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Massachusetts InstituteofTechnologyDepartmentofElectricalEngineeringandComputerScience6.245: MULTIVARIABLECONTROLSYSTEMS

byA.Megretski

SolvingtheH2optimizationproblem1ThereareseveralwaystoderiveasolutiontotheH2optimizationproblem. Thepathdevelopedbelowreliesonreductionoftheoutputfeedbackdesignproblemtoanabstractoptimalprogramcontrolproblem.3.1 ACharacterizationofClosedLoop ImpulseResponsesThissectionprovidesan insight intothe constraints imposedontheclosed loop systembythecoefficientsoftheouputfeedbackdesignplant

x = Ax+B1w+B2u, (3.1)y = C2x+D21w, (3.2)

stabilizedbyafiniteorderstrictlyproperLTIcontrollerxf = Afxf +Bfy, (3.3)

u = Cfxc. (3.4)c1VersionofFebruary17,2004

A.Megretski,2004

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23.1.1 AnAffineParameterizationLetX=X(t)denotetheimpulseresponsematrixfromwtoxintheclosedloopsystem.Similarly,letU =U(t)betheimpulseresponsematrixfromwtou.Theorem3.1 Assume (A, B2) is stabilizable and (C2, A) is detectable. Then a pair(X(t), U(t)) ofmatrix-valuedfunctions can be achieved as implulse responses in a stableclosed loopsystem ifandonly if

X (t)=AX(t)+B2U(t), X(0)=B1, limX(t)=0, (3.5)t

and there exist a realmatrix-valuedfunction V = V(t), each entryofwhich is a linearcombinationoftermsof theformtkest withk {0, 1, . . . },Re(s)>0,such that

U(t)=(t)B1 +V(t)D21, lim(t)=0, (3.6)twhere

(t)=(t)A +V(t)C2, (0)=0. (3.7)Equality (3.5) represents the constraints on the closed loop responses in the case of

full informationcontrol, whem xand w areavailable formeasurement. Whenoutputfeedbackisconsidered,U(t)isnotarbitrary,butisparameterizedby(3.6),(3.7).

Theorem3.1canbegeneralizedtothecaseoffiniteH2normstabilizationbyinfiniteorderLTIcontrollers, inwhichcasetheconditionsrequiringX(t), (t)and(t)toconvergetozeroast shouldbereplacedbytheconditionoftheirsquare integrabilityover{t} =(0, ).Proof For the purpose of driving the H2 optimal controller, we do not need the sufficiencypartofthetheorem. Itwillbeprovenlater,togetherwithaslightmodificationofTheorem3.1concerningtheso-calledQ-parameterization.

ItissufficienttoprovethetheoremforthecasewhenB1 = I 0 , D21 = 0 I ,

i.e. whensystemequationshavetheformx = Ax +B2u +w1, (3.8)y = C2x +w2 (3.9)

withw= [w1;w2]. Then(3.5)followsfrom(3.8). Onewaytoseethisisbyapplyingtheone-sidedLaplacetransform(denotedbytildes)to(3.8)toget

x=A u + s x +B2 w1.

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3Since

w,

w,

w1 = I 0 w,x=X u=U andw isarbitrary,(3.5)follows.

Toprove(3.6)and(3.7),notethat,by linearityofthecontroller, u(t)0whenevery(t)0. Hence

w2 =C2(sIA)1 w1wouldalwaysimply

I

u=U C2(sIA)1 w1 =0

foranarbitraryw1. Therefore U = VC2(sIA)1 V

forsomeV. ConvertingthistotimedomaintermsyieldsU =[ V]where(t)=(t)A+V(t)C2, (0)=0.

3.1.2 ACharacterizationofStateEstimationErrorsAsaspecialcaseofthegeneralclosedloopresponseparameterization,considertheclosedloopresponsefromwtoKxu,whereK isagivenmatrix.Theorem3.2 A matrixfunction G = G(t) can be achieved as a closed loop impulseresponsefrom w to Kxu ifandonly if thereexistsarealmatrix-valuedfunction V =

stV(t), each entry ofwhich is a linear combination of terms of theform tke with k {0,1,...},Re(s)>0,such that

G(t)=(t)B1 +V(t)D21, lim(t)=0, (3.10)t

where(t)=(t)A+V(t)C2, (0)=K. (3.11)

Proof Fornow,wewillonlyprovenecessity(thisisthepartweneedforH2optimization).AccordingtoTheorem3.1, astabilizedclosed loop impulseresponse from w to KxuequalstheresponseH ofsystem

X = AX+B2(B1 +VD21), X(0)=B1, (3.12) = A+VC2, (0)=0, (3.13)H = KXB1 VD21 (3.14)

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4toaninputV whichmakesX()=0and()=0.

