Control Engineering Series Editor WilliamS.Levine UniversityofMaryland Editorial Advisory Board Okko Bosgra DelftUniversity, Netherlands GrahamGoodwin University of Newcastle, Australia Petar Kokotovic UniversityofCalifornia, Santa Barbara Manfred Morar; ETH,Zurich,Switzerland WilliamPowers FordMotor Company, USA Mark Spong University of !/Iinois, lori Hashimoto KyotoUniversity,Japan RobustKalman Filtering for SignalsandSystemswith LargeUncertainties Ian R.Petersen AndreyV.Savkin Birkhauser Boston. Basel. Berlin IanRPetersenAndrey VSavkm Schoolof ElectrIcal Engmeermg AustralIan DefenceForceAcademy Umverslty of NewSouth Wales NorthcottDnve Department of Electncal and ElectromcEngmeenng Umverslty of Western AustralIa Nedlands,WA6907,AustralIa Canberra,ACT 2600AustralIa Library of CongressCataloging-in-PublicationData PetersenIanR RobubtKalmanfiltenngforsignalsandsystems withlarge uncertamtles/IanRPetersen,Andrey VSavkm pem- (Controlengmeermg) Includesbibliographicalreferpn(,pqandmdex ISBN0-8176-4089-4(hardcoveralkpaper) 1Feedbackcontrolsystems2Kalmanfiltermg3Signal processmgISavkm,AndreyVIITitleIIISenesControl engmeermg(Blrkhauser) TJ216 P4621999 629 8'3-DC21 AMSSubjectClassificatIOns49,93 PrInted onaCid-freepaper 1999BlrkhauserBostonBirkhiiuser$ 98-460l0 CIP AllrIghtsreservedThiSworkmaynotbetranqlatedorcopiedmwholeormpartwithout thewnttenpermissionofthepublisher(BlrkhauserBoston,c/oSprmger-VerlagNewYork, Inc,175FifthAvenue,NewYork,NY10010,USA)exceptforbnef excerptsmconnectIOn withreVieWSorscholarlyanalysIsUseIIIconnectIOnwithanyformof mformatlOn,torage and retneval, electromc adaptatIOn, computer softwareor by Similar or diSSimilar methodology nowknownorhereafterdevelopedISforbIdden The use of generaldescnptlve names,trade names,trademarks,etc, mthISpublication,even IftheformerarenotespeCiallyIdentIfiedISnottobetdkenasasignthatsuchnames,as understoodby theTrade MarksandMerchandIse MarksAct,may accordmgly beusedfreely byanyone ISBN0-8176-4089-4 ISBN3-7643-4089-4 SPIN19901575 Typesetby the authorusmg I;\1EX 2E Prmted andboundbyEdwardsBrothers,Inc,AnnArborMI Prmted mtheUmtedStates of Amenca 987654321 Contents Preface i ~1Introduction1 1 1The KalmanFllterI 1 2RobustState EstimatlOn 2 1 3GuaranteedCostState EstllnatlOn ~ 14Set-ValuedState EstllllatlOn3 141Discrete-ContmuousData ; 142StructuredU ncertamty .6 143NonlmearSystems (] 1 5ModelVahdatlOnforUncertamSystems () 1 6RobustH:xJFlltenng7 1 7ApphcatlOnsof RobustKalman Flltenng8 2Continuous-TimeQuadratic GuaranteedCostFiltering11 2 1IntroductlOn11 22AGuaranteedCo 0 forthzssystem.Thenthecorrespondzngaugmentedsystem(2.2.4)wzllbe quadratzcallystableandthesteadystateerrorcovarzanceat tzmetsatzsjies thebound Qll(t)::::Qforalladmzsszbleuncertazntzes,6.(t). Conversely,zfthesystem(2.2.1)zsquadratzcallystable,anystateestz-mator of theform(2.2.3)with A fstablewzllbea quadratzcguaranteedcost stateestimator forthissystemwzthsomecostmatrzx Q > O. Proof Toestablishthefirstpartof thetheorem,suppose(2.2.3)isaquadratic guaranteedcoststateestimatorwithcostmatrixQ>O.It followsthat thematrixA fmustbestable.Also.ithasbeenassumedthatthesystem (2.2.1)isquadratically stable.From thisit isstraightforward to verify that theaugmentedsystem(2.2.4)isquadraticallystable.Thiscanbeshown usingaLyapunovmatrix of the form wherePIdefinesaquadraticLyapunovfunctionfortheuncertainsystem (2.2.1),P2 definesaquadraticLyapunovfunctionforthesystem(2.2.3), anda> 0isasufficientlylargescalingconstant. Nowlet,6.(t)beanyadmissibleuncertaintyandobservethatw(t)is aGaussianwhitenoiseprocesswithidentity covariance.Also,assumethe initial condition random vector for the system (2.2.4)has covariance matrix E=Qo2:o.It now followsfrom astandard result on linear systems driven by whIte noise (e.g., see Theorem 1.52 of [74])that the corresponding statecovariance of the augmentedsystem(2.2.4)at timetisgivenby Qll(t. to) E{x(t)x(t)'}=(t.to)Qo(t. to)' + it (t.T)BoBb(t, T)'dT to where( t. T)isthestatetransitionmatrixassociatedwiththesystem (2.2.4)(withthespecifieduncertainty,6.(t)).Furthermore.usingthefact that the system(2.2.4)isquadratically stable.itfollowsthat lim(t,to)= 0 andhence.the steady state errorcovarianceat time tisgivenby 162.Continuous-Time QuadraticGuaranteedCostFiltering Nowletthe timetbe fixedandlet7]beagivenvectorinR 2n.Then 7]'Ot:..(t)7]=Itex; 7]'if>(t.r)BoBbif>(t, r)'7]dT ItOG7]'( r)BoBb7]( r)dr where '1(T) if>(t, T)'7]isthe solution to thedual state equation r,(r)=-[A + B1l:!.(r)K]'7](r);7](t)= 7].(2.2.9) - t:..-NowwithQdefinedasin(2.2.6),letV(7])=7]'Q7].It followsfrom(2.2.7) and(2.2.9)that - - d 7](r)' BoBb7](r):::dT V(7](r)). Therefore,using the factthat the system (2.2.4)isquadratically stable,the dual system(2.2.9)willbestableinreversetimeandhence Itoo 7]'(r)BoBb7](r)dr:::V(7])- V(7]( -00))= V(7]). Thus,using(2.2.8) 7]' 0 t:..(t)7]:::7]'07] forall7]ER 2n.Fromthisitfollowsthat0 t:..C t):::0foralladmissible uncertaintiesl:!. (t).Nowthe state errorcovariance Q t:.. (t)isthe(1,1)block of the matrix Ot:..(t)andcostmatrix Qisthe(1,1)block of the matrix O. Hence, (2.2.10) foralladmissibleuncertaintiesl:!.(t).Thiscompletestheproof of the first part of the theorem. Conversely,supposethesystem(2.2.1)isquadraticallystableandcon-sideranystate estimatorof theform(2.2.3)withAt stable.Fromthis,it followsthatthecorrespondingaugmentedsystem(2.2.4)isquadratically stable.Hence,thereexistsamatrix 0 > 0suchthat forallmatricesl:!.: l:!.'l:!.:::I.Fromthisinequality,itisstraightforwardto verifythat thereexistsaconstantE> 0such that forallmatricesl:!.:l:!.'l:!.:::I.Thus,thisstateestimatorisaquadratic guaranteed coststate estimator with costmatrix Q0/ E> O.0 2 3 PreliIrunary Results17 Remarks ThISchapterISconcernedwIthconstructmgastateestimatorthatmIm-mizesthe nghthandsIdeof the errorcovanancebound(22 10)However, (22 10)ISamatnxmequahtyToobtamascalarmmimizatlOnproblem, conSIderthe correspondmgboundonthesteady statemean squareerror hmE{e(t)'e(t)}=tr{Q.6-(t)}::;tr{Q} to---+-OO (2211) Thus,theresultsof thISchapterareconcernedwIthmmimIzmg tr{ Q} It shouldbenotedthattheremaybesituatlOnsmwhIchoneISonly concerned wIth estImatmg alImlted number of state vanablesIn thIScase, anoutputvanablez(t)=Hx(t)wouldbeconSIderedandcorrespondmg .6-outputestImatewouldbez(t)=Hx(t)Thecorrespondmgsteadystate mean square errorboundISthen hmE{(z(t) - z(t))'(z(t)- z(t))} =tr{H'Q.6-(t)H}::;tr{H'QH} to---+-CXJ ThusmthIScase,ItISreqUIredto mmimize the quantIty tr{H'QH}How-ever,the approach taken wouldremam vIrtually the