earlysolved

TheH∞controlproblemissolvedbyPierreApkarianandDominikusNollTheH∞controlproblemwasposedbyG.
Zamesin1981[1],andsolvedbyP.
ApkarianandD.
Nollin2006[2].
InthistreatisewepresenttherationalofH∞control,giveashorthistory,andrecallthemilestonesreachedbeforeour2006solution.
WealsodiscusstherecentMatlabfunctionhinfstruct,basedonworkbyApkarian,NollandGahinet(TheMathworks)andavailableintheRobustControlToolboxsinceR2010b.
ONERA,ControlSystemDepartment,Toulouse,FranceUniversitePaulSabatier,InstitutdeMathematiques,Toulouse,France1Contents1TheH∞controlproblem31.
1Somehistory31.
2Formalstatementoftheproblem31.
3Therationale41.
4Controllerstructures61.
5Threebasicexampleswithstructure72ThesolutionoftheH∞-controlproblem92.
1Nonsmoothoptimization92.
2Stabilization102.
3Localversusglobaloptimization113TheH2/H∞-problemisalsosolved134Howtousehinfstruct154.
1Controllerinstate-space154.
2TheH∞/H∞-controlproblem154.
3NonstandarduseofH∞/H∞-synthesis164.
4Controllerastransferfunction164.
5Flightcontrol1184.
6Flightcontrol2194.
7Systemreductionviahinfstruct214.
8ControlofnonlinearsystemswithstructuredH∞-synthesis2421TheH∞controlproblemTheH∞-problemwasframedbyG.
ZamesintwoplenarytalksattheIEEECDCin1976andtheAllertonConferencein1979,andposedformallyinhis1981paper[1].
However,theoriginsoftheH∞-problemaremucholderanddatebacktothe1960s,whenZamesdiscoveredthesmallgaintheorem[3].
Aftermorethan30yearstheH∞-problemwassolvedbyP.
ApkarianandD.
Nollin2006[2].
InthissectionweintroducetheH∞controlproblemformally,discussitsrationale,andpresentthecontextleadingtoour2006solution.
1.
1SomehistoryIntheirseminal1989paper[4]Doyle,Glover,KhargonekarandFrancisshowthattheH∞problemrequiresthesolutionoftwoalgebraicRiccatiequations(AREs).
Doyle[5]discusseshowthismilestoneisreachedandmentionsanearlier1984solution.
In1994P.
GahinetandP.
Apkariangiveasolution[6]oftheH∞problembyreducingittoalinearmatrixinequality(LMI),the1995solution.
HowcanaproblembesolvedseveraltimesWhatdowemeanwhenwesaythatwesolvedtheproblemin2006[2],whentherearethe1984,1989,and1995solutionsalready1.
2FormalstatementoftheproblemTheH∞controlproblemcanbestatedasfollows.
GivenarealrationaltransfermatrixP(s),calledtheplant,andaspaceKofrealrationaltransfermatricesK(s),calledthecontrollerspace,characterizeandcomputeanoptimalsolutionK∈KtothefollowingoptimizationprogramminimizeTw→z(P,K)∞subjecttoKstabilizesPinternallyK∈K(1)HeretheobjectivefunctionistheH∞-normoftheclosed-loopperformancechannelTw→z(P,K),see(2).
AsweshallseethechoiceofthecontrollerspaceKin(1)isthekeyforaproperunderstandingoftheproblem.
(2)-w-zPK-uyLetusrecallthenotionsusedtoformulate(1).
TheplantP(s)hasastate-spacerepresentationoftheformP:˙x=Ax+B1w+B2uz=C1x+D11w+D12uy=C2x+D21w+D22uP(s):AB1B2C1D11D12C2D21D22(3)3wherex∈Rnpisthestate,u∈Rnuthecontrol,y∈Rnythemeasuredoutput,w∈Rnwtheexogenousinput,andz∈Rnztheregulatedoutput.
Similarly,K(s)hasstate-spacerepresentationK:˙xK=AKxK+BKyu=CKxK+DKyK(s):AKBKCKDK(4)withxK∈RkthestateofK.
AssoonasD22=0,theclosed-looptransferchannelTw→z(P,K)in(1)hasthestate-spacerepresentationTw→z(P,K):A(K)B(K)C(K)D(K)(5)whereA(K)=A+B2DKC2B2CKBKC2AK,B(K)=B1+B2DKD12BKD21,C(K)=etc.
(6)andwherethestatedimensionisnp+k.
Finally,forastablerealrationaltransferfunctionT(s),theH∞-normin(1)isdenedasT∞=maxω∈Rσ(T(jω)),(7)whereσ(M)isthemaximumsingularvalueofacomplexmatrixM.
Withthesenotationswecannowgivetherstexplanation.
The1984,1989and1995solutionsoftheH∞problem(1)areallobtainedwithinthespaceKfulloffull-ordercontrollersKfull={K:Khasform(4)withsize(AK)=size(A)}.
