster Irsee and its staff during the workshop. Last but not least, we appreciate the financial support from the European Union which made it all possible.

Augsburg, Frankfurt a.M., September 2001

Fritz Colonius Lars Griine

Contents

I

Bifurcation Problems

Controlling a n Inverted P e n d u l u m with B o u n d e d Controls . . .

Diego M. Alonso, Eduardo E. Paolini, Jorge L. Moiola 1 Introduction 2 Description of the system 3 Bounded control law 4 Local nonlinear analysis 5 Numerical analysis of the global dynamical behavior 6 Desired operating behaviour 7 Conclusions References Bifurcations of Neural Networks with Almost Symmetric Interconnection Matrices Mauro Di Marco, Mauro Forti, Alberto Test 1 Introduction 2 Neural network model and preliminaries 3 Limit cycles in a competitive neural network 4 Hopf bifurcations in sigmoidal neural networks 5 Period-doubling bifurcations in a third-order neural network 6 Conclusion References Bifurcations in Systems with a Rate Limiter Francisco Gordillo, Ismael Alcald, Javier Aracil 1 Introduction 2 Behaviour of rate limiters 3 Describing function of rate limiters 4 Limit cycle analysis of systems with rate limiters 5 Bifurcations in systems with a rate limiter 6 Conclusions References

1 3

3 4 5 7 8 14 15 16 17 17 19 23 26 30 32 32 37 37 38 41 42 43 49 50

Monitoring and Control of Bifurcations Using Probe Signals.. 51 Munther A. Hassouneh, Hassan Yaghoobi, Eyad H. Abed 1 Introduction 51 2 Hopf bifurcation 52 3 Analysis of the effects of near-resonant forcing 54 4 Numerical example 57 5 Combined Stability Monitoring and Control 58

Vm

Table of Contents

6 Detection of Impending Bifurcation in a Power System Model 7 Conclusions References Normal Form, Invariants, and Bifurcations of Nonlinear Control Systems in the Particle Deflection Plane Wei Kang 1 Introduction 2 Problem formulation 3 Normal form and invariants 4 Bifurcation of control systems 5 Bifurcation control using state feedback 6 The cusp bifurcation and hysteresis 7 Other related issues 8 Conclusions References Bifurcations of Reachable Sets Near an Abnormal Direction and Consequences Emmanuel Trelat 1 Setup and definitions 2 Asymptotics of the reachable sets 3 Applications References

II

Stabilization and Robustness

Oscillation Control in Delayed Feedback Systems Fatihcan M. Atay 1 Introduction 2 Perturbations of linear retarded equations 3 The harmonic oscillator under delayed feedback 4 Controlling the amplitude and frequency of oscillations 5 Conclusion References Nonlinear Problems in Friction Compensation Antonio Barreiro, Alfonso Banos, Francisco Gordillo, Javier Aracil 1 Introduction 2 Conic analysis of uncertain friction 3 Harmonic balance 4 Frequencial synthesis using QFT 5 Discussion References

60 64 64 67 67 68 70 75 77 81 83 84 85 89 89 91 94 98

101 103 103 105 106 Ill 115 115 117 117 121 124 127 128 129

Table of Contents

IX

Time-Optimal Stabilization for a Third-Order Integrator: a Robust State-Feedback Implementation 131 Giorgio Bartolini, Siro Pillosu, Alessandro Pisano, Elio Usai 1 Introduction 131 2 Closed loop time-optimal stabilization for a third-order integrator .. 133 3 Sliding-mode implementation of the time-optimal controller 137 4 Simulation results 141 5 Conclusions 143 References 144 Stability Analysis of Periodic Solutions via Integral Quadratic Constraints 145 Michele Basso, Lorenzo Giovanardi, Roberto Genesio 1 Introduction 145 2 A motivating example 146 3 Problem formulation and preliminary results 148 4 Sufficient conditions for stability of periodic solutions 151 5 Application example 154 6 Conclusions 156 References 156 Port Controller Hamiltonian Synthesis Using Evolution Strategies Jose Cesdreo Raimundez Alvarez 1 Introduction 2 Port controlled Hamiltonian systems 3 Controller design 4 Preliminaries on evolution strategies 5 Evolutionary formulation 6 Case study - ball k beam system 7 Conclusions References Feedback Stabilization and l-L^ Control of Nonlinear Systems Affected by Disturbances: the Differential Games Approach . . Pierpaolo Soravia 1 Introduction 2 Differential games approach to nonlinear %oo control 3 Other stability questions 4 Building a feedback solution for nonlinear Hex, control References A Linearization Principle for Robustness with Respect to Time-Varying Perturbations Fabian Wirth 1 Introduction

159 159 160 160 162 165 167 169 170 173 173 175 181 182 188 191 191

X

Table of Contents

2 Preliminaries 3 The discrete time case 4 Continuous time 5 Conclusion References

192 195 197 199 200

III

201

Global Dynamics of Control Systems

On Constrained Dynamical Systems and Algebroids Jesus Clemente-Gallardo, Bernhard M. Maschke, Arjan J. van der Schaft 1 Introduction: Constrained Hamiltonian systems 2 What is a Lie algebroid? 3 Dirac structures and Port Controlled Hamiltonian systems 4 Constrained mechanical systems and algebroids 5 Control of constrained mechanical systems References On the Classification of Control Sets Fritz Colonius, Marco Spadini 1 Introduction 2 Basic definitions 3 Strong inner pairs 4 The dynamic index 5 The index of a control set near a periodic orbit References On the Frequency Theorem for Nonperiodic Systems Roberta Fabbri, Russell Johnson, Carmen Nunez 1 Introduction 2 Nonautonomous Hamiltonian systems 3 Generalization of Yakubovich's theorem References Longtime Dynamics in Adaptive Gain Control Systems Gennady A. Leonov, Klaus R. Schneider 1 Introduction 2 Assumptions and preliminaries 3 Localization of the global attractor 4 Longtime behavior and estimates of the Hausdorff dimension of the global attractor References

