Nonlinear and Adaptive Control of Complex Systems978-94-015-9261-1/1.pdfNonlinear and Adaptive Control ofComplex Systems by Alexander L. Fradkov Institutefor Problems ofMechanical

Nonlinear and Adaptive Control of Complex Systems

Mathematics and ItsApplications

Managing Editor :

M. HAZEWINKEL

Centre Jor Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 491

Nonlinear and AdaptiveControl of Complex Systems

by

Alexander L. FradkovInstitute for Problems ofMechani cal Engineeringof the Russian Academy ofSeien ces.St Petersburg, Russia

Iliya V. MiroshnikSt Petersburg, State Institute of Fine Mechanics and Optics,St Petersburg, Russia

and

Vladimir O. NikiforovSt Petersburg, State Institute of Fine Mechanics and Optics,St Petersburg, Russia

Springer-Science+Business Media, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Printed on acid-free paper

All Rights Reserved

ISBN 978-90-481-5294-0 ISBN 978-94-015-9261-1 (eBook)DOI 10.1007/978-94-015-9261-1

© 1999 Springer Science+BusinessMedia Dordrecht

Originally published by Kluwer Academic Publishers in 1999.

Softcover reprint ofthe hardcover 1st edition 1999

No part of the material protected by this copyright notice may be reproduced orutilized in any form or by any means , electronic or mechanical,including photocopying, recording or by any information storage andretrieval system, without written permission from the copyright owner.

Table of Contents

Preface

Notations and Definitions

1 FACES OF COMPLEXITY1.1 Complexity and Dccomposition

1.2 Mult ivariable Co nt.ro l and Geometry

1.2.1 Coordinati ng control . . ..1.2 .2 Oscilla t ion svn chronization

1.2.3 Spatial motion control

1.2.4 Terminal co nt rol . . .

1.2.5 State tracking sys tem s

1.3 Uncert aintv and Adaptation

xi

xv

11568

1015

1617

2 NONLINEAR SYST EMS:ANALYSIS AND DESIGN TOOLS 252.1 Stabilitv 01' Nonlinea r System s . . . . . . . 25

2.1.1 Corupleteness and stability 252.1.2 Lyapunov Iu nctions and their applications 32

2.1.3 Pa r t ial s t a.bilit y . . . . . . . . . . . . 37

2 .2 Equiva lent Models and Coo rd ina t e Changes . . 44

2 .2.1 Au tonornou s svs terns . . . . . . . . . . . 452.2 .2 Single- input svst erns an d controllabi lity 472.2 .:1 Canonical Iorm s 49

2 .3 Basic Cano nical Form and Linea r ization Techniques 52

2.3 . 1 Basic cano nica l fo r m . 52

2 .3.2 Exact lineari zation . . . . . . 562 .3.:3 Linea r approxi rnation 57

2.4 Equivalcnco 01' Mult i- Input Systems 60

2 .5 Inp ut -O ut p ut Ca. nouical Form s and Stahilization with Re-spect to Output . . . . . . . . . . . . . . . . . . . . . 632 .5.1 Main t ran sformatiou an d linea rized dynamics 642.5 .2 System ze ro dvn amics. . . . . . . . . . . . . . 67

v

VI

2.5.3 Normal form and local stabilization 682.6 Control of Triangular Systems . 712.7 Passivity and Passification . . . . 76

2.7 .1 Passivity and stability . . 762.7.2 Passivity arid dissipativity 782.7 .3 Passivity and Kalman- Yakubovich lemma 802.7.4 Passificatiou and Feedback Kalman-Yakubovich lemma 84

3 SPEED-GRADIENT METHOD ANDPARTIAL STAB ILIZAT ION 913.1 Goal-Oriented Contral Problem Statement. . . . . . . . . . 913.2 Design of Speed-Gradient Algorithms . . . . . . . . . . . . . 94