ObservethatinKfullallentriesinAK,BK,CK,DKarefreevariables.
AltogetherthereareN:=n2p+np(ny+nu)+nynudegreesoffreedomandwehaveKfull=RN.
Inparticular,Kfullisthelargestcontrollerspacewecouldusein(1).
FindingasolutionwithinKfullisthereforeeasiest.
InparticularwithKfullascontrollerspace(1)isconvex,asshownin[6].
WhensmallerandmorepracticalcontrollerspacesKarechosen,problem(1)ismuchhardertosolve.
Our2006solutionaddressesthesedicultcases.
SolutionsoftheH∞-controlprobleminthe1980sand1990srefertothefull-ordercase,whichisessentiallyconvex.
1.
3TherationaleAfterclosingtheloopinthefeedbackscheme(2)wemayconsidertheclosed-loopsystemasalinearoperatorTw→z(P,K)mappinginputwtooutputz.
IfKstabilizesPinternally,thatis,4ifA(K)in(6)isstable,thenTw→z(P,K)mapsw∈L2intoz∈L2.
TheH∞-norm(7)isthennothingelsebuttheL2-L2-operatornorm,thatis,T∞=supw=0Tw2w2=supw=0z2w2.
Inotherwords,foraclosed-loopchannelw→zthenormγ=Tw→z(P,K)∞isthefactorbywhichtheenergyoftheinputsignalisampliedintheoutput.
Inputwwithenergyw22willproduceoutputzwithenergyz22nogreaterthanγ2·w22,aslongascontrollerKisused.
Theoptimizationprogram(1)triestondthecontrollerK∈Kforwhichthisamplicationfactorγissmallest.
Inclosed-loopwithcontrollerKtheinputwwithenergyw22createsoutputzwithenergyz22≤γ2w22,whereγ=Tw→z(P,K)∞.
Thesamerelationholdsforpowersignalsw→z,i.
e.
,powerisampliedbynomorethanγ2.
Thiscanobviouslybeveryuseful.
Allwehavetodoisndcommunicationchannelsw→z,wheresmallnessofanswerztoinputwtellsussomethingusefulaboutthesystem.
Wenowgivethetypicalcontextofloopshaping,wherethisideaisused.
(8)-rc-euy+K-cd-Gcns6-Wu-u-We-e-Wy-yThestandardcontrolscheme(8)featurestheopen-loopsystemG,thecontrollerK,themea-suredoutputy,thecontrolsignalu,thetrackingerrore.
Redsignalsareinputs,ns=sensornoise,d=disturbanceorprocessnoise,andr=referencesignalfory,sometimescalledacommand.
Thebluesignalsarespecicallychosenoutputs,e=Wee,u=Wuu,y=Wyy.
Thisisaspecialcaseof(2),wherew=(r,d,ns)istheinput,z=(e,u,y),andwhereplantPregroupsGandtheltersWe,Wu,Wy.
Theltersmaybedynamic,whichaddsnewstatesintotheplantP.
WhatareusefultransferfunctionsfromredtoblueForinstancethetransferfromreferencertotrackingerroreTr→e(K)=(I+GK)15isatypicalperformancechannel,becauseitdescribeshowfastthesystemfollowsthereferencer.
Asonetypicallywantstotrackonlyinthelowfrequencyrange,Weisalow-passlter.
NowsmallnessofthenormTr→e(K)∞=We(I+GK)1∞meansthatthelowfrequencycomponenteofthetrackingerroreissmall,soyfollowsthereferenceinputrinlowfrequency.
Nextconsideratypicalrobustnesschannel.
Forinstance,theinuenceofsensornoisensonthecontrolsignalu.
Noiseistypicallyofhighfrequency,butthatshouldnotleadtohighfrequencycomponentsinu,asthiscouldleade.
g.
toactuatorfatigue.
ThereforeWuistypicallyahigh-passlteranduarehighfrequencycomponentsofu.
WendTns→u(K)=Wu(I+KG)1KandTns→u(K)∞putsacostonhighfrequencycomponentsinu.
Ifprogram(1)issuccessful,itwillfurnishanoptimalK∈Kwhichmakesthiscostassmallaspossible,therebybuildingrobustnesstosensornoiseintothesystem.
Toconclude,wecanseethatdependingonthespecicapplicationtherewillbeseveralperformanceandrobustnesschannels.
As(2)requiresxingasingleconnectionw→z,wewillhavetodecideonsomespecicweighingbetweenthose.
Settinguptheperformancechannelw→zin(1)couldbeinterpretedasputtingacostonundesirablebehavioroftheclosed-loopsystem.
1.
4ControllerstructuresThereasonwhytheH∞theoryofthe1980sfailedtogripinpracticeisquicklyexplained.
