Linear Matrix Inequalities in System and Control Theory

Linear Matrix Inequalities in
System andControlTheory
SIAM Studies inApplied Mathematics
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Series List
Vol.1Lie-BÄacklundTransformations inApplications
RobertL.Anderson andNailH.Ibragimo v
Vol.2MethodsandApplicationsofIntervalAnalysis
RamonE.Moore
Vol.3Ill-PosedProblemsforIntegrodi®erentialEquations inMechanicsand
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PeterLinz
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N.KikuchiandJ.T.Oden
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RolandGlowinskiandP.LeTallec
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JohnE.Lagnese
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IsaakRubinstein
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MauroFabrizioandAngeloMorro
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YuriiNesterovandArkadiiNemirovskii
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Adelaida B.Vasil'eva,ValentinF.Butuzov,andLeonidV.Kalachev
Vol.15LinearMatrixInequalities inSystemandControlTheory
Stephen Boyd,LaurentElGhaoui, EricFeron,andVenkataramanan
Balakrishnan
Stephen Boyd, LaurentElGhaoui,
EricFeron,andVenkatar amanan Balakrishnan
Linear Matrix Inequalities in
System andControlTheory
SocietyforIndustrial andApplie dMathematics
¨ Philadelphia
Copyrightc°1994bytheSocietyforIndustrial andApplied Mathematics.
Allrightsreserved.Nopartofthisbookmaybereproduced,stored,ortransmitted inany
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Library ofCongress Cataloging-in-Publication Data
Linearmatrixinequalities insystemandcontroltheory/Stephen Boyd
...[etal.].
p.cm.--(SIAMstudiesinappliedmathematics ;vol.15)
Includes bibliographical references andindex.
ISBN0-89871-334-X
1.Controltheory.2.Matrixinequalities. 3.Mathematical
optimization. I.Boyd,Stephen P.II.Series:SIAMstudiesin
appliedmathematics :15.
QA402.3.L489 1994
515'.64--dc20 94-10477
Contents
Preface vii
Acknowledgmen ts ix
1Introduction 1
1.1Overview::::::::::::::::::::::::::::::::::1
1.2ABriefHistoryofLMIsinControlTheory:::::::::::::::2
1.3NotesontheStyleoftheBook::::::::::::::::::::::4
1.4OriginoftheBook:::::::::::::::::::::::::::::5
2SomeStandardProblemsInvolvingLMIs 7
2.1LinearMatrixInequalities :::::::::::::::::::::::::7
2.2SomeStandard Problems :::::::::::::::::::::::::9
2.3Ellipsoid Algorithm :::::::::::::::::::::::::::::12
2.4Interior-PointMethods:::::::::::::::::::::::::::14
2.5StrictandNonstrict LMIs:::::::::::::::::::::::::18
2.6Miscellaneous ResultsonMatrixInequalities ::::::::::::::22
2.7SomeLMIProblems withAnalytic Solutions ::::::::::::::24
NotesandReferences :::::::::::::::::::::::::::::::27
3SomeMatrixProblems 37
3.1Minimizing Condition NumberbyScaling::::::::::::::::37
3.2Minimizing Condition NumberofaPositive-De¯nite Matrix::::::38
3.3Minimizing NormbyScaling:::::::::::::::::::::::38
3.4Rescaling aMatrixPositive-De¯nite :::::::::::::::::::39
3.5MatrixCompletion Problems :::::::::::::::::::::::40
3.6Quadratic Approximation ofaPolytopic Norm:::::::::::::41
3.7Ellipsoidal Approximation:::::::::::::::::::::::::42
NotesandReferences :::::::::::::::::::::::::::::::47
4LinearDi®erentialInclusions 51
4.1Di®erentialInclusions :::::::::::::::::::::::::::51
4.2SomeSpeci¯cLDIs:::::::::::::::::::::::::::::52
4.3Nonlinear SystemAnalysis viaLDIs:::::::::::::::::::54
NotesandReferences :::::::::::::::::::::::::::::::56
5AnalysisofLDIs:StateProperties 61
5.1Quadratic Stability:::::::::::::::::::::::::::::61
5.2InvariantEllipsoids :::::::::::::::::::::::::::::68
v
vi Contents
NotesandReferences :::::::::::::::::::::::::::::::72
6AnalysisofLDIs:Input/Output Properties 77
6.1Input-to-State Properties:::::::::::::::::::::::::77
6.2State-to-Output Properties::::::::::::::::::::::::84
6.3Input-to-Output Properties::::::::::::::::::::::::89
NotesandReferences :::::::::::::::::::::::::::::::96
7State-FeedbackSynthesisforLDIs 99
7.1StaticState-FeedbackControllers:::::::::::::::::::::99
7.2StateProperties::::::::::::::::::::::::::::::100
7.3Input-to-State Properties:::::::::::::::::::::::::104
7.4State-to-Output Properties::::::::::::::::::::::::107
7.5Input-to-Output Properties::::::::::::::::::::::::109
7.6Observer-Based ControllersforNonlinear Systems:::::::::::111
NotesandReferences :::::::::::::::::::::::::::::::112
8Lur'eandMultiplierMethods 119
8.1Analysis ofLur'eSystems:::::::::::::::::::::::::119
8.2IntegralQuadratic Constrain ts::::::::::::::::::::::122
8.3Multipliers forSystems withUnknownParameters :::::::::::124
NotesandReferences :::::::::::::::::::::::::::::::126
9SystemswithMultiplicativeNoise 131
9.1Analysis ofSystems withMultiplicativ eNoise::::::::::::::131
9.2State-FeedbackSynthesis:::::::::::::::::::::::::134
NotesandReferences :::::::::::::::::::::::::::::::136
10Miscellaneous Problems 141
10.1Optimization overanA±neFamilyofLinearSystems:::::::::141
10.2Analysis ofSystems withLTIPerturbations :::::::::::::::143
10.3PositiveOrthantStabilizabilit y::::::::::::::::::::::144
10.4LinearSystems withDelays::::::::::::::::::::::::144
10.5InterpolationProblems:::::::::::::::::::::::::::145
10.6TheInverseProblem ofOptimal Control:::::::::::::::::147
10.7SystemRealization Problems :::::::::::::::::::::::148
10.8Multi-Criterion LQG::::::::::::::::::::::::::::150
10.9NonconvexMulti-Criterion Quadratic Problems :::::::::::::151
NotesandReferences :::::::::::::::::::::::::::::::152
Notation 157
ListofAcronyms 159
Bibliography 161
Index 187
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Preface
Thebasictopicofthisbookissolvingproblems fromsystemandcontroltheoryusing
convexoptimization. Weshowthatawidevarietyofproblems arisinginsystem
andcontroltheorycanbereduced toahandfulofstandard convexandquasiconvex
optimization problems thatinvolvematrixinequalities. Forafewspecialcasesthere
are\analytic solutions" totheseproblems, butourmainpointisthattheycanbe
solvednumerically inallcases.Thesestandard problems canbesolvedinpolynomial-
time(by,e.g.,theellipsoid algorithm ofShor,Nemirovskii,andYudin),andsoare
tractable, atleastinatheoretical sense.Recentlydevelopedinterior-pointmethods
forthesestandard problems havebeenfoundtobeextremely e±cientinpractice.
Therefore, weconsider theoriginalproblems fromsystemandcontroltheoryassolved.
Thisbookisprimarily intendedfortheresearcherinsystemandcontroltheory,
butcanalsoserveasasourceofapplication problems forresearchersinconvexop-
timization. Although webelievethatthemethodsdescribedinthisbookhavegreat
practical value,weshouldwarnthereaderwhoseprimary interestisappliedcontrol
engineering. Thisisaresearchmonograph: Wepresentnospeci¯cexamples ornu-
mericalresults,andwemakeonlybriefcommentsabouttheimplications oftheresults
forpractical controlengineering. Toputitinamorepositivelight,wehopethatthis
bookwilllaterbeconsidered asthe¯rstbookonthetopic,notthemostreadable or
accessible.
Thebackground required ofthereaderisknowledgeofbasicsystemandcontrol
theoryandanexposuretooptimization. Sontag'sbookMathematic alControlThe-
ory[Son90]isanexcellentsurvey.Furtherbackground material iscoveredinthe
textsLinearSystems [Kai80]byKailath, NonlinearSystems Analysis[Vid92]by
Vidyasagar,Optimal Control:LinearQuadraticMethods[AM90]byAnderson and
Moore,andConvexAnalysisandMinimization Algorithms I[HUL93]byHiriart{
UrrutyandLemar¶echal.
Wealsohighlyrecommend thebookInterior-p ointPolynomial Algorithms inCon-
vexProgramming [NN94]byNesterovandNemirovskiiasacompanion tothisbook.
Thereaderwillsoonseethattheirideasandmethodsplayacriticalroleinthebasic
ideapresentedinthisbook.
vii

Acknowledgments
WethankA.Nemirovskiiforencouraging ustowritethisbook.WearegratefultoS.
Hall,M.Gevers,andA.Packardforintroducingustomultiplier methods,realization
theory,andstate-feedbac ksynthesismethods,respectively.Thisbookwasgreatly
improvedbythesuggestions ofJ.Abedor,P.Dankoski,J.Doyle,G.Franklin, T.
Kailath, R.Kosut,I.Petersen,E.Pyatnitskii, M.Rotea,M.Safonov,S.Savastiouk,
A.Tits,L.Vandenberghe,andJ.C.Willems. Itisaspecialpleasure tothankV.
Yakubovichforhiscommentsandsuggestions.
TheresearchreportedinthisbookwassupportedinpartbyAFOSR(under
F49620-92-J-0013), NSF(underECS-9222391), andARPA(underF49620-93-1-0085).
L.ElGhaouiandE.FeronweresupportedinpartbytheD¶el¶egationG¶en¶eralepour
l'Armemen t.E.FeronwassupportedinpartbytheCharlesStarkDraperCareer
DevelopmentChairatMIT.V.Balakrishnan wassupportedinpartbytheControl
andDynamical Systems GroupatCaltechandbyNSF(underNSFDCDR-8803012).
ThisbookwastypesetbytheauthorsusingLATEX,andmanymacrosoriginally
writtenbyCraigBarratt.
Stephen Boyd Stanford,California
LaurentElGhaoui Paris,France
EricFeron Cambridge, Massachusetts
Venkataramanan Balakrishnan CollegePark,Maryland
ix

Linear Matrix Inequalities in
System andControlTheory

Chapter 1
Introduction
1.1Overview
Theaimofthisbookistoshowthatwecanreduceaverywidevarietyofprob-
lemsarisinginsystemandcontroltheorytoafewstandard convexorquasiconvex
optimization problems involvinglinearmatrixinequalities (LMIs). Sincetheseresult-
ingoptimization problems canbesolvednumericallyverye±cientlyusingrecently
developedinterior-pointmethods,ourreduction constitutes asolution totheoriginal
problem, certainly inapractical sense,butalsoinseveralothersensesaswell.Incom-
parison, themoreconventionalapproachistoseekananalytic orfrequency-domain
solution tothematrixinequalities.
Thetypesofproblems weconsider include:
²matrixscalingproblems, e.g.,minimizing condition numberbydiagonal scaling
²construction ofquadratic Lyapunovfunctions forstabilityandperformance anal-
ysisoflineardi®erentialinclusions
²jointsynthesisofstate-feedbac kandquadratic Lyapunovfunctions forlinear
di®erentialinclusions
²synthesisofstate-feedbac kandquadratic Lyapunovfunctions forstochasticand
delaysystems
²synthesisofLur'e-typeLyapunovfunctions fornonlinear systems
²optimization overana±nefamilyoftransfer matrices, including synthesisof
multipliers foranalysis oflinearsystemswithunknownparameters
²positiveorthantstabilityandstate-feedbac ksynthesis
²optimalsystemrealization
²interpolationproblems, including scaling
²multicriterion LQG/LQR
²inverseproblem ofoptimalcontrol
Insomecases,wearedescribing known,published results;inothers,weareextending
knownresults.Inmanycases,however,itseemsthattheresultsarenew.
Byscanning thelistaboveorthetableofcontents,thereaderwillseethatLya-
punov'smethodswillbeourmainfocus.Herewehaveasecondary goal,beyond
showingthatmanyproblems fromLyapunovtheorycanbecastasconvexorquasi-
convexproblems. ThisistoshowthatLyapunov'smethods,whicharetraditionally
1
2 Chapter 1Introduction
appliedtotheanalysis ofsystemstability,canjustaswellbeusedto¯ndboundson
systemperformanc e,providedwedonotinsistonan\analytic solution".
1.2ABriefHistoryofLMIsinControlTheory
ThehistoryofLMIsintheanalysis ofdynamical systems goesbackmorethan100
years.Thestorybeginsinabout1890,whenLyapunovpublished hisseminalwork
introducingwhatwenowcallLyapunovtheory.Heshowedthatthedi®erentialequa-
tion
d
dtx(t)=Ax(t) (1.1)
isstable(i.e.,alltrajectories convergetozero)ifandonlyifthereexistsapositive-
de¯nitematrixPsuchthat
ATP+PA<0: (1.2)
Therequiremen tP>0,ATP+PA<0iswhatwenowcallaLyapunovinequalit yon
P,whichisaspecialformofanLMI.Lyapunovalsoshowedthatthis¯rstLMIcould
beexplicitly solved.Indeed,wecanpickanyQ=QT>0andthensolvethelinear
equationATP+PA=¡QforthematrixP,whichisguaranteedtobepositive-de¯nite
ifthesystem(1.1)isstable.Insummary ,the¯rstLMIusedtoanalyzestabilityofa
dynamical systemwastheLyapunovinequalit y(1.2),whichcanbesolvedanalytically
(bysolvingasetoflinearequations).
Thenextmajormilestone occursinthe1940's.Lur'e,Postnikov,andothersin
theSovietUnionappliedLyapunov'smethodstosomespeci¯cpractical problems
incontrolengineering, especially,theproblem ofstabilityofacontrolsystemwitha
nonlinearit yintheactuator. Although theydidnotexplicitly formmatrixinequalities,
theirstabilitycriteriahavetheformofLMIs.Theseinequalities werereduced to
polynomial inequalities whichwerethenchecked\byhand"(for,needless tosay,small
systems). Nevertheless theywerejusti¯ably excitedbytheideathatLyapunov's
theorycouldbeappliedtoimportant(anddi±cult) practical problems incontrol
engineering. FromtheintroductionofLur'e's1951book[Lur57]we¯nd:
Thisbookrepresentsthe¯rstattempt todemonstrate thattheideasex-
pressed60yearsagobyLyapunov,whichevencomparativ elyrecentlyap-
pearedtoberemotefrompractical application, arenowabouttobecomea
realmedium fortheexamination oftheurgentproblems ofcontemporary
engineering.
Insummary ,Lur'eandotherswerethe¯rsttoapplyLyapunov'smethodstopractical
controlengineering problems. TheLMIsthatresulted weresolvedanalytically ,by
hand.Ofcoursethislimitedtheirapplication tosmall(second, thirdorder)systems.
Thenextmajorbreakthrough cameintheearly1960's,whenYakubovich,Popov,
Kalman, andotherresearcherssucceeded inreducing thesolution oftheLMIsthat
aroseintheproblem ofLur'etosimplegraphical criteria, usingwhatwenowcall
thepositive-real(PR)lemma(seex2.7.2).Thisresulted inthecelebrated Popov
criterion, circlecriterion, Tsypkin criterion, andmanyvariations. Thesecriteriacould
beappliedtohigherordersystems, butdidnotgracefully orusefullyextendtosystems
containingmorethanonenonlinearit y.Fromthepointofviewofourstory(LMIsin
controltheory), thecontribution wastoshowhowtosolveacertainfamilyofLMIs
bygraphical methods.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
1.2ABrief Histor yofLMIs inContr olTheor y 3
TheimportantroleofLMIsincontroltheorywasalreadyrecognized intheearly
1960's,especiallybyYakubovich[Yak62,Yak64,Yak67].Thisisclearsimplyfrom
thetitlesofsomeofhispapersfrom1962{5, e.g.,Thesolution ofcertainmatrix
inequalities inautomatic controltheory(1962),andThemethodofmatrixinequalities
inthestabilitytheoryofnonlinearcontrolsystems(1965;Englishtranslation 1967).
ThePRlemmaandextensions wereintensivelystudiedinthelatterhalfofthe
1960s,andwerefoundtoberelatedtotheideasofpassivity,thesmall-gain criteria
introducedbyZamesandSandberg,andquadratic optimalcontrol.By1970,itwas
knownthattheLMIappearinginthePRlemmacouldbesolvednotonlybygraphical
means,butalsobysolvingacertainalgebraic Riccatiequation (ARE).Ina1971
paper[Wil71b]onquadratic optimalcontrol,J.C.Willems isledtotheLMI
"
ATP+PA+QPB+CT
BTP+C R#
¸0; (1.3)
andpointsoutthatitcanbesolvedbystudying thesymmetric solutions oftheARE
ATP+PA¡(PB+CT)R¡1(BTP+C)+Q=0;
whichinturncanbefoundbyaneigendecomp ositionofarelatedHamiltonian matrix.
(Seex2.7.2fordetails.) Thisconnection hadbeenobservedearlierintheSovietUnion,
wheretheAREwascalledtheLur'eresolving equation (see[Yak88]).
Soby1971,researchersknewseveralmethodsforsolvingspecialtypesofLMIs:
direct(forsmallsystems), graphical methods,andbysolvingLyapunovorRiccati
equations. Fromourpointofview,thesemethodsareall\closed-form" or\analytic"
solutions thatcanbeusedtosolvespecialformsofLMIs.(Mostcontrolresearchers
andengineers consider theRiccatiequation tohavean\analytic" solution, sincethe
standard algorithms thatsolveitareverypredictable intermsofthee®ortrequired,
whichdependsalmostentirelyontheproblem sizeandnottheparticular problem
data.Ofcourseitcannotbesolvedexactlyina¯nitenumberofarithmetic stepsfor
systemsof¯fthandhigherorder.)
InWillems' 1971paperwe¯ndthefollowingstrikingquote:
ThebasicimportanceoftheLMIseemstobelargelyunappreciated. It
wouldbeinteresting toseewhether ornotitcanbeexploited incompu-
tationalalgorithms, forexample.
HereWillems referstothespeci¯cLMI(1.3),andnotthemoregeneralformthat
weconsider inthisbook.Still,Willems' suggestion thatLMIsmighthavesome
advantagesincomputational algorithms (ascompared tothecorrespondingRiccati
equations) foreshado wsthenextchapterinthestory.
Thenextmajoradvance(inourview)wasthesimpleobservationthat:
TheLMIsthatariseinsystemandcontroltheorycanbeformulatedas
convexoptimization problemsthatareamenable tocomputer solution.
Although thisisasimpleobservation,ithassomeimportantconsequences, themost
importantofwhichisthatwecanreliablysolvemanyLMIsforwhichno\analytic
solution" hasbeenfound(orislikelytobefound).
Thisobservationwasmadeexplicitly byseveralresearchers.Pyatnitskii andSko-
rodinskii[PS82]wereperhapsthe¯rstresearcherstomakethispoint,clearlyand
completely .Theyreducedtheoriginalproblem ofLur'e(extended tothecaseofmul-
tiplenonlinearities) toaconvexoptimization problem involvingLMIs,whichthey
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
4 Chapter 1Introduction
thensolvedusingtheellipsoid algorithm. (Thisproblem hadbeenstudiedbefore,but
the\solutions" involvedanarbitrary scalingmatrix.) Pyatnitskii andSkorodinskii
werethe¯rst,asfarasweknow,toformulatethesearchforaLyapunovfunction as
aconvexoptimization problem, andthenapplyanalgorithm guaranteedtosolvethe
optimization problem.
Weshouldalsomentionseveralprecursors. Ina1976paper,HorisbergerandBe-
langer[HB76]hadremarkedthattheexistence ofaquadratic Lyapunovfunction that
simultaneously provesstabilityofacollection oflinearsystems isaconvexproblem
involvingLMIs.Andofcourse,theideaofhavingacomputer searchforaLya-
punovfunction wasnotnew|itappears,forexample, ina1965paperbySchultzet
al.[SSHJ65].
The¯nalchapterinourstoryisquiterecentandofgreatpractical importance: the
developmentofpowerfulande±cientinterior-pointmethodstosolvetheLMIsthat
ariseinsystemandcontroltheory.In1984,N.Karmark arintroducedanewlinear
programming algorithm thatsolveslinearprograms inpolynomial-time, liketheellip-
soidmethod,butincontrasttotheellipsoid method,isalsoverye±cientinpractice.
Karmark ar'sworkspurredanenormous amountofworkintheareaofinterior-point
methodsforlinearprogramming (including therediscoveryofe±cientmethodsthat
weredevelopedinthe1960sbutignored). Essentiallyallofthisresearchactivitycon-
centratedonalgorithms forlinearand(convex)quadratic programs. Thenin1988,
NesterovandNemirovskiidevelopedinterior-pointmethodsthatapplydirectlytocon-
vexproblems involvingLMIs,andinparticular, totheproblems weencounterinthis
book.Although thereremains muchtobedoneinthisarea,severalinterior-point
algorithms forLMIproblems havebeenimplemen tedandtestedonspeci¯cfamilies
ofLMIsthatariseincontroltheory,andfoundtobeextremely e±cient.
Asummary ofkeyeventsinthehistoryofLMIsincontroltheoryisthen:
²1890:FirstLMIappears;analytic solution oftheLyapunovLMIviaLyapunov
equation.
²1940's: Application ofLyapunov'smethodstorealcontrolengineering prob-
lems.SmallLMIssolved\byhand".
²Early 1960's: PRlemmagivesgraphical techniquesforsolvinganotherfamily
ofLMIs.
²Late1960's: ObservationthatthesamefamilyofLMIscanbesolvedbysolving
anARE.
²Early 1980's: Recognition thatmanyLMIscanbesolvedbycomputer via
convexprogramming.
²Late1980's: Developmentofinterior-pointalgorithms forLMIs.
ItisfairtosaythatYakubovichisthefatherofthe¯eld,andLyapunovthegrandfather
ofthe¯eld.Thenewdevelopmentistheabilitytodirectlysolve(general) LMIs.
1.3NotesontheStyleoftheBook
Weuseaveryinformal mathematical style,e.g.,weoftenfailtomentionregularityor
othertechnicalconditions. Everystatemen tistobeinterpreted asbeingtruemodulo
appropriate technicalconditions (thatinmostcasesaretrivialto¯gureout).
Weareveryinformal, perhapsevencavalier,inourreduction ofaproblem to
anoptimization problem. Wesometimes skip\details" thatwouldbeimportantif
theoptimization problem weretobesolvednumerically .Asanexample, itmaybe
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
1.4Origin oftheBook 5
necessary toaddconstrain tstotheoptimization problem fornormalization ortoensure
boundedness. Wedonotdiscussinitialguessesfortheoptimization problems, even
thoughgoodonesmaybeavailable.Therefore, thereaderwhowishestoimplement
analgorithm thatsolvesaproblem considered inthisbookshouldbeprepared to
makeafewmodi¯cations oradditions toourdescription ofthe\solution".
Inasimilarway,wedonotpursueanytheoretical aspectsofreducing aproblem
toaconvexproblem involvingmatrixinequalities. Forexample, foreachreduced
problem wecouldstate,probably simplify,andtheninterpretinsystemorcontrol
theoretic termstheoptimalit yconditions fortheresulting convexproblem. Another
fascinating topicthatcouldbeexplored istherelationbetweensystemandcontrol
theorydualityandconvexprogramming duality.Oncewereduceaproblem arising
incontroltheorytoaconvexprogram, wecanconsider variousdualoptimization
problems, lowerboundsfortheproblem, andsoon.Presumably thesedualproblems
andlowerboundscanbegiveninteresting system-theoretic interpretations.
Wemostlyconsider continuous-time systems, andassumethatthereadercan
translate theresultsfromthecontinuous-time casetothediscrete-time case.We
switchtodiscrete-time systemswhenweconsider systemrealization problems (which
almostalwaysariseinthisform)andalsowhenweconsider stochasticsystems (to
avoidthetechnicaldetailsofstochasticdi®erentialequations).
Thelistofproblems thatweconsider ismeantonlytoberepresentative,and
certainly notexhaustiv e.Toavoidexcessiverepetition,ourtreatmen tofproblems
becomesmoreterseasthebookprogresses. Inthe¯rstchapteronanalysis oflin-
eardi®erentialinclusions, wedescribemanyvariations onproblems (e.g.,computing
boundsonmargins anddecayrates);inlaterchapters, wedescribefewerandfewer
variations, assuming thatthereadercouldworkouttheextensions.
Eachchapterconcludes withasectionentitledNotesandReferences, inwhichwe
hideproofs,precisestatemen ts,elaborations, andbibliograph yandhistorical notes.
Thecompleteness ofthebibliograph yshouldnotbeoverestimated, despiteitssize
(over500entries).Theappendixcontainsalistofnotation andalistofacronyms
usedinthebook.Weapologizetothereaderforthesevennewacronymsweintroduce.
Tolightenthenotation, weusethestandard conventionofdropping thetime
argumen tfromthevariables indi®erentialequations. Thus,_x=Axisourshort
formfordx=dt=Ax(t).HereAisaconstantmatrix;whenweencountertime-
varyingcoe±cients,wewillexplicitly showthetimedependence, asin_x=A(t)x.
Similarly ,wedropthedummyvariablefromde¯niteintegrals,writingforexample,RT
0uTydtforRT
0u(t)Ty(t)dt.Toreducethenumberofparenthesesrequired, weadopt
theconventionthattheoperators Tr(traceofamatrix)andE(expectedvalue)have
lowerprecedence thanmultiplication, transpose,etc.Thus,TrATBmeans Tr¡
ATB¢
.
1.4OriginoftheBook
ThisbookstartedoutasasectionofthepaperMethodofCentersforMinimizing
GeneralizedEigenvalues, byBoydandElGhaoui[BE93],butgrewtoolargetobe
asection. Forafewmonthsitwasamanuscript(thatpresumably wouldhavebeen
submitted forpublication asapaper)entitledGeneralizedEigenvalue ProblemsAris-
inginControlTheory.ThenFeron,andlaterBalakrishnan, startedaddingmaterial,
andsoonitwasclearthatwewerewritingabook,notapaper.Theorderofthe
authors' namesre°ectsthishistory.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Chapter 2
SomeStandardProblems
InvolvingLMIs
2.1LinearMatrixInequalities
Alinearmatrixinequalit y(LMI)hastheform
F(x)¢=F0+mX
i=1xiFi>0; (2.1)
wherex2Rmisthevariableandthesymmetric matricesFi=FT
i2Rn£n,i=
0;:::;m,aregiven.Theinequalit ysymbolin(2.1)meansthatF(x)ispositive-
de¯nite, i.e.,uTF(x)u>0forallnonzerou2Rn.Ofcourse,theLMI(2.1)is
equivalenttoasetofnpolynomial inequalities inx,i.e.,theleadingprincipal minors
ofF(x)mustbepositive.
Wewillalsoencounternonstrict LMIs,whichhavetheform
F(x)¸0: (2.2)
ThestrictLMI(2.1)andthenonstrict LMI(2.2)arecloselyrelated,butaprecise
statemen toftherelation isabitinvolved,sowedeferittox2.5.Inthenextfew
sectionsweconsider strictLMIs.
TheLMI(2.1)isaconvexconstrain tonx,i.e.,thesetfxjF(x)>0gisconvex.
Although theLMI(2.1)mayseemtohaveaspecialized form,itcanrepresenta
widevarietyofconvexconstrain tsonx.Inparticular, linearinequalities, (convex)
quadratic inequalities, matrixnorminequalities, andconstrain tsthatariseincontrol
theory,suchasLyapunovandconvexquadratic matrixinequalities, canallbecastin
theformofanLMI.
Multiple LMIsF(1)(x)>0;:::;F(p)(x)>0canbeexpressed asthesingleLMI
diag(F(1)(x);:::;F(p)(x))>0.Therefore wewillmakenodistinction betweenaset
ofLMIsandasingleLMI,i.e.,\theLMIF(1)(x)>0;:::;F(p)(x)>0"willmean\the
LMIdiag(F(1)(x);:::;F(p)(x))>0".
WhenthematricesFiarediagonal, theLMIF(x)>0isjustasetoflinear
inequalities. Nonlinear (convex)inequalities areconvertedtoLMIformusingSchur
complemen ts.Thebasicideaisasfollows:theLMI
"
Q(x)S(x)
S(x)TR(x)#
>0; (2.3)
7
8 Chapter 2Some StandardProblems InvolvingLMIs
whereQ(x)=Q(x)T,R(x)=R(x)T,andS(x)dependa±nelyonx,isequivalentto
R(x)>0;Q(x)¡S(x)R(x)¡1S(x)T>0: (2.4)
Inotherwords,thesetofnonlinear inequalities (2.4)canberepresentedasthe
LMI(2.3).
Asanexample, the(maximumsingularvalue)matrixnormconstrain tkZ(x)k<1,
whereZ(x)2Rp£qanddependsa±nelyonx,isrepresentedastheLMI
"
IZ(x)
Z(x)TI#
>0
(sincekZk<1isequivalenttoI¡ZZT>0).Notethatthecaseq=1reducestoa
generalconvexquadratic inequalit yonx.
Theconstrain tc(x)TP(x)¡1c(x)<1,P(x)>0,wherec(x)2RnandP(x)=
P(x)T2Rn£ndependa±nelyonx,isexpressed astheLMI
"
P(x)c(x)
c(x)T1#
>0:
Moregenerally ,theconstrain t
TrS(x)TP(x)¡1S(x)<1;P(x)>0;
whereP(x)=P(x)T2Rn£nandS(x)2Rn£pdependa±nelyonx,ishandled by
introducinganew(slack)matrixvariableX=XT2Rp£p,andtheLMI(inxand
X):
TrX<1;"
XS(x)T
S(x)P(x)#
>0:
Manyotherconvexconstrain tsonxcanbeexpressed intheformofanLMI;seethe
NotesandReferences.
2.1.1Matrices asvariables
Wewilloftenencounterproblems inwhichthevariablesarematrices, e.g.,theLya-
punovinequalit y
ATP+PA<0; (2.5)
whereA2Rn£nisgivenandP=PTisthevariable.Inthiscasewewillnotwriteout
theLMIexplicitly intheformF(x)>0,butinsteadmakeclearwhichmatrices are
thevariables. Thephrase\theLMIATP+PA<0inP"meansthatthematrixPis
avariable.(Ofcourse,theLyapunovinequalit y(2.5)isreadilyputintheform(2.1),
asfollows.LetP1;:::;Pmbeabasisforsymmetric n£nmatrices (m=n(n+1)=2).
ThentakeF0=0andFi=¡ATPi¡PiA.)LeavingLMIsinacondensed form
suchas(2.5),inaddition tosavingnotation, mayleadtomoree±cientcomputation;
seex2.4.4.
Asanotherrelatedexample, consider thequadratic matrixinequalit y
ATP+PA+PBR¡1BTP+Q<0; (2.6)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.2Some StandardProblems 9
whereA,B,Q=QT,R=RT>0aregivenmatrices ofappropriate sizes,and
P=PTisthevariable.Notethatthisisaquadraticmatrixinequalit yinthevariable
P.Itcanbeexpressed asthelinearmatrixinequalit y
"
¡ATP¡PA¡QPB
BTP R#
>0:
Thisrepresentationalsoclearlyshowsthatthequadratic matrixinequalit y(2.6)is
convexinP,whichisnotobvious.
2.1.2Linearequalityconstraints
Insomeproblems wewillencounterlinearequalityconstrain tsonthevariables, e.g.
P>0;ATP+PA<0;TrP=1; (2.7)
whereP2Rk£kisthevariable.Ofcoursewecaneliminate theequalityconstrain tto
write(2.7)intheformF(x)>0.LetP1;:::;Pmbeabasisforsymmetric k£kmatrices
withtracezero(m=(k(k+1)=2)¡1)andletP0beasymmetric k£kmatrixwith
TrP0=1.ThentakeF0=diag(P0;¡ATP0¡P0A)andFi=diag(Pi;¡ATPi¡PiA)
fori=1;:::;m.
Wewillrefertoconstrain tssuchas(2.7)asLMIs,leavinganyrequired elimination
ofequalityconstrain tstothereader.
2.2SomeStandardProblems
Herewelistsomecommon convexandquasiconvexproblems thatwewillencounter
inthesequel.
2.2.1LMIproblems
GivenanLMIF(x)>0,thecorrespondingLMIProblem (LMIP)isto¯ndxfeassuch
thatF(xfeas)>0ordetermine thattheLMIisinfeasible. (Byduality,thismeans:
FindanonzeroG¸0suchthatTrGFi=0fori=1;:::;mandTrGF0·0;seethe
NotesandReferences.) Ofcourse,thisisaconvexfeasibilityproblem. Wewillsay
\solving theLMIF(x)>0"tomeansolvingthecorrespondingLMIP.
Asanexample ofanLMIP,consider the\simultaneous Lyapunovstabilityprob-
lem"(whichwewillseeinx5.1):WearegivenAi2Rn£n,i=1;:::;L,andneedto
¯ndPsatisfying theLMI
P>0;AT
iP+PAi<0;i=1;:::;L;
ordetermine thatnosuchPexists.Determining thatnosuchPexistsisequivalent
to¯ndingQ0;:::;QL,notallzero,suchthat
Q0¸0;:::;QL¸0;Q0=LX
i=1¡
QiAT
i+AiQi¢
; (2.8)
whichisanother(nonstrict) LMIP.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
10 Chapter 2Some StandardProblems InvolvingLMIs
2.2.2Eigenvalueproblems
Theeigenvalueproblem (EVP)istominimize themaximumeigenvalueofamatrix
thatdependsa±nelyonavariable,subjecttoanLMIconstrain t(ordetermine that
theconstrain tisinfeasible), i.e.,
minimize ¸
subjectto¸I¡A(x)>0;B(x)>0
whereAandBaresymmetric matrices thatdependa±nelyontheoptimization
variablex.Thisisaconvexoptimization problem.
EVPscanappearintheequivalentformofminimizing alinearfunction subject
toanLMI,i.e.,
minimize cTx
subjecttoF(x)>0(2.9)
withFana±nefunction ofx.InthespecialcasewhenthematricesFiarealldiagonal,
thisproblem reducestothegenerallinearprogramming problem: minimizing the
linearfunctioncTxsubjecttoasetoflinearinequalities onx.
Another equivalentformfortheEVPis:
minimize ¸
subjecttoA(x;¸)>0
whereAisa±nein(x;¸).Weleaveittothereadertoverifythattheseformsare
equivalent,i.e.,anycanbetransformed intoanyother.
Asanexample ofanEVP,consider theproblem (whichappearsinx6.3.2):
minimize °
subjectto"
¡ATP¡PA¡CTCPB
BTP °I#
>0;P>0
wherethematricesA2Rn£n,B2Rn£p,andC2Rm£naregiven,andPand°are
theoptimization variables. Fromourremarks above,thisEVPcanalsobeexpressed
intermsoftheassociatedquadratic matrixinequalit y:
minimize °
subjecttoATP+PA+CTC+°¡1PBBTP<0;P>0
2.2.3Generalized eigenvalueproblems
Thegeneralized eigenvalueproblem (GEVP) istominimize themaximumgeneralized
eigenvalueofapairofmatrices thatdependa±nelyonavariable,subjecttoanLMI
constrain t.ThegeneralformofaGEVPis:
minimize ¸
subjectto¸B(x)¡A(x)>0;B(x)>0;C(x)>0
whereA,BandCaresymmetric matrices thatarea±nefunctions ofx.Wecan
expressthisas
minimize ¸max(A(x);B(x))
subjecttoB(x)>0;C(x)>0
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.2Some StandardProblems 11
where¸max(X;Y)denotesthelargestgeneralized eigenvalueofthepencil¸Y¡X
withY>0,i.e.,thelargesteigenvalueofthematrixY¡1=2XY¡1=2.ThisGEVPis
aquasiconvexoptimization problem sincetheconstrain tisconvexandtheobjective,
¸max(A(x);B(x)),isquasiconvex.Thismeansthatforfeasiblex,~xand0·µ·1,
¸max(A(µx+(1¡µ)~x);B(µx+(1¡µ)~x))·
maxf¸max(A(x);B(x));¸max(A(~x);B(~x))g:
Notethatwhenthematrices arealldiagonal andA(x)andB(x)arescalar,this
problem reducestothegenerallinear-fractional programming problem, i.e.,minimiz-
ingalinear-fractional function subjecttoasetoflinearinequalities. Inaddition,
manynonlinear quasiconvexfunctions canberepresentedintheformofaGEVPwith
appropriate A,B,andC;seetheNotesandReferences.
Anequivalentalternate formforaGEVPis
minimize ¸
subjecttoA(x;¸)>0
whereA(x;¸)isa±neinxfor¯xed¸anda±nein¸for¯xedx,andsatis¯es the
monotonicit ycondition ¸>¹=)A(x;¸)¸A(x;¹).Asanexample ofaGEVP,
consider theproblem
maximize ®
subjectto¡ATP¡PA¡2®P>0;P>0
wherethematrixAisgiven,andtheoptimization variablesarethesymmetric matrix
Pandthescalar®.(Thisproblem arisesinx5.1.)
2.2.4Aconvexproblem
Although wewillbeconcerned mostlywithLMIPs,EVPs,andGEVPs, wewillalso
encounterthefollowingconvexproblem, whichwewillabbreviate CP:
minimize logdetA(x)¡1
subjecttoA(x)>0;B(x)>0(2.10)
whereAandBaresymmetric matrices thatdependa±nelyonx.(Notethatwhen
A>0,logdetA¡1isaconvexfunction ofA.)
Remark :Problem CPcanbetransformed intoanEVP,sincedetA(x)>¸can
berepresentedasanLMIinxand¸;see[NN94,x6.4.3].Aswedowithvariables
whicharematrices, weleavetheseproblems inthemorenaturalform.
Asanexample ofCP,consider theproblem:
minimize logdetP¡1
subjecttoP>0;vT
iPvi·1;i=1;:::;L(2.11)
herevi2RnaregivenandP=PT2Rn£nisthevariable.
Wewillencounterseveralvariations ofthisproblem, whichhasthefollowing
interpretation. LetEdenotetheellipsoid centeredattheorigindetermined byP,
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
12 Chapter 2Some StandardProblems InvolvingLMIs
E¢=©
z¯¯zTPz·1ª
.Theconstrain tsaresimplyvi2E.SincethevolumeofEispro-
portionalto(detP)¡1=2,minimizing logdetP¡1isthesameasminimizing thevolume
ofE.Sobysolving(2.11),we¯ndtheminimumvolumeellipsoid, centeredattheori-
gin,thatcontainsthepointsv1;:::;vL,orequivalently,thepolytopeCofv1;:::;vLg,
where Codenotestheconvexhull.
2.2.5Solvingtheseproblems
Thestandard problems (LMIPs, EVPs,GEVPs, andCPs)aretractable,fromboth
theoretical andpractical viewpoints:
²Theycanbesolvedinpolynomial-time (indeedwithavarietyofinterpretations
fortheterm\polynomial-time").
²Theycanbesolvedinpractice verye±ciently.
By\solvetheproblem" wemean:Determine whether ornottheproblem isfea-
sible,andifitis,compute afeasiblepointwithanobjectivevaluethatexceedsthe
globalminimumbylessthansomeprespeci¯edaccuracy.
2.3EllipsoidAlgorithm
We¯rstdescribeasimpleellipsoid algorithm that,roughlyspeaking,isguaranteed
tosolvethestandard problems. Wedescribeitherebecauseitisverysimpleand,
fromatheoretical pointofview,e±cient(polynomial-time). Inpractice, however,the
interior-pointalgorithms describedinthenextsectionaremuchmoree±cient.
Although moresophisticated versionsoftheellipsoid algorithm candetectin-
feasibleconstrain ts,wewillassumethattheproblem wearesolvinghasatleastone
optimalpoint,i.e.,theconstrain tsarefeasible. (Inthefeasibilityproblem, weconsider
anyfeasiblepointasbeingoptimal.) Thebasicideaofthealgorithm isasfollows.We
startwithanellipsoidE(0)thatisguaranteedtocontainanoptimalpoint.Wethen
compute acuttingplaneforourproblem thatpassesthrough thecenterx(0)ofE(0).
Thismeansthatwe¯ndanonzero vectorg(0)suchthatanoptimalpointliesinthe
half-space©
z¯¯g(0)T(z¡x(0))·0ª
(orinthehalf-space©
z¯¯g(0)T(z¡x(0))<0ª
,
dependingonthesituation). (Wewillexplainhowtodothisforeachofourproblems
later.)Wethenknowthattheslicedhalf-ellipsoid
E(0)\n
z¯¯¯g(0)T(z¡x(0))·0o
containsanoptimalpoint.Nowwecompute theellipsoidE(1)ofminimumvolume
thatcontainsthisslicedhalf-ellipsoid; E(1)isthenguaranteedtocontainanoptimal
point.Theprocessisthenrepeated.
Wenowdescribethealgorithm moreexplicitly .AnellipsoidEcanbedescribed
as
E=©
z¯¯(z¡a)TA¡1(z¡a)·1ª
whereA=AT>0.Theminimumvolumeellipsoid thatcontainsthehalf-ellipsoid
©
z¯¯(z¡a)TA¡1(z¡a)·1;gT(z¡a)·0ª
isgivenby
~E=n
z¯¯¯(z¡~a)T~A¡1(z¡~a)·1o
;
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.3Ellipsoid Algorithm 13
where
~a=a¡A~g
m+1; ~A=m2
m2¡1µ
A¡2
m+1A~g~gTA¶
;
and~g=g=p
gTAg.(Wenotethattheseformulasholdonlyform¸2.Inthecaseof
onevariable,theminimumlengthintervalcontainingahalf-intervalisthehalf-interval
itself;theellipsoid algorithm, inthiscase,reducestothefamiliarbisection algorithm.)
Theellipsoid algorithm isinitialized withx(0)andA(0)suchthatthecorresponding
ellipsoid containsanoptimal point.Thealgorithm thenproceedsasfollows:for
k=1;2;:::
compute ag(k)thatde¯nesacuttingplaneatx(k)
~g:=¡
g(k)TA(k)g(k)¢¡1=2g(k)
x(k+1):=x(k)¡1
m+1A(k)~g
A(k+1):=m2
m2¡1³
A(k)¡2
m+1A(k)~g~gTA(k)´
Thisrecursion generates asequence ofellipsoids thatareguaranteedtocontainan
optimalpoint.Itturnsoutthatthevolumeoftheseellipsoids decreases geometrically .
Wehave
vol(E(k))·e¡k
2mvol(E(0));
andthisfactcanbeusedtoprovepolynomial-time convergenceofthealgorithm. (We
referthereadertothepaperscitedintheNotesandReferences forprecisestatemen ts
ofwhatwemeanby\polynomial-time" andindeed,\convergence," aswellasproofs.)
Wenowshowhowtocompute cuttingplanesforeachofourstandard problems.
LMIPs: Consider theLMI
F(x)=F0+mX
i=1xiFi>0:
IfxdoesnotsatisfythisLMI,thereexistsanonzerousuchthat
uTF(x)u=uTÃ
F0+mX
i=1xiFi!
u·0:
De¯negbygi=¡uTFiu,i=1;:::;m.Thenforanyzsatisfying gT(z¡x)¸0we
have
uTF(z)u=uTÃ
F0+mX
i=1ziFi!
u=uTF(x)u¡gT(z¡x)·0:
Itfollowsthateveryfeasiblepointliesinthehalf-spacefzjgT(z¡x)<0g,i.e.,this
gde¯nesacuttingplanefortheLMIPatthepointx.
EVPs:Consider theEVP
minimize cTx
subjecttoF(x)>0
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
14 Chapter 2Some StandardProblems InvolvingLMIs
Suppose¯rstthatthegivenpointxisinfeasible, i.e.,F(x)6>0.Thenwecancon-
structacuttingplaneforthisproblem usingthemethoddescribedaboveforLMIPs.
Inthiscase,wearediscarding thehalf-spacefzjgT(z¡x)¸0gbecauseallsuch
pointsareinfeasible.
Nowassumethatthegivenpointxisfeasible, i.e.,F(x)>0.Inthiscase,the
vectorg=cde¯nesacuttingplanefortheEVPatthepointx.Here,wearediscarding
thehalf-spacefzjgT(z¡x)>0gbecauseallsuchpoints(whether feasibleornot)
haveanobjectivevaluelargerthanx,andhencecannotbeoptimal.
GEVPs: Weconsider theformulation
minimize ¸max(A(x);B(x))
subjecttoB(x)>0;C(x)>0
Here,A(x)=A0+Pm
i=1xiAi,B(x)=B0+Pm
i=1xiBiandC(x)=C0+Pm
i=1xiCi.
Nowsupposewearegivenapointx.Iftheconstrain tsareviolated, weusethemethod
describedforLMIPstogenerate acuttingplaneatx.
Supposenowthatxisfeasible. Pickanyu6=0suchthat
(¸max(A(x);B(x))B(x)¡A(x))u=0:
De¯negby
gi=¡uT(¸max(A(x);B(x))Bi¡Ai)u;i=1;:::;m:
Weclaimgde¯nesacuttingplaneforthisGEVPatthepointx.Toseethis,wenote
thatforanyz,
uT(¸max(A(x);B(x))B(z)¡A(z))u=¡gT(z¡x):
Hence,ifgT(z¡x)¸0we¯ndthat
¸max(A(z);B(z))¸¸max(A(x);B(x))
whichestablishes ourclaim.
CPs:Wenowconsider ourstandard CP(2.10).Whenthegivenpointxisinfeasible
wealreadyknowhowtogenerate acuttingplane.Soweassumethatxisfeasible. In
thiscase,acuttingplaneisgivenbythegradientoftheobjective
¡logdetA(x)=¡logdetÃ
A0+mX
i=1xiAi!
atx,i.e.,gi=¡TrAiA(x)¡1:Sincetheobjectivefunction isconvexwehaveforall
z,
logdetA(z)¡1¸logdetA(x)¡1+gT(z¡x):
Inparticular, gT(z¡x)>0implieslogdetA(z)¡1>logdetA(x)¡1,andhencez
cannotbeoptimal.
2.4Interior-PointMethods
Since1988,interior-pointmethodshavebeendevelopedforthestandard problems.
Inthissectionwedescribeasimpleinterior-pointmethodforsolvinganEVP,given
afeasiblestarting point.Thereferences describemoresophisticated interior-point
methodsfortheotherproblems (including theproblem ofcomputing afeasiblestarting
pointforanEVP).
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.4Interior-Point Methods 15
2.4.1AnalyticcenterofanLMI
Thenotionoftheanalytic centerofanLMIplaysanimportantroleininterior-point
methods,andisimportantinitsownright.Consider theLMI
F(x)=F0+mX
i=1xiFi>0;
whereFi=FT
i2Rn£n.Thefunction
Á(x)¢=(
logdetF(x)¡1F(x)>0
1 otherwise,(2.12)
is¯niteifandonlyifF(x)>0,andbecomesin¯niteasxapproachestheboundary
ofthefeasiblesetfxjF(x)>0g,i.e.,itisabarrierfunction forthefeasibleset.(We
havealreadyencounteredthisfunction inourstandard problem CP.)
Wesupposenowthatthefeasiblesetisnonempt yandbounded, whichimplies
thatthematricesF1;:::;Fmarelinearlyindependent(otherwise thefeasiblesetwill
containaline).ItcanbeshownthatÁisstrictlyconvexonthefeasibleset,soithas
auniqueminimizer, whichwedenotex?:
x?¢=argmin
xÁ(x): (2.13)
Werefertox?astheanalyticcenteroftheLMIF(x)>0.Equivalently,
x?=argmax
F(x)>0detF(x); (2.14)
thatis,F(x?)hasmaximumdeterminan tamongallpositive-de¯nite matrices ofthe
formF(x).
Remark :Theanalytic centerdependsontheLMI(i.e.,thedataF0;:::;Fm)
andnotitsfeasibleset:WecanhavetwoLMIswithdi®erentanalytic centersbut
identicalfeasiblesets.Theanalyticcenteris,however,invariantundercongruence
oftheLMI:F(x)>0andTTF(x)T>0havethesameanalytic centerprovided
Tisnonsingular.
Wenowturntotheproblem ofcomputing theanalytic centerofanLMI.(This
isaspecialformofourproblem CP.)Newton's method,withappropriate steplength
selection, canbeusedtoe±cientlycomputex?,startingfromafeasibleinitialpoint.
Weconsider thealgorithm:
x(k+1):=x(k)¡®(k)H(x(k))¡1g(x(k)); (2.15)
where®(k)isthedamping factorofthekthiteration, andg(x(k))andH(x(k))denote
thegradientandHessianofÁ,respectively,atx(k).In[NN94,x2.2],Nesterovand
Nemirovskiigiveasimplesteplengthruleappropriate forageneralclassofbarrier
functions (calledself-concordant),andinparticular, thefunctionÁ.Theirdamping
factoris:
®(k):=(
1 if±(x(k))·1=4;
1=(1+±(x(k)))otherwise,(2.16)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
16 Chapter 2Some StandardProblems InvolvingLMIs
where
±(x)¢=q
g(x)TH(x)¡1g(x)
iscalledtheNewtondecrementofÁatx.NesterovandNemirovskiishowthatthis
steplengthalwaysresultsinx(k+1)feasible, i.e.,F(x(k+1))>0,andconvergenceof
x(k)tox?.
Indeed,theygivesharpboundsonthenumberofstepsrequired tocomputex?
toagivenaccuracy usingNewton's methodwiththesteplength(2.16).Theyprove
thatÁ(x(k))¡Á(x?)·²whenever
k¸c1+c2loglog(1=²)+c3³
Á(x(0))¡Á(x?)´
; (2.17)
wherec1,c2,andc3arethreeabsolute constants,i.e.,speci¯cnumbers.The¯rst
andsecondtermsontheright-handsidedonotdependonanyproblem data,i.e.,
thematricesF0;:::;Fm,andthenumbersmandn.Thesecondtermgrowssoslowly
withrequired accuracy²thatforallpractical purposesitcanbelumpedtogether with
the¯rstandconsidered anabsolute constant.Thelasttermontheright-handside
of(2.17)dependsonhow\centered"theinitialpointis.
2.4.2Thepathofcenters
Nowconsider thestandard EVP:
minimize cTx
subjecttoF(x)>0
Let¸optdenoteitsoptimalvalue,soforeach¸>¸opttheLMI
F(x)>0;cTx<¸ (2.18)
isfeasible. WewillalsoassumethattheLMI(2.18)hasaboundedfeasibleset,and
therefore hasananalytic centerwhichwedenotex?(¸):
x?(¸)¢=argmin
xµ
logdetF(x)¡1+log1
¸¡cTx¶
:
Thecurvegivenbyx?(¸)for¸>¸optiscalledthepathofcentersfortheEVP.
Itcanbeshownthatitisanalytic andhasalimitas¸!¸opt,whichwedenotexopt.
Thepointxoptisoptimal(ormoreprecisely,thelimitofaminimizing sequence) since
for¸>¸opt,x?(¸)isfeasibleandsatis¯escTx?(¸)<¸.
Thepointx?(¸)ischaracterized by
@
@xi¯¯¯¯
x?(¸)µ
logdetF(x)¡1+log1
¸¡cTx¶
=¡TrF(x?(¸))¡1Fi+ci
¸¡cTx?(¸)=0;i=1;:::;m:(2.19)
2.4.3Methodofcenters
Themethodofcentersisasimpleinterior-pointalgorithm thatsolvesanEVP,givena
feasiblestartingpoint.Thealgorithm isinitialized with¸(0)andx(0),withF(x(0))>0
andcTx(0)<¸(0),andproceedsasfollows:
¸(k+1):=(1¡µ)cTx(k)+µ¸(k)
x(k+1):=x?(¸(k+1))
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.4Interior-Point Methods 17
whereµisaparameter with0<µ<1.
Theclassicmethodofcentersisobtained withµ=0.Inthiscase,however,x(k)
doesnot(quite)satisfythenewinequalit ycTx<¸(k+1).Withµ>0,however,the
currentiteratex(k)isfeasiblefortheinequalit ycTx<¸(k+1),F(x)>0,andtherefore
canbeusedastheinitialpointforcomputing thenextiteratex?(¸(k+1))viaNewton's
method.Sincecomputing ananalytic centerisitselfaspecialtypeofCP,wecanview
themethodofcentersasawayofsolvinganEVPbysolvingasequence ofCPs(which
isdoneusingNewton's method).
Wenowgiveasimpleproofofconvergence. Multiplying (2.19)by(x(k)
i¡xopt
i)
andsumming overiyields:
TrF(x(k))¡1³
F(x(k))¡F(xopt)´
=1
¸(k)¡cTx(k)cT(x(k)¡xopt):
SinceTrF(x(k))¡1F(xopt)¸0,weconclude that
n¸1
¸(k)¡cTx(k)(cTx(k)¡¸opt):
Replacing cTx(k)by(¸(k+1)¡µ¸(k))=(1¡µ)yields
(¸(k+1)¡¸opt)·n+µ
n+1(¸(k)¡¸opt);
whichprovesthat¸(k)approaches¸optwithatleastgeometric convergence. Notethat
wecanalsoexpresstheinequalit yaboveintheform
cTx(k)¡¸opt·n(¸(k)¡cTx(k));
whichshowsthatthestopping criterion
until³
¸(k)¡cTx(k)<²=n´
guaranteesthatonexit,theoptimalvaluehasbeenfoundwithin².
