Algorithms and Software for LMI Problems in Control Introduction

AlgorithmsandSoftwareforLMIProblemsinControl LievenVandenbergheyandVenkataramananBalakrishnanz
callysolvedbyreformulatingthemasconvexoptimizationproblemswithlinearmatrixcationsofLMIstocontrolsystemanalysisanddesign,therehavebeenfewpublicationsinequality(LMI)constraints.WhilenumerousarticleshaveappearedcatalogingappliAnumberofimportantproblemsfromsystemandcontroltheorycanbenumeriinthecontrolliteraturedescribingthenumericalsolutionoftheseoptimizationprob- numericalalgorithmsforLMIproblems,andoftheavailablesoftware. lems.Thepurposeofthisarticleistoprovideanoverviewofthestateoftheartof
Awidevarietyofproblemsinsystemsandcontroltheorycanbecastorrecastassemidenite programming(SDP)problems1,thatis,problemsoftheform Introduction
minimizebTy
wherey2RmisthevariableandthematricesC=CT2Rnn,andAi=ATi2Rnnare subjecttoC+mXi=1yiAi 0; (1)
given.Theinequalitysigndenotesmatrixinequality,i.e.,thematrixC+PiyiAiisnegative
iscalledalinearmatrixinequality(LMI).Inotherwords,SDPsareconvexoptimization semidenite.Theconstraint
problemswithalinearobjectivefunctionandlinearmatrixinequality(LMI)constraints. C+mXi=1yiAi 0
tothespecialsessionAlgorithmsandSoftwareToolsforLMIProblemsinControlatthe1996IEEECACSD symposiuminDearborn. Angeles,CA90095-1594,telephone:310-206-1259,fax:310-206-4685,email:vandenbe@ee.ucla.edu. yDr.VandenbergheiswiththeDepartmentofElectricalEngineeringattheUniversityofCalifornia,Los Thispaperisanupdatedversionoftheconferencepublication[1],whichwasintendedasanintroduction
WestLafayette,IN47907-1285,telephone:765-494-0728,fax:765-494-3371,email:ragu@ecn.purdue.edu. Fellowship. Dr.Balakrishnan'sresearchissupportedinpartbyONRundercontractN00014-97-1-0640,andaGM 1WeshalluseSDPtomeanboth\semideniteprogramming",aswellasa\semideniteprogram",i.e.,
zDr.BalakrishnaniswiththeSchoolofElectricalandComputerEngineeringatPurdueUniversity,
asemideniteprogrammingproblem. 1

encounteredinsystemsandcontroltheory.Examplesinclude:multicriterionLQG,synthesis state-spacerealizationsoftransfermatrices,normscaling,synthesisofmultipliersforPopov- oflinearstatefeedbackformultipleornonlinearplants(\multi-modelcontrol"),optimal ThoughtheformoftheSDP(1)appearsveryspecialized,itturnsoutthatitiswidely
gain-scheduledcontrollerdesign,andmanyothers. likeanalysisofsystemswithunknowngains,robustnessanalysisandrobustcontrollerdesign,
canbesolvednumericallyveryeciently.Inmanycases|forexample,withmulti-model control[3]|theLMIsencounteredinSDPsinsystemsandcontroltheoryhavetheform fortheonesencounteredwithH2andH1control[2],forexample),butingeneralthey Forafewveryspecialcasesthereare\analyticalsolutions"toSDPs(viaRiccatiequations
ofsimultaneous(coupled)LyapunovoralgebraicRiccatiinequalities;usingrecentinterior-
onthesolutionofalgebraicRiccatiequationstoatheorybasedonthesolutionof(multiple, pointmethodssuchproblemscanbesolvedinatimethatisroughlycomparabletothetime
simultaneous)LyapunovorRiccatiinequalitiesismodest. [3,4].Thereforethecomputationalcostofextendingcurrentcontroltheorythatisbased requiredtosolvethesamenumberof(uncoupled)LyapunovorAlgebraicRiccatiequations