ControllerscomputedviaalgebraicRiccatiequationsarefullorder,orunstructured.
However,forvariousreasons,practitionersprefersimplecontrollerslikePIDs,orcontrolarchitecturescombiningPIDswithlters,andsuchcontrollersarestructured.
ThediscrepancybetweenH∞theoryandcontrolengineeringpracticeishighlightede.
g.
byPIDcontrol.
TothisdayPIDcontrollersaretunedinsteadofoptimized,becausesoftwareforH∞-PIDcontrolwasnotavailable.
Duringthe1990sandearly2000sanewapproachtocontrollerdesignbasedonlinearmatrixinequalities(LMIs)wasdeveloped.
Unfortunately,LMIshaveessentiallythesameshortcomingsasAREs.
H∞controllerscomputedviaLMIsarestillunstructured.
Thesituationonlystartedtoimprovewheninthelate1990stheauthorspioneeredtheinvestigationoffeedbackcontrollersynthesisviabilinearmatrixinequalities(BMIs).
WhiletheLMIeuphoriawasstillinfullprogress,wehadrecognizedthatwhatwasneededwerealgorithmswhichallowedtosynthesizestructuredcontrollers.
Hereistheformaldenitionofstructure(cf.
[2]).
6Denition1.
AcontrollerKoftheform(4)iscalledstructuredifthestate-spacematricesAK,BK,CK,DKdependsmoothlyonadesignparametervectorθvaryinginsomeparameterspaceRn,orinaconstrainedsubsetofRn.
Inotherwords,acontrollerstructureK(·),orK(θ),consistsoffoursmoothmappingsAK(·):Rn→Rk*k,BK(·):Rn→Rk*ny,CK(·):Rn→Rnu*k,andDK(·):Rn→Rnu*ny.
ItisconvenienttoindicatethepresenceofstructureinKbythenotationK(θ),whereθdenotesthefreeparameters.
IntheMatlabfunctionhinfstructonereferstoθasthevectoroftunableparameters.
1.
5ThreebasicexampleswithstructureThestructureconceptisbestexplainedbyexamples.
ThetransferfunctionofarealizablePIDcontrollerisoftheformK(s)=kp+kis+kds1+Tfs=dK+ris+rds+τ,(9)wheredK=kp+kd/Tf,τ=1/Tf,ri=ki,rd=kdT2f.
RealizablePIDsmaythereforeberepresentedinstate-spaceformKpid(θ):00ri0τrd11dK(10)withθ=(ri,rd,dK,τ)∈R4.
Aswecansee,AK(θ)=000τ,BK(θ)=rird,C(K)=[11],DK=dK.
IfweusethePIDstructure(10)withintheH∞framework(1),wecomputeanH∞PIDcon-troller,thatis,aPIDcontrollerwhichminimizestheclosed-loopH∞-normamongallinternallystabilizingPIDcontrollers:Tw→z(P,Kpid)∞≤Tw→z(P,Kpid)∞.
ThecontrollerspaceforthisstructureisKpid=Kpid(θ):asin(10),θ=(ri,rd,dK,τ)∈R4.
ThefactthatPIDisastructureinthesenseofDef.
1meansthatPIDsmaynowbeoptimizedinsteadoftuned.
7Asecondclassicalcontrollerstructure,relatedtothefundamentalworkofKalmaninthe1960s,istheobserver-basedcontroller,whichinstate-spacehastheform:Kobs(θ):A+B2Kc+KfC2KfKc0.
(11)HerethevectoroftunableparametersθregroupstheelementsoftheKalmangainmatrixKfandthestate-feedbackcontrolmatrixKc.
Thatisθ=(vec(Kf),vec(Kc)).
SincetheplantPhasnpstates,nyoutputsandnuinputs,θisofdimensionnp(ny+nu),i.
e.
,n=np(ny+nu)solvedgivesagloballyoptimalsolutionof(1).
(Strictlyspeaking,wemightnotbeabletowritedown(17)directly,butonlytoreachiteventuallybyclimbingupinthehierarchyuntilwegettoi(B).
Thiswouldofcoursespoilthewholeidea.
Butletusassume,asisoftenclaimedintheSOScommunity,thatwecanwritedown(17)explicitly!
).
Doesn'tthissoundniceAfterallwehavebeentoldsincetheearly1990sthatLMIscanbesolvedecientlyinquasi-polynomialtime.
Soallwehavetodoissolve(17)quicklyandgettheglobalminimumof(15),respectively,of(1).
Ofcoursethisisallrubbish.
Weknowthatsolvingproblem(1)globallyisNP-complete.
TheSOSalgorithmisevenprovablyexponential.
ThesizeofLi(B)0growsthereforeexponentiallyinthedatasize(B).
Infact,theseproblemsexplodeextremelyfast.
WewillneedexponentialspaceeventowritedownLi(B)0.
Forsizableplantswemightnotevenbeabletostoretheprobleminthecomputer,letalonesolveit.