203

203 205 208 213 214 216 217 217 218 219 221 224 230 233 233 235 238 240 241 241 242 245 248 253

Table of Contents

XI

Model Reduction for Systems with Low-Dimensional Chaos . . 255 Carlo Piccardi, Sergio Rinaldi 1 Introduction 255 2 Peak-to-peak dynamics 256 3 The control problem 260 4 Examples of application 261 5 Delay-differential systems 263 6 Concluding remarks 265 References 267 Feedback Equivalence to Feedforward Forms for Nonlinear Single-Input Control Systems 269 Issa Amadou Tall, Witold Respondek 1 Introduction 269 2 Definitions and notations 271 3 Feedforward normal form 274 4 m-invariants 275 5 Main results 276 6 Examples 281 7 Feedforward systems in E 4 283 References 285 Conservation Laws in Optimal Control Delfim F. M. Torres 1 Introduction 2 Preliminaries 3 Main results 4 Examples References

287 287 289 291 294 295

List of Participants

297

Bifurcations in Systems with a Rate Limiter Francisco Gordillo, Ismael Alcala , and Javier Aracil Dept. Ingeniera de Sistemas y Automatica, Universidad de Sevilla. Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n. 41092 Sevilla, Spain

Abstract. Limit cycles analysis of feedback systems with rate limiters in the ac-

tuator can be implemented by a classical method in the frequency domain, the harmonic balance method. In this paper, the rate limiter describing function is obtained and applied to the search for limit cycles in such control systems. Three examples with three dierent bifurcations (saddle-node bifurcation of limit cycles, subcritical Hopf bifurcation at in nity and supercritical Hopf-like bifurcation) are included. The method is approximate but its main advantage is that intuition is gained into a dicult problem.

1

Introduction

Nonlinear control of nonlinear plants leads to nonlinear dynamical systems. The last class of systems can display complex dynamical behaviours, and, what is more important, after (even small) changes in parameters or in their system structure, one can observe qualitative changes in their behaviour modes. This is the realm of bifurcation theory [7,9], which supplies tools to study the points where these changes are produced and the archetypical forms of the state portrait changes in these points. These archetypes are of a great value for the control systems designer as they supply a uni ed and global perspective on the behaviour modes of the system (he can see what is expected to happen for all the parameter values involved). Bifurcation theory is a valuable tool for understanding the behaviour richness of nonlinear systems. Roughly speaking, when by moving system parameters one observes a qualitative change in the system response (to be deduced from the state portrait, for instance) it is said that the system undergoes a bifurcation phenomenon. These phenomena can lead to a dierent number of stationary solutions (equilibrium points), to the appearance of oscillations, or even more complex behaviours (chaos, for instance). After a bifurcation analysis it is possible to split the parameter space into several regions with dierent asymptotic dynamics [10]. Furthermore, from a certain point of view, bifurcations are related to robustness issues, since only far from the bifurcation points the system displays behaviours that are structurally stable. In this paper, the presence of a concrete nonlinearity, the rate limiter, in the actuator of control systems is considered. The study of the rate limiter is of a great importance due to fact that this nonlinearity exists in a large type of actuators in which the speed of response is bounded. Furthermore, F. Colonius, L. Grüne (Eds.): Dynamics, Bifurcations, and Control, LNCIS 273, pp. 37−50, 2002. Springer-Verlag Berlin Heidelberg 2002

38

F. Gordillo, I. Alcalá, and J. Aracil

the presence of rate limiters in plants that are controlled with PID gives rise to dicult problems, mainly with unstable plants [16,17]. Anti-windup techniques give good solutions for systems with saturation, but they do not work well in systems with rate limiters. In this paper, the rate limiter is studied with the describing function method [3,5,6,8] which helps to analyze the stability of the system, the appearance of limit cycles and other phenomena such as bifurcations. The method is approximate, but the fact that it allows to analyze not only the local stability but also the global one [1,2,11,15] together with the simplicity and facility of the use of the method, justi es the current study. Furthermore, the analysis leads to detect the occurrence of stable and unstable limit cycles that are organized by the archetypes supplied by several types of bifurcations. The describing function method has been previously applied to bifurcation analysis [4,11,14]. In this paper, three examples of systems with rate limiters are presented. These examples show three dierent kinds of bifurcations: saddle-node bifurcation of limit cycles, subcritical Hopf bifurcation at in nity and supercritical Hopf-like bifurcation. In this way an overall perspective of the behaviour modes displayed by the systems can be reached. This perspective is of a great interest for the control system designer. In Section 2, the dynamical response of rate limiters is described and the dierent behaviour modes when the input is sinusoidal are characterized. This analysis is used in Section 3 to obtain the analytical expression of the describing function of rate limiters. In Section 4 this describing function is used to analyze the existence of limit cycles, while in Section 5 the three examples of bifurcations are presented. Finally, some conclusions are drawn in Section 6.

2 Behaviour of rate limiters In this study the rate-limiter nonlinearity is considered as the block of Fig. 1 with one input and one output. The output attempts to follow its input but with the constraint that the slope of the output is bounded in m. 1

Input

Fig. 1.

1

Rate Limiter (slope m)

Output

Simulink block for a rate limiter.

In this way a saturation is applied not to the input of the nonlinearity but to the derivative of this input. Furthermore, if the output, , is less than the

Bifurcations in Systems with a Rate Limiter

39

input, u, the output will grow at the maximum speed, _ = m, and if > u then _ = m. A sinusoidal input is considered in the describing function method. In this case,

y(t) = a sin !t =) dydt(t) = a ! cos !t =) a! dydt(t) a!: For small values of a and !, the maximum slope of the input does not violate the bound m and, therefore, the output of the rate limiter y(t) follows its input y(t). This behaviour continues while a! m. When this bound is violated, the output of the nonlinearity y(t) is not able to follow exactly the input y(t) during some periods in which the output evolves at the maximum speed (m). Depending on the values of a and ! three qualitatively dierent behaviour modes of the output may exist:

mode (a) The slope of the input y(t) never violates the constraint m. The

input y(t) and the output y(t) are overlapped during the whole period of the input. This behaviour mode is not longer valid when

m !a = 1:

max dydt(t) = a! = m ()

(1)

mode (b) The output in this mode has time periods in which it follows the sinusoidal input and periods where it follows a straight line of slope m (see Fig. 2).

