43rd IEEE Conference on Decision and Control

Atlantis, Paradise Island, Bahamas

FrC13.6

Stabilizinglinearsystemswithsaturation

throughoptimalcontrol

RafalGoebel

Abstract—Weconstructacontinuousfeedbackforasatu-ratedsystemx˙(t)=Ax(t)+Bσ(u(t)).Thefeedbackrendersthesystemasymptoticallystableonthewholesetofstatesthatcanbedrivento0withanopen-loopcontrol.Trajectoriesoftheresultingclosed-loopsystemareoptimalforanauxiliaryoptimalcontrolproblemwithaconvexcostandlineardynam-ics.Thevaluefunctionfortheauxiliaryproblem,whichweshowtobedifferentiable,servesasaLyapunovfunctionforthesaturatedsystem.Relatingthesaturatedsystem,whichisnonlinear,toanoptimalcontrolproblemwithlineardynamicsispossiblethankstothemonotonestructureofsaturation.

Beforeoutliningourapproachtotheproblem,westatethemainresult:

Theorem1.1:(stabilizingfeedback).Considerthesystem

x˙(t)=Ax(t)+Bσ(u(t))

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I.INTRODUCTIONANDMAINRESULT

Globalasymptoticstabilizationofalinearsystemwith

saturatingactuators

x˙(t)=Ax(t)+Bσ(u(t))

(1)

underassumptions(A1)and(A2).LetX0bethesetof

Rnforwhichthereexistsapiecewisecontinuousallx0∈I

controlu:[0,+∞)→IRksuchthatthesolutionof(2)

Rn×nbeanywithx(0)=x0convergesto0.LetQ∈I

symmetricandpositivedefinitematrix.

Then,thereexistsacontinuousmappingF:X0→IRkandaconvex,positivedefinite,anddifferentiablefunction

Rsuchthat,forx0∈X0,thesolutionx(·)toV:X0→I

x˙(t)=Ax(t)+Bσ(F(x(t)))

withx(0)=x0satisfies

1d

V(x(t))≤−x(t)TQx(t)(4)dt2

sothatx(t)→0ast→+∞.

Themaintoolinourapproachisthefollowinglinear-convexregulatorproblem:

󰀁∞

1

x(t)TQx(t)+r(w(t))dtmin

20

(5)LCR(x0):

s.t.x˙(t)=Ax(t)+Bw(t),

x(0)=x0.Thisproblemhasnosaturationbutinformationaboutσisrepresentedthroughtheconvexpenaltyfunctionr.Morespecifically,under(A2),thereexistsaconvexfunctions:IRk→IRwiths(0)=0andwiththegradient∇s=σ.Thenristakentobetheconvexfunctionconjugatetosinthesenseofconvexanalysis,seeRockafellar[11].WeexplainthisindetailinSectionII.

Thestabilizingfeedbackforthesaturatedsystem(1)willturnouttobeverycloselyrelatedtotheoptimalfeedbackfortheLCR.In(5)thecontrolvariableisdenotedw(·)todistinguishitfromthecontrolu(·)in(1)–thesearenotthesame.Theoptimalvaluefunctionfor(5)willbetheLyapunovfunctionmentionedinTheorem1.1.

IntroducingaLCRasanauxiliaryoptimalcontrolprob-lemisanaturalidea.Feedbacksstabilizingalinearsystemx˙(t)=Ax(t)+Bu(t)canbefoundwiththehelpofaLQRproblem.Whenσin(1)isthestandardsaturation,thatisσi(ui)equalsuiif−1≤ui≤1,−1ifui<−1,and1ifui>1,onecanconsideraLQRwithaconstraint|ui|≤1.SuchconstrainedLQRcanbeequivalentlywritteninthe

(3)

cannot,ingeneral,beachievedwithalinearfeedback.Moreover,ifaneigenvalueofAhasapositiverealpartandσisbounded,thesetX0consistingofallstatesthatcanbedrivento0withanopenloopcontrolwillnotequaltothewholestatespace.Ifsucheigenvaluesareexcluded,continuousfeedbacksgloballystabilizing(1)existundermildassumptionsonσ,asshownbySontagandSussmann[15]andSontag,Sussmann,andYang[16].Alsothen,semiglobalstabilizationcanbeachievedwithlinearfeedbackpossessingadditionalpropertieslikerobustnessanddisturbancerejection,seeSaberi,Lin,andTeel[14].Forthegeneralcase,muchworkhasbeendevotedtoestimatingX0andtosemiglobalstabilizationonX0(thatis,toconstructingfeedbackswhichstabilize(1)onanyapriorigivencompactsubsetofX0),seeHuandLin[10]andthenumerousreferencestherein.ApositiveresultonsemiglobalstabilizationwithacontinuousfeedbackofalinearsystemunderbothinputandstateconstraintswasrecentlyshownbyStoorvogel,Saberi,andShi[17].