ThefactthatLMIsaresolvedinpolynomialtimeispointless,becausewearespeakingaboutaproblemofsizepolynomial(exponential).
ButcouldnotsomethingsimilarbesaidabouteveryglobalmethodArewetooseverewhenwecallSOSaredherringIndeed,theproblembeingNP-complete,everyglobalmethodisboundtobeexponential.
ThepointisthatSOSisaparticularlyungainlyglobalmethod,becauseitcommitstwoerrors,whichotherglobalmethodsmayavoid.
Thersterroristhatittransforms(1)toaBMI.
ThisaddsalargenumberofadditionalvariablesX,Y,whichcanbeavoidede.
g.
byournonsmoothapproach.
WehavedemonstratedabundantlythatthepresenceofLyapunovvariablesleadstoseriousill-conditioning.
Towit:Thepoweroscillationdampingcontrolproblemwhichwesolvedin[15]usingnonsmoothoptimizationhasasystemwith90states,3performanceconnections,1input,1output,andacontrollerofreducedorder8.
Thereforedim(θ)=81.
TransformedtoaBMIitrequiresadditional3·90·912=12285Lyapunovvariables.
FortheSOSapproachthisisjustthebottomlinei=1,wheretheLMIhierarchystarts.
TheLMILi(B)0willbeofsizeexponential(12366).
12TheseconderrorintheSOSapproachisthatitonlygoesforglobalminima.
Thatis,itwillnotndlocalminimaof(1)onitswaytowardtheglobal.
Thisisinfelicitous,becauselocalminimaareveryhelpful.
Theymayallowtoimproveboundsinbranch-and-boundmethods,andtheygivegoodpracticalsolutionsasarule.
ThefactthatSOSdoesnotusethisinformation(e.
g.
toinferwhereitisinthehierarchyLi0)isbyitselfalreadysuspicious.
3TheH2/H∞-problemisalsosolvedItbecamealreadyapparentinthe1-DOFscheme(8)thattheL2-L2,respectivelypower-to-power,operatornormisnottheonlypossiblemeasureofsmallnessinachannelw→z.
ConsiderforinstancethetransferTns→ufromsensornoisenstothethehighfrequencypartu=Wuuofthecontrollawu.
Ifwemodelnsaswhitenoise,thenitmakessensetogaugens→ubytheoperatornormfromwhitenoiseattheinputtowardpowerattheoutput.
ThisistheH2-norm.
ForastabletransferoperatorG(s)theH2-normisgivenasG2=∞0TrG(jω)GH(jω)dω1/2,whichmakesitanEuclidiannorminthespaceofstabletransfermatrices.
UnliketheH∞-norm,theH2-normisnotanoperatornorminthetraditionalsense.
Itbecomesoneassoonasstochasticsignalsareconsidered.
wzoperatornormTw→zenergyenergyH∞powerpowerH∞whitenoisepowerH2SobolevW∞,∞L∞worstcaseresponsenormL∞L∞peakgainpastexcitationsystemringHankelInscheme(8)wemightdecidetousetwodierentnorms.
Wemightassessthetrackingerrorr→eintheH∞-norm,andtheinuenceofsensornoiseonthecontrolns→ubytheH2-norm.
ThenweobtainamixedH∞/H2-controlproblemminimizeTr→e(P,K)∞subjecttoTns→u(P,K)2≤γ2KstabilizesPinternallyK=K(θ)hasaxedstructure(18)whereγ2issomethresholdlimitingtheenergyofuinresponsetowhitenoiseintheinputns.
Wemayintroducethefollowingmoreabstractsetting.
ConsideraplantinstatespaceformP:˙xz∞z2y=AB∞B2BC∞D∞0D∞uC200D2uCDy∞Dy20xw∞w2u(19)13wherex∈Rnxisthestate,u∈Rnuthecontrol,y∈Rnytheoutput,andwherew∞→z∞istheH∞,w2→z2theH2performancechannel.
ThisbringustothefollowingstructuredmixedH2/H∞-synthesisproblem.
minimizeTw2→z2(P,K)2subjecttoTw∞→z∞(P,K)∞≤γ∞KstabilizesPinternallyK∈K(20)whereKisastructuredcontrollerspaceasbefore,andγ∞isasuitablethreshold,nowfortheH∞-normintheconstraint.
NoticethattheH2/H∞-andH∞/H2-problemsareequivalentundersuitablechoicesofγ2andγ∞.
ThemixedH2/H∞-synthesisproblemwithstructuredcon-trollersK(θ)isanaturalextensionofH∞-control.
Thisprob-lemhasalsoalonghistory.
ItwasposedforthersttimebyHaddadandBernstein[16]andbyDoyle,Zhou,Boden-heimer[17]in1989.
Wesolvedthisproblemin2008[18].