SDPtothesolutionofsystemandcontrolproblems.Perhapsthemostcomprehensivelistcan befoundinthebook[3].Sinceitspublication,anumberofpapershaveappearedchronicling furtherapplicationsofSDPincontrol;weciteforinstancethesurveyarticle[5]thatappeared Anumberofpublicationscanbefoundinthecontrolliteraturethatsurveyapplicationsof
ControlonLinearMatrixInequalitiesinControlTheoryandApplications,publishedrecently, evidencedbythelargenumberofpublicationsinrecentcontrolconferences. inNovember-December,1996[6].ThegrowingpopularityofLMImethodsforcontrolisalso inthismagazine,andthespecialissueoftheInternationalJournalofRobustandNonlinear
certaineigenvalueminimizationproblemsthatcanbecastasSDPshavebeenusedforobtainingboundsandheuristicsolutionsforcombinatorialoptimizationproblems(see[7,8] and[9,Chapter9]).Theeciencyofrecentinterior-pointmethodsforSDP,whichisdirectly SpecialclassesoftheSDPhavealonghistoryinoptimizationaswell.Forexample,
responsibleforthepopularityofSDPincontrol,hasthereforealsoattractedagreatdealof interestinoptimizationcircles,overshadowingearliersolutionmethodsbasedontechniques
wasprimarilymotivatedbyapplicationsofSDPincombinatorialoptimizationbut,more ference,thereareworkshopsandspecialsessionsdevotedexclusivelytoSDP,andaspecial issueofMathematicalProgramminghasrecentlybeendevotedtoSDP[14].Thisinterest fromnondierentiableoptimization[8,10,11,12,13].Ateverymajoroptimizationcon-
recently,alsobytheapplicationsincontrol. thedenitionandsomebasicpropertiesofthesemideniteprogrammingproblem.We thendescriberecentdevelopmentsininterior-pointalgorithmsandavailablesoftware.We ofnumericalalgorithmsforLMIproblems,andoftheavailablesoftware.Werstreview Theprimarypurposeofthisarticleistoprovideanoverviewofthestateoftheart
concludewithsomeextensionsofSDP.
2

Inthissectionweprovideabriefintroductiontothesemideniteprogrammingproblem.For moreextensivesurveysonthetheoryandapplicationsofSDP,werefertoAlizadeh[15],Boyd Semideniteprogramming
etal.[3],LewisandOverton[16],NesterovandNemirovskii[17,x6.4],andVandenberghe andBoyd[18]. tions,wewillreferto(1)asanSDPininequalityform.TheoptimizationproblemWehavealreadydenedanSDPformallyin(1).Todistinguishitfromotherformula- maximizeTrCX
iscalledanSDPinequalityform.Here,thevariableisthematrixX=XT2Rnn,and subjecttoXTrAiX+bi=0;i=1;:::;m 0 (2)
the`standard'form(althoughtheinequalityformappearstobemoreappropriateforcontrol theory). easilyconvertedinto(2)andvice-versa,soitisamatterofconventionwhatweconsideras Trstandsfortrace,ie.,sumofthediagonalentriesofasquarematrix.TheSDP(1)canbe
eralizationsofseveralimportantoptimizationproblems.Forexample,thelinearprogram (LP) Itturnsoutthatthethesemideniteprograms(1)and(2)canberegardedasgen- maximizecTx
inwhichtheinequalityxsubjecttox 0denotescomponentwiseinequality,canbeexpressedasan aTix+bi=0;i=1;:::;m; 0 (3)
orthantisreplacedbytheconeofpositivesemidenitematrices. inequalitiesbetweenvectorsarereplacedbymatrixinequalities,or,equivalently,therst SDP(2)withAi=diag(ai)andC=diag(c),andX=diag(x).Semideniteprogrammingcanalsoberegardedasanextensionoflinearprogrammingwherethecomponentwise istheoptimalvalueof(2)andu?istheoptimalvalueof(1),thenwehave: Itcanbeshownthatproblems(1)and(2)aredualsofeachother.Moreprecisely,if`?