Wemakeafewcommentshere,andreferthereadertotheNotesandReferences for
furtherelaboration. First,thisvariationonthemethodofcentersisnotpolynomial-
time,butmoresophisticated versionsare.Second,andperhapsmoreimportant,we
notethattwosimplemodi¯cations ofthemethodofcentersasdescribedaboveyield
analgorithm thatisfairlye±cientinpractice. Themodi¯cations are:
²InsteadoftheNesterov{Nemiro vskiisteplength,asteplengthchosentomini-
mizeÁalongtheNewtonsearchdirection (i.e.,anexactline-searc h)willyield
fasterconvergencetotheanalytic center.
²Insteadofde¯ningx?(¸)tobetheanalyticcenteroftheLMIF(x)>0,cTx<¸,
wede¯neittobetheanalytic centeroftheLMIF(x)>0alongwithqcopies
ofcTx<¸,whereq>1isaninteger.Inotherwordsweuse
x?(¸)¢=argmin
xµ
logdetF(x)¡1+qlog1
¸¡cTx¶
:
Usingq>1,sayq¼nwhereF(x)2Rn£n,yieldsmuchfasterreduction of¸
periteration.
Amonginterior-pointmethodsforthestandard problems, themethodofcenters
isnotthemoste±cient.Themoste±cientalgorithms developedsofarappeartobe
primal-dual algorithms (andvariations) andtheprojectivemethodofNemirovskii;see
theNotesandReferences.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
18 Chapter 2Some StandardProblems InvolvingLMIs
2.4.4Interior-pointmethodsandproblemstructure
Animportantfeatureofinterior-pointmethodsisthatproblem structure canbeex-
ploitedtoincrease e±ciency .Theideaisveryroughlyasfollows.Ininterior-point
methodsmostofthecomputational e®ortisdevotedtocomputing theNewtondirec-
tionofabarrierorsimilarfunction. ItturnsoutthatthisNewtondirection canbe
expressed asthesolution ofaweightedleast-squares problem ofthesamesizeasthe
originalproblem. Usingconjugate-gradien tandotherrelatedmethodstosolvethese
least-squares systemsgivestwoadvantages.First,byexploiting problem structure in
theconjugate-gradien titerations, thecomputational e®ortrequired tosolvetheleast-
squaresproblems ismuchsmallerthanbystandard \direct" methodssuchasQRor
Cholesky factorization. Second,itispossibletoterminate theconjugate-gradien titer-
ationsbeforeconvergence, andstillobtainanapproximation oftheNewtondirection
suitableforinterior-pointmethods.SeetheNotesandReferences formorediscussion.
Anexample willdemonstrate thee±ciencies thatcanbeobtained usingthetech-
niquessketchedabove.Theproblem isanEVPthatwewillencounterinx6.2.1.
WearegivenmatricesA1;:::;AL2Rn£n,symmetric matricesD1;:::;DL;E2
Rn£n.Weconsider theEVP
minimize TrEP
subjecttoAT
iP+PAi+Di<0;i=1;:::;L(2.20)
Inthisproblem thevariableisthematrixP,sothedimension oftheoptimization
variableism=n(n+1)=2.WhentheLyapunovinequalities arecombinedintoone
largeLMIF(x)>0,we¯ndthatF(x)2RN£NwithN=Ln.ThisLMIhasmuch
structure: Itisblock-diagonal witheachblockaLyapunovinequalit y.
VandenbergheandBoydhavedevelopeda(primal-dual) interior-pointmethod
thatsolves(2.20),exploiting theproblem structure. Theyprovetheworst-case esti-
mateofO(m2:75L1:5)arithmetic operations tosolvetheproblem toagivenaccuracy.
Incomparison, theellipsoid methodsolves(2.20)toagivenaccuracy inO(m3:5L)
arithmetic operations (moreover,theconstanthiddenintheO(¢)notation ismuch
largerfortheellipsoid algorithm).
Numerical experimentsonfamilies ofproblems withrandomly generated data
andfamiliesofproblems arisinginsystemandcontroltheoryshowthattheactual
performance oftheinterior-pointmethodismuchbetterthantheworst-case estimate:
O(m®L¯)arithmetic operations, with®¼2:1and¯¼1:2.Thiscomplexit yestimate
isremarkablysmall.Forexample, thecostofsolvingLLyapunovequations ofthe
samesizeisO(m1:5L).Therefore, therelativecostofsolvingLcoupled Lyapunov
inequalities, compared tothecostofsolvingLindependentLyapunovinequalities
isO(m0:6L0:2).Thisexample illustrates oneofourpoints:Thecomputational cost
ofsolvingoneofthestandard problems (whichhasno\analytic solution") canbe
comparable tothecomputational costofevaluating thesolution ofasimilarproblem
thathasan\analytic solution".
2.5StrictandNonstrictLMIs
Wehavesofarassumed thattheoptimization problems LMIP,EVP,GEVP,andCP
involvestrictLMIs.Wewillalsoencountertheseproblems withnonstrict LMIs,or
moregenerally ,withamixture ofstrictandnonstrict LMIs.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.5Strict andNonstrict LMIs 19
Asanexample consider thenonstrict versionoftheEVP(2.9),i.e.
minimize cTx
subjecttoF(x)¸0
Intuitionsuggests thatwecouldsimplysolvethestrictEVP(by,say,aninterior-point
method)toobtainthesolution ofthenonstrict EVP.Thisiscorrectinmostbutnot
allcases.
IfthestrictLMIF(x)>0isfeasible, thenwehave
fx2RmjF(x)¸0g=fx2RmjF(x)>0g; (2.21)
i.e.,thefeasiblesetofthenonstrict LMIistheclosureofthefeasiblesetofthestrict
LMI.Itfollowsthat
inf©
cTxjF(x)¸0ª
=inf©
cTxjF(x)>0ª
: (2.22)
Sointhiscase,wecansolvethestrictEVPtoobtainasolution ofthenonstrict EVP.
Thisistruefortheproblems GEVPandCPaswell.
WewillsaythattheLMIF(x)¸0isstrictlyfeasibleifitsstrictversionisfeasible,
i.e.,ifthereissomex02RmsuchthatF(x0)>0.Wehavejustseenthatwhenan
LMIisstrictlyfeasible, wecanreplacenonstrict inequalit ywithstrictinequalit yinthe
problems EVP,GEVP,andCPinordertosolvethem.Inthelanguage ofoptimization
theory,therequiremen tofstrictfeasibilityisa(verystrong)constrain tquali¯cation.
WhenanLMIisfeasiblebutnotstrictlyfeasible, (2.21)neednothold,andthe
EVPswiththestrictandnonstrict LMIscanbeverydi®erent.Asasimpleexample,
considerF(x)=diag(x;¡x)withx2R.Theright-handsideof(2.22)is+1since
thestrictLMIF(x)>0isinfeasible. Theleft-hand side,however,isalways0,since
theLMIF(x)¸0hasthesinglefeasiblepointx=0.Thisexample showsoneofthe
twopathologies thatcanoccur:Thenonstrict inequalit ycontainsanimplicitequality
constrain t(incontrastwithanexplicitequalityconstrain tasinx2.1.2).
Theotherpathology isdemonstrated bytheexampleF(x)=diag(x;0)with
x2Randc=¡1.Onceagain,thestrictLMIisinfeasible sotheright-handside
of(2.22)is+1.Thefeasiblesetforthenonstrict LMIistheinterval[0;1)sothe
right-handsideis¡1.Theproblem hereisthatF(x)isalwayssingular. Ofcourse,
thenonstrict LMIF(x)¸0isequivalent(inthesenseofde¯ningequalfeasiblesets)to
the\reduced" LMI~F(x)=x¸0.NotethatthisreducedLMIsatis¯estheconstrain t
quali¯cation, i.e.,isstrictlyfeasible.
2.5.1Reduction toastrictlyfeasibleLMI
Itturnsoutthatanyfeasiblenonstrict LMIcanbereducedtoanequivalentLMIthat
isstrictlyfeasible, byeliminating implicitequalityconstrain tsandthenreducing the
resulting LMIbyremovinganyconstantnullspace.
Theprecisestatemen tis:LetF0;:::;Fm2Rn£nbesymmetric. Thenthere
isamatrixA2Rm£pwithp·m,avectorb2Rm,andsymmetric matrices
~F0;:::;~Fp2Rq£qwithq·nsuchthat:
F(x)¸0()x=Az+bforsomez2Rpwith
~F(z)¢=~F0+pX
i=1zi~Fi¸0
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
20 Chapter 2Some StandardProblems InvolvingLMIs
wheretheLMI~F(z)¸0iseitherinfeasible orstrictlyfeasible. SeetheNotesand
References foraproof.
ThematrixAandvectorbdescribetheimplicitequalityconstrain tsfortheLMI
F(x)¸0.Similarly ,theLMI~F(z)¸0canbeinterpreted astheoriginalLMIwith
itsconstantnullspace removed(seetheNotesandReferences). Inmostoftheprob-
lemsencounteredinthisbook,therearenoimplicitequalityconstrain tsornontrivial
common nullspace forF,sowecanjusttakeA=I,b=0,and~F=F.
Usingthisreduction wecan,atleastinprinciple, alwaysdealwithstrictlyfeasible
LMIs.Forexample wehave
inf©
cTx¯¯F(x)¸0ª
=infn
cT(Az+b)¯¯¯~F(z)¸0o
=infn
cT(Az+b)¯¯¯~F(z)>0o
sincetheLMI~F(z)¸0iseitherinfeasible orstrictlyfeasible.
2.5.2Example: Lyapunovinequality
Toillustrate theprevious ideasweconsider thesimplest LMIarisingincontroltheory,
theLyapunovinequalit y:
ATP+PA·0;P>0; (2.23)
whereA2Rk£kisgiven,andthesymmetric matrixPisthevariable.Notethatthis
LMIcontainsastrictinequalit yaswellasanonstrict inequalit y.
WeknowfromsystemtheorythattheLMI(2.23)isfeasibleifandonlyifall
trajectories of_x=Axarebounded, orequivalently,iftheeigenvaluesofAhave
nonpositiverealpart,andthosewithzerorealpartarenondefectiv e,i.e.,correspond
toJordanblocksofsizeone.WealsoknowfromsystemtheorythattheLMI(2.23)is
strictlyfeasibleifandonlyifalltrajectories of_x=Axconvergetozero,orequivalently,
alltheeigenvaluesofAhavenegativerealpart.
Consider theinteresting casewheretheLMI(2.23)isfeasiblebutnotstrictly
feasible. Fromtheremarks above,weseethatbyachangeofcoordinates wecanput
Aintheform
~A¢=T¡1AT
=diagÃ"
0!1Ik1
¡!1Ik10#
;:::;"
0!rIkr
¡!rIkr0#
;0kr+1;Astab!
where0<!1<¢¢¢<!r,0kr+1denotesthezeromatrixinRkr+1£kr+1,andallthe
eigenvaluesofthematrixAstab2Rs£shavenegativerealpart.Roughly speaking,
wehaveseparated outthestablepartofA,thepartcorrespondingtoeachimaginary
axiseigenvalue,andthepartassociatedwitheigenvaluezero.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.5Strict andNonstrict LMIs 21
Usingstandard methodsofLyapunovtheoryitcanbeshownthat
©
P¯¯ATP+PA·0;P>0ª
=
8
>>>>>>>>>>>><
>>>>>>>>>>>>:T¡T~PT¡1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯~P=diagÃ"
P1Q1
QT
1P1#
;:::;"
PrQr
QT
rPr#
;Pr+1;Pstab!
Pi2Rki£ki;i=1;:::;r+1;QT
i=¡Qi;i=1;:::;r
"
PiQi
QT
iPi#
>0;i=1;:::;r;Pr+1>0;Pstab>0
AT
stabPstab+PstabAstab·09
>>>>>>>>>>>>=
>>>>>>>>>>>>;:
Fromthischaracterization wecan¯ndareduced LMIthatisstrictlyfeasible. We
cantakethesymmetric matricesP1;:::;Pr+1;Pstabandtheskew-symmetric matrices
Q1;:::;Qrasthe\free"variablez;thea±nemapping fromzintoxsimplymaps
thesematrices into
P=T¡TdiagÃ"
P1Q1
QT
1P1#
;:::;"
PrQr
QT
rPr#
;Pr+1;Pstab!
T¡1:(2.24)
Putanotherway,theequalityconstrain tsimplicitintheLMI(2.23)arethatPmust
havethisspecialstructure.
Nowwesubstitute Pintheform(2.24)backintotheoriginalLMI(2.23).We
¯ndthat
ATP+PA=TTdiag¡
0;AT
stabPstab+PstabAstab¢
T
wherethezeromatrixhassize2k1+¢¢¢+2kr+kr+1.Weremovetheconstantnullspace
toobtainthereduced version,i.e.,AT
stabPstab+PstabAstab·0.Thus,thereduced
LMIcorrespondingto(2.23)is
"
PiQi
QT
iPi#
>0;i=1;:::;r;Pr+1>0;
Pstab>0;AT
stabPstab+PstabAstab·0:(2.25)
ThisreducedLMIisstrictlyfeasible, sincewecantakeP1;:::;Pr+1asidentitymatri-
ces,Q1;:::;Qraszeromatrices, andPstabasthesolution oftheLyapunovequation
AT
stabPstab+PstabAstab+I=0.
Insummary ,theoriginalLMI(2.23)hasonesymmetric matrixofsizekasvariable
(i.e.,thedimension oftheoriginal variablexism=k(k+1)=2).Thereduced
LMI(2.25)hasasvariablethesymmetric matricesP1;:::;Pr+1;Pstabandtheskew-
symmetric matricesQ1;:::;Qr,sothedimension ofthevariablezinthereducedLMI
is
p=rX
i=1k2
i+kr+1(kr+1+1)
2+s(s+1)
2<m:
TheoriginalLMI(2.23)involvesamatrixinequalit yofsizen=2k(i.e.,twoinequali-
tiesofsizek).ThereducedLMI(2.25)involvesamatrixinequalit yofsizek+s<n.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
22 Chapter 2Some StandardProblems InvolvingLMIs
2.6Miscellaneous ResultsonMatrixInequalities
2.6.1Elimination ofsemide¯nite terms
IntheLMIP
¯ndxsuchthatF(x)>0; (2.26)
wecaneliminate anyvariablesforwhichthecorrespondingcoe±cientinFissemidef-
inite.Supposeforexample thatFm¸0andhasrankr<n.Roughly speaking,by
takingxmextremely large,wecanensurethatFispositive-de¯nite ontherangeof
Fm,sotheLMIPreducestomakingFpositive-de¯nite onthenullspace ofFm.
Moreprecisely,letFm=UUTwithUfull-rank, andlet~Ubeanorthogonal
complemen tofU,e.g.,UT~U=0and[U~U]isofmaximumrank(whichinthiscase
justmeansthat[U~U]isnonsingular). Ofcourse,wehave
F(x)>0()h
~UUiT
F(x)h
~UUi
>0:
Since
h
~UUiT
F(x)h
~UUi
="~UT~F(~x)~U ~UT~F(~x)U
UT~F(~x)~UUT~F(~x)U+xm(UTU)2#
;
where~x=[x1¢¢¢xm¡1]Tand~F(~x)¢=F0+x1F1+¢¢¢+xm¡1Fm¡1,weseethat
F(x)>0ifandonlyif~UT~F~U(~x)>0,andxmislargeenoughthat
UT~F(~x)U+xm(UTU)2>UT~F(~x)~U³
~UT~F(~x)~U´¡1~UT~F(~x)U:
Therefore, theLMIP(2.26)isequivalentto
¯ndx1;:::;xm¡1suchthat~UT~F(x1;:::;xm¡1)~U>0:
2.6.2Elimination ofmatrixvariables
Whenamatrixinequalit yhassomevariablesthatappearinacertainform,wecan
deriveanequivalentinequalit ywithoutthosevariables. Consider
G(z)+U(z)XV(z)T+V(z)XTU(z)T>0; (2.27)
wherethevectorzandthematrixXare(independent)variables, andG(z)2Rn£n,
U(z)andV(z)donotdependonX.Matrixinequalities oftheform(2.27)arisein
thecontrollersynthesisproblems describedinChapter 7.
Supposethatforeveryz,~U(z)and~V(z)areorthogonal complemen tsofU(z)
andV(z)respectively.Then(2.27)holdsforsomeXandz=z0ifandonlyifthe
inequalities
~U(z)TG(z)~U(z)>0;~V(z)TG(z)~V(z)>0 (2.28)
holdwithz=z0.Inotherwords,feasibilityofthematrixinequalit y(2.27)with
variablesXandzisequivalenttothefeasibilityof(2.28)withvariablez;wehave
eliminated thematrixvariableXfrom(2.27)toform(2.28).NotethatifU(z)or
V(z)hasranknforallz,thenthe¯rstorsecondinequalit yin(2.28)disappears.We
provethislemmaintheNotesandReferences.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.6Miscellaneous Resul tsonMatrixInequalities 23
Wecanexpress(2.28)inanotherformusingFinsler's lemma(seetheNotesand
References):
G(z)¡¾U(z)U(z)T>0;G(z)¡¾V(z)V(z)T>0
forsome¾2R.
Asanexample, wewillencounterinx7.2.1theLMIPwithLMI
Q>0;AQ+QAT+BY+YTBT<0; (2.29)
whereQandYarethevariables. ThisLMIPisequivalenttotheLMIPwithLMI
Q>0;AQ+QAT<¾BBT;
wherethevariablesareQand¾2R.ItisalsoequivalenttotheLMIP
Q>0;~BT¡
AQ+QAT¢~B<0;
withvariableQ,where~Bisanymatrixofmaximumranksuchthat~BTB=0.Thus
wehaveeliminated thevariableYfrom(2.29)andreducedthesizeofthematrices in
theLMI.
2.6.3TheS-procedure
Wewilloftenencountertheconstrain tthatsomequadratic function (orquadratic
form)benegativewheneversomeotherquadratic functions (orquadratic forms)are
allnegative.Insomecases,thisconstrain tcanbeexpressed asanLMIinthedata
de¯ning thequadratic functions orforms;inothercases,wecanformanLMIthatis
aconservativebutoftenusefulapproximation oftheconstrain t.
TheS-procedureforquadraticfunctionsandnonstrictinequalities
LetF0;:::;Fpbequadratic functions ofthevariable³2Rn:
Fi(³)¢=³TTi³+2uT
i³+vi;i=0;:::;p;
whereTi=TT
i.Weconsider thefollowingcondition onF0;:::;Fp:
F0(³)¸0forall³suchthatFi(³)¸0;i=1;:::;p: (2.30)
Obviouslyif
thereexist¿1¸0;:::;¿p¸0suchthat
forall³;F0(³)¡pX
i=1¿iFi(³)¸0;(2.31)
then(2.30)holds.Itisanontrivialfactthatwhenp=1,theconverseholds,provided
thatthereissome³0suchthatF1(³0)>0.
Remark :IfthefunctionsFiarea±ne,then(2.31)and(2.30)areequivalent;
thisistheFarkaslemma.
Notethat(2.31)canbewrittenas
"
T0u0
uT
0v0#
¡pX
i=1¿i"
Tiui
uT
ivi#
¸0:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
24 Chapter 2Some StandardProblems InvolvingLMIs
TheS-procedureforquadraticformsandstrictinequalities
WewilluseanothervariationoftheS-procedure,whichinvolvesquadratic formsand
strictinequalities. LetT0;:::;Tp2Rn£nbesymmetric matrices. Weconsider the
followingcondition onT0;:::;Tp:
³TT0³>0forall³6=0suchthat³TTi³¸0;i=1;:::;p: (2.32)
Itisobviousthatif
thereexists¿1¸0;:::;¿p¸0suchthatT0¡pX
i=1¿iTi>0; (2.33)
then(2.32)holds.Itisanontrivialfactthatwhenp=1,theconverseholds,provided
thatthereissome³0suchthat³T
0T1³0>0.Notethat(2.33)isanLMIinthevariables
T0and¿1;:::;¿p.
Remark :The¯rstversionoftheS-proceduredealswithnonstrict inequalities
andquadratic functions thatmayincludeconstantandlinearterms.Thesecond
versiondealswithstrictinequalities andquadratic formsonly,i.e.,quadratic
functions withoutconstantorlinearterms.
Remark :SupposethatT0,u0andv0dependa±nelyonsomeparameterº.Then
thecondition (2.30)isconvexinº.Thisdoesnot,however,meanthattheproblem
ofcheckingwhether thereexistsºsuchthat(2.30)holdshaslowcomplexit y.On
theotherhand,checkingwhether (2.31)holdsforsomeºisanLMIPinthe
variablesºand¿1;:::;¿p.Therefore, thisproblem haslowcomplexit y.Seethe
NotesandReferences forfurtherdiscussion.
S-procedureexample
InChapter 5wewillencounterthefollowingconstrain tonthevariableP:
forall»6=0and¼satisfying¼T¼·»TCTC»,"
»
¼#T"
ATP+PAPB
BTP0#"
»
¼#
<0:(2.34)
Applying thesecondversionoftheS-procedure,(2.34)isequivalent totheexistence
of¿¸0suchthat
2
4ATP+PA+¿CTCPB
BTP¡¿I3
5<0:
Thustheproblem of¯ndingP>0suchthat(2.34)holdscanbeexpressed asan
LMIP(inPandthescalarvariable¿).
2.7SomeLMIProblemswithAnalyticSolutions
Thereareanalytic solutions toseveralLMIproblems ofspecialform,oftenwithim-
portantsystemandcontroltheoretic interpretations. Webrie°ydescribesomeofthese
resultsinthissection. Atthesametimeweintroduceandde¯neseveralimportant
termsfromsystemandcontroltheory.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
2.7Some LMIProblems withAnalyticSolutions 25
2.7.1Lyapunov'sinequality
WehavealreadymentionedtheLMIPassociatedwithLyapunov'sinequalit y,i.e.
P>0;ATP+PA<0
wherePisvariableandA2Rn£nisgiven.LyapunovshowedthatthisLMIisfeasible
ifandonlythematrixAisstable,i.e.,alltrajectories of_x=Axconvergetozeroas
t!1,orequivalently,alleigenvaluesofAmusthavenegativerealpart.Tosolvethis
LMIP,wepickanyQ>0andsolvetheLyapunovequationATP+PA=¡Q,which
isnothingbutasetofn(n+1)=2linearequations forthen(n+1)=2scalarvariables
inP.Thissetoflinearequations willbesolvableandresultinP>0ifandonlyif
theLMIisfeasible. Infactthisprocedurenotonly¯ndsasolution whentheLMIis
feasible; itparametrizes allsolutions asQvariesoverthepositive-de¯nite cone.
2.7.2Thepositive-reallemma
Another importantexample isgivenbythepositive-real(PR)lemma,whichyields
a\frequency-domain" interpretation foracertainLMIP,andundersomeadditional
assumptions, anumerical solution procedureviaRiccatiequations aswell.Wegivea
simpli¯ed discussion here,andreferthereadertotheReferences formorecomplete
statemen ts.
TheLMIconsidered is:
P>0;"
ATP+PAPB¡CT
BTP¡C¡DT¡D#
·0; (2.35)
whereA2Rn£n,B2Rn£p,C2Rp£n,andD2Rp£paregiven,andthematrix
P=PT2Rn£nisthevariable.(WewillencounteravariationonthisLMIinx6.3.3.)
NotethatifD+DT>0,theLMI(2.35)isequivalenttothequadratic matrix
inequalit y
ATP+PA+(PB¡CT)(D+DT)¡1(PB¡CT)T·0: (2.36)
WeassumeforsimplicitythatAisstableandthesystem(A;B;C)isminimal.
Thelinkwithsystemandcontroltheoryisgivenbythefollowingresult.The
LMI(2.35)isfeasibleifandonlyifthelinearsystem
_x=Ax+Bu;y=Cx+Du (2.37)
ispassive,i.e.,
ZT
0u(t)Ty(t)dt¸0
forallsolutions of(2.37)withx(0)=0.
Passivitycanalsobeexpressed intermsofthetransfermatrixofthelinearsys-
tem(2.37),de¯nedas
H(s)¢=C(sI¡A)¡1B+D
fors2C.PassivityisequivalenttothetransfermatrixHbeingpositive-real,which
meansthat
H(s)+H(s)¤¸0forallRes>0:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
26 Chapter 2Some StandardProblems InvolvingLMIs
Whenp=1thiscondition canbecheckedbyvariousgraphical means,e.g.,plotting
thecurvegivenbytherealandimaginary partsofH(i!)for!2R(calledtheNyquist
plotofthelinearsystem). Thuswehaveagraphical, frequency-domain condition for
feasibilityoftheLMI(2.35).Thisapproachwasusedinmuchoftheworkinthe1960s
and1970sdescribedinx1.2.
Withafewfurthertechnicalassumptions, includingD+DT>0,theLMI(2.35)
canbesolvedbyamethodbasedonRiccatiequations andHamiltonian matrices.
Withtheseassumptions theLMI(2.35)isfeasibleifandonlyifthereexistsareal
matrixP=PTsatisfying theARE
ATP+PA+(PB¡CT)(D+DT)¡1(PB¡CT)T=0; (2.38)
whichisjustthequadratic matrixinequalit y(2.36)withequalitysubstituted forin-
equality.NotethatP>0.
TosolvetheARE(2.38)we¯rstformtheassociatedHamiltonian matrix
M="
A¡B(D+DT)¡1CB(D+DT)¡1BT
¡CT(D+DT)¡1C¡AT+CT(D+DT)¡1BT#
:
Thenthesystem(2.37)ispassive,orequivalently,theLMI(2.35)isfeasible, ifand
onlyifMhasnopureimaginary eigenvalues.ThisfactcanbeusedtoformaRouth{
Hurwitz (Sturm) typetestforpassivityasasetofpolynomial inequalities inthedata
A,B,C,andD.
WhenMhasnopureimaginary eigenvalueswecanconstruct asolutionPareas
follows.PickV2R2n£nsothatitsrangeisabasisforthestableeigenspace of
M,e.g.,V=[v1¢¢¢vn]wherev1;:::;vnareasetofindependenteigenvectorsofM
associatedwithitsneigenvalueswithnegativerealpart.PartitionVas
V="
V1
V2#
;
whereV1andV2aresquare;thensetPare=V2V¡1
1.ThesolutionParethusobtained
istheminimal elementamongthesetofsolutions of(2.38):ifP=PTsatis¯es(2.38),
thenP¸Pare.Muchmorediscussion ofthismethod,including theprecisetechnical
conditions, canbefoundintheReferences.
2.7.3Thebounded-real lemma
Thesameresultsappearinanotherimportantform,thebounded-reallemma. Here
weconsider theLMI
P>0;"
ATP+PA+CTCPB+CTD
BTP+DTCDTD¡I#
·0: (2.39)
whereA2Rn£n,B2Rn£p,C2Rp£n,andD2Rp£paregiven,andthematrix
P=PT2Rn£nisthevariable. ForsimplicityweassumethatAisstableand
(A;B;C)isminimal.
ThisLMIisfeasibleifandonlythelinearsystem(2.37)isnonexpansive,i.e.,
ZT
0y(t)Ty(t)dt·ZT
0u(t)Tu(t)dt
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 27
forallsolutions of(2.37)withx(0)=0,Thiscondition canalsobeexpressed in
termsofthetransfermatrixH.Nonexpansivit yisequivalenttothetransfermatrix
Hsatisfying thebounded-realcondition, i.e.,
H(s)¤H(s)·IforallRes>0:
Thisissometimes expressed askHk1·1where
kHk1¢=supfkH(s)kjRes>0g
iscalledtheH1normofthetransfermatrixH.
Thiscondition iseasilycheckedgraphically ,e.g.,byplottingkH(i!)kversus!2R
(calledasingularvalueplotforp>1andaBodemagnitude plotforp=1).
OnceagainwecanrelatetheLMI(2.39)toanARE.Withsomeappropriate tech-
nicalconditions (seetheNotesandReferences) includingDTD<I,theLMI(2.39)
isfeasibleifandonlytheARE
ATP+PA+CTC+(PB+CTD)(I¡DTD)¡1(PB+CTD)T=0(2.40)
hasarealsolutionP=PT.Onceagainthisisthequadratic matrixinequalit y
associatedwiththeLMI(2.39),withequalityinsteadoftheinequalit y.
Wecansolvethisequation byformingtheHamiltonian matrix
M="
A+B(I¡DTD)¡1DTCB(I¡DTD)¡1BT
¡CT(I¡DDT)¡1C¡AT¡CTD(I¡DTD)¡1BT#
:
Thenthesystem(2.37)isnonexpansiv e,orequivalently,theLMI(2.39)isfeasible,
ifandonlyifMhasnoimaginary eigenvalues.Inthiscasewecan¯ndtheminimal
solution to(2.40)asdescribedinx2.7.2.
2.7.4Others
SeveralotherLMIproblems haveanalytic solutions, e.g.,theonesthatariseinthe
synthesisofstate-feedbac kforlinearsystems, orestimator gainsforobservers.Asa
simpleexample consider theLMI
P>0;AP+PAT<BBT
whereA2Rn£nandB2Rn£paregivenandPisthevariable.ThisLMIisfeasible
ifandonlyifthepair(A;B)isstabilizable, i.e.,thereexistsamatrixK2Rp£nsuch
thatA+BKisstable.Therearesimplemethodsfordetermining whether thisisthe
case,andwhenitis,ofconstructing anappropriate P.Wewilldescribethisandother
resultsinChapter 7.
Severalstandard resultsfromlinearalgebracanalsobeinterpreted asanalytic
solutions tocertainLMIproblems ofspecialform.Forexample, itiswell-known
thatamongallsymmetric positive-de¯nite matrices withgivendiagonal elements,
thediagonal matrixhasmaximumdeterminan t.Thisgivesananalytic solution toa
special(andfairlytrivial)CP(seex3.5).
Notes andReferences
LMIs
Theterm\LinearMatrixInequalit y"wascoinedbyJ.C.Willems andiswidelyusednow.
Aswementionedinx1.2,thetermwasusedinseveralpapersfromthe1970storefertothe
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
28 Chapter 2Some StandardProblems InvolvingLMIs
speci¯cLMI(1.3).Thetermisalsoconsisten twiththetitleofanearlypaperbyBellman
andKyFan:OnSystems ofLinearInequalities inHermitian MatrixVariables[BF63](see
below).
Amoreaccurate termmightbe\A±neMatrixInequalit y",sincethematrixisana±ne
andnotlinearfunction ofthevariable.Wethinkoftheterm\LinearMatrixInequalit y"as
analogous totheterm\LinearInequalities" usedtodescribeaT
ix·bi,i=1;:::;p.
Whenm=1,anLMIisnothingbutamatrixpencil.Thetheoryofmatrixpencilsisveryold
andquitecomplete, andthereisanextensiveliterature (see,e.g.,GolubandVanLoan[GL89]
andthereferences therein). Forthecaseofm=1allofourstandard problems arereadily
solved,e.g.,bysimultaneous diagonalization ofF0andF1byanappropriate congruence.
Thecasem=1doesnotariseonitsowninanyinteresting problem fromsystemorcontrol
theory,butitdoesariseinthe\line-searc h(sub)problem", i.e.,whenwerestrictoneofour
standard problems toaspeci¯cline.Inthiscontextthevariablerepresentsthe\steplength"
tobetakeninaniteration ofanalgorithm, anditusefultoknowthatsuchproblems canbe
solvedextremely e±ciently.
Schurcomplementsfornonstrictinequalities
TheSchurcomplemen tresultofsectionx2.1canbegeneralized tononstrict inequalities as
follows.SupposeQandRaresymmetric. Thecondition
"
QS
STR#
¸0 (2.41)
isequivalentto
R¸0;Q¡SRyST¸0;S(I¡RRy)=0;
whereRydenotestheMoore{PenroseinverseofR.
Toseethis,letUbeanorthogonal matrixthatdiagonalizes R,sothat
UTRU="
§0
00#
;
where§>0anddiagonal. Inequalit y(2.41)holdsifandonlyif
"
I0
0UT#"
QS
STR#"
I0
0U#
=2
64QS1S2
ST
1§0
ST
2003
75¸0;
where[S1S2]=SU,withappropriate partitioning. WemustthenhaveS2=0,whichholds
ifandonlyifS(I¡RRy)=0,and
"
QS1
ST
1§#
¸0;
whichholdsifandonlyifQ¡SRyST¸0.
FormulatingconvexproblemsintermsofLMIs
TheideathatLMIscanbeusedtorepresentawidevarietyofconvexconstrain tscanbe
foundinNesterovandNemirovskii'sreport[NN90b](whichisreallyapreliminary version
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 29
oftheirbook[NN94])andsoftwaremanual[NN90a].Theyformalize theideaofa\positive-
de¯niterepresentable"function; seex5.3,x5.4,andx6.4oftheirbook[NN94].Theidea
isalsodiscussed inthearticle[Ali92b]andthesis[Ali91]ofAlizadeh. In[BE93],Boyd
andElGhaouigivealistofquasiconvexfunctions thatcanberepresentedasgeneralized
eigenvaluesofmatrices thatdependa±nelyonthevariable.
InfeasibilitycriterionforLMIs
TheLMIF(x)>0isinfeasible meansthea±nesetfF(x)jx2Rmgdoesnotintersectthe
positive-de¯nite cone.Fromconvexanalysis, thisisequivalenttotheexistence ofalinear
functionalÃthatispositiveonthepositive-de¯nite coneandnonpositiveonthea±neset
ofmatrices. Thelinearfunctionals thatarepositiveonthepositive-de¯nite coneareofthe
formÃ(F)=TrGF,whereG¸0andG6=0.FromthefactthatÃisnonpositiveonthe
a±nesetfF(x)jx2Rmg,wecanconclude thatTrGFi=0,i=1;:::;mandTrGF0·0.
Theseareprecisely theconditions forinfeasibilit yofanLMIthatwementionedinx2.2.1.
ForthespecialcaseofmultipleLyapunovinequalities, theseconditions aregivenBellman
andKyFan[BF63]andKamenetskii andPyatnitskii [KP87a,KP87b].
Itisstraightforwardtoderiveoptimalit ycriteriafortheotherproblems, usingconvexanal-
ysis.Somegeneralreferences forconvexanalysis arethebooks[Roc70,Roc82]andsurvey
article[Roc93]byRockafellar.Therecenttext[HUL93]givesagoodoverviewofconvex
analysis; LMIsareusedasexamples inseveralplaces.
Complexityofconvexoptimization
Theimportantroleofconvexityinoptimization isfairlywidelyknown,butperhapsnot
wellenoughappreciated, atleastoutsidetheformerSovietUnion.Inastandard (Western)
treatmen tofoptimization, ourstandard problems LMIP,EVP,GEVP,andCPwouldbe
considered verydi±cult sincetheyarenondi®eren tiableandnonlinear. Theirconvexity
properties,however,makethemtractable, bothintheoryandinpractice.
In[Roc93,p194],Rockafellarmakestheimportantpoint:
Onedistinguishing ideawhichdominates manyissuesinoptimization theoryis
convexity...Animportantreasonisthefactthatwhenaconvexfunction is
minimized overaconvexseteverylocallyoptimalsolution isglobal.Also,¯rst-
ordernecessary conditions turnouttobesu±cient.Avarietyofotherproperties
conduciv etocomputation andinterpretation ofsolutions rideonconvexityas
well.Infactthegreatwatershed inoptimization isn'tbetweenlinearityand
nonlinearit y,butconvexityandnonconvexity.
Detailed discussion ofthe(low)complexit yofconvexoptimization problems canbefound
inthebooksbyGrÄotschel,Lov¶asz,Schrijver[GLS88],NemirovskiiandYudin[NY83],and
Vavasis[Vav91].
Complete anddetailedworst-case complexit yanalyses ofseveralalgorithms forourstandard
problems canbefoundinChapters 3,4,and6ofNesterovandNemirovskii[NN94].
Ellipsoidalgorithm
Theellipsoid algorithm wasdevelopedbyShor,Nemirovskii,andYudininthe1970s[Sho85,
NY83].ItwasusedbyKhachiyanin1979toprovethatlinearprograms canbesolvedin
polynomial-time [Kha79,GL81,GLS88].Discussion oftheellipsoid method,aswellas
severalextensions andvariations, canbefoundin[BGT81]and[BB91,ch14].Adetailed
historyofitsdevelopment,including EnglishandRussianreferences, appearsinchapter3of
AkgÄul[Akg84].
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
30 Chapter 2Some StandardProblems InvolvingLMIs
OptimizationproblemsinvolvingLMIs
OneoftheearliestpapersonLMIsisalso,inouropinion, oneofthebest:Onsystemsof
linearinequalities inHermitian matrixvariables ,byBellman andKyFan[BF63].Inthis
paperwe¯ndmanyresults,e.g.,adualitytheoryforthemultipleLyapunovinequalit yEVP
andatheorem ofthealternativ eforthemultipleLyapunovinequalit yLMIP.Thepaper
concentratesontheassociatedmathematics: thereisnodiscussion ofhowtosolvesuch
problems, oranypotentialapplications, although theydocommentthatsuchinequalit y
systems ariseinLyapunovtheory.GivenBellman's legendary knowledgeofsystemand
controltheory,andespeciallyLyapunovtheory,itistempting toconjecture thatBellman
wasawareofthepotentialapplications. Butsofarwehavefoundnoevidence forthis.
Relevantworkincludes Cullumetal.[CDW75],CravenandMond[CM81],Polakand
Wardi[PW82],Fletcher[Fle85],Shapiro [Sha85],Friedland etal.[FNO87],Gohand
Teo[GT88],Panier[Pan89],Allwright[All89],OvertonandWomersley [Ove88,Ove92,
OW93,OW92],Ringertz [Rin91],FanandNekooie[FN92,Fan93]andHiriart{Urrut yand
Ye[HUY92].
LMIPsaresolvedusingvariousalgorithms forconvexoptimization inBoydandYang[BY89],
Pyatnitskii andSkorodinskii[PS83],Kamenetskii andPyatnitskii [KP87a,KP87b].
Asurveyofmethodsforsolvingproblems involvingLMIsusedbyresearchersincontroltheory
canbefoundinthepaperbyBeck[Bec91].Thesoftwarepackages[BDG+91]and[CS92a]
useconvexprogramming tosolvemanyrobustcontrolproblems. Boyd[Boy94]outlineshow
interior-pointconvexoptimization algorithms canbeusedtobuildrobustcontrolsoftware
tools.Convexoptimization problems involvingLMIshavebeenusedincontroltheorysince
about1985:Gilbert'smethod[Gil66]wasusedbyDoyle[Doy82]tosolveadiagonal scaling
problem; anotherexample isthemusolprogram ofFanandTits[FT86].
Amongothermethodsusedincontroltosolveproblems involvingmultipleLMIsisthe
\methodofalternating convexprojections" describedinthearticlebyGrigoriadis andSkel-
ton[GS92].Thismethodisessentiallyarelaxation method,andrequires ananalytic ex-
pression fortheprojectionontothefeasiblesetofeachLMI.Itisnotapolynomial-time
algorithm; itscomplexit ydependsontheproblem data,unliketheellipsoid methodorthe
interior-pointmethodsdescribedin[NN94],whosecomplexities dependontheproblem size
only.However,thealternating projectionsmethodisreportedtoworkwellinmanycases.
Interior-pointmethodsforLMIproblems
Interior-pointmethodsforvariousLMIproblems haverecentlybeendevelopedbyseveralre-
searchers.The¯rstwereNesterovandNemirovskii[NN88,NN90b,NN90a,NN91a,NN94,
NN93];othersincludeAlizadeh [Ali92b,Ali91,Ali92a],Jarre[Jar93c],Vandenbergheand
Boyd[VB93b],Rendl,Vanderbei,andWolkowocz[RVW93],andYoshise[Yos94].
Ofcourse,generalinterior-pointmethods(andthemethodofcentersinparticular) havea
longhistory.Earlyworkincludes thebookbyFiaccoandMcCormic k[FM68],themethod
ofcentersdescribedbyHuardetal.[LH66,Hua67],andDikin'sinterior-pointmethod
forlinearprogramming [Dik67].Interestininterior-pointmethodssurgedin1984when
Karmark ar[Kar84]gavehisinterior-pointmethodforsolvinglinearprograms, whichappears
tohaveverygoodpractical performance aswellasagoodworst-case complexit ybound.Since
thepublication ofKarmark ar'spaper,manyresearchershavestudiedinterior-pointmethods
forlinearandquadratic programming. Thesemethodsareoftendescribedinsuchaway
thatextensions tomoregeneral(convex)constrain tsandobjectivesarenotclear.However,
NesterovandNemirovskiihavedevelopedatheoryofinterior-pointmethodsthatappliesto
moregeneralconvexprogramming problems, andinparticular, everyproblem thatarises
inthisbook;see[NN94].Inparticular, theyderivecomplexit yboundsformanydi®erent
interior-pointalgorithms, including themethodofcenters,Nemirovskii'sprojectivealgorithm
andprimal-dual methods.Themoste±cientalgorithms seemtobeNemirovskii'sprojective
algorithm andprimal-dual methods(forthecaseoflinearprograms, see[Meh91]).
Otherrecentarticlesthatconsider interior-pointmethodsformoregeneralconvexprogram-
mingincludeSonnevend[Son88],Jarre[Jar91,Jar93b],Kortanek etal.[KPY91],Den
Hertog,Roos,andTerlaky[DRT92],andthesurveybyWright[Wri92].
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 31
Interior-pointmethodsforGEVPsaredescribedinBoydandElGhaoui[BE93](variation
onthemethodofcenters),andNesterovandNemirovskii[NN91b]and[NN94,x4.4](a
variation onNemirovskii'sprojectivealgorithm). SinceGEVPsarenotconvexproblems,
devising areliablestopping criterion ismorechallenging thanfortheconvexproblems LMIP,
EVP,andCP.Adetailed complexit yanalysis (inparticular, astatemen tandproofofthe
polynomial-time complexit yofGEVPs) isgivenin[NN91b,NN94].Seealso[Jar93a].
Severalresearchershaverecentlystudiedthepossibilityofswitchingfromaninterior-point
methodtoaquadratically convergentlocalmethodinordertoimproveonthe¯nalconver-
gence;see[Ove92,OW93,OW92,FN92,NF92].
Interior-pointmethodsforCPsandotherrelatedextremal ellipsoid volumeproblems canbe
foundin[NN94,x6.5].
Interior-pointmethodsandproblemstructure
Manyresearchershavedevelopedalgorithms thattakeadvantageofthespecialstructure
oftheleast-squares problems arisingininterior-pointmethodsforlinearprogramming. As
farasweknow,the¯rst(andsofar,only)interior-pointalgorithm thattakesadvantage
ofthespecial(Lyapunov)structure ofanLMIproblem arisingincontrolisdescribedin
VandenbergheandBoyd[VB93b,VB93a].Nemirovskii'sprojectivemethodcanalsotake
advantageofsuchstructure; thesetwoalgorithms appeartobethemoste±cientalgorithms
developedsofarforsolvingtheLMIsthatariseincontroltheory.
SoftwareforsolvingLMIproblems
Gahinet andNemirovskiihaverecentlydevelopedasoftwarepackagecalledLMI-Lab [GN93]
basedonanearlierFORTRANcode[NN90a],whichallowstheusertodescribeanLMI
problem inahigh-levelsymbolicform(notunliketheformulasthatappearthroughout this
book!).LMI-Lab thensolvestheproblem usingNemirovskii'sprojectivealgorithm, taking
advantageofsomeoftheproblem structure (e.g.,blockstructure, diagonal structure ofsome
ofthematrixvariables).
Recently,ElGhaouihasdevelopedanothersoftwarepackageforsolvingLMIproblems. This
noncommercial package,calledLMI-tool,canbeusedwithmatlab.Itisavailableviaanony-
mousftp(formoreinformation, sendmailtoelghaoui@ensta.fr ).Another versionofLMI-
tool,developedbyNikoukhahandDelebecque,isavailableforusewiththematlab-like
freewarepackagescilab;inthisversion,LMI-toolhasbeeninterfaced withNemirovskii's
projectivecode.scilabcanbeobtained viaanonymousftp(formoreinformation, sendmail
toScilab@inria.fr ).
Acommercial softwarepackagethatsolvesafewspecialized controlsystemanalysis and
designproblems viaLMIformulation, calledoptin,wasrecentlydevelopedbyOlasand
Associates;see[OS93,Ola94].
ReductiontoastrictlyfeasibleLMI
Inthissectionweprovethefollowingstatemen t.LetF0;:::;Fm2Rn£nbesymmetric
matrices. ThenthereisamatrixA2Rm£pwithp·m,avectorb2Rm,andsymmetric
matrices ~F0;:::;~Fp2Rq£qwithq·nsuchthat:
F(x)¸0ifandonlyifx=Az+bforsomez2Rpand~F(z)¢=~F0+pX
i=1zi~Fi¸0:
Inaddition, iftheLMIF(x)¸0isfeasible, thentheLMI~F(z)¸0isstrictlyfeasible.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
32 Chapter 2Some StandardProblems InvolvingLMIs
Consider theLMI
F(x)=F0+mX
i=1xiFi¸0; (2.42)
whereFi2Rn£n,i=0;:::;m.LetXdenotethefeasiblesetfxjF(x)¸0g.
IfXisempty,thereisnothingtoprove;wecantakep=m,A=I,b=0,~F0=F0;:::;~Fp=
Fp.
Henceforth weassumethatXisnonempt y.IfXisasingleton, sayX=fx0g,thenwith
p=1,A=0,b=x0,~F0=1,and~F1=0,thestatemen tfollows.
Nowconsider thecasewhenXisneitheremptynorasingleton. Then,thereisana±ne
subspaceAofminimal dimensionp¸1thatcontainsX.Leta1;:::apabasisforthelinear
partofA.Theneveryx2Acanbewrittenasx=Az+bwhereA=[a1¢¢¢ap]isfull-rank
andb2Rm.De¯ningG0=F(b),Gi=F(ai)¡F0,i=1;:::;p,andG(z)=G0+Pp
i=1ziGi,
weseethatF(x)¸0ifandonlyifthereexistsz2Rpsatisfyingx=Az+bandG(z)¸0.
LetZ¢=fz2RpjG(z)¸0g.Byconstruction, Zhasnonempt yinterior.Letz0apointin
theinteriorofZandletvlieinthenullspaceofG(z0).Now,vTG(z)visanonnegativ ea±ne
function ofz2ZandiszeroataninteriorpointofZ.Therefore, itisidenticallyzeroover
Z,andhenceoverRp.Therefore, vbelongstotheintersectionBofthenullspaces ofthe
Gi,i=0;:::;p.Conversely,anyvbelonging toBwillsatisfyvTG(z)v=0foranyz2Rp.
Bmaytherefore beinterpreted asthe\constan t"nullspace oftheLMI(2.42).Letqbethe
dimension ofB.
Ifq=n(i.e.,G0=¢¢¢=Gp=0),thenobviouslyF(x)¸0ifandonlyifx=Az+b.Inthis
case,thestatemen tissatis¯ed with~F0=I2Rp£p,~Fi=0,i=1;:::;p.
Ifq<n,letv1;:::;vqabasisforB.Complete thebasisv1;:::;vqbyvq+1;:::;vnto
obtainabasisofRn.De¯neU=[vq+1¢¢¢vn],~Fi=UTGiU,i=1;:::;pand~F(z)=
~F0+Pp
i=1zi~Fi.Then,theLMIG(z)¸0isequivalenttotheLMI~F(z)¸0,andby
construction, thereexistsz0suchthat~F(z0)>0.Thisconcludes theproof.
Eliminationprocedureformatrixvariables
Wenowprovethematrixelimination resultstatedinx2.6.2:GivenG,U,V,thereexistsX
suchthat
G+UXVT+VXTUT>0 (2.43)
ifandonlyif
~UTG~U>0;~VTG~V>0 (2.44)
holds,where~Uand~Vareorthogonal complemen tsofUandVrespectively.
Itisobviousthatif(2.43)holdsforsomeX,sodoinequalities (2.44).Letusnowprovethe
converse.
Supposethatinequalities (2.44)arefeasible. Supposenowthatinequalit y(2.43)isnotfeasible
foranyX.Inotherwords,supposethat
G+UXVT+VXTUT6>0;
foreveryX.Byduality,thisisequivalenttothecondition
thereexistsZ6=0withZ¸0,VTZU=0,andTrGZ·0. (2.45)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 33
NowletusshowthatZ¸0andVTZU=0implythat
Z=~VHHT~VT+~UKKT~UT(2.46)
forsomematricesH,K(atleastoneofwhichisnonzero, sinceotherwiseZ=0).This
will¯nishourproof,since(2.44)impliesthatTrGZ>0forZoftheform(2.46),which
contradicts TrGZ·0in(2.45).
LetRbeaCholesky factorofZ,i.e.Z=RRT.TheconditionVTZU=0isequivalentto
VTRRTU=(VTR)(UTR)T=0:
ThismeansthatthereexistsaunitarymatrixT,andmatricesMandNsuchthat
VTRT=[0M]andUTRT=[N0];
wherethenumberofcolumns ofMandNadduptothatofR.Inotherwords,RTcanbe
writtenas
RT=[AB]
forsomematricesA;BsuchthatVTA=0,UTB=0.Fromthede¯nition of~Vand~U,
matricesA,BcanbewrittenA=~VH,B=~UKforsomematricesH;K,andwehave
Z=RRT=(RT)(RT)T=AAT+BBT=~VHHT~VT+~UKKT~UT;
whichisthedesiredresult.
Theequivalencebetweentheconditions
~U(z)TG(z)~U(z)>0
and
G(z)¡¾U(z)U(z)T>0forsomerealscalar¾
isthroughFinsler's lemma[Fin37](seealsox3.2.6of[Sch73]),whichstatesthatifxTQx>0
forallnonzeroxsuchthatxTAx=0,whereQandAaresymmetric, realmatrices, then
thereexistsarealscalarsuchthatQ¡¾A>0.
Finsler's lemmahasalsobeendirectlyusedtoeliminate variablesincertainmatrixinequali-
ties(seeforexample [PH86,KR88,BPG89b ]);itiscloselyrelatedtotheS-procedure.We
alsonoteanotherresultonelimination ofmatrixvariables, duetoParrott[Par78].
Theelimination lemmaisrelatedtoamatrixdilationproblem considered in[DKW82 ],which
wasusedincontrolproblems in[PZPB91,PZP+92,Gah92,Pac94,IS93a,Iwa93](see
alsotheNotesandReferences ofChapter 7).
TheS-procedure
Theproblem ofdetermining ifaquadratic formisnonnegativ ewhenotherquadratic formsare
nonnegativ ehasbeenstudiedbymathematicians foratleastseventyyears.Foracomplete
discussion andreferences, wereferthereadertothesurveyarticlebyUhlig[Uhl79].Seealso
thebookbyHestenes [Hes81,p354-360 ]andHornandJohnson [HJ91,p78-86]forproofs
ofvariousS-procedureresults.
Theapplication oftheS-proceduretocontrolproblems datesbackto1944,whenLur'e
andPostnikovusedittoprovethestabilityofsomeparticular nonlinear systems. The
name\S-procedure"wasintroducedmuchlaterbyAizerman andGantmacherintheir1964
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
34 Chapter 2Some StandardProblems InvolvingLMIs
book[AG64];sincethenYakubovichhasgivenmoregeneralformulations andcorrected
errorsinsomeearlierproofs[Yak77,FY79].
TheproofofthetwoS-procedureresultsdescribedinthischaptercanbefoundinthe
articlesbyYakubovich[FY79,Yak77,Yak73].Recentandimportantdevelopmentsabout
theS-procedurecanbefoundin[Yak92]andreferences therein.
ComplexityofS-procedurecondition
SupposethatT0,u0,andv0dependa±nelyonsomevariableº;thecondition (2.30)is
convexinº.Indeed,for¯xed³,theconstrain tF0(³)¸0isalinearconstrain tonºandthe
constrain t(2.30)issimplyanin¯nitenumberoftheselinearconstrain ts.Wemighttherefore
imaginethattheproblem ofcheckingwhether thereexistsºsuchthat(2.30)holdshaslow
complexit ysinceitcanbecastasaconvexfeasibilityproblem. Thisiswrong.Infact,merely
verifyingthatthecondition (2.30)holdsfor¯xeddataTi,ui,andvi,isashardassolving
ageneralinde¯nite quadratic program, whichisNP-complete. Wecoulddetermine whether
thereexistsºsuchthat(2.30)holdswithpolynomially manystepsoftheellipsoid algorithm;
theproblem isthat¯ndingacuttingplane(whichisrequired foreachstepoftheellipsoid
algorithm) isitselfanNP-complete problem.
Positive-realandbounded-reallemma
In1961,PopovgavethefamousPopovfrequency-domain stabilitycriterion fortheabsolute
stabilityproblem [Pop62].Popov'scriterion couldbecheckedviagraphical means,byverify-
ingthattheNyquistplotofthe\linearpart"ofthenonlinear systemwascon¯ned toaspeci¯c
regioninthecomplex plane.Yakubovich[Yak62,Yak64]andKalman [Kal63a,Kal63b]
established theconnection betweenthePopovcriterion andtheexistence ofapositive-
de¯nitematrixsatisfying certainmatrixinequalities. ThePRlemmaisalsoknownbyvarious
namessuchastheYakubovich{Kalman{P opov{Anderson LemmaortheKalman{Y akubovich
Lemma. ThePRlemmaisnowstandard material, describedinseveralbooksoncontroland
systemstheory,e.g.,Narendra andTaylor[NT73],Vidyasagar[Vid92,pp474{478 ],Faurre
andDepeyrot[FD77],Anderson andVongpanitlerd [AV73,ch5{7],Brockett[Bro70],and
Willems [Wil70].