Notethat

the

zero

input

response

of

system

(3.12),

(3.13)

(i.e.

with

V

0)

has

0

andhenceH(t)=H0(t)=KeAtB1. Thisresponsecanbeachieved in(3.10),(3.11)withV 0. On the other hand, according to Theorem 3.1, every zero-state response of thesystem (i.e. when X(0)=B1 is replaced by X(0)=0) must coincidewithazero-stateresponseof(3.10),(3.11).3.2 AbstractH2OptimizationWe will use the term abstract H2 optimization to refer to an auxiliary optimizationproblem: an optimal program control in completelydeterministic settings. Itturns outthatageneralH2feedbackdesignproblemreducestoapairofabstractH2optimizations,and the optimal controller, as well as a parameterization of suboptimal controllers, canbeobtainedeasilyfromsolutionsofthetwoauxiliaryproblems.3.2.1 FormalDefinitionsandRelationtoOutputFeedbackDesignAnabstractH2optimizationproblemisdefinedbyastabilizableLTIsystem

a b

S = .c d

andrequestsfinding,foreveryfixedvectorp0 inthestatespace,afunctionoftimeq=q(t)(boundedfort [0,)andconvergingtozeroast )suchthatthesolutionp=p(t)of

p(t)=ap(t)+bq(t), p(0)=p0 (3.15)satisfies

limp(t)=0, (3.16)t

and,subjecttothisconstraints,minimizesthequadraticintegral(u())= |cp(t)+dq(t)|2dt min. (3.17)

0One motivation for considering abstract H2 optimization is the full information

versionofthestandardH2outputfeedbackdesign. AccordingtoTheorem3.1,theclosedloopH2normcanbeexpressedastheintegral

2fi = C1X(t)+D12U(t)Fdt,0

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5where

M2F =trace(MM)

denotes the square of the Frobenius norm of matrix M, and X,U are the closed loopimpulseresponsesfromwtoxandurespectively,constrainedonlyby

X =AX+B2U, X(0)=B1, X()=0, U()=0.IfXi,Ui,B1i denotethei-thcolumnofX,U,andB1 respectively,fi canbewrittenas

fi = |C1Xi(t)+D12Ui(t)|2dt.

0iSinceevery3-typle(Xi,Ui,B1i isindependentlyconstrainedby

X i(t)=AXi(t)+B2Ui(t), Xi(0)=B1i,thetaskofminimizingfi decomposesintoindependentabstractH2minimizationswith

a=A, b=B2, c=C1, d=D12, p0 =B1i.AsimilarmotivationforabstractH2optimizationcomesfromanattempttominimize

the closed loop H2 norm from w to Kxu, where K is a given matrix. According toTheorem3.2,theclosedloopimpulseresponsecanbechosenaccordingto

H=B1 +VD21, =A+VC2, (0)=K.Leti,Vi,Ki bethei-throwof,V,Krespectively. TheH2normofHequalsthesumoftheintegrals

|i(t)B1 +Vi(t)D21|2dt,0

minimizingwhichleadstosolvingafamilyofindependentabstractH2optimizationproblemswith

a=A, b=C2, c=B1, d=D21, p0 =Ki.3.2.2 TheeasyversionoftheKYPLemmaThe so-called Kalman-Yakubovich-Popov Lemma (also frequently referred to as thePositive Real Lemma provides, among other things, a complete solution to abstractH2 optimization. The much simplified version of the KYP Lemma we need here willbe complemented by a full statement used extensively in H-Infinity optimization andoptimizationrelyingonsemidefiniteprogramming.

