Complementarity methods in the analysis of piecewise linear dynamical systems

Tilburg University

Complementarity methods in the analysis of piecewise linear dynamical systems

Camlibel, M.K.

Publication date:

2001

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Camlibel, M. K. (2001). Complementarity methods in the analysis of piecewise linear dynamical systems.

CentER, Center for Economic Research.

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Complementarity Methods in

the Analysis

of

Piecewise

Complementarity Methods in

the Analysis of

Piecewise

Linear Dynamical

Systems

Proefschrift

ter verkrijging van de graad vandoctor aan de

Katholieke Universiteit Brabant,

op gezag van de

Rector _{Magnificus, prof. dr. F.A. van der} Duyn Schouten,

in het openbaar te verdedigen ten overstaan

van een door _{het college voor promoties}

aangewezen commissie in de aula van de _{Universiteit op}

maandag 28 mei 2001 om 16.15 uur

door

Mehmet _{Kanat Camlibel}

geboren op 17 mei 1970

te Istanbul, Turkije.

to my parents

Acknowledgement

As a master's student, I became aware of the Dutch _{Systems and} Control community

while I was reading the book Three Decades of Mathematical System Theory edited by

Henk Nijmeijer andHansSchumacher at the occasion of the 50thbirthday ofJanWillems.

It was then that I decided tostudy in the Netherlands. After meeting Hans Schumacher

at a workshop in Istanbul and obtaining a NATO fellowship, I started working with him

at CWIin Amsterdam. Fouryearspassed and I amnowwriting the last lines of my Ph. D.

dissertation which would not exist without contributions, help, advice and support of a

number _{of people that I would like}tothank inwhat follows.

First and foremost, I wish toexpress my deep gratitude to my promotor Hans

Schu-macher, for hisconstant support from the first days, for the_{inspiring/stimulating}

discus-Sionsand_{ideas, and for}hiseffortstoimprovemy writingskills. I havelearnt a lot from his

professionalism and his way oflooking at mathematical problems. I feel both privileged

and fortunate to have had such an advisor. Thank you Hans!

I am also very much indebted toMaurice Heemels. Our pleasant and

fruitful

coopera-tionenormouslycontributed to thisthesis. Ido remember that hewas alwaysready when

I needed help. Once, he even had toimprovise a talk atthe Benelux meeting since I could

not show up. Hartetijk bedankt Maurice!

Further, I would liketo thankArjan van der Schaft who madean _{important impact on}

myresearch first atregular CWI meetingsand lateron every occasionwe could discuss.

I profited a lot

from numerous discussions that I had with my former teacher Kiilmiz

Cevik during his one year stay in the Netherlands. Besides our academic collaboration,

therewere chess games and lots of fun. Te#ekktirter Kulmiz!

I am also _{grateful to} the members of my promotion committee Bart De Moor, Jacob

Engwerda, Henk Nijmeijer, Arjan van der Schaft, Stef Tijs, and Jan Willems _{for reading}

thedraftversion andmaking valuable comments.

It is also a _{great pleasure for me to} thank those people with whom I shared so many

things and from whom I received a lot of help: Stefi Cavallar,Tamas Fleiner, EbruAngun,

Bram van den Broek, Kaifeng Chen, Gul Gurkan, Norbert Hari,Attila Korpos, and Amol

Viii _{ACKNOWLEDGEAIENT}

Last but not least, Ishouldmention _{UlviyeBa$er, Kulmiz Cevik,} ibrahim Eksin, Vasfi

Eldem, Cem _{Gdknar, Leyla Gtiren, Mujde Guzelkaya,} Kadri _{Ozqaldiran, Kemal Sarioglu,}

and Hasan Selbuz who encouraged and helped me to come to the Netherlands.

Finally, I acknowledge NWO (Nederlandse Organisatie voor Wetenschappelijk

Onder-zoek) and

TUBiTAK

(The Scientific and Technical Research Council ofTurkey) for their

financial support ofmyresearch.

Contents

Acknowledgement Vii

1 Introduction and Preliminaries 5

1.1 Introduction. 5 1.1.1 Outline ofthe thesis . . . . . 8

1.1.2 Origins of thechapters . . . . . . . . . . . . . . 9

1.2 Preliminaries . . . . . . . . 9 1.2.1

Notation . . . . . . . 13

... ... 9 1.2.2 _{Linear complementarity p r o b l e m. . . .} 1.2.3 Solutionc o n c e p t s. . . 14

References . . . . . . . . . .

. . . .1 5

I Well-posedness 21

2 Well-posedness of Linear Complementarity Systems 23

2.1 Introduction . . . . . . 23

2.2 Linear Complementarity

Systems . . . . . . 24

2.3 Main Results . . . . . . . . . . . . 30

2.4 Conclusions . . . .

31

2.5 Proofs . . . . .

. . . 31

2.5.1 _{Lipschitzian properties of LCP . . . . . . . . 31}

2.5.2 Rational matrices with index 1 . . . . . . 33

2.5.3 Towards to theproofofTheorem 2.3.3 . . . . . . . 37

2.5.4 Proofs ofTheorem 2.3.3 and Theorem 2.3.4 . . . . . . . . 40

References . . . . . 44

3

Linear

Passive

_{Complementarity}

_Systems 47

3.1 Introduction . . . .

47

2 CONTENTS

3.3 Linear Passive Complementarity

Systems . . . . . . . . . . . 50

3.4 Passifiability by Pole

Shifting . . . . . . . . . 53

3.5 Zeno

_{Behavior . . . . . . . . . . . . . 54}

3.6 Nonregular

Initial

States . . . . . . . . . 55

3.7 Conclusions... . . . . , . . . .5 8

3.8 Proofs . . . . . . , . . . . , . 59

3.8.1 Somefacts frommatrix

theory . . . . . . . 59

3.8.2 Someimplicationsof passivity . . . . . . . . 61

3.8.3 ProofsforSection 3.3 . . . . . . . . . . . . . . 64

3.8.4 Proofsfor

_{Section 3.4 . . . . . . . . . . 67}

3.8.5 Proofsfor _{Section 3.5 . . . . . . . . . 69}

3.8.6 On quadratic

programming . . . 71

3.8.7 Proofsfor Section 3.6 . . . . . . 71

References . . . , . . . , . . .