3.2.1 Speed-Gradient algorithms for local objective func-tionals . . . . . . . . . . . . . . . . . . . . . . . . .. 94

3.2.2 Speed-Gradient algorithms for integral objective func-tionals . . . . . . . . . . . . . . . . . . . . 95

3.2 .3 Speed-Gradient algorithms in fini te form. 963.2.4 Combined algorithms 97

3.3 Convergence of the Speed-Gradient Algorithms 973.3.1 Regulation and tracking . . . . . . . . . 973.3 .2 Partial stabilization 105

3.4 Identifying Properti es of Speed-Gradient Algorithms 1113.5 Robustness of Speed-Gradient Algorithms with Respect to

Disturbances 1133.6 Speed-Gradient Cont.rol of Harnilt onia n Systems 118

3.6.1 Co ntrol of energy . . . . 1183.6.2 Control of Iirst Integrals . . . . . . . . . . 122

4 NONLINEAR CONTROL OF MULTIVARIABLESYSTEMS 1274.1 Multivariab le Control and Geometrie approach 1274.2 Equilibrium Stabilization 129

4.2 .1 State regu lation problerns . . . . . . . 1294.2.2 Stabilization via exact linearization . . 1314 .2.3 Stabilization via linear approximation 133

4.3 Attracting Sets . . . . . . . . . . . 1354.3.1 Attractivity and invariance . . . . . . 1354 .3 .2~eighborhood propertics , . . . . . . . 1404.3.:3 Equivalent dynamics 01" autonomaus systems 1464.3.4 Co uditio us 01" at.tractivity . . . . . . . . . . 148

4.4 Set Stabilization 1.534.4 .1 Co utrol problerus and invariance conditions 153

4.4 .2 Problem de cornposit ion and invariant contral

4.4 .3 Equivalent dvnarn ics4 .4.4 System s t. ab ilizat io n

4.5 State T racking Con t ro l ..4 .5 .1 Tracking problern . .

4 .5.2 Inva ria nt. cont.ro l ..

4 .5 .3 System st.ahiliza t ion

VII

157

163

167

173173

175

177

5 NONLINEAR CO NTROL OF MIM O SYSTEMS 1835 .1 Problems 01' Out.pu t. Cont ro l a nd Coordination of MIM O

Systems . . . . . . . . . . . 1835.2 O u t pu t Rpg ula t iol l . . . . . . 185

5 .2 .1 Regul a t.i o n pro ble rn 185

5 .2 .2 :vI ain transfor urat io ns 186

5.2 .3 Systems wit hout zera dyua mics 19 1

5.2.4 Zero dvnamics submanifold . . 193

5.2 .5 Sy st em eq uivalen t dynamics . . 194

5 .2 .6 Iss tres 01' eq uiva lence and st a bility 199

5.3 Output Coo rdination . . . . . . . . . . . 212

5 .3 .1 Coo rdin atiou rondit.ion s . . . . . 212

5 .3 .2 Prob lems 01" coordinating contral 2165 .3 .3 Transfo rmal ion to normal form . 2195 .3.4 Tra nsformal ion to task-oriented form. 223

5.4 Coordinating Cont.rol . . . . . . . . . . . . . . 2305.4.1 Cont.ro l prohlerns in st ate space . . . . 230

5 .4 .2 Basic cont.ro l law and partial decoupling 231

5.4 .3 Coni .rol bv using irnplicit mod els 2325 .4.4 Co utro l by us in g referon ce model 237

5 .5 Sp atial Mot ion Co utrol . . . . . . . . . . 2425. 5 . 1 Spl st abili za t ion in ou t put spac e 2435.5 .2 Ca uoni ca l rr-prcsentat.ions and d ifferential forms . 2465 .5 .:3 System cq uivalont dynarnics and contral design 257

6 ADAPTIVE A ND ROBUST CONTROL DESIGN 2656.1 State-Fecdback Co utrol 266

6.1.1 Cert.ain ty cq uivalence design 266

6.1.2 Recu rsive <lPsig n procedures . 283

6.2 Output-Feedback C'ont.ro l . . . . . . 297

6.2 .1 Coni.ro l of s t rict.ly passiv e Cl nd st r ictly minimum phasesyster n-, . . . . . . . . . . . . . 298