100

80

60

Input and Output

40

20

0

−20

−40

0

−60

−80

−100

0

10.5

11

11.5

12

12.5

13

Time

Fig. 2. Response of the rate limiter (solid line) to sinusoidal inputs (dashed) in mode (b).

40

F. Gordillo, I. Alcalá, and J. Aracil

Assume that the steady state has been reached. Let and be the values of !t when y(t) reaches y(t) (in Fig. 2). Notice that = + . Analogously, let and be the values of !t when y(t) leaves y(t), at these points. 0

0

0

d y(t) dy(t) dt = dt =

(

m for ! t = m for ! t = :

(2)

0

Notice that = + . It is evident that y(t) = y(t) in [; ] [ [ ; ]. During [ ; ] y(t) is a straight line slope equal to m and during [ ; + 2] y(t) is a straight line with slope equal to m. mode (c) In this case the output y(t) is too slow and is not able to follow the input y(t). Therefore , y(t) evolves at the maximum rate in both directions m. Figure 3 shows how, each time that the output reaches the input, the slope of the output changes. In the frontier between modes (b) and (c) = and = . Furthermore, into this critical case, the slope of the input is equal to m in !t = and equal to m in !t = . Thus, 0

0

0

0

0

0

0

0

m ( + ) sin = sin + !a a! cos = m

m = cos = p 2 : = arctan 2 =) !a 4 + 2

100

80

60

Input and Output

40

20

0

−20

−40

0

−60

−80

−100

5.2

5.4

5.6

5.8

6

6.2

6.4

Fig. 3. Response of the rate limiter (solid line) to sinusoidal inputs (dashed) in mode (c). Time

Bifurcations in Systems with a Rate Limiter

41

Notice the delay of y(t) with respect to y(t) in modes (b) and (c). This delay has an important eect in the global behaviour of the system. As it can be seen, the value of the adimensional parameter m=(!a) de nes the qualitative behaviour mode of the nonlinearity, as appears in Table 1. Modes (a) and (c) are particular cases of mode (b). Indeed, in mode (a) = 0 and = while in mode (c) = . This fact will be used in the next section to obtain the describing function of rate limiters. Modes

m Range of !a m (x) @@H x yR G>R (x) @@H x

#

(3)

with uR = RyR for some positive semi-de nite symmetric matrix R. Incorporating in (1) leads to models with the structure (

R(x)] @@Hx (x) + G(x)u : x_ = [J >(x) @ H y = G (x) @ x (x)

(4)

3 Controller design Considering the closed loop dynamics for the pair plant-controller given by x_ = [Jd(x) Rd (x)]

@Hd @x

(5)

The problem is to nd a static feed-back control u = (x) in (4) responsible for the new closed loop energy shaping. De ning

Jd (x) = J (x) + Ja (x) Rd (x) = R(x) + Ra (x)

Port Controller Hamiltonian Synthesis Using Evolution Strategies

Hd (x) = H(x) + Ha (x)

161

(6)

and considering x_ = [Jd (x) Rd (x)] @@Hxd @H = [J + Ja (R + Ra )] @@H x + @ xa @H = [J R] @@H x + [Ja Ra ] @ x + [Jd (x) @ H = [J R] @ x + G(x) (x) we conclude

Rd

(x)] @@Hxa

(7)

= G(x) (x) (8) )] Hxa + [ a a ] H x which is the basic relationship involving (x) and Ha . Considering K (x) = @@Hxa , the controller synthesis problem can be stated as: Given (x) (x) H G(x) and the desired equilibrium to be stabilized x 2 (x) + a (x) = [ (x) + a (x)]> 0 2. Integrability K (x) is the gradient of a scalar function: J

J

R

R

J

R

J

R

> @K K ( x) = ( x) @x @x 3. Equilibrium Assignment K (x) at x veri es @

K (x ) =

@

H (x )

x 4. Lyapunov Stability The Jacobian of K (x) at x satis es the bound K (x ) @x

@

@

@

2

@

H (x )

x2

(9)

162

J.C. Raimúndez Álvarez

Under these conditions, the closed-loop system u = (x) will be a Port Controlled Hamiltonian (PCH) system with dissipation of the form (5), where Hd (x) is given by (6) and @ Ha (x) = K (x) @x

(10)

Supposing that such a K (x) can be found, the control can be calculated using the formula h

i 1

(x) = G> (x)G(x) G> (x) n [Jd (x) Rd (x)] K (x) + [Ja (x)

o

Ra (x)] @@H x (x)

(11)

4 Preliminaries on evolution strategies 4.1

Evolutionary basics

Evolution Strategies (ES) belongs to the class of stocastic optimization techniques, commonly described as evolutionary algorithms. Simulated Evolution is based on the collective learning processes within a population of individuals, in the quest for survival [1]. Each individual represents a search point in the space of potential solutions to a given problem. There are currently three main lines of research strongly related but independently developed in simulated evolution : Genetic Algorithms (GA), Evolution Strategies (ES), and Evolutionary Programming (EP). In each of these methods, the population of individuals is arbitrarily initialized and evolves towards better regions of the search space by means of a stochastic process of selection, mutation, and recombination if appropriate. These methods dier in the speci c representation, mutation operators and selection procedures. While genetic algorithms emphasize chromosomal operators based on observed genetic mechanisms (e.g., cross-over and bit mutation), evolution strategies and evolutionary programming emphasize the adaptation and diversity of behavior from parent to ospring over successive generations. Evolution is the result of interplay between the creation of new genetic information and its evaluation and selection. A single individual of a population is aected by other individuals of the population as well as by the environment. The better an individual performs under these conditions the greater is the chance for the individual to survive for a longer while and generate ospring, which inherit the parental genetic information. The main contributions in the evolutionary computation approach are:

Model regularity independence (Applicable to nonsmooth problems).