. . . .7 7 4 Systems with Piecewise Linear Elements 81

4.1 Introduction . . . . . . 81

4.2 _{Motivational Examples . . . . . .} . . . 83

4.3 Piecewise Linear

Characteristics . . . . . . 85

4.4 Complementarity

_{Problems . . . . . . . . . . . 88}

4.5 Piecewise Linear

Systems . . . . . . 89

4.6 Examples . . . . . . . . .9 1 4.6.1 Linear complementaritysystems . . . . . . . . 91

4.6.2 Linear relay systems . . . . . . . . . . 92

4.6.3 Linear _{systemswith saturation . . . .9 3}

4.7 Conclusions . . . . . . .

94

4.8 Proofs . . . .

. . . .9 5 4.8.1 Some_Lipschitzian results on HLCP . . . 95

4.8.2 On the _{invertibility of rational matrices . . . . . . 97}

4.8.3

Initial

solutionsand their

characterizations . . . . . . . . . 97

4.8.4 ProofofTheorem 4.5.4 . . . . . . . . . . 106

4.8.5 ProofofTheorem 4.5.5 . . . . . . . . . 108

4.8.6 Proofs _{for Section 4.6 . . . . . . . . . . . . . . . 112}

References . . . . . . .

. . . . 114

II

_{Approximations}

117

CONTENTS 3

5.1 Introduction . . . . . . . 119

5.2 Preliminaries . . . . . . . . 121

5.3 LinearComplementarity Systems . . . . . . . . . . . 122

5.4 Continuityof

Solutions . . . . . . . 123

5.4.1 Structured approximations . . . . . . . . . . 123

5.4.2 Unstructured approximations . . . . 127

5.5 Nonregular

Initial States . . . . . . . . . . . . .

. . . 128

5.6 Conclusions . . . . . . . . . . 129

5.7 Proofs . . . . . .

. . . 130

5.7.1 Topological complementarity

problem . . . . 130

5.7.2 ProofsforSection 5.4 . . . 132

. . . 137

References . . . . 6 Consistency of Backward Euler Method 139 6.1 Introduction . . . 139

6.2 Preliminaries . . . . . . . 141

6.3 _{The Backward Euler Time-stepping} Method . . . 143

6.4 Main Results for LPCS . . . . . . . . . . 147

6.5 Conclusions . . . . . . . . . 148

6.6 Proofs . . . . . . . . . . . 149

6.6.1 Preliminaries . . . 149

6.6.2 ProofofTheorem6.3.4 items 1 and 2 . . . . . . . . . 150

6.6.3 _{Topologicalcomplementarityproblem...} . . 152

6.6.4 The time-stepping method in aTCP

formulation . . . . . . 153

6.6.5 Convergence ofsolutions to TCPs . . . . . . . . . . . . . 156

6.6.6 Completing the proof ofTheorem 6.3.4 . . . . . 158

6.6.7 Some results on LCPs . . . 160

6.6.8 ProofofTheorem 6.4.1 . . . . . . . 164

6.6.9 ProofofTheorem 6.4.2 . . . . . . . 165

References . . . . . . . . . . . . . 167

7 A Time-stepping Method for Relay Systems 171 7.1 Introduction . . . . . . . 171

7.2 Linear Relay Systems . . . . . . . . . . 173

7.3 Example . . . . . . . . . . 173

7.4 _{The Backward Euler Time-stepping}Method . . . . . . . 174

7.5 _{Complementarity Framework . . . . . . . 175}

4 CONTENTS

7.5.2 Numerical

s c h e m e. . . , . . . , .

176

7.6 _{Linear Complementarity}

_{Systems . . .}

_{. . . .}

₁₇₇

7.7 Consistency ofTime-steppingfor RelaySystems . . . 178

7.8 Example . . . 180

7.9 Lemke's

_{Method . . . . . . . . . . . . . . . 182}

7.10 Conclusions . . . . . . . . . . . 183

7.11 Proofs . . . . . . _{. . . 184}

7.11.1 Proofof_{Theorem 7.5.1 . . . . . . . . . . . . . 184}

7.11.2 _{The remaining proofs} . . . 186

. . . 187

References . . . . . . , . . . . 8 Conclusions 191 8.1 Contributions . . . 191

8.2 FurtherResearch Topics

. . . .

. . . 192

Summary 195

Chapter 1

Introduction

and

Preliminaries

1.1 Introduction

Piecewiselinear_{modeling has been}a _widely used_{technique in many engineering areas for}

a long time. By means ofpiecewise linear models, nonlinear phenomena can be

approxi-mated as accurately as desired. In _{general, the cost} is_{sacrificing the} smoothness _and/or

having largemodels. However, the propertiesoffered bylinearity, even ina piecewise

man-ner, still make it one of the most natural _{options. Other ways in which piecewise} linear

systems may emerge include for instance gain scheduling type of_{controllers [39,58,591,}

variable structure _{systems 1661} and_{bang-bang control [9,431.}

Of

course, piecewise linear

systems form a very general class. Inevitably, one sometimes has to sacrifice generality

and consider specific subclasses _{in order to establish reasonably significant results. By}

following this idea, our treatment

will

focus ona subclasswhich allows us toemploy

com-plementaritymethodsofmathematical_{programming. With} a_slight abuse of_terminology,

we sometimes use the term complementarity systems (see 33,38,55,561) for thissubclass

ofpiecewise linear systems that can bedealt with by means of_{complementarity methods.}

It is possible to find lots of applicationareasin various fields such aselectrical engineering,

mechanical _{systems, operations research,} economics etc. We _{refer to 132,33,571 for more}

detailed discussion of (potential) application areas. Since our treatmentis based on

com-plementarity theory, wecan roughly say that our work lies in the

junction of

the system

theory and the mathematical programming. To

put/fit

thisthesis into a place within the

existing literature, we discuss related areasand approaches in what follows.

Motivated, to a great extent, by the applications in mechanical systems (see for

in-stance 142,521 for classical treatments of unilateral constraints, and see also 81 for a

survey on _{nonsmooth mechanics), and in} circuit theory and control systems theory (see

e.g. _{19,43,53,661),}discontinuous dynamicalsystems have beenstudied extensivelysince the

6 1.1. INTRODUCTION

differential _equations with discontinuous right hand sides have been under consideration

withan emphasis on the existence and uniquenessofsolutions in thesenseofCarathbodory

In the first part of_{this thesis, we}

will

address similarquestions for complementarity

sys-tems. Our development differs from Filippov's since complementaritysystems do not fit

into _{the framework of 1281 in}general. The work on differentialinclusions (see e.g. 11) is

another branchofresearch on_{discontinuous dynamical systems.} The combinations of

dif-ferential equationsand inequalities, and hence piecewiselinear systems, can be easily cast

as differential inclusions _{which usually} have nonunique solutions by their nature. On the

other hand, the uniqueness ofsolutions is ofgreat importance from our model validation

perspective.