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6Theorem3.3 Assume thatpair (a,b) is stabilizable. Then thefollowingconditionsareequivalent:

(a) auniqueoptimalcontrolexistsforeveryp0 intheabstractH2optimizationproblem

(3.15)-(3.17);(b) matrix

ajI bM(j)=

c dis left invertibleforall Randat =(whichmeans that d is left invertibleaswell);

(c)

d

d>

0,

and

matrix

ab(dd)1dc b(dd)1b

H= c ccd(dd)1dc a +cd(dd)1b

hasnoeigenvalueson the imaginaryaxis;(d) dd>0and thereexist (unique)matrices = 0and k such that a+bk isa

Hurwitzmatrix,and|cp+dq|2 +2p (ap+bq)=(qkp)dd(qkp) (3.18)

forallvectorsp,q.Ifconditions(a)-(d)aresatisfiedthentheoptimalq isdefinedbytherelationq=kp,i.e.

q(t)=ke(a+bk)tp0,andtheminimalcostequalsp0p0.Moreover,thedimensionofthestrictlystableinvariant

subspaceV+ ofHwillequalthenumbernofcomponentsofp,andcanbefoundfrom

1= 1 2 ... n x1 x2 ... xn ,

where x1 x2 xn, ,1 2 n

isanarbitrarybasis inV+.

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7Matrix H is usually called theHamiltonianmatrix associated with the abstract H2

optimization.

Condition

(c)

provides

a

convenient

way

of

checking

the

non-singularity

imposed by (b). Condition (d) defines a completion of squares procedure, which is anaturalandintuitivewayofsolvingtheabstractH2optimizationproblem. Bycomparingthecoefficientsonbothsidesof(3.18),onecanderiveaquadraticalgebraicequationfor,calledthealgebraicRiccatiequation(ARE):

++=,where,, arethecoefficientsoftheHamiltinianmatrix:

H= .

Asolutionforwhicha+bk=

isaHurwitzmatrix iscalledastabilizing solutionoftheARE.Astabilizingsolution, ifitexists,isunique. Itdefinestheoptimalcontrollergaink inq=kp,though,formally,abstractH2optimizationisaboutprogramcontroloptimization.

WewillpostponeprovingtheKYPLemmauntilitsfullversionisformulated.3.2.3 SolutionofFeedbackH2OptimizationInordertosolvethestandardH2feedbackoptimizationproblem,considerthetwoauxiliary abstract H2 optimization setups, the full information control version, definedwith

a=A, b=B2, c=C1, d=D12, (3.19)andthestateestimationversion,definedby

a=A, b=C2, c=B1, d=D21. (3.20)Itiseasytoverifybyinspectionthatabsenseofcontrolsingularityintheoriginalfeedbackdesignsetupisequivalenttosatisfyingcondition(b)ofTheorem3.3intheabstractsetupdefinedby(3.19),andabsenseofsensorsingularity isequivalenttosatisfying(b) intheabstract setup defined by (3.19). Hence, in the non-singular case, the correspondingcompletion of squares is possible, with matrices = Pfi, k = Kfi in the control setup,=Pse andk=Kse inthestateestimationsetup.

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8Theorem3.4 TheH2optimalcontroller isdefinedby

u(t) = Kfix(t), (3.21)dx(t)

dt = Ax(t)+B2u(t)+Kse(C2x(t)y(t)), (3.22)andprovidestheminimalH2norm(squared)of

J =trace(B 1PfiB1 +trace(D12KfiPseKfeD12).Proof TheoverallclosedloopH2norm(squared)isgivenby

J=

C1X(t)

+

D12U

(t)

2Fdt.0

AccordingtothedefinitionofKfi andPfi,thisintegralequals

J =trace(B1PfiB1 + D12(UKfiX)2Fdt.0

Thiscanbe interpretedassayingthat,intheH2optimalclosedloopsystem,umustbethebestestimateofKfix,inthesensethattheH2normfromwtoD12(uKfix)shouldbe minimal. According to the definition of Kse and Pse, the minimal estimation errorequals

K

D

trace(D12KfiPse fe 12),andisachievedwhentheimpulseresponseGfromwtoKfixuisgivenby

G(t)=(t)B1 +V(t)D21,where

(t)=(t)A+V(t)C2, (0)=K,and

V(t)=(t)Kse.By inspection,thisoptimal impulseresponse isachievedbythecontroller inthe formulationofthetheorem.