strongduality:If(1)isstrictlyfeasible(i.e.,thereexistsaywithC+PiyiAi0withTrAiX+bi=0),then `?;
orythatdoesnotrequirestrictfeasibilitywasrecentlydevelopedbyRamana,TuncelandTheresultfollowsfromstandardconvexoptimizationduality.(Astrongerdualitythe- u?=`?.
in[20]. Wolkowicz[19].)SomeconnectionsbetweenSDPdualityanddualityincontrolareexplored Ifweassumethatboth(1)and(2)arestrictlyfeasible,thentheoptimalvaluesinboth
3

problemsareattained,andthesolutionsarecharacterizedbytheoptimalityconditions TrAiX+bi=0;i=1;:::;m XZ+C+Pmi=1yiAi=0 XZ=0: 0;Z 0
TherstthreeconditionsstatefeasibilityofX,Zandy.Thelastconditioniscalled (4)
complementaryslackness.
Briefhistory Interior-pointmethods
methodforLP[24].In1988NesterovandNemirovskii[25]showedthatthoseinterior-point estinthemwasrevivedin1984,whenKarmarkarintroducedapolynomial-timeinterior-point Theideasunderlyinginterior-pointmethodsforconvexoptimizationcanbetracedbackto
methodsforlinearprogrammingcan,inprinciple,begeneralizedtoallconvexoptimization thesixties;seee.g.,FiaccoandMcCormick[21],LieuandHuard[22],andDikin[23]).Inter-
therefore,interior-pointmethodsareapplicable. ertycalledself-concordance.Linearmatrixinequalitiesareanimportantclassofconvex constraintsforwhichreadilycomputableself-concordantbarrierfunctionsareknown,and, problems.Thekeyelementistheknowledgeofabarrierfunctionwithacertainprop28]generalizedinterior-pointmethodsfromlinearprogrammingtosemideniteprogram-
methodsforlinearprogramminghavebeenextendedtosemideniteprogramming.This ming.Vastprogresshasbeenmadeinthelasttwoyears,andtodayalmostallinterior-point IndependentlyofNesterovandNemirovskii,Alizadeh[26]andKamathandKarmarkar[27,
theexcellentperformanceofprimal-dualinterior-pointmethodsforlarge-scalelinearpro- recentresearchhaslargelyconcentratedonprimal-dualmethodsinthehopeofemulating
astheprojectivealgorithmandthemethodofcenters.Wereferto[5,p.80]or[3,x2]for gramming[29,30].Theremainderofthissectionwillconcentrateonthisrecentwork.We
surveysoftheseearliermethods. algorithm,themethodofalternatingprojections,andprimalinterior-pointmethodssuch shouldmentionhoweverthatothermethodshavebeenusedsuccessfully,e.g.,theellipsoid
Primal-dualmethodsforSDP Themostpromisingmethodsforsemideniteprogrammingsolvethetwoproblems(1) and(2)simultaneously.Theseprimal-dualmethodsareusuallyinterpretedasmethodsfor y()ofthenonlinearequationsX followingtheprimal-dualcentralpath,whichisdenedasthesetofsolutionsX(),Z(),
TrAiX+bi=0;i=1;:::;m Z+C+Pmi=1yiAi=0 XZ=I; 0;Z 0 (5)
4

uniquefor complementaryslacknessconditionXZ=0.Itcanbeshownthatthesolutionof(5)is conditions(4).Theonlydierenceisthelastcondition,XZ=I,whichreplacesthe where 0isaparameter.Notethattheseconditionsareverysimilartotheoptimality
X()minimizes approachoptimalityifgoestozero. ThecentralpointsX(),y(),Z()arealsotheminimizersoftwoconvexfunctions: >0(assumingstrictprimalanddualfeasibility),andthatX(),Z(),y()