Initsoriginalform,thePRLemmastatesthatthetransfer functionc(sI¡A)¡1bofthe
single-input single-output minimal system(A;b;c)ispositive-real,i.e.
Rec(sI¡A)¡1b¸0forallRes>0 (2.47)
ifandonlyifthereexistsP>0suchthatATP+PA·0andPb=cT.Thecondition (2.47)
canbecheckedgraphically .
Anderson [And67,And66b,And73]extended thePRlemmatomulti-input multi-output
systems, andderivedsimilarresultsfornonexpansiv esystems[And66a].Willems [Wil71b,
Wil74a]describedconnections betweenthePRlemma,certainquadratic optimal control
problems andtheexistence ofsymmetric solutions totheARE;itisin[Wil71b]thatwe¯nd
Willems' quoteontheroleofLMIsinquadratic optimalcontrol(seepage3).
Theconnections betweenpassivity,LMIsandAREscanbesummarized asfollows.We
considerA,B,C,andDsuchthatalleigenvaluesofAhavenegativerealpart,(A;B)is
controllable andD+DT>0.Thefollowingstatemen tsareequivalent:
1.Thesystem
_x=Ax+Bu;y=Cx+Du;x(0)=0
ispassive,i.e.,satis¯es
ZT
0u(t)Ty(t)dt¸0
foralluandT¸0.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 35
2.ThetransfermatrixH(s)=C(sI¡A)¡1B+Dispositive-real,i.e.,
H(s)+H(s)¤¸0
forallswithRes¸0.
3.TheLMI"
ATP+PAPB¡CT
BTP¡C¡(D+DT)#
·0 (2.48)
inthevariableP=PTisfeasible.
4.ThereexistsP=PTsatisfying theARE
ATP+PA+(PB¡CT)(D+DT)¡1(PB¡CT)T=0: (2.49)
5.ThesizesoftheJordanblockscorrespondingtothepureimaginary eigenvaluesofthe
Hamiltonian matrix
M="
A¡B(D+DT)¡1CB(D+DT)¡1BT
¡CT(D+DT)¡1C¡AT+CT(D+DT)¡1BT#
arealleven.
Theequivalenceof1,2,3,and4canbefoundinTheorem 4of[Wil71b]andalso[Wil74a].
Theequivalenceof5isfoundin,forexample, Theorem 1of[LR80].Originsofthisresultcan
betracedbacktoReid[Rei46]andLevin[Lev59];itisexplicitly statedinPotter[Pot66].
Anexcellentdiscussion ofthisresultanditsconnections withspectralfactorization theory
canbefoundin[AV73];seealso[Fra87,BBK89,Rob89].Connections betweentheex-
tremalpointsofthesetofsolutions oftheLMI(2.48)andthesolutions oftheARE(2.49)
areexplored in[Bad82].Themethodforconstructing Parefromthestableeigenspace of
M,describedinx2.7.2,isin[BBK89];thismethodisanobviousvariation ofLaub'salgo-
rithm[Lau79,AL84],orVanDooren's[Doo81].Alessstablenumerical methodcanbefound
in[Rei46,Lev59,Pot66,AV73].Forotherdiscussions ofstrictpassivity,seethearticles
byWen[Wen88],Lozano{Leal andJoshi[LLJ90].Itisalsopossibletocheckpassivity(or
nonexpansivit y)ofasystemsymbolicallyusingSturmmethods,whichyieldsRouth{Hurwitz
likealgorithms; seeSiljak[Sil71,Sil73]andBoyd,Balakrishnan, andKabamba[BBK89].
ThePRlemmaisusedinmanyareas,e.g.,interconnected systems [MH78]andstochastic
processesandstochasticrealization theory[Fau67,Fau73,AM74].
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Chapter 3
SomeMatrixProblems
3.1MinimizingConditionNumberbyScaling
Thecondition numberofamatrixM2Rp£q,withp¸q,istheratioofitslargest
andsmallest singular values,i.e.,
·(M)¢=µ¸max(MTM)
¸min(MTM)¶1=2
forMfull-rank, and·(M)=1otherwise. Forsquareinvertiblematrices thisreduces
to·(M)=kMkkM¡1k.
Weconsider theproblem:
minimize ·(LMR)
subjecttoL2Rp£p,diagonal andnonsingular
R2Rq£q,diagonal andnonsingular(3.1)
whereLandRaretheoptimization variables, andthematrixM2Rp£qisgiven.
Wewillshowthatthisproblem canbetransformed intoaGEVP.Weassumewithout
lossofgeneralit ythatp¸qandMisfull-rank.
Letus¯x°>1.Thereexistnonsingular, diagonalL2Rp£pandR2Rq£q
suchthat·(LMR)·°ifandonlyiftherearenonsingular, diagonalL2Rp£pand
R2Rq£qand¹>0suchthat
¹I·(LMR)T(LMR)·¹°2I:
Sincewecanabsorbthefactor1=p¹intoL,thisisequivalenttotheexistence of
nonsingular, diagonalL2Rp£pandR2Rq£qsuchthat
I·(LMR)T(LMR)·°2I;
whichisthesameas
(RRT)¡1·MT(LTL)M·°2(RRT)¡1: (3.2)
Thisisequivalenttotheexistence ofdiagonalP2Rp£p,Q2Rq£qwithP>0,
Q>0and
Q·MTPM·°2Q: (3.3)
37
38 Chapter 3Some MatrixProblems
Toseethis,¯rstsupposethatL2Rp£pandR2Rq£qarenonsingular anddiagonal,
and(3.2)holds.Then(3.3)holdswithP=LTLandQ=(RRT)¡1.Conversely,
supposethat(3.3)holdsfordiagonalP2Rp£pandQ2Rq£qwithP>0,Q>0.
Then,(3.2)holdsforL=P1=2andR=Q¡1=2.
Hencewecansolve(3.1)bysolvingtheGEVP:
minimize °2
subjecttoP2Rp£panddiagonal;P>0;
Q2Rq£qanddiagonal;Q>0
Q·MTPM·°2Q
Remark :Thisresultisreadilyextended tohandlescalingmatrices thathave
agivenblock-diagonal structure, andmoregenerally ,theconstrain tthatoneor
moreblocksareequal.Complex matrices arereadilyhandledbysubstituting M¤
(complex-conjugate transpose)forMT.
3.2MinimizingConditionNumberofa
Positive-De¯niteMatrix
Arelatedproblem isminimizing thecondition numberofapositive-de¯nite matrix
Mthatdependsa±nelyonthevariablex,subjecttotheLMIconstrain tF(x)>0.
Thisproblem canbereformulatedastheGEVP
minimize °
subjecttoF(x)>0;¹>0;¹I<M(x)<°¹I(3.4)
(thevariablesherearex,¹,and°).
WecanreformulatethisGEVPasanEVPasfollows.Suppose
M(x)=M0+mX
i=1xiMi;F(x)=F0+mX
i=1xiFi:
De¯ning thenewvariablesº=1=¹,~x=x=¹,wecanexpress(3.4)astheEVPwith
variablesº=1=¹,~x=x=¹and°:
minimize °
subjecttoºF0+mX
i=1~xiFi>0;I<ºM0+mX
i=1~xiMi<°I
3.3MinimizingNormbyScaling
Theoptimaldiagonally scalednormofamatrixM2Cn£nisde¯nedas
º(M)¢=inf©°°DMD¡1°°¯¯D2Cn£nisdiagonal andnonsingularª
;
wherekMkisthenorm(i.e.,largestsingular value)ofthematrixM.Wewillshow
thatº(M)canbecomputed bysolvingaGEVP.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
3.4Rescaling aMatrixPositive-Definite 39
Notethat
º(M)=inf(
°¯¯¯¯¯¡
DMD¡1¢¤¡
DMD¡1¢
<°2I
forsomediagonal, nonsingular D)
=inf©
°¯¯M¤D¤DM·°2D¤Dforsomediagonal, nonsingular Dª
=inf©
°¯¯M¤PM·°2PforsomediagonalP=PT>0ª
:
Therefore º(M)istheoptimalvalueoftheGEVP
minimize °
subjecttoP>0anddiagonal;M¤PM<°2P
Remark :Thisresultcanbeextended inmanyways.Forexample, themore
generalcaseofblock-diagonal similarity-scaling, withequalityconstrain tsamong
theblocks,isreadilyhandled. Asanotherexample, wecansolvetheproblem of
simultaneous optimaldiagonal scalingofseveralmatricesM1;:::;Mp:Tocom-
pute
inf(
max
i=1;:::;p°°DMiD¡1°°¯¯¯¯¯D2Cn£nisdiagonal
andnonsingular)
;
wesimplycompute theoptimalvalueoftheGEVP
minimize °
subjecttoP>0anddiagonal;M¤
iPMi<°2P;i=1;:::;p
Another closelyrelatedquantity,whichplaysanimportantroleinthestability
analysis oflinearsystemswithuncertain parameters, is
inf(
°¯¯¯¯¯P>0;M¤PM+i(M¤G¡GM)<°2P;
PandGdiagonal andreal)
; (3.5)
whichcanalsobecomputed bysolvingaGEVP.
3.4RescalingaMatrixPositive-De¯nite
WearegivenamatrixM2Cn£n,andaskwhether thereisadiagonal matrixD>0
suchthattheHermitian partofDMispositive-de¯nite. Thisistrueifandonlyif
theLMI
M¤D+DM>0;D>0 (3.6)
isfeasible. Sodetermining whether amatrixcanberescaled positive-de¯nite isan
LMIP.
WhenthisLMIisfeasible, wecan¯ndafeasiblescalingDwiththesmallest
condition numberbysolvingtheEVP
minimize ¹
subjecttoDdiagonal;M¤D+DM>0;I<D<¹2I
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
40 Chapter 3Some MatrixProblems
Remark :TheLMI(3.6)canalsobeinterpreted asthecondition fortheexistence
ofadiagonal quadratic Lyapunovfunction thatprovesstabilityof¡M;seethe
NotesandReferences andx10.3.
3.5MatrixCompletionProblems
Inamatrixcompletion problemsomeentriesofamatrixare¯xedandtheothersare
tobechosensotheresulting (\completed") matrixhassomeproperty,e.g.,isToeplitz
andpositive-de¯nite. Manyoftheseproblems canbecastasLMIPs. Thesimplest
example ispositive-de¯nite completion: Someentriesofasymmetric matrixaregiven,
andwearetocomplete thematrixtomakeitpositive-de¯nite. Evidentlythisisan
LMIP.
Wecaneasilyrecoversomeknownresultsaboutthisproblem. Wecanassumethat
thediagonal entriesarespeci¯ed(otherwise wesimplymakethemverylarge,i.e.,elim-
inatethem;seex2.6.1).Thenthesetofallpositive-de¯nite completions, ifnonempt y,
isbounded. HencetheassociatedLMIhasananalytic centerA?(seex2.4.1).The
matrixA?ischaracterized byTrA?¡1B=0foreveryB=BTthathaszeroentries
whereAhasspeci¯edentries(seex2.4.2).ThusthematrixA?¡1hasazeroentry
ineverylocationcorrespondingtoanunspeci¯edentryintheoriginalmatrix,i.e.,
ithasthesamesparsitystructure astheoriginalmatrix. Dependingonthepattern
ofspeci¯edentries,thiscondition canleadtoan\analytic solution" oftheproblem.
Butofcourseanarbitrary positive-de¯nite completion problem isreadilysolvedasan
LMIP.
Letusmentionafewsimplegeneralizations ofthismatrixcompletion problem
thathavenotbeenconsidered intheliterature butarereadilysolvedasLMIPs,and
mightbeusefulinapplications.
Supposethatforeachentryofthematrixwespecifyanupperandlowerbound,
inotherwords,weconsider an\intervalmatrix". Forentriesthatarecompletely
speci¯ed,thelowerandupperboundcoincide; forentriesthatarecompletely free
thelowerboundis¡1andtheupperboundis+1.Theproblem ofcompleting
thematrix,subjecttotheboundsoneachentry,tomakeitpositive-de¯nite (ora
contraction, i.e.,havenormlessthanone)isreadilyformulatedasanLMIP.
Asanotherexample, supposewearegivenasetofmatrices thathavesomegiven
sparsitypattern. Wewanttodetermine amatrixwiththecomplemen tarysparsity
patternthat,whenaddedtoeachofthegivenmatrices, resultsinapositive-de¯nite
(orcontractive,etc.)matrix.Wemightcallthisthe\simultaneous matrixcompletion
problem" sinceweseekacompletion thatservesformultipleoriginalmatrices. This
problem isalsoreadilycastasanLMIP.
Asamorespeci¯cexample, supposewearegivenupperandlowerboundsfor
someofthecorrelation coe±cientsofastationary time-series, andmustdetermine
somecomplemen tarycorrelation coe±cientsthatareconsisten twiththem.Inother
words,wehaveaToeplitzmatrixwithsomeentrieswithingivenintervals,whilethe
othersarecompletely free(unspeci¯ed). Wemustdetermine thesefreeentriessothat
theToeplitzmatrixispositive-de¯nite, foranychoiceofthespeci¯edelementsinthe
intervals.ThisisanLMIP.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
3.6Quadra ticApproximationofaPolytopic Norm 41
3.6QuadraticApproximationofaPolytopicNorm
Consider apiecewise linearnormk¢kplonRnde¯nedby
kzkpl¢=max
i=1;:::;p¯¯aT
iz¯¯;
whereai2Rnfori=1;:::;p.
ForanyP>0,thequadratic normde¯nedbykzkP¢=p
zTPz=kP1=2zksatis¯es
1=p®kzkP·kzkpl·p®kzkPforallz;
forsomeconstant®¸1.Thus,thequadratic normp
zTPzapproximateskzkplwithin
afactorof®.Wewillshowthattheproblem of¯ndingPthatminimizes ®,i.e.,the
problem ofdetermining theoptimal quadratic normapproximation ofk¢kpl,isan
EVP.
Letv1;:::;vLbetheverticesoftheunitballBplofk¢kpl,sothat
Bpl¢=fzjkzkpl·1g=Cofv1;:::;vLg
andletBPdenotetheunitballofk¢kP.Then
1=p®kzkP·kzkpl·p®kzkP (3.7)
isequivalentto
1=p®BPµBplµp®BP:
The¯rstinclusion, 1=p®BPµBpl,isequivalentto
aT
iP¡1ai·®;i=1;:::;p;
andthesecond,Bplµp®BP,isequivalentto
vT
iPvi·®;i=1;:::;L:
Therefore wecanminimize®suchthat(3.7)holdsforsomeP>0bysolvingtheEVP
minimize ®
subjecttovT
iPvi·®;i=1;:::;L;"
Pai
aT
i®#
¸0;i=1;:::;p
Remark :ThenumberofverticesLoftheunitballofk¢kplcangrowexponen-
tiallyinpandn.Therefore theresultisuninteresting fromthepointofviewof
computational complexit yandoflimitedpractical useexceptforproblems with
lowdimensions.
Theoptimalfactor®inthisproblem neverexceedspn.Indeed,bycomputing
themaximumvolumeellipsoid thatliesinsidetheunitballofk¢kpl,(whichdoes
notrequireusto¯ndtheverticesvi,i=1;:::;L),wecan¯nd(inpolynomial-
time)anormforwhich®islessthanpn.SeetheNotesandReferences formore
details.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
42 Chapter 3Some MatrixProblems
3.7EllipsoidalApproximation
Theproblem ofapproximating somesubsetofRnwithanellipsoid arisesinmany
¯eldsandhasalonghistory;seetheNotesandReferences. Toposesuchaproblem
precisely weneedtoknowhowthesubsetisdescribed,whether weseekaninneror
outerapproximation, andhowtheapproximation willbejudged(volume,majoror
minorsemi-axis, etc.).
Insomecasestheproblem canbecastasaCPoranEVP,andhencesolved
exactly.Asanexample consider theproblem of¯ndingtheellipsoid centeredaround
theoriginofsmallest volumethatcontainsapolytopedescribedbyitsvertices.We
sawinx2.2.4thatthisproblem canbecastasaCP.
Inothercases,wecancompute anapproximation oftheoptimal ellipsoid by
solvingaCPoranEVP.Insomeofthesecases,theproblem isknowntobeNP-hard,
soitisunlikelythattheproblem canbereduced(polynomially) toanLMIproblem.
Italsosuggests thattheapproximations obtained bysolvingaCPorEVPwillnotbe
goodforallinstances ofproblem data,although theapproximations maybegoodon
\typical"problems, andhenceofsomeuseinpractice.
Asanexample consider theproblem of¯ndingtheellipsoid ofsmallest volume
thatcontainsapolytopedescribedbyasetoflinearinequalities, i.e.,fxjaT
ix·
bi;i=1;:::;pg.(Thisisthesameproblem asdescribedabove,butwithadi®erent
description ofthepolytope.)Thisproblem isNP-hard. Indeed,consider theproblem
ofsimplyverifyingthatagiven,¯xedellipsoid containsapolytopedescribedbya
setoflinearinequalities. Thisproblem isequivalenttothegeneralconcavequadratic
programming problem, whichisNP-complete.
Inthissectionweconsider subsetsformedfromellipsoidsE1;:::;Epinvarious
ways:union,intersection, andaddition. WeapproximatethesesetsbyanellipsoidE0.
WedescribeanellipsoidEintwodi®erentways.The¯rstdescription usesconvex
quadratic functions:
E=fxjT(x)·0;g;T(x)=xTAx+2xTb+c; (3.8)
whereA=AT>0andbTA¡1b¡c>0(whichensuresthatEisnonempt yanddoes
notreducetoasinglepoint).Notethatthisdescription ishomogeneous, i.e.,wecan
scaleA,bandcbyanypositivefactorwithouta®ectingE.
ThevolumeofEisgivenby
vol(E)2=¯det¡
(bTA¡1b¡c)A¡1¢
where¯isaconstantthatdependsonlyonthedimension ofx,i.e.,n.Thediameter
ofE(i.e.,twicethemaximumsemi-axis length)is
2q
(bTA¡1b¡c)¸max(A¡1):
Wewillalsodescribeanellipsoid astheimageoftheunitballunderana±ne
mapping withsymmetric positive-de¯nite matrix:
E=©
x¯¯(x¡xc)TP¡2(x¡xc)·1ª
=fPz+xcjkzk·1g; (3.9)
whereP=PT>0.(Thisrepresentationisunique,i.e.,Pandxcareuniquely
determined bytheellipsoidE.)ThevolumeofEisthenproportional todetP,and
itsdiameter is2¸max(P).
Eachofthesedescriptions ofEisreadilyconvertedintotheother.Givenarepre-
sentation(3.8)(i.e.,A,b,andc)weformtherepresentation(3.9)with
P=p
bTA¡1b¡cA¡1=2;xc=¡A¡1b:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
3.7Ellipsoid alApproximation 43
Giventherepresentation(3.9),wecanformarepresentation(3.8)ofEwith
A=P¡2;b=¡P¡2xc;c=xT
cP¡2xc¡1:
(ThisformsonerepresentationofE;wecanformeveryrepresentationofEbyscaling
theseA,b,cbypositivefactors.)
3.7.1Outerapproximation ofunionofellipsoids
WeseekasmallellipsoidE0thatcoverstheunionofellipsoidsE1;:::;Ep(orequiva-
lently,theconvexhulloftheunion,i.e.,CoSp
i=1Ei).Wewilldescribetheseellipsoids
viatheassociatedquadratic functionsTi(x)=xTAix+2xTbi+ci.Wehave
E0¶p[
i=1Ei (3.10)
ifandonlyifEiµE0fori=1;:::;p.Thisistrueifandonlyif,foreachi,everyx
suchthatTi(x)·0satis¯esT0(x)·0.BytheS-procedure,thisistrueifandonlyif
thereexistnonnegativ escalars¿1;:::;¿psuchthat
foreveryx;T0(x)¡¿iTi(x)·0;i=1;:::;p;
or,equivalently,suchthat
"
A0b0
bT
0c0#
¡¿i"
Aibi
bT
ici#
·0;i=1;:::;p:
SinceourrepresentationofE0ishomogeneous, wewillnownormalize A0,b0andc0
inaconvenientway:suchthatbT
0A¡1
0b0¡c0=1.Inotherwordsweset
c0=bT
0A¡1
0b0¡1 (3.11)
andparametrizeE0byA0andb0alone.Thusourcondition becomes:
"
A0b0
bT
0bT
0A¡1
0b0¡1#
¡¿i"
Aibi
bT
ici#
·0;i=1;:::;p:
UsingSchurcomplemen ts,weobtaintheequivalentLMI
2
64A0b00
bT
0¡1bT
0
0b0¡A03
75¡¿i2
64Aibi0
bT
ici0
0003
75·0;i=1;:::;p; (3.12)
withvariablesA0,b0,and¿1,...,¿p.Tosummarize, thecondition (3.10)isequivalent
totheexistence ofnonnegativ e¿1;:::;¿psuchthat(3.12)holds.(SinceA0>0,it
followsthatthe¿imust,infact,bepositive.)
Withthenormalization (3.11),thevolumeofE0isproportional toq
detA¡1
0.
Thuswecan¯ndthesmallest volumeellipsoid containingtheunionofellipsoidsE1,
...,EpbysolvingtheCP(withvariablesA0,b0,and¿1,...,¿p)
minimize logdetA¡1
0
subjecttoA0>0;¿1¸0;:::;¿p¸0;(3:12)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
44 Chapter 3Some MatrixProblems
Thisellipsoid iscalledtheLÄowner{John ellipsoid forSp
i=1Ei,andsatis¯es several
niceproperties(e.g.,a±neinvariance, certainboundsonhowwellitapproximates
CoSEi);seetheNotesandReferences.
Wecan¯ndtheellipsoid (orequivalently,sphere)ofsmallest diameter thatcon-
tainsSEibyminimizing ¸max(A¡1
0)subjecttotheconstrain tsoftheprevious CP,
whichisequivalenttosolvingtheEVP
maximize ¸
subjecttoA0>¸I;¿1¸0;:::;¿p¸0;(3:12)
Theoptimaldiameter is2=p
¸opt,where¸optisanoptimalvalueoftheEVPabove.
3.7.2Outerapproximation oftheintersection ofellipsoids
Simplyverifyingthat
E0¶p\
i=1Ei (3.13)
holds,givenE0,E1,...,Ep,isNP-complete, sowedonotexpecttorecastitasone
ofourLMIproblems. Afortiori,wedonotexpecttocasttheproblem of¯ndingthe
smallest volumeellipsoid coveringtheintersection ofsomeellipsoids asaCP.
Wewilldescribethreewaysthatwecancompute suboptimalellipsoids forthis
problem bysolvingCPs.The¯rstmethodusestheS-proceduretoderiveanLMI
thatisasu±cientcondition for(3.13)tohold.
Onceagainwenormalize ourrepresentationofE0sothat(3.11)holds.Fromthe
S-procedureweobtainthecondition: thereexistpositivescalars¿1;:::;¿psuchthat
"
A0b0
bT
0bT
0A¡1
0b0¡1#
¡pX
i=1¿i"
Aibi
bT
ici#
·0;
whichcanbewrittenastheLMI(invariablesA0,b0,and¿1;:::;¿p):
2
64A0b00
bT
0¡1bT
0
0b0¡A03
75¡pX
i=1¿i2
64Aibi0
bT
ici0
0003
75·0: (3.14)
ThisLMIissu±cient butnotnecessary for(3.13)tohold,i.e.,itcharacterizes some
butnotalloftheellipsoids thatcovertheintersection ofE1,...,Ep.Wecan¯ndthe
bestsuchouterapproximation (i.e.,theoneofsmallest volume)bysolvingtheCP
(withvariablesA0,b0,and¿1;:::;¿p)
minimize logdetA¡1
0
subjecttoA0>0;¿1¸0;:::;¿p¸0;(3:14)(3.15)
Remark :TheintersectionTp
i=1Eiisemptyorasinglepointifandonlyifthere
existnonnegativ e¿1;:::;¿p,notallzero,suchthat
pX
i=1¿i"
Aibi
bT
ici#
¸0:
Inthiscase(3.15)isunboundedbelow.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
3.7Ellipsoid alApproximation 45
Another methodthatcanbeusedto¯ndasuboptimalsolutionfortheproblem of
determining theminimumvolumeellipsoid thatcontainstheintersection ofE1,...,Ep
isto¯rstcompute themaximum volumeellipsoid thatiscontainedintheintersection,
whichcanbecastasaCP(seethenextsection). Ifthisellipsoid isscaledbyafactor
ofnaboutitcenterthenitisguaranteedtocontaintheintersection. SeetheNotes
andReferences formorediscussion.
Yetanothermethodforproducingasuboptimalsolution isbasedontheideaof
theanalytic center(seex2.4.1).SupposethatE1,...,Eparegivenintheform(3.9),
withx1;:::;xpandP1,...,Pp.TheLMI
F(x)=diagÃ"
IP¡1
1(x¡x1)
(x¡x1)TP¡1
1 1#
;:::
:::;"
IP¡1
p(x¡xp)
(x¡xp)TP¡1
p 1#!
>0
inthevariablexhasasfeasiblesettheinteriorofTp
i=1Ei.Weassumenowthatthe
intersectionTp
i=1Eihasnonempt yinterior(i.e.,isnonempt yandnotasinglepoint).
Letx?denoteitsanalytic center,i.e.,
x?=argmin
F(x)>0logdetF(x)¡1
andletHdenotetheHessianoflogdetF(x)¡1atx?.Thenitcanbeshownthat
E0¢=©
x¯¯(x¡x?)TH(x¡x?)·1ª
µp\
i=1Ei:
InfactwecangiveaboundonhowpoorlyE0approximatestheintersection. Itcan
beshownthat
©
x¯¯(x¡x?)TH(x¡x?)·(3p+1)2ª
¶p\
i=1Ei:
Thus,whenE0isenlarged byafactorof3p+1,itcoversTp
i=1Ei.SeetheNotesand
References formorediscussion ofthis.
3.7.3Innerapproximation oftheintersection ofellipsoids
TheellipsoidE0iscontainedintheintersection oftheellipsoidsE1;:::;Epifandonly
ifforeveryxsatisfying
(x¡x0)TP¡2
0(x¡x0)·1;
wehave
(x¡xi)TP¡2
i(x¡xi)·1;i=1;:::;p:
UsingtheS-procedure,thisisequivalenttotheexistence ofnonnegativ e¸1;:::;¸p
satisfying
foreveryx;(x¡xi)TP¡2
i(x¡xi)
¡¸i(x¡x0)TP¡2
0(x¡x0)·1¡¸i;i=1;:::;p:(3.16)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
46 Chapter 3Some MatrixProblems
WeproveintheNotesandReferences sectionthat(3.16)isequivalentto
2
64¡P2
ixi¡x0P0
(xi¡x0)T¸i¡10
P0 0¡¸iI3
75·0;i=1;:::;p: (3.17)
Therefore, weobtainthelargestvolumeellipsoid containedintheintersection of
theellipsoidsE1;:::;EpbysolvingtheCP(withvariablesP0,x0,¸1;:::;¸p)
minimize logdetP¡1
0
subjecttoP0>0;¸1¸0;:::;¸p¸0;(3:17)
Amongtheellipsoids containedintheintersection ofE1,...,Ep,wecan¯ndone
thatmaximizes thesmallest semi-axis bysolving
minimize ¸
subjectto¸I>P0>0;¸1¸0;:::;¸p¸0;(3:17)
Equivalently,thisyieldsthesphereoflargestdiameter containedintheintersection.
3.7.4Outerapproximation ofthesumofellipsoids
ThesumoftheellipsoidsE1;:::;Episde¯nedas
A¢=fx1+¢¢¢+xpjx12E1;:::;xp2Ep:g
WeseekasmallellipsoidE0containingA.
TheellipsoidE0containsAifandonlyifforeveryx1;:::;xpsuchthatTi(xi)·0,
i=1;:::;p,wehave
T0ÃpX
i=1xi!
·0:
Byapplication oftheS-procedure,thiscondition istrueif
thereexist¿i¸0,i=1;:::;psuchthat
foreveryxi,i=1;:::;p;T0ÃpX
i=1xi!
¡pX
i=1¿iTi(xi)·0:(3.18)
Onceagainwenormalize ourrepresentationofE0sothat(3.11)holds.Letx
denotethevectormadebystackingx1;:::;xp,i.e.,
x=2
664x1
...
xp3
775:
ThenbothT0(Pp
i=1xi)andPp
i=1¿iTi(xi)arequadratic functions ofx.Indeed,let
Eidenotethematrixsuchthatxi=Eix,andde¯ne
E0=pX
i=1Ei;~Ai=ET
iAiEi;~bi=ET
ibi;i=0;:::;p:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 47
Withthisnotation, weseethatcondition (3.18)isequivalentto
thereexist¿i¸0,i=1;:::;psuchthat
"~A0~b0
~bT
0bT
0A¡1
0b0¡1#
¡pX
i=1¿i"~Ai~bi
~bT
ici#
·0:(3.19)
Thiscondition isreadilywrittenasanLMIinvariablesA0,b0,¿1;:::;¿pusingSchur
complemen ts:
2
64~A0~b00
~bT
0¡1bT
0
0b0¡A03
75¡pX
i=1¿i2
64~Ai~bi0
~bT
ici0
0003
75·0: (3.20)
Therefore, wecompute theminimumvolumeellipsoid, proventocontainthe
sumofE1;:::;EpviatheS-procedure,byminimizing logdetA¡1
0subjecttoA0>0
and(3.20).ThisisaCP.
Notes andReferences
Matrixscalingproblems
Theproblem ofscalingamatrixtoreduceitscondition numberisconsidered inForsythe
andStraus[FS55](whodescribeconditions forthescalingTtobeoptimal intermsofa
newvariableR=T¡¤T¡1,justaswedo).Otherreferences ontheproblem ofimproving
thecondition numberofamatrixviadiagonal orblock-diagonal scalingare[Bau63,GV74,
Sha85,Wat91].Weshouldpointoutthattheresultsofx3.1arenotpractically usefulin
numerical analysis, sincethecostofcomputing theoptimalscalings(viatheGEVP)exceeds
thecostofsolvingtheproblem forwhichthescalingsarerequired (e.g.,solvingasetoflinear
equations).
Aspecialcaseoftheproblem ofminimizing thecondition numberofapositive-de¯nite matrix
isconsidered in[EHM92,EHM93].Finally,KhachiyanandKalantari([KK92])consider the
problem ofdiagonally scalingapositivesemide¯nite matrixviainterior-pointmethodsand
giveacomplete complexit yanalysis fortheproblem.
Theproblem ofsimilarity-scaling amatrixtominimize itsnormappearsoftenincontrol
applications (seeforexample [Doy82,Saf82]).Arecentsurveyarticleonthistopicisby
PackardandDoyle[PD93].Arelatedquantityistheone-sided multivariablestabilitymargin,
describedin[TSC92];theproblem ofcomputing thisquantitycanbeeasilytransformed
intoaGEVP.Thequantity(3.5),usedintheanalysis oflinearsystems withparameter
uncertain ties,wasintroducedbyFan,TitsandDoylein[FTD91](seealso[BFBE92 ]).Some
closelyrelatedquantitieswereconsidered in[SC93,SL93a,CS92b];seealsoChapter 8in
thisbook,whereadi®erentapproachtothesameproblem anditsextensions isconsidered.
WealsomentionsomeotherrelatedquantitiesfoundinthearticlesbyKouvaritakisand
Latchman[KL85a,KL85b,LN92],andRoteaandPrasanth[RP93,RP94],whichcanbe
computed viaGEVPs.
DiagonalquadraticLyapunovfunctions
Theproblem ofrescaling amatrixMwithadiagonal matrixD>0suchthatitsHer-
mitianpartispositive-de¯nite isconsidered inthearticlesbyBarkeretal.[BBP78]and
Khalil[Kha82].Theproblem isthesameasdetermining theexistence ofdiagonal quadratic
Lyapunovfunction foragivenlinearsystem;see[Ger85,Hu87,BY89,Kra91].In[KB93],
Kaszkurewicz andBhayagiveanexcellentsurveyonthetopicofdiagonal Lyapunovfunc-
tions,including manyreferences andexamples frommathematical ecology,powersystems,
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
48 Chapter 3Some MatrixProblems
anddigital¯lterstability.Diagonal Lyapunovfunctions ariseintheanalysis ofpositive
orthantstability;seex10.3.
Matrixcompletionproblems
Matrixcompletions problems areaddressed inthearticleofDymandGohberg[DG81],in
whichtheauthorsexamine conditions fortheexistence ofcompletions ofbandmatrices such
thattheinverseisalsoabandmatrix. In[GJSW84 ],theirresultsareextended toamore
generalclassofpartially speci¯edmatrices. Inbothpapersthe\sparsitypattern" result
mentionedinx3.5appears.In[BJL89],Barrettetal.provideanalgorithm for¯ndingthe
completion maximizing thedeterminan tofthecompleted matrix(i.e.,theanalytic centerA?
mentionedinx3.5).In[HW93],HeltonandWoerdeman consider theproblem ofminimum
normextension ofHankelmatrices. In[GFS93],Grigoriadis, FrazhoandSkeltonconsider
theproblem ofapproximating agivensymmetric matrixbyaToeplitzmatrixandproposea
solution tothisproblem viaconvexoptimization. Seealso[OSA93].
Dancis([Dan92,Dan93]andJohnson andLunquist [JL92])studycompletion problems in
whichtherankorinertiaofthecompleted matrixisspeci¯ed. Asfarasweknow,such
problems cannotbecastasLMIPs.
Aclassicpaperonthecontractivecompletion problem(i.e.,completing amatrixsoitsnorm
islessthanone)isDavis,Kahan,andWeinbergerin[DKW82 ],inwhichananalytic solution
isgivenforcontractivecompletion problems withaspecialblockform.Theresultinthis
paperiscloselyrelatedto(andalsomoregeneralthan)thematrixelimination procedure
giveninx2.6.2.See[Woe90,NW92,BW92]formorereferences oncontractivecompletion
problems.
Maximumvolumeellipsoidcontainedinasymmetricpolytope
Inthissection,weestablish somepropertiesofthemaximumvolumeellipsoid containedin
asymmetric polytopedescribedas:
P=©
z¯¯¯¯aT
iz¯¯·1;i=1;:::;pª
:
Consider anellipsoid describedbyE=fx¯¯xTQ¡1x·1gwhereQ=QT>0.Itsvolumeis
proportionaltopdetQ.TheconditionEµPcanbeexpressedaT
iQai·1fori=1;:::;p.
Henceweobtainthemaximumvolumeellipsoid containedinPbysolvingtheCP
minimize logdetQ¡1
subjecttoQ>0;aT
iQai·1;i=1;:::;p
SupposenowthatQisoptimal. Fromthestandard optimalit yconditions, therearenonneg-
ative¸1;:::;¸psuchthat
Q¡1=pX
i=1¸i
aT
iQaiaiaT
i;
and¸i=0ifaT
iQai=1.Multiplying byQ1=2ontheleftandrightandtakingthetrace
yieldsPp
i=1¸i=n.
Foranyx2Rn,wehave
xTQ¡1x=pX
i=1¸i(xTai)2
aT
iQai:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 49
Foreachi,either¸i=0oraT
iQai=1,soweconclude
xTQ¡1x=pX
i=1¸i(xTai)2:
SupposenowthatxbelongstoP.Then
xTQ¡1x·pX
i=1¸i=n:
Equivalently,PµpnE,sowehave:
EµPµpnE;
i.e.,themaximumvolumeellipsoid containedinPapproximatesPwithinthescalefactorpn.
Ellipsoidalapproximation
Theproblem ofcomputing theminimumvolumeellipsoid inRnthatcontainsagivenconvex
setwasconsidered byJohnandLÄowner,whoshowedthattheoptimalellipsoid, whenshrunk
aboutitscenterbyafactorofn,iscontainedinthegivenset(see[Joh85,GLS88]).Such
ellipsoids arecalledLÄowner{John ellipsoids. Inthecaseofsymmetric sets,thefactorncan
beimprovedtopn(usinganargumen tsimilartotheoneintheNotesandReferences section
above).Acloselyrelatedproblem is¯ndingthemaximumvolumeellipsoid containedina
convexset.Similarresultsholdfortheseellipsoids.
Specialtechniqueshavebeendevelopedtocompute smallest spheresandellipsoids containing
agivensetofpointsinspacesoflowdimension; seee.g.,Preparata andShamos[PS85,p.255]
orPost[Pos84].
Minimumvolumeellipsoids containingagivensetariseoftenincontroltheory.Schweppe
andSchlaepfer [Sch68,SS72]useellipsoids containingthesumandtheintersection oftwo
givenellipsoids intheproblem ofstateestimation withnorm-boundeddisturbances. Seealso
thearticlesbyBertsekasandRhodes[BR71],Chernousk o[Che80a,Che80b,Che80c]and
Kahan[Kah68].In[RB92],ellipsoidal approximations ofrobotlinkagesandworkspace are
usedtorapidlydetectoravoidcollisions.
Minimumvolumeellipsoids havebeenwidelyusedinsystemidenti¯cation; seethear-
ticlesbyFogel[Fog79],FogelandHuang[FH82],Deller[Del89],Belforte, Bonaand
Cerone[BBC90],Lau,KosutandBoyd[LKB90,KLB90,KLB92],Cheung, Yurkovich
andPassino[CYP93]andthebookbyNorton[Nor86].
Intheellipsoid algorithm weencounteredtheminimumvolumeellipsoid thatcontainsagiven
half-ellipsoid; thereferences ontheellipsoid algorithm describevariousextensions.
E±cientinterior-pointmethodsforcomputing minimumvolumeouterandmaximumvolume
innerellipsoids forvarioussetsaredescribedinNesterovandNemirovskii[NN94].See
also[KT93].
Ellipsoidalapproximationviaanalyticcenters
Ellipsoidal approximations viabarrierfunctions ariseinthetheoryofinterior-pointalgo-
rithms,andcanbetracedbacktoDikin[Dik67]andKarmark ar[Kar84]forpolytopes
describedbyasetoflinearinequalities. ForthecaseofmoregeneralLMIs,seeNesterovand
Nemirovskii[NN94],BoydandElGhaoui[BE93],andJarre[Jar93c].
Theseapproximations havethepropertythattheyareeasytocompute, andcomewith
provableboundsonhowsuboptimal theyare.Theseboundshavethefollowingform:if
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
50 Chapter 3Some MatrixProblems
theinnerellipsoid isstretchedaboutitscenterbysomefactor®(whichdependsonlyon
thedimension ofthespace,thetypeandsizeoftheconstrain ts)thentheresulting ellipsoid
coversthefeasibleset.Thepaperscitedabovegivedi®erentvaluesof®;onemaybebetter
thananother, dependingonthenumberandtypeofconstrain ts.
Incontrast,themaximumvolumeinnerellipsoid, i.e.,theLÄowner{John ellipsoid, ismoredif-
¯culttocompute (stillpolynomial, however)butthefactor®dependsonlyonthedimension
ofthespaceandnotonthetypeornumberofconstrain ts,andisalwayssmallerthanthe
onesobtained viabarrierfunctions. Soanellipsoidal approximation fromabarrierfunction
canserveasa\cheapsubstitute" fortheLÄowner{John ellipsoid.
Proofoflemmainx3.7.3
Weprovethatthestatemen t
foreveryx,(x¡x1)TP¡2
1(x¡x1)¡¸(x¡x0)TP¡2
0(x¡x0)·1¡¸ (3.21)
whereP0,P1arepositivematrices,¸¸0,isequivalenttothematrixinequalit y
2
64¡P2
1x1¡x0P0
(x1¡x0)T¸¡10
P0 0¡¸I3
75·0: (3.22)
Themaximumoverxof
(x¡x1)TP¡2
1(x¡x1)¡¸(x¡x0)TP¡2
0(x¡x0) (3.23)
is¯niteifandonlyifP2
0¡¸P2
1·0(whichimpliesthat¸>0)andthereexistsx¤satisfying
P¡2
1(x¤¡x1)=¸P¡2
0(x¤¡x0); (3.24)
or,equivalently,satisfying
(P2
0=¸¡P2
1)P¡2
1(x¤¡x1)=x1¡x0: (3.25)
Inthiscase,x¤isamaximizer of(3.23).Using(3.24),condition (3.21)implies
(x¤¡x1)TP¡2
1(P2
1¡P2
0=¸)P¡2
1(x¤¡x1)·1¡¸:
UsingSchurcomplemen ts,thislastcondition isequivalentto
"
¸¡1 (x¤¡x1)TP¡2
1(P2
0=¸¡P2
1)
(P2
0=¸¡P2
1)P¡2
1(x¤¡x1) P2
0=¸¡P2
1#
·0:
Using(3.25),thisimplies
"
¸¡1(x1¡x0)T
x1¡x0P2
0=¸¡P2
1#
·0;
whichisequivalentto(3.22).Conversely,itiseasytoshowthat(3.22)implies(3.21).
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Chapter 4
LinearDi®erentialInclusions
4.1Di®erentialInclusions
Adi®erentialinclusion (DI)isdescribedby:
_x2F(x(t);t);x(0)=x0; (4.1)
whereFisaset-valuedfunction onRn£R+.Anyx:R+!Rnthatsatis¯es(4.1)is
calledasolutionortrajectoryoftheDI(4.1).Ofcourse,therecanbemanysolutions
oftheDI(4.1).Ourgoalistoestablish thatvariouspropertiesaresatis¯ed byall
solutions ofagivenDI.Forexample, wemightshowthateverytrajectoryofagiven
DIconvergestozeroast!1.
Byastandard resultcalledtheRelaxation Theorem, wemayaswellassume
F(x;t)isaconvexsetforeveryxandt.TheDIgivenby
_x2CoF(x(t);t);x(0)=x0 (4.2)
iscalledtherelaxedversionoftheDI(4.1).SinceCoF(x(t);t)¶F(x(t);t),every
trajectoryoftheDI(4.1)isalsoatrajectoryofrelaxedDI(4.2).Veryroughlyspeak-
ing,theRelaxation Theorem statesthatformanypurposestheconverseistrue.(See
theReferences forprecisestatemen ts.)Asaspeci¯candsimpleexample, itcanbe
shownthatforeveryDIweencounterinthisbook,thereachableorattainable setsof
theDIanditsrelaxedversioncoincide, i.e.,foreveryT¸0
fx(T)jxsatis¯es(4:1)g=fx(T)jxsatis¯es(4:2)g:
InfactwewillnotneedtheRelaxation Theorem, orrather,wewillalwaysgetit
\forfree"|ev eryresultweestablish inthenexttwochaptersextendsimmediately to
therelaxedversionoftheproblem. Thereasonisthatwhenaquadratic Lyapunov
function isusedtoestablish somepropertyfortheDI(4.1),thenthesameLyapunov
function establishes thepropertyfortherelaxedDI(4.2).
4.1.1Lineardi®erentialinclusions
Alineardi®erentialinclusion (LDI)isgivenby
_x2x;x(0)=x0; (4.3)
whereisasubsetofRn£n.WecaninterprettheLDI(4.3)asdescribing afamily
oflineartime-varyingsystems. EverytrajectoryoftheLDIsatis¯es
_x=A(t)x;x(0)=x0; (4.4)
51
52 Chapter 4Linear Differential Inclusions
forsomeA:R+!.Conversely,foranyA:R+!,thesolution of(4.4)isa
trajectoryoftheLDI(4.3).Inthelanguage ofcontroltheory,theLDI(4.3)mightbe
describedasan\uncertain time-varyinglinearsystem," withthesetdescribing the
\uncertain ty"inthematrixA(t).
4.1.2Ageneralization tosystems
Wewillencounterageneralization oftheLDIdescribedabovetolinearsystemswith
inputsandoutputs. Wewillconsider asystemdescribedby
_x=A(t)x+Bu(t)u+Bw(t)w;x(0)=x0;
z=Cz(t)x+Dzu(t)u+Dzw(t)w(4.5)
wherex:R+!Rn,u:R+!Rnu,w:R+!Rnw,z:R+!Rnz.xisreferred
toasthestate,uisthecontrolinput,wistheexogenous inputandzistheoutput.
Thematrices in(4.5)satisfy
"
A(t)Bu(t)Bw(t)
Cz(t)Dzu(t)Dzw(t)#
2: (4.6)
forallt¸0,whereµR(n+nz)£(n+nu+nw).Wewillbemorespeci¯cabouttheform
ofshortly.
Insomeapplications wecanhaveoneormoreoftheintegersnu,nw,andnzequal
tozero,whichmeansthatthecorrespondingvariableisnotused.Forexample, the
LDI_x2xresultswhennu=nw=nz=0.Inordernottointroduceanotherterm
todescribethesetofallsolutions of(4.5)and(4.6),wewillcall(4.5)and(4.6)a
systemdescribedbyLDIsorsimply,anLDI.
4.2SomeSpeci¯cLDIs
Wenowdescribesomespeci¯cfamiliesofLDIsthatwewillencounterinthenexttwo
chapters.
4.2.1Lineartime-invariantsystems
Whenisasingleton, theLDIreducestothelineartime-invariant(LTI)system
_x=Ax+Buu+Bww;x(0)=x0;
z=Czx+Dzuu+Dzww;
where
=("
ABuBw
CzDzuDzw#)
: (4.7)
Although mostoftheresultsofChapters 5{7arewell-knownforLTIsystems, some
arenew;wewilldiscusstheseindetailwhenweencounterthem.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
4.2Some Specific LDIs 53
4.2.2PolytopicLDIs
Whenisapolytope,wewillcalltheLDIapolytopicLDIorPLDI.Mostofour
resultsrequirethatbedescribedbyalistofitsvertices,i.e.,intheform
Co("
A1Bu;1Bw;1
Cz;1Dzu;1Dzw;1#
;:::;"
ALBu;LBw;L
Cz;LDzu;LDzw;L#)
: (4.8)
wherethematrices (4.8)aregiven.
Ifinsteadisdescribedbyasetofllinearinequalities, thenthenumberof
vertices,i.e.,L,willgenerally increase veryrapidly(exponentially)withl.Therefore
resultsforPLDIsthatrequirethedescription (4.8)areoflimitedinterestforproblems
inwhichisdescribedintermsoflinearinequalities.
4.2.3Norm-boundLDIs
Another specialclassofLDIsistheclassofnorm-boundLDIs(NLDIs), describedby
_x=Ax+Bpp+Buu+Bww;x(0)=x0;
q=Cqx+Dqpp+Dquu+Dqww;
z=Czx+Dzpp+Dzuu+Dzww;
p=¢(t)q;k¢(t)k·1:(4.9)
where¢:R+!Rnp£nq,withk¢(t)k·1forallt.Wewilloftenrewritethe
conditionp=¢(t)q,k¢k·1intheequivalentformpTp·qTq.
FortheNLDI(4.9)thesethastheform
=n
~A+~B¢(I¡Dqp¢)¡1~C¯¯¯k¢k·1o
;
where
~A="
ABuBw
CzDzuDzw#
;~B="
Bp
Dzp#
;~C=h
CqDquDqwi
:(4.10)
Thus,istheimageofthe(matrixnorm)unitballundera(matrix) linear-fractional
mapping. Notethatisconvex.Thesetiswellde¯nedifandonlyifDT
qpDqp<I,
inwhichcasewesaythattheNLDIiswell-posed.Inthesequelwewillconsider only
well-posedNLDIs.
Theequations (4.9)canbeinterpreted asanLTIsystemwithinputsu,w,output
z,andatime-varyingfeedbackmatrix¢(t)connected betweenqandp.
Remark :Inthesequel,weoftendropsometermsin(4.9):DqwandDzpwill
alwaysbezero,andveryoftenDqp,Dzw,DzuandDquwillbeassumed zero.
Theseassumptions greatlysimplify theformulasandtreatmen t.Itisnothardto
workoutthecorrespondingformulasandresultsforthemoregeneralcase.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
54 Chapter 4Linear Differential Inclusions
4.2.4Diagonal norm-boundLDIs
AusefulvariationontheNLDIisthediagonal norm-boundLDI(DNLDI), inwhich
wefurtherrestrictthematrix¢tobediagonal. Itisgivenby
_x=Ax+Bpp+Buu+Bww;x(0)=x0;
q=Cqx+Dqpp+Dquu+Dqww;
z=Czx+Dzpp+Dzuu+Dzww;
pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq:(4.11)
Wewilloftenexpressthecondition pi=±i(t)qi,j±i(t)j·1inthesimplerform
jpij·jqij.Herewehavethecomponentwiseinequalitiesjpij·jqijinsteadofthesingle
inequalit ykpk·kqkappearingintheNLDI.Inthelanguage ofcontroltheory,we
describeaDNLDIasalinearsystemwithnqscalar,uncertain, time-varyingfeedback
gains,eachofwhichisboundedbyone.TheDNLDIisdescribedbytheset
=n
~A+~B¢(I¡Dqp¢)¡1~C¯¯¯k¢k·1;¢diagonalo
; (4.12)
where~A,~Band~Caregivenby(4.10).
Thecondition ofwell-posedness foraDNLDIisthatI¡Dqp¢shouldbeinvertible
foralldiagonal ¢withk¢k·1.Thisisequivalenttodet(I¡Dqp¢)>0when¢
takesonits2nqextreme values,i.e.,j¢iij=1.Although thiscondition isexplicit, the
numberofinequalities growsexponentiallywithnq.Infact,determining whether a
DNLDIiswell-posedisNP-hard; seetheNotesandReferences.
ItturnsoutthatforaDNLDI, Coisapolytope.Infact,
Co=Con
~A+~B¢(I¡Dqp¢)¡1~C¯¯¯¢diagonal;j¢iij=1o
;
i.e.,theverticesofCoareamong2nqimagesoftheextreme pointsof¢underthe
matrixlinear-fractional mapping. Therefore aDNLDIcanbedescribedasaPLDI,at
leastinprinciple. Butthereisaveryimportantdi®erence betweenthetwodescriptions
ofthesameLDI:thenumberofverticesrequired togivethePLDIrepresentationof
anLDIincreases exponentiallywithnq(seetheNotesandReferences ofChapter 5
formorediscussion).
Remark :Moreelaboratevariations ontheNLDIaresometimes useful.Doyle
introducedtheideaof\structured feedback,"inwhichthematrix¢isrestricted
tohaveagivenblock-diagonal form,andinaddition, therecanbeequalitycon-
straintsamongsomeoftheblocks.Whenwehaveasingleblock,wehavean
NLDI;whenitisdiagonal (withnoequalityconstrain ts),wehaveaDNLDI.
MostoftheresultswepresentforNLDIsandDNLDIs canbeextended tothese
moregeneraltypesofLDIs.
4.3NonlinearSystemAnalysisviaLDIs
Muchofourmotivationforstudying LDIscomesfromthefactthatwecanusethem
toestablish variouspropertiesofnonlinear, time-varyingsystems usingatechnique
knownas\globallinearization". Theideaisimplicitintheworkonabsolute stability
originating intheSovietUnioninthe1940s.
Consider thesystem
_x=f(x;u;w;t);z=g(x;u;w;t): (4.13)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
4.3Nonlinear System AnalysisviaLDIs 55
Supposethatforeachx,u,w,andtthereisamatrixG(x;u;w;t)2suchthat
"
f(x;u;w;t)
g(x;u;w;t)#
=G(x;u;w;t)2
64x
u
w3
75 (4.14)
whereµR(n+nz)£(n+nu+nw).Thenofcourseeverytrajectoryofthenonlinear
system(4.13)isalsoatrajectoryoftheLDIde¯nedby.Ifwecanprovethatevery
trajectoryoftheLDIde¯nedbyhassomeproperty(e.g.,convergestozero),then
afortioriwehaveprovedthateverytrajectoryofthenonlinear system(4.13)hasthis
property.
4.3.1Aderivativecondition
Conditions thatguaranteetheexistence ofsuchaGaref(0;0;0;0)=0,g(0;0;0;0)=
0,and
2
664@f
@x@f
@u@f
@w
@g
@x@g
@u@g
@w3
7752forallx;u;w;t: (4.15)
Infactwecanmakeastronger statemen tthatlinksthedi®erence betweenapair
oftrajectories ofthenonlinear systemandthetrajectories oftheLDIgivenby.
Supposewehave(4.15)butnotnecessarily f(0;0;0;0)=0,g(0;0;0;0)=0.Thenfor
anypairoftrajectories (x;w;z)and(~x;~w;~z)wehave
"
_x¡_~x
z¡~z#
2Co2
64x¡~x
u¡~u
w¡~w3
75;
i.e.,(x¡~x;w¡~w;z¡~z)isatrajectoryoftheLDIgivenbyCo.
Theseresultsfollowfromasimpleextension ofthemean-valuetheorem: ifÁ:
Rn!Rnsatis¯es
@Á
@x2
throughout Rn,thenforanyxand~xwehave
Á(x)¡Á(~x)2Co(x¡~x): (4.16)
Toseethis,letc2Rn.Bythemean-valuetheorem wehave
cT(Á(x)¡Á(~x))=cT@Á
@x(³)(x¡~x)
forsome³thatliesonthelinesegmentbetweenxand~x.Sincebyassumption
@Á
@x(³)2;
weconclude that
cT(Á(x)¡Á(~x))·sup
A2CocTA(x¡~x):
Sincecwasarbitrary ,weseethatÁ(x)¡Á(~x)isineveryhalf-space thatcontainsthe
convexsetCo(x¡~x),whichproves(4.16).