same as If the quantIty tr{ Q}weretobemmimizedThus,throughoutthesequelItWIllbeas-sumed that the quantIty tr{Q}ISto be mimmized 2.3PrelImmaryResults ThISsectlOncontamsanumberof Importantresultsrelatmgtouncertam systems wIth norm boundeduncertamty,HOGcontrol theory,andalgebraIc RiccatiequatlOnswhIchWIllbeusefulwhenaddressmgtheproblemof optImalguaranteedcostfiltenngforthe uncertamsystem(22 1) ThefollowmglemmaISaverSlOnofthesmallgamtheoremwhIchre-latestherobuststabIlItyofanuncertamsystemtoanHOGnormbound conditlOnThISverSlOnof the smallgam theoremwasongmally presented m[68] Lemma2.3.1Theunceriamsystem(22 1)zsquadratzcallystablezf and onlyzf thefollowmgtwocondztzonsaresatzsjied (z)ThematrzxAzsstable(thatzs,allof ztsezgenvaluehemtheopen lefthalf of thecomplexplane), (zz)TheHOGnormboundIIK(sI - A)-I BIII=< 1zssatzsjied Proof See[68]0 Thefollowmgresultsrelate to the algebraICRlccatlequatlOnandsome of ItSproperties 182.Continuous-Time Quadratic Guaranteed Cost Filtering Notatzon A symmetric matrix p+ issaid to be a stabilizing solution to the Riccati equation A' P+ P A- PM P+ N= 0 IfitsatisfiestheRiccatiequationandthematrixA- AIP+isstable. SImilarlyasymmetricmatrixP+issaidtobeastrongsolution tothis Riccati equation if it satisfies the Riccati equation and the matrix A - M p+ hasallof its eigenvaluesintheclosedlefthalf of thecomplexplane.Note thatanystabilizingsolutiontotheRiccatiequationWIllalsobeastrong solution. The followingcomparison resultforstabilizing solutIOns to the algebraic Riccatiequationwasfirstpresentedin[109].It isfrequentlyusedinthe theory of quadraticstabilizationandH= control. Lemma 2.3.2ConszderthealgebrazcRzccatzequatzon A' P+ P A - P]\;1 P+ f.r= 0 (2.3.1) whereM ;:::0and(A, B)zsstabzl2zable.SupposethzsRzccatieqtI.atwnhas asymmetrzcsolutzonP anddefinethematrzx - [f.rA' _]. H=A--M AlsoconszdertheRzccatzequatzon A' P+ P A- PM P+ N=0(2.3.2) whereM;:::0anddefinethematrzx H=[ ~ ~ ~ 1 ] If H;:::H,thentheRzccatzequatzon{2.3.2}willIuwectUf&iquestrong solutzonP+whzchsatzsfiesp+;:::P. Proof SeeTheorems2.1and2.2of [109].0 Thefollowingcorollarytothislemmaisusedinthesequelandinthe proof of the StrictBoundedRealLemma givenbelow. Corollary2.3.3SupposeAzsstableandtheRzccatzequation (2.3.3) hasasymmetrzcsolutzonP.Furthermore,supposeQ ;:::Q;:::O.Thenthe Rzccatzequatzon A' P+ P A+ P BB' P+ Q =0(2.3.4) wzllhaveaunzquestrongsolutzonPandmoreover,0:::;P:::;P. 2 3PrelImmary Results19 Proof LetX =-PHence,RlccatlequatlOn(233)canbe rewntten as A' X + X BB' X - Q = 0 Furthermore, smce A 1Sstablethe palI (A, B) must be stabilizableHence, USIngLemma 23 2,Itfollowsthat theRlccatlequatlOn A' X+ XA- XBB' X- Q = 0 WIllhaveaumquestrongsolutlOnX::::XNowletP=-XIt follows llnmedlately that P::;F ISthe umque strong solutlOn to(23 4)Moreover, USIngastandard result on Lyapunov equatlOns,Itfollowsfrom(234) that P::::0seeLemma121 of [158]D Theproof of the themaInresultof thISchaptermakesextensIveuseof thefollOWIngStnctBoundedRealLemmaThIS\erSlOnof thISresultwas ongInallypresentedIn[97] Lemma2.3.4(Stnet Bounded Real Lemma)The followmgstatements are equzvalent (z)AzsstableandIIC(8I- A)1 BII= 0such that A' F+FA+FBB' P+C'C < 0 (zzz)TheRzccatzequatwn A'P+PA+PBB'P+C'C =0(23 5) hasastabzlzzzngsolutwnP::::0 Furthermore,zf thesestatementshold,thenP0suchthatQ+::;S+.Also,thestate estzmator iCt)(A + EQ+ K' K) x(t) +(EQ+C'+ BIDD (EV+ D1DD-I (y(t)- Cx(t)) (2.4.3) hasthefollowzngproperty:Gwenanyt5>0,thereexzstsamatnx Q >0 suchthatQ+::;Q 0suchthat Rzccatzequatzons(2.4.1)and(2.4.2)havestabzlzzzngsolutzonsS+> 0and Q+> 0,respectwely,and Q+ 0suchthat Therefore,Lemma2.3.4impliesthatforallEE(0, E*),the followingcon-ditionshold: 242Contmuous-TImeQuadratICGuaranteedCost FIlterIng (1)There eXIstsamatnx S> 0suchthat - - - - 1 AS + SA' + ESK'KS ++ W< 0 E (244) (u)RiccatlequatlOn(241)hasastablhzmgsolutlOnS+ 0 Furthermore, smce WISposItlve-defimte, It followsImmedIately from(241) that S+ISposltlVe-defil1lte NowletEE(0, E*)be gIwnandconsIdermequahty(244)It ISdesIred toapplyLemma2 3 5tothISmequahtyInordertodothIS,consIderthe correspondmg Hoo controlproblemdefinedby the system xA'x + VEK'w + VEC'U, z [ ..LB'1[ D'1 1X+U (245) ItISstratghtforwardtovenfythatthISsystemlSasystemoftheform (2315)whIchsatlsfiesassumptlOns(1)and(u)of Lemma235Suppose thestatefeedbackcontrolu:=0lSappliedtothlSsystemUsmgLemma 23 5Itfollowsfrommequahty(244)thatRlccatlequatlOn(242)hasa stablhzmg solutlOnQ+2:0Furthermore,W>0lmpliesQ+> 0 -1 Nowcompare Rlccatl equatlOns(241) and(242)Lettmg=(S+)- , ItISstratghtforwardto venfythatf:> 0ISthestabIlIzmgsolutIOntothe RiccatiequatlOn - - 1- - - --- --- EK'K =0 E (246) 1- Also,let =(Q+)- andmtroducethenotatlOnV=EV+ It ISstraightforwardtovenfythat >0ISthestablhzmgsolutlOntothe RlccatlequatlOn (247) Lemma2 3 2wIllbeapplIedtocompareRIccatiequatIOns(24 6)and (247)Thus,consIderthe matnx - [-EK'K M= -A assocIatedWIthRlccatl equatIOn(24 6)andthematnx - (A-BIDiV-IC)'1 (I - V-I Dl) Bi- W 2.4OptimalGuaranteedCostFilter Design25 associatedwiththeRiccatiequation(2.4.7).Hence, M-M= [ EC'V-lCC'V-l DlBi] V-lC V-I DlBi [ ]V-I [VEC)DlBi] >O. Therefore,itfollowsfromLemma 2.3.2thatI: ;:::tandhenceQ+::;S+. Nowconsidertheaugmentedsystem(2.2.4)obtainedwhenthestate estimator(2.4.3)isappliedtotheuncertainsystem(2.2.1).Inthiscase, thematricesA,Eo,131,andK areasfollows: [A- (EQ+C' + BlDi) V-lC-EQ+ K' K] (EQ+C'+BlDDv-1CA+EQ+K'K' [:! -(EQ+C'+BlDDV-lV!] (EQ+C' + BlDD V-IV!' [ Bl - (EQ+C' + V-I Dl] (EQ+C' + BlDi) V-I Dl' [KKJ. Withthesedefinitions,itisstraightforwardbuttedioustoverifythat the matrix Q[Qo+S+ Q+] ;:::0 satisfiestheRiccatiequation -- -- ---- 1-- --AQ + QA' + EQK' KQ ++ BoBb= o. E (2.4.8) Furthermore,the matrix A+EQk'k [A- (EQ+C' + BlDi) V-lC + EQ+ K' K0] (EQ+C' + BlDU V-lC + E(S+- Q+)K' KA+ ES+ K' K is stable sinceQ+isthe stabilizing solution to Riccati equation(2.4.2)and S+isthestabilizingsolutiontoRiccatiequation(2.4.1).Thus,Q isthe stabilizing solution toRiccatiequation(2.4.8).UsingLemma 2.3.4,itnow followsthatA isstableand Lemma 2.3.4also implies that there existmatrices Ql> Q and if> 0 such that 262.Continuous-TimeQuadratic Guaranteed CostFiltering Thus givenanyp E(0, 1), -- -- ---- 1-- -- -AQI + QlA' + fQIK' KQI ++ + pV< 0 f andtherefore,Lemma2.3.4impliesthattheRiccatiequation (2.4.9) hasastabilizingsolutionQp> o.