Tosummarize,theexistenceofacontinuousfeedbackthatrenders(1)asymptoticallystableonX0hasnotbeenpreviouslyestablished.Weproveithere,assuming:

Rn×k,is(A1)Thepair(A,B),whereA∈IRn×n,B∈I

controllable.

Rkhastheform(A2)Thesaturationfunctionσ:IRk→I

σ(u)=(σ1(u1),σ2(u2),...,σk(uk)),

Rwhereσ(0)=0,andeachσiisnondecreasingonI

andstrictlyincreasingonaneighborhoodof0.

Forafullversion,seeGoebel[6]

CenterforControlEngineeringandComputation,UniversityofCalifor-nia,SantaBarbara.Currentaddress:P.O.Box15172,Seattle,WA98115.

rafal@ece.ucsb.edu

0-7803-8682-5/04/$20.00 ©2004 IEEE5517

LCRform(5),withrbeingquadraticifusatisfiestheconstraint,andequalto+∞otherwise(thisisawell-knowntechniqueinoptimization).TheuseofvaluefunctionsofauxiliaryproblemsasLyapunovfunctionsispossibleforgeneralnonlinearsystems,andClarke,Ledyaev,Rifford,Stern[4].Theexpectedlackofdifferentiabilityofvaluefunctionsforsuchgeneralproblems,andtheconsequentlackofcontinuityofoptimalfeedbacks,wereapartofthemotivationforanalternateapproachtostabilizationofasaturatedsystemin[16].

ThespecialstructureofLCRhasimportantconsequencesforthevaluefunctionV(x0)definedastheoptimalvaluein(5).Mostimportantly,Visaconvexfunction.Itispositivedefinite,hasfinitevaluesontheopenandconvexsetX0whileV(x0)=+∞ifx0∈X0,anditssublevelsets{x|V(x)≤α}arecompactforeachα≥0.Finally,weproveitisdifferentiableonX0,andthencontinuityof∇VonX0(whichwillbethekeytocontinuityofthestabilizingfeedbackfor(1))followsfromageneralpropertyofconvexfunctions.DetailsareprovidedinSectionIII.

WiththedifferentiabilityofVestablished,standarddynamicprogrammingargumentsshowthattheoptimalfeedbackfortheLCRis

w=FLCR(x)=∇s(−B∗∇V(x)).

󰀂󰀃

Equivalently,w=argmaxw−∇V(x)TBw−r(w).Optimaltrajectoriesx(·)resultingfromapplyingthisop-dV(x(t))≤timalfeedbacktothelinearsystemsatisfydt1−2x(t)TQx(t),andhencex(t)→0ast→∞.

Now,therelationshipbetween(1)andLCRshouldbecomeclear.Since∇s=σ,thenonsaturatedlinearsystemwiththefeedbackw=∇s(−B∗∇V(x))isexactlythesameasthesaturatedsystem(1)withthefeedback

u=F(x)=−B∗∇V(x).

ThismeansthatFisastabilizingfeedbackforthesaturatedsystem.Moreover,thevaluefunctionforLCRservesasaclassicalLyapunovfunctionforthesaturatedsystem.WemakethispreciseinSectionIV.

II.SATURATIONFUNCTIONSASGRADIENTS

Thekeytoourapproachisexpressingthesaturationfunctionσofthesaturatedlinearsystem(1)asagradientofaconvexfunction.AstandardreferencefortheconvexanalysisfactsweusebelowisthebookbyRockafellar[11].Example2.1:Letσ:IR→IRbeacontinuousandnondecreasingfunction,withσ(0)=0.Then

󰀁u

σ(t)dts(u)=

0

12

δu2if|u|<󰀞,by−δ󰀞u−1isboundedbelowby22δ󰀞2

ifu≤−󰀞,andbyδ󰀞u−12δ󰀞if󰀞u.-IfσisLipschitzwithconstantl,thens(u)≤2Here,theimportantrelationshipisbetweenstrictconvexityofsonaneighborhoodof0andσbeingstrictlyincreasingonsuchaneighborhood.