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
56 Chapter 4Linear Differential Inclusions
4.3.2Sectorconditions
Sometimes thenonlinear system(4.13)canbeexpressed intheform
_x=Ax+Bpp+Buu+Bww;
q=Cqx+Dqpp+Dquu+Dqww;
z=Czx+Dzpp+Dzuu+Dzww;
pi=Ái(qi;t);i=1;:::;nq;(4.17)
whereÁi:R£R+!Rsatisfythesectorconditions
®iq2·qÁi(q;t)·¯iq2
forallqandt¸0,where®iand¯iaregiven.Inwords,thesystemconsistsofalinear
parttogether withnqtime-varyingsector-boundedscalarnonlinearities.
Thevariablespiandqicanbeeliminated from(4.17)providedthematrixI¡Dqp¢
isnonsingular foralldiagonal ¢with®i·¢ii·¯i.Inthiscase,wesaythesystem
iswell-posed.
Assumenowthatthesystem(4.17)iswell-posed,andlet(4.13)betheequations
obtained byeliminating thevariablespandq.De¯ne
=n
~A+~B¢(I¡Dqp¢)¡1~C¯¯¯¢diagonal, ¢ii2[®i;¯i]o
;
where~A,~Band~Caregivenby(4.10).Then,thecondition (4.14)holds.
Notes andReferences
Di®erentialinclusions
ThebooksbyAubinandCellina[AC84],Kisielewicz [Kis91],andtheclassictextbyFilip-
pov[Fil88]coverthetheoryofdi®erentialinclusions. SeealsothearticlebyRoxin[Rox65]
andreferences therein. Background material, e.g.,set-valuedanalysis, canbefoundinthe
bookbyAubinandFrankowska[AF90].
Thetermlineardi®erentialinclusion isabitmisleading, sinceLDIsdonotenjoyanyparticu-
larlinearityorsuperpositionproperties.Weusethetermonlytopointouttheinterpretation
asanuncertain time-varyinglinearsystem. ManyauthorsrefertoLDIsusingsomesubset
ofthephrase\uncertain lineartime-varyingsystem". Amoreaccurate term,suggested to
usandusedbyA.FilippovandE.Pyatnitskii, isselector-lineardi®erentialinclusion .
PLDIscomeupinseveralarticlesoncontroltheory,e.g.,[MP89,KP87a,KP87b,BY89].
Integralquadraticconstraints
Thepointwisequadratic constrain tp(t)Tp(t)·q(t)Tq(t)thatoccurinNLDIscanbegener-
alizedtotheintegralquadratic constrain tRt
0pTpd¿·Rt
0qTqd¿;seex8.2andx8.3.
Globallinearization
Theideaofreplacing anonlinear systembyatime-varyinglinearsystemcanbefoundinLiu
etal.[Liu68,LSL69].Theycallthisapproachgloballinearization. Ofcourse,approximating
thesetoftrajectories ofanonlinear systemviaLDIscanbeveryconservative,i.e.,thereare
manytrajectories oftheLDIthatarenottrajectories ofthenonlinear system. Wewillsee
inChapter 8howthisconservatismcanbereducedinsomespecialcases.
Theideaofgloballinearization isimplicit intheearlySovietliterature ontheabsolute
stabilityproblem, e.g.,Lur'eandPostnikov[LP44,Lur57]andPopov[Pop73].
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 57
ModelingsystemsasPLDIs
Oneofthekeyissuesinrobustcontroltheoryishowtomodelormeasure plant\uncertain ty"
or\variation". Weproposeasimplemethodthatshouldworkwellinsomecases.Atthe
least,themethodisextremely simpleandnatural.
Supposewehavearealsystemthatisfairlywellmodeledasalinearsystem. Wecollect
manysetsofinput/output measuremen ts,obtained atdi®erenttimes,underdi®erentop-
eratingconditions, orperhapsfromdi®erentinstances ofthesystem(e.g.,di®erentunits
fromamanufacturing run).Itisimportantthatwehavedatasetsfromenoughplantsor
plantconditions tocharacterize oratleastgivesomeideaoftheplantvariationthatcanbe
expected.
Foreachdatasetwedevelopalinearsystemmodeloftheplant.Tosimplify theproblem
wewillassumethatthestateinthismodelisaccessible, sothedi®erentmodelsrefertothe
samestatevector.Thesemodelsshouldbefairlyclose,butofcoursenotexactlythesame.
Wemight¯ndforexample thatthetransfermatrices ofthesemodelsats=0di®erbyabout
10%,butathighfrequencies theydi®erconsiderably more.Moreimportantly,thiscollection
ofmodelscontainsinformation aboutthe\structure" oftheplantvariation.
WeproposetomodelthesystemasaPLDIwiththeverticesgivenbythemeasured or
estimated linearsystemmodels.Inotherwords,wemodeltheplantasatime-varyinglinear
system,withsystemmatrices thatcanjumparoundamonganyofthemodelsweestimated.
Itseemsreasonable toconjecture thatacontrollerthatworkswellwiththisPLDIislikely
toworkwellwhenconnected totherealsystem. Suchcontrollerscanbedesigned bythe
methodsdescribedinChapter 7.
Well-posednessofDNLDIs
See[BY89]foraproofthataDNLDIiswell-posedifandonlyifdet(I¡Dqp¢)>0forevery
¢withj¢iij=1;theideabehindthisproofcanbetracedbacktoZadehandDesoer[ZD63,
x9.17];seealso[Sae86].Anequivalentcondition isintermsofP0matrices (amatrixisP0
ifeveryprincipal minorisnonnegativ e;see[FP62,FP66,FP67]):ADNLDIiswell-posedif
andonlyifI+Dqpisinvertible,and(I+Dqp)¡1(I¡Dqp)isaP0matrix[Gha90].Another
equivalentcondition isthat(I+Dqp;I¡Dqp)isaW0pair(see[Wil70]).Theproblem
ofdetermining whether ageneralmatrixisP0isthoughttobeverydi±cult, i.e.,ofhigh
complexit y[CPS92,p149].
Standard branch-and-boundtechniquescanbeappliedtotheproblem ofdetermining well-
posedness ofDNLDIs; seeforexample [Sd86,GS88,SP89,BBB91,BB92].Ofcoursethese
methodshavenotheoretical advantageoversimplycheckingthatthe2nqdeterminan tsare
positive.Butinpracticetheymaybemoree±cient,especiallywhenusedwithcomputation-
allycheapsu±cientconditions (seebelow).
Thereareseveralsu±cientconditions forwell-posedness ofaDNLDIthatcanbecheckedin
polynomial-time. Fan,Tits,andDoyle[FTD91]showthatiftheLMI
DT
qpPDqp<P;Pdiagonal;P>0; (4.18)
isfeasible, thentheDNLDIiswell-posed.Condition (4.18)isequivalenttotheexistence ofa
diagonal (scaling) matrixRsuchthatkRDqpR¡1k<1.In[FTD91],theauthorsalsogivea
su±cientcondition forwell-posedness ofmoregeneral\structured" LDIs,e.g.,whenequality
constrain tsareimposedonthematrix¢in(4.11).Theseconditions canalsobeobtained
usingtheS-procedure;see[Fer93].
RepresentingDNLDIsasPLDIs
Forawell-posedDNLDI, Coisapolytope,whichmeansthattheDNLDI(4.12)canalso
berepresentedasaPLDI.Morespeci¯cally,wehave
Co=Co©~A+~B¢(I¡Dqp¢)¡1~C¯¯¢diagonal,j¢iij=1ª
;
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
58 Chapter 4Linear Differential Inclusions
where~A,~Band~Caregivenby(4.10).Notethatthesetontheright-handsideisapolytope
withatmost2nqvertices.Wenowprovethisresult.
De¯ne
K=©~A+~B¢(I¡Dqp¢)¡1~C¯¯¢diagonal;j¢iij=1ª
;
whichisasetcontainingatmost2nqpoints.Byde¯nition
Co=Co©~A+~B¢(I¡Dqp¢)¡1~C¯¯¢diagonal,j¢iij·1ª
:
ClearlywehaveCo¶CoK.Wewillnowshowthateveryextreme pointofCois
containedinCoK,whichwillimplythatCoK¶Co,completing theproof.
LetWextbeanextreme pointofCo.ThenthereexistsalinearfunctionÃsuchthatWext
istheuniquemaximizer ofÃoverCo.Infact,Wext2,sinceotherwise, thehalf-space
fWjÃ(W)<Ã(Wext)¡²g,where²>0issu±cientlysmall,containsbutnotWext,which
contradictsWext2Co.
Now,forW2,Ã(W)isaboundedlinear-fractional function ofthe¯rstdiagonal entry
¢11of¢,for¯xed¢22;:::;¢nqnq,i.e.,itequals(a+b¢11)=(c+d¢11)forsomea,b,cand
d.Moreover,thedenominator c+d¢11isnonzero for¢112[¡1;1],sincetheDNLDIis
well-posed.Therefore, forevery¯xed¢22;:::;¢nqnq,thefunctionÃachievesamaximumat
anextreme valueof¢11,i.e.,¢11=§1.Extending thisargumen tto¢22;:::;¢nqnqleads
ustothefollowingconclusion: Wext=~A+~BS(I¡DqpS)¡1~C,whereSsatis¯esSii=§1.
Inotherwords,Wext2CoK,whichconcludes ourproof.
Wehavealreadynotedthatthedescription ofaDNLDIasaPLDIwillbemuchlargerthan
itsdescription asaDNLDI; seetheNotesandReferences ofChapter 5,andalso[BY89]and
[PS82,KP87a,Kam83].
ApproximatingPLDIsbyNLDIs
ItisoftenusefultoconservativelyapproximateaPLDIasanNLDI.Onereason,among
many,isthepotentiallymuchsmallersizeofthedescription asanNLDIcompared tothe
description ofthePLDI.Weconsider thePLDI
_x=A(t)x;A(t)2PLDI¢=CofA1;:::;ALg; (4.19)
withx(t)2Rn,andtheNLDI
_x=(A+Bp¢(t)Cq)x;k¢(t)k·1; (4.20)
withassociatedsetNLDI¢=fA+Bp¢Cqjk¢k·1g.Ourgoalisto¯ndA,Bp,andCq
suchthatPLDIµNLDIwiththesetNLDIassmallaspossibleinsomeappropriate sense.
Thiswillgiveane±cientouterapproximation ofthePLDI(4.19)bytheNLDI(4.20).Since
PLDIµNLDIimpliesthateverytrajectoryofthePLDIisatrajectoryoftheNLDI,every
resultweestablish foralltrajectories oftheNLDIholdsforalltrajectories ofthePLDI.For
Lverylarge,itismucheasiertoworkwiththeNLDIthanthePLDI.
Forsimplicity,wewillonlyconsider thecasewithBp2Rn£nandinvertible.Evidentlythere
issubstantialredundancy inourrepresentationoftheNLDI(4.20):WecanreplaceBpby
®BpandCqbyCq=®where®6=0.(WecanalsoreplaceBpandCqwithBpUandVCq
whereUandVareanyorthogonal matrices, without a®ecting thesetNLDI,butwewill
notusethisfact.)
WehavePLDIµNLDIifforevery»andk=1;:::;L,thereexists¼suchthat
Bp¼=(Ak¡A)»;¼T¼·»TCT
qCq»:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 59
Thisyieldstheequivalentcondition
(Ak¡A)TB¡T
pB¡1
p(Ak¡A)·CT
qCq;k=1;:::;L;
which,inturn,isequivalentto
BpBT
p>0;"
CT
qCq(Ak¡A)T
Ak¡ABpBT
p#
¸0;k=1;:::;L:
IntroducingnewvariablesV=CT
qCqandW=BpBT
p,wegettheequivalentLMIinA,V
andW
W>0;"
V(Ak¡A)T
Ak¡AW#
¸0;k=1;:::;L: (4.21)
ThuswehavePLDIµNLDIifcondition (4.21)holds.
Thereareseveralwaystominimize thesizeofNLDI¶PLDI.Themostobviousisto
minimize TrV+TrWsubjectto(4.21),whichisanEVPinA,VandW.Thisobjective
isclearlyrelatedtoseveralmeasures ofthesizeofNLDI,butwedonotknowofanysimple
interpretation.
Wecanformulatetheproblem ofminimizing thediameter ofNLDIasanEVP.Wede¯nethe
diameter ofasetµRn£nasmaxfkF¡GkjF;G2g.Thediameter ofNLDIisequal
to2p
¸max(V)¸max(W).Wecanexploitthescalingredundancy inCqtoassumewithoutloss
ofgeneralit ythat¸max(V)·1,andthenminimize¸max(W)subjectto(4.21)andV·I,
whichisanEVP.Itcanbeshownthattheresulting optimalVoptsatis¯es¸max(Vopt)=1,
sothatwehaveinfactminimized thediameter ofNLDIsubjecttoPLDIµNLDI.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Chapter 5
AnalysisofLDIs:
StateProperties
Inthischapterweconsider propertiesofthestatexoftheLDI
_x=A(t)x;A(t)2; (5.1)
whereµRn£nhasoneoffourforms:
²LTIsystems: LTIsystemsaredescribedby_x=Ax.
²Polytopic LDIs:PLDIsaredescribedby_x=A(t)x,A(t)2CofA1;:::;ALg.
²Norm-bound LDIs:NLDIsaredescribedby
_x=Ax+Bpp;q=Cqx+Dqpp;p=¢(t)q;k¢(t)k·1;
whichwewillrewriteas
_x=Ax+Bpp;pTp·(Cqx+Dqpp)T(Cqx+Dqpp): (5.2)
Weassumewell-posedness, i.e.,kDqpk<1.
²Diagonal Norm-bound LDIs:DNLDIs aredescribedby
_x=Ax+Bpp;q=Cqx+Dqpp;
pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq:(5.3)
whichcanberewritten as
_x=Ax+Bpp;q=Cqx+Dqpp;jpij·jqij;i=1;:::;nq:
Again,weassumewell-posedness.
5.1QuadraticStability
We¯rststudystability oftheLDI(5.1),thatis,weaskwhether alltrajectories of
system(5.1)convergetozeroast!1.Asu±cientcondition forthisistheexistence
ofaquadratic functionV(»)=»TP»,P>0thatdecreases alongeverynonzero
trajectoryof(5.1).IfthereexistssuchaP,wesaytheLDI(5.1)isquadratically
stableandwecallVaquadraticLyapunov function.
61
62 Chapter 5AnalysisofLDIs: StateProperties
Since
d
dtV(x)=xT¡
A(t)TP+PA(t)¢
x;
anecessary andsu±cientcondition forquadratic stabilityofsystem(5.1)is
P>0;ATP+PA<0forallA2: (5.4)
Multiplying thesecondinequalit yin(5.4)ontheleftandrightbyP¡1,andde¯ning
anewvariableQ=P¡1,wemayrewrite(5.4)as
Q>0;QAT+AQ<0forallA2: (5.5)
Thisdualinequalit yisanequivalentcondition forquadratic stability.Wenowshow
thatconditions forquadratic stabilityforLTIsystems, PLDIs,andNLDIscanbe
expressed intermsofLMIs.
²LTIsystems: Condition (5.4)becomes
P>0;ATP+PA<0: (5.6)
Therefore, checkingquadratic stabilityforanLTIsystemisanLMIPinthevariable
P.Thisisprecisely the(necessary andsu±cient)Lyapunovstabilitycriterion forLTI
systems. (Inotherwords,alinearsystemisstableifandonlyifitisquadratically sta-
ble.)Alternativ ely,using(5.5),stabilityofLTIsystemsisequivalenttotheexistence
ofQsatisfying theLMI
Q>0;AQ+QAT<0: (5.7)
Ofcourse,eachofthese(veryspecial)LMIPscanbesolvedanalytically bysolvinga
Lyapunovequation (seex1.2andx2.7).
²Polytopic LDIs:Condition (5.4)isequivalentto
P>0;AT
iP+PAi<0;i=1;:::;L: (5.8)
Thus,determining quadratic stabilityforPLDIsisanLMIPinthevariableP.The
dualcondition (5.5)isequivalenttotheLMIinthevariableQ
Q>0;QAT
i+AiQ<0;i=1;:::;L: (5.9)
²Norm-bound LDIs:Condition (5.4)isequivalenttoP>0and
"
»
¼#T"
ATP+PAPBp
BT
pP 0#"
»
¼#
<0 (5.10)
forallnonzero»satisfying
"
»
¼#T2
4¡CT
qCq¡CT
qDqp
¡DT
qpCqI¡DT
qpDqp3
5"
»
¼#
·0: (5.11)
InordertoapplytheS-procedureweshowthattheset
A=f(»;¼)j»6=0;(5:11)g
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
5.1Quadra ticStability 63
equalstheset
B=f(»;¼)j(»;¼)6=0;(5:11)g:
Itsu±cestoshowthatf(»;¼)j»=0;¼6=0;(5:11)g=;.Butthisisimmediate:
If¼6=0,thencondition (5.11)cannotholdwithouthaving»6=0,sinceI¡DT
qpDqp>0.
Therefore, thecondition thatdV(x)=dt<0forallnonzero trajectories isequivalent
to(5.10)beingsatis¯ed foranynonzero(»;¼)satisfying (5.11).(Thisargumen trecurs
throughout thischapterandwillnotberepeated.)UsingtheS-procedure,we¯nd
thatquadratic stabilityof(5.2)isequivalenttotheexistence ofPand¸satisfying
P>0;¸¸0;
2
4ATP+PA+¸CT
qCqPBp+¸CT
qDqp
(PBp+¸CT
qDqp)T¡¸(I¡DT
qpDqp)3
5<0:(5.12)
Thus,determining quadratic stabilityofanNLDIisanLMIP.SinceLMI(5.12)implies
that¸>0,wemay,byde¯ning ~P=P=¸,obtainanequivalentcondition
~P>0;2
4AT~P+~PA+CT
qCq~PBp+CT
qDqp
(~PBp+CT
qDqp)T¡(I¡DT
qpDqp)3
5<0; (5.13)
anLMIinthevariable~P.Thus,quadratic stabilityoftheNLDIhasthefrequency-
domaininterpretation thattheH1normoftheLTIsystem
_x=Ax+Bpp;q=Cqx+Dqpp
islessthanone.
WiththenewvariablesQ=P¡1,¹=1=¸,quadratic stabilityofNLDIsisalso
equivalenttotheexistence of¹andQsatisfying theLMI
¹¸0;Q>0;
"
AQ+QAT+¹BpBT
p¹BpDT
qp+QCT
q
(¹BpDT
qp+QCT
q)T¡¹(I¡DqpDT
qp)#
<0:(5.14)
Remark :Notethatourassumption ofwell-posedness isinfactincorporatedin
theLMI(5.12)sinceitimpliesI¡DqpDT
qp>0.Therefore theNLDI(5.2)is
well-posedandquadratically stableifandonlyiftheLMIP(5.12)hasasolution.
Intheremainder ofthischapterwewillassumethatDqpin(5.2)iszero;the
readershouldbearinmindthatallthefollowingresultsholdfornonzeroDqpaswell.
²Diagonal Norm-bound LDIs: FortheDNLDI(5.3),wecanobtainonlya
su±cientcondition forquadratic stability.TheconditiondV(x)=dt<0forallnonzero
trajectories isequivalentto
»T(ATP+PA)»+2»TPBp¼<0
forallnonzero(»;¼)satisfying
¼2
i·(Cq;i»+Dqp;i¼)T(Cq;i»+Dqp;i¼);i=1;:::;L;
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
64 Chapter 5AnalysisofLDIs: StateProperties
wherewehaveusedCq;iandDqp;itodenotetheithrowsofCqandDqprespec-
tively.UsingtheS-procedure,weseethatthiscondition isimpliedbytheexistence
ofnonnegativ e¸1;:::;¸nqsuchthat
»T(ATP+PA)»+2»TPBp¼
+LX
i=1¸i³
(Cq;i»+Dqp;i¼)T(Cq;i»+Dqp;i¼)¡¼2
i´
<0
forallnonzero(»;¼).With¤=diag(¸;:::;¸nq),thisisequivalentto
"
ATP+PA+CT
q¤CqPBp+CT
q¤Dqp
BT
pP+DT
qp¤CqDT
qp¤Dqp¡¤#
<0: (5.15)
Therefore, ifthereexistP>0anddiagonal ¤¸0satisfying (5.15),thenthe
DNLDI(5.3)isquadratically stable.Checkingthissu±cientcondition forquadratic
stabilityisanLMIP.Notethatfrom(5.15),wemusthave¤>0.
Equivalently,quadratic stabilityisimpliedbytheexistence ofQ>0,M=
diag(¹1;:::;¹nq)>0satisfying theLMI
"
AQ+QAT+BpMBT
pQCT
q+BpMDT
qp
CqQ+DqpMBT
p¡M+DqpMDT
qp#
<0; (5.16)
anotherLMIP.
Remark :Quadratic stabilityofaDNLDIviatheS-procedurehasasimple
frequency-domain interpretation. Denoting byHthetransfer matrixH(s)=
Dqp+Cq(sI¡A)¡1Bp,andassuming (A;Bp;Cq)isminimal, quadratic stabil-
ityisequivalenttothefactthat,forsomediagonal, positive-de¯nite matrix¤,
k¤1=2H¤¡1=2k1<1.¤¡1=2canthenbeinterpreted asascaling.(SeetheNotes
andReferences formoredetailsontheconnection betweentheS-procedureand
scaling.)
Remark :Heretooourassumption ofwell-posedness isinfactincorporated
intheLMI(5.15)or(5.16).Thebottomrightblockisprecisely thesu±cient
condition forwell-posedness mentionedonpage57.Therefore theDNLDI(5.2)
iswell-posedandquadratically stableiftheLMI(5.15)or(5.16)isfeasible.
NotethesimilaritybetweenthecorrespondingLMIsfortheNLDIandtheDNLDI;
theonlydi®erence isthatthediagonal matrixappearingintheLMIassociatedwith
theDNLDIis¯xedasthescaledidentitymatrixintheLMIassociatedwiththeNLDI.
Ingeneralitisstraightforwardtoderivethecorresponding(su±cien t)condition for
DNLDIs fromtheresultforNLDIs.Intheremainder ofthebook,wewilloftenrestrict
ourattentiontoNLDIs,andpayonlyoccasional attentiontoDNLDIs; weleaveto
thereaderthetaskofgeneralizing alltheresultsforNLDIstoDNLDIs.
Remark :AnLTIsystemisstableifandonlyifitisquadratically stable.For
moregeneralLDIs,however,stabilitydoesnotimplyquadratic stability(seethe
NotesandReferences attheendofthischapter). ItistruethatanLDIis
stableifandonlyifthereexistsaconvexLyapunovfunction thatprovesit;see
theNotesandReferences.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
5.1Quadra ticStability 65
5.1.1Coordinatetransformations
Wecangiveanotherinterpretation ofquadratic stabilityintermsofstate-space coor-
dinatetransformations. Consider thechangeofcoordinatesx=T¹x,wheredetT6=0.
Inthenewcoordinate system,(5.1)isdescribedbythematrix¹A(t)=T¡1A(t)T.
Weaskthequestion: DoesthereexistTsuchthatinthenewcoordinate system,all
(nonzero) trajectories oftheLDIarealwaysdecreasing innorm,i.e.,dk¹xk=dt<0?
Itiseasytoshowthatthisistrueifandonlyifthereexistsaquadratic Lyapunov
functionV(»)=»TP»fortheLDI,inwhichcasewecantakeanyTwithTTT=P¡1,
forexample,T=P¡1=2(theseT'sareallrelatedbyrightorthogonal matrixmulti-
plication). Withthisinterpretation, itisnaturaltoseekacoordinate transformation
matrixTthatmakesallnonzero trajectories decreasing innorm,andhasthesmall-
estpossiblecondition number·(T).ThisturnsouttobeanEVPforLTIsystems,
polytopic andnorm-boundLDIs.
Toseethis,letP=T¡TT¡1,sothat·(T)2=·(P).Minimizing thecondition
numberofTsubjecttotherequiremen tthatdk¹xk=dt<0forallnonzerotrajectories is
thenequivalenttominimizing thecondition numberofPsubjecttod(xTPx)=dt<0.
(AnyTwithTTT=P¡1
optwherePoptisanoptimalsolution, isoptimalfortheoriginal
problem.) Foreachofoursystems (LTIsystems, PLDIsandNLDIs), thechangeof
coordinates withsmallest condition numberisobtained bysolvingthefollowingEVPs:
²LTIsystems: TheEVPinthevariablesPand´is
minimize ´
subjectto(5:6);I·P·´I
(Inthisformulation,wehavetakenadvantageofthehomogeneit yoftheLMI(5.6)in
thevariableP.)
²Polytopic LDIs:TheEVPinthevariablesPand´is
minimize ´
subjectto(5:8);I·P·´I
²Norm-bound LDIs:TheEVPinthevariablesP,´and¸is
minimize ´
subjectto(5:12);I·P·´I
5.1.2Quadratic stabilitymargins
Quadratic stabilitymargins giveameasure ofhowmuchthesetcanbeexpanded
aboutsomecenterwiththeLDIremaining quadratically stable.
²Polytopic LDIs:ForthePLDI
_x=(A0+A(t))x;A(t)2®CofA1;:::;ALg;
wede¯nethequadraticstabilitymarginasthelargestnonnegativ e®forwhichitis
quadratically stable.Thisquantityiscomputed bysolvingthefollowingGEVPinP
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
66 Chapter 5AnalysisofLDIs: StateProperties
and®:
maximize ®
subjecttoP>0;®¸0;
AT
0P+PA0+®¡
AT
iP+PAi¢
<0;i=1;:::;L
²Norm-bound LDIs:Thequadratic stabilitymarginofthesystem
_x=Ax+Bpp;pTp·®2xTCT
qCqx;
isde¯nedasthelargest®¸0forwhichthesystemisquadratically stable,andis
computed bysolvingtheGEVPinP,¸and¯=®2:
maximize ¯
subjecttoP>0;¯¸0;"
ATP+PA+¯¸CT
qCqPBp
BT
pP¡¸I#
<0
De¯ning ~P=P=¸,wegetanequivalentEVPin~Pand¯:
maximize ¯
subjectto~P>0;¯¸0;"
AT~P+~PA+¯CT
qCq~PBp
BT
p~P¡I#
<0
Remark :Thequadratic stabilitymarginobtained bysolvingthisEVPisjust
1=kCq(sI¡A)¡1Bpk1.
5.1.3Decayrate
Thedecayrate(orlargestLyapunovexponent)oftheLDI(5.1)isde¯nedtobethe
largest®suchthat
lim
t!1e®tkx(t)k=0
holdsforalltrajectoriesx.Equivalently,thedecayrateisthesupremumof
liminf
t!1¡logkx(t)k
t
overallnonzerotrajectories. (Stabilitycorrespondstopositivedecayrate.)
Wecanusethequadratic LyapunovfunctionV(»)=»TP»toestablish alower
boundonthedecayrateoftheLDI(5.1).IfdV(x)=dt·¡2®V(x)foralltrajectories,
thenV(x(t))·V(x(0))e¡2®t,sothatkx(t)k·e¡®t·(P)1=2kx(0)kforalltrajectories,
andtherefore thedecayrateoftheLDI(5.1)isatleast®.
²LTIsystems: Thecondition thatdV(x)=dt·¡2®V(x)foralltrajectories is
equivalentto
ATP+PA+2®P·0: (5.17)
Therefore, thelargestlowerboundonthedecayratethatwecan¯ndusingaquadratic
Lyapunovfunction canbefoundbysolvingthefollowingGEVPinPand®:
maximize ®
subjecttoP>0;(5:17)(5.18)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
5.1Quadra ticStability 67
Thislowerboundissharp,i.e.,theoptimalvalueoftheGEVP(5.18)isthedecayrate
oftheLTIsystem(whichisthestabilitydegreeofA,i.e.,negativeofthemaximum
realpartoftheeigenvaluesofA).
Asanalternate condition, thereexistsaquadratic Lyapunovfunction proving
thatthedecayrateisatleast®ifandonlyifthereexistsQsatisfying theLMI
Q>0;AQ+QAT+2®Q·0: (5.19)
²Polytopic LDIs:Thecondition thatdV(x)=dt·¡2®V(x)foralltrajectories is
equivalenttotheLMI
AT
iP+PAi+2®P·0;i=1;:::;L: (5.20)
Therefore, thelargestlowerboundonthedecayrateprovableviaquadratic Lyapunov
functions isobtained bymaximizing ®subjectto(5.20)andP>0.ThisisaGEVP
inPand®.
Asanalternate condition, thereexistsaquadratic Lyapunovfunction proving
thatthedecayrateisatleast®ifandonlyifthereexistsQsatisfying theLMI
Q>0;AiQ+QAT
i+2®Q·0;i=1;:::;L: (5.21)
²Norm-bound LDIs: Applying theS-procedure,thecondition thatdV(x)=dt·
¡2®V(x)foralltrajectories isequivalenttotheexistence of¸¸0suchthat
"
ATP+PA+¸CT
qCq+2®PPBp
BT
pP ¡¸I#
·0: (5.22)
Therefore, weobtainthelargestlowerboundonthedecayrateof(5.2)bymaximizing
®overthevariables®,Pand¸,subjecttoP>0,¸¸0and(5.22),aGEVP.
Thelowerboundhasasimplefrequency-domain interpretation: De¯nethe®-
shiftedH1normofthesystemby
kHk1;®¢=supfkH(s)kjRes>¡®g:
ThentheoptimalvalueoftheGEVPisequaltothelargest®suchthatkHk1;®<1.
Thiscanbeseenbyrewriting (5.22)as
"
(A+®I)TP+P(A+®I)+¸CT
qCqPBp
BT
pP ¡¸I#
·0;
andnotingthattheH1normofthesystem(A+®I;Bp;Cq)equalskHk1;®.
Analternate necessary andsu±cientcondition fortheexistence ofaquadratic
Lyapunovfunction provingthatthedecayrateof(5.2)isatleast®isthatthere
existsQand¹satisfying theLMI
Q>0;¹¸0;
"
AQ+QAT+¹BpBT
p+2®QQCT
q
CqQ ¡¹I#
·0:(5.23)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
68 Chapter 5AnalysisofLDIs: StateProperties
5.2InvariantEllipsoids
Quadratic stabilitycanalsobeinterpreted intermsofinvariant ellipsoids.ForQ>0,
letEdenotetheellipsoid centeredattheorigin
E=©
»2Rn¯¯»TQ¡1»·1ª
:
TheellipsoidEissaidtobeinvariantfortheLDI(5.1)ifforeverytrajectoryxofthe
LDI,x(t0)2Eimpliesx(t)2Eforallt¸t0.Itiseasilyshownthatthisisthecase
ifandonlyifQsatis¯es
QAT+AQ·0;forallA2;
orequivalently,
ATP+PA·0;forallA2; (5.24)
whereP=Q¡1.Thus,forLTIsystems, PLDIsandNLDIs,invarianceofEischarac-
terizedbyLMIsinQorP,itsinverse.
Remark :Condition (5.24)isjustthenonstrict versionofcondition (5.4).
²LTIsystems: ThecorrespondingLMIinPis
P>0;ATP+PA·0; (5.25)
andtheLMIinQis
Q>0;AQ+PAT·0; (5.26)
²Polytopic LDIs:ThecorrespondingLMIinPis
P>0;AT
iP+PAi·0;i=1;:::;L; (5.27)
andtheLMIinQis
Q>0;QAT
i+AiQ·0;i=1;:::;L: (5.28)
²Norm-bound LDIs: Applying theS-procedure,invarianceoftheellipsoidEis
equivalenttotheexistence of¸suchthat
P>0;¸¸0;
2
4ATP+PA+¸CT
qCqPBp+¸CT
qDqp
(PBp+¸CT
qDqp)T¡¸(I¡DT
qpDqp)3
5·0:(5.29)
InvarianceoftheellipsoidEisequivalenttotheexistence of¹suchthat
¹¸0;Q>0;and
"
AQ+QAT+¹BpBT
p¹BpDT
qp+QCT
q
(¹BpDT
qp+QCT
q)T¡¹(I¡DqpDT
qp)#
·0:(5.30)
Insummary ,thecondition thatEbeaninvariantellipsoid canbeexpressed in
eachcaseasanLMIinQoritsinverseP.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
5.2Invariant Ellipsoids 69
5.2.1Smallestinvariantellipsoidcontainingapolytope
Consider apolytopedescribedbyitsvertices,P=Cofv1;:::;vpg.TheellipsoidE
containsthepolytopePifandonlyif
vT
jQ¡1vj·1;j=1;:::;p:
Thiscondition maybeexpressed asanLMIinQ
"
1vT
j
vjQ#
¸0;j=1;:::;p; (5.31)
orasanLMIinP=Q¡1as
vT
jPvj·1;j=1;:::;p: (5.32)
Minimumvolume
ThevolumeofEis,uptoaconstantthatdependsonn,pdetQ.Wecanminimize
thisvolumebysolvingappropriate CPs.
²LTIsystems: TheCPinPis
minimize logdetP¡1
subjectto(5:25);(5:32)
²Polytopic LDIs:TheCPinPis
minimize logdetP¡1
subjectto(5:27);(5:32)(5.33)
²Norm-bound LDIs:TheCPinPand¸is
minimize logdetP¡1
subjectto(5:29);(5:32)
Minimumdiameter
Thediameter oftheellipsoidEis2p
¸max(Q).Wecanminimize thisquantityby
solvingappropriate EVPs.
²LTIsystems: TheEVPinthevariablesQand¸is
minimize ¸
subjectto(5:26);(5:31);Q·¸I
²Polytopic LDIs:TheEVPinQand¸is
minimize ¸
subjectto(5:28);(5:31);Q·¸I(5.34)
²Norm-bound LDIs:TheEVPinQ,¸and¹is
minimize ¸
subjectto(5:30);(5:31);Q·¸I
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
70 Chapter 5AnalysisofLDIs: StateProperties
Remark :Theseresultshavethefollowinguse.ThepolytopePrepresentsour
knowledgeofthestateattimet0,i.e.,x(t0)2P(thisknowledgemayre°ect
measuremen tsorpriorassumptions). AninvariantellipsoidEcontainingPthen
givesaboundonthestatefort¸t0,i.e.,wecanguaranteethatx(t)2Eforall
t¸t0.
5.2.2Largestinvariantellipsoidcontainedinapolytope
Wenowconsider apolytopedescribedbylinearinequalities:
P=©
»2Rn¯¯aT
k»·1;k=1;:::;qª
:
TheellipsoidEiscontainedinthepolytopePifandonlyif
max©
aT
k»j»2Eª
·1;k=1;:::;q:
Thisisequivalentto
aT
kQak·1;k=1;:::;q; (5.35)
whichisasetoflinearinequalities inQ.Themaximumvolumeofinvariantellipsoids
containedinPcanbefoundbysolvingCPs:
²LTIsystems: ForLTIsystems, theCPinthevariableQis
minimize logdetQ¡1
subjectto(5:26);(5:35)
(SincethevolumeofEis,uptoaconstant,pdetQ,minimizing logdetQ¡1will
maximize thevolumeofE).
²Polytopic LDIs:ForPLDIs,theCPinthevariableQis
minimize logdetQ¡1
subjectto(5:28);(5:35)
²Norm-bound LDIs:ForNLDIs,theCPinthevariablesQand¹is
minimize logdetQ¡1
subjectto(5:30);(5:35)
Wealsocan¯ndthemaximumminordiameter (thatis,thelengthoftheminor
axis)ofinvariantellipsoids containedinPbysolvingEVPs:
²LTIsystems: ForLTIsystems, theEVPinthevariablesQand¸is
maximize ¸
subjectto(5:26);(5:35);¸I·Q
²Polytopic LDIs:ForPLDIs,theEVPinthevariablesQand¸is
maximize ¸
subjectto(5:28);5:35);¸I·Q
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
5.2Invariant Ellipsoids 71
²Norm-bound LDIs:ForNLDIs,theEVPinthevariablesQ,¹and¸is
maximize ¸
subjectto(5:30);(5:35);¸I·Q
Remark :Theseresultscanbeusedasfollows.ThepolytopePrepresentsthe
allowable(orsafe)operatingregionforthesystem. Theellipsoids foundabove
canbeinterpreted asregionsofsafeinitialconditions, i.e.,initialconditions for
whichwecanguaranteethatthestatealwaysremainsinthesafeoperatingregion.
5.2.3Boundonreturntime
ThereturntimeofastableLDIforthepolytopePisde¯nedasthesmallestTsuch
thatifx(0)2P,thenx(t)2Pforallt¸T.Upperboundsonthereturntimecan
befoundbysolvingEVPs.
²LTIsystems: Letusconsider apositivedecayrate®.IfQ>0satis¯es
QAT+AQ+2®Q·0;
thenEisaninvariantellipsoid, andmoreoverx(0)2Eimpliesthatx(t)2e¡®tEfor
allt¸0.Therefore ifTissuchthat
e¡®TEµPµE;
wecanconclude thatifx(0)2P,thenx(t)2Pforallt¸T,sothatTisanupper
boundonthereturntime.Ifweusebothrepresentationsofthepolytope,
P=©
x2Rn¯¯aT
kx·1;k=1;:::;qª
=Cofv1;:::;vpg;
theconstrain te¡®TEµPisequivalenttotheLMIinQand°
aT
iQai·°;k=1;:::;q; (5.36)
where°=e2®T,andtheproblem of¯ndingthesmallest such°andtherefore the
smallestT(fora¯xed®)isanEVPinthevariables°andQ:
minimize °
subjectto(5:19);(5:31);(5:36)
²Polytopic LDIs:ForPLDIs,thesmallest boundonthereturntimeprovablevia
invariantellipsoids, withagivendecayrate®,isobtained bysolvingtheEVPinthe
variablesQand°
minimize °
subjectto(5:21);(5:31);(5:36)
²Norm-bound LDIs:ForNLDIs,thesmallest boundonthereturntimeprovable
viainvariantellipsoids, withagivendecayrate®,isobtained bysolvingtheEVPin
thevariablesQ,°and¹
minimize °
subjectto(5:23);(5:31);(5:36)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
72 Chapter 5AnalysisofLDIs: StateProperties
Notes andReferences
QuadraticstabilityforNLDIs
Lur'eandPostnikov[LP44]gaveoneoftheearlieststabilityanalyses forNLDIs: They
considered thestabilityofthesystem
_x=Ax+bpp;q=cqx;p=Á(q;t)q; (5.37)
wherepandqarescalar.Thisproblem cametobepopularly knownastheproblem of
\absolute stabilityinautomatic control".IntheoriginalsettingofLur'eandPostnikov,Áwas
assumed tobeatime-invariantnonlinearit y.Subsequen tly,variousadditional assumptions
weremadeaboutÁ,generating di®erentspecialcasesandsolutions. (Thefamilyofsystemsof
theform(5.37),whereÁ(q;t)canbeanysector[¡1;1]nonlinearit y,isanNLDI.)Pyatnitskii,
in[Pya68],pointsoutthatby1968,over200papershadbeenwrittenaboutthesystem(5.37).
Amongthese,theonesmostrelevanttothisbookareundoubtedly fromYakubovich.As
farasweknow,Yakubovichwasthe¯rsttomakesystematic useofLMIsalongwiththe
S-proceduretoprovestabilityofnonlinear controlsystems(seereferences [Yak62,Yak64,
Yak67,Yak66,Yak77,Yak82]).ThemainideaofYakubovichwastoexpressthere-
sultingLMIsasfrequency-domain criteriaforthetransferfunctionG(s)=cq(sI¡A)¡1bp.
(Yakubovichcallsthismethodthe\methodofmatrixinequalities".) Thesecriteriaaremost
usefulwhendealingwithexperimentaldataarisingfromfrequency responsemeasuremen ts;
theyaredescribedindetailinthebookbyNarendra andTaylor[NT73].Seealso[Bar70b].
Popov[Pop73]andWillems [Wil71b,Wil74a]outlinetherelationship betweentheproblem
ofabsolute stabilityofautomatic controlandquadratic optimalcontrol.
Thecasewhenpandqarevector-valuedsignalshasbeenconsidered muchmorerecently;
seeforinstance, [ZK88,ZK87,KR88,BH89].In[KPZ90,PD90],theauthorsremark
thatquadratic stabilityofNLDIsisequivalenttotheH1condition (5.13).However,as
statedin[PD90],thereisnofundamen taldi®erence betweentheseandtheolderresults
ofYakubovichandPopov;thenewerresultscanberegarded asextensions oftheworks
ofYakubovichandPopovtofeedbacksynthesisandtothecaseof\structured perturba-
tions"[RCDP93,PZP+92].
WhilealltheresultspresentedinthischapterarebasedonLyapunovstabilitytechniques,
theyhaveverystrongconnections withtheinput-output approachtostudythestability
ofnonlinear, uncertain systems. ThisapproachwassparkedbyPopov[Pop62],whoused
ittostudytheLur'esystem. Othermajorcontributors includeZames[Zam66a,Zam66b,
ZF67],Sandberg[San64,San65a,San65b]andBrockett[Bro65,Bro66,BW65].See
alsothepapersandthebookbyWillems [Wil69a,Wil69b,Wil71a].Theadvantageof
theinput-output approachisthatthesystems arenotrequired tohave¯nite-dimensional
state[Des65,DV75,Log90];thecorrespondingstabilitycriteriaareusuallymosteasily
expressed inthefrequency domain, andresultinin¯nite-dimensional convexoptimization
problems. ModernapproachesfromthisviewpointincludeDoyle's¹-analysis [Doy82]and
Safonov'sKm-analysis [Saf82,SD83,SD84],implemen tedintherobustness analysis soft-
warepackages¹-toolsandKm-tools[BDG+91,CS92a],whichapproximately solvethese
in¯nite-dimensional problems. Seealso[AS79,SS81].
QuadraticstabilityforPLDIs
ThistopicismuchmorerecentthanNLDIs,thelikelyreasonbeingthattheLMIexpressing
quadratic stabilityofageneralPLDI(whichisjustanumberofsimultaneous Lyapunov
inequalities (5.8))cannotbeconvertedtoaRiccatiinequalit yorafrequency-domain criterion.
Oneofthe¯rstoccurrences ofPLDIsisin[HB76],whereHorisbergerandBelanger note
thattheproblem ofquadratic stabilityforaPLDIisconvex(theauthorswritedownthe
correspondingLMIs).Forotherresultsandcomputational proceduresforPLDIssee[PS82,
PS86,BY89,KP87a,Kam83,EZKN90,KB93,AGG94];thearticle[KP87a],wherethe
authorsdevelopasubgradien tmethodforprovingquadratic stabilityofaPLDI,whichis
furtherre¯nedin[Mei91];thepaper[PS86],wherethediscrete-time counterpartofthis
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 73
problem isconsidered. SeealsothearticlesbyGuetal.[GZL90,GL93b,Gu94],Garofalo
etal.[GCG93],andthesurveyarticlebyBarmish etal.[BK93].
Necessaryandsu±cientconditionsforLDIstability
Muchattentionhasalsofocusedonnecessary andsu±cientconditions forstabilityofLDIs
(asopposedtoquadratic stability);seeforexample [PR91b]or[MP86].Earlierreferences on
thistopicinclude[Pya70b]whichconnects theproblem ofstabilityofanLDItoanoptimal
controlproblem, and[Pya70a,BT79,BT80],wherethediscrete-time counterpartofthis
problem isconsidered.
VectorLyapunovfunctions
ThetechniqueofvectorLyapunovfunctions alsoyieldssearchproblems thatcanbeexpressed
asLMIPs. Oneexample isgivenin[CFA90].Herewealreadyhavequadratic Lyapunov
functions foranumberofsubsystems; theproblem isto¯ndanappropriate positivelinear
combination thatproves,e.g.,stabilityofaninterconnected system.
Finally,letuspointoutthatquadratic Lyapunovfunctions havebeenusedtodetermine
estimates ofregionsofstabilityforgeneralnonlinear systems. Seeforexample [Gha94,
BD85].
StableLDIsthatarenotquadraticallystable
AnLTIsystemisstableifandonlyifitisquadratically stable;thisisjustLyapunov's
stabilitytheorem forlinearsystems(seee.g.,[Lya47,p277]).Itispossible,however,foran
LDItobestablewithoutbeingquadratically stable.HereisaPLDIexample:
_x=A(t)x;A(t)2CofA1;A2g;
A1="
¡1000
0¡1#
;A2="
8¡9
120¡18#
:
From(2.8),thisPLDIisnotquadratically stableifthereexistQ0¸0,Q1¸0andQ2¸0,
notallzero,suchthat
Q0=A1Q1+Q1AT
1+A2Q2+Q2AT
2:
Itcanbeveri¯edthatthematrices
Q0="
5:22
224#
;Q1="
0:13
390#
;Q2="
2:71
11#
satisfythesedualityconditions.
However,thepiecewise quadratic Lyapunovfunction
V(x)=max©
xTP1x;xTP2xª
;
P1="
14¡1
¡11#
;P2="
00
01#
;(5.38)
provesthatthePLDIisstable.Toshowthis,weusetheS-procedure. Anecessary and
su±cientcondition fortheLyapunovfunctionVde¯nedin(5.38)toprovethestabilityof
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
74 Chapter 5AnalysisofLDIs: StateProperties
thePLDIistheexistence offournonnegativ enumbers¸1,¸2,¸3,¸4suchthat
AT
1P1+P1A1¡¸1(P2¡P1)<0;
AT
2P1+P1A2¡¸2(P2¡P1)<0;
AT
1P2+P2A1+¸3(P2¡P1)<0;AT
2P2+P2A2+¸4(P2¡P1)<0:(5.39)
Itcanbeveri¯edthat¸1=50,¸2=0,¸3=1,¸4=100aresuchnumbers.
Forotherexamples ofstableLDIsthatarenotquadratically stable,seeBrockett[Bro65,
Bro77,Bro70]andPyatnitskii [Pya71].In[Bro77],BrockettusestheS-procedureto
provethatacertainpiecewise quadratic formisaLyapunovfunction and¯ndsLMIssimilar
to(5.39);seealso[Vid78].
Finally,anLDIisstableifandonlyifthereisaconvexLyapunovfunction thatprovesit.
OnesuchLyapunovfunction is
V(»)=supfkx(t)kjxsatis¯es(5.1);x(0)=»;t¸0g:
SeeBraytonandTong[BT79,BT80],andalso[Zub55,MV63,DK71,VV85,PR91b,
PR91a,Rap90,Rap93,MP89,Mol87].Ingeneral,computing suchLyapunovfunctions is
computationally intensive,ifnotintractable.
Nonlinearsystemsandfadingmemory
Consider thenonlinear system
_x=f(x;w;t); (5.40)
with
@f
@x2forallx;w;t
whereisconvex.SupposetheLDI_x2xisstable,i.e.,alltrajectories convergetozeroas
t!1.Thisimpliesthesystemhasfadingmemory,thatis,for¯xedinputw,thedi®erence
betweenanytwotrajectories of(5.40)convergestozero.Inotherwords,thesystem\forgets"
itsinitialcondition. (See[BC85].)
Fixaninputwandletxand~xdenoteanytwosolutions of(5.40).Usingthemean-value
theorem fromx4.3.1,wehave
d
dt(x¡~x)=f(x;w;t)¡f(~x;w;t)=A(t)(x¡~x)
forsomeA(t)2.SincetheLDIisstable,x(t)¡~x(t)convergestozeroast!1.
RelationbetweenS-procedureforDNLDIsandscaling
Wediscusstheinterpretation ofthediagonal matrix¤in(5.15)asascalingmatrix.
EverytrajectoryoftheDNLDI(5.3)isalsoatrajectory(andviceversa)oftheDNLDI
_x=Ax+BpT¡1p;
q=TCqx+TDqpT¡1p;
pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq(5.41)
foranydiagonal nonsingular T.Therefore, ifDNLDI(5.41)isstableforsomediagonal
nonsingular T,thensoisDNLDI(5.3).Tisreferredtoasascalingmatrix.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 75
TreatingDNLDI(5.41)asanNLDI,andapplying thecondition (5.13)forquadratic stability,
werequire,forsomediagonal nonsingular T,
P>0;"
ATP+PA+CT
qT2CqPBpT¡1+CT
qT2DqpT¡1
(PBpT¡1+CT
qT2DqpT¡1)T¡(I¡T¡1DT
qpT2DqpT¡1)#
<0;
Anobviouscongruence, followedbythesubstitution T2=¤,yieldstheLMIcondition in
P>0anddiagonal ¤>0:
"
ATP+PA+CT
q¤CqPBp+Cq¤Dqp
BT
PP+DT
qp¤CqDT
qp¤Dq¡¤#
<0:
Thisisprecisely LMI(5.15),thecondition forstabilityoftheDNLDI(5.3)obtained using
theS-procedure.Thisrelationbetweenscalingandthediagonal matrices arisingfromthe
S-procedureisdescribedinBoydandYang[BY89].
Pyatnitskii andSkorodinskii[PS82,PS83],andKamenetskii [Kam83]reducetheproblem of
numerical searchforappropriate scalingsandtheassociatedLyapunovfunctions toaconvex
optimization problem. SaekiandAraki[SA88]alsoconclude convexityofthescalingproblem
andusethepropertiesofM-matrices toobtainasolution.
RepresentingaDNLDIasaPLDI:Implicationsforquadraticstability
WesawinChapter 4thataDNLDIcanberepresentedasaPLDI,wherethenumberLof
verticescanbeaslargeas2nq.Fromtheresultsofx5.1,checkingquadratic stabilityforthis
PLDIrequiresthesolution ofLsimultaneous Lyapunovinequalities inP>0(LMI(5.8)).If
thesystemisrepresentedasaDNLDI, asu±cientcondition forquadratic stabilityisgiven
bytheLMI(5.15),inthevariablesP>0and¤>0.
Comparing thetwoconditions forquadratic stability,weobservethatwhilethequadratic
stabilitycondition (5.8)forLDI(5.3),obtained byrepresentingitasaPLDI,isbothnecessary
andsu±cient(i.e.,notconservative),thesizeoftheLMIgrowsexponentiallywithnq,the
sizeof¢.Ontheotherhand,thesizeoftheLMI(5.15)growspolynomial lywithnq,but
thisLMIyieldsonlyasu±cientcondition forquadratic stabilityofLDI(5.3).Discussion of
thisissuecanbefoundinKamenetskii [Kam83]andPyatnitskii andSkorodinskii[PS82].
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Chapter 6
AnalysisofLDIs:
Input/Output Properties
6.1Input-to-State Properties
We¯rstconsider input-to-state propertiesoftheLDI
_x=A(t)x+Bw(t)w;[A(t)Bw(t)]2; (6.1)
wherewisanexogenous inputsignal.
²LTIsystems: Thesystemhastheform_x=Ax+Bww.
²Polytopic LDIs:Herethedescription is
_x=A(t)x+Bw(t)w;[A(t)Bw(t)]2Cof[A1Bw;1];:::;[ALBw;L]g:
²Norm-bound LDIs:NLDIshavetheform
_x=Ax+Bpp+Bww;q=Cqx;p=¢(t)q;k¢(t)k·1;
orequivalently,
_x=Ax+Bpp+Bww;q=Cqx;pTp·qTq:
²Diagonal Norm-bound LDIs:DNLDIs havetheform
_x=Ax+Bpp+Bww;q=Cqx;
pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq;
orequivalently,
_x=Ax+Bpp+Bww;q=Cqx;jpij·jqij;i=1;:::;nq:
6.1.1Reachablesetswithunit-energy inputs
LetRuedenotethesetofreachablestateswithunit-energy inputsfortheLDI(6.1),
i.e.,
Rue¢=8
>><
>>:x(T)¯¯¯¯¯¯¯¯x,wsatisfy(6.1);x(0)=0
ZT
0wTwdt·1;T¸09
>>=
>>;:
77
78 Chapter 6AnalysisofLDIs: Input/Output Properties
WewillboundRuebyellipsoids oftheform
E=©
»j»TP»·1ª
; (6.2)
whereP>0.
SupposethatthefunctionV(»)=»TP»,withP>0,satis¯es
dV(x)=dt·wTwforallx;wsatisfying (6.1): (6.3)
Integrating bothsidesfrom0toT,weget
V(x(T))¡V(x(0))·ZT
0wTwdt:
NotingthatV(x(0))=0sincex(0)=0,weget
V(x(T))·ZT
0wTwdt·1;
foreveryT¸0,andeveryinputwsuchthatRT
0wTwdt·1.Inotherwords,the
ellipsoidEcontainsthereachablesetRue.
²LTIsystems: Condition (6.3)isequivalenttotheLMIinP
P>0;"
ATP+PAPBw
BT
wP¡I#
·0; (6.4)
whichcanalsobeexpressed asanLMIinQ=P¡1
Q>0;AQ+QAT+BwBT
w·0: (6.5)
Remark :Foracontrollable LTIsystem, thereachablesetistheellipsoid
f»j»TW¡1
c»·1g,whereWcisthecontrollabilityGramian,de¯nedby
Wc¢=Z1
0eAtBwBT
weATtdt:
SinceWcsatis¯estheLyapunovequation
AWc+WcAT+BwBT
w=0; (6.6)
weseethatQ=Wcsatis¯es(6.5).Thustheellipsoidal boundissharpforLTI
systems.