Port Controller Hamiltonian Synthesis Using Evolution Strategies

163

Parallelization to cope with intensive cost tness computation. Population search versus individual search (classical). General meta-heuristics. Good convergence properties.

Evolutionary algorithms mimic the process of neo-Darwinian organic evolution and involves concepts such as: t Time or epoch. ; Individual. Exogenous parameters. (Search Space). Endogenous parameters. (Adaptation). P Population. P = 1 ; 1 ; : : : ; n ; n + (P ) Fitness. (P ) = (( 1 ); : : : ; (n )) ( ) : operators (Mutation, Selection, Variation, etc.) where ni is the number of individuals in the population. A simple evolutionary algorithm follows: t 0 initialize P evaluate (P ) f

g

ff

g

f

i gg

i

j

ospring and the best are selected as the next population [Q = ]. ( + ). Selects the survivors from the union of parents and ospring, such that a monotonic course of evolution is guaranteed [Q = P ] i

k

k

;

4.2

Fitness evaluation

Each individual is characterized by a set of exogenous and endogenous parameter values and respectively. The exogenous parameters are inherited from the response function. The endogenous parameters also called strategic parameters, they do not in uence the tness measure. Each individual represents a set of independent paths over the phase space beginning at dierent initial conditions, under the in uence of the same controller. This set of ns orbits should cover conveniently the phase space and can be randomly generated at the very beginning, being common to all the individuals of the population. Care must be taken in the process of initial conditions generation. The set of initial conditions must spread over the expected attraction basin and to avoid over tting [9] a minimum must be imposed over ns and to avoid prohibitive computational costs a maximum must be imposed over ns . Given an open-loop system represented as x_ = F (x) + G(x)u

(15)

and a stabilizing controller u = (; x) in which represents a set of parameters to be xed, the controller stabilizing behaviour can be measured taking the set of ns orbits described by the closed-loop system, beginning at a

Port Controller Hamiltonian Synthesis Using Evolution Strategies

165

set of initial conditions spread over a region of interest. Under the stabilizing controller action, the resultant set of orbits should approach the origin considered as an equilibrium point. The tness should detect and measure this performance to serve as learning factor. Thus being xk (0); k = 1; ; ns initial conditions, each orbit starting at xk (0) under the in uence of the controller parameters can be represented as: f

X k (; t) = X (; xk (0); t)

g

(16)

Settling time performance is measured through a function (X ) > 0 which normally has one of the following structures:

max X t (X ) = R tmax8t k

0

(17)

k

X k t dt

k

with > 1. A typical tness measure can be obtained as: f () = k1

X k

(X k ()) + k2

X k

b(X k ()) + k3 g()

(18)

k1 ; k2 and k3 being positive scale factors and g() is a measure of closeness from the parameters to the origin, as a means to guarantee regularity [9] in the approximator. Usually g() = and b(xk ()) is a barrier function [2] which penalizes unwanted states or control eorts. k

k

5 Evolutionary formulation Our individual will be represented by a set of exogenous parameters which are the controller parameters. (The endogenous parameters are related to the search process). As can be seen in (11), the controller depends on a , Ja and Ra . In this paper the Ja and Ra values will be heuristically chosen according to Remark 1 later. The evolutionist process will act only in a de ned according to (19). Under the controller action, a set of independent orbits started at previously de ned points xk (0); k = 1; ; ns (initial conditions), will reach the equilibrium point x and will remain there, assuring asymptotic stability. The asymptotic convergence task is performed by the evolutionist process unsupervised learning capabilities, through a behaviour measure ( tness) minimization. The controller derives from an energy function a whose gradient eld is modulated according (10),(11). Our purpose is then to nd a vector eld K (x) which is the gradient of an energy function a (x) with structure H

H

f

g

H

H

166

J.C. Raimúndez Álvarez

Ha (x) = > ( 1 ; 2 ; x) 0 ( 1 ; 2 ; x)

(19)

where ( 1 ; 2 ; x) is a neural net with n inputs, n outputs where n = dimfxg, with the structure ( 1 ; 2 ; x) = ( 1 ( 2 x)) and (x) = 2=(1 + exp( 2x)) 1. 0 is a positive de nite matrix of weights. ( k = fijk g; k = 0; 1; 2) are square matrices, so the vector of design parameters is obtained by putting inside a vector all the independent coecients of the square matrices 0 ; 1 ; 2 of size n n each, giving n = 3n(n +1)=2 Adopting for Ha (x) the structure (19) implies to assume x = 0. If rank( k ) = n; k = 1; 2 then x = x is the only point which obeys ( 1 ( 2x)) = x In order to characterize the controller performance, a set of initial conditions xk (0); fk = 1; ; ns g spread over the desired attraction basin are given, remaining constant during the calculations. The feasible controllers are those which obey the conditions stated in items 1 to 4. For the feasible controllers, the minimization process involves a measure over the plant-controller behaviour. The better the controller, the smaller the measure ( tness). This measure is achieved through the steps:

For a given calculate (x) (; x) according to (11) with K (x) K (x) = 2> ( 1 ; 2 ; x) 0

@ ( 1; 2; x) @x

Integrate for t 2 [0; tmax] the dierential equation @H x_ = [J R] + G(x) (; x)

(20)

@x

Being x(0) = fx1 (0); : : : ; xns (0)ig the set of ns initial conditions de ning the needed attraction basin, X (; t) the closed loop path with initial conditions xi (0) and 1, calculate the tness index

X i ( ; t) t i () = (X i ()) = t2[0max ;tmax ]

(21)

with a suitable norm kk so the tness is calculated as

=

ns X i

i())

(22)

Incorporating the feasibility search in the tness calculation gives the following procedure: function fitness n f

f = min eigen

@ > 0 @ @2H @xi @xj + @x @x (x ); eigen

0 ;

Port Controller Hamiltonian Synthesis Using Evolution Strategies

if

167

(f < 0) f = k1 + k2 jj;

else;

f =

end return

g

P

ns i

i ();

f;

This procedure applies a penalization to () in the case that the desired positive-de niteness conditions fail. k1 and k2 are large positive numbers.