Onemayimmediatelythinkaboutothermulti-objectiveextensionsof(1).
Forinstance,combiningtheH∞-normwithtime-domainconstraintslikeinIFT,orH∞/H∞-control.
Fortherstthemewereferthereadertooursolutionpresentedin[19,20],whileH∞/H∞-controlwillbeaddressedinthenextsection.
144HowtousehinfstructThenewMatlabfunctionhinfstructbasedonourseminalpaper[2]allowsalargevarietyofpracticalapplications.
ThissectionpresentsseveralexampleswhichwillmotivatetheinterestedusertointegratestructuredH∞-synthesisintohisorhermenuofcontroldesignmethods.
Formoreinformationonhowtousehinfstructseealsohttp://pierre.
apkarian.
free.
fr/NEWALGORITHMS.
htmlhttp://www.
math.
univ-toulouse.
fr/noll/http://www.
mathworks.
fr/help/toolbox/robust/ref/hinfstruct.
html4.
1Controllerinstate-spaceThemostgeneralformtorepresentacontrollerK(θ)isparametrizedinstate-space.
ThisisjustaccordingtoDenition1.
TheMatlabR2011bdocumentationoftheRobustControlToolboxgivesthesimpleexampleK(θ)=1a+b3.
00ab1.
50.
300,˙x1=x1+(a+b)x23.
0y˙x2=abx2+1.
5yu=0.
3x1(21)whereθ=(a,b)∈R2iswhatiscalledthevectoroftunableparameters,andwhatintheoptimizationprogram(1)aretheunknownvariables.
Thecommandstodenethisstructurearea=realp('a',-1);%aisaparameterinitializedas-1b=realp('b',3);A=[1a+b;0a*b];B=[-3.
0;1.
5];C=[0.
30];D=0;Ksys=ss(A,B,C,D);4.
2TheH∞/H∞-controlproblemItisimportantthatthestate-spacestructureincludesthepossibilityofrepetitionsoftheθi.
Forinstance,in(21)bothaandbarerepeated.
ThisallowsustosolveH∞programswithseveralchannels.
Forinstance,themixedH∞/H∞-problemcanbeseenasaspecialcaseof(1).
SupposewehavetwoplantsP1andP2withperformancechannelswi→zi,i=1,2.
AssumethattheoutputsyiandinputsuiintoPihavethesamedimension,i.
e.
,dim(y1)=dom(y2)anddim(u1)=dim(u2).
Thenwemayconnectthesamecontrollerui=K(θ)yitobothplantssimultaneously.
Thatis,wemaysolveaprogramoftheformminimizeTw1→z1(P1,K)∞subjecttoTw2→z2(P2,K)∞≤γ2KstabilizesP1andP2K=K(θ)isstructured(22)15Itturnsoutthatwemaytransform(22)favorablyintoaprogramoftheformminimizemax{Tw1→z1(P1,K(θ))∞,βTw2→z2(P2,K(θ))∞}subjecttoK(θ)stabilizesP1andP2(23)whichissometimescalledamulti-diskproblem[8].
Forsuitablechoicesofγ2andβthesetwoprogramsareequivalent.
However,sincethemaximumoftwoH∞-normsisagainanH∞-normofanaugmentedplant,wecansolve(23)directlyvia(1)withanewspecicstructure,whichconsistinrepeatingK(θ).
Schematically(24)P1P2ppppppppppppppppppppppppppppppppppppppppppppppppppppppppK(θ)K(θ)pppppppppppppppppppppppppppppppppppppppppppppppppppppppp-z1-w1y1β--w2βz2zwy2-u2-u1andtheonlyconnectionbetweenthetwodiagonalpartsisthefactthatthediagonalblockofKisrepeated.
Theobjectiveof(23)isthenthechannelw=(w1,w2)→z=(z1,βz2)oftheaugmentedplant.
Wemaynowhavetoupdateβinordertosolve(24)foraspecicγ2.
4.
3NonstandarduseofH∞/H∞-synthesisThestandardwaytousemultipleH∞criteriaiscertainlyinH∞-loopshaping,andthedoc-umentationofhinfstructmakesthisastrongpoint.
However,therearesomelessobviousideasinwhichonecanuseaprogramoftheform(22).
Twoheuristicsforparametricrobustcontrol,whichweproposedin[21]and[22],canindeedbesolvedviahinfstruct.
4.
4ControllerastransferfunctionInmanysituationsitmaybepreferabletoavoidstate-spacerepresentationsofKandusethetransferfunctiondirectly.
Considerthefollowingsituation16(25)-reuyF(s)g+--L(s)-G(s)-H(s)6SupposeG(s)andH(s)aregivenblocks,G(s)=1(s+1)2andH(s)=5s+4.
TheunknowncontrollerK(s)regroupstheblockL(s)andtheprelterF(s).