²Polytopic LDIs:ForPLDIs,condition (6.3)holdsifandonlyif
P>0and"
A(t)TP+PA(t)PBw(t)
Bw(t)TP¡I#
·0forallt¸0. (6.7)
Inequalit y(6.7)holdsifandonlyif
P>0;"
AT
iP+PAiPBw;i
BT
w;iP¡I#
·0;i=1;:::;L: (6.8)
ThustheellipsoidEcontainsRueiftheLMI(6.8)holds.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.1Input-to-St ateProperties 79
Alternativ ely,wecanwrite(6.8)as
Q>0;QAT
i+AiQ+Bw;iBT
w;i·0;i=1;:::;L; (6.9)
whereQ=P¡1.Notethatthiscondition impliesthatEisinvariant.
²Norm-bound LDIs:Condition (6.3)holdsifandonlyifP>0and
»T(ATP+PA)»+2»TP(Bp¼+Bw!)¡!T!·0
holdforevery!andforevery»and¼satisfying
¼T¼¡»TCT
qCq»·0:
UsingtheS-procedure,thisisequivalenttotheexistence ofPand¸satisfying
P>0;¸¸0;2
64ATP+PA+¸CT
qCqPBpPBw
BT
pP¡¸I0
BT
wP 0¡I3
75·0:(6.10)
Equivalently,de¯ningQ=P¡1,condition (6.3)holdsifandonlyifthereexistQand
¹satisfying
Q>0;¹¸0;"
AQ+QAT+BwBT
w+¹BpBT
pQCT
q
CqQ ¡¹I#
·0:(6.11)
²Diagonal Norm-bound LDIs: Condition (6.3)holdsifandonlyifP>0and
forall»and¼thatsatisfy
¼2
i·(Cq;i»)T(Cq;i»);i=1;:::;nq;
wehaveforall!,
»T(ATP+PA)»+2»TP(Bp¼+Bw!)¡!T!·0:
ItfollowsfromtheS-procedurethatthiscondition holdsifthereexistPand¤=
diag(¸;:::;¸nq)satisfying
P>0;¤¸0;2
64ATP+PA+CT
q¤CqPBpPBw
BT
pP¡¤0
BT
wP 0¡I3
75·0:
Equivalently,de¯ningQ=P¡1,condition (6.3)holdsifthereexistQandadiagonal
matrixMsuchthat
Q>0;M¸0;"
AQ+QAT+BwBT
w+BpMBT
pQCT
q
CqQ ¡M#
·0:(6.12)
Fortheremainder ofthissectionweleavetheextension toDNLDIs tothereader.
Theresultsdescribedabovegiveusasetofellipsoidal outerapproximations of
Rue.Wecanoptimize overthissetinseveralways.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
80 Chapter 6AnalysisofLDIs: Input/Output Properties
Smallestouterapproximations
²LTIsystems: ForLTIsystems, wecanminimize thevolumeoverellipsoidsEwith
Psatisfying (6.3)byminimizing logdetP¡1overthevariablePsubjectto(6.4).This
isaCP.
If(A;Bw)isuncontrollable, thevolumeofthereachablesetiszero,andthisCP
willbeunboundedbelow.If(A;Bw)iscontrollable, thentheCP¯ndstheexact
reachableset,whichcorrespondstoP=W¡1
c.
²Polytopic LDIs: ForPLDIs,theminimumvolumeellipsoid oftheform(6.2)
withPsatisfying (6.3),isfoundbyminimizing logdetP¡1subjectto(6.8).Thisisa
CP.
²Norm-bound LDIs:ForNLDIs,theminimumvolumeellipsoid oftheform(6.2),
satisfying (6.3),isobtained byminimizing logdetP¡1overthevariablesPand¸
subjectto(6.10).ThisisagainaCP.
Wecanalsominimize thediameter ofellipsoids oftheform(6.2),satisfying (6.3),
byreplacing theobjectivefunction logdetP¡1intheCPsbytheobjectivefunction
¸max(P¡1).ThisyieldsEVPs.
Testingifapointisoutsidethereachableset
Thepointx0liesoutsidethereachablesetifthereexistsanellipsoidal outerapprox-
imationofRuethatdoesnotcontainit.ForLTIsystems,x0doesnotbelongtothe
reachablesetifthereexistsPsatisfying xT
0Px0>1and(6.4).ThisisanLMIP.Of
course,if(A;Bw)iscontrollable, thenx0doesnotbelongtothereachablesetifand
onlyifxT
0W¡1
cx0>1,whereWcisde¯nedbytheequation (6.6).
ForPLDIs,asu±cientcondition forapointx0tolieoutsidethereachableset
RuecanbecheckedviaanLMIPinthevariableP:xT
0Px0>1and(6.8).ForNLDIs,
x0liesoutsidethereachablesetRueifthereexistPand¸satisfying xT
0Px0>1
and(6.10),anotherLMIP.
Testingifthereachablesetliesoutsideahalf-space
ThereachablesetRueliesoutsidethehalf-spaceH=©
»2Rn¯¯aT»>1ª
ifandonly
ifanellipsoidEcontainingRueliesoutsideH,thatis,theminimumvalueof»TQ¡1»
over»satisfyingaT»>1exceedsone.Weeasilyseethat
min©
»TQ¡1»¯¯aT»>1ª
=1=(aTQa):
Therefore, forLTIsystems,Hdoesnotintersectthereachablesetifthereexists
QsatisfyingaTQa<1and(6.5).ThisisanLMIP.If(A;Bw)iscontrollable, thenH
doesnotintersectthereachablesetifandonlyifaTWca<1,whereWcisde¯nedby
theequation (6.6).
ForPLDIs,asu±cientcondition forHnottointersectthereachablesetRue
canbecheckedviaanLMIPinthevariableQ:aTQa<1and(6.9).ForNLDIs,H
doesnotintersectthereachablesetRueifthereexistsQand¹satisfying aTQa<1
and(6.11).ThisisanLMIP.
Testingifthereachablesetiscontainedinapolytope
Sinceapolytopecanbeexpressed asanintersection ofhalf-spaces (thesedetermine
its\faces"), wecanimmediately usetheresultsoftheprevious subsection inorderto
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.1Input-to-St ateProperties 81
checkifthereachablesetiscontainedinapolytope.(Noteweonlyobtainsu±cient
conditions.) Weremarkthatwecanusedi®erentellipsoidal outerapproximations to
checkdi®erentfaces.
6.1.2Reachablesetswithcomponentwiseunit-energy inputs
Wenowturntoavariationoftheprevious problem; weconsider thesetofreachable
stateswithinputswhosecomponentshaveunit-energy ,thatis,weconsider theset
Ruce¢=8
><
>:x(T)¯¯¯¯¯¯¯x,wsatisfy(6.1);x(0)=0;ZT
0w2
idt·1;i=1;:::;nw;T¸09
>=
>;:
Supposethereisapositive-de¯nite quadratic functionV(»)=»TP»,andR=
diag(r1;:::;rnw)suchthat
P>0;R¸0;TrR=1;
d
dtV(x)·wTRwforallxandwsatisfying (6.1):(6.13)
ThentheellipsoidEgivenby(6.2)containsthereachableset.Toprovethis,we
integratebothsidesofthelastinequalit yfrom0toT,toget
V(x(T))¡V(x(0))·ZT
0wTRwdt:
NotingthatV(x(0))=0sincex(0)=0,weget
V(x(T))·ZT
0wTRwdt·1:
Letusnowconsider condition (6.13)forthevariousLDIs:
²LTIsystems: ForLTIsystems, condition (6.13)isequivalenttotheLMIinP
andR:
P>0,R¸0anddiagonal, TrR=1,
"
ATP+PAPBw
BT
wP¡R#
·0:(6.14)
Therefore, ifthereexistPandRsuchthat(6.14)issatis¯ed, thentheellipsoidE
containsthereachablesetRuce.
Alternativ ely,usingthevariableQ=P¡1,wecanwriteanequivalentLMIinQ
andR:
Q>0,R¸0anddiagonal, TrR=1,
"
QAT+AQBw
BT
w¡R#
·0:(6.15)
Remark :ItturnsoutthatwithLTIsystems, thereachablesetisequaltothe
intersection ofallsuchellipsoids. (Formoredetailsonthis,seetheNotesand
References.)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
82 Chapter 6AnalysisofLDIs: Input/Output Properties
²Polytopic LDIs:ForPLDIs,condition (6.13)holdsifandonlyif
P>0,R¸0anddiagonal, TrR=1,
"
AT
iP+PAiPBw;i
BT
w;iP¡R#
·0;i=1;:::;L:(6.16)
Therefore, ifthereexistPandRsuchthat(6.16)holds,thentheellipsoidE=©
»j»TP»·1ª
containsRuce.
WithQ=P¡1,condition (6.13)isequivalentto
Q>0,R¸0anddiagonal, TrR=1,
"
QAT
i+AiQBw;i
BT
w;i¡R#
·0;i=1;:::;L:(6.17)
²Norm-bound LDIs:ForNLDIs,applying theS-procedure,condition (6.13)holds
ifandonlyifthereexistP,¸andRsuchthat
P>0,R¸0anddiagonal, TrR=1,¸¸0,
2
64ATP+PA+¸CT
qCqPBpPBw
BT
pP¡¸I0
BT
wP 0¡R3
75·0:(6.18)
Therefore, ifthereexistsP,¸andRsuchthat(6.18)issatis¯ed, thentheellipsoidE
containsthereachablesetRuce.
WithQ=P¡1,condition (6.13)isalsoequivalenttotheexistence ofQ,Rand¹
suchthat
Q>0,R¸0anddiagonal, TrR=1,¹¸0,
2
64AQ+QAT+¹BpBT
pQCT
qBw
CqQ¡¹I0
BT
w 0¡R3
75·0:(6.19)
Asintheprevious section,wecan¯ndtheminimumvolumeandminimumdi-
ameterellipsoids amongallellipsoids oftheform(6.2),satisfying condition (6.13),by
solvingCPsandEVPs,respectively.Wecanalsocheckthatapointdoesnotbelong
tothereachablesetorthatahalf-space doesnotintersectthereachableset,bysolving
appropriate LMIPs.
6.1.3Reachablesetswithunit-peakinputs
Weconsider reachablesetswithinputswthatsatisfywTw·1.Thus,weareinter-
estedintheset
Rup¢=(
x(T)¯¯¯¯¯x,wsatisfy(6.1);x(0)=0;
wTw·1;T¸0)
:
Supposethatthereexistsaquadratic functionV(»)=»TP»with
P>0;anddV(x)=dt·0forallx,w
satisfying (6.1),wTw·1andV(x)¸1:(6.20)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.1Input-to-St ateProperties 83
Then,theellipsoidEgivenby(6.2)containsthereachablesetRup.Letusconsider
nowcondition (6.20)forthevariousLDIs:
²LTIsystems: ForLTIsystems, condition (6.20)isequivalenttoP>0and
»T(ATP+PA)»+»TPBww+wTBT
wP»·0
forany»andwsatisfying
wTw·1and»TP»¸1:
UsingtheS-procedure,weconclude thatcondition (6.20)holdsifthereexist®¸0
and¯¸0suchthatforallxandw,
"
x
w#T"
ATP+PA+®PPBw
BT
wP¡¯I#"
x
w#
+¯¡®·0; (6.21)
orequivalently
2
64ATP+PA+®PPBw0
BT
wP¡¯I0
0 0¯¡®3
75·0: (6.22)
Clearlywemusthave®¸¯.Next,if(6.22)holdsforsome(®0;¯0),thenitholdsfor
all®0¸¯¸¯0.Therefore, wecanassumewithoutlossofgeneralit ythat¯=®,and
rewrite(6.21)as
"
ATP+PA+®PPBw
BT
wP¡®I#
·0: (6.23)
Therefore, ifthereexistsPand®satisfying
P>0;®¸0;and(6:23); (6.24)
thentheellipsoidEcontainsthereachablesetRup.Notethatinequalit y(6.23)isnot
anLMIin®andP;however,for¯xed®itisanLMIinP.
Alternativ ely,condition (6.23)isequivalenttothefollowinginequalit yinQ=P¡1
and®:"
QAT+AQ+®QBw
BT
w¡®I#
·0: (6.25)
²Polytopic LDIs:Condition (6.20)holdsifP>0andthereexists®¸0satisfying
"
AT
iP+PAi+®PPBw;i
BT
w;iP¡®I#
·0;i=1;:::;L: (6.26)
Therefore, ifthereexistsPand®satisfying
P>0;®¸0;and(6:26); (6.27)
thentheellipsoidEcontainsthereachablesetRup.Onceagain,wenotethatcondi-
tion(6.26)isnotanLMIin®andP,butisanLMIinPfor¯xed®.Wecanalso
rewritecondition (6.26)intermsofQ=P¡1astheequivalentinequalit y
"
QAT
i+AiQ+®QBw;i
BT
w;i¡®I#
·0;i=1;:::;L: (6.28)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
84 Chapter 6AnalysisofLDIs: Input/Output Properties
²Norm-bound LDIs:ForNLDIs,condition (6.20)holdsifandonlyifP>0and
»T(ATP+PA)»+2»TP(Bw!+Bp¼)·0
forevery!,»and¼satisfying
!T!·1;»TP»¸1;¼T¼¡»TCT
qCq»·0:
Therefore, usingtheS-procedure,asu±cientcondition for(6.20)toholdisthatthere
exist®¸0and¸¸0suchthat
2
64ATP+PA+®P+¸CT
qCqPBpPBw
BT
pP ¡¸I0
BT
wP 0¡®I3
75·0: (6.29)
IfthereexistsP,®and¸satisfying
P>0;®¸0;¸¸0;and(6:29); (6.30)
theellipsoidEP¡1containsthereachablesetRup.Wenoteagainthatinequalit y(6.29)
isnotanLMIinPand®,butisanLMIinPfor¯xed®.
De¯ningQ=P¡1,condition (6.20)isalsoimpliedbytheexistence of¹¸0and
®¸0suchthat
2
64QAT+AQ+®Q+¹BT
pBpQCT
qBw
CqQ ¡¹I0
BT
w 0¡®I3
75·0: (6.31)
(Infact,theexistence of¸¸0and®¸0satisfying (6.29)isequivalenttotheexistence
of¹¸0and®¸0satisfying (6.31).)
For¯xed®,wecanminimize thevolumeordiameter amongallellipsoidsEsatis-
fyingcondition (6.24)forLTIsystems, condition (6.27)forPLDIsandcondition (6.30)
forNLDIsbyformingtheappropriate CPs,orEVPs,respectively.
Remark :Inequalit y(6.23)canalsobederivedbycombiningtheresultsofx6.1.1
withanexponentialtime-weighting oftheinputw.SeetheNotesandReferences
fordetails.
Asinx6.1.2,itispossibletodetermine outerellipsoidal approximations ofthe
reachablesetofLTIsystems, PLDIsandNLDIs,subjecttocomponentwiseunit-
peakinputs.
6.2State-to-Output Properties
Wenowconsider state-to-output propertiesfortheLDI
_x=A(t)x;z=Cz(t)x (6.32)
where
"
A(t)
Cz(t)#
2: (6.33)
²LTIsystems: LTIsystemshavetheform_x=Ax,z=Czx.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.2State-to-Output Properties 85
²Polytopic LDIs:PLDIshavetheform_x=A(t)x,z=Cz(t)x,where
"
A(t)
Cz(t)#
2Co("
A1
C1#
;:::;"
AL
CL#)
:
²Norm-bound LDIs:NLDIshavetheform
_x=Ax+Bpp;q=Cqx;z=Czxp=¢(t)q;k¢(t)k·1;
orequivalently
_x=Ax+Bpp;q=Cqx;z=Czx;pTp·qTq:
²Diagonal Norm-bound LDIs:DNLDIs havetheform
_x=Ax+Bpp;q=Cqx;
z=Czx;pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq;
orequivalently
_x=Ax+Bpp;q=Cqx;
z=Czx;jpij·jqij;i=1;:::;nq:
6.2.1Boundsonoutputenergy
Weseekthemaximumoutputenergygivenacertaininitialstate,
max½Z1
0zTzdt¯¯¯¯_x=A(t)x;z=Cz(t)x¾
; (6.34)
wherex(0)isgiven,andthemaximumistakenoverA(t),Cz(t)suchthat(6.33)holds.
Supposethereexistsaquadratic functionV(»)=»TP»suchthat
P>0andd
dtV(x)·¡zTz;foreveryxandzsatisfying (6.32): (6.35)
Then,integrating bothsidesofthesecondinequalit yin(6.35)from0toT,weget
V(x(T))¡V(x(0))·¡ZT
0zTzdt:
foreveryT¸0.SinceV(x(T))¸0,weconclude thatV(x(0))=x(0)TPx(0)isan
upperboundonthemaximumenergyoftheoutputzgiventheinitialconditionx(0).
WenowderiveLMIsthatprovideupperboundsontheoutputenergyforthe
variousLDIs.
²LTIsystems: InthecaseofLTIsystems, condition (6.35)isequivalentto
P>0;ATP+PA+CT
zCz·0: (6.36)
Therefore, weobtainthebestupperboundontheoutputenergyprovableviaquadratic
functions bysolvingtheEVP
minimize x(0)TPx(0)
subjecttoP>0;(6:36)(6.37)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
86 Chapter 6AnalysisofLDIs: Input/Output Properties
Inthiscasethesolutioncanbefoundanalytically ,anditisexactlyequaltotheoutput
energy,whichisx(0)TWox(0),whereWoistheobservability Gramianofthesystem,
de¯nedby
Wo¢=Z1
0eATtCT
zCzeAtdt:
SinceWosatis¯estheLyapunovequation
ATWo+WoA+CT
zCz=0; (6.38)
itsatis¯es(6.35).
WithQ=P¡1,condition (6.35)isequivalentto
Q>0;"
AQ+QATQCT
z
CzQ¡I#
·0: (6.39)
IfQsatis¯es(6.39),thenanupperboundontheoutputenergy(6.34)isx(0)TQ¡1x(0).
²Polytopic LDIs:ForPLDIs,condition (6.35)isequivalentto
P>0;A(t)TP+PA(t)+Cz(t)TCz(t)·0forallt¸0. (6.40)
Inequalities (6.40)holdifandonlyifthefollowingLMIinthevariablePholds:
P>0;AT
iP+PAi+CT
z;iCz;i·0;i=1;:::;L: (6.41)
Therefore theEVPcorrespondingto¯ndingthebestupperboundontheoutput
energyprovableviaquadratic functions is
minimize x(0)TPx(0)
subjecttoP>0;(6:41)(6.42)
De¯ningQ=P¡1,condition (6.35)isalsoequivalenttotheLMIinQ
Q>0;"
AiQ+QAT
iQCT
z;i
Cz;iQ¡I#
·0:
²Norm-bound LDIs:ForNLDIs,condition (6.35)isequivalenttoP>0and
»T(ATP+PA+CT
zCz)»+2»TPBp¼·0
forevery»and¼thatsatisfy¼T¼·»TCT
qCq».UsingtheS-procedure,anequivalent
condition istheexistence ofPand¸suchthat
P>0;¸¸0;
"
ATP+PA+CT
zCz+¸CT
qCqPBp
BT
pP ¡¸I#
·0:(6.43)
Therefore weobtainthesmallest upperboundontheoutputenergy(6.34)provable
viaquadratic functions satisfying condition (6.35)bysolvingthefollowingEVPinthe
variablesPand¸
minimize x(0)TPx(0)
subjecttoP>0;¸¸0;(6:43)(6.44)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.2State-to-Output Properties 87
WithQ=P¡1,condition (6.35)isalsoequivalenttotheexistence of¹¸0satisfying
Q>0;2
64AQ+QAT+¹BpBT
pQCT
zQCT
q
CzQ ¡I0
CqQ 0¡¹I3
75·0:
²Diagonal Norm-bound LDIs:ForDNLDIs, condition (6.35)isequivalentto
»T(ATP+PA+CT
zCz)»+2»TPBp¼·0;
forevery»and¼thatsatisfy
¼2
i·(Cq;i»)T(Cq;i»);i=1;:::;nq:
UsingtheS-procedure,asu±cientcondition isthatthereexistsadiagonal ¤¸0
suchthat
"
ATP+PA+CT
zCz+CT
q¤CqPBp
BT
pP ¡¤#
·0: (6.45)
Therefore weobtainthesmallest upperboundontheoutputenergy(6.34)provable
viaquadratic functions satisfying condition (6.35)andtheS-procedurebysolvingthe
followingEVPinthevariablesPand¤:
minimize x(0)TPx(0)
subjecttoP>0;¤¸0anddiagonal;(6:45)
WithQ=P¡1,condition (6.35)isalsoequivalenttotheexistence ofadiagonal matrix
M¸0satisfying
2
64AQ+QAT+BpMBT
pQCT
zQCT
q
CzQ ¡I0
CqQ 0¡M3
75·0:
Onceagain,weleaveittothereadertoextendtheresultsintheremainder ofthe
sectiontoDNLDIs.
Maximumoutputenergyextractablefromaset
Asanextension, weseekboundsonthemaximumextractable energyfromx(0),which
isknownonlytolieinapolytopeP=Cofv1;:::;vpg.
We¯rstnotethatsinceP>0,x(0)TPx(0)takesitsmaximumvalueononeof
theverticesv1;:::;vp.Therefore, thesmallest upperboundontheextractable energy
fromPprovableviaquadratic Lyapunovfunctions foranyofourLDIsiscomputed
bysolving(6.37),(6.42)or(6.44)ptimes,successiv elysettingx(0)=vi,i=1;:::;p.
Themaximumofthepresulting optimalvaluesisthesmallest upperboundsought.
Asavariation, supposethatx(0)isonlyknowntobelongtotheellipsoidE=
f»j»TX»·1g.Then,thesmallest upperboundonthemaximumoutputenergy
extractable fromEprovableviaquadratic functions satisfying (6.35)isobtained by
replacing theobjectivefunctionx(0)TPx(0)intheEVPs(6.37),(6.42),and(6.44)by
theobjectivefunction¸max(X¡1=2PX¡1=2),andsolvingthecorrespondingEVPs.For
LTIsystems, theexactvalueofthemaximumextractable outputenergyisobtained.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
88 Chapter 6AnalysisofLDIs: Input/Output Properties
Finally,ifx0isarandomvariablewithEx0xT
0=X0,thenweobtainthesmall-
estupperboundontheexpectedoutputenergyprovableviaquadratic functions by
replacing theobjectivefunctionx(0)TPx(0)intheEVPs(6.37),(6.42),and(6.44)by
theobjectivefunction TrX0PandsolvingthecorrespondingEVPs.
6.2.2Boundsonoutputpeak
Itispossibletoderiveboundsonkz(t)kusinginvariantellipsoids.
Assume¯rstthattheinitialconditionx(0)isknown.SupposeE=f»j»TP»·1g
isaninvariantellipsoid containingx(0)fortheLDI(6.32).Then
z(t)Tz(t)·max
»2E»TCz(t)TCz(t)»
forallt¸0.Wecanexpressmax»2E»TCz(t)TCz(t)»asthesquarerootofthe
minimumof±subjectto
"
PCT
z
Cz±I#
¸0: (6.46)
²LTIsystems: Thesmallest boundontheoutputpeakthatcanbeobtained via
invariantellipsoids isthesquarerootoftheoptimalvalueoftheEVPinthevariables
Pand±
minimize ±
subjecttoP>0;x(0)TPx(0)·1;
(6:46);ATP+PA·0(6.47)
²Polytopic LDIs:ForPLDIs,weobtainthesmallest upperboundonkzkprovable
viainvariantellipsoids bytakingthesquarerootoftheoptimalvalueofthefollowing
EVPinthevariablesPand±
minimize ±
subjecttoP>0;x(0)TPx(0)·1;(6:46);
AT
iP+PAi·0;i=1;:::;L
²Norm-bound LDIs: ForNLDIs,weobtainthesmallest upperboundonkzk
provableviainvariantellipsoids bytakingthesquarerootoftheoptimalvalueofthe
followingEVPinthevariablesP,¸and±
minimize ±
subjecttoP>0;x(0)TPx(0)·1;¸¸0;(6:46);
2
4ATP+PA+¸CT
qCqPBp
BT
pP¡¸I3
5·0
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.3Input-to-Output Properties 89
Remark :Asvariationofthisproblem, wecanimposeadecayrateconstrain ton
theoutput,thatis,given®>0,wecancompute anupperboundonthesmallest
°suchthattheoutputzsatis¯es
kz(t)k·°e¡®t;
forallt¸0.ThisalsoreducestoanEVP.
6.3Input-to-Output Properties
We¯nallyconsider theinput-output behavioroftheLDI
_x=A(t)x+Bw(t)w;x(0)=x0;
z=Cz(t)x+Dzw(t)w;(6.48)
where
"
A(t)Bw(t)
Cz(t)Dzw(t)#
2: (6.49)
6.3.1Hankelnormbounds
Inthissection,weassumethatDzw(t)=0forallt,andconsider thequantity
Á¢=max8
<
:Z1
TzTzdt¯¯¯¯¯¯RT
0wTwdt·1;x(0)=0;
w(t)=0;t>T¸09
=
;;
andthemaximumistakenoverA(t),B(t)andC(t)satisfying (6.49).ForanLTI
system,pÁequalstheHankelnorm,anamethatweextendforconveniencetoLDIs.
AnupperboundforÁcanbecomputed bycombiningtheellipsoidal boundson
reachablesetsfromx6.1.1andtheboundsontheoutputenergyfromx6.2.1.Suppose
thatP>0andQ>0satisfy
d
dt¡
xTPx¢
·wTw;d
dt¡
xTQx¢
·¡zTz (6.50)
forallx,wandzsatisfying (6.48).Then,fromtheargumen tsinx6.1.1andx6.2.1,we
conclude that
Á·sup©R1
TzTzdt¯¯x(T)TPx(T)·1ª
·sup©
x(T)TQx(T)¯¯x(T)TPx(T)·1ª
=¸max(P¡1=2QP¡1=2):
²LTIsystems: Wecompute thesmallest upperboundontheHankelnormprovable
viaquadratic functions satisfying (6.50)bycomputing thesquarerootoftheminimum
°suchthatthereexistP>0andQ>0satisfying
ATP+PA+PBwBT
wP·0;
ATQ+QA+CT
zCz·0;
°P¡Q¸0:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
90 Chapter 6AnalysisofLDIs: Input/Output Properties
ThisisaGEVPinthevariables°,P,andQ.Itispossibletotransform itintoanEVP
byintroducingthenewvariable~Q=Q=°.ThecorrespondingEVPinthevariables
°,P,and~Qisthentominimize°subjectto
ATP+PA+PBwBT
wP·0;
AT~Q+~QA+CT
zCz=°·0;
P¡~Q¸0
Inthiscase,theoptimalvalueisexactlytheHankelnorm.Itcanbefoundanalytically
as¸max(WcWo)whereWcandWoarethecontrollabilit yandobservabilityGramians,
i.e.,thesolutions oftheLyapunovequations (6.6)and(6.38),respectively.
²Polytopic LDIs: Wecompute thesmallest upperboundontheHankelnorm
provableviaquadratic functions satisfying (6.50)bycomputing thesquarerootofthe
minimum°suchthatthereexistP>0andQ>0satisfying
AT
iP+PAi+PBw;iBT
w;iP·0;
AT
iQ+QAi+CT
z;iCz;i·0;i=1;:::;L
°P¡Q¸0:
ThisisaGEVPinthevariables°,P,andQ.De¯ning ~Q=Q=°,anequivalentEVP
inthevariables°,P,and~Qistominimize°subjectto
AT
iP+PAi+PBw;iBT
w;iP·0;
AT
i~Q+~QAi+CT
z;iCz;i=°·0;i=1;:::;L
P¡~Q¸0
²Norm-bound LDIs: Similarly ,thesmallest upperboundontheHankelnorm
provableviaquadratic functions satisfying (6.50)iscomputed bytakingthesquare
rootoftheminimum°suchthatthereexistP>0,Q>0,¸¸0,and¹¸0satisfying
2
64ATP+PA+¸CT
qCqPBpPBw
BT
pP¡¸I0
BT
wP 0¡I3
75·0;
"
ATQ+QA+CT
zCz+¹CT
qCqQBp
BT
pQ ¡¹I#
·0;
°P¡Q¸0:
ThisisaGEVPinthevariables°,P,Q,¸,and¹.Ifweintroducethenewvariables
~Q=Q=°,º=¹=°andequivalentEVPinthevariables°,P,~Q,¸,andºisto
minimize°suchthat
2
64ATP+PA+¸CT
qCqPBpPBw
BT
pP¡¸I0
BT
wP 0¡I3
75·0;
"
AT~Q+~QA+CT
zCz=°+ºCT
qCq~QBp
BT
p~Q ¡ºI#
·0;
P¡~Q¸0:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.3Input-to-Output Properties 91
6.3.2 L2andRMSgains
WeassumeDzw(t)=0forsimplicityofexposition. Wede¯netheL2gainofthe
LDI(6.48)asthequantity
sup
kwk26=0kzk2
kwk2
wheretheL2normofuiskuk2
2=R1
0uTudt,andthesupremumistakenover
allnonzero trajectories oftheLDI,starting fromx(0)=0.TheLDIissaidtobe
nonexpansiveifitsL2gainislessthanone.
Now,supposethereexistsaquadratic functionV(»)=»TP»,P>0,and°¸0
suchthatforallt,
d
dtV(x)+zTz¡°2wTw·0forallxandwsatisfying (6.48): (6.51)
ThentheL2gainoftheLDIislessthan°.Toshowthis,weintegrate(6.51)from0
toT,withtheinitialconditionx(0)=0,toget
V(x(T))+ZT
0¡
zTz¡°2wTw¢
dt·0:
sinceV(x(T))¸0,thisimplies
kzk2
kwk2·°:
Remark :Itcanbeeasilycheckedthatif(6.51)holdsforV(»)=»TP»,P>0,
then°isalsoanupperboundontheRMSgainoftheLDI,wheretheroot-mean-
square(RMS)valueof»isde¯nedas
RMS(»)¢=µ
limsup
T!11
TZT
0»T»dt¶1=2
;
andtheRMSgainisde¯nedas
sup
RMS(w)6=0RMS(z)
RMS(w):
Nowreconsider condition (6.51)forLDIs.
²LTIsystems: Condition (6.51)isequivalentto
"
ATP+PA+CT
zCzPBw
BT
wP¡°2I#
·0: (6.52)
Therefore, wecompute thesmallest upperboundontheL2gainoftheLTIsystem
provableviaquadratic functions byminimizing °overthevariablesPand°satisfying
conditions P>0and(6.52).ThisisanEVP.
Remark :Assuming that(A;Bw;Cz)isminimal, thisEVPgivestheexactvalue
oftheL2gainoftheLTIsystem,whichalsoequalstheH1normofitstransfer
matrix,kCz(sI¡A)¡1Bwk1.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
92 Chapter 6AnalysisofLDIs: Input/Output Properties
AssumeBw6=0andCz6=0;thentheexistence ofP>0satisfying (6.52)is
equivalenttotheexistence ofQ>0satisfying
"
AQ+QAT+BwBT
w=°2QCT
z
CzQ ¡I#
·0:
²Polytopic LDIs:Condition (6.51)isequivalentto
"
AT
iP+PAi+CT
z;iCz;iPBw;i
BT
w;iP¡°2I#
·0;i=1;:::;L: (6.53)
Assumethereexistsi0forwhichBw;i06=0,andj0forwhichCz;j06=0.Thenthere
existsP>0satisfying (6.53)ifandonlyifthereexistsQ>0satisfying
"
QAT
i+AiQ+Bw;iBT
w;iQCT
z;i
Cz;iQ¡°2I#
·0;i=1;:::;L:
Wegetthesmallest upperboundontheL2gainprovableviaquadratic functions by
minimizing °(over°andP)subjectto(6.53)andP>0,whichisanEVP.
²Norm-bound LDIs:ForNLDIs,condition (6.51)isequivalentto
»T(ATP+PA+CT
zCz)»+2»TP(Bp¼+Bww)¡°2wTw·0
forall»and¼satisfying
¼T¼·»TCT
qCq»:
Thisistrueifandonlyifthereexists¸¸0suchthat
2
64ATP+PA+CT
zCz+¸CT
qCqPBpPBw
BT
pP ¡¸I0
BT
wP 0¡°2I3
75·0: (6.54)
Therefore, weobtainthebestupperboundontheL2gainprovableviaquadratic
functions byminimizing °overthevariables°,Pand¸,subjecttoP>0,¸¸0
and(6.54).
IfBw6=0andCz6=0,thentheexistence ofP>0and¸¸0satisfying (6.54)is
equivalenttotheexistence ofQ>0and¹¸0satisfying
2
64QAT+AQ+BwBT
w+¹BpBT
pQCT
qQCT
z
CqQ ¡¹I0
CzQ 0¡°2I3
75·0:
²Diagonal Norm-bound LDIs:ForDNLDIs, condition (6.51)isequivalentto
»T(ATP+PA+CT
zCz)»+2»TP(Bp¼+Bww)¡°2wTw·0
forall»and¼satisfying
¼T
i¼i·(Cq;i»i)T(Cq;i»i);i=1;:::;nq:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.3Input-to-Output Properties 93
UsingtheS-procedure,asu±cientcondition isthatthereexistadiagonal ¤¸0such
that
2
64ATP+PA+CT
zCz+CT
q¤CqPBpPBw
BT
pP ¡¤0
BT
wP 0¡°2I3
75·0: (6.55)
Weobtainthebestsuchupperboundbyminimizing °overthevariables°,Pand¸,
subjecttoP>0,¸¸0and(6.55).
IfBw6=0andCz6=0,thentheexistence ofP>0and¸¸0satisfying (6.54)is
equivalenttotheexistence ofQ>0andM=diag(¹1;:::;¹nq)¸0satisfying
2
64QAT+AQ+BwBT
w+BpMBT
pQCT
qQCT
z
CqQ ¡M0
CzQ 0¡°2I3
75·0:
6.3.3Dissipativit y
TheLDI(6.48)issaidtobepassiveifeverysolutionxwithx(0)=0satis¯es
ZT
0wTzdt¸0
forallT¸0.Itissaidtohavedissipation ´if
ZT
0¡
wTz¡´wTw¢
dt¸0 (6.56)
holdsforalltrajectories withx(0)=0andallT¸0.Thuspassivitycorrespondsto
nonnegativ edissipation. Thelargestdissipation ofthesystem,i.e.,thelargestnumber
´suchthat(6.56)holds,willbecalleditsdissipativity.
Supposethatthereisaquadratic functionV(»)=»TP»,P>0,suchthat
forallxandwsatisfying (6.48);d
dtV(x)¡2wTz+2´wTw·0: (6.57)
Then,integrating (6.57)from0toTwithinitialconditionx(0)=0yields
V(x(T))¡ZT
0¡
2wTz¡2´wTw¢
dt·0:
SinceV(x(T))¸0,weconclude
ZT
0¡
wTz¡´wTw¢
dt¸0;
whichimpliesthatthedissipativit yoftheLDIisatleast´.Nowreconsider condi-
tion(6.57)forthevariousLDIs.
²LTIsystems: ForLTIsystems, condition (6.57)isequivalentto
"
ATP+PAPBw¡CT
z
BT
wP¡Cz2´I¡(DT
zw+Dzw)#
·0: (6.58)
We¯ndthelargestdissipation thatcanbeguaranteedusingquadratic functions by
maximizing ´overthevariablesPand´,subjecttoP>0and(6.58),anEVP.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
94 Chapter 6AnalysisofLDIs: Input/Output Properties
Assuming thatthesystem(A;Bw;Cz)isminimal, theoptimalvalueofthisEVP
isexactlyequaltothedissipativit yofthesystem,whichcanbeexpressed intermsof
thetransfermatrixH(s)=Cz(sI¡A)¡1Bw+Dzwas
inf
Res>0¸min(H(s)+H(s)¤)
2: (6.59)
ThisfollowsfromthePRlemma.
De¯ningQ=P¡1,condition (6.58)isequivalentto
"
QAT+AQBw¡QCT
z
BT
w¡CzQ2´I¡(DT
zw+Dzw)#
·0: (6.60)
²Polytopic LDIs:ForPLDIs,condition (6.57)isequivalentto
"
AT
iP+PAiPBw;i¡CT
z;i
BT
w;iP¡Cz;i2´I¡(DT
zw;i+Dzw;i)#
·0;i=1;:::;L: (6.61)
We¯ndthelargestdissipation thatcanbeguaranteedviaquadratic functions by
maximizing ´overthevariablesPand´satisfyingP>0and(6.61).ThisisanEVP;
itsoptimalvalueisalowerboundonthedissipativit yofthePLDI.
Ifwede¯neQ=P¡1,condition (6.57)isequivalentto
"
QAT
i+AiQBw;i¡QCT
z;i
BT
w;i¡Cz;iQ2´I¡(DT
zw;i+Dzw;i)#
·0;i=1;:::;L: (6.62)
²Norm-bound LDIs: ForNLDIs,usingtheS-procedure,thecondition (6.57)is
trueifandonlyifthereexistsanonnegativ escalar¸suchthat
2
64ATP+PA+¸CT
qCqPBw¡CT
zPBp
BT
wP¡Cz 2´I¡(DT
zw+Dzw)0
BT
pP 0¡¸I3
75·0: (6.63)
Therefore wecan¯ndthelargestdissipation provablewithaquadratic Lyapunov
function bymaximizing ´(over´andP),subjecttoP>0and(6.63).Thisisan
EVP.De¯ningQ=P¡1,condition (6.57)isequivalenttotheexistence of¹¸0
satisfying
2
64QAT+AQ+¹BpBT
pBw¡QCT
zQCT
q
BT
w¡CzQ 2´I¡(DT
zw+Dzw)0
CqQ 0¡¹I3
75·0: (6.64)
6.3.4Diagonal scalingforcomponentwiseresults
WeassumeforsimplicitythatDzw(t)=0.Assuming thatthesystem(6.48)hasas
manyinputsasoutputs, weconsider thesystem
_x=A(t)x+Bw(t)T¡1~w;x(0)=0;
~z=TCz(t)x:(6.65)
whereTisapositive-de¯nite diagonal matrix,whichhastheinterpretation ofascal-
ing.Wewillseethatscalingenablesustoconclude \componentwise"resultsforthe
system(6.65).
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
6.3Input-to-Output Properties 95
Consider forexample, the\scaled L2gain"
®=inf
Tdiagonal;
T>0sup
k~wk26=0k~zk2
k~wk2:
Thescaled L2gainhasthefollowinginterpretation:
max
i=1;:::;n zsup
kwik26=0kzik
kwik·®:
Weshowthisasfollows:Forevery¯xedscalingT=diag(t1;:::;tnz),
supk~wk26=0k~zk2
2
k~wk2
2=supkwk26=0Pnz
it2
ikzik2
2Pnz
it2
ikwik2
2
¸supkwik26=0kzik2
2
kwik2
2:
Sincetheinequalit yistrueforalldiagonal nonsingular T,thedesiredinequalit y
follows.Wenowshowhowwecanobtainboundsonthescaled L2gainusingLMIs
forLTIsystemsandLDIs.
²LTIsystems: TheL2gainfortheLTIsystemscaledbyTisguaranteedtobe
lessthan°ifthereexistsP>0suchthattheLMI
"
ATP+PA+CT
zSCzPBw
BT
wP¡°2S#
<0
holdswithS=TTT.Thesmallest scaled L2gainofthesystemcantherefore be
computed asaGEVP.
²Polytopic LDIs:ForPLDIs,thescaled L2gainisguaranteedtobelowerthan
°ifthereexistsS>0,withSdiagonal, andP>0whichsatisfy
"
AT
iP+PAi+CT
z;iSCz;iPBw;i
BT
w;iP¡°2S#
<0;i=1;:::;L: (6.66)
Theoptimalscaled L2gain°istherefore obtained byminimizing °over(°,Pand
S)subjectto(6.66),P>0,S>0andSdiagonal. ThisisaGEVP.
²Norm-bound LDIs:ForNLDIs,thescaled L2gain(withscalingT)islessthan°
ifthereexist¸¸0,P>0,andS>0,Sdiagonal, suchthatthefollowingLMIholds:
2
64ATP+PA+CT
zSCz+¸CT
qCqPBpPBw
BT
pP ¡¸I0
BT
wP 0¡°2S3
75·0: (6.67)
Therefore, minimizing °2subjecttotheconstrain t(6.67)isanEVP.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
96 Chapter 6AnalysisofLDIs: Input/Output Properties
Notes andReferences
Integralquadraticconstraints
Inx8.2(andx10.9)weconsider ageneralization oftheNLDIinwhichthepointwisecondition
pTp·qTqisreplaced bytheintegralconstrain t
Z1
0pTpdt·Z1
0qTqdt:
ManyoftheresultsderivedforNLDIsandDNLDIs thenbecomenecessary andsu±cient.
Integralquadratic constrain tswereintroducedbyYakubovich.Forageneralstudyofthese,
wereferthereadertothearticlesofYakubovich(see[Yak92]andreferences therein), and
alsoMegretsky [Meg93,Meg92a,Meg92b ].
Reachablesetsforcomponentwiseunit-energyinputs
Westudyingreaterdetailthesetofstatesreachablewithcomponentwiseunit-energy inputs
forLTIsystems(seex6.1.2).Webeginbyobserving thatapointx0belongstothereachable
setifandonlyiftheoptimalvalueoftheproblem
minimize max
iZ1
0w2
idt
subjectto_x=¡Ax+Bww;x(0)=x0;lim
t!1x(t)=0
islessthanone.Thisisamulti-criterion convexquadratic problem, considered inx10.8.In
thiscase,theproblem reducestocheckingwhether theLMI
xT
0Px0<1;"
ATP+PAPB
BTP¡R#
·0;
P>0;R>0;diagonal;TrR=1
isfeasible. Thisshowsthatthereachablesetistheintersection ofellipsoids©
»j»TP»·1ª
satisfying equation (6.14).
Avariation ofthisproblem isconsidered in[SZ92],wheretheauthorsconsider inputsw
satisfyingR1
0wwTdt·Q,orequivalently
Z1
0wTWwdt·TrWQforeveryW¸0.
Theygetnecessary conditions characterizing thecorrespondingreachableset.Wenowderive
LMIconditions thatarebothnecessary andsu±cient.
Asbefore,x0belongstothereachablesetifandonlytheoptimalvalueoftheproblem
minimizeZ1
0wTWwdt
subjectto_x=¡Ax+Bww;x(0)=x0;lim
t!1x(t)=0
islessthanTrWQforeveryW¸0.Thisisequivalenttoinfeasibilit yoftheLMI(inthe
variablesPandW):
xT
0Px0¡TrWQ¸0;P>0;"
ATP+PAPB
BTP¡W#
·0:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 97
Theseareprecisely theconditions obtained in[SZ92].
Reachablesetswithunit-peakinputs
Schweppe[Sch73,x4.3.3]considers themoregeneralproblem oftime-varying ellipsoidal ap-
proximations ofreachablesetswithunit-peakinputsfortime-varyingsystems; thepositive-
de¯nitematrixdescribing theellipsoidal approximation satis¯esamatrixdi®erentialequa-
tion,termsofwhichcloselyresemblethoseinthematrixinequalities describedinthisbook.
SabinandSummers [SS90]studytheapproximation ofthereachablesetviaquadratic func-
tions.AsurveyoftheavailabletechniquescanbefoundinthearticlebyGayek[Gay91].
Seealso[FG88].
ThetechniqueusedtoderiveLMI(6.23)canbeinterpreted asanexponentialtime-weighting
procedure. Fix®>0.ThenforeveryT>0,rewriteLDI(6.1),withnewexponentially
time-weightedvariablesxT(t)=e®(t¡T)=2x(t)andv(t)=e®(t¡T)=2w(t)as
_xT=³
A(t)+®
2I´
xT+Bw(t)v;xT(0)=0:
Sincew(t)Tw(t)·1,wehaveRT
0vTvd¿·1=®.
NowsupposethatP>0satis¯escondition (6.23),whichwerewriteforconvenienceas
"
(A+®I=2)TP+P(A+®I=2)PBw
BT
wP¡®I#
·0: (6.68)
Theresultsofx6.1.1implythatxT(T)satis¯esxT(T)TPxT(T)·1;therefore,x(T)TPx(T)·
1.SinceLMI(6.68)isindependentofT,theellipsoid©
xjxTPx·1ª
containsthereachable
setwithunit-peakinputs.
Boundsonovershoot
Theresultsofx6.1.3canbeusedto¯ndaboundonthestepresponsepeakforLDIs.Consider
asingle-input single-output LDI,subjecttoaunit-step input:
_x=A(t)x+bw(t);x(0)=0;
s=cz(t)x;"
A(t)bw(t)
cz(t)0#
2:
Sinceaunit-step isalsoaunit-peakinput,anupperboundonmaxt¸0js(t)j(i.e.,theover-
shoot)canbefoundbycombiningtheresultsonreachablesetswithunit-peakinputs(x6.1.3)
andboundsonoutputpeak(x6.2.2).
SupposethereexistQ>0and®>0satisfying (6.25),
A(t)Q+QA(t)T+®Q+1
®bw(t)bw(t)T<0:
Then,fromx6.2.2,weconclude thatmaxt¸0js(t)jdoesnotexceedM,whereM2=maxt¸0
cz(t)TQcz(t).Thiscanbeusedto¯ndLMI-based boundsonovershootforLTIsystems,
PLDIsandNLDIs. Wenotethattheresulting boundscanbequiteconservative.These
LMIscanbeusedtodetermine state-feedbac kmatrices aswell(seetheNotesandReferences
ofChapter 7).
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
98 Chapter 6AnalysisofLDIs: Input/Output Properties
BoundsonimpulseresponsepeakforLTIsystems
Theimpulseresponseh(t)=ceAtbofthesingle-input single-output LTIsystem
_x=Ax+bu;x(0)=0;y=cx
isjusttheoutputywithinitialconditionx(0)=bandzeroinput.Therefore, wecancom-
puteaboundontheimpulseresponsepeakbycombiningtheresultsoninvariantellipsoids
fromx5.2withthoseonthepeakvalueoftheoutputfromx6.2.2.Theboundobtained this
waycanbequiteconservative.Forinstance, consider theLTIsystemwithtransferfunction
H(s)=1
s+s1nY
i=2s¡si
s+si;
wheresi+1Àsi>0,i.e.,thedynamics ofthissystemarewidelyspaced. Theboundon
thepeakvalueoftheimpulse responseofthesystemcomputed viatheEVP(6.47)turns
outtobe2n¡1timestheactualmaximumvalueoftheimpulseresponse,wherenisthe
dimension ofaminimal realization ofthesystem[Fer93].Weconjecture thatthebound
canbenomoreconservative,thatis,foreveryLTIsystem,theboundonthepeakofthe
impulseresponsecomputed viatheEVP(6.47)canbenomorethan2n¡1timestheactual
maximumvalueoftheimpulseresponse.
Examples forwhichtheboundobtained viatheEVP(6.47)issharpincludethecaseofpassive
LTIsystems. Inthiscase,weknowthereexistsapositive-de¯nitePsuchthatATP+PA·0,
Pb=cT,sothatthemaximumvalueoftheimpulseresponseiscbandisattained fort=0.
Thisisanillustration ofthefactthatpassivesystemshavenice\peaking"properties,which
isusedinnonlinear control[Kok92].
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Chapter 7
State-FeedbackSynthesisforLDIs
7.1StaticState-FeedbackControllers
Weconsider theLDI
_x=A(t)x+Bw(t)w+Bu(t)u;z=Cz(t)x+Dzw(t)w+Dzu(t)u;
"
A(t)Bw(t)Bu(t)
Cz(t)Dzw(t)Dzu(t)#
2;(7.1)
wherehasoneofourspecialforms(i.e.,singleton, polytope,imageofaunitball
underamatrixlinear-fractional mapping). Hereu:R+!Rnuisthecontrolinput
andw:R+!Rnwistheexogenousinputsignal.
LetK2Rnu£n,andsupposethatu=Kx.Sincethecontrolinputisalinear
function ofthestate,thisiscalled(linear,constant)state-feedback,andthematrixK
iscalledthestate-feedbackgain.Thisyieldstheclosed-loopLDI
_x=(A(t)+Bu(t)K)x+Bw(t)w;
z=(Cz(t)+Dzu(t)K)x+Dzw(t)w:(7.2)
Inthischapterweconsider thestate-feedbac ksynthesisproblem, i.e.,theproblem of
¯ndingamatrixKsothattheclosed-lo opLDI(7.2)satis¯es certainpropertiesor
speci¯cations, e.g.,stability.
Remark :Usingtheideaofgloballinearization describedinx4.3,themethodsof
thischaptercanbeusedtosynthesizealinearstate-feedbac kforsomenonlinear,
time-varyingsystems. Asanexample, wecansynthesizeastate-feedbac kforthe
system
_x=f(x;w;u;t);z=g(x;w;u;t);
providedwehave
2
64@f
@x@f
@w@f
@u
@g
@x@g
@w@g
@u3
75(x;w;u;t)2;
andf(0;0;0;t)=0,g(0;0;0;t)=0,forallx,t,wandu.
99
100 Chapter 7State-Feedba ckSynthesis forLDIs
7.2StateProperties
Weconsider theLDI
_x=A(t)x+Bu(t)u;[A(t)Bu(t)]2: (7.3)
²LTIsystems: ForLTIsystems, (7.3)becomes
_x=Ax+Buu: (7.4)
²Polytopic LDIs:PLDIsaregivenby
_x=A(t)x+Bu(t)u;[A(t)Bu(t)]2Cof[A1Bu;1];::::[ALBu;L]g:(7.5)
²Norm-bound LDIs:ForNLDIs,(7.3)becomes
_x=Ax+Buu+Bpp;q=Cqx+Dquu+Dqpp
p=¢(t)q;k¢(t)k·1:(7.6)
Equivalently,wehave
_x=Ax+Buu+Bpp;q=Cqx+Dqpp+Dquu;pTp·qTq:
²Diagonal Norm-bound LDIs:ForDNLDIs, equation (7.3)becomes
_x=Ax+Buu+Bpp;q=Cqx+Dqpp+Dquu;
pi=±i(t)qi;j±i(t)j·1;i=1;:::;L:(7.7)
Equivalently,wehave
_x=Ax+Buu+Bpp;q=Cqx+Dqpp+Dquu;
jpij·jqij;i=1;:::;nq:
7.2.1Quadratic stabilizabilit y
Thesystem(7.1)issaidtobequadraticallystabilizable (vialinearstate-feedbac k)if
thereexistsastate-feedbac kgainKsuchthattheclosed-lo opsystem(7.2)isquadrat-
icallystable(hence,stable).Quadratic stabilizabilit ycanbeexpressed asanLMIP.
²LTIsystems: Letus¯xthematrixK.TheLTIsystem(7.4)is(quadratically)
stableifandonlyifthereexistsP>0suchthat
(A+BuK)TP+P(A+BuK)<0;
orequivalently,thereexistsQ>0suchthat
Q(A+BuK)T+(A+BuK)Q<0: (7.8)
Neitheroftheseconditions isjointlyconvexinKandPorQ,butbyasimplechange
ofvariableswecanobtainanequivalentcondition thatisanLMI.
De¯neY=KQ,sothatforQ>0wehaveK=YQ¡1.Substituting into(7.8)
yields
AQ+QAT+BuY+YTBT
u<0; (7.9)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
7.2StateProperties 101
whichisanLMIinQandY.Thus,thesystem(7.4)isquadratically stabilizable if
andonlyifthereexistQ>0andYsuchthattheLMI(7.9)holds.IfthisLMIis
feasible, thenthequadratic functionV(»)=»TQ¡1»proves(quadratic) stabilityof
system(7.4)withstate-feedbac ku=YQ¡1x.
Remark :WewillusethesimplechangeofvariablesY=KQmanytimesin
thesequel.Itallowsustorecasttheproblem of¯ndingastate-feedbac kgain
asanLMIPwithvariablesthatincludeQandY;thestate-feedbac kgainKis
recoveredasK=YQ¡1.
Analternate equivalentcondition for(quadratic) stabilizabilit y,involvingfewer
variables, canbederivedusingtheelimination ofmatrixvariablesdescribedinx2.6.2:
ThereexistQ>0andascalar¾suchthat
AQ+QAT¡¾BuBT
u<0: (7.10)
Sincewecanalwaysassume¾>0inLMI(7.10),andsincetheLMIishomogeneous
inQand¾,wecanwithoutlossofgeneralit ytake¾=1,thusreducing thenumberof
variablesbyone.IfQ>0satis¯estheLMI(7.10),astabilizing state-feedbac kgainis
givenbyK=¡(¾=2)BT
uQ¡1.
Fromtheelimination procedureofx2.6.2,anotherequivalentcondition is
~BT
u¡
AQ+QAT¢~Bu<0; (7.11)
where~Buisanorthogonal complemen tofBu.ForanyQ>0satisfying (7.11),a
stabilizing state-feedbac kgainisK=¡(¾=2)BT
uQ¡1,where¾isanyscalarsuch
that(7.10)holds(condition (7.11)impliesthatsuchascalarexists).
ForLTIsystems, theseLMIconditions arenecessary andsu±cientforstabiliz-
ability.Intermsoflinearsystemtheory,stabilizabilit yisequivalenttothecondition
thateveryunstable modebecontrollable; itisnothardtoshowthatthisisequivalent
tofeasibilityoftheLMIs(7.9),(7.11),or(7.10).