(Thepositive-definitenessof Qpfollows fromthe positive-definiteness of v.) Furthermore, comparing Riccati equa-tions(2.4.8)and(2.4.9),itfollowsfromCorollary2.3.3that Qp Q.Now since the stabilizing solution to Riccati equation (2.4.9) willbe a continuous functionof theparameterp,itfollowsthat (2.4.10) Hence,(2.4.9)and(2.4.10)imply that givenany 6> 0 there exists amatrix Qp> 0such that Q:::Qp< Q + Mand (2.4.11) Also,letting Q > 0be definedasthe(1,1)blockof thematrix Qp,then it followsthat Q+:::Q < Q++ M. Nowletthematrix begivensuchthat :::I .Usingastandard matrix inequality,Lemma2.4.11impliesthat (A +Qp+ Qp(A + +-- -- ---- 1- -- O.Hence, there existsamatrix - [Q Q=2.4Optimal Guaranteed CostFilter Design27 such that forallmatrices :.:;1.HerethematricesA,Bo,fh,andKdefine thecorrespondingaugmenteduncertainsystemasin(2.2.4)and(2.2.5). UsingLemma 3.1andObservation3.1of[96].thisimpliesthat x'(AQ+ QA' + BoBb)x +< 0 forallx of.O.Thus.AQ + QA' + BoBb< 0and (x'[AQ+ QA' + BoBb]x)2> 4x' KQx forallxof.O.Hence,usingTheorem4.7of [98],itfollowsthat there exists anE> 0such that (2.4.12) Now pre-multiply this inequality by the matrix[11] and post-multiply bythe matrix[1I] '.It isstraightforwardto verify that this yieldsthe inequality 1 AIT + ITA' + EITK' KIT ++ W< 0 E whereIT Q + QI2 ++ Q22> O.HenceusingLemma 2.3.4,it follows immediately that Riccati equation (2.4.1)has astabilizing solution S+2:O. Furthermore,W> 0impliesS+> O. Nowpre-multiply(2.4.12)bythematrix[I-QI2Q2l]andpost-multiplyby thematrix[1-QI2Q221]'.It isstraightforwardtoverify that thisyieldstheinequality [A- (I + QI2Q2n BfC]+[A- (I + QI2Q2n BfC]' + (1 + Q12Q:;l) BfVBj (I + Q12Q221)'+ W++[BI- (I + Q12Q2l) BfDd[BI- (I + QI2Q:;l) BfD1]'< 0 E (2.4.13) whereQ ->O.Hence, .:;Q.Furthermore,itfollows from(2.4.13)that satisfiesthe inequality [A+ y'ELCj+[A+ y'ELC],+ + ELVL' + W +BI+ LDI] BI+ LDI]'< 0(2.4.14) 282.Continuous-TimeQuadraticGuaranteedCostFiltering whereL - JE(I + Q12Q2l) BfItisdesiredtoapplyLemma2.3.5to thisinequality.Todothis,considerastate feedbackHOCcontrolproblem definedbythesystem(2.4.5).Inthiscase.supposethestatefeedback control u=Ix isapphed to this system. Using Lemma 2.3.5 it followsfrom inequality(2414)thatRiccatiequation(2.4.2)hasastabilizingsolution Q+2':0 such that Q+0,theR2ccatiequatIOn(2.4.1) hasasolutIOnS> 0and theR2ccat2equatIOn(2.4.2)hasasolution Q> O. ThenforanyEEO(0. f'),theR2ccat2equatIOn(2.4.1)wzllhaveastab2l2zing solutIOnS+>0andtheR2ccat2equatIOn(2.4.2)w211haveastab21Izmg solutIOnQ+> O. Proof SupposetheRiccatiequation(2.4.1)hasasolutionS>0andRiccati equation(2.4.2)hasasolutionQ>0withE =f'.LetIT tQ-l>O.It followsthat ITsatisfiestheRiccatiequation IT(A - v-Ie) + (A - V-Ie)' IT + K' K- e'v-1e +ITBI(I - V-I Dl) + f'ITWIT=0(2.4.15) A__ where V=f'V+Also,let =f'S> O.It followsthat satisfiesthe Riccatiequation (2.4.16) 2.4OptimalGuaranteedCostFilterDesign29 NowletEE(0, E)begivenandconsidertheRiccatiequations fr(A -+ (A - fr+ K'K - C'V-1C +frBl(1 - V-I Dl) fr+ EfrWfr=0(2.4.17) and (2.4.18) ,Ll. whereV= EV+ Lemma2.3.2willbeappliedtocompareRiccatiequations(2.4.15)and (2.4.17).Thus,consider the matrix - (A - B 1 V -1 C) ,1 -BI (1 - - EW associatedwiththeRiccatiequation(2.4.15)andthematrix - (A - B 1 V -1 C)'1 -Bl (1 - - EW associatedwithRiccatiequation(2.4.17).It followsbyastraightforward algebraicmanipulationthat H- if =[BC1D' 'I](V-I- V-I)[C + [00(0)]. E-E W (2.4.19) UsingthefactthatE O. Toestablish that II+isin factastabilizing solutionto(2.4.17),observe thatRiccatiequation(2.4.15)can bere-writtenintheform - [1fr] if [A] =o. (2.4.20) Similarly,Riccati equation(2.4.17)canbere-writtea in the form 302.Continuous-TimeQuadraticGuaranteedCostFiltering Thus using(2.4.19)and(2.4.20),it followsthat fI> 0 satisfies the inequal-ity -[1fIJH[AJ - [1fIJ H [A] - [1fIJ (H - H)[ J 0suchthat fI(A-+ (A- fI+K'K - c'v-1e +fIBl(1 - V-I Dl) +EfIwfI +N= 0(2.4.21) Nowsubtracting Riccati equation (2.4.21)from(2.4.17),it followsthat the matrixZ II+- fI:2:0satisfiesthe equation AZ+ZA' +V = 0 where and NZ{Bl(I -+ EW}Z + N> o. A standardLyapunovargumentnowimpliesthatthematrix A isstable; seeLemma12.2in[158].Hence,II+isastabilizingsolutionto(2.4.17). Nowlet Q+ > O. E Itisstraightforwardto verifyusingRiccatiequation(2.4.17)thatQ+isa stabilizing solutionto Riccatiequation(2.4.2). Inordertocompletetheproof of thistheorem,Corollary2.3.3isnow applied to compare Riccati equations (2.4.16)and(2.4.18).From this corol-lary,it followsthat Riccati equation(2.4.18)hasastrong solutionsuch that 0::::;::::;t. AlsosinceW> 0,itfollowsfrom(2.4.18)that> O. Now,itfollowsfrom(2.4.16)thattsatisfiesthe Riccatiequation (2.4.22) where N=(E - E)W> O.Subtracting (2.4.18)from(2.4.22),it followsthat thematrix 2 4OptimalGuaranteed Cost Filter DesIgn31 satisfiesthe equatIOn where A = A+ 'L,+K'KandN =(I:- 'L,+)K'K(I:- 'L,+)+N > 0 AstandardLyapunovargumentnowImphesthatthematnxA ISstable, seeLemma122m[158]Hence,L,+ISastablhzmgsolutIOnto(24 18) ThiScompletestheproof of the theorem0 Theorem2.4.3SupposeRzccatzequatwn(242)hasaposztzve-defimte stabzlmngsolutwn Q+(t)foreachtmthemterval(OJ)Thentr(Q+(t)) zsaconvex functwnof tover (0, E) Proof If Q+> 0ISastablhzmgsolutIOnto(242)ItISstraightforwardto venfy that n+ (Q+) -I > 0ISthe stablhzmg solutIOnto theRlccatI equatIOn - n (A- (tV +e) - (A- (tV +e)' n 1(,(,)-1)'II - 1- DltV + DIDIDlBl- IIwn - EK'K + Ee' (EV+ D1DD-1 e =0(2423) Nowletn+ f. n+,andn+n+DlfferentIatmgRlccatlequatIOn (2423) tWiceleads to the followmgequatIOn(after asome straightforward but tedIOUSalgebraicmampulatlOns) -An+ - n+A' + 2[(d + V-Iv-Ie -+x(d + V-I DJ}-l[(d + V-I v-Ie -+ =0 where However.smcethe matnx A ISstable,Itfollowsfromastandardresulton the Lyapunov equatIOnthatn+::;0e g,seeLemma 121of [158] c.dc.d2 NowletQ+=-Q+andQ+=-Q+Then de'de2 Q+=-Q+n+Q+andQ+=2Q+n+Q+n+Q+- Q+n+Q+ Thus smce n+::;0,Itfollowsthat Q+ 0Hence,tr( Q+)Willbe aconvex functIOnof t0 322.Continuous-Time QuadraticGuaranteed Cost Filtering Remarks It followsfromtheabovetheoremthatinminimizing tr(Q+)withrespect toE,anylocalminimumwillalsobeaglobalminimumoftr(Q+)and efficientnumericalmethodscanbefoundtoperformthe minimization. Using the above results,it followsthat the optimal guaranteed cost state estimatorcanbeobtainedbychoosingE>0tominimizetr(Q+)where Q+isthestabilizingsolutiontoRiccatiequation(2.4.2).[Thisconvex minimization problem isalso subject to the constraint that Riccati equation (2.4.1)hasastabilizing solutionS+> 0.]Therequiredoptimal estimator isthengivenby(2.4.3).Notehoweverthatthereisnoguaranteethata minimumwillexist. 