LetussayL(s)=kp+kis+kds1+Tfs,F(s)=as+a,whereθ=(a,kp,ki,kd,Tf)isthevectoroftunableparameters.
ThenwehavethefollowingcommandstosetupthecontrollerG=tf(1,[121]);H=tf(5,[14]);a=realp('a',10);F=tf(a,[1a]);kp=realp('kp',0);ki=realp('ki',0);kd=realp('kd',0);Tf=realp('Tf',1);L=tf([kd+Tfki*Tf+1ki],[Tf10]);T=feedback(G*L,H)*F;OrwemayrecognizeKtobeaPIDcontroller,whichallowsustouseapredenedstructureundertheformL=ltiblock.
pid('L','PID');ThecommandT.
Blockswillshowthedierence.
ThecontrollerKconsistingoftheblocksFandLcouldalsobewritteninthestandardform(2)asfollows.
Introducer=Fr,y=Hy,e=ry,u=Le,y=Guin(25),thenF:˙ξ3=aξ3+arr=ξ3F(s):aa10andL:˙ξ1=rie˙ξ2=τξ2+rdeu=ξ1+ξ2+deL(s):00ri0τrd11d17withtherelations(9),(10).
SotogetherthecontrollerKwiththreestatesξ1,ξ2,ξ3,inputsyandr,andoutputu,couldberepresentedinstate-spaceasK(θ):00riri00τrdrd000a0a11dd0whereu(s)=K(θ,s)(y(s),r(s))T.
Herewehaveswitchedtoθ=(a,τ,ri,rd,d),whichisequivalenttoθ=(a,kp,ki,kd,Tf)via(9),(10).
Noticeagainthatinbothrepresentationssomeofthecontrollergainsarerepeated.
ThecorrespondingplantPisfoundasfollows.
WehaveH:˙x3=4x3+5yy=x3andG:˙x1=x2˙x2=x12x2+uy=x1ThereforeP:˙x1=x2˙x2=x12x2+u˙x3=5x14x3y=x3towhichwewouldaddaperformancechannelw→z.
4.
5Flightcontrol1Anexamplediscussedin[23]andasrctairframe1inthedocumentationofhinfstructcon-cernsightcontrolofanaircraft.
Thecontrolarchitectureisgivenbythefollowingscheme-e-eazrefPI-qrefe-1s-QQ-ed--en66systemazqqGainudqwzplant-azref-1md-1LSn-ePIqGainqrefq-delta-azLS-eHavingextractedtheplantfromthesimulinkmodel,onemarkstheinputsandoutputs18io=getlinio('rctairframe1');TunedBlocks={'rctairframe1/azControl';'rctairframe1/qGain'};P=linlft('rctairframe1',io,TundedBlocks);P.
InputName={'azref','d','n','qref','delta'};P.
OutputName={'e','az','ePI','qInt'};Pdes=blkdiag(LS,eye(3))*P*blkdiag(1/LS,1/m,eye(3));PIO=ltiblock.
pid('azControl','pi');qGain0=ltiblock.
gain('qGain',0);Afterdeningoptions,onerunsC=hinfstruct(Pdes,{PIO,qGain0},opt);TheinformationisretrievedbyPI=pid(C{1});qGain=ss(C{2});andtheclosed-loopLFTisobtainedfromCL=lft(P,blkdiag(PI,qGain));Inthisexampleitistrivialtoobtainthecontrollerinstate-space.
Wehave(PI)˙x1=kieqref=x1+kpeforthePI-block.
Then(qGain)˙x2=qGain·dqu=x2However,ifmoreSISOcontrollerblocksarecombined,itmaybepreferableandmorenaturaltoworkwithtransferfunctions.
4.
6Flightcontrol2Amoreinterestingsituationisthefollowingcontrolarchitecturefrom[24],alsointhedomainofightcontrol,whereaPI-block,again,andalterarecombined.
Theoverallcontrolarchitectureisasfollows19Extractingthecontrollergivesthefollowingstructure,wherethetunableparametersareθ=(kp,ki,kv,a,b).
Noticethenoveltyhere,weareconsideringthelterasunknownandthereforeaspartofthecontroller.
Standardprocedureswoulddesignthelterrstandthentuneki,kp,kv.
Whenvariouselementsofthistypearecombined,wespeakaboutacontrolarchitecture.
Nowifwewritedownthestate-spacerepresentationofthisschemeinastraightforwardway,thenwemayendupwithK(θ):˙x1˙x2˙x3dm=01000abaakpakv000.
001ki010000x1x2x3dNzq(26)whichisaccordingtodenition1,butgivesanonlinearparametrization.
IfweaugmenttheplantParticiallyintoaplantP,wemayobtainanequivalentaneparametrizationofthecontroller:K(θ):˙x1˙x2˙x3dme=010000ab0a00000.
0010ki01000000010kpkvx1x2x3edNzq(27)20Thisrequirespassingethroughtheplantasindicatedbythefollowinggure,whichexplainsthemeaningoftheaugmentedplantP.