²Polytopic LDIs: ForthePLDI(7.5),thesameargumen tapplies. Quadratic
stabilizabilit yisequivalenttotheexistence ofQ>0andYwith
QAT
i+AiQ+Bu;iY+YTBT
u;i<0;i=1;:::;L: (7.12)
²Norm-bound LDIs: Withu=Kx,theNLDI(7.6)isquadratically stable(see
LMI(5.14))ifthereexistQ>0and¹>0suchthat
2
64Ã
AQ+QAT+BuKQ
+QKTBT
u+¹BpBT
p!
¹BpDT
qp+Q(Cq+DquK)T
¹DqpBT
p+(Cq+DquK)Q¡¹(I¡DqpDT
qp)3
75<0:
Thiscondition hasasimplefrequency-domain interpretation: TheH1normofthe
transfer matrixfromptoqfortheLTIsystem_x=(A+BuK)x+Bpp,q=(Cq+
DquK)x+Dqppislessthanone.
WithY=KQ,weconclude thattheNLDIisquadratically stabilizable ifthere
existQ>0,¹>0andYsuchthat
2
64Ã
AQ+QAT+¹BpBT
p
+BuY+YTBT
u!
¹BpDT
qp+QCT
q+YTDT
qu
¹DqpBT
p+CqQ+DquY¡¹(I¡DqpDT
qp)3
75<0:(7.13)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
102 Chapter 7State-Feedba ckSynthesis forLDIs
Usingtheelimination procedureofx2.6.2,weobtaintheequivalentconditions:
2
664Ã
AQ+QAT
¡¾BuBT
u+¹BpBT
p!
QCT
q+¹BpDT
qp¡¾BuDT
qu
CqQ+¹DqpBT
p¡¾DquBT
u¡¹(I¡DqpDT
qp)¡¾DquDT
qu3
775<0;
¡¹(I¡DqpDT
qp)<0:(7.14)
Thelattercondition isalwayssatis¯ed sincetheNLDIiswell-posed.Byhomogeneit y,
wecanset¹=1.IfQ>0and¾>0satisfy(7.14)with¹=1,astabilizing
state-feedbac kgainisK=(¡¾=2)BT
uQ¡1.
Analternate, equivalentcondition forquadratic stabilizabilit yisexpressed in
termsoftheorthogonal complemen tof[BT
uDT
qu]T,whichwedenoteby~G:
~GT"
AQ+QAT+¹BpBT
pQCT
q+¹BpDT
qp
CqQ+¹DqpBT
p¡¹(I¡DqpDT
qp)#
~G<0: (7.15)
(Again,wecanfreelyset¹=1inthisLMI.)
Intheremainder ofthischapterwewillassumethatDqpin(7.6)iszeroinorder
tosimplify thediscussion; allthefollowingresultscanbeextended tothecasein
whichDqpisnonzero.
²Diagonal Norm-bound LDIs:Withu=Kx,theDNLDI(7.7)isquadratically
stable(seeLMI(5.16))ifthereexistQ>0,M=diag(¹1;:::;¹nq)>0satisfying
2
64Ã
AQ+QAT+BuKQ
+QKTBT
u+BpMBT
p!
BpMDT
qp+Q(Cq+DquK)T
DqpMBT
p+(Cq+DquK)Q¡(M¡DqpMDT
qp)3
75<0:
Thiscondition leadstoasimplefrequency-domain interpretation forquadratic stabi-
lizabilityofDNLDIs: LetHdenotethetransfermatrixfromptoqfortheLTIsystem
_x=(A+BuK)x+Bpp,q=(Cq+DquK)x+Dqpp.ThentheDNLDIisquadratically
stabilizable ifthereexistsKsuchthatforsomediagonal positive-de¯nite matrixM,
kM¡1=2HM1=2k1·1.Thereisnoanalytical methodforcheckingthiscondition.
WithY=KQ,weconclude thattheNLDIisquadratically stabilizable ifthere
existQ>0,M>0anddiagonal, andYsuchthat
2
64Ã
AQ+QAT+BpMBT
p
+BuY+YTBT
u!
BpMDT
qp+QCT
q+YTDT
qu
DqpMBT
p+CqQ+DquY¡(M¡DqpMDT
qp)3
75<0:
Asbefore,wecaneliminate thevariableYusingtheelimination procedureofx2.6.2,
andobtainequivalentLMIsinfewervariables.
7.2.2Holdable ellipsoids
Justasquadratic stabilitycanalsobeinterpreted intermsofinvariantellipsoids, we
caninterpretquadratic stabilizabilit yintermsofholdable ellipsoids. Wesaythatthe
ellipsoid
E=©
»2Rn¯¯»TQ¡1»·1ª
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
7.2StateProperties 103
isholdable forthesystem(7.3)ifthereexistsastate-feedbac kgainKsuchthatE
isinvariantforthesystem(7.3)withu=Kx.Therefore theLMIsdescribedinthe
previous sectionalsocharacterize holdable ellipsoids forthesystem(7.3).
Withthisparametrization ofquadratic stabilizabilit yandholdable ellipsoids as
LMIsinQ>0andY,wecansolvevariousoptimization problems, suchas¯nding
thecoordinate transformation thatminimizes thecondition numberofQ,imposing
normconstrain tsontheinputu=Kx,etc.
Remark :InChapter 5,wesawthatquadratic stabilityandinvariantellipsoids
werecharacterized byLMIsinavariableP>0andalsoitsinverseQ=P¡1.
Incontrast,quadratic stabilizabilit yandellipsoid holdabilit ycanbeexpressed
asLMIsonlyinvariablesQandY;itisnotpossibletorewritetheseLMIsas
LMIswithQ¡1asavariable.Thisrestricts theextension ofsomeoftheresults
fromChapter 5(andChapter 6)tostate-feedbac ksynthesis.Asaruleofthumb,
resultsintheanalysis ofLDIsthatareexpressed asLMIsinthevariableQcan
beextended tostate-feedbac ksynthesis,withafewexceptions; resultsexpressed
asLMIsinParenot.Forexample, wewillnotbeabletocompute theminimum
volumeholdable ellipsoid thatcontainsagivenpolytopeP(seeproblem (5.33))
asanoptimization problem overLMIs;however,wewillbeabletocompute the
minimumdiameter holdable ellipsoid (seeproblem (5.34))containingP.
7.2.3Constrain tsonthecontrolinput
Whentheinitialcondition isknown,wecan¯ndanupperboundonthenormof
thecontrolinputu(t)=Kx(t)asfollows.PickQ>0andYwhichsatisfythe
quadratic stabilizabilit ycondition (either(7.9),(7.12)or(7.13)), andinaddition
x(0)TQ¡1x(0)·1.Thisimpliesthatx(t)belongstoEforallt¸0,andconse-
quently,
max
t¸0ku(t)k=maxt¸0kYQ¡1x(t)k
·maxx2EkYQ¡1xk
=¸max(Q¡1=2YTYQ¡1=2):
Therefore, theconstrain tku(t)k·¹isenforced atalltimest¸0iftheLMIs
"
1x(0)T
x(0)Q#
¸0;"
QYT
Y¹2I#
¸0 (7.16)
hold,whereQ>0andYsatisfythestabilizabilit yconditions (either(7.9),(7.12)
or(7.13)).
Wecanextendthistothecasewherex(0)liesinanellipsoid orpolytope.For
example supposewerequire(7.16)toholdforallkx(0k·1.Thisiseasilyshownto
beequivalentto
Q¸I;"
QYT
Y¹2I#
¸0:
Asanotherextension wecanhandleconstrain tson
ku(t)kmax¢=max
ijui(t)j
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
104 Chapter 7State-Feedba ckSynthesis forLDIs
inasimilarmanner:
max
t¸0ku(t)kmax=maxt¸0kYQ¡1x(t)kmax
·maxx2EkYQ¡1xkmax
=maxi¡
YQ¡1YT¢
ii:
Therefore, theconstrain tku(t)kmax·¹fort¸0isimpliedbytheLMI
"
1x(0)T
x(0)Q#
¸0;"
XY
YTQ#
¸0;Xii·¹2;
whereonceagain,Q>0andYsatisfythestabilizabilit yconditions (either(7.9),
(7.12)or(7.13)).
7.3Input-to-State Properties
Wenextconsider theLDI
_x=A(t)x+Bw(t)w+Bu(t)u: (7.17)
²LTIsystems: ForLTIsystems, equation (7.17)becomes_x=Ax+Bww+Buu.
²Polytopic LDIs: PLDIsaregivenby_x=A(t)x+Bw(t)w+Bu(t)u,where
[A(t)Bw(t)Bu(t)]2Cof[A1Bw;1Bu;1];:::;[ALBw;LBu;L]g.
²Norm-bound LDIs:ForNLDIs,equation (7.17)becomes
_x=Ax+Buu+Bpp+Bww;q=Cqx+Dquu
p=¢(t)q;k¢(t)k·1
whichwecanalsoexpressas
_x=Ax+Buu+Bpp+Bww;q=Cqx+Dquu;pTp·qTq:
²Diagonal Norm-bound LDIs:ForDNLDIs, equation (7.17)becomes
_x=Ax+Buu+Bpp+Bww;q=Cqx+Dquu
pi=±i(t)qi;j±i(t)j·1;i=1;:::;L:
Equivalently,wehave
_x=Ax+Buu+Bpp+Bww;q=Cqx+Dquu;
jpij·jqij;i=1;:::;nq:
7.3.1Reachablesetswithunit-energy inputs
Forthesystem(7.17)withu=Kx,thesetofstatesreachablewithunitenergyis
de¯nedas
Rue¢=8
><
>:x(T)¯¯¯¯¯¯¯x,w,usatisfy(7.17);u=Kx;x(0)=0;ZT
0wTwdt·1;T¸09
>=
>;:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
7.3Input-to-St ateProperties 105
Wewillnowderiveconditions underwhichthereexistsastate-feedbac kgainKguar-
anteeingthatagivenellipsoidE=f»2Rn¯¯»TQ¡1»·1gcontainsRue.
²LTIsystems: Fromx6.1.1,theellipsoidEcontainsRueforthesystem(7.4)for
somestate-feedbac kgainKifKsatis¯es
AQ+QAT+BuKQ+QKTBT
u+BwBT
w·0:
SettingKQ=Y,weconclude thatE¶RueforsomeKifthereexistYsuchthat
QAT+AQ+BuY+YTBT
u+BwBT
w·0:
ForanyQ>0andYsatisfying thisLMI,thestate-feedbac kgainK=YQ¡1issuch
thattheellipsoidEcontainsthereachablesetfortheclosed-lo opsystem.
Usingtheelimination procedureofx2.6.2,weeliminate thevariableYtoobtain
theLMIinQ>0andthevariable¾:
QAT+AQ¡¾BuBT
u+BwBT
w·0: (7.18)
ForanyQ>0and¾satisfying thisLMI,acorrespondingstate-feedbac kgainisgiven
byK=¡(¾=2)BT
uQ¡1.Another equivalentcondition forEtocontainRueis
~BT
u¡
AQ+QAT+BwBT
w¢~Bu·0; (7.19)
where~Buisanorthogonal complemen tofBu.
²Polytopic LDIs: ForPLDIs,EcontainsRueforsomestate-feedbac kgainKif
thereexistQ>0andYsuchthatthefollowingLMIholds(seeLMI(6.9)):
QAT
i+AiQ+Bu;iY+YTBT
u;i+Bw;iBT
w;i<0;i=1;:::;L:
²Norm-bound LDIs:ForNLDIs,EcontainsRueforsomestate-feedbac kgainK
ifthereexist¹>0andYsuchthatthefollowingLMIholds(seeLMI(6.11)):
"
QAT+AQ+BuY+YTBT
u+BwBT
w+¹BpBT
p(CqQ+DquY)T
CqQ+DquY ¡¹I#
·0:
WecaneliminateYtoobtaintheLMIinQ>0and¾thatguaranteesthatEcontains
thereachablesetRueforsomestate-feedbac kgainK:
2
64Ã
AQ+QAT¡¾BuBT
u
+¹BpBT
p+BwBT
w!
QCT
q+¹BpDT
qp¡¾BuDT
qu
CqQ+¹DqpBT
p¡¾DquBT
u¡¹(I¡DqpDT
qp)¡¾DquDT
qu3
75<0:(7.20)
Thecorrespondingstate-feedbac kgainisK=(¡¾=2)BT
uQ¡1.
Anequivalentcondition isexpressed intermsof~G,theorthogonal complemen t
of[BT
uDT
qu]T:
~GT2
64Ã
AQ+QAT
+¹BpBT
p+BwBT
w!
QCT
q+¹BpDT
qp
CqQ+¹DqpBT
p¡¹(I¡DqpDT
qp)3
75~G<0:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
106 Chapter 7State-Feedba ckSynthesis forLDIs
²Diagonal Norm-bound LDIs: ForDNLDIs,EcontainsRueforsomestate-
feedbackgainKifthereexistQ>0,M>0anddiagonal, andYsuchthatthe
followingLMIholds(seeLMI(6.12)):
"
QAT+AQ+BuY+YTBT
u+BwBT
w+BpMBT
p(CqQ+DquY)T
CqQ+DquY ¡M#
·0:
Usingtheelimination procedureofx2.6.2,wecaneliminate thevariableYtoobtain
anequivalentLMIinfewervariables.
UsingtheseLMIs(seex6.1.1fordetails):
²Wecan¯ndastate-feedbac kgainKsuchthatagivenpointx0liesoutsidethe
setofreachablestatesforthesystem(7.3).
²Wecan¯ndastate-feedbac kgainKsuchthatthesetofreachablestatesforthe
system(7.3)liesinagivenhalf-space. Thisresultcanbeextended tocheckif
thereachablesetiscontainedinapolytope.Inthiscase,incontrastwiththe
resultsinChapter 6,wemustusethesameouterellipsoidal approximation to
checkdi®erentfaces.(Thisisduetothecoupling induced bythenewvariable
Y=KQ.)
7.3.2Reachablesetswithcomponentwiseunit-energy inputs
Withu=Kx,thesetofreachablestatesforinputswithcomponentwiseunit-energy
forthesystem(7.17)isde¯nedas
Ruce¢=8
><
>:x(T)¯¯¯¯¯¯¯x,w,usatisfy(7.17);u=Kx;x(0)=0;ZT
0wT
iwidt·1;T¸09
>=
>;:
Wenowconsider theexistence ofthestate-feedbac kgainKsuchthattheellipsoid
E=f»2Rn¯¯»TQ¡1»·1gcontainsRuceforLTIsystemsandLDIs.
²LTIsystems: FromLMI(6.15),E¶RucefortheLTIsystem(7.4)ifthereexists
diagonalR>0withunittracesuchthattheLMI
"
QAT+AQ+BuY+YTBT
uBw
BT
w ¡R#
·0:
²Polytopic LDIs: ForPLDIs,E¶Ruceforsomestate-feedbac kgainKifthere
existQ>0,Y(=KQ),andadiagonalR>0withunittrace(seeLMI(6.17)),such
that"
QAT
i+AiQ+Bu;iY+YTBT
u;iBw;i
BT
w;i ¡R#
<0;i=1;:::;L:
²Norm-bound LDIs:ForNLDIs,E¶Ruceforsomestate-feedbac kgainKifthere
existQ>0,¹>0,YandadiagonalR>0withunittrace(seeLMI(6.19)),such
that
2
64QAT+YTBT
u+AQ+BuY+¹BpBT
pQCT
q+YTDT
quBw
CqQ+DquY ¡¹I 0
BT
w 0¡R3
75·0:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
7.4State-to-Output Properties 107
Remark :ForLTIsystemsandNLDIs,itispossibletoeliminate thevariableY.
7.3.3Reachablesetswithunit-peakinputs
Fora¯xedstate-feedbac kgainK,thesetofstatesreachablewithinputswithunit-
peakisde¯nedas
Rup¢=(
x(T)¯¯¯¯¯x,w,usatisfy(7.17);u=Kx;x(0)=0;
w(t)Tw(t)·1;T¸0)
:
²LTIsystems: Fromcondition (6.25),EcontainsRucefortheLTIsystem(7.4)if
thereexists®>0suchthat
AQ+QAT+BuY+YTBT
u+BwBT
w=®+®Q·0;
whereY=KQ.
²Polytopic LDIs: ForPLDIs,E¶RuceifthereexistQ>0andY(seecondi-
tion(6.28))suchthat
AiQ+QAT
i+Bu;iY+YTBT
u;i+®Q+Bw;iBT
w;i=®·0
fori=1;:::;L.
²Norm-bound LDIs: FromChapter 6,EcontainsRuceifthereexistQ>0,Y,
®>0and¹>0suchthat
2
64Ã
AQ+QAT+®Q
+BuY+YTBu+¹BpBT
p+BwBT
w=®!
(CqQ+DquY)T
CqQ+DquY ¡¹I3
75·0:
Thiscondition isanLMIfor¯xed®.
Remark :Again,forLTIsystems andNLDIs,itispossibletoeliminate the
variableY.
7.4State-to-Output Properties
Weconsider state-feedbac kdesigntoachievecertaindesirable state-to-output prop-
ertiesfortheLDI(4.5).Wesettheexogenous inputwtozeroin(4.5),andconsider
_x=A(t)x+Bu(t)u;z=Cz(t)x:
²LTIsystems: ForLTIsystems, thestateequations are
_x=Ax+Buu;z=Czx+Dzuu: (7.21)
²Polytopic LDIs:ForPLDIswehave
"
A(t)Bu(t)
Cz(t)Dzu(t)#
2Co("
A1Bu;1
Cz;1Dzu;1#
;:::;"
ALBu;L
Cz;LDzu;L#)
:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
108 Chapter 7State-Feedba ckSynthesis forLDIs
²Norm-bound LDIs:NLDIsaregivenby
_x=Ax+Buu+Bpp;z=Czx+Dzuu
q=Cqx+Dquup=¢(t)q;k¢(t)k·1;
whichcanberewritten as
_x=Ax+Buu+Bpp;z=Czx+Dzuu
q=Cqx+Dquu;pTp·qTq:
²Diagonal Norm-bound LDIs:Finally,DNLDIs aregivenby
_x=Ax+Buu+Bpp;z=Czx+Dzuu;q=Cqx+Dquu;
pi=±i(t)qi;j±i(t)j·1;i=1;:::;L:
whichcanberewritten as
_x=Ax+Buu+Bpp;z=Czx+Dzuu;q=Cqx+Dquu;
jpij·jqij;i=1;:::;nq:
7.4.1Boundsonoutputenergy
We¯rstshowhowto¯ndastate-feedbac kgainKsuchthattheoutputenergy(as
de¯nedinx6.2.1)oftheclosed-lo opsystemislessthansomespeci¯edvalue.We
assume¯rstthattheinitialconditionx0isgiven.
²LTIsystems: Weconclude fromLMI(6.39)thatforagivenstate-feedbac kgain
K,theoutputenergyofsystem(7.21)doesnotexceedxT
0Q¡1x0,whereQ>0and
Y=KQsatisfy"
AQ+QAT+BuY+YTBT
u(CzQ+DzuY)T
CzQ+DzuY ¡I#
·0; (7.22)
Regarding Yasavariable,wecanthen¯ndastate-feedbac kgainthatguaranteesan
outputenergylessthan°bysolvingtheLMIPxT
0Q¡1x0·°and(7.22).
Ofcourse,inequalit y(7.22)iscloselyrelatedtotheclassical Linear-Quadratic
Regulator (LQR)problem; seetheNotesandReferences.
²Polytopic LDIs:Inthiscase,theoutputenergyisboundedabovebyxT
0Q¡1x0,
whereQsatis¯estheLMI2
664Ã
AiQ+QAT
i
+Bu;iY+YTBT
u;i!
(Cz;iQ+Dzu;iY)T
Cz;iQ+Dzu;iY ¡I3
775·0;i=1;:::;L
forsomeY.
²Norm-bound LDIs: InthecaseofNLDIs,theoutputenergyisboundedabove
byxT
0Q¡1x0,foranyQ>0,Yand¹¸0suchthat
2
66664Ã
AQ+QAT+BuY
+YTBT
u+¹BpBT
p!
(CzQ+DzuY)T(CqQ+DquY)T
CzQ+DzuY ¡I 0
CqQ+DquY 0 ¡¹I3
77775·0:(7.23)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
7.5Input-to-Output Properties 109
Remark :Asbefore,wecaneliminate thevariableYfromthisLMI.
²Diagonal Norm-bound LDIs:Finally,inthecaseofDNLDIs, theoutputenergy
isboundedabovebyxT
0Q¡1x0,foranyQ>0,YandM¸0anddiagonal suchthat
2
66664Ã
AQ+QAT+BuY
+YTBT
u+BpMBT
p!
(CzQ+DzuY)T(CqQ+DquY)T
CzQ+DzuY ¡I 0
CqQ+DquY 0 ¡M3
77775·0:
Givenaninitialcondition, ¯ndingastate-feedbac kgainKsoastominimize the
upperboundontheextractable energyforthevariousLDIsistherefore anEVP.We
canextendtheseresultstothecasewhenx0isspeci¯edtolieinapolytopeoran
ellipsoid (seex6.2.1).Ifx0isarandomvariablewithEx0xT
0=X0,EVPsthatyield
state-feedbac kgainsthatminimize theexpectedvalueoftheoutputenergycanbe
derived.
7.5Input-to-Output Properties
We¯nallyconsider theproblem of¯ndingastate-feedbac kgainKsoastoachievede-
siredpropertiesbetweentheexogenous inputwandtheoutputzforthesystem(7.1).
AsmentionedinChapter 6,thelistofproblems considered hereisfarfromexhaustiv e.
7.5.1 L2andRMSgains
Weseekastate-feedbac kgainKsuchthattheL2gain
sup
kwk2=1kzk2=sup
kwk26=0kzk2
kwk2
oftheclosed-lo opsystemislessthanaspeci¯ednumber°.
²LTIsystems: Asseenin6.3.2,theL2gainforLTIsystemsisequaltotheH1
normofthecorrespondingtransfermatrix.Fromx6.3.2,thereexistsastate-feedbac k
gainKsuchthattheL2gainofanLTIsystemislessthan°,ifthereexistKand
Q>0suchthat,
2
64Ã
(A+BuK)Q+Q(A+BuK)T
+BwBT
w!
Q(Cz+DzuK)T
(Cz+DzuK)Q ¡°2I3
75·0: (7.24)
IntroducingY=KQ,thiscanberewritten as
"
AQ+QAT+BuY+YTBT
u+BwBT
w(CzQ+DzuY)T
CzQ+DzuY ¡°2I#
·0:(7.25)
InthecaseofLTIsystems, thiscondition isnecessary andsu±cient.Assuming
CT
zDzu=0andDT
zuDzuinvertible,itispossibletosimplify theinequalit y(7.25)
andgettheequivalentRiccatiinequalit y
AQ+QAT¡Bu(DT
zuDzu)¡1BT
u+BwBT
w+QCT
zCzQ=°2·0: (7.26)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
110 Chapter 7State-Feedba ckSynthesis forLDIs
ThecorrespondingRiccatiequation isreadilysolvedviaHamiltonian matrices. The
resulting controller,withstate-feedbac kgainK=¡(DT
zuDzu)¡1BT
u,yieldsaclosed-
loopsystemwithH1-normlessthan°.
²Polytopic LDIs: Fromx6.3.2,thereexistsastate-feedbac kgainsuchthatthe
L2gainofaPLDIislessthan°ifthereexistYandQ>0suchthat
2
64Ã
AiQ+QAT
i+Bu;iY
+YTBT
u;i+Bw;iBT
w;i!
(Cz;iQ+Dzu;iY)T
Cz;iQ+Dzu;iY ¡°2I3
75·0: (7.27)
²Norm-bound LDIs:ForNLDIs,theLMIthatguaranteesthattheL2gainisless
than°forsomestate-feedbac kgainKis
2
6666666640
B@AQ+QAT
+BuY+YTBT
u
+BwBT
w+¹BpBT
p1
CA(CzQ+DzuY)T(CqQ+DquY)T
CzQ+DzuY ¡°2I 0
CqQ+DquY 0 ¡¹I3
777777775·0:(7.28)
Wecantherefore ¯ndstate-feedbac kgainsthatminimize theupperboundonthe
L2gain,provablewithquadratic Lyapunovfunctions, forthevariousLDIsbysolving
EVPs.
7.5.2Dissipativit y
Wenextseekastate-feedbac kgainKsuchthattheclosed-lo opsystem(7.2)ispassive;
moregenerally ,assumingDzw(t)isnonzero andsquare,wewishtomaximize the
dissipativit y,i.e.,´satisfying
ZT
0¡
wTz¡´wTw¢
dt¸0
forallT¸0.
²LTIsystems: Substituting Y=KQ,andusingLMI(6.60),weconclude thatthe
dissipation oftheLTIsystemisatleast´iftheLMIin´,Q>0andYholds:
"
AQ+QAT+BuY+YTBT
uBw¡QCT
z¡YTDT
zu
BT
w¡CzQ¡DzuY 2´I¡(Dzw+DT
zw)#
·0:
²Polytopic LDIs: From(6.62),thereexistsastate-feedbac kgainsuchthatthe
dissipation exceeds´ifthereexistQ>0andYsuchthat
2
664Ã
AiQ+QAT
i
+Bu;iY+YTBT
u;i!
Bw;i¡QCT
z;i¡YTDzu;i
BT
w;i¡Cz;iQ¡Dzu;iY2´I¡(Dzw;i+DT
zw;i)3
775·0:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
7.6Obser ver-Based Contr ollers forNonlinear Systems 111
²Norm-bound LDIs:From(6.64),thereexistsastate-feedbac kgainsuchthatthe
dissipation exceeds´ifthereexistQ>0andYsuchthat
2
66666664Ã
AQ+QAT+BuY
+YTBT
u+¹BqBT
q!Ã
Bw¡QCT
z
¡YTDT
zu!
(CqQ+DquY)T
Ã
BT
w¡CzQ
¡DzuY!
¡(Dzw+DT
zw¡2´I) 0
CqQ+DquY 0 ¡¹I3
77777775·0
Wecantherefore ¯ndstate-feedbac kgainsthatmaximize thelowerboundonthe
dissipativit y,provablewithquadratic Lyapunovfunctions, forthevariousLDIsby
solvingEVPs.
Remark :Aswithx6.3.4,wecanincorporatescalingtechniquesintomanyofthe
resultsabovetoderivecomponentwiseresults.SincethenewLMIsthusobtained
arestraightforwardtoderive,wewillomitthemhere.
7.5.3Dynamic versusstaticstate-feedbac kcontrollers
Adynamic state-feedbac kcontrollerhastheform
_¹x=¹A¹x+¹Byx;u=¹Cu¹x+¹Duyx; (7.29)
where¹A2Rr£r,¹By2Rr£n,¹Cu2Rnu£r,¹Duy2Rnu£n.Thenumberriscalled
theorderofthecontroller.Notethatbytakingr=0,thisdynamic state-feedbac k
controllerreducestothestate-feedbac kwehaveconsidered sofar(whichiscalled
staticstate-feedbac kinthiscontext).
Itmightseemthatthedynamic state-feedbac kcontroller(7.29)allowsustomeet
morespeci¯cations thancanbemetusingastaticstate-feedbac k.Forspeci¯cations
basedonquadratic Lyapunovfunctions, however,thisisnotthecase.Forexam-
ple,supposetheredoesnotexistastaticstate-feedbac kgainthatyieldsclosed-lo op
quadratic stability.Inthiscasewemightturntothemoregeneraldynamic state-
feedback.Wecouldworkoutthemorecomplicated LMIsthatcharacterize quadratic
stabilizabilit ywithdynamic state-feedbac k.Wewould¯nd,however,thatthesemore
generalLMIsarealsoinfeasible. SeetheNotesandReferences.
7.6Observer-BasedControllersforNonlinearSystems
Weconsider thenonlinear system
_x=f(x)+Buu;y=Cyx; (7.30)
wherex:R+!Rnisthestatevariable,u:R+!Rpisthecontrolvariable,
andy:R+!Rqisthemeasured orsensedvariable. Weassumethefunction
f:Rn!Rnsatis¯esf(0)=0and
@f
@x2CofA1;:::;AMg
whereA1;:::;ALaregiven,whichisthesameas(4.15).
Welookforastabilizing observer-b asedcontrolleroftheform
_¹x=f(¹x)+Buu+L(Cy¹x¡y);u=K¹x; (7.31)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
112 Chapter 7State-Feedba ckSynthesis forLDIs
i.e.,wesearchforthematricesK(theestimated-statefeedbackgain)andL(the
observer gain)suchthattheclosed-lo opsystem
"
_x
_¹x#
="
f(x)+BuK¹x
¡LCyx+f(¹x)+(BuK+LCy)¹x#
(7.32)
isstable.Theclosed-lo opsystem(7.32)isstableifitisquadratically stable,whichis
trueifthereexistsapositive-de¯nite matrix~P2R2n£2nsuchthatforanynonzero
trajectoryx,¹x,wehave:
d
dt"
x
¹x#T
~P"
x
¹x#
<0:
IntheNotesandReferences, weprovethatthisistrueifthereexistP,Q,Y,andW
suchthattheLMIs
Q>0;AiQ+QAT
i+BuY+YTBT
u<0;i=1;:::;M (7.33)
and
P>0;AT
iP+PAi+WCy+CT
yWT<0;i=1;:::;M (7.34)
hold.ToeveryP,Q,Y,andWsatisfying theseLMIs,therecorrespondsastabilizing
observer-based controlleroftheform(7.31),obtained bysettingK=YQ¡1and
L=P¡1W.
Usingtheelimination procedureofx2.6.2,wecanobtainequivalentconditions in
whichthevariablesYandWdonotappear.Theseconditions arethatsomeP>0
andQ>0satisfy
AiQ+QAT
i<¾BuBT
u;i=1;:::;M (7.35)
and
AT
iP+PAi<¹CT
yCy;i=1;:::;M (7.36)
forsome¾and¹.Byhomogeneit ywecanfreelyset¾=¹=1.ForanyP>0,Q>0
satisfying theseLMIswith¾=¹=1,wecanobtainastabilizing observer-based
controlleroftheform(7.31),bysettingK=¡(1=2)BT
uQ¡1andL=¡(1=2)P¡1Cy.
Another equivalentcondition isthat
~BT
u¡
AiQ+QAT
i¢~Bu<0;i=1;:::;M
and
~Cy¡
AT
iP+PAi¢~CT
y<0;i=1;:::;M
holdforsomeP>0,Q>0,where~Buand~CT
yareorthogonal complemen tsofBuand
CT
y,respectively.IfPandQsatisfytheseLMIs,thentheysatisfytheLMIs(7.36)
and(7.35)forsome¾and¹.Theobserver-based controllerwithK=¡(¾=2)BT
uQ¡1
andL=¡(¹=2)P¡1Cystabilizes thenonlinear system(7.30).
Notes andReferences
Lyapunovfunctionsandstate-feedback
Theextension ofLyapunov'smethodstothestate-feedbac ksynthesisproblem hasalong
history;seee.g.,[Bar70a,Lef65].Itisclearlyrelatedtothetheoryofoptimalcontrol,in
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 113
whichthenaturalLyapunovfunction istheBellman{P ontryagin\min-cost-to-go" orvalue
function (see[Pon61,Pon62]).Wesuspectthatthemethodsdescribedinthischaptercan
beinterpreted asasearchforsuboptimal controlsinwhichthecandidate valuefunctions
arerestricted toaspeci¯cclass,i.e.,quadratic. Inthegeneraloptimal controlproblem,
the(exact)valuefunction satis¯es apartialdi®erentialequation andishardtocompute;
byrestricting oursearchtoquadratic approximatevaluefunctions weendupwithalow
complexit ytask,i.e.,aconvexproblem involvingLMIs.
SeveralapproachesuseLyapunov-likefunctions tosynthesizenonlinear state-feedbac kcontrol
laws.Oneexample isthe\LyapunovMin-Max Controller"describedin[Gut79,GP82,
Cor85,CL90,Wei94].Seealso[Zin90,SC91,Rya88,GR88].
Quadraticstabilizability
Theterm\quadratic stabilizabilit y"seemstohavebeencoinedbyHollotandBarmish
in[HB80],wheretheauthorsgivenecessary andsu±cientconditions forit;seealso[Lei79,
Gut79,BCL83,PB84,Cor85].Petersen[Pet85],showsthatthereexistLDIsthatare
quadratically stabilizable, thoughnotvialinearstate-feedbac k;seealso[SP94e].Hol-
lot[Hol87]describesconditions underwhichaquadratically stabilizable LDIhasin¯nite
stabilizabilit ymargin;seealso[SRC93].Wei[Wei90]derivesnecessary andsu±cient\sign-
pattern" conditions ontheperturbations fortheexistence ofalinearfeedbackforquadratic
stabilityforaclassofsingle-input uncertain lineardynamical systems. Petersen[Pet87b]
derivesaRiccati-based approachforstabilizing NLDIs,whichisinfactthestate-feedbac k
H1equation (7.28)(thisisshownin[Pet87a]).
ThechangeofvariablesQ=P¡1andY=KP¡1,whichenablestheextension ofmany
oftheresultsofChapters 5and6tostate-feedbac ksynthesis,isduetoBernussou,Peres
andGeromel [BPG89a ].Seealso[GPB91,BPG89b,BPG89a,PBG89,PGB93].This
changeofvariables, thoughnotexplicitly described,isusedinthepaperbyThorpand
Barmish [TB81];seealso[HB80,Bar83,Bar85].Forarecentandbroadreviewabout
quadratic stabilization ofuncertain systems, see[Cor94].
In[Son83],Sontagformulatesageneraltheorem givingnecessary andsu±cientLyapunov
typeconditions forstabilizabilit y.ForLTIsystems hiscriteriareducetotheLMIsinthis
chapter.
QuadraticLyapunovfunctionsforprovingperformance
Theaddition ofperformance considerations intheanalysis ofLDIscanbetracedbackto
1955andevenearlier,whenLetov[Let61]studiedtheperformance ofunknown,nonlinear
andpossiblytime-varyingcontrolsystemsandcallsit\theproblem ofcontrolquality".At
thattime,theperformance criteriaweredecayrateandoutputpeakdeviations forsystems
subjecttoboundedpeakinputs(see[Let61,p.234]fordetails).
InthecaseofLTIsystems, addingperformance indicesallowsonetorecovermanyclassical
resultsofautomatic control.Theproblem ofmaximizing decayrateisdiscussed ingreat
detailin[Yan92];otherreferences ondecayratesandquadratic stabilitymarginsare[Gu92b,
Gu92a,GPB91,Gu92b,Sko91b,PZP+92,EBFB92,Cor90].Arzelier etal.[ABG93]
consider theproblem ofrobuststabilization withpoleassignmen tforPLDIsviacutting-plane
techniques.
Theproblem ofapproximating reachablesetsforLTIsystemsundervariousassumptions on
theenergyoftheexogenous inputhasbeenconsidered forexample bySkelton[SZ92]and
references therein;seealsotheNotesandReferences oftheprevious chapter.Seealso[OC87].
Theideathattheoutputvarianceforwhitenoiseinputs(andrelatedLQR-likeperformance
measures) canbeminimized usingLMIscanbefoundin[BH88a,BH89,RK91,KR91,
KKR93,BB91,FBBE92,RK93,SI93].Peres,SouzaandGeromel [PSG92,PG93]con-
sidertheextension toPLDIs. ThecounterpartforNLDIsisinPetersen, McFarlaneand
Rotea[PM92,PMR93],aswellasinStoorvogel[Sto91].
Theproblem of¯ndingastate-feedbac kgaintominimize thescaled L2gainforLTIsys-
temsisdiscussed in[EBFB92 ].TheL2gainofPLDIshasbeenstudiedbyObradovicand
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
114 Chapter 7State-Feedba ckSynthesis forLDIs
Valavani[OV92],byPeres,Geromel andSouza[PGS91]andalsobyOhara,Masubuc hiand
Suda[OMS93].NLDIshavebeenstudiedinthiscontextbyPetersen[Pet89],andDeSouza
etal.[XFdS92,XdS92,WXdS92,dSFX93 ],whopointoutexplicitly thatquadratic sta-
bilityofNLDIswithanL2gainboundisequivalenttothescaled H1condition (6.54);see
also[ZKSN92,Gu93].ForLTIsystems, itisinteresting tocompare theinequalit y(7.25)
whichprovidesanecessary andsu±cientcondition forasystemtohaveL2gainlessthan
°withthecorrespondingmatrixinequalit yfoundinthearticlebyStoorvogelandTrentel-
man[ST90].Inparticular, thequadratic matrixinequalit yfoundthere(expressed inour
notation) is
"
ATP+PA+°2PBwBT
wP+CT
zCzPBu+CT
zDzu
BT
uP+DT
zuCz DT
zuDzu#
¸0;
whichdoesnotpossessanyobviousconvexityproperty.WenotethattheRiccatiinequal-
ity(7.26)isalsoencounteredinfull-state feedbackH1control[Pet87a].Itsoccurrence in
controltheoryismucholder;itappearsverbatiminsomearticlesonthetheoryofnonzero-
sumdi®erentialgames;seeforexample StarrandHo[SH68],Yakubovich[Yak70,Yak71]
andMageirou [Mag76].
Kokotovic[Kok92]considers stabilization ofnonlinear systems, andprovidesamotivationfor
rendering anLTIsystempassiveviastate-feedbac k(seex6.3.3).In[PP92],theauthorsgive
analytic conditions formakinganLTIsystempassiveviastate-feedbac k;seealso[SKS93].
Finally,wementionaninstance ofaLyapunovfunction withabuilt-inperformance criterion.
InthebookbyAubinandCellina[AC84,x6],we¯nd:
Weshallinvestigate whether di®erentialinclusions _x2F(x(t));x(0)=x0
havetrajectories satisfying theproperty
8t>s;V(x(t))¡V(x(s))+Zt
sW(x(¿);_x(¿))d¿·0: (7.37)
WeshallthensaythatafunctionV[¢¢¢]satisfying thiscondition isaLyapunov
function forFwithrespecttoW.
Similarideascanbefoundin[CP72].WeshallencountersuchLyapunovfunctions inx8.2.
Inequalit y(7.37)isoftencalleda\dissipation inequalit y"[Wil71b].
Theproblem ofminimizing thecontrole®ortgivenaperformance boundontheclosed-lo op
systemcanbefoundinthearticlesofSchÄomig,SznaierandLy[SSL93],andGrigoriadis,
Carpenter,ZhuandSkelton[GCZS93 ].
LMIformulationofLQRproblem
FortheLTIsystem_x=Ax+Buu,z=Czx+Dzuu,theLinear-Quadratic Regulator (LQR)
problem is:Givenaninitialconditionx(0),¯ndthecontrolinpututhatminimizes theoutput
energyR1
0zTzdt.Weassumeforsimplicitythat(A;B;C)isminimal,DT
zuDzuisinvertible
andDT
zuCz=0.
Itturnsoutthattheoptimalinputucanbeexpressed asaconstantstate-feedbac ku=Kx,
whereK=¡(DT
zuDzu)¡1BT
uPareandPareistheuniquepositive-de¯nite matrixthatsatis¯es
thealgebraic Riccatiequation
ATPare+PareA¡PareBu(DT
zuDzu)¡1BT
uPare+CT
zCz=0: (7.38)
Theoptimaloutputenergyforaninitialconditionx(0)isthengivenbyx(0)TParex(0).With
Qare=P¡1
are,wecanrewrite(7.38)as
AQare+QareAT¡Bu(DT
zuDzu)¡1BT
u+QareCT
zCzQare=0; (7.39)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 115
andtheoptimaloutputenergyasx(0)TQ¡1
arex(0).
Insectionx7.4.1,weconsidered thecloselyrelatedproblem of¯ndingastate-feedbac kgain
Kthatminimizes anupperboundontheenergyoftheoutput,givenaninitialcondition.
ForLTIsystems, theupperboundequalstheoutputenergy,andweshowedthatinthiscase,
theminimumoutputenergywasgivenbyminimizing x(0)TQ¡1x(0)subjecttoQ>0and
"
AQ+QAT+BuY+YTBT
u(CzQ+DzuY)T
CzQ+DzuY ¡I#
·0: (7.40)
Ofcourse,theoptimalvalueofthisEVPmustequaltheoptimaloutputenergygivenvia
thesolutiontotheRiccatisolution(7.39);theRiccatiequation canbethusbeinterpreted as
yieldingananalyticsolutiontotheEVP.WecanderivethisanalyticsolutionfortheEVPvia
thefollowingsteps.First,itcanbeshown,usingasimplecompletion-of-squares argumen t
thatLMI(7.40)holdsforsomeQ>0andYifandonlyifitholdsforY=¡(DT
zuDzu)¡1BT
u;
inthiscase,Q>0mustsatisfy
QAT+AQ+QCT
zCzQ¡Bu¡
DT
zuDzu¢¡1BT
u·0: (7.41)
Next,itcanbeshownbystandard manipulations thatifQ>0satis¯es(7.41),thenQ·Qare.
Therefore foreveryinitialconditionx(0),wehavex(0)TQ¡1
arex(0)·x(0)TQ¡1x(0),and
therefore theoptimalvalueoftheEVPisjustx(0)TQ¡1
arex(0),andtheoptimalstate-feedbac k
gainisKopt=YoptQ¡1
are=¡(DT
zuDzu)¡1BT
uPare.
Itisinteresting tonotethatwithP=Q¡1,theEVPisequivalenttominimizing x(0)TPx(0)
subjectto
ATP+PA+CT
zCz¡PBu¡
DT
zuDzu¢¡1BT
uP·0; (7.42)
whichisnotaconvexconstrain tinP.However,theproblem ofmaximizing x(0)TPx(0)
subjectP>0andtheconstrain t
ATP+PA¡PBu(DT
zuDzu)¡1BT
uP+CzCT
z¸0;
whichisnothingotherthan(7.42),butwiththeinequalit yreversed,isanEVP;itiswell-
known(seeforexample [Wil71b])thatthisEVPisanotherformulationoftheLQRproblem.
Staticversusdynamicstate-feedback
AnLTIsystemcanbestabilized usingdynamic state-feedbac kifandonlyifitcanbestabi-
lizedusingstaticstate-feedbac k(seeforexample, [Kai80]).Infact,similarstatemen tscan
bemadeforallthepropertiesandforalltheLDIsconsidered inthischapter,providedthe
propertiesarespeci¯edusingquadratic Lyapunovfunctions. Wewilldemonstrate thisfact
ononesimpleproblem considered inthischapter.
Consider theLTIsystem
_x=Ax+Buu+Bww;x(0)=0;z=Czx: (7.43)
Fromx7.5.1,theexistsastaticstate-feedbac ku=KxsuchthattheL2gainfromwtoz
doesnotexceed°iftheLMIinvariablesQ>0andYisfeasible:
"
AQ+QAT+BuY+YTBT
u+BwBT
wQCT
z
CzQ ¡°2I#
·0:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
116 Chapter 7State-Feedba ckSynthesis forLDIs
(Infactthiscondition isalsonecessary ,butthisisirrelevanttothecurrentdiscussion.)
Eliminating Y,wegettheequivalentLMIinQ>0andscalar¾:
AQ+QAT+BwBT
w+QCT
zCzQ
°2·¾BuBT
u: (7.44)
Wenextconsider adynamic state-feedbac kforthesystem:
_xc=Acxc+Bcx;xc(0)=0;u=Ccxc+Dcx: (7.45)
WewillshowthatthereexistmatricesAc,Bc,CcandDcsuchthattheL2gainofthe
system(7.43,7.45) doesnotexceed°ifandonlyiftheLMI(7.44)holds.Thiswillestablish
thatthesmallest upperboundontheL2gain,provableviaquadratic Lyapunovfunctions,
isthesameirrespectiveofwhether astaticstate-feedbac koradynamic state-feedbac kis
employed.
Wewillshowonlythepartthatisnotobvious.Withstatevectorxcl=[xTxT
c]T,the
closed-lo opsystemis
_xcl=Aclxcl+Bclu;xcl(0)=0;z=Cclxcl;
whereAcl=Abig+BbigKbigwith
Abig="
A0
00#
;Bbig="
0Bu
I0#
;Kbig="
BcAc
DcCc#
;
Bcl="
Bw
0#
;Ccl=[Cz0]:
Fromx6.3.2,theL2gainoftheclosed-lo opsystemdoesnotexceed°iftheLMIinPbig>0
isfeasible:
"
AT
bigP+PbigA+PBbigKbig+KT
bigBT
bigP+CT
clCclPBcl
BT
clP ¡°2I#
·0:
Eliminating KbigfromthisLMIyieldstwoequivalentLMIsinPbig>0andascalar¿.
"
AT
bigP+PbigA¡¿I+CT
clCclPBcl
BT
clP¡°2I#
·0;
"
AT
bigP+PbigA¡¿PBbigBT
bigP+CT
clCclPBcl
BT
clP ¡°2I#
·0:
Itiseasytoverifythatthe¯rstLMIholdsforlargeenough¿,whilethesecondreducesto
¿>0and
AQ11+Q11AT+BwBT
w
°2+Q11CzCT
zQ11·¿BuBT
u;
whereQ11istheleadingn£nblockofP¡1
big.ThelastLMIisprecisely (7.44)withthechange
ofvariablesQ=°2Q11and¾=°2¿.
Asimilarresultisestablished [PZP+92];otherreferences thatdiscussthequestion ofwhen
dynamic ornonlinear feedbackdoesbetterthanstaticfeedbackincludePetersen, Khar-
gonekarandRotea[Pet85,KPR88,RK88,KR88,Pet88,RK89].
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 117
Dynamicoutput-feedbac kforLDIs
Anaturalextension totheresultsofthischapteristhesynthesisofoutput-feedbac ktomeet
variousperformance speci¯cations. Wesuspect,however,thatoutput-feedbac ksynthesis
problems havehighcomplexit yandaretherefore unlikelytoberecastasLMIproblems.
Severalresearchershaveconsidered suchproblems: SteinbergandCorless[SC85],Galimidi
andBarmish [GB86],andGeromel, Peres,deSouzaandSkelton[PGS93];seealso[HCM93].
Onevariationofthisproblem isthe\reduced-order controllerproblem", thatis,¯ndingdy-
namicoutput-feedbac kcontrollerswiththesmallest possiblenumberofstates.Itisshown
in[PZPB91,EG93,Pac94]thatthisproblem canbereducedtotheproblem ofminimizing
therankofamatrixP¸0subjecttocertainLMIs.Thegeneralproblem ofminimizing the
rankofamatrixsubjecttoanLMIconstrain thashighcomplexit y;infact,aspecialrank
minimization problem canbeshowntobeequivalenttotheNP-hard zero-one linearpro-
gramming problem (seee.g.[Dav94]).Nevertheless severalresearchershavetriedheuristic,
localalgorithms forsuchproblems andreportpractical success. Othersstudying thistopic
includeIwasaki,Skelton,Geromel anddeSouza[IS93b,ISG93,GdSS93,SIG93].
Observer-basedcontrollersfornonlinearsystems
WeprovethattheLMIs(7.33)and(7.34)(inthevariablesP>0,Q>0,WandY)ensure
theexistence ofanobserver-based controlleroftheform(7.31)whichmakesthesystem(7.30)
quadratically stable.
Westartbyrewriting thesystem(7.32)inthecoordinates ¹x,¹x¡x:
"
_¹x
_¹x¡_x#
="
LCy(¹x¡x)+f(¹x)+BuK¹x
f(¹x)¡f(x)+LCy(¹x¡x)#
: (7.46)
Usingtheresultsofx4.3,wecanwrite
"
_¹x
_¹x¡_x#
2Co©~Aij¯¯1·i·M;1·j·Mª"
¹x
¹x¡x#
with
~Aij="
Ai+BuKLCy
0Aj+LCy#
;i;j=1;:::;M:
Simplestate-space manipulations showthesystem(7.32)isquadratically stableifandonlyif
thesystem(7.46)isquadratically stable.Therefore, thesystem(7.32)isquadratically stable
ifthereexists~P>0suchthat
~AT
ij~P+~P~Aij<0;i;j=1;:::;M: (7.47)
WewillnowshowthattheLMI(7.47)hasasolutionifandonlyiftheLMIs(7.33)and(7.34)
do.
Firstsupposethat(7.47)hasasolution ~P.With~Ppartitioned asn£nblocks,
~P="
P11P12
PT
12P22#
;
weeasilycheckthattheinequalit y(7.47)implies
(Ai+BuK)TP11+P11(Ai+BuK)<0;i=1;:::;M: (7.48)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
118 Chapter 7State-Feedba ckSynthesis forLDIs
WiththechangeofvariablesQ=P¡1
11andY=BuQ,theinequalit y(7.48)isequivalent
to(7.33).
De¯ne~Q=~P¡1.Theinequalit y(7.47)becomes
~Aij~Q+~Q~AT
ij<0;i;j=1;:::;M:
Denoting byQ22thelower-rightn£nblockof~Q,weseethatthisinequalit yimplies
(Ai+LCy)Q22+Q22(Ai+LCy)T<0;i=1;:::;M: (7.49)
WithP=Q¡1
22andW=PL,theinequalit y(7.49)isequivalentto(7.34).
Conversely,supposethatP,Q,YandWsatisfytheLMIs(7.34)and(7.33)andde¯ne
L=P¡1W,K=YQ¡1.Wenowprovethereexistsapositive¸suchthat
~P="
¸Q¡10
0P#
satis¯es(7.47)andtherefore provestheclosed-lo opsystem(7.32)isquadratically stable.We
compute
~AT
ij~P+~P~Aij=2
6664¸Ã
(Ai+BuK)TQ¡1
+Q¡1(Ai+BuK)!
¸Q¡1LCy
¸(LCy)TQ¡1 (Aj+LCy)TP
+P(Aj+LCy)3
7775;
fori;j=1;:::;M.Therefore, usingSchurcomplemen ts,theclosed-lo opsystem(7.32)is
quadratically stableif¸>0satis¯es
¸(Q¡1LCy((Aj+LCy)TP+P(Aj+LCy))¡1(LCy)TQ¡1)
¡(A+BuK)TQ¡1¡Q¡1(A+BuK)>0
fori;j=1;:::;M.Thiscondition issatis¯ed forany¸>0suchthat
¸min
1·j·M¹j>max
1·i·Mºi;
where
¹j=¸min(Q¡1LCy((Aj+LCy)TP+P(Aj+LCy))¡1(LCy)TQ¡1);j=1;:::;M
and
ºi=¸max((Ai+BuK)TQ¡1+Q¡1(Ai+BuK));i=1;:::;M:
Since(7.34)and(7.33)aresatis¯ed, sucha¸alwaysexists.
Problems ofobserver-based controllerdesignalongwithquadratic Lyapunovfunctions that
provestabilityhavebeenconsidered byBernstein andHaddad [BH93]andYaz[Yaz93].
Gain-scheduledorparameter-dep endentcontrollers
Severalresearchershaveextended theresultsonstate-feedbac ksynthesistohandlethecase
inwhichthecontrollerparameters candependonsystemparameters. Suchcontrollersare
calledgain-scheduled orparameter-dep endent.Wereferthereadertothearticlescitedfor
preciseexplanations ofwhatthesecontrollawsare,andwhichdesignproblems canberecast
asLMIproblems. Lu,ZhouandDoylein[LZD91]andBeckerandPackard[Bec93,BP91]
consider theproblem inthecontextofNLDIs. Otherrelevantreferences include[PB92,
Pac94,PBPB93,BPPB93,GA94,IS93a,IS93b,AGB94].
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Chapter 8
Lur'eandMultiplier Methods
8.1AnalysisofLur'eSystems
Weconsider theLur'esystem
_x=Ax+Bpp+Bww;q=Cqx;
z=Czx;pi(t)=Ái(qi(t));i=1;:::;np;(8.1)
wherep(t)2Rnp,andthefunctionsÁisatisfythe[0;1]sectorconditions
0·¾Ái(¾)·¾2forall¾2R; (8.2)
or,equivalently,
Ái(¾)(Ái(¾)¡¾)·0forall¾2R:
Thedatainthisproblem arethematricesA,Bp,Bw,CqandCz.Theresultsthat
followwillholdforanynonlinearities Áisatisfying thesectorconditions. Incases
wheretheÁiareknown,however,theresultscanbesharpened.
Itispossibletohandlethemoregeneralsectorconditions
®i¾2·¾Ái(¾)·¯i¾2forall¾2R;
where®iand¯iaregiven.Suchsystems arereadilytransformed totheformgiven
in(8.1)and(8.2)byalooptransformation (seetheNotesandReferences).
Animportantspecialcaseof(8.1),(8.2)occurswhenthefunctionsÁiarelinear,
i.e.,Ái(¾)=±i¾,where±i2[0;1].Intheterminology ofcontroltheory,thisisreferred
toasasystemwithunknown-but-constan tparameters. Itisimportanttodistinguish
thiscasefromthePLDIobtained with(8.1),Ái(¾)=±i(t)¾and±i(t)2[0;1],which
isreferredtoasasystemwithunknown,time-varyingparameters.
Ouranalysis willbebasedonLyapunovfunctions oftheform
V(»)=»TP»+2npX
i=1¸iZCi;q»
0Ái(¾)d¾; (8.3)
whereCi;qdenotestheithrowofCq.Thusthedatadescribing theLyapunovfunction
arethematrixPandthescalars¸i,i=1;:::;np.WerequireP>0and¸i¸0,
whichimpliesthatV(»)¸»TP»>0fornonzero».