Another way of looking at piecewise linear systems is toconsider them as asubfamily

of the_{huge family ofhybrid systems. Indeed, piecewiselinear systems can} be _{regarded as}

hybridsystems _{(what cannot be?) just by}translatingthe piecewiselinearity tothe language

of hybridsystems. Embedding thepiecewise linear nature intoa hybridautomatonmodel

would be one ofsuchtranslations. Suppose that the piecewiselinearsystem isgiven in the

following explicit form

i· = A'. + b' if x € X,

fori= 1,2'...,m. For the corresponding hybrid automaton model, one can choose m

modes in the natural way. The state space partition determined by the sets X' directly

indicates the inuariants and guards. 1 It is hard to come upwith tractableanalysis methods

forgeneral hybrid systems. Naturally, some researches have focused on _{special subclasses}

of hybriddynamical systems. Inparticular, the work that has been done onmixed logical

dynamical systems ( 13,41), first order linear hybrid systems with saturation ( 24 ) and

piecewise affine systems ( 60,61 ) is closely related to complementaritysystems. Indeed,

in a recent report 31l it has been shown for discrete systems that these subclasses and complementarity systems are equivalent urider certain assumptions.

Among the fields that stimulated the work on piecewise linear systems, circuit theory

has a special place _{because of the fact that} the piecewise linear modeling idea comes up

rather naturally in this context. Onebranchof_{research (see e.g. 15,19-22,29,40,41,45,67))}

is_mainly focusedon_{canotiical representations ofpiecewise}linear_{characteristics/functions.}

In the cited referencesonly analysis of static piecewise linear systems _(resistive piecewise

linear circuits in network theoretical terminology) has been considered. The main goal of

those works was to represent resistive piecewise linear circuits in a canonical form and to

propose methods to find the solutions (driving points) of the circuit. The employment of

the complementarity setting separates 15.40,41,45,671 from the others. The

first part

ofour thesis can l,e viewed as the contiriiiation ofthisstrand ofwork towards dynamical

1. INTRODUCTION AND

PRELIMINARIES 7

systerns.

Anotherdirection ofresearch in thecommunityofcircuittheory which our work can be

connected with is the simulation of switchingcircuits (see e.g. 12,5,27,44,45,48,54,681).

Roughlyspeaking, therearethree main approaches, namely event-tracking methods,

time-steppingmethods andsmoothingmethods.2 While_{the papers 12,48}are}examples of work on

event-tracking methods, [5,44,45,541 give examplesofstudiesontime-stepping methods.

At

this point, weshould mention the work on time-stepping methods that has been done

for unilaterallyconstrained mechanical_systemswithfrictionphenomena 147,49,51,62-641.

It seems that the questionofconvergence forthese methodsisusuallynotconsidered in the

literature of circuit theory. With the inspiration ofthe cited work onmechanical systems,

we have attempted to emphasize the need of justification of the time-stepping methods

for switching circuits in the last two chapters of the second part ofthe thesis. Thefirst

chapter ofthe secondpart dealswith _{smoothing methods.} As related work in the context

ofmechanicalsystems, one can refer to 18, Chapter 21 andreferencestherein.

After their introduction by Dupuis and _{Nagurney [251 (see also [501} for further

de-velopment), projected dynamical systems have been used for studying the behavior of

oligopolistic_{markets, urban} transportation networks,tramcnetworks, international trade,

agricultural and energy markets. Variational _{inequalities have} been employed to

charac-terize the stationary points of the projected dynamical systems. The well-known close

relationship (see e.g. 1301) between complementarity problemsandvariational inequalities

suggests that complementarity systems and projected dynamical systems are related to

eachother. Indeed, this relation has been addressed in 33, Chapter 61

In the operations research community, several variations/extensions/generalizations

of complementarity problems have been under consideration. Among all those

varia-tions/extensions/generalizations, the topological complementarity problem (TCP) (see

16,71) isof considerable importance for us. In the second part of thethesis,weemploy TCP

as a generalframework to investigate theconvergence ofapproximations. Well-posedness

of_{complementarity systems can}be _{formulated in a pure TCP} framework as well. _Indeed,

finding asolution ofa _{complementarity system} is _{nothing butfinding} a solution of a

cer-tainTCP. However, theavailable conditions which guaranteesolvability of TCPs are very

restrictive and are not satisfied in general by the systems we are looking at in this thesis.

In thisrespect, our well-posedness results providesolvability conditions for aspecialclass

of TCPs.

In aninfinite-dimensionalsystems setting, the book 261 addresses well-posedness issues

as well as convergence of smoothing and time-stepping methods for partial diferential

inequatities that arisefrom mechanics and physics. Since we work in a finite dimensional

8 1.1. INTRODUCTION

framework here, thetreatment in thecited reference isclearlymore general. _{However, its}

development hasbeen based on some coerciveness conditionandhence it has_implications

for only a rather restrictivesubclassof linear _{passive complementaritysystems.}

1.1.1 Outline of the

thesis

The thesis isdivided into two parts each containing three chapters. While Part I deals

with

the well-posedness of complementarity systems, Part II investigates convergence of

approximations ofcomplementaritysystems.

In Chapter 2we consider the well-posedness (in the sense ofexistence and uniqueness

of solutions) of linear complementaritysystems with external inputswhere the _underlying

linearsystem isofindex 1 asdefinedin Definition 2.3.1.

Linear passive complementarity systems (LPCS) are the objects of Chapter 3. The

properties thatare offered bypassivity make it possible to derive strongerwell-posedness

results in the sense that the solutionsare unique in largerspaces. The chapter contains

comparisons ofseveral solution concepts for LPCS. All the results that are obtained for

LPCS will

be extended to the class of systems that are passifiable by pole shifting (see

Definition 3.4.2). After investigating Zeno behavior ofthis _{newly introducedclass of}

sys-tems, we will pass to thediscussion onnonregular

initial

states. Finally, the chapter will be

closed

with

results on well-posedness for distributionalversions of two previously defined

solution _concepts.

Chapter 4isdevoted toa classofpiecewise systems that can be formulated in a

com-plementarity setting. Its main goal is to establish well-posedness results for this class of

systems. It will beshown that linear _{complementarity systems and linear relay systems}

can be treated withinthe framework used in this_chapter.