6.2.2 Aug lTl e llt ed crror based design 303

6.2 .3 Hig h-o rde r t.unor based design . 309

viii

6.3 Output-Feedback Adaptive Systems with Implicit ReferenceModel . . . . . . . . . . . . . . . . . 3206.3.1 Design 01' adaptive controller . . . . 3206.3.2 Shunting mct.hod . . . . . . . . . . . . . . . . . . . . 323

6.4 Output-Feedback Coutrol 01' Uncertain Linear Plants underIdeal Co nditions 3286.4.1 Problem statement . . . . . . 3296.4 .2 Plant model parametrizations 3296.4.3 Cert.ainty equivalence design 3376.4.4 Dynamic certainty equivalence design 3426.4 .5 Nonlinear adaptive design 1'01' linear plants . 3496.4.6 Discussion ... ... ...... ....... 369

6.5 Output-feedback Control of Uncertain Linear Plants in thePresence 01' External Disturbances . . . . . . . . 3736.5.1 Plallt model parametrizations . . . . . . . . . . . 3736.5.2 Robust cont.rollers with high-erder tuners . . . . 3756.5.3 Robust controller with nonl inear damping terms 381

7 DECOMPOSITION OF ADAPTIVE SYSTEMS 3917.1 Separation of Metions in Adaptive Systems . . . . . . . . . 392

7.1.1 TIH' first sehetue of motion separation for continuoustime systerns . . . . . . . . . . . . . . . . . . . . . . 392

7.1.2 Tho first sehetue of rnotion separation for discrete-time systcms . . . . . . . . . . . . . . . . . . . . . . 399

7.1.3 Tlte secend scheme of motion separation . . . . . . . 4017.2 Conditions of Applicability and Estinration of Accuracy of

the Motion Separation Schemes in Adaptive Systems . . . . 4037.2.1 Applicabilitv ofthe Speed-Gradient algorithms 1.0sin­

gul arly peri.urbed systems . . . . . . . . . . . . . . . 4057.2.2 Discretization of the Speed-Gradient algorithms. . . 417

7.3 Adaptive Deccntrali zcd Control of Interconnected NonlinearSystems . . . . . . . . . . . . . . . . . . . . . . . 4197.3.1 Problem statement and cont rol algorithm 4197.3.2 Properfies of tbe control systern .. . 421

8 CONTROL OF MECHANICAL SYSTEMS 4298.1 Spatial Motion Control of Rigid Body . . . . 429

8.1.1 Dynamics and kinematical properties . 4298.1.2 Mass-point control 4338.1.3 Control of rotation. . . . . . . . . . . 436

8.2 Robot Motion Control . . . . . . . . . . . . . 4388.2 .1 Robot modcl and the problern statement. 439

ix

8. 2.2 Differ en ti al relations a nd the main approach . 4428 .2.3 Traj ectory co nt. rol st ra t egv 446

8 .3 Mobile Robot Cont.rol . . . . . . . 4498 .3.1 An alysis 01' ro bot dynamics 44 98 .3.2 Cont.ro l prob lc m statcme nt 4538.3 .3 Con trol law design . . . . . 455

8.4 Contral 01' Oscilla t ory Syste ms .. 4598.4 .1 Ene rgy cont 1'01 o f a pendu lum with contralIed pivot 4598.4 .2 \Vheeling a ca l' out of a ditch . . . . . . . . . . . . . 464

9 PHYSICS AND CONTROL9.1 Feedb ack Heso na nc<' in a Non linear Oscillator9.2 "Supero pt. imal" Esca pe frorn Pot ent ial Wells9.3 St abilizat ion 01' Uust. ab le Modcs .9.4 Feedback Spcrtroscopv .9 .5 Exc ita bility Index .9 .6 Speed-G rad ient Law s of Dvn ami cs9.7 Onzagger Equat io us9.8 Discu ssion .

A APPENDIX

References

Index

469469473475476477479482483

485

493

507

PREFACE

T his book presen ts a t. hoo ret ica l framewe rk and cont ro l methodologyfor a class of complcx dyna.m ical systenis cha racterized by high st a t e spacedimension , multiple inpu t. s a nrl o ut puts . sig nificant nonlinearity, parametricun cer t ainty a nd unmodellod dyu arni cs .