2.5IllustrativeExample This sectionpresentsan example to illustratethe theory developedinthis chapter. Indeed,consider the uncertain system described by the state equa-tions x( t) yet) ]x(t) + wet): -11x(t) + vet); where isa scalar uncertain parameter subject to the bound:::;1, andwet)and vet)are zeromeanGaussian white noiseprocesseswith joint covariancematrix Thissystemisof the form(2.2.1)where A K !l],C=[l [051' W=[-1], Bl =[ ], Dl =0, ], V=1. Forthissystem,itisstraightforwardtoverifythatconditions(i)and(ii) of Lemma 2.3.1aresatisfied.Hence.thissystemisquadratically stable. ConsidertheproblemofestimatingXl (t),thefirstcomponentofthe state of thissystem.In ordertoconstructtherequiredoptimalquadratic guaranteed cost state estimator, it is required to find the value of the param-eter E> 0whichminimizesthemeansquareerrorboundH1Q+ Hfwhere Hl =[10].HereQ+isthe stabilizing solutiontoRiccatiequation(2.4.2). 2.5IllustrativeExample33 Riccatiequations(2.4.1)and(2.4.2)werefoundtohavepositive-definite stabilizingsolutionsforEintherange(0,4x10-4).AplotofH1 Q+ H{ versusEisshowninFigure 2.5.1.Fromthisplot,the opt1malvalueof fis foundtobef1=2 88X10-4. 240,--,--,------,------,-----,------,------,------,-----0 en ""0 220 200 o .0 ~g ~ JBO (ij :J 0"" en c ro160 Q) ~140 , '. , H1Q+H1' H2Q+H2' - tr(Q+) / _. I I I I 1 2 0 ~ ~ ~ L ~ ~ ~ L ~o05152253354 Parameter EJ(1.0-4 FIGURE2.5.1.Meansquare errorbounds versusE. Figure 2.5.1alsoshowsaplotof the mean square errorboundH2Q+~whereH2=[0.707 O. 707J.Thisbound wouldbe usedif the output variable z(t)=H2X(t)weretobeestimated.Inthiscase,theoptimalvalueof f isfoundtobef2=1. 76X10-4.Thus,itcanbeseenthatdependingon thestate variablestobeestimated,adifferentoptimum valueof fwillbe obtained.Athirdcasewhichisconsideredisthecaseinwhichthetotal state vectoristobeestimated.In thiscase,themean squareerrorbound isgivenby trace( Q+).Aplot of trace ( Q+)versusfisalsoshown in Figure 2.5.1.The corresponding optimal value of fisfoundto be f3=2.31X10-4. With to=f3=2.31X10-4 ,the stabilizing solutions to Riccati equations (2.4.1)and(2.4.2)are s+= 103[2.25510.0367]Q+=[125.4738 x0.03670.0606> 0,59.7607 59.7607]O. 60.3444> Thecorrespondingvalueof themeansquareerrorboundistrace(Q+)= 185.8181.Also,equation(2.4.3)givestheoptimalquadraticguaranteed 342.Continuous-TimeQuadraticGuaranteed Cost Filtering cost state estimator of the form(2.2.3)with A=[-66.7130 f0.5836 67.0582]B=[65.7130] -1.2351'f-0.5836' Toillustratetherobustfilteringpropertiesof thequadraticguaranteed coststateestimator,Figure2.5.2showsaplotofthemeansquareerror asafunctionof the uncertainparameter6..From thisplot.itcanbe seen thatforalladmissiblevaluesof theuncertainparameter,thecorrespond-ing mean square error islessthan the calculated quadratic guaranteedcost bound.Notethatinthisplot,onlyconstantvaluesoftheuncertainpa-rameterwereconsidered.However,thequadraticguaranteedcostbound alsoholds fortime-varying uncertain parameters. If time varying uncertain parameters wereallowed,one would expect the boundto be somewhat less conservative. Forthesakeof comparison,Figure2.5.2alsoshowsaplotof themean squareerrorwhichwouldbe obtained if astandardKalmanFilter(based on the system without uncertainty)wereapplied.From this plot,it can be seen that inthe worstcase,the robustfilterissignificantly better than the KalmanFilter. 500,----,-----,----,-----,-----,----,-----,-----r----,-----, ... o 450 400 350 t:300 0forthzssystem.Thenthecorrespondmgaugmentedsystem (3.2.4)wzllbequadratzcallystableandthesteadystateerrorcovarzance matrzxattzmeksatzsfiesthebound Qc,.(k):s:Q(3.2.8) foralladmzsszbleuncertamtzesConversely,anystateestzmator of theform(3.2.3)withAISchur stable wzllbeaquadratzcguaranteedcoststateesizmator forthesystem(3.2.1) wzthsomecostmatrtx Q > O. Proof Toestablishthefirstpartofthetheorem,suppose(3.2.3)isaquadratic guaranteed cost state estimator with costmatrix Q> O.It followsthat the matrix A Imustbe Schur stable.Also,it hasbeenassumedthat the uncer-tainsystem(3.2.1)isquadraticallystable.Fromthisitisstraightforward toverifythattheaugmentedsystem(3.2.4)willbequadraticallystable. ThisfactcanbeestablishedusingtheLyapunovargumentmentionedin theproof of Theorem2.2.3orbysimplynotingthattheaugmentedsys-tem isformedby cascadingaquadratically stable uncertain system witha stable lineartime-invariantsystem. NowletbeanyadmissibleuncertaintyandobservethatWkisa Gaussianwhitenoiseprocesswithidentitycovariance.Also,assumethat theinitialconditionrandomvectorforthesystem(3.2.4)hascovariance matrix E{x(ko)x(ko)'}=Qo:::::O. 3.2Discrete-TimeQuadraticGuaranteedCostFiltering39 Itnowfollowsthatattimek.thecorrespondingstatecovariancematrix forthe augmentedsystem(3.2.4)isgivenby k Qc,,(k, ko)~ E {x(k)x(k)'}=if>(k,ko)Qoif>(k, ko)' + Lif>(k,j)BB'if>(k,j)' )=ko whereif>(k, J)isthestatetransitionmatrixassociatedwiththesystem (3.2.4)[with the specifieduncertainty realization ~ k ) ] ; e.g.,seeChapter 2 of[2].Furthermore,usingthe factthat the system(3.2.4)isquadratically stable,itfollowsthat limif>(k.ko)=0 ko---+-oo andhence,thesteadystate errorcovariancematrix attimekisgivenby k Qc,,(k)~ limQ6(k,ko)=Lif>(k,j)BB'if>(k,j)'. ko---+-oo )=-00 Nowlet Hk~ Q - (A + Bl8.(k)K)Q(A + Bl8.(k)K)' - BB' > 0 and Y(k, ko)~ Q - Qc,,(k, ko). It isstraightforward to verify that Y (k, ko) satisfies the Lyapunov difference equation Nowsincethesystem(3.2.4)isquadratically stable,itfollowsthat limY(k. ko)= Q - QD.(k)~ 0 ko---+-oo independentlyof theinitialconditionY(ko. ko).HenceQc,,(k):::;Qforall admissibleuncertainties8.(k).However,thestate errorcovarianceQc,,(k) isthe(1,1)block of the matrix Qc,,(k)and costmatrix Q isthe(1,1)block of the matrix Q.Hence, QD.(k):::;Q(3.2.9) foralladmissibleuncertainties~ k ) . Thiscompletestheproof of thefirst part of the theorem. To establish the secondpart of the theorem,consider any state estimator of the form(3.2.3)with Af Schur stable.Asabove,since the system (3.2.1) isquadraticallystable,itisstraightforwardtoverifythattheaugmented system(3.2.4)willbe quadratically stable.Hence,thereexistsamatrix 403.Discrete-Time QuadraticGuaranteed CostFiltering suchthat (A+ Ih6.K) Q (A + Ih6.K)' - Q < 0 forallmatrices6.: 6.'6.:::;1.Therefore,there existsaconstant f> 0such that (A + fh6.K)Q (A+ fhf1K)' - Q + BE' < 0 EE forallmatrices6.:6.'6.:::;I.Thus,thisstateestimatorisaquadratic guaranteedcost state estimator withthecostmatrix QnlE0 Remarks Themainresultof thischapterisconcernedwithconstructingastate es-timatorthatminimizestherighthandsideof theerrorcovariancebound (3.2.9).However,(3.2.9)isamatrixinequality.Toobtainascalarmini-mizationproblem,wethereforeconsiderthecorrespondingboundonthe steadystate mean square error: limE{e(k)'e(k)} =tr{Q,c,.