6 Case study - ball & beam system r x Jb

Jp

Fig. 1.

Ball & Beam diagram

Consider the Ball & Beam plant as can be depicted in Figure 1. The beam is made to rotate in a vertical plane by applying a torque at the center of rotation, and the ball is free to roll along the beam which is one-dimensional. The ball must remain in contact with the beam and the rolling must occur without slipping, which imposes a constraint on the rotational acceleration of the beam as well as in the friction coecient. This plant is a well known example of a nonlinear system which is neither feedback linearizable nor minimum phase. Controllers for tracking purposes can be found in [6],[5]. Our goal is to drive the ball to the rest position over a set of initial conditions with values taken on a neighborhood of the origin given by 0:5 xi 0:5; i = 1; : : : ; 4 on the phase space, which characterize the needed attraction basin. Let the moment of inertia of the beam be Jp , the mass and moment of inertia of the ball be mb and Jb respectively, the radius of the ball be r and the

168

J.C. Raimúndez Álvarez

acceleration of gravity be g. Choosing the beam angle and the ball position over the beam x as generalized coordinates for the system and according with the above diagram, the kinetic energy is given by

T (; x; ;_ x_ ) = 12 Jp + mb x2 _2 + Jr2b + mb x_ 2 and the potential by

V (; x) = mb gx sin transforming to the hamiltonian formalism in which

x1 = x x2 = x3 = px x4 = p

(23)

the total system energy is given by

H(x) = 12 J =rx + m + J +xm x + mb gx sin x b b p b 2 3

2 4

2

2 1

1

2

(24)

and the movement has the description:

0 x_ BB x_ @ x_

1 2 3

x_ 4

1 0 CC = BB A @

0 0 1 0

0 @H 1 0 1 @x C 0 10 B C B BB @@xH CC + G(x)u = BB 0 0 1C C B@ 0 0 0AB @H C C B @x 100 @ A 1 2 3

@H @x4

x3 Jb =r2 +mb x4 J2p +mb x21 mb x1 x4 mb g sin x2 2 (Jp +mb x2 1) mb gx1 cos x2 +u

1 CC CA (25)

calculating with the values

r = 0:04 Jb = 0:001 Jp = 2 mb g = 4:9

Remark 1 The choice of Ja and Ra should accomplish the conditions previ-

ously stated within section 3. There are no general rules for their determination. The considered Ball & Beam plant has no dissipation so it is convenient

Port Controller Hamiltonian Synthesis Using Evolution Strategies

169

to de ne Ra 0 in order to inject dissipation through the controller action. Concerning Ja , their contribution should be considered for interconnection enhancement in plants which have poor interconnection between states. An example of interconnection enhancement between mechanical and electrical parts using the participation of Ja can be seen in [3]. For our plant we choose

0 B Ja = B @

1

0 101 1 0 0 0C C 0 0 0 0A; 1000

04 0 0 01 B 0 4 0 0 CC Ra = B @0 0 0 0A

(26)

0000

Processing in a Pentium II at 600Mz after 20 minutes we obtained: PI0 = [ 6.966073 -5.319661 -2.699448 -4.489864 PI1 = [ 0.379863 -0.460202 0.004198 0.172915

-0.011644 -1.817432 -0.208203 -0.018196

-1.027616 -0.597628 -0.054679 -0.085693

-5.319661 7.976571 3.093717 1.910575

-0.185061 -0.615594 0.459153 -0.274629 ]';

-2.699448 3.093717 1.860503 0.797894

-4.489864 1.910575 0.797894 7.964292 ]';

PI2 = [ -0.882976 0.259639 -0.547722 -1.013208

-2.350738 -0.259648 0.3397 -0.782236

-0.035688 0.056688 -0.572971 -0.46785

0.296927 1.268853 -1.985099 2.375783 ]';

In Figures 2,3 the local potential and kinetic energy modi cations introduced by Ha can be observed. In Figure 4 a set of four orbits with initial conditions over the needed attraction basin are shown characterizing the controller eectiveness. 7

Conclusions

The main contribution of this technique is associated to the vector eld

K (x) determination, avoiding the cumbersome resolution of a partial dierential equation. The saturated nature of neural nets also facilitates the bounding of the control action as practically desired. The vector eld K (x) maximum strength is xed previously, depending on the desired attraction basin, after a judicious eort consideration. The maximum strength is controlled as a barrier on the maximum eigenvalue of 0 and limiting a suitable norm over 1 and 2 (18). The controller performance depends on the choice of Ja ,Ra which models state interconnections and dissipation and which models closed-loop settling time. The plant model can be described with class C 0 or even piecewise C 0 functions and state space restrictions are also allowed (anti slippage for instance)

170

J.C. Raimúndez Álvarez

H

H

1

5

0.8

4

0.6 3

0.4

2

0.2

1 0

0

−1

−0.2 −2

−0.4

−3 −4

−0.6 1 0.5

−5 −1

−0.5

−0.8

0 0

0.5

−0.5 1

−1

H + Ha

−1 −1

−0.8

−0.6

−0.4

−0.2

−0.8

−0.6

−0.4

−0.2

1

14

0.8

12

0.6

10

0.4

H + Ha 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.2

8

0 6

−0.2 4

−0.4 2

−0.6 1 0.5

0 −1

−0.5

0 0

0.5

−0.8

−0.5 1

−1

−1 −1

Comparison of H and Hd = H + Ha for f 1 = 0 4 = 0. (Potential).

Fig. 2.

1g

; x3

x1

1g f 1 ;

x2

; x

As future work we are also introducing the automatic con guration of and Ra using an additional neural net that will evolve jointly with the controller.