The book st art.s wit.h an in l.rod uct.orv Chapte r 1 where t he peculiari­ti es of cont rol problcrns Ior co mplex sys te ms are discussed and motivatingexam ples from diffe rent fi olds of seience and technology are given.

Cha pter 2 prcscnts SO Il I(' rcsults of no nlinear control theory which assistin reading subseque nt chaptors. The main notions and co ncept s of stabilitytheory are int ro duced. a nd pro blems of no nlinear transformation of sys­tem coordinates an' discus sod . On this basis, we consider different designtechniques and approach es t 0 linearization . stabilizat ion and passificationof nonlinear dynam ical SySt(' II IS.

Cha pter 3 gives a n cx posi t.ion of th e Speed-Gradient method and its ap­plications to nonlinear au d adapt ive co nt rol. Conve rge nce and robustnessproperties a re ex am iued . I ~ ro blcm s of rcgu lat ion , t racking, partial stabiliza­t ion and con trol o f 11 a miIto nia. n syste rns are con sid ered .

In C ha pter ,I W(' in t.rod uco th e main not.ions related to t he propertiesof regular hyp ersurfa cos of bcing a n invari ant set a nd nontrivial attractorof a dyn amical sys t.om. T hcn. we presen t a methodology of system analysisin the st ate spare a nd dcsi).!; 11 t.ools for solving the problerns of equilib riumand set stabiliza t. io n. as weil as t.rackin g co ntrol, for nonlinear multivariablesyst em s having scvoral co ntro lling inpu ts.

In C ha pter .') wo st udy mul ti-dim ensional prohlems of outputs regula­tion , coordinating control a nd curve - (s urfa ce-) following , having the evi ­dent geometrie nat.u re simila r t. o t hat af t.he problerns considered in Chapter4. However , unli kc t hc prcv ious parts , t. he emphasis is here placed on theoutput space where t. hc majority of t he real problerns are originally stated .

In Chapter 6 tho basi c dosign methods of adapt ive, rob ust adaptive androbust nonlinear control of unce rt.a in pla nts are pr esented in the form ofuniversal design tools. Va.rious rnethodologies (ind uding recursive design ,augmented err o r hascd dcsign , high -e rder tuner based design and reducedorder referen ce mod el dosign ). which allow one 1,0 overcom e structural ob ­st acles ca used bv vio latio n or t.he mat ching condition 01' by high re lative

Xll

degree , are considercd in t.he chapte r. The pract ical applicability of theintroduced design tools is illnstratcd hy the example of output-feedbackcont rol of uncertain single-inpu t/singlc-output linear systems.

Chapter 7 is dovotcd to decomposition methods in adaptive controlbased on separation of slow and fast rnotions in t.he system . Convergenceand accuracy of decomposit ion for singularly perturbed and discretized sys­tems are examined . Thc Spcod-Cradient approach t.o decentralized adaptivecontrol of nonlinear systeru s is presentcd.

In Chapter 8 we study applied nonlinear control problems of providingthe required spatial motion 01' complex mechanical systems described bythe Newton, Euler and Lagrange equations. The pr esentation begins withinvestigating the problern of motion 01' a rigid body, which is the basis forfurther consideration 01' mult.i-body mechanical systems such as multi-linkmanipulation robots and multi-drive wheeled mechanisms. Also applica­tions 1,0 control of oscilla.tor v rnechanical systems, based on the material ofChapter 3, are presentcd .

Finally, in Chaptcr 9 thc relation s between control and physics are dis­cussed . New concepts 01' " Iocdback rosonance" , "excit ability index" areintroduced with tho purpese 1.0 bet.ter understand behavior of nonlinearnearly conserva ti ve syst.cm s under feed back action. The Speed-Gradientmethod of Chapter ;3 is appl ied both 1.0 organize resonant system behaviorand 1.0 reforrnulate I.IL<' laws 01' dyna mics for a wide dass of physical sys ­tems . Applications 1.0 escap « from a potential well, stabilization of unstablemodes, feedback spcctrosco py and deriva.tion of the Onzagger principle aregiven. The chapter ou tli IWS a. ncw Iield of researcli that may be called cy­bernetical phsjsics.