(k)}:::;tr{Q}. k o ~ o o(3.2.10) Thus, the results of this chapter are concerned with constructing a quadratic guaranteedcoststate estimator that minimizesthe quantity tr{ Q}. As in Chapter 2,it should be noted that there may be situations in which notallof the statevariablesarerequiredtobeestimated.Instead,itmay be requiredto estimate the value of an output variablez(k)=Hx(k).The solutiontothisproblemwillbeanoutputestimateoftheformi(k)~Hx(k).Furthermore.thecorrespondingsteadystatemeansquareerror boundwillbe asfollows: limE{(z(k) - i(k))'(z(k) - i(k))}=tr{H'Q,c,.(k)H}:::;tr{H'QH}. ko-'/>-cc Thusaquadratic guaranteedcoststate estimator wouldbe constructed to minimizethe quantity tr{H'QH}.Throughoutthe sequel,wewillassume thatthestateestimatoristobeconstructedtominimizethequantity tr{ Q}.However,itcan beseenthat onlyaminormodificationisrequired if insteadaquantity suchastr{H'QH}istobe minimized. 3.3PreliminaryResults Thissectioncontainsacollectionof resultsrelatingtodiscrete-timeun-certainsystemswithnormboundeduncertaintyandrelatedresultson discrete-timeHoocontrolandalgebraicRiccatiequations.Mostofthe resultspresentedinthissectionareanalogoustocorrespondingresults presentedinSection2.3. The followingresultisadiscrete-time versionof thesmallgaintheorem 2.3.1. 3.3PreliminaryResults41 Lemma 3.3.1Theuncertamsystem(3.2.1)zsquadratzcallystablezf and onlyzf thefollowingtwocondztwnsaresatzsfied: (z)ThematrzxAzsSchurstable. (zz)Thedzscrete-tzmeHOO normboundI/K(zI -A)-lBllIoo < 1is satis-fied. Proof See[90,103,104].0 Thefollowingmatrixinversionlemma isausefulmatrix identitywhich isextensivelyusedinproblemsinvolvingdiscrete-timelinearsystems. Lemma 3.3.2(MatrzxInverswnLemma).If AERnxn and CERmxm arenonsmgularmatrzces,thenformatricesBand Dof approprzatedz-menswns: Proof Thisidentitycanbeverifiedbysubstitutionusingthedefinitionofthe matrix inverse.0 Notation AsymmetricmatrixP+ issaid to be astrong solution to thediscrete-timealgebraicRiccatiequation APA' - P- APC'(M-1 + CPC')-lCPA' + N=0 (3.3.1) if itsatisfiesthe Riccatiequationandthe matrix (3.3.2) hasallof its eigenvaluesin the closedunitdisk.The solutionP+issaid to be astabilizing solution if the matrix (3.3.2)isSchur stableFurthermore if P> 0,then it can be shown using the Matrix Inversion Lemma(Lemma 3.3.2)thatRiccatiequation(3.3.1)isequivalentto the Riccatlequation A(P-1 + C'MC)-l A' - P+ N= O.(3.3.3) Inthiscase,asymmetricmatrixP+>0isthestabilizingsolutionto Riccatiequation(3.3.3)ifitsatisfiesthisRiccatiequationandthematrix (I + C' MCp+)-l A'isSchurstable. The followingcomparisonresultforstabilizing solutionsto the discrete-time algebraic Riccati equation isthe discrete-time analog of Lemma 2.3.2. Lemma 3.3.3Conszderthedzscrete-tzmealgebrazcRzccatzequatzon A'PA - P- A'PB(M + B'PB)-lB'PA + IV= (3.3.4) 423DIscrete-TImeQuadraticGuaranteedCostFIltermg whereif::;>0and(A, B)zsstabzhzable.SupposethzsRzccatiequationhas asymmetrzcsolutwnF suchthat 111+ B'FB > 0 anddefinethematrzx AlsoconszdertheRzccatzequatwn A'PA - P- A'PB(M + B'PB)-lB'PA+N= 0 (3.3.5) where111::;>0anddefinethematrzx IfH::;>iI,thentheRzccatzequatwn(3.3.5)wzllhaveaumquestrong solutwnP+suchthat if +B'P+B > 0 whzchsatzsfiesp+::;>F. Proof SeeTheorems3.1and3.2of [109]0 The followingresultisadiscrete-timeversionof the strict boundedreal lemma(Lemma2.3.4). Lemma 3.3.4(Dzscrete-TzmeStrzctBoundedRealLemma,see(35)for proof.)Thefollowmgthreecondztwnsareequwalent: (z)Thematrzx AzsSchurstableand[[G(zI- A)-lB[[oo< 1. (n)ThereexzstsamatrzxP> 0suchthatp-1 - BB' > 0and A'(P-1 - BB')-l A- P+ G'G< 0 (m)ThealgebrazcRzccatzequatzon A'PA - P+ A'PB(I - B'PB)-l B'PA + G'G = 0 hasastabzlzzzngsolutwnP+::;>0suchthatI- B' p+ B> o. Furthermore,zf thesecondztzonshold,then P+zsumqueand satzsfiesp+< PwhereP> 0zsanymatrzxsatzsjymgcondztzon(zz). 3.4DIE>crete-TimeOptimal Guaranteed CostFilter Design43 Theproofofthemainresultofthischapterwillusearesultonthe standard discrete-tirne state feedback HOO control problem.The underlying linearsystem forthisproblemisasfollows: x(k + 1) z(k) Ax(k) + B1W(k)+ B2U(k); C1x(k) + D12U(k). This system isassurnedtosatisfy the followingassumptions: (i) ~ D12=0: (ii)DbD12>0; (iii)q C1 > o. (3.3.6) Lemma 3.3.5Conszderasystemof theform(3.3.6)satisfymgassump-tzons(z)- (zzz)above.Furthermore,supposethereexzstsastatefeedback controllawu(k)=Fx(k)thatsolvestheHOOcontrolproblemdefinedby thesystem(3.3.6).Thatzs,thefollowmgcondztzonsaresatzsfied: 1.Thematrzx A+ B2FzsSchurstable; ThentheRzccatzequation \,;).,;).7) hasastabzlzzmgsolutwnp+ > 0whzchsatzsfiestheconditwn (3.3.8) Proof SeeTheorem 3.7 of [13)orTheorem3.1of [35]. o 3.4Discrete-TimeOptimalGuaranteedCostFilter Design This section presents the main result of this chapter. This isaRiccati equa-tionapproachtoconstructingthequadraticguaranteedcoststate estima-torwhichminimizestherighthandsideof themeansquareerrorbound (3.2.10). 443.Discrete-TImeQuadraticGuaranteedCostFIltering Theorem 3.4.1Supposetheuncertam system(3.21) zsquadratzcallysta-ble.Thenthereexzstsaconstantf*>0suchthatforallf E(0, f*),the Rzccatzequatwn hasa stabzlzzmgsolutwnS+> 0satzsfymgthecondztzon (S+)-1- fK' K> O. For anysuch f,theRzccatlequatwn A(Q-l + C'V-1C- fK' K)-1 A' - Q+B I B ~ + w;::: f hasastabzlzzmgsolutwn Q+> 0whzchsatzsfiestheconddwn andfurthermore,Q+::;S+.Also,thestateestzmator (3.4.1 ) (3.42) (3.4.3) (3.4.4) x(k + 1) A(I + f((Q+)-1 + C'V-1C- fK' K)-1 K' K)x(k) +A((Q+)-l + C'V-1C - fK' K)-IC'V-I(y(k) - Cx(k)) (3.4.5) hasthefollowmgproperty:Gwenanyb>0,thereexzstsamatrzx Q > 0 suchthatQ+::;Q 0suchthat Rzccatzequatwns(3.4.1)and(3.4.3)havestabzlzzzngsolutwnsS+> 0and Q+> 0satzsfymgcondztwns(3.4.2)and(3.4.4)andQ+::;Q. Proof Inordertoestablishthefirstpartof the theorem,notethat Lemma3.3.1 and the quadratic stability of the system(3.2.1)imphes that the matrix A isSchurstableandIIK(zI - A)-lB11I= 0such that Ilvf* K(zI - A)-1[v ~ BlW ~ ] 11=< 1 Therefore,Lemma3.3.4impliesthatforallfE(0, f*),thefollowingcon-ditionshold: (i)There existsamatrix5 > 0such that5-1 - fK' K> 0and A(5-1 - EK'K)-IA' - 5 +B I B ~ + W< O. f (3.4.6) 3.4Discrete-TimeOptimal Guaranteed Cost Filter Design45 (ii)The Riccatiequation ASA' - S+ EASK'(I - EKSK')-l KSA' +E I E ~ + W=0 E(3.4.7) hasastabilizing solutionS+2:0suchthat 1- EK S+ K' > O. Furthermore,sinceW>0,itfollowsfrom(3.4.7)thatS+>O.Hence, using(3.4.7)and the Matrix InversionLemma(Lemma 3.3.2). wecan con-cludethatS+isalsothestabilizingsolutiontoRiccatiequation(3.4.1). Moreover,since1- EK S+ K' > 0,itfollowsimmediately thatS+satisfies