NotethattheLyapunovfunction (8.3)dependsonthe(notfullyspeci¯ed)non-
linearities Ái.ThusthedataPand¸ishouldbethoughtofasprovidingarecipefor
119
120 Chapter 8Lur'eandMultiplier Methods
constructing aspeci¯cLyapunovfunction giventhenonlinearities Ái.Asanexam-
ple,consider thespecialcaseofasystemwithunknown-but-constan tparameters, i.e.,
Ái(¾)=±i¾.TheLyapunovfunction willthenhavetheform
V(»)=»T¡
P+CT
q¢¤Cq¢
»
where¢=diag(±1;:::;±np)and¤=diag(¸1;:::;¸np).Inotherwords,wearereally
synthesizing aparameter-dep endentquadratic Lyapunovfunction forourparameter-
dependentsystem.
8.1.1Stability
Wesettheexogenous inputwtozero,andseekPand¸isuchthat
dV(x)
dt<0forallnonzeroxsatisfying (8.1)and(8.2). (8.4)
Since
dV(x)
dt=2Ã
xTP+npX
i=1¸ipiCi;q!
(Ax+Bpp);
condition (8.4)holdsifandonlyif
Ã
»TP+npX
i=1¸i¼iCi;q!
(A»+Bp¼)<0
forallnonzero»satisfying
¼i(¼i¡ci;q»)·0;i=1;:::;np: (8.5)
Itiseasilyshownthat
f(»;¼)j»6=0;(8:5)g=f(»;¼)j»6=0or¼6=0;(8:5)g:
TheS-procedurethenyieldsthefollowingsu±cientcondition for(8.4):TheLMIin
P>0,¤=diag(¸1;:::;¸np)¸0andT=diag(¿1;:::;¿np)¸0
"
ATP+PAPBp+ATCT
q¤+CT
qT
BT
pP+¤CqA+TCq¤CqBp+BT
pCT
q¤¡2T#
<0 (8.6)
holds.
Remark :Whenweset¤=0weobtaintheLMI
"
ATP+PAPBp+CT
qT
BT
pP+TCq¡2T#
<0;
whichcanbeinterpreted asacondition fortheexistence ofaquadratic Lyapunov
function fortheLur'esystem,asfollows.Performalooptransformation onthe
Lur'esystemtobringthenonlinearities tosector[¡1;1].ThentheLMIabove
isthesame(towithinascalingofthevariableT)asonethatyieldsquadratic
stabilityoftheassociatedDNLDI; seex5.1.
Remark :Condition (8.6)isonlyasu±cientcondition fortheexistence ofaLur'e
Lyapunovfunction thatprovesstabilityofsystem(8.1).Itisalsonecessary when
thereisonlyonenonlinearit y,i.e.,whennp=1.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
8.1AnalysisofLur'eSystems 121
8.1.2Reachablesetswithunit-energy inputs
Weconsider thesetreachablewithunit-energy inputs,
Rue¢=8
>><
>>:x(T)¯¯¯¯¯¯¯¯x,wsatisfy(8.3);x(0)=0
ZT
0wTwdt·1;T¸09
>>=
>>;:
Theset
F=(
»¯¯¯¯¯»TP»+2npX
i=1¸iZCi;q»
0Ái(¾)d¾·1)
containsRueif
d
dtV(x)·wTwforallxandwsatisfying (8.1)and(8.2). (8.7)
FromtheS-procedure,thecondition (8.7)holdsifthereexistsT=diag(¿1;:::;¿np)¸
0suchthat
2
64ATP+PAPBp+ATCT
q¤+CT
qTPBw
BT
pP+¤CqA+TCq¤CqBp+BT
pCT
q¤¡2T¤CqBw
BT
wP BT
wCT
q¤ ¡I3
75·0 (8.8)
holds,where¤=diag(¸1;:::;¸np)¸0.
Ofcourse,theellipsoidE=f»2Rn¯¯»TP»·1gcontainsF,sothatE¶
F¶Rue.Therefore,Egivesanouterapproximation ofRueifthenonlinearities
arenotknown.Ifthenonlinearities Áiareknown,Fgivesapossiblybetterouter
approximation. Wecanusetheseresultstoprovethatapointx0doesnotbelongto
thereachablesetbysolvingappropriate LMIPs.
8.1.3Outputenergybounds
Weconsider thesystem(8.1)withinitialconditionx(0),andcompute upperbounds
ontheoutputenergy
J=Z1
0zTzdt
usingLyapunovfunctionsVoftheform(8.3).
If
d
dtV(x)+zTz·0forallxsatisfying (8.1); (8.9)
thenJ·V(x(0)).Thecondition (8.9)isequivalentto
2Ã
»TP+npX
i=1¸i¼iCi;q!
(A»+Bp¼)+»TCT
zCz»·0;
forevery»satisfying
¼i(¼i¡ci;q»)·0;i=1;:::;np:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
122 Chapter 8Lur'eandMultiplier Methods
UsingtheS-procedure,weconclude thatthecondition (8.9)holdsifthereexists
T=diag(¿1;:::;¿np)¸0suchthat
"
ATP+PA+CT
zCzPBp+ATCT
q¤+CT
qT
BT
pP+¤CqA+TCq¤CqBp+BT
pCT
q¤¡2T#
·0: (8.10)
SinceV(x(0))·x(0)T(P+CT
q¤Cq)x(0),anupperboundonJisobtained
bysolvingthefollowingEVPinthevariablesP,¤=diag(¸1;:::;¸np)andT=
diag(¿1;:::;¿np):
minimize x(0)T¡
P+CT
q¤Cq¢
x(0)
subjectto(8:10);T¸0;¤¸0;P>0
Ifthenonlinearities Áiareknown,wecanobtainpossiblybetterboundsonJby
modifyingtheobjectiveintheEVPappropriately .
8.1.4 L2gain
WeassumethatDzw=0forsimplicity.IfthereexistsaLyapunovfunction ofthe
form(8.3)and°¸0suchthat
d
dtV(x)·°2wTw¡zTzforallxandwsatisfying (8.1); (8.11)
thentheL2gainofthesystem(8.1)doesnotexceed°.Thecondition (8.11)is
equivalentto
2Ã
»TP+npX
i=1¸i¼iCi;q!
(A»+Bp¼)·°2wTw¡»TCT
zCz»
forany»satisfying¼i(¼i¡ci;q»)·0;i=1;:::;np.UsingtheS-procedure,thisis
satis¯ed ifthereexistsT=diag(¿1;:::;¿np)¸0suchthat
2
64ATP+PA+CT
zCzPBp+ATCT
q¤+CT
qTPBw
BT
pP+¤CqA+TCq¤CqBp+BT
pCT
q¤¡2T¤CqBw
BT
wP BT
wCT
q¤¡°2I3
75·0:(8.12)
Thesmallest upperboundontheL2gain,provableusingLur'eLyapunovfunctions,
istherefore obtained byminimizing °over°,P,¤andTsubjecttoP>0,¤=
diag(¸1;:::;¸np)¸0,T=diag(¿1;:::;¿np)¸0and(8.12).ThisisanEVP.
8.2IntegralQuadraticConstraints
Inthissectionweconsider animportantvariation ontheNLDI,inwhichthepoint-
wiseconstrain tp(t)Tp(t)·q(t)Tq(t)isreplaced withaconstrain tontheintegralsof
p(t)Tp(t)andq(t)Tq(t):
_x=Ax+Bpp+Buu+Bww;
q=Cqx+Dqpp+Dquu+Dqww;
z=Czx+Dzpp+Dzuu+DzwwZt
0p(¿)Tp(¿)d¿·Zt
0q(¿)Tq(¿)d¿:(8.13)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
8.2Integral Quadra ticConstraints 123
SuchasystemiscloselyrelatedtotheNLDIwiththesamedatamatrices; indeed,
everytrajectoryoftheassociatedNLDIisatrajectoryof(8.13).Thissystemis
not,however,adi®erentialinclusion. Incontroltheoryterms,thesystem(8.13)is
describedasalinearsystemwith(dynamic) nonexpansiv efeedback.Animportant
example iswhenpandqarerelatedbyalinearsystem,i.e.,
_xf=Afxf+Bfq;p=Cfxf+Dfq;xf(0)=0;
wherekDf+Cf(sI¡Af)¡1Bfk1·1.
Wecanalsoconsider ageneralization oftheDNLDI, inwhichwehavecompo-
nentwiseintegralquadratic constrain ts.Moreover,wecanconsider integralquadratic
constrain tswithdi®erentsectorbounds;asanexample wewillencounterconstrain ts
oftheformRt
0p(¿)Tq(¿)d¿¸0inx8.3.
WewillnowshowthatmanyoftheresultsonNLDIs(andDNLDIs) fromChap-
ters5{7generalize tosystems withintegralquadratic constrain ts.Wewilldemon-
stratethisgeneralization forstabilityanalysis andL2gainbounds,leavingothersto
thereader.
AsinChapters 5{7,ouranalysiswillbebasedonquadratic functions ofthestate
V(»)=»TP»,androughlyspeaking,theverysameLMIswillarise.However,the
interpretation ofVisdi®erenthere.Forexample, instabilityanalysis, theVof
Chapters 5{7decreases monotonically tozero.Here,theverysameVdecreases to
zero,butnotnecessarily monotonically .Thus,VisnotaLyapunovfunction inthe
conventionalsense.
8.2.1Stability
Consider system(8.13)withoutwandz.SupposeP>0and¸¸0aresuchthat
d
dtxTPx<¸¡
pTp¡qTq¢
; (8.14)
or
d
dtµ
xTPx+¸Zt
0¡
q(¿)Tq(¿)¡p(¿)Tp(¿)¢
d¿¶
<0:
Notethatthesecondtermisalwaysnonnegativ e.Usingstandard argumen tsfrom
Lyapunovtheory,itcanbeshownthatlimt!1x(t)=0,orthesystemisstable.
Now,letusexamine thecondition (8.14).Itisexactlythesameasthecondition
obtained byapplying theS-proceduretothecondition
d
dtxTPx<0;wheneverpTp·qTq;
whichinturn,leadstotheLMIcondition forquadratic stabilityofNLDI(5.3)
P>0;¸¸0;
2
4ATP+PA+¸CT
qCqPBp+¸CT
qDqp
(PBp+¸CT
qDqp)T¡¸(I¡DT
qpDqp)3
5<0:
8.2.2 L2gain
Weassumethatx(0)=0and,forsimplicity,thatDzw=0andDqp=0.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
124 Chapter 8Lur'eandMultiplier Methods
SupposeP>0and¸¸0aresuchthat
d
dtxTPx<¸¡
pTp¡qTq¢
+°2wTw¡zTz: (8.15)
Integrating bothsidesfrom0toT,
x(T)TPx(T)+¸ZT
0¡
q(¿)Tq(¿)¡p(¿)Tp(¿)¢
d¿<ZT
0¡
°2wTw¡zTz¢
dt:
ThisimpliesthattheL2gainfromwtozforsystem(8.13)doesnotexceed°.
Condition (8.15)leadstothesameLMI(6.55)whichguaranteesthattheL2gain
oftheNLDI(4.9)fromwtozdoesnotexceed°.
Remark :AlmostalltheresultsdescribedinChapters 5,6and7extendimme-
diatelytosystems withintegralquadratic constrain ts.Theonlyexceptions are
theresultsoncoordinate transformations (x5.1.1,alsox7.2.2),andonreachable
setswithunit-peakinputs(x6.1.3andx7.3.3).
8.3MultipliersforSystemswithUnknownParameters
Weconsider thesystem
_x=Ax+Bpp+Bww;q=Cqx+Dqpp;
pi=±iqi;i=1;:::;np;z=Czx+Dzww;(8.16)
wherep(t)2Rnp,and±i,i=1;:::;npareanynonnegativ enumbers.Wecanconsider
themoregeneralcasewhen®i·±i·¯iusingalooptransformation (seetheNotes
andReferences).
Forreasonsthatwillbecomeclearshortly,webeginbyde¯ningnpLTIsystems
withinputqiandoutputqm;i,whereqm;i(t)2R:
_xm;i=Am;ixm;i+Bm;iqi;xm;i(0)=0;
qm;i=Cm;ixm;i+Dm;iqi;(8.17)
whereweassumethatAm;iisstableand(Am;i;Bm;i)iscontrollable. Wefurther
assumethat
Zt
0qi(¿)qm;i(¿)d¿¸0forallt¸0.
Thislastcondition isequivalenttopassivityoftheLTIsystems(8.17),whichinturn
isequivalenttotheexistence ofPi¸0,i=1;:::;np,satisfying
"
AT
m;iPi+PiAm;iPiBm;i¡CT
m;i
BT
m;iPi¡Cm;i¡¡
Dm;i+DT
m;i¢#
·0;i=1;:::;np: (8.18)
(Seex6.3.3.)
Thus,qandqm¢=[qm;1;:::;qm;n p]Taretheinputandoutputrespectivelyofthe
diagonal systemwithrealization
Am=diag(Am;i);Bm=diag(Bm;i);Cm=diag(Cm;i);Dm=diag(Dm;i);
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
8.3Multipliers forSystems withUnkno wnParameters 125
De¯ning
~A="
A0
BmCqAm#
;~Bp="
Bp
BmDqp#
;~Bw="
Bw
0#
;
~Cq=[DmCqCm];~Dqp=DmDqp;~Cz=[Cz0]:
consider thefollowingsystemwithintegralquadratic constrain ts:
d~x
dt=~A~x+~Bpp+~Bww;
qm=~Cq~x+~Dqpp;
z=~Cz~x+~Dqpp+Dzww;Zt
0pi(¿)qm;i(¿)d¿¸0;i=1;:::;np:(8.19)
Itiseasytoshowthatifxisatrajectoryof(8.16),then[xTxT
m]Tisatrajectory
of(8.19)forsomeappropriate xm.Therefore, conclusions about(8.16)haveimplica-
tionsfor(8.19).Forinstance, ifsystem(8.19)isstable,soissystem(8.16).
8.3.1Stability
Consider system(8.19)withoutwandz.SupposeP>0issuchthat
d
dt~xTP~x+2npX
i=1piqm;i<0; (8.20)
or
d
dtÃ
~xTP~x+2npX
i=1Zt
0pi(¿)qm;i(¿)d¿!
<0:
Then,itcanbeshownusingstandard argumen tsthatlimt!1~x(t)=0,i.e.,the
system(8.19)isstable.
Condition (8.20)isjusttheLMIinP>0,CmandDm:
"~ATP+P~AP~Bp+~CT
q
~BT
pP+~Cq~Dqp+~DT
qp#
<0: (8.21)
Remark :LMI(8.21)canalsobeinterpreted asapositivedissipation con-
ditionfortheLTIsystemwithtransfer matrixG(s)=¡W(s)H(s),where
W(s)=Cm(sI¡Am)¡1Bm+DmandH(s)=Cq(sI¡A)¡1Bp+Dqp.Therefore,
weconclude thatsystem(8.16)isstableifthereexistsapassiveWsuchthat
¡W(s)H(s)haspositivedissipation. HenceWisoftencalleda\multiplier". See
theNotesandReferences fordetails.
8.3.2Reachablesetswithunit-energy inputs
Thesetofreachablestateswithunit-energy inputsforthesystem
_x=Ax+Bpp+Bww;q=Cqx+Dqpp;
p=±q;±¸0;(8.22)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
126 Chapter 8Lur'eandMultiplier Methods
is
Rue¢=8
>><
>>:x(T)¯¯¯¯¯¯¯¯x,wsatisfy(8.22);x(0)=0
ZT
0wTwdt·1;T¸09
>>=
>>;:
Wecanobtainasu±cientcondition forapointx0tolieoutsidethereachablesetRue,
usingquadratic functions fortheaugmentedsystem.SupposeP>0issuchthat
d
dt~xTP~x+2npX
i=1piqm;i<wTw; (8.23)
or
~xTP~x+2npX
i=1Zt
0pi(¿)qm;i(¿)d¿<Zt
0w(¿)Tw(¿)d¿:
thenthesetE=©
~x¯¯~xTP~x·1ª
containsthereachablesetforthesystem(8.19).
Condition (8.23)isequivalenttotheLMIinP>0,CmandDm:
2
64~ATP+P~AP~BwP~Bp+~CT
q
~BT
wP¡I 0
~BT
pP+~Cq0~Dqp+~DT
qp3
75<0: (8.24)
Now,thepointx0liesoutsidethesetofreachablestatesforthesystem(8.22)if
thereexistsPsatisfying (8.23)andthereexistsnozsuchthat
"
x0
z#T
P"
x0
z#
·1:
Partitioning Pconformally withthesizesofx0andzas
P="
P11P12
PT
12P22#
;
wenotethat
minz"
x0
z#T"
P11P12
PT
12P22#"
x0
z#
=xT
0(P11¡P12P¡1
22PT
12)x0:
Therefore, x0doesnotbelongtothereachablesetforthesystem(8.22)ifthereexist
P>0,Pi¸0,¸¸0,Cm,andDMsatisfying (8.18),(8.24)and
"
xT
0P11x0¡1xT
0P12
PT
12x0P22#
>0:
ThisisanLMIP.
Notes andReferences
Lur'esystems
TheNotesandReferences ofChapters 5and7arerelevanttothischapteraswell.Lur'e
andPostnikovwerethe¯rsttoproposeLyapunovfunctions consisting ofaquadratic form
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 127
plustheintegralofthenonlinearit y[LP44].Thebook[Lur57]followedthisarticle,inwhich
theauthorshowedhowtosolvetheproblem of¯ndingsuchLyapunovfunctions analyti-
callyforsystemsoforder2or3.Another earlybookthatusessuchLyapunovfunctions is
Letov[Let61].WorkonthistopicwascontinuedbyPopov[Pop62],Yakubovich[Yak64].
Tsypkin [Tsy64d,Tsy64c,Tsy64a,Tsy64b],Szego[Sze63]andMeyer[Mey66].Jury
andLee[JL65]consider thecaseofLur'esystemswithmultiplenonlinearities andderivea
correspondingstabilitycriterion. Seealso[IR64].
Theellipsoid methodisusedtoconstruct Lyapunovfunctions fortheLur'estabilityproblem
inthepapersbyPyatnitskii andSkorodinskii[PS82,PS83].Seealso[Sko91a,Sko91b]for
generalnumerical methodstoprovestabilityofnonlinear systems viathePopovorcircle
criterion. Aprecursor isthepaperbyKarmark arandSiljak[KS75],whousedalocallycon-
vergentalgorithm todetermine margins foraLur'e-Postnikovsystemwithonenonlinearit y.
Indeed,intheearlybooksbyLur'e,Postnikov,andLetovtheproblem ofconstructing Lya-
punovfunctions reducestothesatisfaction ofsomeinequalities. Insomespecialcasesthey
evenpointoutgeometrical propertiesoftheregionsde¯nedbytheinequalities, i.e.,thatthey
formaparallelogram. Butasfarasweknow,convexityoftheregionsisnotnotedinthe
earlywork.Inanycase,itwasnotknowninthe1940'sand1950'sthatsolution ofconvex
inequalities ispractically andtheoretically tractable.
ConstructionofLyapunovfunctionsforsystemswithintegralquadraticconstraints
Wesawinx8.2.1thatthequadratic positivefunctionV(»)=»TP»isnotnecessarily a
Lyapunovfunction intheconventionalsenseforthesystem(8.13)withintegralquadratic
constrain ts;althoughV(x(t))!0ast!1,itdoesnotdosomonotonically .Wenowshow
howwecanexplicitly construct aLyapunovfunction givenP,incaseswhenpandqare
relatedby
_xf=f(xf;q;t);p=g(xf;q;t); (8.25)
wherexf(t)2Rnf.
SinceRt
0pTpd¿·Rt
0qTqd¿,itcanbeshown(seeforexample [Wil72])thatVf:Rnf!R+
givenby
Vf(»f)¢=sup½
¡ZT
0¡
qTq¡pTp¢
dt¯¯¯¯xf(0)=»f;xfsatis¯es(8.25);T¸0¾
isaLyapunovfunction thatprovesstabilityofsystem(8.25). Moreover,theLyapunov
functionV:Rn£Rnf!R+givenby
V(»;»f)¢=»TP»+Vf(»f)
provesstabilityoftheinterconnection
_x=Ax+Bpp;q=Cqx+Dqpp;
_xf=f(xf;q;t);p=g(xf;q;t):
Notethatthisconstruction alsoshowsthat»TP»provesthestabilityoftheNLDI(5.2).
Integralquadratic constrain tshaverecentlyreceivedmuchattention,especiallyintheformer
SovietUnion.Seeforexample [Yak88,Meg92a,Meg92b,Yak92,Sav91,SP94b,SP94h,
SP94d].
Systemswithconstant,unknownparameters
Problems involvingsystem(8.16)arewidelystudiedinthe¯eldofrobustcontrol,sometimes
underthename\real-¹problems"; seeSiljak[Sil89]forasurvey.Theresultsofx8.3were
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
128 Chapter 8Lur'eandMultiplier Methods
derivedbyFan,TitsandDoylein[FTD91]usinganalternate approach,involvingtheap-
plication ofano®-axiscirclecriterion (seealsox3.3).Wealsomention[TV91],whereTesi
andVicinostudyahybridsystem,consisting ofaLur'esystemwithunknownparameters.
Thereareelegantanalytic solutions forsomeveryspecialproblems involvingparameter-
dependentlinearsystems, e.g.,Kharitono v'stheorem [Kha78]anditsextensions [BHL89].
AnicesurveyoftheseresultscanbefoundinBarmish [Bar93].
Complexityofstabilizationproblems
Thestability(andperformance) analysis problems considered inx8.3havehighcomplexit y.
Forexample, checkingif(8.16)isstableisNP-hard (seeCoxsonandDemarco [CD91],Braatz
andYoung[BYDM93 ],PoljakandRohn[PR94],andNemirovskii[Nem94]).Weconjecture
thatcheckingwhether ageneralDNLDIisstableisalsoNP-hard.
Likewise,manystabilization problems foruncertain systemsareNP-hard. Forinstance, the
problem ofcheckingwhether system(8.16)isstabilizable byaconstantstate-feedbac kcan
beshowntobeNP-hard usingthemethodof[Nem94].
Arelatedresult,duetoBlondel andGevers[BG94],statesthatcheckingwhether there
existsacommon stabilizing LTIcontrollerforthreeLTIsystemsisundecidable. Incontrast,
checkingwhether thereexistsastatic,output-feedbac kcontrollawforasingleLTIsystem
isrationally decidable [ABJ75].
Multipliermethods
See[Wil69a,NT73,DV75,Sil69,Wil70,Wil76]fordiscussion ofandbibliograph yon
multiplier theoryanditsconnections toLyapunovstabilitytheory.Ingeneralmultiplier
theoryweconsider thesystem
_x=Ax+Bpp+Bww;q=Cqx+Dqpp;p(t)=¢(q;t):
ThemethodinvolvescheckingthatHm=W¡
Cq(sI¡A)¡1Bp+Dqp¢
satis¯es
Hm(j!)+Hm(j!)¤¸0forevery!2R;
forsomechoiceofWfromasetthatisdetermined bythepropertiesof¢.When¢is
anonlinear time-invariantmemoryless operatorsatisfying asectorcondition (i.e.,whenwe
haveaLur'esystem), themultipliersWareoftheform(1+qs)forsomeq¸0.Thisis
thefamousPopovcriterion [Pop62,Pop64];wemustalsociteYakubovich,whoshowed,
usingtheS-procedure[Yak77],thatfeasibilityofLMI(8.6)isequivalenttotheexistence of
qnonnegativ esuchthat(1+qs)(cq(sI¡A)¡1bp)+1=khaspositivedissipation.
Inthecasewhen¢(q;t)=±q(t)forsomeunknownrealnumber±,themultiplierWcanbe
anypassivetransfermatrix.ThisobservationwasmadebyBrockettandWillems in[BW65];
givenatransferfunctionHqptheyshowthatthetransferfunctionHqp=(1+kHqp)isstablefor
allvaluesofkin(0;1)ifandonlyifthereexistsapassivemultiplierWsuchthatWHqpis
alsopassive.Thiscasehasalsobeenconsidered in[CS92b,SC93,SL93a,SLC94]wherethe
authorsdeviseappropriate multipliers forconstantrealuncertain ties.Thepaper[BHPD94 ]
byBalakrishnan etal.,showshowvariousstabilitytestsforuncertain systems canbered-
erivedinthecontextofmultiplier theory,andhowthesetestscanbereducedtoLMIPs.See
also[LSC94].
SafonovandWyetzneruseaconvexparametrization ofmultipliers foundbyZamesand
Falb[ZF68,ZF67]toprovestabilityofsystemssubjectto\monotonic" or\odd-monotonic"
nonlinearities (see[SW87]fordetails;seealso[GG94]).HallandHow,alongwithBern-
steinandHaddad, generalize theuseofquadratic Lyapunovfunctions alongwithmultipliers
forvariousclassesofnonlinearities, andapplythemtoAerospace problems; seeforex-
ample[HB91a,How93,HH93b,HHHB92,HH93a,HHH93a,HCB93,HB93b,HB93a,
HHH93b ].Seealso[CHD93].
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 129
MultiplenonlinearitiesforLur'esystems
Ouranalysis methodforLur'esystems withmultiplenonlinearities inx8.1canbeconser-
vative,sincetheS-procedurecanbeconservativeinthiscase.Rapoport[Rap86,Rap87,
Rap88]devisesawayofdetermining ifthereexistsLyapunovfunctions oftheform(8.3)
nonconserv ativelybygenerating appropriate LMIs.Wenote,however,thatthenumberof
theseLMIsgrowsexponentiallywiththenumberofnonlinearities. Kamenetskii [Kam89]
andRapoport[Rap89]derivecorrespondingfrequency-domain stabilitycriteria. Kamenet-
skiicallshisresults\convolutionmethodforsolvingmatrixinequalities".
Looptransformations
Consider theLur'esystemwithgeneralsectorconditions, i.e.,
_x=Ax+Bp;q=Cx+Dp;
pi(t)=Ái(qi(t));®i¾2·¾Ái(¾)·¯i¾2:(8.26)
Herewehavedroppedthevariableswandzandthesubscripts onB,CandDtosimplify
thepresentation.Weassumethissystemiswell-posed,i.e.,det(I¡D¢)6=0foralldiagonal
¢with®i·¢ii·¯i.
De¯ne
¹pi¢=1
¯i¡®i(Ái(qi)¡®iqi)=¹Ái(qi):
Itisreadilyshownthat0·¾¹Ái(¾)·¾2forall¾.Let¤and¡denotethediagonal matrices
¤=diag(®1;:::;®nq);¡=diag(¯1¡®1;:::;¯nq¡®nq);
sothat¹p=¡¡1(p¡¤q).Wenowsubstitutep=¡¹p+¤qinto(8.26)and,usingourwell-
posedness assumption, solvefor_xandqintermsofxand¹p.Thisresultsin
_x=¡
A+B¤(I¡D¤)¡1C¢
x+B(I¡¤D)¡1¡¹p;
q=(I¡D¤)¡1Cx+(I¡D¤)¡1D¡¹p:
Wecantherefore express(8.26)as
_x=¹Ax+¹B¹p;q=¹Cx+¹D¹p;
¹pi(t)=¹Ái(qi(t));0·¾¹Ái(¾)·¾2:(8.27)
where
¹A=A+B¤(I¡D¤)¡1C;¹B=B(I¡¤D)¡1¡;
¹C=(I¡D¤)¡1C;¹D=(I¡D¤)¡1D¡:
Notethat(8.27)isinthestandard Lur'esystemform.SotoanalyzethemoregeneralLur'e
system(8.26),wesimplyapplythemethodsofx8.1totheloop-transformed system(8.27).
Inpractice, i.e.,inanumerical implemen tation,itisprobably bettertoderivetheLMIs
associatedwiththemoregeneralLur'esystemthantolooptransform tothestandard Lur'e
system.
Theconstruction aboveisalooptransformation thatmapsnonlinearities insector[®i;¯i]
intothestandard sector,i.e.,[0;1].Similartransformations canbeusedtomapanysetof
sectorsintoanyother,including so-called in¯nitesectors,inwhich,say,¯i=+1.
Thetermlooptransformation comesfromasimpleblock-diagram interpretation oftheequa-
tionsgivenabove.Fordetailed descriptions oflooptransformations, seethebookbyDesoer
andVidyasagar[DV75,p50].
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Chapter 9
SystemswithMultiplicativ eNoise
9.1AnalysisofSystemswithMultiplicativeNoise
9.1.1Mean-square stability
We¯rstconsider thediscrete-time stochasticsystem
x(k+1)=Ã
A0+LX
i=1Aipi(k)!
x(k); (9.1)
wherep(0),p(1),:::,areindependent,identicallydistributed randomvariableswith
Ep(k)=0;Ep(k)p(k)T=§=diag(¾1;:::;¾L): (9.2)
Weassumethatx(0)isindependentoftheprocessp.
De¯neM(k),thestatecorrelation matrix,as
M(k)¢=Ex(k)x(k)T:
Ofcourse,Msatis¯esthelinearrecursion
M(k+1)=AM(k)AT+LX
i=1¾2
iAiM(k)AT
i;M(0)=Ex(0)x(0)T:(9.3)
Ifthislinearrecursion isstable,i.e.,regardless ofx(0),limk!1M(k)=0,wesay
thesystemismean-squarestable.Mean-square stabilityimplies, forexample, that
x(k)!0almostsurely.
Itcanbeshown(seetheNotesandReferences) thatmean-square stabilityis
equivalenttotheexistence ofamatrixP>0satisfying theLMI
ATPA¡P+LX
i=1¾2
iAT
iPAi<0: (9.4)
Analternativ e,equivalentcondition isthatthedualinequalit y
AQAT¡Q+LX
i=1¾2
iAiQAT
i<0 (9.5)
holdsforsomeQ>0.
131
132 Chapter 9Systems withMultiplica tiveNoise
Remark :ForP>0satisfying (9.4),wecaninterpretthefunctionV(»)=»TP»
asastochasticLyapunov function:EV(x)decreases alongtrajectories of(9.1).
Alternativ ely,thefunctionV(M)=TrMPisa(linear)Lyapunovfunction for
thedeterministic system(9.3)(seetheNotesandReferences).
Remark :Mean-square stabilitycanbeveri¯eddirectlybysolvingtheLyapunov
equation inP
ATPA¡P+LX
i=1¾2
iAT
iPAi+I=0
andcheckingwhetherP>0.ThustheLMIPs(9.4)and(9.5)o®ernocomputa-
tional(ortheoretical) advantagesforcheckingmean-square stability.
Wecanconsider variations onthemean-square stabilityproblem, forexample,
determining themean-square stabilitymargin. Here,wearegivenA0;:::;AL,and
askedto¯ndthelargest°suchthatwith§<°2I,thesystem(9.1)ismean-square
stable. FromtheLMIPcharacterizations ofmean-square stability,wecanderive
GEVPcharacterizations thatyieldtheexactmean-square stabilitymargin. Again,
theGEVPo®ersnocomputational ortheoretical advantages,sincethemean-square
stabilitymargincanbeobtained bycomputing theeigenvalueofa(large)matrix.
9.1.2Statemeanandcovarianceboundswithunit-energy inputs
Wenowaddanexogenous inputwtoourstochasticsystem:
x(k+1)=Ax(k)+Bww(k)+LX
i=1(Aix(k)+Bw;iw(k))pi(k);x(0)=0:(9.6)
Weassumethattheexogenous inputwisdeterministic, withenergynotexceeding
one,i.e.,
1X
k=0w(k)Tw(k)·1:
Let¹x(k)denotethemeanofx(k),whichsatis¯es¹x(k+1)=A¹x(k)+Bww(k),andlet
X(k)denotethecovarianceofx(k),i.e.,X(k)=E(x(k)¡¹x(k))(x(k)¡¹x(k))T.We
willdevelopjointboundson¹x(k)andX(k).
SupposeV(»)=»TP»whereP>0satis¯es
EV(x(k+1))·EV(x(k))+w(k)Tw(k) (9.7)
foralltrajectories. Then,
EV(x(k))·kX
j=0w(j)Tw(j);k¸0;
whichimpliesthat,forallk¸0,
Ex(k)TPx(k)=¹x(k)TP¹x(k)+E(x(k)¡¹x(k))TP(x(k)¡¹x(k))
=¹x(k)TP¹x(k)+TrX(k)P
·1:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
9.1AnalysisofSystems withMultiplica tiveNoise 133
Letusnowexamine condition (9.7).Wenotethat
EV(x(k+1))="
¹x(k)
w(k)#T
M"
¹x(k)
w(k)#
+TrÃ
ATPA+LX
i=1¾2
iAT
iPAi!
X(k);
where
M="
AT
BT
w#
Ph
ABwi
+LX
i=1¾2
i"
AT
i
BT
w;i#
Ph
AiBw;ii
:
Next,theLMIconditions
"
AT
BT
w#
Ph
ABwi
+LX
i=1¾2
i"
AT
i
BT
w;i#
Ph
AiBw;ii
·"
P0
0I#
(9.8)
and
ATPA+LX
i=1¾2
iAT
iPAi<P (9.9)
imply
EV(x(k+1))·¹xT(k)P¹x(k)+w(k)Tw(k)
+TrÃ
ATPA+LX
i=1¾2
iAT
iPAi!
X(k)
·¹xT(k)P¹x(k)+w(k)Tw(k)
+E(x(k)¡¹x(k))TP(x(k)¡¹x(k))
=EV(x(k))+w(k)Tw(k);k¸0:
Ofcourse,LMI(9.8)impliesLMI(9.9);thus,weconclude thatLMI(9.8)im-
plies(9.7).Therefore, weconclude that¹x(k)andX(k)mustbelongtotheset
©
(¹x(k);X(k))¯¯¹x(k)TP¹x(k)+TrX(k)P·1ª
foranyP>0satisfying theLMI(9.8).Forexample, wehavetheboundTrM(k)P·1
onthestatecovariancematrix.
WecanderivefurtherboundsonM(k).Forexample, sincewehaveTrM(k)P·
¸min(P)TrM(k)andtherefore TrM(k)·1=¸min(P),wecanderiveboundsonTrM
bymaximizing ¸min(P)subjectto(9.8)andP>0.ThisisanEVP.Asanother
example, wecanderiveanupperboundon¸max(M)bymaximizing TrPsubject
to(9.8)andP>0,whichisanotherEVP.
9.1.3BoundonL2gain
Wenowconsider thesystem
x(k+1)=Ax(k)+Bww(k)+LX
i=1(Aix(k)+Bw;iw(k))pi(k);x(0)=0;
z(k)=Czx(k)+Dzww(k)+LX
i=1(Cz;ix(k)+Dzw;iw(k))pi(k):(9.10)
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
134 Chapter 9Systems withMultiplica tiveNoise
withthesameassumptions onp.Weassumethatwisdeterministic.
Wede¯netheL2gain´ofthissystemas
´2¢=sup(
E1X
k=0z(k)Tz(k)¯¯¯¯¯1X
k=0w(k)Tw(k)·1)
: (9.11)
SupposethatV(»)=»TP»,withP>0,satis¯es
EV(x(k+1))¡EV(x(k))·°2w(k)Tw(k)¡Ez(k)Tz(k): (9.12)
Then°¸´.Thecondition (9.12)canbeshowntobeequivalenttotheLMI
"
ABw
CzDzw#T"
P0
0I#"
ABw
CzDzw#
¡"
P0
0°2I#
+LX
i=1¾2
i"
AiBw;i
Cz;iDzw;i#T"
P0
0I#"
AiBw;i
Cz;iDzw;i#
·0:(9.13)
Minimizing °subjecttoLMI(9.13),whichisanEVP,yieldsanupperboundon´.
Remark :Itisstraightforwardtoapplythescalingmethoddevelopedinx6.3.4
toobtaincomponentwiseresults.
9.2State-FeedbackSynthesis
Wenowaddacontrolinpututooursystem:
x(k+1)=Ax(k)+Buu(k)+Bww(k)+
LX
i=1(Aix(k)+Bu;iu(k)+Bw;iw(k))pi(k);
z(k)=Czx(k)+Dzuu(k)+Dzww(k)+
LX
i=1(Cz;ix(k)+Dzu;iu(k)+Dzw;iw(k))pi(k):(9.14)
Weseekastate-feedbac ku(k)=Kx(k)suchthattheclosed-lo opsystem
x(k+1)=(A+BuK)x(k)+Bww(k)+
LX
i=1((Ai+Bu;iK)x(k)+Bw;iw(k))pi(k);
z(k)=(Cz+DzuK)x(k)+Dzww(k)+
LX
i=1((Cz;i+Dzu;iK)x(k)+Dzw;iw(k))pi(k);(9.15)
satis¯esvariousproperties.
9.2.1Stabilizabilit y
WeseekKandQ>0suchthat
(A+BuK)Q(A+BuK)T¡Q
+LX
i=1¾2
i(Ai+Bu;iK)Q(Ai+Bu;iK)T<0:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
9.2State-Feedba ckSynthesis 135
WiththechangeofvariableY=KQ,weobtaintheequivalentcondition
(AQ+BuY)Q¡1(AQ+BuY)T¡Q
+LX
i=1¾2
i(AiQ+Bu;iY)Q¡1(AiQ+Bu;iY)T<0;(9.16)
whichiseasilywrittenasanLMIinQandY.
Thus,system(9.14)ismean-square stabilizable (withconstantstate-feedbac k)if
andonlyiftheLMIQ>0,(9.16)isfeasible. Inthiscase,astabilizing state-feedbac k
gainisK=YQ¡1.
Asanextension, wecanmaximize theclosed-lo opmean-square stabilitymargin.
System(9.15)ismean-square stablefor§·°2Iifandonlyif
(AQ+BuY)Q¡1(AQ+BuY)T¡Q
+°2LX
i=1(AiQ+Bu;iY)Q¡1(AiQ+Bu;iY)T<0(9.17)
holdsforsomeYandQ>0.Therefore, ¯ndingKthatmaximizes °reducestothe
followingGEVP:
maximize °
subjecttoQ>0;(9:17)
9.2.2Minimizing theboundonL2gain
Wenowseekastate-feedbac kgainKwhichminimizes theboundonL2gain(as
de¯nedby(9.11))forsystem(9.15).´·°forsomeKifthereexistP>0andK
suchthat
LX
i=1¾2
i"
Ai+Bu;iKBw;i
Cz;i+Dzu;iKDzw;i#T
~P"
Ai+Bu;iKBw;i
Cz;i+Dzu;iKDzw;i#
+"
A+BuKBw
Cz+DzuKDzw#T
~P"
A+BuKBw
Cz+DzuKDzw#
·"
P0
0°2I#
where
~P="
P0
0I#
:
WemakethechangeofvariablesQ=P¡1,Y=KQ.Applying acongruence
with~Q=diag(Q;I),weget
LX
i=1¾2
i"
AiQ+Bu;iYBw;i
Cz;iQ+Dzu;iYDzw;i#T
~Q¡1"
AiQ+Bu;iYBw;i
Cz;iQ+Dzu;iYDzw;i#
+"
AQ+BuYBw
CzQ+DzuYDzw#T
~Q¡1"
AQ+BuYBw
CzQ+DzuYDzw#
·"
Q0
0°2I#
:
Thisinequalit yisreadilytransformed toanLMI.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
136 Chapter 9Systems withMultiplica tiveNoise
Notes andReferences
Systemswithmultiplicativenoise
Systems withmultiplicativ enoiseareaspecialcaseofstochasticsystems. Foranintroduction
tothistopicseeArnold[Arn74]orºAstrÄom[ºAst70].
Arigorous treatmen tofcontinuous-time stochasticsystemsrequiresmuchtechnicalmachin-
ery(see[EG94]).Infacttherearemanywaystoformulatethecontinuous-time versionofa
systemsuchas(9.10);seeIt^o[Ito51]andStratono vitch[Str66].
Stabilityofsystemwithmultiplicativenoise
Forasurveyofdi®erentconcepts ofstabilityforstochasticsystems see[Koz69]andthe
references therein. Thede¯nition ofmean-square stabilitycanbefoundin[Sam59].Mean-
squarestabilityisastrongformofstability.Inparticular, itimpliesstabilityofthemean
Ex(k)andthatalltrajectories convergetozerowithprobabilit yone[Kus67,Wil73].Forthe
systemsconsidered here,itisalsoequivalenttoL2-stability,whichisthepropertythat(see
forexample, [EP92])EPT
t=0kx(t)k2hasa(¯nite)limitasT!1forallinitialconditions.
Theequation forthestatecorrelation matrix(9.3)canbefoundin[McL71,Wil73].Analge-
braiccriterion formean-square stabilityofasystemwithmultiplicativ ewhitenoiseisgiven
in[NS72].Itisreminiscen toftheclassical (deterministic) Hurwitz criterion. Frequency-
domaincriteriaaregivenin[WB71].TherelatedLyapunovequation wasgivenbyKlein-
mann[Kle69]forthecaseL=1(i.e.,asingleuncertain ty).Thecriterion thenreduces
toanH2normcondition onacertaintransfermatrix. (Thisisinparallelwiththeresults
of[EP92],seebelow.)
KatsandKrasovskiiintroducedin[KK60]theideaofastochasticLyapunovfunction and
provedarelatedLyapunovtheorem. Athorough stabilityanalysis of(nonlinear) stochastic
systems wasmadebyKushner in[Kus67]usingthisideaofastochasticLyapunovfunc-
tion.Kushner usedthisideatoderiveboundsforreturntime,expectedoutputenergy,etc.
However,thisapproachdoesnotprovideawaytoconstruct theLyapunovfunctions.
Proofofstabilitycriterion
Wenowprovethatcondition (9.4)isanecessary andsu±cientcondition formean-square
stability.We¯rstprovethatcondition (9.4)issu±cient.LetP>0satisfy(9.4).Introduce
the(linearLyapunov)functionV(M)=TrMP,whichispositiveontheconeofnonnegativ e
matrices. FornonzeroM(k)satisfying (9.3),
V(M(k+1))=TrÃ
AM(k)AT+LX
i=1¾2
iAiM(k)AT
i!
P
=TrM(k)Ã
ATPA+LX
i=1¾2
iAT
iPAi!
<TrM(k)P=V(M(k)):
Afurtherstandard argumen tfromLyapunovtheoryprovesthatM(k)convergestozeroas
k!1.
Toprovethatcondition (9.4)isalsonecessary ,weassumethatthesystemismean-square
stable.ThismeansthatforeverypositiveinitialconditionM(0)¸0,thesolution tothe
linearrecursion (9.3)goestozeroask!0.Linearityofsystem(9.3)impliesthat,foran
arbitrary choiceofM(0),thecorrespondingsolutionM(k)convergesto0,i.e.,(9.3)isstable.
Wenowwrite(9.3)asm(k+1)=Am(k),wherem(k)isavectorcontainingthen2elements
ofM(k),andAisarealmatrix. (Thismatrixcanbeexpressed viaKroneckerproducts
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 137
involvingA0;:::;AL.)StabilityofAisequivalenttothatofAT,andthecorresponding
matrixdi®erence equation,
N(k+1)=ATN(k)A+LX
i=1¾2
iAT
iN(k)Ai; (9.18)
isalsostable.
NowletN(0)>0,andletN(k)bethecorrespondingsolution of(9.18).SinceN(k)satis¯es
astable¯rst-order lineardi®erence equation withconstantcoe±cients,thefunction
P(k)=kX
j=0N(j)
hasa(¯nite)limitfork!1,whichwedenotebyP.ThefactthatN(0)>0impliesP>0.
Now(9.18)implies
P(k+1)=N(0)+ATP(k)A+LX
i=1¾2
iAT
iP(k)Ai
Takingthelimitk!1showsthatPsatis¯es(9.4).
Mean-squarestabilitymargin
System(9.10)canbeviewedasalinearsystemsubjecttorandomparameter variations. For
systems withdeterministic parameters onlyknowntolieinranges,thecomputation ofan
exactstabilitymarginisanNP-hard problem [CD92].Thefactthatastochasticanalog
ofthedeterministic stabilitymarginiseasiertocompute shouldnotbesurprising; weare
dealingwithaveryspecialformofstability,whichonlyconsiders theaveragebehaviorofthe
system.
Anexactstabilitymargincanalsobecomputed forcontinuous-time systemswithmultiplica-
tivenoiseprovidedtheIt^oframeworkisconsidered; see[EG94].Incontrast,thecomputation
ofanexactstabilitymarginseemsdi±cultforStratono vitchsystems.
In[EP92],ElBouhtouriandPritchardprovideacomplete robustness analysis inaslightly
di®erentframeworkthanours,namelyforso-called \block-diagonal" perturbations. Roughly
speaking,theyconsider acontinuous-time It^osystemoftheform
dx(t)=Ax(t)dt+LX
i=1Bi¢i(Cx(t))dpi(t);
wheretheoperators¢iareLipschitzcontinuous,with¢i(0)=0;theprocessespi,i=
1;:::;L,arestandard Brownianmotions (which,roughly speaking,areintegralsofwhite
noise).
Wenotethatourframeworkcanberecoveredbyassuming thattheoperators¢iarerestricted
toadiagonal structure oftheform¢i(z)=¾iz.
Measuring the\size"oftheperturbation termby¡PL
i=1k¢ik2¢1=2,wherek¢kdenotesthe
Lipschitznorm,leadstheauthorsof[EP92]totherobustness margin
minimize °
subjecttoP>0;ATP+PA+CTC<0;BT
iPBi·°;i=1;:::;L
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
138 Chapter 9Systems withMultiplica tiveNoise
ForL=1,andB1andCarevectors,thatis,whenasinglescalaruncertain parameter
perturbsanLTIsystem,boththisresultandourscoincide withtheH2normof(A;B1;C).
Thisisconsisten twiththeearlierresultsofKleinmann [Kle69]andWillems [WB71](see
also[BB91,p114]forarelatedresult). ForL>1however,applying theirresultstoour
frameworkwouldbeconservative,sinceweonlyconsider \diagonal" perturbations with\re-
peatedelements",¢i=¾iI.NotethattheLMIapproachcanalsosolvethestate-feedbac k
synthesisproblem intheframeworkof[EP92].ElBouhtouriandPritchardprovideasolution
ofthisproblem in[EP94].
Stabilityconditions forarbitrary noiseintensities aregiveningeometric termsin[WW83].
Inourframework,thishappensifandonlyiftheoptimalvalueofthecorrespondingGEVP
iszero.
Boundsonstatecovariancewithnoiseinput
Weconsider system(9.6)inwhichwisaunitwhitenoiseprocess,i.e.,w(k)areindependent,
identicallydistributed withEw(k)=0,Ew(k)w(k)T=W,andwisindependentofp.
Inthiscase,ofcourse,thestatemean¹xiszero.Wederiveanupperboundonthestate
correlation (orcovariance) ofthesystem.
Thestatecorrelation M(k)satis¯esthedi®erence equation
M(k+1)=AM(k)AT+BwWBT
w+LX
i=1¾2
i¡
AiM(k)AT
i+Bw;iWBT
w;i¢
;
withM(0)=0.
Sincethesystemismean-square stable,thestatecorrelation M(k)hasa(¯nite)limitM1
whichsatis¯es
M1=AM1AT+BwWBT
w+LX
i=1¾2
i¡
AiM1AT
i+Bw;iWBT
w;i¢
:
Infact,thematrixM1canbecomputed directlyasthesolution ofalinearmatrixequation.
However,theaboveLMIformulationextendsimmediately tomorecomplicated situations (for
instance, whenthewhitenoiseinputwhasacovariancematrixwithunknowno®-diagonal
elements),whilethe\direct" methoddoesnot.SeetheNotesandReferences forChapter 6
fordetailsonhowthesemorecomplicated situations canbehandled.
Extensiontootherstochasticsystems
TheLur'estabilityproblem considered inx8.1canbeextended tothestochasticframework;
seeWonham[Won66].Thisextension canbecastintermsofLMIs.
Itisalsopossibletoextendtheresultsinthischapterto\uncertain stochasticsystems", of
theform
x(k+1)=Ã
A(k)+LX
i=1¾iAipi(k)!
x(k);
wherethetime-varyingmatrixA(k)isonlyknowntobelongtoagivenpolytope.
State-feedbacksynthesis
Conditions forstabilizabilit yforarbitrary noiseintensities aregiveningeometric terms
in[WW83].Apartfromthiswork,thebulkofearlierresearchonthistopicconcentrated
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 139
onLinear-Quadratic Regulator theoryforcontinuous-time It^osystems, whichaddresses the
problem of¯ndinganinpututhatminimize aperformance indexoftheform
J(K)=EZ1
0¡
x(t)TQx(t)+u(t)TRu(t)¢
dt
whereQ¸0andR>0aregiven.Thesolution ofthisproblem, asfoundin[McL71]
or[Ben92],canbeexpressed asa(linear,constant)state-feedbac ku(t)=Kx(t)wherethe
gainKisgivenintermsofthesolution ofanon-standard Riccatiequation. (See[Kle69,
Won68,FR75,BH88b]foradditional details.) Theexisting methods(seee.g.[Phi89,
RSH90])usehomotop yalgorithms tosolvetheseequations, withnoguaranteeofglobalcon-
vergence. Wecansolvethesenon-standard Riccatiequations reliably(i.e.,withguaranteed
convergence), byrecognizing thatthesolutionisanextremal pointofacertainLMIfeasibility
set.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Chapter 10
Miscellaneous Problems
10.1OptimizationoveranA±neFamilyofLinearSystems
Weconsider afamilyoflinearsystems,
_x=Ax+Bww;z=Cz(µ)x+Dzw(µ)w; (10.1)
whereCzandDzwdependa±nelyonaparameter µ2Rp.WeassumethatAis
stableand(A;Bw)iscontrollable. Thetransferfunction,
Hµ(s)¢=Cz(µ)(sI¡A)¡1Bw+Dzw(µ);
dependsa±nelyonµ.
Severalproblems arisinginsystemandcontroltheoryhavetheform
minimize '0(Hµ)
subjectto'i(Hµ)<®i;i=1;:::;p(10.2)
where'iarevariousconvexfunctionals. Theseproblems canoftenberecastasLMI
problems. Todothis,wewillrepresent'i(Hµ)<®iasanLMIinµ,®i,andpossibly
someauxiliary variables³i(foreachi):
Fi(µ;®i;³i)>0:
Thegeneraloptimization problem (10.2)canthenbeexpressed astheEVP
minimize ®0
subjecttoFi(µ;®i;³i)>0;i=0;1;:::;p
10.1.1 H2norm
TheH2normofthesystem(10.1),i.e.,
kHµk2
2=1
2¼TrZ1
0Hµ(j!)¤Hµ(j!)d!;
is¯niteifandonlyifDzw(µ)=0.Inthiscase,itequals TrCzWcCT
z,whereWc>0is
thecontrollabilit yGramian ofthesystem(10.1)whichsatis¯es(6.6).Therefore, the
H2normofthesystem(10.1)islessthanorequalto°ifandonlyifthefollowing
conditions onµand°2aresatis¯ed:
Dzw(µ)=0;TrCz(µ)WcCz(µ)T·°2:
Thequadratic constrain tonµisreadilycastasanLMI.
141
142 Chapter 10Miscellaneous Problems
10.1.2 H1norm
Fromthebounded-real lemma,wehavekHµk1<°ifandonlyifthereexistsP¸0
suchthat
"
ATP+PAPBw
BT
wP¡°2I#
+"
Cz(µ)T
Dzw(µ)T#h
Cz(µ)Dzw(µ)i
·0:
Remark :Herewehaveconvertedtheso-called \semi-in¯nite" convexconstrain t
kHµ(i!)k<°forall!2Rintoa¯nite-dimensional convex(linearmatrix)
inequalit y.
10.1.3Entropy
The°-entropyofthesystem(10.1)isde¯nedas
I°(Hµ)¢=8
<
:¡°2
2¼Z1
¡1logdet(I¡°2Hµ(i!)Hµ(i!)¤)d!;ifkHµk1<°,
1; otherwise.
Whenitis¯nite,I°(Hµ)isgivenbyTrBT
wPBw,wherePisasymmetric matrix
withthesmallest possiblemaximumsingular valueamongallsolutions oftheRiccati
equation
ATP+PA+Cz(µ)TCz(µ)+1
°2PBwBT
wP=0:
Forthesystem(10.1),the°-entropyconstrain tI°·¸istherefore equivalenttoan
LMIinµ,P=PT,°2,and¸:
Dzw(µ)=0;2
64ATP+PAPBwCz(µ)T
BT
wP¡°2I0
Cz(µ)0¡I3
75·0;TrBT
wPBw·¸:
10.1.4Dissipativit y
Supposethatwandzarethesamesize.Thedissipativit yofHµ(see(6.59))exceeds
°ifandonlyiftheLMIinthevariablesP=PTandµholds:
"
ATP+PA PBw¡Cz(µ)T
BT
wP¡Cz(µ)2°I¡Dzw(µ)¡Dzw(µ)T#
·0:
Weremindthereaderthatpassivitycorrespondstozerodissipativit y.
10.1.5Hankelnorm
TheHankelnormofHµislessthan°ifandonlyifthefollowingLMIinQ,µand°2
holds:
Dzw(µ)=0;ATQ+QA+Cz(µ)TCz(µ)·0;
°2I¡W1=2
cQW1=2
c¸0;Q¸0:
(Wcisthecontrollabilit yGramian de¯nedby(6.6).)
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
10.2 AnalysisofSystems withLTIPerturba tions 143
10.2AnalysisofSystemswithLTIPerturbations
Inthissection,weaddresstheproblem ofdetermining thestabilityofthesystem
_x=Ax+Bpp;q=Cqx+Dqpp;pi=±i¤qi;i=1;:::;np;(10.3)
wherep=[p1¢¢¢pnp]T,q=[q1¢¢¢qnp]T,and±i;i=1;:::;np,areimpulse
responsesofsingle-output single-output LTIsystemswhose H1normisstrictlyless
thanone,and¤denotesconvolution. Thisproblem hasmanyvariations: Someofthe
±imaybeimpulseresponsematrices, ortheymaysatisfyequalityconstrain tssuchas
±1=±2.Ouranalysis canbereadilyextended tothesecases.