A unique feature 01' t.he a.u thors ' approach is the combination of rigorousconcepts and methods of modern non linear control such as goal sets, invari­ant and attracting submau ilolds, Lyapunov functions , exact linearizationand passification , the Kahuan- Yakubovich lem ma and so on, with approx­imate decomposition bascd methodologies related 1.0 partial linear approx­imation, averaging and siugular perturbation techniques.

The authors prcscnt a number 01' original concepts and methods: set(submanifold) stabilizat.iou and coordinating control, Speed-Gradient con­1,1'01 and adaptation algorit.luns, systoms with implicit reference models,simplified robust modificat ions of high-erder tuners and so on. Also someresults published prcviouslv in the Russian literature and not well known inthe West are exposcd. Partirularly, thc book presonts the most importantresults given in thc aut.hors ' previous publications :

• Fomin, V.K .. 1\.1.. Fradkov a nd V.A . Yakubovich (1981) AdaptiveControl 01 Dynamic Ohjrd8, Mos cow, Nauka, (in Russian};

Xll1

• Fradkov . A.1. (1990) Adaptive Con trol in Gomplex Sys tems, Moscow .Na uka , (in Russian );• Dr ozdov, V.N ., LV. Miros hnik and V.I. Scorubsky (1989) A utom atieControl System" wi th Microcompu iers, Leningr ad , Mashinostroe nie, (inRu ssian );• Mirosh nik, I.V. ( 1990) Goordinuting Conirol of Mult ivariable S ystems.Lening rad , Energoatomizdat , (in Ru ssian );• Control of Complex Systems ( 1995) Fradkov. A.1. and A.A . St otsky(Eds.), St. Petersburg , Inst itu te for Problem s of Mechanical Enginee ring ;• Proceedinqs o] the Laboraioru of Gybemciics un d Coutrol S ys tems( 1996) Miros hn ik, LV. an d V.O. Nikiforov (Eds .), St. Petersbur g, Insti ­t ute of F ine Mechan ics a nd Op tics.

T he prosp ect ive reader should have some degr ee of familiarity with st an ­dard uni ver sity cou rses of ca lculus, linear alg ebra a nd ordiuary differentialeqnat ions . Knowledge of t he basic course on linear cont ro l theory and themain conee pts 01' differ en ti al geo metry is also desir abl e. T he book will beuseful Ior resear cher s. enginee rs, univer sity lectu rers , and postgr aduate stu­dents speeializing in t he fields 01' automatie control, rneehani es and appliedmat hematies.

The effor ts 01" th e authors when writ ing t he book have been shared inthe following way :

• A .L. Fradkov wrote Chapters :3, 7 and 9, Sections 6.4, 8.4 , and Ap­pendix ;• I. V. Mirosh nik wrote Chapters 4 and 5, Sections 1.1, 1.2, 8 .1-8.3;• V.O. Nikiforov wrote Cha pter 6, Seetions 1.3, 7.1 and 7.2.

Chapter 2 was written by all aut hors in elose eoo peration.T hc autho rs would like to acknow ledge t he valua ble help 01' t hcir col­

leagu es associated wit h t he La bo ratory "C ontrol 01' Co mplex Syst em s"01' t he Insti tute for P roblem s 01' Meehanical Eng ineering 01' t he Ru ssianAca de my 01' Seiences and t he Laboratory "Cybe rnet ics and Cont rol Sys­tems" of the Saint -Pe ters burg State Institute 01' Fin e Mechanics and Optics(Technieal Univers ity ): B.R. Andrievsky, M.V. Dru zhinina , P.Yu. Guzenko.S.M. Kor olev , A.V. Lyamin , A. Yu. Pogromsky, LG . Polushi n, V.A . Shiegin ,A.S. Shiriaev , 0.1. Koroleva , K.V . Voronov .

We are also grate ful to G .G . Levin a , M.1. Miroshnik a nd D.A . Tomehinfor t hei r help in pr ep a ration 01' t he manuscrip t.