Noticethatwhatwehighlightedbyredandbluein(27)isadecentralizedcontrollerstructure,theonementionedinsection2.
3.
NoticethatiftheproblemisenteredviatheTFstructure,theuserwillnotnoticethedierencebetween(26)and(27).
Itshouldalsobeclearthateveryrationalparametrizationlike(26)canbeshuedintoananeoneusingthesametrick.
4.
7SystemreductionviahinfstructAnideaalreadyputforwardinourpaper[2]isH∞-systemreduction.
ConsiderastablesystemG=ABCDwithsize(A)=n*n.
SupposenislargeandwewanttocomputeareducedstablesystemGred=AredBredCredDredofsmallerstatedimensionsize(Ared)=knwhichrepresentsGasaccuratelyaspossible.
w--GGred6e+-eWe-z(28)Themodelmatchingerrorise=(GGred)w,andafteraddingasuitablelterWe,wemightwanttohavew→zsmallinasuitablenorm.
TheHankelnormreductionmethodminimizes21We(GGred)HintheHankelnorm·H,theadvantagebeingthatthesolutioncanbeobtainedbylinearalgebra.
AmorenaturalnormwouldbetheH∞-norm,buttheclassicalbalancedreductionmethodgivesonlyupperboundsofWe(GGred)∞.
ButwecansolvetheH∞-normreductionproblemdirectlyasaspecialcaseof(1).
Inthecasez=ewithoutlterwecanpasstothestandardformbyconsideringtheplantP:AB0CDI0I0=AB1B2C1D11D12C2D210(29)thenGredisthecontroller,whichisofxedreduced-order.
Theapproachcanbeputtowayasfollows.
Wehavetestedthiswitha15thorderRolls-RoyceSpeygasturbineenginemodel,decribedin[27,Chapter11.
8,p.
463].
ThedataareavailablefordownloadonI.
Postlethwaites'shomepageasaero0.
mat.
loadaero0G=Geng;A=G.
a;B=G.
b;C=G.
c;D=G.
d;Denetheplantaccordingto(29):[nbout,nbin]=size(D);Aplant=A;Bplant=[Bzeros(n,nbout)];Cplant=[Czeros(nbin,n)];Dplant=[D-eye(nbout)eye(nbin)zeros(nbin,nbout)];Plant=ss(Aplant,Bplant,Cplant,Dplant);Nowdenethestructureofthe"controller",whichinthiscaseisnothingelsebutthereduced-ordersystemGredin(28).
Letthereducedorderbek≤n.
(Intheexamplewehaven=15,k=6.
)ThenRed=ltiblock.
ss('reduced',k,nbout,nbin);Nowwearereadytorunhinfstruct.
Wecouldforinstancedothefollowing.
Increasethemaximumnumberofiterationsto900(defaultis300),andallow6randomrestarts.
Opt=hinfstructOptions('MaxIter',900,'RandomStart',6);[Gred,gam,info]=hinfstruct(Plant,Red,Opt);Theresultisthefollowingoutput.
Final:Peakgain=0.
17,Iterations=743Final:Peakgain=0.
382,Iterations=900Final:Peakgain=0.
808,Iterations=55322Final:Peakgain=0.
768,Iterations=550Final:Peakgain=0.
65,Iterations=422Final:Peakgain=0.
77,Iterations=591Final:Peakgain=0.
236,Iterations=900Final:Peakgain=0.
44,Iterations=900Final:Peakgain=0.
169,Iterations=794Final:Peakgain=0.
535,Iterations=900Final:Peakgain=0.
794,Iterations=538Final:Peakgain=0.
176,Iterations=900Final:Peakgain=0.
638,Iterations=561Final:Peakgain=0.
555,Iterations=900Final:Peakgain=0.
43,Iterations=788Final:Peakgain=0.
486,Iterations=900Final:Peakgain=0.
169,Iterations=845Final:Peakgain=0.
419,Iterations=634Final:Peakgain=0.
49,Iterations=900Final:Peakgain=0.
742,Iterations=900Final:Peakgain=0.
169,Iterations=758Aswecansee,thebesterrorisGGred∞=0.
169,butvariousotherlocalminimaarefound,sowithouttestingseveralinitialguesses(hereatrandom),wecouldnotrelyonasinglerun.
However,systemreductionisasituationwherewecandomuchbetter.
Whynotinitializetheoptimizationusingoneofthestandardreductions,liketheHankelnormreduction,orabalancedtruncationHereishowtodoit.
[Gb,hsig]=balreal(G);[Gh,HankInfo]=hankelmr(Gb,k);Ah=Gh.
a;Bh=Gh.
b;Ch=Gh.
c;Dh=Gh.
d;Sofarwehavethestate-spaceformofthe6thorderHankelreducedmodelGh.