condition(3.4.2). NowletEE(0, E*)begivenandconsiderinequality(3.4.6).Associated with thisinequality isastate feedbackHOCcontrol problem definedby the system: x(k + 1)A'x(k) + JEK'w(k) + JEC'u(k); z(k) [ ...LE'1[ 01 t! 1x(k) +.E V ~ u(k). (3.4.8) Usinginequality(3.4.6)andLemma3.3.4,itisstraightforwardtoverify thatthecontrollawu(k)==0solvestheH=controlproblemassociated withthissystem.Hence,usingLemma3.3.5,itfollowsthatRiccatiequa-tion (3.4.3)has a stabilizing solution Q+> 0 such that (Q+)-l-EK' K> O. Also,bycomparingRiccatiequations(3.4.1)and(3.4.3),itisstraightfor-wardtoverifyusingthedifferentialgameinterpretationof theseRiccati equationsthat Q+:SS+;seeTheorem3.7of[13]. Wenowconsider the augmented system(3.2.4)obtained when the state estimator(3.4.5)isappliedtotheuncertainsystem(3.2.1).Inthiscase, thematrices ..4,B,BI,andKaregivenby: where [A(I - MC'V-IC) AMC'V-1C [ ~-AMC'V-! AMC'V-! [KK1 -EAMK'K] A (1+ EMK'K), ]:Bl=[~ ]; M~ (Q+)-l + C'V-1C- EK'K)-I. Withthese definitions,itisstraightforwardbuttedioustoverify that the matrix - ~ [Q+0] Q=0S+_Q+2:0 463.Discrete-TimeQuadratic GuaranteedCostFiltering satisfies the Riccatiequation AQA' - Q + fAQR' (I - fRQR') -1 RQA' ++ EE' = o. c(3.4.9) Furthermore, A + cQR' (I - cRQR') -1 R [A (I + Q+[C'V-1C- EK'KJ)-l =A((1 - cS+ K' K)-l - (I + Q+[C'V-1C- eK'K]) -1) - fS+K'K)-l]. However,thismatrixisSchurstablesinceQ+isthestabilizingsolution toRiccatiequation(3.4.3)andS+isthestabilizingsolutiontoRiccati equation(3.4.1).Thus,Q isthestabilizingsolutiontoRiccatiequation (3.4.9).UsingLemma 3.3.4,itnowfollowsthatA isSchurstableand Lemma 3.3.4 also implies that there exist matrices Q1>Q and V >0such that A (Ql1 - fR'R)-l A' - Q1++ EE' + V =0 c and Ql1- cR'R > O.ThusgivenanypE(0,1), A (Ql1 - fR'Rr1 A' - Q1++ BB' + pV< o. c ThereforeusingLemma3.3.4,itfollowsthat theRiccatiequation A - cR'R)-l A' - Qp++ EB' + pV=0 c(3.4.10) has astabilizing solution Qp> 0 such that Q;l - cR'R > O.Furthermore, usingtheMatrixInversionLemma(Lemma3.3.2),thisRiccatiequation canbere-writtenas AQpA'- Qp+ cAQpR'(I- cRQpR') -1 RQpA' 1- - -- -+ BB' + pV =O.(3.4.11) c Now,comparingRiccatiequations(3.4.9)and(3.4.11),itfollowsfrom Lemma3.3.3thatQp?::Q.Also,sincethestabilizingsolutiontoRiccati equation(3.4.10)isacontinuousfunctionoftheparameterp,itfollows 3.4Discrete-TimeOptimalGuaranteedCostFilter Design47 that limp-->oQp= Q.Hence,using(3.4.10)itfollowsthat givenanyr5> 0, there existsamatrix Qp> 0suchthat (3.4.12) (3.4.13) andQ ::;Qp < Q + r5I.Also,if weletQ > 0bedefinedasthe(1,1)block of the matrixQp,wemusthaveQ+::;Q 0 and hence I -EKIIK' >O. Therefore,II-1 - EK' K> O.Thus,usingLemma 3.3.4,itfollowsimmedi-ately that theRiccatiequation ASA' - S+ f.ASK'(1- f.KSK,)-lKSA' +B 1 B ~ + W=0 f. 3.4Discrete-TimeOptimalGuaranteedCostFilter Design49 hasastabilizingsolutionS+:::::0suchthat1- EKS+ K'>o.However, sinceW> 0,itfollowsthatS+> 0andhenceusingtheMatrix Inversion Lemma (Lemma 3.3.2),wecan conclude that S+isthe stabilizing solution toRiccatiequation(3.4.1)andcondition(3.4.2)issatisfied. Wenowreturn to(3.4.17)andpre-multiplybythe matrix andpost-multiplybythematrix It isstraightforward to verify that thisyieldstheinequality (A - BfC) - EK'Kr1 (A - BfG)' - 1--+ BfVBi + W< 0 E where Hence,E:::::Q.Also,wecanwrite [ (>-1=_Q-1Q'2212 andtherefore(3.4.16)implies Q-1] - 1222 Q-1+ Q-1Q' 1QQ-1 222212L.1222 (3.4.18) [ _cK' K - EK' K] - EK' KQ2l +EK' K> O. Taking the(1,1)blockof thisinequality,weconclude _EK'K >o. Usingthisfact,isstraightforwardto verifythat where (A - BfC) - EK' K) -1 (A - BfC), - E 1--+ BfVBi + W E + C'V-1C- EK' K)-l A' -++ W E TV=-Bi + - EK' K)-lC' + V] -1- EK' K)-l A'. 503.Discrete-Time Quadratic Guaranteed Cost Filtering Hence,(3.4.18)implies A(L:-1 + C'V-1C - fK'K)-lA' - L:+ I ~ + W::::o. f That is.there existsaZ~ 0suchthat A(L:-1 + C'V-1C - EK'K)-lA' - L:+ I ~ + W+ Z=O. f Wenowcompare thisRiccatiequation withRiccatiequation(3.4.3)using the differential game interpretation of these Riccati equations. Indeedusing thefactthatL:-1 - fK' K>0,itfollowsfromTheorem3.7of[13]that Riccati equation(3.4.3)hasasolution Q> 0suchthat condition(3.4.4)is satisfiedandQ 0,thereenstsamatrzx Q > 0suchthat Q< Q < Q + 5Iand thestate estzmatorisaquadratzcguaranteedcoststateestzmatorwzthcostmatrzx Q. HenceTheorem3.2.3zmplzesthatsteadystateerrorcovarzancematrzx satzsfiesthebound Q6.(k)::::Q < Q + 5I foralladmzsszbleuncertamtzesll(k).However,thzsboundholdsforall 5> O.Hence,wemusthave Qdk):::: Q foralladmzsszbleuncertamtzesll(k). Remarks Using the above results. wecan now see that the optimal quadratic guaran-teed coststate estimator can be obtained by choosingtheparameterf> 0 tominimizetr( Q+)whereQ+isthestabilizingsolutiontoRiccatiequa-tion(3.4.3).Thisoptimizationistobecarriedoutforfcontainedinthe 3.5IllustrativeExample51 interval(0, E*)over which Riccati equation(3.4.1)has astabilizing solution satisfyingcondition(3.4.2).Therequiredoptimalestimatoristhengiven by (3.4.5).Note however,that there isno guarantee that the minimum will actuallybe achieved. It isof interest to note that the equations defining the optimal quadratic guaranteed cost state estimator (3.4.5) can also be re-written in a'predictor-corrector"formasiscommonforthestandardKalman filter;e.g.,see[2]. Thispredictor-correctorformisasfollows: Predictor:x(k + 11k)=Ax(klk) Corrector: x(klk)=(I + EM K' K)x(klk - 1)+ MC'V-1(1I(k)- Cx(klk - 1)) whereM=((Q+)-l + C'V-IC - EK' K)-l. 3.5IllustrativeExample Inthissection,wepresentanexamplethatillustratestheapproachto discrete-time optimal guaranteed cost state estimation which has been pre-sentedinthischapter.Theexampleweconsiderisobtainedbydiscretiz-ingthecontinuous-timeexampleconsideredinSection2.5.Indeed,the continuous-timeuncertainsystemconsideredinSection2.5canberepre-sentedasintheblockdiagramshowninFigure3.5.1. ~ t )w(t) Continuous-time Nominalsystem z(t) y(t) v(t) FIGURE3.5.1.Continuous-timeuncertainsystem. Inthisuncertainsystem,thenominalsystemisdescribedbythe state equations XAx + BIt. + w, zKx, yCx +v, 523.Discrete-Time QuadraticGuaranteedCost Filtering where A [!l],C=[l-1], Bl =[ ],Dl =0, K [05],W=[ 100 ],v =1. 0 Here E{[ :][WI u'] }=[: and=where :::;1.We nowapproximate this continuous-time un-certain system with adiscrete-time uncertain system with sampling period h= 0.028asshowninFigure3.5.2. f,(kh) ZOH w(t) f,( t) A Nominal System z(kh) z(t) Sample y(t)y(kh) I;Sample v(t) FIGURE3.5.2.Approximatediscrete-timeuncertainsystem. Thisapproximateuncertainsystem willbe describedby state equations of theform x(k + 1) y(k) where(e.g.,see[7]) C (.4.