overallX>0withTrAiX+bi=0;y()minimizes 'd(;y)=bTy�logdet 'p(;X)=�TrCX�logdetX�1
self-concordant,whichallowsustoapplyNesterovandNemirovskii'stheoryforproving overallywithC+Pmi=1yiAi0,Z>0,ybethecurrentiterate.Forsimplicityweassumethatthesepoints
Z+mXi=1yiAi=0 TrAiX=0;i=1;:::;m
IfweeliminateZfromthesecondandthirdequations,weobtain X+Z�1ZZ�1=Z�1�X:
��1ZXZ+mXi=1yiAi=�1ZXZ�Z: TrAiX=0;i=1;:::;m (7) (6)
diag(z),X=diag(x),and Thisisasetofm+n(n+1)=2equationsinthem+n(n+1)=2variablesy,X=XT2Rnn. InthespecialcaseoftheLP(3),whereallmatricesarediagonal,wecanwriteZ= " ��1Z2A#"xy#=" AT 0 5
�1Z2x�z#: 0

whereA=[a1:::am]. whichleadsto AsecondpossibilityforlinearizingtheequationXZ=IistowriteitasZ=X�1,
Z+mXi=1yiAi=0 TrAiX=0;i=1;:::;m
EliminatingZ,weobtainX�1XX�1+Z=�Z+X�1: �X�1XX�1+mXi=1yiAi=Z�X�1: TrAiX=0;i=1;:::;m
Specializedtolinearprogrammingtheequationsbecome "
TherstSDPmethodswerebasedontheseprimalordualscalings(seeforexample, �X�2A#"xy#=" AT 0 z�X�1e#: 0
Yoshise[31].Forlinearprogrammingtheresultingequationsforthesearchdirectionshave oneusuallyprefersaprimal-dualsymmetricscalingintroducedbyKojima,Mizunoand programming,however,theprimalanddualscalingsarerarelyusedinpractice.Instead, NesterovandNemirovskii[17],Alizadeh[26],andVandenbergheandBoyd[4]).Inlinear
theform TheseequationsareobtainedbylinearizingXZ=Ias " �X�1ZA#"xy#=" AT 0 z�X�1e:#: 0 (8)
scalingcanachieveahigheraccuracythanmethodsbasedonthetheprimalordualscaling Severalresearchershavedemonstratedthatmethodsthatusethisprimal-dualsymmetric XZ+XZ=I�XZ: (9)
(seeforexampleWright[32]),andthereforethesymmetricscalingisthebasisofallpractical LPinterior-pointmethods. linearization(9)leadstoalinearsystemTrAiX=0;i=1;:::;m Theextensionofthissymmetricprimal-dualscalingtoSDPisnotstraightforward:The
�X�1XZ+mXi=1yiAi=Z�X�1 (11) (10)
researchinSDPhasthereforeconcentratedonextendingtheprimal-dualsymmetricscaling butunfortunatelythesolutionXisnotsymmetricingeneral.Muchofthemostrecent 6

seemtobethemostpromising.Helmberg,Rendl,Vanderbei,andWolkowicz[33],Kojima, ShidohandHara[34],andMonteiro[35]solve(10)and(11)andlinearizetheresultingX. lasttwoyears.Amongtheproposedsymmetricprimal-dualalgorithms,threevariations fromLPtoSDP,and,asaresultofthiseort,veryrapidprogresshasbeenmadeinthe
Alizadeh,HaeberlyandOverton[36]rstwriteXZ=IasXZ+ZX=2Iandthen
andrecentlySturmandZhang[39],havedenedathirddirection,obtainedasfollows.First TheresultingXandZareautomaticallysymmetric.Finally,NesterovandTodd[37,38], linearizethisas
amatrixRiscomputedsuchthatRTXR=1=2andRTZ�1R=1=2,whereisadiagonal XZ+XZ+ZX+ZX=2I�XZ�ZX:
matrixwithasdiagonalelementstheeigenvaluesofXZ.Onethensolvestheequations �RRTXRRT+mXi=1yiAi=Z�X�1: TrAiX=0;i=1;:::;m (13) (12)
toobtainthesearchdirectionsX,Z,y.Numericaldetailsonthismethodcanbefound inTodd,TohandTutuncu[40].Finally,Kojima,ShindohandHara[34],Monteiro[35],and MonteiroandZhang[41]havepresentedunifyingframeworksforprimal-dualmethods.