We considersome_{continuityproperties}of_{linear complementaritysystemsin Chapter 5.}

The idea is toreplacethe non-Lipschitziancomplementaritycharacteristic byaLipschitzian

characteristic and investigate the convergence of the sequence of trajectories produced by

approximatingsystems that have Lipschitziancharacteristic as theLipschitzian

character-istic tends to the _{non-Lipschitzian complementarity} characteristic. Wewill present

suffi-cient conditions forthe convergenceof approximating trajectories tothetrajectories of the

actual_system. Thechapter willbe_{closed by}adiscussion on_{more generalapproximations.}

In Chapter 6 we will show that a _{time-stepping method, namely the backward Euler}

method, is _{consistent (in the sense that the approximations generated by the method}

converge to the actual solution ofthe original system in a _{suitable sense) for LPCS. As a}

side result, it willbe proven that the solutions depend on the

initial

datacontinuously for

that class ofsystems.

inves-1. INTRODUCTION AND

_{PRE:LIMIN.ARIE:S 9}

tigate the consistency of the backward Euler method for relaysystems in Chapter 7. This

chapter will be _{followed by the conclusions} in _{Chapter 8.}

1.1.2 Origins of the chapters

Chapter 2 is mainly based on _{15 ,} which has been presented at the 14th International

SymposiumofMathematical TheoryofNetworks_{and Systemsin Perpignan (France), with}

slight changes. The only addition is Theorem 2.3.4 which provides a necessary condition

for well-posedness of the systems under consideration.

The material of Chapter 3 is a cocktail of 10.13,34,361 Indeed, the results on the

existence and uniqueness of solutions to LPCS were presented, for the

first time, at the

38thIEEEConference on Decision and Control in_{Phoenix (USA) (see 13l} where one can

also find the characterizationofregularinitial states). The notion of passifiability by pole

shifting (PPS) has been introduced in ₁₁₂₁ which has been presented at the 39th IEEE

Conference on Decision and Control in Sydney (Australia). The _{necessary and} sufficient

conditions for PPS property are again due to [121. The results on Zeno behavior can be

found in _118J Section 3.6 is based on

Ilq

which is an improved version of the paper

1341 that has been presented at the 4th International Conference on Automation of Mixed

Processes: _{Hybrid DynamicSystems} in Dortmund _(Germany).

Chapter 4is_basically based on 111 which isanoutgrowth ofthe paper 1461. Anearly

attempt, with weaker results, in this direction was presented at the European Control

Conference'99 _{in Karlsruhe (Germany)(see 117 )}

Chapter 5 is an extendedversion of the _{paper 1121.}

The report 1161, aftera minorrevision, has beenincludedasChapter 6. It has already

been submitted toIEEE Transactionson Circuitsand _{Systems. For a}less _technical

(with-out _{proofs) exposition,} werefer to 1141which has been presented at the 4th International

Conference on Automation ofMixed Processes: _{Hybrid Dynamic Systems} in Dortmund

(Germany).

The paper 1351, which was presented at the 39th IEEE Conference on Decision and

Controlin Sydney _{(Australia), has} been appended asChapter7after includingthe proofs.

1.2 Preliminaries

1.2.1 Notation

Every text thatcontains a bitofmathematics, like this thesis,iswritten in two languages.

10 1.2. PRELIMINARIES

mathematical notations. In Mathesis Biceps vetus et nova _(1670), Johann Caramue13

writes 102 = 857 where the sign '=' is employed as the separatrix in decimal fractions.

Althoughsuchseverecomplications are very unlikelyto_arise,we devotethissubsection to

the second language: mathematical notations.

Sets

The symbols R, R+, R++, _{R(s) and} C denote the sets ofreal numbers, nonnegative real

numbers, positive real numbers, real coefficient rational functions and _{complex numbers,}

respectively. For a given integer n, we _{write n for the set {1,2, . . . ,n} . Let A be a set.}

The notations

X

71 X m where n and m are integers denote the sets ofn-tuples and n x m

matrices ofthe elements of X. The set

of

subsets of X will be denoted by 2A'. We write

XI for

the number ofelements of X.

Matrices

Let A € Xnx™ be a matrix ofthe elements of the set X. We write A. · for the (i, j)th₄

element of A. The transpose of Aisdenoted by AT. For J c n, and K c m, _AJK denotes

the submatrix

{A,j}jEJ,k€K· If J = n (K = iii), we

also _{write A.K}

(Aj.). In order to

avoid_{bulky notation, we} use_{A.IK and A.71} instead

of

_{(AJK)-r and (AJK)-1,} respectively.

Given two

matrices A f X

na x m

and B E Xn,xm, the matrix obtained by stacking A over B i s denoted by col(A, B). The diagonal matrix withthe diagonalelement al, a2, · · · ,a n i s

denoted bydiag(al, a2,···,an)

A rational matrix A(s) f Rn*„' (s) is said to be _proper if lim,_,coA(s) is

finite. If

lims»ooA(s) = 0 it is said to be strictlyproper. Asquare rational matrixA(s) c Rnxm (s)

is_{called biproper if it}is_{proper, invertible as}arational matrix and itsinverse is also_proper.

Mappings

Given a mapping f:U- * V. we denote the image of

f b y i m f: = {v€V I U=

f (u) for some u €U} andthe kernel of A b y ker

_{f: = {1 1€u l f(U) = O}.}

_flwwill

denote

the restriction of f to W C U

Functionspaces

The notation F(U,

V) stands for the functions

defined from U to V. When U c R, we

definethe reverse operator rev[t'.1"} : · ([t', t"}, v) -+ F([t', t"l, v) by

(revlt#,t"j u)(t) = 21(t' + t" - t)

1. INTRODUCTION AND PRELIMINARIES 11

The most often utilized function space will be the space of Bohl functions. A function

f

:R+-*Ris

called Boht function if it has a rational Laplace transform. Every Bohl

function is of the form He,2. Gfor some matrices FE R;Nxn,G E Rn*t and H E Rixn. The

set of all Bohl functions will be denoted by B. As one can expect from _{their definition,}

Bohl functions arerelatedtolinear constant coefficienthomogeneousdifferentialequations

and hence linear _{(time-invariant)} dynamical systems. In our treatmentofpiecewise linear

dynamical systems, piecewise Bohl functions play a similar role to the one is played in

the study oflinear systems by Bohl functions. A

_{function f : R+ -* R is said to be a}

piecewise Boht function if for each

_{t e R+}

there

exist >0 and a

Bohl function g such

that flit, +,) = g l_{to,e). The set of all}such functionsis _{denoted by PB. Note that PB is not}

closed under timereversal. Since Bohl functions are real-analytic, the corresponding Bohl

function to a piecewise Bohl function for a given _(time

_{instant) t}

is uniquely determined

and the quantity max{E >0 1 fltu+A=gl to,4i s well-defined. For_convenience, we define

a: PBnx R -+Bnas

a(f, t) = g

and B:PB" x R -* Rp, u {co} as

B(f, t)

=max{€ >0 1 flit,t+E, = glio,£,}

where theBohlfunction g is suchthat _{flIt,t+p) = glp,p) for some p > O. The set}ofbounded

piecewise Bohl functions, denoted by

PBB,

consists of piecewise Bohl functions that are

bounded on [0, T] for each T > 0.