NowweinitializethetunablestructureRedasthisreduced-ordermodelGh.
ThatisdoneviathestructureValue.
Red.
a.
Value=Ah;Red.
b.
Value=Bh;Red.
c.
Value=Ch;Red.
d.
Value=Dh;Opt=hinfstructOptions('MaxIter',900);[Gred,gam,info]=hinfstruct(Plant,Red,Opt);Thistimetheresultismuchmoreecient.
TheoutputisState-spacemodelwith3outputs,3inputs,and6states.
Final:Peakgain=0.
169,Iterations=25Thatmeans,theseeminglyglobalminimumwithGGred∞=0.
169isreachedveryfast,andnorandomrestartsareused.
23Thisisasomewhatsurprisingapplicationofhinfstruct,be-causewehaveanH∞-lteringproblem,notanH∞-controlproblem.
TheplantPisnotcontrollable,anditisstabiliz-ableonlybecauseAisstable.
4.
8ControlofnonlinearsystemswithstructuredH∞-synthesisInthissectionwediscussasomewhatunexpectedapplicationofstructuredH∞-synthesisinthecontrolofnonlinearsystems.
TheclassofsystemswehaveinmindareoftheformP(y):˙x=A(y)x+B1(y)w+B2(y)uz=C1(y)x+D11(y)w+D12(y)uy=C2(y)x+D21(y)w+D22(y)u(30)wherethesystemmatricesdependsmoothlyonthemeasuredoutputy.
ItappearsthereforenaturaltodeviseacontrolleroftheformK(y):˙xK=AK(y)xK+BK(y)yu=CK(y)xK+DK(y)y(31)whichusesthesamemeasurementyavailableinrealtime.
Anaturalidea,goingbackto[25],istoconsiderylikeatime-varyingexternalparameterpandpre-computeK(p)forP(p)foralargesetp∈Πofpossibleparametervalues.
InightcontrolforinstanceΠistheightenvelopep=(h,V)∈R2,indexedbyaltitudehandgroundspeedV,orsometimesbyMachnumberanddynamicpressure.
Wenowproposethefollowingcontrolstrategy.
Inarststepwepre-computetheoptimalH∞controllerK(p)foreveryp∈Πusingprogram(1):minimizeTw→z(P(p),K)∞subjecttoKstabilizesP(p)internallyK∈K(32)ThesolutionK(p)of(32)hasthestructureK.
Intheterminologyof[25]isthebestwaytocontrolthesystemP(p)frozenatp(t)=y(t)instantaneously.
Inotherwords,atinstantt,weapplythecontrollawK(y(t))basedonthereal-timemeasurementy(t).
Ifwecoulddoreal-timestructuredH∞-synthesis,thencon-trollerK(y(t))wouldbecomputedandappliedinstanta-neouslyattimetusing(32)andthemeasurementy(t)avail-ableatinstantt.
Aslongasthisisimpossible,wemaypre-computeK(p)foralargesetofpossibleparameterval-uesp∈Π,andassoonasy(t)becomesavailableattimet,lookK(y(t))upinthetable{K(p):p∈Π},andapplyitinstantaneously.
24Therearetwolimitationstothisidealapproach.
Firstly,theidealtable{K(p):p∈Π},computedbyhinfstruct,maybetoolarge.
Andsecondly,thebehaviorofK(p)asafunctionofpmaybequiteirregular.
Infact,thisiswhathasstoppedthisideainthepast2.
WithstructuredcontrollawsK(θ)thesituationissubstantiallyimproved,becauseonecanusefewerdegreesoffreedominθ.
Whatwehavetestedin[26]isacompromisebetweenoptimalityofK(p)inthesenseofprogram(32),thenecessitytoavoidirregularbehaviorofthecurvesp→K(p),andthestoragerequirementofsuchalaw.
Weusethefollowingdenition.
Acontrollerparametrizationp→K(p)ofthegivenstructureKisadmissibleforthecontrolofP(y)ifthefollowingholds.
K(p)stabilizesP(p)internallyforeveryp∈Π,andTw→z(P(p),K(p))∞≤(1+α)Tw→z(P(p),K(p))∞(33)foreveryp∈Π,whereαissomexedthreshold.
Typically,α=0.
1%.
WenowseekaparametrizationK(p)whichisclosetotheidealH∞-parametrizationK(p)inthesensethat(33)isrespected,butotherwiseiseasytostore(toembed)andshowsasregularabehavioraspossible.
Noticethat(33)allowsK(p)tolagbehindK(p)inperformancebynomorethan100α%.
AcknowledgementTheauthorsacknowledgefundingbyAgenceNationaledeRecherche(ANR)undergrantsGuidageandControvert,byFondationd'EntrepriseEADSundergrantsSolvingChallengingProblemsinControlandTechnicom,andbyFondationdeRecherchepourl'Aeronautiqueetl'EspaceundergrantSurvol.
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