Statementsjustmadecanbeeasilyverifiedforthestandardsaturationfunctionσ:IR→[−1,1],whichisthederivativeofthefollowingconvexfunction:

⎧

foru<−1,⎨−u−1212

(6)s(u)=ufor−1≤u≤1,

⎩21for1wecanassociateaconvex󰀇functionsiasinExample2.1.

k

Thenσ=∇sfors(u)=i=1si(ui),thisisofcourseaconvexfunction.Growthofscanbeanalyzedintermsof

Rkthatofσi’s.Inparticular,sisstrictlyconvexaround0∈I

ififandonlyifeachσiisstrictlyincreasingaround0∈IR.Now,weexplainhowtheconvexfunctionr,representingthecontrolcostinthelinear-convexregulator(5),isrelatedtoσ.Givenaconvexfunctionswith∇s=σands(0)=0,wesetrtobetheconvexfunctionconjugatetosinthesenseofconvexanalysis:

󰀂󰀃

r(w)=supwTu−s(u).(7)

u∈IRk

Thisfunctionisalwaysconvexandlowersemicontinuous.

Itneednotbefiniteeverywhere–forsomew,wemayhaver(w)=+∞.Also,rneednotbedifferentiable–it’ssubdifferential∂ristheset-valuedinverseof∇s(whichequalsσ,andneednotbeinvertibleintheclassicalsense).Inmanycasesofpracticalinterest,rcanbefounddi-󰀇k

rectly.First,notethatifs(u)=i=1si(ui)asinExample

󰀇k

2.2,bydefinition(7)wehaver(w)=i=1ri(wi),whereriistheconvexconjugateofsi.Thatis,rcanbefoundcomponentwise.Wenowgivesomeexamples.

Example2.3:(standardsaturation).Considerthestan-dardsaturationσ,shownbelowongraph(b).Thefunctionsgivenby(6),andshownbelowongraph(a),canbeusedtocalculaterdirectlyfromthedefinition(7).Alternateapproachistolookattheset-valuedinverseofσ,equalto∂r,whichisshownongraph(c).Then,itremainsto“integrate”∂rtoobtainr,shownongraph(d).

(a)(b)definesadifferentiableconvexfunctions:IR→IR,with

󰀈

s(0)=0,s≥0,and,ofcourse,s=σ.Otheroftenassumedpropertiesofσreflectinthoseofsasfollows:-Ifσ(u)=0onlyforu=0,sispositivedefinite.

u)

-Ifliminfu→0σ(u>0–equivalently,ifforsome󰀞>0,δ>0,wehaveuσ(u)≥δu2for|u|<󰀞–thens(u)

(c)(d)5518

Thesubdifferential∂risgivenbyr(w)=∅ifw<−1,r(−1)=(−∞,1],r(w)=wfor−11.Theexplicitformulaforris:

󰀈12

wforw∈[−1,1],

r(w)=2+∞forw∈[−1,1].

√

Example2.4:Considerσ(u)=u/u2+1,whichisa√

derivativeofs(u)=u2+1−1.Theconjugatercanbefoundthrough(7).Alternatively,σ−1(w)=r󰀈(w)=√wforw∈(−1,1),whileforw∈(−1,1),σ−1(w)=1−w2r󰀈(w)=∅.Then,r(w)canbefound,forany√w∈[−1,1],

󰀈

byintegratingr.Thisleadstor(w)=1−1−w2on[−1,1],whiler(w)=+∞forw∈[−1,1].Graph(a)showss,(b)showsσ,(c)displaysσ−1,andrison(d).

(a)(b)III.THEVALUEFUNCTIONFORLCR

Thevaluefunctionofthelinear-convexregulator,

⎧󰀁∞⎫

1T⎪⎪⎪x(t)Qx(t)+r(w(t))dt⎪⎨⎬20

(8)V(x0)=min

s.t.x˙(t)=Ax(t)+Bw(t),⎪⎪⎪⎪⎩⎭

x(0)=x0,withtheminimizationcarriedoutoveralllocallyintegrablecontrolsu:[0,+∞),isobviouslypositivedefinite.Itmayoccurthatforsomex0∈IRn,V(x0)=+∞;thisisthecasewhennocontrolmakestheintegralin(8)finite.