+ + w(k); Cx(k) + v(k) Ah=[0.98020.0196]. e00.9802' [1- 1];K=[05];V= 1. 3.5IllustrativeExample53 Also, where(e.g.,see[7])Wisthesolutiontothe Lyapunovequation AW + WA' = eAhWeA'h- W. Inthisexample,weobtain W =[1.960800195] 0.019519605. Furthermore.lJ.(k)isascalaruncertainparametersubjecttothebound 1lJ.(k)1:s:1.Tofindthe optimal quadratic guaranteed coststate estimator, wesolve Riccati equations(3.4.1)and(3.4 3)foraseriesof values of E> 0 andplottr(Q+)versusEThisplotisshowninFigure3.53Fromthis 20,-,--,,----,-----,-----,-----,-----,-----,-----, "0 c: 18 16 14 :::l12 o .c e 10 lii Q) Ca8 :::l g c:6 t1l Q) ::!:4 o L - - - ~ L - - - ~ - - - - ~ ~ - - ~ - - - - ~ - - - - ~ - - - - ~ - - - ~ .o05152253354 Parameter Ex 10-' FIGURE3.5.3.tr(Q+)versusEo plot.wedetermine the optimal valueof Eto be E=5.98X10-6.With this valueof E,therequiredstabilizingsolutionsto Riccati equationsare S+=103 [1. 77370.0441]0 x0.04410.0662> 543.Discrete-Time QuadraticGuaranteedCostFiltering Q+=[134.9989 65.2477 65.2477J 65.9187> O. Then,usingequation(3.4.5),weobtain theoptimalquadratic guaranteed coststate estimator -(k1)=[0.3301 x+0.0063 0.6796J -(k)[0.6501](k) 0.9837x+-0.0063Y. Toillustratetherobustfilteringpropertiesofthisquadraticguaranteed coststateestimator,Figure3.5.4showsaplotofthemeansquareerror asafunctionof theuncertainparameter~ From thisplot,itcanbe seen that forall admissible values of the uncertain parameter, the corresponding meansquareerrorislessthanthecalculatedquadraticguaranteedcost bound.Notethatinthisplot,wehaveonlyconsideredconstantvaluesof the uncertain parameter ~ However,the quadratic guaranteed costbound alsoholdsfortime-varyinguncertainparameters.If uncertainparameter wasallowedtobetime-varying,onewouldexpectthemeansquareerror boundtobesomewhatlessconservative. Forthe sakeof comparison,Figure3.5.4alsoshowsaplotof themean squareerrorwhichwouldbe obtainedif astandardKalmanFilter(based on thenominalsystemwithoutuncertainty)wereapplied.Fromthisplot, wecanseethatintheworstcase,therobustfilterissignificantlybetter than the standardKalmanFilter. 3.5IllustrativeExample55 400 300 250 8 E 200- -'" 0-150 '" c: --- Robust Filter - - Kalman Filter - - - Quadratic CostBound , / / ;I _.L / / / / / I / I / ____________________________50 __ Uncertain Parameter ll. FIGURE 3.5.4.Mean square estimation error versus the uncert8.in parameter fl.. 4 Continuous-TimeSet-ValuedState EstimationandModelValidation 4.1Introduction Inthepreviouschapters,theproblemof robuststateestimationwasad-dressedbyextendingthe standard steady stateKalmanFilter tothecase inwhichtheunderlyingsignalmodelisanuncertainsystem.Theuncer-tainsystemsbeingconsideredwereuncertain systemswithnormbounded uncertaintyandsubjecttostochasticwhitenoisedisturbances.Inpartic-ular.therobustKalmanFiltersofChapters2and3areconcernedwith constructingastateestimatorwhichboundsthemeansquareestimation error.However,theseresultsmaybeconservativeinthatonlyanupper bound isobtained forthe mean square estimation error.Also,theseresults donotextendinastraightforwardwaytothecaseof finitetimehorizon state estimation problemsor robuststate estimationproblemswith struc-tureduncertainty. A somewhat differentapproach to the robust state estimation problem is the approach of[120].Reference[120]builds on the deterministicinterpre-tationof KalmanFilteringpresentedin[16].Thisdeterministicapproach toKalmanFilteringalsoformsthelaunchingpointfortheresultsof this chapterandthe remainingchaptersof thebook. In[16],thefollowingdeterministicstateestimationproblemisconsid-ered:Givenoutputmeasurementsfromatime-varyinglinearsystemwith noiseinputs subjecttoanL2normbound,findthe set of allstates consis-tent with these measurements.Such aproblem isreferred to asaset-valued stateestzmatwnproblem.Thesolutiontothisproblemwasfoundtobe 584.Continuous-TimeSet-ValuedStateEstimation anellipsoidinstate space whichisdefinedbythestandardKalmanFilter equations.Thus,theresults of [16]giveanalternative interpretation of the standard Kalman Filter. In [57],an attempt wasmade to extend the results of[16]tothecaseofuncertainsystemscontainingnormboundeduncer-tainties.Themainresultof[57]leadstotheconstructionofasetwhich over boundsthe true setof possiblestates.Thissetisalsoconstructedvia KalmanFilterlikeequations.However.becausethesetof statesobtained usingthemethodsof[57]isonlyanoverboundof thetruesetof possible states, the results may be conservative(and no indication isgivenas to the degreeof conservatism). Theresultspresentedinthischapteroriginallyappearedinthepa-pers[120,121,126,127].Inparticular,weconsideranapproachtorobust set-valuedstateestimationwhichbuildsontheresultsof[16].However following[120],weconsideradifferentclassof uncertainsystemstothat consideredin[57].Theclassof time-varyinguncertainsystemsconsidered inthischaptercontainuncertaintywhichisdefinedbyacertainIntegral QuadraticConstraint(IQC);e.g.,see[165-168].Thisclassofuncertain systems originated in the work of Yakubovichandisaparticularly richun-certainty classallowingfornonlinear,time-varying,dynamicuncertainties. Furthermore,anumber of newrobustcontrol system designmethodologies have recently been developedforuncertain systems with integral quadratic constraints;e.g.,see[115,117,119,123,136].Theintegralquadraticcon-straintapproachtorobuststate estimationconsideredinthischapterex-tendsthe resultsof [16]toincludeuncertainty of thistype.Aninteresting featureof thisset-valuedstateestimationapproachisthat italsoleadsto thesolutiontoacertainproblemof modelvalidation.Thismodelvalida-tionproblemisconcernedwithdeterminingif anuncertainsystemmodel isconsistentwithagivensetof input - outputmeasurements. Theproblemof modelvalidationforuncertainsystemsismotivatedby theproblemofsystemidentificationforuncertainsystemsinwhichan uncertain systemmodelistobe constructedon thebasisof acollectionof measured input - output data.An integral part of any system identification processismodelvalidation.Modelvalidationisaprocessofdetermining if giveninput - outputmeasurementsarecompatiblewithagivenmodel. Model validationcan actually onlydetermineif agivenmodelisinvalidin that it isincompatible with agivendata set.Inpractice,model validation canbeusedtochoosebetweenacollectionofmodelsandtodetermine boundsonuncertainparameters. Thesignificantadvancesbeingmadeinthefieldof robustcontrolthe-oryhavemotivatedanumberofauthorstostudythemodelvalidation problemforuncertainsystems:e.g.,see[107,143,144,171].Theapproach