Another class of functions that appears later is the space of one-variable real-valued

(locally) square integrablefunctions. In the standard way, we say a _Lebesguemeasurable

function f : Q -+ R" is square integrable if

Q f-r(T)f (7-) d·r < 00

holds where Q c R. This class will be denoted by £2(Q, Rn). It is well-known that

£2(Q, Rn) is a Hilbert space with the inner product

(f, 9> = fT(7')g(T) d'r

where f, g E £2(Q, Rn). The norm that induced by this inner product can be given by

Ilfll=

12 1.2. PRELIMINARIES

A sequence {fn} C £2(Q, Rn) is said to converge (strongly) to f E £2(Q, Rn) if

lim Hfn - fll = 0,

n-*00

and it is said to convergeweakly to f E £2(Q, Rn) if

Jit<f.,g> = <f, g>

for all g € £2(fl, lin)

Two _particularsubspaces ofdistributions will be

of

interest. We denote the set of

dis-tributions thataresupported ona

point {t} by D;. It

iswell-known from thedistributional

theory (see e.g. _{[65, Theorem 24.6 ) that v E D; is of the form}

N

v=

Z vi8(,) i=0

where N i s a natural number, v' is a real

number for all i G N and

8(i) denotes the ith

derivative of the Dirac distribution 6 with the convention 8(0) = 8. _{Later on,} we restrict

our attentionto ratherspecial classesofdistributions, more_{specifically direct sums of D&}

and some function spaces. With an abuse of terminology, we say a distribution v is aBoht

distribution if it is of the form v = vimp

+

Ureg where the impulsive part vimP E Di and

the regular part v_reg E B. The set of all such distributionsis denoted by 86. Note that

Ba = 130 e B. The leading coefficient of the impulsive part of a Bohl

distribution v is

defined by

f O if vimp = O,

lead(uimp) = C

l IN if vimp = Elo v,8(i) with uN 96 0

We say that aBohl distribution uis initially nonnegative if

(lead(vimp) > 0) or (lead(vimp) = 0 and vre,(t) 2 0 for all t c [0, c) for some € > 0).

It is known ( [37, Lemma 5.31) that v is

initially

nonnegative if and only

if

0 (a) 2 0 for all

sufiiciently largea where 0(s) is its Laplace transform.

In parallel to thedefinitionofpiecewise Bohl_{distributions,}wedefinethespace ( ([0, T],

R) consisting of distributions v = vimp

+

Ureg where the impulsive part Uimp f I) and

1. INTRODUCTION AND

PRE;LIMINARIES 13

Miscellaneous

The notations {In} and

[lilli

denotes the sequence Il, I2, · · · and the ordered set of the

elements 71,1/2, · · ·, 1/k, respectively.

All _{inequalities involving} vectors must be _{understood componentwise. For two vectors}

1, y C Rn, max(I, y) and min(I, y) denote the componentwise maximum and minimum,

respectively. The nonnegativeand nonpositiveparts ofa vector s are denoted by x+ and

x-, i.e., I+ = max(z, 0) and z- - -min(z, 0). Note that I+ 2 0, I- 2 0 and x+ 1 x-.

We say thataproposition

_P(a)

holds forallsufficiently small (large) a f R+

if

there exists

ao € R+ > 0 such that P(a) holds for all 0 < a 5 ao (ao 5 a).

1.2.2

Linear

complementarity

problem

We_brieflyrecallthe_{linear complementarityproblem (LCP) of}mathematical programming.

For an extensive _{survey on the problem, the} reader is_{referred to 1231.}

Problem 1.2.1 (LCP(q, M)) Given q € Rm and M € Rmxm, find z E R™ such that

120

(1.la)

7+Mz,0

(1.lb)

IT (q + Mi) =0. _(1.lc)

We say that z isfeasibte if it satisfies

_{(1.la)-(1.lb).}

Similarly, we sayz _{solves LCP(q, M)}

if it satisfies (1.1). The set of all solutions of LCP(q, M) will bedenoted by SOL(q, M).

In general, SOL(q, M) may be the empty set. The notation K (M) denotes the set {q I

SOL(q, M) 0 0}. It is easy to see that IR

c

_{K(M) for all M}

E xm. The following

fact on the closedness of KCM) will be used several times in the sequel.

Fact 1.2.2 The set

_KIM)

(possibly empty) is closed for any matrix M.

The LCP leads to thestudy ofasubstantial number ofmatrixclassesthat relatetoseveral

aspects of theproblem suchas_{feasibility,solvability, unique solvability. The following ones}

will be of particular interest forourpurposes.

Definition 1.2.3

A matrix M € Rmxm is called

• nondegenerate if all its principal matrices arenonzero.

• a P-matrix if all its principal minors are _positive.

14 PRELIMINARIES • positive (nonnegative) definite if ITMz > 0 (2 0) for all 0 9,6 z E R"'.

• copositive if Il-Mz 2 0 for all z 2 0.

• copositive-plus if it iscopositive and thefollowing implicationholds:

sTMI =0 and I 2 0= * (M+ MT)I =0.

For a given nonempty set S, we say that the set {v I v.rw 2 0 for all w € S} is the dual

cone of S. It is_{denoted by S'. The next lemma states some of the standard results on the}

matrix classesdefined above.

Lemma 1.2.4 Let M f R"'*"' be given. The fottowing statements hold.

1. [23, Theorem 3.3.71 LCP(q, M) has a unique solution for all q € R™ if and only if

M is a P-matrix.

2. [23, Corollary 3.8.101 If M is copositive-plus then K(M) = (SOL(0, M))*.

Note that the last_implicationholds in particular when M is_{nonnegative definite.}

1.2.3

_{Solution concepts}

It is

already well-known that the selection of universum, the space where all possible

solutions live, is ofgreat importance for the existence and uniqueness issues. We aim

to illustrate this fact by means ofan example in this subsection. Consider the following

example due to Filippov 128, p. 1161

Il = Sgn It - 2sgn I2 I2 = 2sgn Il + Sgn I2

where sgn I is theset-valued function given by

-1 ift<0

SgnI= [-1,11 ifT-0.