Softwarepackages Someotherimportantrecentarticlesandreportsarelistedinthereferencesofthispaper2.
implementationofaninterior-pointmethodforSDPwasbyNesterovandNemirovskiiin [65],usingtheprojectivealgorithm[17].Matlab'sLMIControlToolbox[66]isbasedon Severalresearchershavemadeavailablesoftwareforsemideniteprogramming.Therst thesamealgorithm,andoersagraphicaluserinterfaceandextensivesupportforcontrol applications.ThecodeSP[67]isbasedonaprimal-dualpotentialreductionmethodwith theNesterovandToddscaling.ThecodeiswritteninCwithcallstoBLASandLAPACK interfacestoSPthatsimplifythespecicationofSDPswherethevariableshavematrix andincludesaninterfacetoMatlab.SDPSOL[68]andLMITOOL[69]oeruser-friendly structure.TheInduced-NormControlToolbox[70]isatoolboxforrobustandoptimal control,inturnbasedonLMITOOL.
Wolkowicz[33].SDPHA[73]isaMatlabimplementationofahomogeneousformulationof CSDP[72]isaCimplementationofthealgorithmofHelmberg,Rendl,Vanderbei,and SDPA[71]isaC++code,basedonthealgorithmofKojima,ShindohandHara[34]. Severalimplementationsofthemostrecentprimal-dualmethodsarealsoavailablenow.
thedierentprimal-dualmethodsdescribedabove.SDPT3[74]isaMatlabimplementation
linearconstraintsdirectly. ofthemostimportantinfeasibleprimal-dualpath-followingmethods.SDPPACK[75]isan inCtoincreasetheeciency.Italsoprovidestheusefulfeatureofhandlingquadraticand implementationofthealgorithmof[36].ItiswritteninMatlab,withcriticalpartswritten
gonneNationalLaboratory(http://www.mcs.anl.gov/home/otc/InteriorPoint/index.html).Helmberg(http://www.zib-berlin.de/~bzfhelmb/semidef.html)andtheinterior-pointarchiveatAr- 2MostrecentpapersareavailableatthesemideniteprogramminghomepagemaintainedbyCristoph
7

Thedeterminantmaximizationproblem Extensions
thefollowingform: theSDP(1),whichwasdiscussedinmoredetailin[76].Thisextensioncanbewrittenin IntheirsurveyofLMIproblemsincontrol,Boydetal.[3]alsoconsideredanextensionof
minimizebTy�logdet subjecttoC+mXi=1yiAi �D�mXi=1yiBi!
D+mXi=1yiBi0 (15)
informationandcommunicationtheory.Thereforethetheorybehindtheirsolutionisof and(14)areduals.
aninterior-pointmethodforthemaxdet-problemaredescribedin[76].Softwareforsolving greatinterest,andtheresultingalgorithmshavewideapplication.Alistofapplicationsand Maxdet-problemsariseinmanyelds,includingcomputationalgeometry,statistics,and
maxdet-problemsisavailablein[77],andhasbeenincorporatedinSDPSOL[68]. Thegeneralizedeigenvalueminimizationproblem
theirmaximumgeneralizedeigenvalue,wecansolvetheoptimizationproblem posewehaveapairofmatrices(A(x);B(x)),bothanefunctionsofx.InordertominimizeAthirdstandardproblemfrom[3]isthegeneralizedeigenvalueminimizationproblem.Sup- minimizet
3TheproblemwasdenotedCPin[3]. subjecttotB(x)�A(x) B(x) 0: 0 (16)
8

Thisiscalledageneralizedlinear-fractionalproblem.Itincludesthelinearfractionalproblem
subjecttoAx+b minimize eTx+f cTx+d
asaspecialcase. Problem(16)isnotasemideniteprogram,however,becauseofthebilineartermtB(x). 0;eTx+f>0
Itisaquasi-convexproblem,andcanstillbeecientlysolved.SeeBoydandElGhaoui[78], izedalgorithms,and[3]forapplicationsofthisproblemincontrol.AnimplementationofHaeberlyandOverton[79],andNesterovandNemirovskii[17,80,81]fordetailsonspecial- theNesterovandNemirovskiialgorithmisalsoprovidedintheLMIControltoolbox[66].