AkeypropertyofVisthatitisaconvexfunctiononnIR.Thisisaconsequenceofageneralprinciplethatvaluefunctionsforconvexoptimizationproblemsareconvex,seeRockafellar[13].Itcanalsobeverifieddirectly.

Asaconsequenceofconvexity,thelevelsetsofV,being{x∈IRn|V(x)≤r},areconvexandbounded,foreachr∈IR.Thiscaninturnbeusedtoshowthatanyprocess(¯x(·),u¯(·))forwhichtheintegralin(8)isfinitesatisfiesx¯(t)→0ast→∞.Finally,theset

domV={x∈IRn|V(x)<+∞}

isopen.Thiscanbearguedfromcontinuityofrat0andcontrollabilityof(A,B).

Theorem3.1:(differentiabilityofV).Thevaluefunc-tionVisdifferentiableateverypointofdomVand󰀕∇V(xi)󰀕→+∞foranysequenceofpointsxi∈domVconvergingtoapointnotindomV.Thegradient∇ViscontinuousondomV.ThefunctionVisstrictlyconvex.Theproof,whichwedonotincludehere,reliesonthedescriptionofVasaconvexconjugateofavaluefunctionofacertaindualoptimalcontrolproblem.SuchdescriptionispossiblethroughgeneraldualityresultsinGoebel[5](foraconferenceversion,see[8]).Wenotethatsomeresultsonsmoothnessofvaluefunctionsinsimilarsettingsexist,forexampleBenvenisteandScheinkman[2],andGotaandMontrucchio[9],Rockafellar[12],andBarbu[1],butdonotapplyhere.Forapplicationsofgeneraldualityresultstoregularityofvaluefunctionsforfinitehorizonproblems,seeGoebel[7].

Corollary3.2:(optimalfeedbackforLCR).Themap-RkdefinedbypingFLCR:domV→I

FLCR(x)=∇s(−B∗∇V(x))

istheoptimalfeedbackforLCR.Thatis,foranyx0∈

domV,theprocess(¯x(·),w¯(·))withx¯(·)beingasolutionto

˙(t)=Ax¯(t)+Bw¯(t)andw¯(t)=FLCR(¯x(t)),x¯(0)=x0,x

isoptimalforLCR(x0).

Weoutlinethestandardargument.ThevaluefunctionVsatisfiestheHamilton-Jacobiequation

H(x,−∇V(x))=0forallx∈domV,

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T∗

whereH(x,p)=pTAx−12xQx+s(Bp).Fromthedefinitionofrintermsofsin(7),onecanseethat

(c)(d)Thesetdomr={w∈IRn|r(w)<+∞}neednotequalIRn.Infactr(w)=+∞wheneverw∈rgeσ(theclosureoftherangeofσ).Infinitevaluesofrintroduceacontrolconstrainttothelinear-convexregulator–feasiblecontrolsmustsatisfyw(t)∈domr.Forthestandardsaturation,asexpected,thismeansw(t)∈domr=rgeσ=[−1,1].Ingeneralrgeσ⊂domr.Fordetails,seethebeginningofSection24inRockafellar[11].

Finally,westressthatseveral”flatsegments”inσdonotleadtodiscontinuityinr(thelatterisaconvexfunction,andassuch,continuousontheinteriorofthesetwhereitisfinite).Suchsegmentsdoleadto”corners”inr,thatis,pointswhererisnotdifferentiable.Anexampleissketchedbelow,withσandrgivenon(a),(b),respectively.

(a)(b)Tosummarizethissection,westatethefollowing.Lemma2.5:(saturationandconvexfunctions).GivenasaturationfunctionσasinAssumption(A2),thereexistconvexfunctionss:IRk→[0,+∞)andr:IRk→[0,+∞]relatedtoeachotherby(7)andsuchthat:

(i)sisdifferentiable,∇s=σ,s(0)=0,andsisstrictly

convexonsomeneighborhoodof0;

(ii)rispositivedefiniteandonsomeneighborhoodof0,

ithasfinitevalues.

r(∇s(u))=∇s(u)Tu−s(u).