(1 i f y>0

Its time-reversed version can be_{given by}

1/1 = -sgn 11 + 2sgn 72

1. INTRODUCTION AND PRELIMINARIES 15

Solutions of the time-reversed version are spiralingtowards the origin, which is an

equi-2 15-D 05 0 - -0.5--0.5 0 05 1 15 2 2.5 3 ._1

Figure 1.1: Trajectory with

initial

_{state (2,2)T.}

librium. Since (Iyl(t)I + 11/2(t)1) =

-2 when y(t) 76 0 along trajectories z of the

sys-tem, solutions reach the origin in finite time _(see Figure 1.1 for a trajectory). Therefore,

time-reversals of all these trajectories qualify as a solution (starting from the

origin) to

the original system in the sense ofDefinition 3.3.8 belowfor which the universum is

£2-functions that are defined on a bounded interval. However, if one requires solutions to

be right _{continuous (as in Definition} 3.3.1 _{below) then there is} a unique solution, namely

the zero solution. As this example shows, a system might be well-posed forone solution

concept but not for another one.

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Part I

Chapter 2

Well-posedness

of

Linear

Complementarity

Systems

with Inputs:

Low Index Case

2.1 Introduction

The appropriateness ofaproposed mathematical model fora_{given physical system can be}

tested in various ways. A very basic test is the

following: if

the physical system that is

being modeledisdeterministic inthe sense thatit showsidenticalbehavior underidentical

circumstances, then the mathematical model should have thesame property. Modelvalidity

would be putintoseriousdoubt ifit would turn out thatthe equations of the mathematical

model allow _{multiple solutions for} some

initial data. With

any model formulation for a

deterministic physical system it istherefore important to establish well-posedness of the

model, i.e.,existence and uniqueness ofsolutionsfor feasible initialconditions.

Thischapter considersthe well-posedness ofaclass of linearcomplementaritysystems,

i.e., linear systems coupled to complementarity conditions. The most typical examples of these systems arelinear electrical networkswithideal diodes. In the engineering_literature,

mathematicalmodels that make use of the ideal diode characteristicare _{routinely used for}

such networks. Remarkably enough, it seems that the well-posedness ofsuch models has

notbeenrigorously establishedbefore. Althoughgeneralresults from the theoryof ordinary

differentialequations may be used to establishwell-posednessofnetworkmodelscontaining

elements with _{Lipschitzian characteristics (see for instance 1121) or in} special ca.ses even

for non-Lipschitzian _{characteristics (see} for instance l2,71),such results do not cover the

ideal diode characteristicsinceitcannot bereformulated asa _{current or voltage-controlled}

resistor. Neither does itseem possible to derive general well-posednessresultsfornetwork

24 2.2. LINEAR COMPLEMENTARITY SYSTEMS

hand sides _131,whichinnetworkterminologyisconcernedwithmodels_involvingideal relay

elements. Thetheory thatwedevelopbelow willbe based onthe theoryofcomplementarity

systems that has been worked out ina seriesofrecent papers I4-6,9,101, see also Illl

It is easy to come up with examples of mathematical models involving ideal diode

characteristics (which are equivalent to complementarity _{conditions) that are not}

well-posed; seefor instance [91. Therefore, somerestrictions need tobeimposed. Wewill study

this classofmodels in the more general setting ofcomplementarity conditions coupled to

linear_{dynamical systems with} a _{special zero} structure at _infinity. Some _{might say that}

it is "intuitively clear" that such_{network models are well-posed; nevertheless,} idealdiodes

are only approximations to real diodes and so the fact that actual networks with diodes

behave deterministically does not make it evident that the corresponding mathematical

modelswithidealized elements have_{unique solutions. Rather,} as_arguedabove,oneshould

consider well-posedness as a test of model validity.

The chapteris_organizedasfollows. In Section 2 wefirst of alldevelopa_precisenotion of

solutionfor linear complementarity systems. Then in Section 3webrieflydiscussthe linear

complementarity problem (LCP) of mathematicalprogramming that plays an important

role in _{our development. The main results follow}in Section 4. The _{chapter will}beclosed

by conclusions in Section 5 and _proofsin Section 6.

2.2

_{Linear Complementarity}

_Systems

As interconnection of a continuous, time-invariant, linear system and complementarity

conditions, a _{linear complementarity system can}be _{given by}

2(t) = Az(t) + Bu(t) +

Ew(t)

(2.la)

7(t) = Cz(t) +

Du(t)

(2.lb)

0 5 u(t) 1 y(t) 2

0. (2.lc)

where x(t) E lf, u(t) € R™, 7(t) c R™, w(t) c RP, and A, B, C, D and Earematrices with

appropriate sizes. We denote the _{above system by LCS(A, B, C, D, E). For} the previous

study onthisclassofsystems, the reader is _{referred to [4-6,9,10l. From a} hybrid system

point of view, one can distinguish 2"' modes depending on complementarity conditions

(2.lc). Everyindex set K c mdetermines oneofthesemodes byimposing the constraints

YK = 0 and u#1K = 0. Associated to each mode

K,

there area linear dynamics given by

2(t) = Ax(t) + Bu(t) + Ew(t)

7(t) = Cz(t) + Du(t)

2. WELL-POSEDNESS OF LINEAR COMPLEMENTARITY

SYSTEMS 25

and aset called invariants _{given by}

1/m\K (1) 2 0, UK (t) 2 0 (2.2)

Starting at a given mode, the system trajectories must obey the dynamics corresponding

to this mode as long as they belong to the _{invariant set, i.e.,} satisfy the inequalities (2.2).

Time instants at which the state variables tend to leave the invariant set arecalled event

times. Whenever _{an event occurs,} another mode

will

become active depending on the

state variables z andinputs w at theevent time. Beforegiving a precise definition of the

solutionconcept, we illustratetheabovefeatures of thesystems underconsideration in the

followingexample.

R

a ./t.

D,A

C=

L yD2

Figure 2.1: RLC circuitwith ideal diodes

Example 2.2.1 Consider the linear RLC

_{circuit (with R =1 Ohm,}

L=1 Henry and

C = 1 Farad) coupled to two idealdiodesasshown in Figure 2.1. By choosing thevoltage

across the capacitor and the current through the inductor as the state variables and by

taking into account the ideal diode characteristic depicted in Figure 2.2, the governing

equations ofthe network can be given by

C vc = iL - iD, +

11)2 (2.3a)

L iL - -vc - RiL -

R:DY (2.3b) ViC)1 = VC (2.3c)

UD2 - -vc - RlL -

RiD, (2.3d) 0 5 101 1 -UD, 2 0 (2.3e)

0 5 102 1 -vD, 2

0 _(2.3f)

26 2.2.

LINEAR COMPLEMENTARITY SYSTEMS

ZD

-VD

Figure 2.2: Ideal diode characteristic

• Mode BB. In this mode, both diodes are blocking, i. e., 10, = 102 = 0. Hence, the

conditions (2.3e)-(2.3f) yield

0 - iDi

-UDi 20

0 = iD, - VD, 2 0.