T he book contai ns t he resul ts of research suppo rted by t he Ru ssianFound at ion for Basir Research (g rants 96-01-011 .51. 97-01-0432,99-0100672,99-01-007( 1) an d bv th e St. Petersburg Seient ific and Educational Centerfor t hc P roblems of Maeh ine-Build ing , Mec hanics an d Control Processes(P roject 2.1-589 of t he Ru ssian Fede ral P rogr am "Integra t ion" ). T he work

XIV

of A.L. Fradkov was also supported in part by the Dutch Organization forPure Research (N\tVO) . Important contributions stem from contacts withthe students and from i.ho lecture courses deliverod by the authors in theSaint-Petersburg St ato Institute of Fine Mechanics and Optics (Techni­cal University), Baltic St.ato Tcchnical University ("Voenmekh") and theSaint-Petersburg SUite Univorsit.y in 199,1-1998.

Saint- Petersburq, Russin, 1999

NOTATIONS AND DEFINITIONS

Throughout the book we use the following notations and definitions.The set of real numbers is denot ed as IR 01' IR} , while IRn stands for the

n-dimens ion al voctor spae e. An element of IRn is t he eolumn veetor com­posed of X l ,X 2 , . .. , X n and denoted as x = eol(x} ,x2, .. . , xn ) 01' x = {x;} ,i = 1,2 , .. . , n . The set of positive real numbers and zero is denoted as IR+or [0 ,00) .

Euclidean nortn of the vector x E IRn is denoted as

Let A E IRn x IRn be a real n x n matrix. The eigenvalues of Aare denotedas >-dA} , i = 1,2 , . . . , n , and lAI means a matrix norm indueed by theEu elidean vector norm , i.e. ,

Let P be a symmetrie real n X n matrix and x T P.L is a qu adratie form. Ifx T Px > °for any 1: =I- 0, then the matrix P is ealled positive definite anddenoted a.s P > O. Matrices satisfying nonstrict inequ a.lity x T P x ~ 0, fora11 x E IRn

, are called posit iv e se midefi nite 01' nonnegative. The notationIxlp is used for the weighted Euclidean tiorm of x , i.e.,

Ixlp = VxT P x .

Let f(t) be a measurabl e veetor function c1efined on IR+, i.e., f: IR+­IRn

. The c; norrn , where 1 <p < 00. is introdueed as

while [, 00 norrn is defined as

Ilfll oo = ess sup If(t)l ,t

xvi

where " ess sup" is taken over IR+ with possible exception 01' a set 01' zeroLebesgue measure. If the norm Ilfllp is finite, we write f E {,p, The spaces01' all functions that are globally bounded and square-integrable on [0,00)are denoted by {, OO and {,2, respectively. The vector space 01' continuousfunctions f : IR+ ~ IRn with the uniform norm

Ilfllc = sup If(t)1t

is denoted C[O, 00).A scalar function v : IRn

~ IR+ is called positive definite if '0(0) = 0 andv(x) > 01'01' all .7: =J O. A scalar function v : IRn

X IR+ ~ IR is called mdiallyunbounded if

lim infv(x,t) = 00.Ixl--+oo t~O

A function , : IR+ ~ IR+ is called a /C-function if it is continuous, strictlyincreasing and ,(0) = 0; it is referred 1,0 as a /Coo-function if it is a radiallyunbounded /C-function .

The function f( x, t) : IRnX IR+ ~ IRn is called Lipschitz in x in the

set D = X X T c IRnX IR+ uniformly in t if there exists a constant

L = L(X) > 0 such that 1'01' all (x, t) E D and (x*, t) E D the followinginequality is valid

If(x ,t) - (J(x*,t)1 ::; L Ix - z "], (N .!)

and the constant L does not depend on t E T. The function f(x, t) iscalled locally Lipschitz in x uniformly in t if it is Lipschitz in D = X X IR+uniformly in t 1'01' any compact set X C IRn

. Finally, the function f( x, t)is called globally Lipschitz in x (01' , simply, Lipschitz) if it is Lipschitz inIRn X IR+ , i.e ., inequality (N .1) holds 1'01' all x ERn , x* E Rn and t ~ 0,while the constant L does not depend on t .