Ja

Acknowledgements This work is supported by CICYT, under project TAP99-0926-C04-03.

References 1. Schwefel H.-P., Rudolph, G. (1995), Contemporary Evolution Strategies. In: Moran F., Moreno A., Merelo J.J., Chacon, P. (eds.) Advances in Arti cial Life. Third International Conference on Arti cial Life, vol. 929 of Lecture Notes in Computer Science, 893{907, Springer, Berlin 2. Fiacco A.V., McCormick, G.P. (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley and Sons, Inc. 3. Ortega R., van der Schaft A.J., Mareels I., Maschke B. (2001), Putting Energy Back in Control. Control Systems Magazine, 21, 18{33

Port Controller Hamiltonian Synthesis Using Evolution Strategies

H

H

1

0.7

0.8

0.6

0.6

171

0.4

0.5

0.2 0.4

0 0.3

−0.2 0.2

−0.4 0.1

−0.6 1 0.5

0 −1

−0.8

0

−0.5

0

0.5

−0.5 1

−1

H + Ha

−1 −1

−0.8

−0.6

−0.4

−0.2

−0.8

−0.6

−0.4

−0.2

1

12

H + Ha 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.8

0.6

10

0.4 8

0.2 6

0

−0.2

4

−0.4 2

−0.6 1 0.5

0 −1

0

−0.5

0

0.5

−0.8

−0.5 1

−1

−1 −1

Fig. 3. Comparison of H and Hd = H + Ha for 1g f 1 4 1g. (Kinetic). ;

x1

=0

; x2

=0 f 1 ;

x3

x

4. Ortega R., Spong M.W. (2000), Stabilization of Underactuated Mechanical Systems Via Interconnection and Damping Assignment. CNRS-SUPELEC, University of Illinois, Proceedings of Lagrangian and Hamiltonian Methods for Nonlinear Control, 1, 69{74 5. Liu P., Zinober A.S.I (1994), Recursive Interlacing Regulation of Flat and NonFlat Systems. School of Mathematics and Statistics, University of Sheeld. 6. Hauser J., Sastry S., Kokotovich P. (1992), Nonlinear Control Via Approximate Input-Output Linearization: The Ball and Beam Example. School of Mathematics and Statistics, University of Sheeld, Centre for Systems & Control, University of Glasgow | Internal Report, 37, 392{398 7. van der Schaft A.J. (1996), L2 Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Science, Vol. 218, Springer Verlag, London 8. Back T., Hammel U., Schwefel H.P. (1997), Evolutionary Computation: Comments on the History and Current State. In: IEEE Trans. on Evolutionary Computation, vol. 1 n 1 9. Sjoberg J., Ljung L. (1992), Overtraining, regularization and searching for minimum in neural networks. In: 4th IFAC Symposium on Adaptive Systems in Control and Signal Processing, Grenoble, France, 669{674

172

J.C. Raimúndez Álvarez

0.5

0.4

0.3

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5 −1.5

−1

−0.5

0

0.5

1

1.5

6

4

2

0

−2

−4

−6

0

Fig. 4. x1

1

2

3

4

5

6

x2 phase space projection (upper) and u(t) (lower)

7

8

A Linearization Principle for Robustness with Respect to Time-Varying Perturbations ? Fabian Wirth Zentrum fur Technomathematik, Universitat Bremen, 28334 Bremen, Germany, fabian@math.uni-bremen.de

Abstract. We study nonlinear systems with an asymptotically stable xed point

subject to time-varying perturbations that do not perturb the xed point. Based on linearization theory we show that in discrete time the linearization completely determines the local robustness properties at exponentially stable xed points of nonlinear systems. In the continuous time case we present a counterexample for the corresponding statement. Sucient conditions for the equality of the stability radii of nonlinear respective linear systems are given. We conjecture that they hold on an open and dense set.

1

Introduction

A natural question in perturbation or robustness theory of nonlinear systems concerns the information that the linearization of a nonlinear system at a singular point contains with respect to local robustness properties. This question has been treated for time-invariant perturbations in [8] for continuous time, (see the references therein for the discrete time case). The result obtained in these papers was that generically the linearization determines the local robustness of the nonlinear system, where genericity is to be understood in the sense of semi-algebraic geometry (on the set of linearizations). Speci cally, the objects under consideration are the local stability radius of the nonlinear system and the stability radius of the linear system, where as usual the stability radius of a system is the in mum of the norms of destabilizing perturbations in a prescribed class. The question is then, whether these two quantities are equal or, more precisely, when this is case, see also [4, Chapter 11]. In this paper we treat this problem for nonlinear systems subject to timevarying perturbations. Our analysis is based on recent results on the generalized spectral radius of linear inclusions. In particular, we see a surprising dierence between continuous and discrete time. While the linearization always determines the robustness of the nonlinear system if the nominal system is exponentially stable this fails to be true for continuous time. On the other hand we are able to give a sucient condition which guarantees equality between linear and nonlinear stability radius on an open set of systems. As it is known from [9] that the Lebesgue measure of those linearizations for which ?

Research supported by the European Nonlinear Control Network.

F. Colonius, L. Grüne (Eds.): Dynamics, Bifurcations, and Control, LNCIS 273, pp. 191−200, 2002. Springer-Verlag Berlin Heidelberg 2002

192

F. Wirth

it is possible that the nonlinear stability radius is dierent from the linear is zero it seems therefore natural to conjecture that the set of systems where these two quantities coincide is open and dense. We proceed as follows. In Section 2 we recall the de nition of the stability radius for nonlinear systems with time varying perturbations and state some relevant results from the theory of linear inclusions. In particular, we recall upper and lower bounds of the stability radius of the nonlinear system in terms of the stability radius and the strong stability radius of the linearization. In Section 3 we develop a local robustness theory based on the linearization of the system for the discrete time case. It is shown that the two linear stability radii coincide under weak conditions, demonstrating that one need only consider the linearization in order to determine the local nonlinear robustness properties of a system. The continuous time case is treated in Section 4. We rst present a counterexample showing that analogous statements to the discrete time case cannot be expected in continuous time. We then present a sucient condition for the equality of the two linear stability radii on an open set. Concluding remarks are found in Section 5.