The activities, or circuittopology (see Figure 2.3 (a)) as it is called in network theory

terminology, can be_{given by}

C uc = iL

LliiL - -uc - RiL·

The correspondinginvariants _(theconditionsthatensurethediodes tokeepblocking

state) are

-VD -uc 2 0

-UD2 = vc + RiL 2 0.

• Mode BC: Thefirst diode isblocking while thesecond oneisconducting, i. e., iD, =

UD, - 0 in thismode. Hence, the conditions (2.3e)-(2.3f) yield

0 = iD,

- vol 2 0

0 5 102 vD, = 0.

The activities can be given by

C i t Llc = i L + 1 D,

LliL - -Dc - RZL - RiD,

2. WELL-POSEDNESS OF LINEAR

COMPLEMENTARITY SYSTEMS 27

The corresponding circuit topology is shown in Figure 2.3 (b). The invariants, as

being the conditionsthat ensurethe firstdiode tokeep _{blocking state and the}second

conducting state, are

-UDi = -vc 2 0

iD2 = - VC - iL 2 0.

• Mode CB: The first diode isconducting and the second one is blocking, i. e., vDI

-iD, = 0 in this

mode. Hence, the _{conditions (2.3e)-(2.3f) yield}

0 iD, vot = 0

0 _=iD2

_{-UD, 20.}

The activities can begiven by

Cluc = iL - iDI

L iL = -vc -REL

1)Di = VC = 0.

The corresponding circuit topologyisshown in Figure 2.3 (c). The invariants are

iDi = 4 2 0

-VD2 = RiL 2 0

• Mode CC:In this mode both diodes are conducting, i. e., vol = UD2 - 0. Hence, the

conditions (2.3e)-(2.3f)yield

0 5 toi UD, = 0

0 5 iD, UD, - 0.

The correspondingcircuit topologyis depicted in Figure 2.3 (d) and the activities of

the mode can be given by

Ct VC = LL - iD, + iD,

L iL = -vc - RiL - RiDe

UDl

i UC = 0

28 2.2.

LINEAR COMPLEMENTARITY SYSTEMS

The invariants can be obtained as

ED, = 0

102 = -iL 2 0.

R R

-lva

C L C L

(a) Mode BB (b) Mode BC

R R

n o 9\A' T

C L C= L

(c) ModeCB _{(d) Mode CC}

Figure 2.3: Circuit topologies for the modes

We investigate the behaviour of the network for the

initial

_{condition (vc(0), iL(0)) =}

(-e, 1). Note that the first diode must be blocking initially_{since vD, (0) = vc(0) 9, 0 and}

the second one must be conducting_{initially since vc(0) + iL(0) < 0. Then, the mode BC}

is active at thebeginning. It can be checked that thedynamics of this modeyields

1-t

vc(t) = -e

iL (t) = 1.

The first inequality ofthosedescribing the invariants of thismode holds for all twhile the

second one holds only if t c _{[0,11 Therefore, tl = 1 is} the first event time. At the event

time, the state ofthe system is

given by uc(1) = -1 and iL(1) = 1. In the next mode,

the first diode still must be blocking initially _{since vD, (1) = Uc(1) 96 0 but the} second

2. WELL-POSE;DNESS OF LINEAR

COMPLEMENTARITY SYSTEMS 29

should be the mode BB. It can be computed that the dynamics ofthisniode yields

vc(t) = -e-2(t-1)[cos( · (t - 1)) - 9 sin(30(t - 1))}

iL(t) =

e-&(t-13[cos

(9(t - 1)) + 9 sin(30(t - 1))]

for t 2 1.I t can be verified that vc(t) + iL(t) 2 0 and vc(t) 5 0 for all 1 5 1 5 1+ 2 71-,

and also that vc(1

+

2e7r) = 0 and

(1 + 2 ,r) >

0. Consequently, the first diode

cannot be blocking anymore and this means that the second event takes place at event

time t2

=1+

T. At the event time, thestate ofthe system can begiven by vc(t2) = 0

and it;(t2) = e=f". The next

mode should be the mode CB and its dynamics result in VC(t) = 0

2v/5

iL(t) = et-(1--91)

fort 2 tl · It can

be easily verified that invariants of this mode are

_{satisfied for all t 2 tl,}

i.e., there will be no modechange anymore. The trajectories aredepicted in Figure 2.4.

2 1 '

--I

0- --1 -T-E-Tz -1- 2 2 --3 1 0 0.5 1 1.5 2 2.5 3 3.5 4 1 1 0 -1

VD'

-'0 2 --3 0 0.5 1 1 5 2 2 5 3 3.5 21 '

1»\

- VD ie' 0---1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 time

Figure2.4: Trajectories forthe

initial

_{state (-e, 1).}

Later on, we

will

employ'hybrid system' thinkingto construct solutions to LCSs.

How-ever, the concept ofsolution will be clarified first. In what follows, we propose a solution

notion by keeping in mind the hybrid features of the system. Indeed, the 'universum' we

consider,namely the space ofpiecewise_{Bohl functions,} isasymmetric in time inthe sense

30 2.3.

MAIN RESULTS

Definition 2.2.2 A triple (u,

_{I,y) E PBmtntm is a solution on [0, T] of LCS(A,B,C,D,E)}

for the input wE PBBp andthe

initial

state zo ifthe followingconditions hold

ft

x(t) = zo + / [Ax(s) + Bu(s) + Ew(s)] ds

Jo

y(t) = Cz(t) + Du(t)

0 5 u(t) ly(t) 2 0

for all t € [O, Tl

Notice that I-trajectory is continuous by

definition. In

the sequel, we

will

derive

suf-ficient _{conditions under which linear complementarity}systems have unique solutions.

Be-fore doing this, we

will

review some facts from complementarity theory in order to be

self-contained.

2.3

Main Results

In thissection,wepresent sufficientconditions for well-posedness, in thesenseofexistence

and uniquenessofsolutions, oflinear complementarity systems. One of our main

assump-tions will be on the index ofthe underlying system. The following definitions will make

clear what is_{meant by the index of}a linear_system.

Definition 2.3.1

Arationalmatrix_{H(s) E Rixt (s) is said to be of index k if it}isinvertible

as arational matrix and s-kH-1 (s)is proper.

Definition 2.3.2

A rational matrix H(s) € R x' (s) is said to be totally of index k if all

its principal submatrices are of index k.

Now, we can state the main result concerning the well-posedness of the linear

comple-mentarity systems.