Let x E X. where X C IRn is an open set, and a scalar real-valuedfunction f( 1:) = f( Xl, x2,' .. , xn ) is the mapping X ~ IR. The function f isa function of a dass c- , k = 1,2, .. . ,00 (01' f( x) ECk) when it is k timescontinuously differentiable. It is smooth when f E COO 01' f E c- , where kis the necessary order 01' its derivatives.

The vector function f(x) = COIUl(X), h(x), . .. , fm(x)), 01' the mappingX ~ IRm

, is smooth. (j E C'" 01' f E c» 1'01' some large k) when all scalarfunctions fj are 01' the dass COO 01' c-, respectively.

Let x EX. X C IRn and Y C IRm be open set.s. A Jacobian matrix 01'the smooth vector function f( x), 01' 01' the mapping f : X ~ Y, is an m Xn

XVll

rnatrix of it s partial derivatives 81j/8xi defined as

81d8xl 8h / 8x2 8h/8xn

81 8h / 8xl 8h / 8x2 8h /8x n-8x

8fm/ 8x I 81m/ 8x2 81m/ 8xn

The smoot h mapping 1 is called tionsinqular at ihc point x = x* E X whenrank f( x*) = m , i .e.,

rank 81 I = m ,8x x '

Let m = n . The smoot h mapping 1 : X --+ Y is called a diffeomorphismwhen it is on e-to-on e and there exists a smooth inverse mapping 1-1 : Y--+X .

A smooth rnapping .f: X --+ IRm that assigns to eac h point x E X C IRn

a vector .f E IRm is called a smooth oector field. Let gl(X),g2( X) , ... ,gAx )be th e smoot h vector fields defined on the set X. A mapping 9( x) assigningto each point :r E X a vector space that spans gl , !J2, . . . ,gv, or

9( x) = span{gl(x) ,g2(X), . .. , !Jv(x )}

is called a sm ool li distribuiion on the set X .Let 1>(x ) E ('I be a scalar function IRn

--+ IR. Then V 1>( x) denotes thecolumn vector of its first derivatives calc ulated as

8<jY T

V <jY( x) = (- )8x

If :r = col( :r I , .1:2 ) and cp is a fun ction of two vector var iables . th en

Let f( x ) be a srnoot h vector fi eld defined on X C IRn. The scalar func­

tio u X --+ 1R1 int rod uced as

is a derivati ve 01' (/) alo ng I , often called a (scalar) Lie deriuaiiue. Let f( x )and g(:/:) be smoot h vector fields defined on ,1" . T he mapping [f,g] : X X

X --+ IRn (a vector l.ie der'ivative) introduced as

. 8g 81[./( x) ,g( x) ] = Ljg(X) = 8.1J(X) - 8:J;g(x)

xviii

is called a Lie bracket . Throughout the book the following notations arealso used

LgLJcP(X) = :x(LJ cP) g(x) , LJcP(X) = : x(L,-I cP) g(x)

ad~g(x ) = g(x), ad}g(x) = [J(x) ,g(x)), adJg(x) = [J(x) ,ad,-Ig(x)),

where k = 2,3, . . ..A smooth distribution 9( x) = span {gI(x) ,g2(:I:) , . .. ,gl/(x)} defined on

th e set X is called tionsinqular when dirn 9( x) = m = const for all x EX,and in volutive when

[gi(X),9j(X)] E 9(x)

for all vectors gi(X),gj(x) E 9( x).A polynomial ß(p) is called Hurwitz if all roots of the equation ß(p) = 0

have negative real parts . Areal n X n matrix A is called Hurwitz if all itseigenvalues AdA} , i = 1,2, ... , n, have negative real parts.

The deqree of a polynomial ß(p) is denoted as n = deg ß(p). The relativedeqree of a rational function ß(p)/ a(p) is the integer p = dega(p) - degß(p) .The rational function is called:

i) proper if p ~ 0;ii) strictly proper if p > 0;iii) minimum phase if ß (p) is Hurwitz;iv) asymptotically stable if a (p) is Hurwitz.