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This,andtheHamilton-Jacobiequation,showthatd1V(¯x(t))=−x¯(t)TQx¯(t)−r(w¯(t)),(10)dt2

whichimpliesboththatx¯(t)→0ast→0andthatx(·)isoptimalforLCR(x0).Thelatterfollowsfromintegrating(10)on[0,+∞)andcomparingtheresultwiththedefinitionofV(x0).Additionally,thisshowsthattheoptimalcontrolw¯(·)iscontinuousandw¯(t)→0ast→∞.

IV.PROOFOFTHEMAINRESULT

Asmaybenowexpected,justificationofTheorem1.1hingesupontranslatingtheoptimalfeedbackforLCRtoastabilizingfeedbackforthesaturatedsystem.First,wenotethatthesetwherethevaluefunctionVisfinite,domV,isequaltoX0.This,andtheresultsofSectionIII,letusproveTheorem1.1.

Proof:Giventhesystem(2)andamatrixQasassumed,letVbethevaluefunction(8)withtheconvexfunctionrgivenby(7)andssuchthats(0)=0,∇s=σ.Corollary3.2andthediscussionfollowingitshowthatforanypointx0∈domV=X0,anysolutionx(·)to

x˙(t)=Ax(t)+BFLCR(x(t))

withx(0)=x0satisfies(4).Asbyconstruction∇s=σ,

RkdefinedbythemappingF:X0→I

F(x)=−B∗∇V(x)

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dynamicsprovideddirectlybythesaturatedsystem,

doesnotleadtoaconvexproblemandisunlikelytoyieldaregularfeedbackorevenaregularvaluefunction(whichneedsnotbeconvexinsuchacase).Toconclude,wepointoutthatthecomponentwisestruc-tureofσasinassumption(A2)isnotnecessaryforourmainresult.Infact,theconclusionsofTheorem1.1holdforanyσsuchthatfunctionss,rasdescribedinLemma2.5exist.Anexampleofsuchσ(whichneednothavethecomponentwisestructure)istheprojectionPContoanonempty,closed,andconvexsetC.WhenCistheunitballinIRk,PCisanidentityforpointsinC,andaradialprojectionontotheunitsphereforpointsoutsideit(thatis,PC(u)=u/󰀕u󰀕).Thens(u)=12

2󰀕u󰀕for󰀕u󰀕≤1,󰀕u󰀕−1/2for󰀕u󰀕>1.Notethatthississtrictlyconvexaround0,infactthispropertyispresentwhenever0isintheinteriorofC.Finally,

󰀈1󰀕w󰀕2forw∈C,2r(w)=

+∞forw∈C.

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satisfiestheconclusionsofTheorem1.1.ContinuitywasestablishedinTheorem3.1.

V.COMMENTSANDEXTENSIONS

WenowmakeseveralcommentsregardingTheorem1.1,andtheconstructionsleadinguptoit.

(i)ThestabilizingfeedbackFforthesaturatedsystem

isnotthesameastheoptimalfeedbackforLCR.However,byconstruction,trajectoriesofthesaturatedsystemwithu(t)=F(x(t))agreewithoptimaltrajectoriesforthelinear-convexregulator.

(ii)TheoptimalfeedbackFLCRforthelinear-convex

regulatorisrelatedtothestabilizingfeedbackFbyFLCR(x)=σ(F(x)),andwhenσisinvertible,F(x)=σ−1(FLCR(x)).Whenσisnotinvertible,therelationshipF(x)=σ−1(FLCR(x))isnotvalidevenintheset-valuedsense.

(iii)TheconstructionofFdoesnotrelyonconsidering

σ−1,notevenonasubsetofrgeσonwhichσisinvertible(thiswas,forexample,theapproachof[15]).Furthermore,wedonotrequestthatσbeLipschitz,differentiableat0,orbounded.

(iv)LCRisaconvexoptimizationproblem.Fromthe

numericalcomputationviewpoint,suchproblemshavemanyadvantagesovertheirnonconvexcounterparts,seethebookbyBoydandVandenberghe[3].Aseeminglymoreobviouschoiceofanauxiliarycontrolproblem,withaconvexorevenquadraticcostandthe

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