LetHqp(s)=Cq(sI¡A)¡1Bp+Dqp.Supposethatthereexistsadiagonal transfer
matrixW(s)=diag(W1(s);:::;Wnp(s)),whereW1(s);:::;Wnp(s)aresingle-input
single-output transfer functions withnopolesorzerosontheimaginary axissuch
that
sup
!2R°°W(i!)Hqp(i!)W(i!)¡1°°<1 (10.4)
Thenfromasmall-gain argumen t,thesystem(10.3)isstable.SuchaWisreferred
toasafrequency-dep endentscaling.
Condition (10.4)isequivalenttotheexistence of°>0suchthat
Hqp(i!)¤W(i!)¤W(i!)Hqp(i!)¡W(i!)¤W(i!)+°I·0;
forall!2R.Fromspectralfactorization theory,itcanbeshownthatsuchaW
existsifandonlyifthereexistsastable,diagonal transfer matrixVandpositive¹
suchthat
Hqp(i!)¤(V(i!)+V(i!)¤)Hqp(i!)¡(V(i!)+V(i!)¤)+°I·0;
V(i!)+V(i!)¤¸¹I;(10.5)
forall!2R.
Inordertoreduce(10.5)toanLMIcondition, werestrictVtobelongtoana±ne,
¯nite-dimensional set£ofdiagonal transfermatrices, i.e.,oftheform
Vµ(s)=DV(µ)+CV(µ)(sI¡A)¡1B;
whereDVandCVarea±nefunctions areµ.
Then,Hqp(¡s)TVµ(s)Hqp(s)¡Vµ(s)hasastate-space realization (Aaug,Baug,
Caug(µ),Daug(µ))whereCaugandDaugarea±neinµ,and(Aaug;Baug)iscontrollable.
Fromx10.1.4,thecondition thatVµ(i!)+Vµ(i!)¤¸¹Iforall!2Risequivalent
totheexistence ofP1¸0,µand°>0suchthattheLMI
"
AT
VP1+P1AV P1BV¡CV(µ)T
(P1BV¡CV(µ)T)T¡(DV(µ)+DV(µ)T)+¹I#
·0
holds.
TheconditionHqp(i!)¤(Vµ(i!)+Vµ(i!)¤)Hqp(i!)¡(Vµ(i!)+Vµ(i!)¤)+°I·0
isequivalenttotheexistence ofP2=PT
2andµsatisfying theLMI
"
AT
augP2+P2Aaug P2Baug+Caug(µ)T
(P2Baug+Caug(µ)T)TDaug(µ)+Daug(µ)T+°I#
·0: (10.6)
(SeetheNotesandReferences fordetails.) Thusprovingstabilityofthesystem(10.3)
usinga¯nite-dimensional setoffrequency-dep endentscalingsisanLMIP.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
144 Chapter 10Miscellaneous Problems
10.3PositiveOrthantStabilizability
TheLTIsystem_x=Axissaidtobepositiveorthantstableifx(0)2Rn
+impliesthat
x(t)2Rn
+forallt¸0,andx(t)!0ast!1.Itcanbeshownthatthesystem
_x=AxispositiveorthantstableifandonlyifAij¸0fori6=j,andthereexists
adiagonalP>0suchthatPAT+AP<0.Therefore checkingpositiveorthant
stabilityisanLMIP.
Wenextconsider theproblem of¯ndingKsuchthatthesystem_x=(A+BK)x
ispositiveorthantstable.Equivalently,weseekadiagonalQ>0andKsuchthat
Q(A+BK)T+(A+BK)Q<0,withtheo®-diagonal entriesofA+BKbeing
nonnegativ e.SinceQ>0isdiagonal, thelastcondition holdsifandonlyifall
theo®-diagonal entriesof(A+BK)Qarenonnegativ e.Therefore, withthechangeof
variablesY=KQ,provingpositiveorthantstabilizabilit yforthesystem_x=Ax+Bu
isequivalentto¯ndingasolution totheLMIPwithvariablesQandY:
Q>0;(AQ+BY)ij¸0;i6=j;QAT+YTBT+AQ+BY<0:(10.7)
Remark :Themethodcanbeextended toinvarianceofmoregeneral(polyhedral)
sets.Positiveorthantstabilityandstabilizabilit yofLDIscanbehandled ina
similarway.SeetheNotesandReferences.
10.4LinearSystemswithDelays
Consider thesystemdescribedbythedelay-di®eren tialequation
d
dtx(t)=Ax(t)+LX
i=1Aix(t¡¿i); (10.8)
wherex(t)2Rn,and¿i>0.Ifthefunctional
V(x;t)=x(t)TPx(t)+LX
i=1Z¿i
0x(t¡s)TPix(t¡s)ds; (10.9)
whereP>0,P1>0;:::;PL>0,satis¯esdV(x;t)=dt<0foreveryxsatisfying (10.8),
thenthesystem(10.8)isstable,i.e.,x(t)!0ast!1.
Itcanbeveri¯edthatdV(x;t)=dt=y(t)TWy(t),where
W=2
66666666664ATP+PA+LX
i=1PiPA1¢¢¢PAL
AT
1P¡P1¢¢¢0
............
AT
LP 0¢¢¢¡PL3
77777777775;y(t)=2
66664x(t)
x(t¡¿1)
...
x(t¡¿L)3
77775:
Therefore, wecanprovestabilityofsystem(10.8)usingLyapunovfunctionals ofthe
form(10.9)bysolvingtheLMIPW<0,P>0,P1>0;:::;PL>0.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
10.5 Interpola tionProblems 145
Remark :Thetechniquesforprovingstabilityofnorm-boundLDIs,discussed
inChapter 5,canalsobeusedonthesystem(10.8);itcanbeveri¯edthatthe
LMI(5.15)thatwegetforquadratic stabilityofaDNLDI, inthecasewhenthe
statexisascalar,isthesameastheLMIW<0above.Wealsonotethatwe
canregardthedelayelementsofsystem(10.8)asconvolutionoperatorswithunit
L2gain,andusethetechniquesofx10.2tocheckitsstability.
Next,weaddaninpututothesystem(10.8)andconsider
d
dtx(t)=Ax(t)+Bu(t)+LX
i=1Aix(t¡¿i): (10.10)
Weseekastate-feedbac ku(t)=Kx(t)suchthatthesystem(10.10)isstable.
Fromthediscussion above,thereexistsastate-feedbac kgainKsuchthataLya-
punovfunctional oftheform(10.9)provesthestabilityofthesystem(10.10),if
W=2
66666666664(A+BK)TP+P(A+BK)+LX
i=1PiPA1¢¢¢PAL
AT
1P ¡P1¢¢¢0
............
AT
LP 0¢¢¢¡PL3
77777777775<0
forsomeP,P1;:::;PL>0.
Multiplying everyblockentryofWontheleftandontherightbyP¡1andsetting
Q=P¡1,Qi=P¡1PiP¡1andY=KP¡1,weobtainthecondition
X=2
666666666664AQ+QAT+BY+YTBT+LX
i=1QiA1Q¢¢¢ALQ
QAT
1 ¡Q1¢¢¢0
............
QAT
L 0¢¢¢¡QL3
777777777775<0:
Thus,checkingstabilizabilit yofthesystem(10.8)usingLyapunovfunctionals ofthe
form(10.9)isanLMIPinthevariablesQ,Y,Q1;:::;QL.
Remark :Itispossibletousetheelimination procedureofx2.6.2toeliminate
thematrixvariableYandobtainanequivalentLMIPwithfewervariables.
10.5InterpolationProblems
10.5.1TangentialNevanlinna-Pic kproblem
Given¸1;:::;¸mwith¸i2C+¢=fsjRes>0ganddistinct,u1;:::;um,with
ui2Cqandv1;:::;vm,withvi2Cp,i=1;:::;m,thetangentialNevanlinna-Pic k
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146 Chapter 10Miscellaneous Problems
problem isto¯nd,ifpossible,afunctionH:C¡!Cp£qwhichisanalytic inC+,
andsatis¯es
H(¸i)ui=vi;i=1;:::;m,withkHk1·1. (10.11)
Problem (10.11)arisesinmulti-input multi-output H1controltheory.
FromNevanlinna-Pic ktheory,problem (10.11)hasasolution andonlyifthePick
matrixNde¯nedby
Nij=u¤
iuj¡v¤
ivj
¸¤
i+¸j)
ispositivesemide¯nite. Ncanalsobeobtained asthesolution oftheLyapunov
equation
A¤N+NA¡(U¤U¡V¤V)=0:
whereA=diag(¸1;:::;¸m),U=[u1¢¢¢um],V=[v1¢¢¢vm].Forfuturereference,
wenotethatN=Gin¡Gout,where
A¤Gin+GinA¡U¤U=0;A¤Gout+GoutA¡V¤V=0:
SolvingthetangentialNevanlinna-Pic kproblem simplyrequirescheckingwhetherN¸
0.
10.5.2Nevanlinna-Pic kinterpolationwithscaling
Wenowconsider asimplevariation ofproblem (10.11):Given¸1;:::;¸mwith¸i2
C+,u1;:::;um,withui2Cpandv1;:::;vm,withvi2Cp,i=1;:::;m,theproblem
isto¯nd
°opt=inf(
kDHD¡1k1¯¯¯¯¯Hisanalytic inC+;D=D¤>0
D2D;H(¸i)ui=vi;i=1;:::;m)
;(10.12)
whereDisthesetofm£mblock-diagonal matrices withsomespeci¯edblockstruc-
ture.Problem (10.12)correspondsto¯ndingthesmallest scaled H1normofall
interpolants.Thisproblem arisesinmulti-input multi-output H1controlsynthesis
forsystemswithstructured perturbations.
WithachangeofvariablesP=D¤Dandthediscussion intheprevious section,
itfollowsthat°optisthesmallest positive°suchthatthereexistsP>0,P2Dsuch
thatthefollowingequations andinequalit yhold:
A¤Gin+GinA¡U¤PU=0;
A¤Gout+GoutA¡V¤PV=0;
°2Gin¡Gout¸0:
ThisisaGEVP.
10.5.3Frequency responseidenti¯cation
Weconsider theproblem ofidentifyingthetransfer functionHofalinearsystem
fromnoisymeasuremen tsofitsfrequency responseatasetoffrequencies. WeseekH
satisfying twoconstrain ts:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
10.6 TheInverse Problem ofOptimal Contr ol 147
²Consistency withmeasurements. Forsomeniwithjnij·²,wehavefi=
H(j!i)+ni.Here,fiisthemeasuremen tofthefrequency responseatfrequency
!i,niisthe(unknown)measuremen terror,and²isthemeasuremen tprecision.
²Priorassumption. ®-shifted H1normofHdoesnotexceedM.
FromNevanlinna-Pic ktheory,thereexistHandnwithH(!i)=fi+nisatisfying
theseconditions ifandonlyifthereexistniwithjnij·²,Gin>0andGout>0such
that
M2Gin¡Gout¸0;
(A+®I)¤Gin+Gin(A+®I)¡e¤e=0;
(A+®I)¤Gout+Gout(A+®I)¡(f+n)¤(f+n)=0;
whereA=diag(j!1;:::;j!m),e=[1¢¢¢1],f=[f1¢¢¢fm],andn=
[n1¢¢¢nm].Itcanbeshownthattheseconditions areequivalentto
M2Gin¡Gout¸0;
(A+®I)¤Gin+Gin(A+®I)¡e¤e=0;
(A+®I)¤Gout+Gout(A+®I)¡(f+n)¤(f+n)¸0;
withjnij·².
Withthisobservation,wecanansweranumberofinteresting questions infre-
quencyresponseidenti¯cation bysolvingEVPsandGEVPs.
²For¯xed®and²,minimizeM.SolvingthisGEVPanswersthequestion \Given
®andaboundonthenoisevalues,whatisthesmallest possible®-shifted
H1normofthesystemconsisten twiththemeasuremen tsofthefrequency
response?"
²For¯xed®andM,minimize².SolvingthisEVPanswersthequestion \Given
®andaboundon®-shifted H1normofthesystem,whatisthe\smallest"
possiblenoisesuchthatthemeasuremen tsareconsisten twiththegivenvalues
of®andM.
10.6TheInverseProblemofOptimalControl
Givenasystem
_x=Ax+Bu;x(0)=x0;z="
Q1=20
0R1=2#"
x
u#
;
with(A;B)isstabilizable, (Q;A)isdetectable andR>0,theLQRproblem isto
determine uthatminimizes
Z1
0zTzdt:
Thesolution ofthisproblem canbeexpressed asastate-feedbac ku=Kxwith
K=¡R¡1BTP,wherePistheuniquenonnegativ esolution of
ATP+PA¡PBR¡1BTP+Q=0:
Theinverseproblem ofoptimalcontrolisthefollowing.GivenamatrixK,determine
ifthereexistQ¸0andR>0,suchthat(Q;A)isdetectable andu=Kxisthe
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
148 Chapter 10Miscellaneous Problems
optimalcontrolforthecorrespondingLQRproblem. Equivalently,weseekR>0and
Q¸0suchthatthereexistsPnonnegativ eandP1positive-de¯nite satisfying
(A+BK)TP+P(A+BK)+KTRK+Q=0;BTP+RK=0
andATP1+P1A<Q.ThisisanLMIPinP,P1,RandQ.(Thecondition involving
P1isequivalentto(Q;A)beingdetectable.)
10.7SystemRealizationProblems
Consider thediscrete-time minimal LTIsystem
x(k+1)=Ax(k)+Bu(k);y(k)=Cx(k); (10.13)
wherex:Z+!Rn,u:Z+!Rnuy:Z+!Rny.Thesystemrealization problemis
to¯ndachangeofstatecoordinatesx=T¹x,suchthatthenewrealization
¹x(k+1)=T¡1AT¹x(k)+T¡1Bu(k)y(k)=CT¹x(k); (10.14)
satis¯estwocompetingconstrain ts:First,theinput-to-state transfermatrix,T¡1(zI¡
A)¡1Bshouldbe\small", inordertoavoidstateover°owinnumerical implemen ta-
tion;andsecond,thestate-to-output transfermatrix,C(zI¡A)¡1Tshouldbe\small",
inordertominimize e®ectsofstatequantizationattheoutput.
SupposeweassumethattheRMSvalueoftheinputisboundedby®andrequire
theRMSvalueofthestatetobelessthanone.Thisyieldsthebound
°°T¡1(zI¡A)¡1B°°
1·1=®; (10.15)
wheretheH1normofthetransfermatrixH(z)isde¯nedas
kHk1=supfkH(z)kjjzj>1g:
Next,supposethatthestatequantizationnoiseismodeledasawhitenoisesequence
w(k)withEw(k)Tw(k)=´,injected directly intothestate,anditse®ectonthe
outputismeasured byitsRMSvalue,whichisjust´timestheH2normofthe
state-to-output transfermatrix:
pnoise=´°°C(zI¡A)¡1T°°
2: (10.16)
Ourproblem isthentocomputeTtominimize thenoisepower(10.16)subjecttothe
over°owavoidanceconstrain t(10.15).
Theconstrain t(10.15)isequivalenttotheexistence ofP>0suchthat
"
ATPA¡P+T¡TT¡1ATPB
BTPA BTPB¡I=®2#
·0:
Theoutputnoisepowercanbeexpressed as
p2
noise=´2TrTTWoT;
whereWoistheobservabilityGramian oftheoriginalsystem(A;B;C),givenby
Wo=1X
k=0(AT)kCTCAk:
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
10.7 System Realiza tionProblems 149
Woisthesolution oftheLyapunovequation
ATWoA¡Wo+CTC=0:
WithX=T¡TT¡1,therealization problem is:Minimize´2TrWoX¡1subject
toX>0andtheLMI(inP>0andX)
"
ATPA¡P+XATPB
BTPABTPB¡I=®2#
·0: (10.17)
Thisisaconvexproblem inXandP,andcanbetransformed intotheEVP
minimize ´2TrYWc
subjectto(10:17);P>0;"
YI
IX#
¸0
Similarmethodscanbeusedtohandleseveralvariations andextensions.
Input uhasRMSvalueboundedcomponentwise
SupposetheRMSvalueofeachcomponentuiislessthan®(instead oftheRMSvalue
ofubeinglessthan®)andthattheRMSvalueofthestateisstillrequired tobeless
thanone.Then,usingthemethodsofx10.9,anequivalentcondition istheLMI(with
variablesX,PandR):
"
ATPA¡P+XATPB
BTPABTPB¡R=®2#
·0;P>0; (10.18)
whereR>0isdiagonal, withunittrace,sothattheEVP(overX,Y,PandR)is
minimize ´2TrYWc
subjectto(10:18);P>0;X>0
R>0;Risdiagonal;TrR=1"
YI
IX#
¸0
Minimizingstate-to-output H1norm
Alternativ ely,wecanmeasure thee®ectofthequantizationnoiseontheoutputwith
theH1norm,thatis,wecanchooseTtominimizekC(zI¡A)¡1Tk1subjecttothe
constrain t(10.15). Theconstrain tkC(zI¡A)¡1Tk1·°isequivalenttotheLMI
(overY>0andX=T¡TT¡1>0):
"
ATYA¡Y+CTCATY
YA Y¡X=®2#
·0: (10.19)
Therefore choosingTtominimizekC(zI¡A)¡1Tk1subjectto(10.15)istheGEVP
minimize °
subjecttoP>0;Y>0;X>0;(10:17);(10:19)
Anoptimal statecoordinate transformation Tisanymatrixthatsatis¯esXopt=
(TTT)¡1,whereXoptisanoptimalvalueofXintheGEVP.
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
150 Chapter 10Miscellaneous Problems
10.8Multi-Criterion LQG
Consider theLTIsystemgivenby
_x=Ax+Bu+w;y=Cx+v;z="
Q1=20
0R1=2#"
x
u#
;
whereuisthecontrolinput;yisthemeasured output;zistheexogenous output;
wandvareindependentwhitenoisesignalswithconstant,positive-de¯nite spectral
densitymatricesWandVrespectively;wefurtherassumethatQ¸0andR>0,
that(A;B)iscontrollable, andthat(C;A)and(Q;A)areobservable.
Thestandard Linear-Quadratic Gaussian (LQG)problem istominimize
Jlqg=lim
t!1Ez(t)Tz(t)
overu,subjecttothecondition thatu(t)ismeasurable ony(¿)for¿·t.Theoptimal
costisgivenby
J¤
lqg=Tr(XlqgU+QYlqg);
whereXlqgandYlqgaretheuniquepositive-de¯nite solutions oftheRiccatiequations
ATXlqg+XlqgA¡XlqgBR¡1BTXlqg+Q=0;
AYlqg+YlqgAT¡YlqgCTV¡1CYlqg+W=0;(10.20)
andU=YlqgCTV¡1CYlqg.
Inthemulti-criterion LQGproblem, wehaveseveralexogenous outputsofinterest
givenby
zi="
Q1=2
i0
0R1=2
i#"
x
u#
;Qi¸0;Ri>0;i=0;:::;p:
Weassumethat(Q0;A)isobservable.Foreachzi,weassociateacostfunction
Ji
lqg=lim
t!1Ezi(t)Tzi(t);i=0;:::;p: (10.21)
Themulti-criterion LQGproblem istominimizeJ0
lqgoverusubjecttothemeasurabil-
itycondition andtheconstrain tsJi
lqg<°i;i=1;:::;p.Thisisaconvexoptimization
problem, whosesolution isgivenbymaximizing
Tr(XlqgU+QYlqg)¡pX
i=1°i¿i;
overnonnegativ e¿1;:::;¿p,whereXlqgandYlqgarethesolutions of(10.20)with
Q=Q0+pX
i=1¿iQiandR=R0+pX
i=1¿iRi:
NotingthatXlqg¸XforeveryX>0thatsatis¯es
ATX+XA¡XBR¡1BTX+Q¸0;
weconclude thattheoptimalcostisthemaximumof
TrÃ
XU+(Q0+pX
i=1¿iQi)Ylqg!
¡pX
i=1°i¿i;
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
10.9 Nonconvex Multi-Criterion Quadra ticProblems 151
overX,¿1;:::;¿p,subjecttoX>0,¿1¸0;:::;¿p¸0and
ATX+XA¡XBÃ
R0+pX
i=1¿iRi!¡1
BTX+Q0+pX
i=1¿iQi¸0:
Computing thisisanEVP.
10.9NonconvexMulti-Criterion QuadraticProblems
Inthissection,weconsider theLTIsystem_x=Ax+Bu;x(0)=x0;where(A;B)is
controllable. Foranyu,wede¯neasetofp+1costindicesJ0;:::;Jpby
Ji(u)=Z1
0[xTuT]"
QiCi
CT
iRi#"
x
u#
dt;i=0;:::;p:
Herethesymmetric matrices
"
QiCi
CT
iRi#
;i=0;:::;p;
arenotnecessarily positive-de¯nite. Theconstrained optimalcontrolproblem is:
minimize J0;
subjecttoJi·°i;i=1;:::;p;x(t)!0ast!1(10.22)
Thesolution tothisproblem proceedsasfollows:We¯rstde¯ne
Q=Q0+pX
i=1¿iQi;R=R0+pX
i=1¿iRi;C=C0+pX
i=1¿iCi;
where¿1¸0;:::;¿p¸0.Next,with¿=[¿1¢¢¢¿p],wede¯ne
S(¿;u)¢=J0+pX
i=1¿iJi¡pX
i=1¿i°i:
Then,thesolution ofproblem (10.22),whenitisnot¡1,isgivenby:
sup
¿inf
ufS(¿;u)jx(t)!0ast!1g
(SeetheNotesandReferences.)
Forany¯xed¿,thein¯mumoveruofS(¿;u),whenitisnot¡1,iscomputed
bysolvingtheEVPinP=PT
maximize x(0)TPx(0)¡pX
i=1¿i°i
subjectto"
ATP+PA+QPB+CT
BTP+C R#
¸0(10.23)
Therefore, theoptimalcontrolproblem issolvedbytheEVPinP=PTand¿:
maximize x(0)TPx(0)¡pX
i=1¿i°i
subjectto"
ATP+PA+QPB+CT
BTP+C R#
¸0;¿1¸0;:::;¿p¸0:
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
152 Chapter 10Miscellaneous Problems
Notes andReferences
Optimizationoverana±nefamilyoflinearsystems
Optimization overana±nefamilyoftransfermatrices arisesinlinearcontrollerdesignvia
Youla'sparametrization ofclosed-lo optransfermatrices; seeBoydandBarratt[BB91]and
thereferences therein. Otherexamples include¯nite-impulse-resp onse(FIR)¯lterdesign(see
forexample [OKUI88 ])andantennaarrayweightdesign[KRB89].
In[Kav94]Kavrano¸gluconsiders theproblem ofapproximating intheH1normagiven
transfermatrixbyaconstantmatrix,whichisaproblem oftheformconsidered inx10.1.He
caststheproblem asanEVP.
StabilityofsystemswithLTIperturbations
Forreferences aboutthisproblem, wereferthereadertotheNotesandReferences inChap-
ter3.LMI(10.6)isderivedfromTheorems 3and4inWillems [Wil71b].
Positiveorthantstabilizability
Theproblem ofpositiveorthantstabilityandholdabilit ywasextensivelystudiedbyBerman,
Neumann andStern[BNS89,x7.4].Theysolvethepositiveorthantholdabilit yproblem
whenthecontrolinputisscalar;ourresultsextendtheirstothemulti-input case.Thestudy
ofpositiveorthantholdabilit ydrawsheavilyfromthetheoryofpositivematrices, references
forwhichare[BNS89]andthebookbyBerman andPlemmons [BP79,ch6].
Diagonal solutions totheLyapunovequation playacentralroleinthepositiveorthant
stabilizabilit yandholdabilit yproblems. ArelatedpaperisbyGeromel [Ger85],whogives
acomputational procedureto¯nddiagonal solutions oftheLyapunovequation. Diagonal
quadratic Lyapunovfunctions alsoariseinthestudyoflarge-scale systems [MH78];see
alsox3.4.Asurveyofresultsandapplications ofdiagonal Lyapunovstabilityisgivenin
Kaszkurewicz andBhaya[KB93].
Ageneralization oftheconceptofpositiveorthantstabilityisthatofinvarianceofpolyhedral
sets.Theproblem ofcheckingwhetheragivenpolyhedron isinvariantaswellastheassociated
state-feedbac ksynthesisproblem havebeenconsidered bymanyauthors, seee.g.,[Bit88,
BBT90,HB91b]andreferences therein. TheLMImethodofx10.3canbeextended tothe
polyhedral case,usingtheapproachof[CH93].Finally,wenotethattheresultsofx10.3are
easilygeneralized totheLDIsconsidered inChapter 4.
Stabilizationofsystemswithtimedelays
TheintroductionofLyapunovfunctionals oftheform(10.9)isduetoKrasovskii[Kra56].
Skorodinskii[Sko90]observedthattheproblem ofprovingstabilityofasystemwithdelays
viaLyapunov{Krasovskiifunctionals isaconvexproblem. Seealso[PF92,WBL92,SL93b,
FBB92].
Itisinteresting tonotethatthe\structured stabilityproblem" or\¹-analysis problem"
canbecastintermsoflinearsystems withdelays.Consider astableLTIsystemwith
transfermatrixHandLinputsandoutputs, connected infeedbackwithadiagonal transfer
matrix¢withH1normlessthanone.Thissystemisstableforallsuch¢ifandonlyif
sup!2R¹(H(j!))<1,where¹denotesthestructured singular value(see[Doy82]).Itcan
beshown(see[Boy86])thatifthefeedbacksystemisstableforall¢whicharediagonal,
with¢ii(s)=e¡sTi,Ti>0(i.e.,delays)thenthesystemisstableforall¢withk¢k1·1.
Thus,verifyingsup!2R¹(H(j!))<1canbecastascheckingstabilityofalinearsystem
withLarbitrary ,positivedelays.Inparticular weseethatKrasovskii'smethodfrom1956
canbeinterpreted asaLyapunov-based¹-analysis method.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 153
Interpolationproblems
Theclassical Nevanlinna-Pic kproblem datesbackatleastto1916[Pic16,Nev19].There
aretwomatrixversionsofthisproblem: thematrixNevanlinna-Pic kproblem [DGK79]
andthetangentialNevanlinna-Pic kproblem [Fed75].Acomprehensiv edescription ofthe
Nevanlinna-Pic kproblem, itsextensions andvariations canbefoundinthebookbyBall,
GohbergandRodman[BGR90].
Anumberofresearchershavestudiedtheapplication ofNevanlinna-Pic ktheorytoproblems
insystemandcontrol:Delsarte, GeninandKampdiscusstheroleofthematrixNevanlinna-
Pickproblem incircuitandsystemstheory[DGK81].ZamesandFrancis,in[ZF83],study
theimplications ofinterpolationconditions incontrollerdesign;alsosee[OF85].Chang
andPearson[CP84]solveaclassofH1-optimal controllerdesignproblems usingmatrix
Nevanlinna-Pic ktheory.Safonov[Saf86]reducesthedesignofcontrollersforsystemswith
structured perturbations tothescaledmatrixNevanlinna-Pic kproblem. In[Kim87],Kimura
reducesaclassofH1-optimal controllerdesignproblems tothetangentialNevanlinna-Pic k
problem; seealso[Bal94].
Systemidenti¯cation
Thesystemidenti¯cation problem considered inx10.5.3canbefoundin[CNF92].Other
systemidenti¯cation problems canbecastintermsofLMIproblems; seee.g.,[PKT+94].
Theinverseproblemofoptimalcontrol
Thisproblem was¯rstconsidered byKalman [Kal64],whosolveditwhenthecontroluis
scalar;seealso[And66c].Itisdiscussed ingreatdetailbyAnderson andMoore[AM90,
x5.6],FujiiandNarazaki [FN84]andthereferences therein; theysolvetheproblem when
thecontrolweightingmatrixRisknown,bycheckingthatthereturndi®erence inequalit y
¡
I¡BT(¡i!I¡AT)¡1K¢
R¡
I¡KT(i!I¡A)¡1B¢
¸R
holdsforall!2R.OurresulthandlesthecaseofunknownR,i.e.,themostgeneralform
oftheinverseoptimalcontrolproblem.
Systemrealizationproblems
Foradiscussion oftheproblem ofsystemrealizations see[MR76,Thi84],whichgivean-
alyticalsolutions viabalanced realizations forspecialrealization problems. Thebookby
GeversandLi[GL93a]describesanumberofdigital¯lterrealization problems. Liu,Skelton
andGrigoriadis discusstheproblem ofoptimal ¯niteword-length controllerimplemen ta-
tion[LSG92].SeealsoRoteaandWilliamson [RW94b,RW94a].
Weconjecture thattheLMIapproachtosystemrealization canalsoincorporatetheconstrain t
ofstability,i.e.,thatthe(nonlinear) systemnotexhibitlimitcyclesduetoquantizationor
over°ow;see,e.g.,[Wil74b].
WealsonotethatanH1normconstrain tontheinput-to-state transfer matrixyields,
indirectly ,aboundonthepeakgainfrominputtostate;BoydandDoyle[BD87].Thiscan
beusedtoguaranteenoover°owgivenamaximumpeakinputlevel.
Multi-criterion LQG
Themulti-criterion LQGproblem isdiscussed in[BB91];seealsothereferences citedthere.
Rotea[Rot93]callsthisproblem \generalized H2control".Heshowsthatonecanrestrict
attentiontoobserver-based controllerssuchas
d
dt^x=(A+BK)^x+YCTV¡1(y¡C^x);u=K^x;
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
154 Chapter 10Miscellaneous Problems
whereYisgivenby(10.20). Thecovarianceoftheresidualstate^x¡xisYandthecovariance
Pofthestate^xisgivenbythesolution oftheLyapunovequation:
(A+BK)P+P(A+BK)T+YCTV¡1CY=0: (10.24)
ThevalueofeachcostindexJi
lqgde¯nedby(10.21)isthen
Ji
lqg=TrRiKPKT+TrQi(Y+P):
Now,takingZ=KPasanewvariable,wereplace(10.24)bythelinearequalityinWand
P
AP+PAT+BZ+ZTBT+YCTV¡1CY=0
andeachcostindexbecomes
Ji
lqg=TrRiZP¡1ZT+TrQi(Y+P):
Themulti-criterion problem isthensolvedbytheEVPinP,ZandX
minimize TrR0X+TrQ0(P+Y)
subjectto"
XZ
ZTP#
¸0;
AP+PAT+BZ+ZTBT+YCTV¡1CY=0;
TrRiX+TrQi(P+Y)·°i;i=1;:::;L
Thisformulationofthemulti-criterion LQGproblem isthedualofourformulation;thetwo
formulationscanbeusedtogether inane±cientprimal-dual method(see,e.g.,[VB93b]).We
alsomentionanalgorithm devisedbyZhu,RoteaandSkelton[ZRS93]tosolveasimilarclass
ofproblems. Foravariationonthemulti-criterion LQGproblem discussed here,see[Toi84];
seealso[TM89,Mak91].
Nonconvexmulti-criterionquadraticproblems
ThissectionisbasedonMegretsky [Meg92a,Meg92b,Meg93]andYakubovich[Yak92].
Inourdiscussion weomittedimportanttechnicaldetails,suchasconstrain tregularity,which
iscoveredinthesearticles. Also,Yakubovichgivesconditions underwhichthein¯mum
isactually aminimum(thatis,thereexistsanoptimalcontrollawthatachievesthebest
performance). TheEVP(10.23)isderivedfromTheorem 3inWillems [Wil71b].
Manyofthesu±cientconditions forNLDIsinChapters 5{7canbeshowntobenecessary
andsu±cientconditions whenweconsider integralconstrain tsonpandq,byanalyzing them
asnonconvexmulti-criterion quadratic problems. Thisisillustrated inthefollowingsection.
Mixed H2{H1problem
Weconsider theanalogofanNLDIwithintegralquadratic constrain t:
_x=Ax+Bpp+Buu;q=Cqx;z="
Q1=20
0R1=2#"
x
u#
Z1
0pTpdt·Z1
0qTqdt
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Notes andReferences 155
De¯nethecostfunctionJ=R1
0zTzdt.Ourgoalistodetermine minKmaxpJ,overall
possiblelinearstate-feedbac kstrategiesu=Kx.(Notethatthisisthesameastheprob-
lemconsidered inx7.4.1withintegralconstrain tsonpandqinsteadofpointwiseintime
constrain ts.)
For¯xedK,¯ndingthemaximumofJoveradmissiblep'sisdoneusingtheresultsofx10.9.
Thismaximumisobtained bysolvingtheEVP
minimize x(0)TPx(0)
subjectto2
64(A+BuK)TP+P(A+BuK)+
Q+KTRK+¿CT
qCqPBp
BT
pP ¡¿I3
75·0;¿¸0
Theminimization overKisnowsimplydonebyintroducingthenewvariablesW=P¡1,
Y=KW,¹=1=¿.Indeed,inthisparticular case,weknowwecanaddtheconstrain t
P>0.ThecorrespondingEVPisthen
minimize x(0)TW¡1x(0)
subjectto2
64Ã
WAT+YBT
u+AW+BuY+
WQW+YTRY+WCT
qCqW=¹!
¹Bp
¹BT
p ¡¹I3
75·0;¹¸0
whichisthesameastheEVP(7.23).
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Notation
R;Rk;Rm£nTherealnumbers,realk-vectors,realm£nmatrices.
R+ Thenonnegativ erealnumbers.
C Thecomplex numbers.
Re(a) Therealpartofa2C,i.e.,(a+a¤)=2.
S TheclosureofasetSµRn.
Ik Thek£kidentitymatrix.Thesubscript isomittedwhenkisnot
relevantorcanbedetermined fromcontext.
MTTransposeofamatrixM:(MT)ij=Mji.
M¤Complex-conjugate transposeofamatrixM:(M¤)ij=M¤
ji,where
®¤denotesthecomplex-conjugate of®2C.
TrM TraceofM2Rn£n,i.e.,Pn
i=1Mii.
M¸0 Missymmetric andpositivesemide¯nite, i.e.,M=MTand
zTMz¸0forallz2Rn.
M>0 Missymmetric andpositive-de¯nite, i.e.,zTMz>0forallnonzero
z2Rn.
M>N MandNaresymmetric andM¡N>0.
M1=2ForM>0,M1=2istheuniqueZ=ZTsuchthatZ>0,Z2=M.
¸max(M) ThemaximumeigenvalueofthematrixM=MT.
¸max(P;Q)ForP=PT,Q=QT>0,¸max(P;Q)denotesthemaximumeigen-
valueofthesymmetric pencil(P;Q),i.e.,¸max(Q¡1=2PQ¡1=2).
¸min(M) TheminimumeigenvalueofM=MT.
kMk ThespectralnormofamatrixorvectorM,i.e.,p
¸max(MTM).
Reduces totheEuclidean norm,i.e.,kxk=p
xTx,foravectorx.
diag(¢¢¢) Block-diagonal matrixformedfromtheargumen ts,i.e.,
diag(M1;:::;Mm)¢=2
664M1
...
Mm3
775:
157
158 Notation
CoS ConvexhullofthesetSµRn,givenby
CoS¢=(pX
i=1¸ixi¯¯¯¯¯xi2S;p¸0)
:
(Without lossofgeneralit y,wecantakep=n+1here.)
Ex Expectedvalueof(therandomvariable)x.
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Acronyms
AcronymMeaning Page
ARE Algebraic RiccatiEquation 3
CP ConvexMinimization Problem 11
DI Di®erentialInclusion 51
DNLDI Diagonal Norm-boundLinearDi®erentialInclusion 54
EVP EigenvalueMinimization Problem 10
GEVP Generalized EigenvalueMinimization Problem 10
LDI LinearDi®erentialInclusion 51
LMI LinearMatrixInequalit y 7
LMIP LinearMatrixInequalit yProblem 9
LQG Linear-Quadratic Gaussian 150
LQR Linear-Quadratic Regulator 114
LTI LinearTime-invariant 52
NLDI Norm-boundLinearDi®erentialInclusion 53
PLDI Polytopic LinearDi®erentialInclusion 53
PR Positive-Real 25
RMS Root-Mean-Square 91
159

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Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Index
A
Absolute stability,54,56,72
Acronyms
apologyfortoomany,5
list,159
A±ne
familyoftransfermatrices, 141
matrixinequalit y,27
Algebraic Riccatiequation, 3,26,110,115
Algorithm
conjugate-gradien ts,18
ellipsoid, 49
forLyapunovinequalities, 18
interior-point,30
Karmark ar's,4
methodofcenters,16,30
NesterovandNemirovskii's,4
Newton, 15
projective,30
Almostsureconvergence, 131
systemwithmultiplicativ enoise,136
®-shifted H1norm,67
Analytic center,15
ellipsoidal approximation, 45
ellipsoidal bound,49
Analytic solution, 2,62,115
LMI,24
Approximation
ellipsoidal, 49
ofPLDI,58
ARE,3,26,110,115
Augmentedsystem,124
B
Barrierfunction, 15
Bellman function, 112
Bounded-real lemma,26
Brownianmotion,137
C
Centralpath,16
Cheapellipsoid approximation, 50
Circlecriterion, 2
Closed-lo opsystem
dynamic state-feedbac k,116
staticstate-feedbac k,99
Youlaparametrization, 152
Co,12,51Complemen t
orthogonal, 22
Schur,7
Completion problem, 40,48
Complexit y
convexoptimization, 18,29
S-procedure,24
stabilization problem, 128
Componentwiseresults
usingscaling,111,134
viascaling,94
Componentwiseunit-energy input
LDI,81,96
state-feedbac k,106
Concavequadratic programming, 42
Condition number,37
coordinate transformation, 65,103
Conjugate-gradien ts,18
Constrained optimalcontrolproblem, 151
Constrain t
equality,9,19
integralquadratic vs.pointwise,96,
122
oncontrolinput,103
quali¯cation, 19
Contractivecompletion problem, 48
Controlinput,52
normconstrain t,103
Controllabilit yGramian, 78
Controller
dynamic feedback,111
gain-scheduled, 118
order,111
reduced-order, 117
state-feedbac k,99
Convex
Lyapunovfunction, 74
Convexfunction
LMIrepresentation,29
Lyapunov,64
Convexhull,12,51,158
Convexoptimization
complexit y,18,29
duality,5
ellipsoid algorithm, 12
interior-pointmethod,14
problem structure, 18
software,31
Convexproblem, 11
187
188 Index
Convolution,143
Coordinate transformation
condition number,65
digital¯lterrealization, 148
state-feedbac k,103
CP,11
Cuttingplane,12
D
Decayrate,66
Delay,144
DI,51
Diagonal
Lyapunovfunction, 40,47,144
norm-boundLDI,54
Diameter, 69
invariantellipsoid ofLDI,80
NLDI,59
Di®erentialinclusion, 51
selector-linear, 56
Dissipativit y
LDI,93
state-feedbac k,110
DNLDI, 54
well-posedness, 57
Duality,5,9,29,49
Dynamic state-feedbac k,111
E
E,11
Eigenvalueproblem, 10
Elimination
inLMI,22
ofvariable,9
procedure,48
Ellipsoid
algorithm, 12,29,49
approximating intersection, 44
approximating polytope,42
approximating reachableset,78,104,
106,121,125,132
approximating sum,46
approximating union,43
approximation, 49
diameter, 69,80
extractable energy,87,109
holdable, 102
invariant,68
LÄowner{John, 44,49
minimumvolume,11,44
poorman's,50
viaanalytic center,45,49
Entropy,142
Equalityconstrain t,19
Euclidean norm,8
EVP,10
Exogenous input,52
E,5
Expectedoutputenergy
LDI,87state-feedbac k,109
Exponentialtime-weighting,84,97
Extractable energy
LDI,87
state-feedbac k,109
F
Fadingmemory,74
Familyoftransfermatrices
examples, 152
parametrization, 143
Feedback
diagonal norm-bound,54
nonexpansiv e,123
structured, 54
time-varying,53
Finsler's lemma,22
Frequency measuremen ts,146
Function
quasiconvex,29
value,112
G
Gain
L2orRMS,91
state-feedbac k,99
Gain-scheduledcontroller,118
°-entropy,142
Generalized eigenvalueproblem, 10
GEVP,10
Globallinearization, 54
state-feedbac k,99
Gramian
controllabilit y,78
observability,85
H
Half-space
cuttingplane,12
Hamiltonian matrix,26
Hankelnorm
LDI,89
transfermatrix,142
H1norm,27,91,142
Holdabilit y,102
Holdable ellipsoid
approximating reachableset,106
LDI,102
outputenergy,108
H2norm,141
H2{H1problem, 154
I
Identi¯cation, 146
I°,142
Implicit equalityconstrain t,19
Impulse response,98
Inequalit y
return-di®erence, 153
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Index 189
Infeasible LMI,29
Initialcondition
extractable energy,85,108
safe,71
Input
controlorexogenous, 52
unit-energy ,77,104,121,125,132
unit-peak,82,107
Input-to-output properties
LDI,89
Lur'esystem,122
state-feedbac k,109
systemwithmultiplicativ enoise,133
Input-to-state properties
¯lterrealization, 148
LDI,77
Lur'esystem,121
state-feedbac k,104
systemwithmultiplicativ enoise,132
systemwithunknownparameters, 125
Integralquadratic constrain t,96,122
Interior-pointmethod,4,14,30
primal-dual, 18
Intervalmatrix,40
Invariance
ofpositiveorthant,152
Invariantellipsoid
approximating reachableset,78,104,
106
extractable energy,87
forLDI,68
outputenergy,108
Inverseproblem ofoptimalcontrol,147
It^ostochasticsystem,136
K
·,37
Kalman-Y akubovich-Popovlemma,2
Karmark ar'salgorithm, 4
L
LÄowner{John ellipsoid, 49
LDI
bettername,56
componentwiseunit-energy input,81
decayrate,66
de¯nition, 51
diagonal norm-bound,54
dissipativit y,93,110
expectedoutputenergy,87
Hankelnorm,89
invariantellipsoid, 68
L2gain,91,109
norm-bound,53
outputenergy,85,108
outputpeak,88
polytope,53
positiveorthantstabilizabilit y,144
quadratically stabilizable, 100
quadratic stability,61reachableset,77,104
returntime,71
scaling,94,111
stability,61
stabilitymargin,65
stable,notquadratically stable,64,
73
state-feedbac k,99
unit-energy input,77
unit-peakinput,82
Lemma
bounded-real, 26
Finsler,22
PR,25,93
Linear
inequalit y,7
matrixinequalit y,7
program, 10
Lineardi®erentialinclusion, 51
Linear-fractional
mapping, 53
programming, 11
Linearization
global,54
Linear-Quadratic Regulator, 108
Line-searc h,28
LMI
analytic center,15
analytic solution, 24
convexfunction representation,29
de¯nition, 7
equalityconstrain t,9
father,4
feasibilityproblem, 9
grandfather, 4
graphical condition, 2
history,2
infeasible, 29
interior-pointmethod,4,14
multiple,7
nonstrict, 7,18
optimization, 30
reduction, 19,31
Riccatiequation solution, 3
semide¯nite terms,22
slackvariable,8
software,31
standard problems, 9
strict,18
LMI-Lab, 31
LMIP,9
LMI-tool,31
Looptransformation, 119
sectorcondition, 129
LÄowner{John ellipsoid, 44
LP,10
LQG
multi-criterion, 150
systemwithmultiplicativ enoise,138
LQR,108,114
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
190 Index
LTIsystem
de¯nition, 52
input-to-state properties,78
overshoot,97
realization, 148
stability,62
transfermatrix,91
withdelay,144
withnonparametric uncertain ty,143
L2gain
LDI,91
LTIsystem,142
Lur'esystem,122
scaled,94
state-feedbac k,109
systemwithmultiplicativ enoise,133
Lur'eresolving equation, 3
Lur'esystem
absolute stability,56
de¯nition, 119
history,126
Lur'eterm
inLyapunovfunction, 119
Lyapunov
equation, 2,25,62,78,85
exponent,66
functional, 144
inequalit y,2,8
Lyapunovfunction
convex,64,74
diagonal, 40,47,144
integralquadratic constrain ts,123,127
Lur'e,119
parameter-dep endent,119
quadratic, 61
stochastic,132,136
Lyapunovinequalit y
reduction toastrictone,20
M
Margin
quadratic stability,65
quadratic stabilizabilit y,113
systemwithmultiplicativ enoise,132
matlab,31
Matrix
completion problem, 40
elimination ofvariable,22,48
Hamiltonian, 26
inequalit y,7
interval,40
linear-fractional mapping, 53
M,75
Moore{Penrose inverse,28
norm,8
P0,57
pencil,28
Pick,146
problems, 37
quadratic inequalit y,8transfer, 25
Maximumsingular value,8
Mean-square stability
de¯nition, 132
state-feedbac k,135
Methodofcenters,16,30
Minimumsizeellipsoid, 69
Minimumvolumeellipsoid, 11
Mixed H2{H1problem, 154
M-matrix, 75
Moore{Penrose inverse,28
¹-analysis, 152
Multi-criterion LQGproblem, 96,150,153
Multiple LMI,7
Multiplicativ enoise,131
Multiplier
nonparametric uncertain ty,143
systemwithunknownparameters, 124
theory,128
N
NesterovandNemirovskii'salgorithm, 4
Nevanlinna-Pic kinterpolation,145
Nevanlinna-Pic ktheory,146
Newton's method,15
NLDI
approximating PLDI,58
de¯nition, 53
input-to-state properties,79
quadratic stability,62
state-to-output properties,84
well-posedness, 53,63
Noise
ininterpolationproblems, 146
multiplicativ e,131
Nonconvexquadratic problem
mixedLQR{L2problem, 154
multi-criterion LQG,151
Nonexpansiv e,26
feedback,123
LDI,91
Nonlinear system
fadingmemory,74
state-feedbac k,99
Nonparametric uncertain ty
inLTIsystem,143
Nonstrict LMI,7,18
Norm
®-shifted H1,67
Euclidean, 8
Hankel,89
H1,27,91
H2,141
matrix,8
maximumsingular value,8
ofLTIsystem,148
quadratic approximation ofpiecewise
linear,41
scaledsingular value,38
Norm-boundLDI,53
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Index 191
Notation
di®erentialequations, 5
table,157
NP-complete, 42
NP-hard, 128
well-posedness ofDNLDI, 54,57
O
ObservabilityGramian, 85
,51
Optimal control,112,151
Optimization
ellipsoid algorithm, 12
interior-pointmethod,14
linear-fractional, 11
linearprogram, 10
LMI,30
quasiconvex,11
optin,31
Orderofcontroller,111
Orthant
invariant,152
stability,144
Orthogonal complemen t,22
Outerapproximation ofreachableset
state-feedbac k,106
Output, 52
Outputenergy
LDI,85
Lur'esystem,121
state-feedbac k,108
Outputpeak
LDI,88
Outputvarianceforwhitenoiseinput,113,
138
Over°ow,148
Overshoot
LTIsystem,97
P
P0matrix,57
Parameter-dep endent
controller,118
Lyapunovfunction, 119
Parametric uncertain ty,124
scalingmatrixproblem, 39
Parametrization
familyoftransfermatrices, 143
Youla,152
Passive,25
Passivity,3
dissipativit y,93
state-feedbac k,110
Pathofcenters,16
Pencil,28
Penroseinverse,28
Performance
LQGcost,150
ofLDI,113
systemwithmultiplicativ enoise,133Perturbation
LTI,143
norm-bound,53
stochastic,137
structured, 39,54
Pickmatrix,146
Piecewise linearnorm,41
PLDI,53
approximation byNLDI,58
input-to-state properties,78
quadratic stability,62
state-to-output properties,84
Pointwiseconstrain t,96,122
Polyhedron
invariant,152
Polynomial-time, 12
Polytope,12
approximated byinvariantellipsoid,
69
containingreachableset,80
ellipsoid approximation, 42
extractable energy,87,109
LDI,53
returntime,71
symmetric, 48
Polytopic LDI,53
Polytopic norm,41
Popovcriterion, 2,119
Positive-de¯nite
completion, 40
matrix,7
representable,29
Positiveorthantstabilizabilit y,144,152
Positive-real
lemma,25
transfermatrix,25
Primal-dual interior-pointmethod,18
Procedure
elimination ofvariable,22
S,23
Projectivealgorithm, 30
PRlemma,2,25,93
Q
Quadratic
Lyapunovfunction, 61
matrixinequalit y,8,114
Quadratic approximation
ofpolytopic norm,41
ofreachableset,105
Quadratic programming
inde¯nite, 42
Quadratic stability
invariantellipsoid, 68
LDI,61
margin,65
nonlinear system,74
vs.stability,64
Quadratic stabilizabilit y,100
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.
192 Index
Quali¯cation
constrain t,19
Quantization, 148
Quasicon vex
function, 29
problem, 11
R
Reachableset
LDI,77
Lur'esystem,121
relaxedDI,51
state-feedbac k,104
systemwithmultiplicativ enoise,132
systemwithunknownparameters, 125
Ruce,81
Rue,77
Rup,82
Realization ofsystem,148
Reduced-order controller,117
Reduction
Lyapunovinequalit y,20
strictlyfeasibleLMI,19
Regularit yconditions, 4
Relaxed versionofaDI,51
Response
steporimpulse, 97
Return-di®erence inequalit y,153
Returntime
LDI,71
Riccatiequation, 3,26,115
Riccatiinequalit y,110,115
RMSgain
LDI,91
state-feedbac k,109
RMSnorm,148
Root-Mean-Square, 91
Routh{Hurwitz algorithm, 35
S
Safeinitialcondition, 71
Scaled L2gain
LDI,94
state-feedbac k,111
systemwithmultiplicativ enoise,134
Scaledsingular value,38
Scaling
andS-procedure,64
componentwiseresults,94
forcomponentwiseresults,111,134
inmatrixproblem, 37
interpolation,145
Schurcomplemen t
nonstrict, 28
strict,7
scilab,31
Sectorcondition, 56
looptransformation, 129
Selector-linear
di®erentialinclusion, 56Self-concordan tbarrier,15
Simultaneous matrixcompletion, 40
Singular valueplot,27
SlackvariableinLMIs,8
Small-gain theorem, 3,143
SoftwareforLMIproblems, 31
Sparsitypattern, 40
Sphere
smallest orlargest,44,46
S-procedure
andscaling,64
description, 23
Stability
absolute, 56,72
degree,67
LDI,61
LTIsystem,62
Lur'esystem,120
positiveorthant,144
systemwithmultiplicativ enoise,132
systemwithunknownparameters, 125
Stabilitymargin
LDI,65
mean-square, 132,137
Stabilizabilit y
LDI,100
positiveorthant,144
systemwithmultiplicativ enoise,134
Stabilization problem
complexit y,128
Standard LMIproblems, 9
¤,143
State,52
State-feedbac k
controller,99
globallinearization, 99
inlinearsystemwithdelay,145
LDI,99
nonlinear system,99
positiveorthantstability,144
scaling,111
synthesis,99
systemwithmultiplicativ enoise,134
Stateproperties
LDI,61
Lur'esystem,120
state-feedbac k,100
systemwithmultiplicativ enoise,131
State-to-output properties
¯lterrealization, 148
LDI,84
Lur'esystem,121
state-feedbac k,107
Staticstate-feedbac k,99
Step
length,28
response,97
Stochasticstabilitymargin,132
Stochasticsystem
continuous-time, 136
Copyright c°1994bytheSocietyforIndustrial andAppliedMathematics.
Index 193
discrete-time, 131
Stopping criterion
methodofcenters,17
Stratono vitchstochasticsystem,136
Strictfeasibility
reduction to,19
StrictLMI,18
Structured
feedback,54,152
singular value,152
Structured uncertain ty
interpolation,146
matrixproblem, 39
Sturmmethod,26,35
Symmetric polytope,48
Synthesis
output-feedbac k,117,150
state-feedbac k,99,134
System
realization, 148
stochastic,136
uncertain, 56
withmultiplicativ enoise,131
Systemtheoryduality,5
Systemwithunknownparameters
matrixproblem, 39
multipliers for,124
T
TangentialNevanlinna-Pic kinterpolation,
145
Time-varying
feedbackmatrix,53
system,51
Tr,5
Trace,5
Transfermatrix,25
LTIsystem,64,91
optimization, 141
positive-real,25
Transformation
loop,119
Tsypkin criterion, 2
U
Uncertain ty
LTI,143
parametric, 124
random, 131
time-varying,51
Unit-energy input
LDI,77
Lur'esystem,121
state-feedbac k,104
systemwithmultiplicativ enoise,132
systemwithunknownparameters, 125
Unit-intensitywhitenoise,138
Unit-peakinput,97
LDI,82
state-feedbac k,107Unknownparameters
systemwith,124
V
Valuefunction, 112
Variable
elimination, 9
matrix,8
slack,8
Vertex
ofpolytope,41
vol,13
Volume
ellipsoid algorithm, 13
invariantellipsoid, 69
ofellipsoid, 41
W
Well-posedness
DNLDI, 54,57
NLDI,53,63
Whitenoise
Brownianmotion,137
uncertain ty,131
Y
Youlaparametrization, 152
Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistribute d.

Stephen Boyd
Laurent El Ghaoui
Eric Feron
Venkataramanan Balakrishnan