2 Preliminaries Consider nominal discrete and continuous time nonlinear systems of the form x(t + 1) = f0 (x(t)) ; t 2 N ; (1a) x_ (t) = f0 (x(t)) ; t 2 R+ ; (1b) which are exponentially stable at a xed point which we take to be 0. By this we mean that there exists a neighborhood U of 0 and constants c > 1; < 0 such that the solutions '(t; x; 0) of (1a),(1b) satisfy k'(t; x; 0)k ce tkxk for all x 2 U . As the concepts we will discuss do not dier in continuous and discrete time we will summarize our notation by writing T = N ; R + for the time-scale and x+ (t) := x_ (t); x(t + 1) according to the time-scale we are working on. Assume that (1a),(1b) are subject to perturbations of the form

x+ (t) = f0 (x(t)) +

Xm di(t)fi(x(t)) =: F (x(t); d(t)) ; i=1

(2)

where the perturbation functions fi leave the xed point invariant, i.e. fi (0) = 0; i = 0; 1; : : : ; m. We assume that the fi are continuously dierentiable in 0 (and locally Lipschitz in the case T = R+ ). The unknown perturbation function d is assumed to take values in D Rm ,

d : T ! D ;

A Linearization Principle for Robustness

193

where in the case T = R+ we impose that d is measurable. Here > 0 describes the perturbation intensity, which we intend to vary in the sequel, while the perturbation set D is xed. Thus structural information about the perturbations one wants to consider can be included in the functions fi ; i = 1; : : : ; m and in the set D. For the perturbation set D Rm we assume that it is compact, convex, with nonempty interior, and 0 2 int D. Solutions to the initial value problem (2) with x(0) = x0 for a particular time-varying perturbation d will be denoted '(t; x0 ; d). The question we are interested in concerns the critical perturbation intensity at which the system (2) becomes unstable. The stability radius is thus de ned as

rnl (f0 ; (fi )) := inf f > 0 j 9d : T ! D : x+ (t) = F (x(t); d (t)) is not asymptotically stable at 0g :

(3)

By linearizing the perturbed system in (2) we are led to the system

x+ (t) = A0 +

m

X

i=1

!

di (t)Ai x(t) ; t 2 T :

(4)

This is a (discrete or dierential) linear inclusion, which is in principle determined by the set (

M(A0 ; : : : ; Am ; ) := A0 +

m

X

i=1

)

di Ai kdk :

If the matrices Ai are xed we will denote this set by M() for the sake of succinctness. The inclusion (4) is called exponentially stable, if there are constants M 1; < 0 such that

k (t)k Me tk (0)k ; 8t 2 T for all solutions of (4). Exponential stability is characterized by the number

(M(A0 ; : : : ; Am ; )) := sup lim sup k (t)k1=t ; t!1

where the supremum is taken over all solutions of (4). Namely, (4) is exponentially stable i (M(A0 ; : : : ; Am ; )) < 1. Again we will write () if there is no fear of confusion. In the discrete time case the number is known as the joint or the generalized spectral radius. We refer to [2,10] for further characterizations of this number and for further references. In the continuous time case it is more

194

F. Wirth

customary to consider the quantity () := log (), which is known under the name of maximal Lyapunov exponent, see [4] and references therein. As in the nonlinear case we now de ne stability radii by rLy (A0 ; (Ai )) := inf f 0 j () 1g ; rLy (A0 ; (Ai )) := inf f 0 j () > 1g : The relation between the linear and the nonlinear stability radii is indicated by the following result which is contained in [3] for the continuous and in [7] for the discrete time case. Lemma 1. Let T = N; R + and consider system (2) and its linearization (4), then T (A0 ; (Ai )) rT (f0 ; (fi )) rT (A0 ; (Ai )) : rLy Ly nl

It is the aim of this paper to obtain further results on the information the linear stability radii contain for the nonlinear system. The following set of matrix sets will play a vital role in our analysis. Recall that a set of matrices M is called irreducible if only the trivial subspaces of Rn are invariant under all A 2 M. We de ne

I (Rnn ) := fM Rnn j M compact and irreducibleg : Note that this set is open and dense in the set of compact subsets of Rnn endowed with the usual Hausdor metric. The proof of the following statements can be found in [10]. They are the foundation for our analysis of linearization principles. Theorem 1. (i) The generalized spectral radius is locally Lipschitz continuous on I (Rnn ). (ii) The maximal Lyapunov exponent is locally Lipschitz continuous on I (Rnn ). Furthermore in the discrete time case a strict monotonicity property can be shown to hold, under the assumption that the following condition can be satis ed. Given A 2 Rnn we denote by PA the reducing projection corresponding to the eigenvalues 2 (A) with jj = r(A). Property 1. The set M K(Rnn ) is said to have Property 1 if n = 1; 2 or if there exists an A 2 conv M such that

r(A) < (M) ; or rank PA 6= 2 ; or ((I PA )A) 6= f0g : It is easy to construct a set M that does not possess Property 1 (just take a set of matrices with entries A11 = A22 = 1 and A12 = c and zero

elsewhere). The interesting question, however, is whether it is possible to do

A Linearization Principle for Robustness

195

this in a way so that M is irreducible. We would assume that this is not the case, but this matter remains unresolved for the moment. In any case, the negation of Property 1 is highly nongeneric, as it requires that the spectrum of all A 2 M are contained in f0g [ fz 2 C j jz j = (M)g. Modulo this point we now state a monotonicity property of the generalized spectral radius valid in the discrete time case. In the following we denote the ane subspace generated by a set M Rnn by a M while int a M Y denotes the interior of Y with respect to this ane subspace. Proposition 1. Let M1; M2 2 I (Rnn ) satisfy M1 6= M2 and M1 int a M2 conv M2 : (5) Assume that M1 has Property 1 then (M1 ) < (M2 ) :