Theorem 2.3.3 Consider a matrix quintupte (A, B, C, D, E). Suppose that G(s)

=D+

C(sI - A)-1B is totally

_of index 1 and

_{G(a) is a P-matrix for all}

stdiciently Large a. Then, thefollowing two statements are equivalent.

1. For each w € PBBp, there exists a unique solution on [0,00) of LCS(A, B, C, D, E)

forthe

input w and the initial state zo

2. WELL-POSE;DNESS OF LINEAR COMPLEMENTARITY

SYSTEMS 31

Note that G(s) is totally of index 1 if and only if D + CBs -1 is. _{Since det(·) is}

a continuous

_{function, if D}

+

CBs-1 is

of index 1 then we have sign(det(GJJ(a_{))) =}

sign(det(DJJ

+ Cj•B.ja-1)) for

all _{sufficiently large a. This} means that the P-matrix

assumption on the transfer matrix holds if D

+

CBa-1 is

a

P-matrix for

all _sufficiently

large 0. In general, there are noexplicitcharacterizations of theset K(D). However, if D

is copositive-plus the set _{K(D) can} be characterized _explicitly as stated in Lemma 1.2.4

item 2. Note that all nonnegativedefinite matrices arecopositive-plus.

Theabove theorem provides sufficient conditions for well-posedness. In the next

theo-rem wewill present a_necessary condition.

Theorem

2.3.4 Consider a matrix quintuple (A, B, C, D, E). Suppose that D is

nonde-generate and C is offull row rank. If D is not a Po-matrix then for some zo € Rn and

T > 0,

there exist at least two _{diferent solutions on [O, T] of LCS(A, B, C, D, E) for the}

zero input and the initial state so.

2.4 Conclusions

We showed that a class of_{linear complementarity systems including} electrical networks

with diodes as typical examples passes the validity test ofwell-posedness. Using

comple-mentaritytheory, we were abletoprove the existence anduniquenessofsolutiontrajectories

under a condition on the zero structure ofthe underlying statespace description. As an

additional result we gave an explicit characterization of the regular states, i.e., the

ini-tial states for which the linear complementarity systems admit solutions in the sense of

Definition 2.2.2.

2.5 Proofs

This section isdevoted tothe proofs ofthe presented results.

2.5.1

_{Lipschitzian properties of LCP}

We beginwith statingsome results on _{Lipschitzian properties of LCP. For} our_purposes,

it is important to relate the index of the system and Lipschitzian properties of a series

of LCPs involving the transferfunction of the system. First, we present a rather general

32 2.5. PROOFS

Lemma 2.5.1 Let the sets Q, c R" for i = 1.2....,p be such that Q' _{is closed and convez,}

and

15 Q'-R" /=1

Assume that f. Rn _* Rm is a continuous function which is Lipschitz on each set Qi with

aLipschitz constant a'. Then, f is Lipschitz continuous with the Lipschitz constant max a'

Proof: Let z. and zb f

Rn. Consider the linesegment Isa, Ibl in R'i. Sincethe number of

Q's is finite and they are allclosed convex sets. one can find finitenumber ofpoints in Rn,

say xs =: xi, 3,2,. .,Zi := 1,6. such that for each i c Z- 1 theline segment [zi, I,+1] c Qi·

for some j,. Note that due to thecontinuity of f we have

Ilf(zi) - f(Ii)11 5 Ilf(Ii) - j.(3:2)11 + Ilf(I2) - f(Za)11 + . . . + 11,/(:EL-1) - f(Zi)11 5 011 ZI - I.211+ 0121'I2 - 13311+ . . . + c¥j,-111It-1 - I'll

5 (max n)(11/1 - I.211 + 11.Z,2 - I311 +...+11:r'-1 - XIII).

Since all ts are on thelinesegment [Ii, It], itis_{obvious that Z:21 'Izi -z,+111 = 111:1 -1,1 11.}

Consequently, we_{get If(:rt) - f(:rt)11 < (niax at')11.1,1 - .211.}

In thesequel, foragiven nondegenerate _{matrix M € Rnx„, d(M)} is defined as follows:

d(AI) = (max Iljlfi,111).

JC:

It is known (see 1, Theorem 7.3.101) that if LCP(q. M) is uniquely solvable for each

q then the mapping q »+ z where z is the unique solution of LCP(q, M) is Lipschitz

continuous. However, tocompute the Lipschitz constant

given in Ill is not so easy. By

making use ofabovelemma, we will show that thequantity d(AI) canbetakenasLipschitz

constant for the LCP(q,_{AI) whenever AI is} aP-matrix.

Lemma

2.5.2 Assume that Al e R..n

is a P-matrix. Let z' be the unique solution of

LCP(q'..11) for i = 1.2. Then. we have

2' - 2.211 <d(M)'Iqi- q211.

Proof:

Since M is a _{P-matrix, I,emma 1.2.4 item 1 implies that LCP(q, M) is uniquely}

solvable for allq. Consider the ftinctioll q e-+ Zwhere z is the uniquesolution of LCP(q, Al)

For a given _{index set J f ii. define the set Q'' as}

2. WELL-POSEDNESS OF LINEAR COMPLEMENTARITY

SYSTEMS 33

i. Clearly, QJ is closed andconvex for each J.

ii. Note that LCP(q, M) is _{solvable for all q € Rn, and if z is the unique solution of}

LCP(q, M) and J = {j € n I zj > 0} then q € QJ. Thus, we have U Qi = Rn.

JCn

iii. Note that if q E Q then z with zi = -Mljq., and zii\J = 0 is the (unique) solution

of LCP(q, M). Then, the function q »+ z can begiven by

z = Alq if q €QJ

where

Ajj = -M.N and A(L =0 for J n K n

L=0.

Moreover, it iscontinuous due

to the uniqueness of thesolution ofthecorresponding LCPand Lipschitz continuous

on QJ with the _{constant IIAJ 11.}

iv . N otice t ha t A J 11 = 11 Myj 11.

The facts

i-iv

enables us to get the required result byapplying

Lemma 2.5.1. I

2.5.2

Rational matrices with index 1

We

will

characterize the index ofa rationalmatrix in terms of its power series expansion

around

infinity in

the following Lemma.

Lemma 2.5.3 Let H(s) C R'xt (s) be given and let its power series expansion around infinity be given by

H(s) =H o + His-1 + . . . .

Then, thefollowingstatement are equivalent.

1. H (s) isofindex 1.

2. Ho + Hi 8-1 is _{of indeI 1.}

3. im Ho e Hi (ker Ho) = Rf.

4. There exist matrices p E Rpxt and Q e RCE-P)*t such that

[p] and Iptio]

LQj LQI,t1