WenallyconsideranextensionoftheSDP(1),obtainedbyreplacingthelinearmatrix Thebilinearmatrixinequalityproblem inequalityconstraintsbyaquadraticmatrixinequality, minimizebTy
Thisproblemisnonconvex,butitisextremelygeneral.Forexample,ifthematricesC,Ai, subjecttoC+mXi=1yiAi+mX i;j=1yiyjBij 0: (17)
andBijarediagonal,theconstraintin(17)reducestoasetofn(possiblyindenite)quadratic constraintsinx.Problem(17)thereforeincludesallquadraticoptimizationproblems.It alsoincludesallpolynomialproblems(sincebyintroducingnewvariables,onecanreduce
wesplitthevariablesintwovectorsxandy,andreplacetheconstraintbyabilinear(or etc.Incontroltheory,amorerestrictedbilinearformseemstobegeneralenough.Here anypolynomialinequalitytoasetofquadraticinequalities),allf0;1gandintegerprograms,
bi-ane)matrixinequality(BMI): minimizecTx+bTy
Theproblemdataarethevectorsc2Rmandb2RlandthesymmetricmatricesAi,Bk, subjecttoD+mXi=1yiAi+lXk=1xkBk+mXi=1lXk=1xiykCik 0: (18)
andCik2Rnn.
mentaldierencewithLMIsisthatBMIproblemsarenon-convex,andnonon-exponentialGoh,andothers[82,83,84,85,86,87],ElGhaouiandBalakrishnan[88],etc).Thefundacertainty,xed-ordercontrollerdesign,decentralizedcontrollersynthesisetc.(seeSafonov,BMIsincludeawidevarietyofcontrolproblems,includingsynthesiswithstructuredun- referencesareeitherlocalmethodsthatalternatebetweenminimizingoverxandy,orglobal timealgorithmsfortheirsolutionareknowntoexist.Thealgorithmsdescribedintheabove (branchandbound)techniquesbasedonthesolutionofasequenceofLMIproblems. 9

ThecurrentstateofresearchonLMIsincontrolcanbesummarized: Conclusion
thelattercase,bilinearmatrixinequalities(BMIs)havebeenrecognizedasauseful intermsofLMIs,andthoseforwhichanLMIformulationisunlikelytoexist.In Therehasbeenintensiveresearchonidentifyingcontrolproblemsthatcanbecast
rapidprogressininterior-pointalgorithmsforsolvingSDPs,focusingonlocalconver- formulation.
gencerates,worst-casecomplexity,etc.,andonextendingtoSDPthesophisticated Thecombinedactivityinmathematicalprogrammingandcontroltheoryhasledtovery andecientprimal-dualinterior-pointmethodsdevelopedforlinearprogramming.
ThusLMIsarebecomingbasictoolsincontrol,muchthewayRiccatiequationsbecame available.Thesecodeshaveprovenusefulforsmalltomedium-sizedproblems. Severalbasicsoftwareimplementationsofinterior-pointmethodsforSDPhavebecome
problemsthatariseincontrol.Weexpectthatthisresearchwillleadtoasecondgeneration ofgeneral-purposeLMIcodes,whichwillexploitmoreproblemstructure(e.g.,sparsity)to calprogrammingcommunityisleadingtomorepowerfulalgorithmsfortheLMIandBMIbasictoolsinthe1960s.Atthesametime,thecurrentstronginterestinthemathematipurposesoftwareforSDPwouldhaveasimilareect:itwouldmakeitpossibletoroutinely Toalargeextent,linearprogrammingowesitssuccesstotheexistenceofgeneral-purpose softwareforlargesparseLPs.Onamoremodestscale,theavailabilityofecientgeneral- increasetheeciency.Theanalogywithlinearprogrammingillustratestheramications.
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