3 The discrete time case In discrete time the situation turns out to be particularly simple. In fact, if Property 1 holds then we can immediately conclude the following linearization principle. Theorem 2. Let T = N and consider the discrete-time system (1a) and the perturbed system (2) along with its linearization (4). If for some < rLy (A0 ; (Ai )) the set M( ) is irreducible and satis es Property 1, then rLy (A0 ; (Ai )) = rnl (f0 ; (fi )) = rLy (A0 ; (Ai )) : Proof. The assumptions guarantee that the map 7! () is strictly increasing on [ ; 1) by Proposition 1. This implies rLy (A0 ; (Ai )) = rLy (A0 ; (Ai )). The assertion now follows from Lemma 1. The situation simpli es even more if we assume that the unperturbed system (1a) is exponentially stable. We can use this natural assumption to replace the somewhat awkward condition concerning Property 1. The reason for this is simple. Exponential stability implies that r(A0 ) < 1. The stability radius rLy (A0 ; (Ai )) equals only if (M(A0 ; A1 ; : : : ; Am ; )) = 1 > r(A0 ). These two things enforce that M(A0 ; A1 ; : : : ; Am ; ) has Property 1. Corollary 1. Let T = N and consider the discrete-time system (1a) and the perturbed system (2) along with its linearization (4). If the point x = 0 is exponentially stable for the unperturbed system x(t + 1) = f0 (x(t)) then rLy (A0 ; (Ai )) = rnl (f0 ; (fi )) = rLy (A0 ; (Ai )) :

196

F. Wirth

Proof. There exists a similarity transformation T such that all Ai , i = 0, : : : ,

m are similar to matrices of the form 2 i i 3 A11 A12 : : : : : : Ai1d 66 0 Ai22 Ai23 : : : Ai2d 7 66 .. 777 i 0 0 A . 7; 33 TA T 1 = 6 i

66 .. . . . . . . .. 77 . 7 64 . 5 i 0 : : : 0 Add where each of the sets Mj := fAijj j i = 0; : : : ; mg; j = 1; : : : ; d is irreducible. It holds that () = maxj=1;::: ;d (Mj ()). Thus it is sucient to consider the blocks individually to determine rLy , resp. rLy . Under the assumption of exponential stability we have r(A0 ) < 1. Hence for each j we have r(A0jj ) < 1 and the set Mj () has Property 1 for all > 0 such that (Mj ()) > r(A0 ). Now the result follows from Theorem 2. Corollary 2. Let T = N . The stability radius of linear systems with respect to time-varying perturbations rLy is continuous on the set f(A0 ; : : : ; Am ) 2 (Rnn )m+1 j r(A0 ) 6= 1g : Furthermore, the set f(A0 ; : : : ; Am ) 2 (Rnn )m+1 j rLy (A0 ; : : : ; Am ) 6= rLy (A0 ; : : : ; Am )g is contained in a lower dimensional algebraic set. Proof. It was shown in [7] that rLy ; rLy are upper respectively lower semicontinuous on (Rnn )m+1 . The preceding Corollary 1 shows that these two functions coincide if r(A0 ) < 1, which shows continuity in this case. If r(A0 ) > 1 the statement is obvious as both functions are equal to 0. The second statement now follows because a necessary condition for the condition rLy (A0 ; : : : ; Am ) 6= rLy (A0 ; : : : ; Am ) is r(A0 ) = 1. The latter condition de nes a lower dimensional algebraic set. The result for the linear stability radii extends to the case of nonlinear systems as follows. First, denote by C 1 (Rn ; Rn ; 0) the set of continuously dierentiable maps from Rn to itself satisfying f (0) = 0. This space may be endowed with the C 1 topology inherited from the topologies on the space C 1 (Rn ; Rn ), (see [6, Chapter 17]). Corollary 3. Given n; m 2 N , the set W of functions (f0 ; f1 ; : : : ; fm ) 2 C 1 (Rn ; Rn ; 0)m+1 for which rnl (f0 ; (fi )) = rLy (A0 ; (Ai )) (6) contains an open and dense subset of C 1 (Rn ; Rn ; 0)m+1 with respect to both the coarse and the ne C 1 topology. Proof. This is immediate from the de nition of the C 1 topology.

A Linearization Principle for Robustness

197

4 Continuous time A natural question is if statements similar to those of Theorem 2 and Corollary 1 hold in continuous time. The fundamental tool for this results is the monotonicity property given by Proposition 1. This statement is unfortunately in general false in continuous time, as any subset M1 of the skewsymmetric matrices generates a linear inclusion whose system semigroup is a subset of the orthogonal group and for which the maximal Lyapunov exponent is therefore equal to 0. Taking a set M2 which contains M1 in its interior (with respect to the skew-symmetric matrices) does not yield a Lyapunov exponent larger than zero, so that the strict monotonicity property fails to hold. This example leaves still some hope that maybe a statement corresponding to Corollary 1 remains true in continuous time. The following example shows that even such expectations are unfounded. Example 1. Consider the matrices A(d) := 0d 2d+ d : It is easy to see that A (d) + A(d) 0 for all d 2 ( 1; 2). Hence for D ( 1; 2) it is immediate that (D) 0 as the Euclidean unit ball is forward invariant under the associated time-varying linear system. On the other hand while for the spectral abscissa (A(0)) = 0, we have (A(d)) < 0 for all d 2 (0; 2), see Figure 1.

spectral abscissa

2 1 0 −1 −2 −3 Fig. 1.

−2

−1

0 d

1

2

3

The spectral abscissa of A(d) in dependence of d.

The consequence of this is the following. If we de ne A0 = A(1=2) and A1 := 01 11 ; then 0 < rLy (A0 ; A1 ) 21 < 32 = rLy (A0 ; A1 ) ;

198

F. Wirth

because at least A0 1=2A1 = A(0) is not asymptotically stable. While on the other hand for

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close