of[143,144]involvesanuncertainsystemwithstructureduncertaintyof thetypewhicharisesintheJ-Lframework.Forthisclassof uncertainsys-tems,[143,144]appliesafrequency domainapproachtoconvertthemodel validation problem intoaJ-Lproblem which can be solvedvia numerical op-4.2 Model Validation and Set-Valued State Estimation Problem Statements59 timization techniques.The approach of [107]involvesaclass of single input singleoutputdiscrete-timeuncertainsystemswithnormboundeduncer-tainty.Forthisclassofuncertainsystems.reference[107Jappliesatime domainapproachtoconvertthemodelvalidationproblemintoaconvex optimization problem.Asimilarapproachisalsoconsideredin[171 J. Themainresultspresentedinthischapterrelatetotheproblemsof modelvalidationandrobuststateestimationforcontinuoustimeuncer-tain systemsin which the uncertainty isdescribedby an integral quadratic constraint.Theseresultswereoriginallypresentedinthepapers[118.120, 121,126.127, 135J.The resultsof thischapter involvethe solutionof aRic-catidifferentialequationandthesolutionof asetof filteringstateequa-tions.This isin contrast to the existing model validation results mentioned abovewhicharebasedonconvexoptimization.Inthissense.theresults of thischapteraremorecloselyrelatedtoresultsonobserverbasedfault detection:e.g ..see[92J.However,thefilterstateequationsarisinginthis chapterare not standardKalman Filter state equations.Instead they take theformoftherelatedrobuststateestimator.Thismayleadtoacon-siderablecomputationaladvantageascomparedtoexistingtechniquesfor modelvalidation. Inadditiontothecomputationaladvantagesmentionedabove.ourap-proach to model validation has anumber of other advantagesoverexisting methods.The firstpointtomentionisthat ourapproachdoesnotrequire zeroinitialconditionsontheplantfromwhich the data ismeasured.Also. incontrastto[143,144],ourapproachrequiresinput- outputdatade-finedonlyoverafinitetimeinterval.Furthermore.theintegralquadratic constraintuncertaintydescriptionconsideredinourapproachallowsfor nonlineartime-varyinguncertainties. Therobuststateestimationresultspresentedinthischaptermaybe preferableoverthoseof Chapter2insituationsinwhichafinitehorizon state estimationproblemisto be solved.Also,the IQCuncertainty model consideredin this chapter allowsforaricherclass of uncertainties than the normboundeduncertainty modelsconsideredinChapters2and3. 4.2ModelValidationandSet-ValuedState EstimationProblem Statements Considerthe time-varyinguncertainsystem: x( t) z( t) y(t) A(t)x(t) + Bl(t)W(t) + B2(t)U(t); K(t)x(t) + G(t)u(t): C(t)x(t) + v(t)(4.2.1) wherex(t)ERnisthestate.w(t)ERPandv(t)ERlare theuncertainty inputs.u(t)ERhisaknowncontrolmput.z(t)ERqistheuncertainty 604Contmuous-TimeSet-Valued StateEstImatIOn outputandy(t)ERlISthemeasuredoutput,A(),BI (), B2(), K(), G() andC( ) areboundedplece'WlsecontInUOUSmatnx functIOns SystemUncertamty TheuncertaIntyIntheabovesystemISdescnbedbyanequationof the form [ w(t)] v(t)= (t,x()) wherethefolloWIngIntegralQuadraticConstraIntISsatisfiedLetN= N>0beagivenmatnx,XoERnbeagivenvector,d>0beagiven constant,Q()=Q( )'andR()=R()'begivenboundedpiecewisecon-tInUOUSmatnx weightIng functIOnssatisfYIng the followmgconditIOnthere eXistsaconstantb > 0 such that Q(t) M,R(t) Mforall tForagiven fimtetimeInterval[0,s],weWillconsidertheuncertamtyInputsw()and v()andImtIalconditionsx(O)suchthat (x(O)- xo)' N(x(O)- xo)+ 18 (w(t)'Q(t)w(t) + v(t)' R(t)v(t))dt :::::d + 18 Ilz(t)112dt(422) HereIIIIdenotesthe standardEuchdeannorm Theuncertamty Inthe uncertaIn system(42 1),(422)canberegarded as a feed back mterconnectIOn between the nomInal lInear system (421)and anuncertamtyblockwhichtakestheuncertaIntyoutputzandproduces the uncertaIntyInputswand vThiSISIllustratedmFigure4 2 1 Uncertamty Nommal System z FIGURE42 1BlockdiagramrepresentatIonof uncertamsystem NotethattheaboveuncertaIntydescnptIOnallowsforuncertaIntiesIn 'WhichtheuncertaIntymputsw()andv()dependdynamicallyonthe uncertaInty outputz()In thiScase,the constant dmaybe Interpretedas 4.2 Model Validation and Set-Valued State Estimation Problem Statements61 ameasureof theSIzeof theinitialconditionsonthenominalsystemand uncertaintydynamics. It isclear that the uncertain system (4.2.1),(4.22)allows foruncertainty satisfyingastandardnormboundconstraint.InthIScase.theuncertain system wouldbedescribedby thestate equations x( t) y(t) [A(t)+ BI(t)6.I(t)K(t)]x(t) +[B2(t) + Bdt)6.I(t)G(t)]u(t); [C(t)+ 6.2(t)K(t)]x(t)+ 6.2(t)G(t)u(t); 11[6.I(t)'6.2 (t)'] II:::;1(4.2.3) where6.1(t)and6.2(t)aretheuncertaintymatricesand1111denotesthe standardinducedmatrixnorm.Also.theinitialconditionswouldbere-quiredto satisfy the inequality (x(O)- xo)' N(x(O)- xo):::;d. Toverifythatsuchuncertaintyisadmissiblefortheuncertainsystem (4.2.1),(422),letw(t)= 6.1(t)z(t).v(t)=6.2(t)z(t)where 11[6.I(t)'6.2(t)']11:::;1 foralltE[0, T].Thencondition(4.2.2)issatisfiedwithQ(.) R()==I. Notatzon Iand Letu(t)=uo(t)beafixedcontrolinputandy(t)=yo(t)beafixed measuredoutputoftheuncertainsystem(4.2.1),andletthefinitetime interval[0.s]be given.Then, denotesthe setof allpossiblestatesx( s)attimesfortheuncertainsys-tem(4.2.1)withuncertaintyinputsandinitialconditionssatisfyingthe constraint(4.2.2) Definition4.2.1Theunceriam system (4.2.1),(4.2.2)zssazd tobestrict-ly verifiableon[0. T],if foranyvector XoERn,anytzmesE(0. T],any constant d> 0,any fixedcontrol mput u(t)= uo(t)andany fixedmeasured outputy(t)=yo(t).theset Xs[xo, d]zsbounded. Definition4.2.2LetXoERn andd>0begwen.Also,letuo(-)and Yo(-)begwenvector functzonsdefinedover a gwentzmemterval [0.s].The input- outputpazr[uo(-). Yo(-)]zssazdtoberealizablewzthparameters Xoanddif thereexzst[xU, w(),v()]satzsfyingcondztzons(4.2.1),(4.2.2) wzthu(t)=uo(t)andy(t)=yo(t). 624.Continuous-Time Set-ValuedState Estimation ModelValidationProblem We consider the following problem: Given an input - output pair [uoe) , YoO], determine if thispairisrealizableforthe uncertain system(4.2.1),(4.2.2). Remarks Therestrictionof strictverifiabilityasdefinedaboveisareasonableone since if itdoesnothold,then it isunlikely that any reasonable conclusions can be obtained about the system model from measured data. In particular, notethatwithinourintegralquadraticconstraintframework,if thestrict verifiability condition isnot satisfied, then the model may be such that any pairUo (-), Yo (-)ispossible. Set- ValuedStateEstimationProblem The main results of this chapter alsoconcern the followingset-valued state estimationproblem:Givenaninput- outputpair[uaU. yoU]'forany SE[0, Tj,findthe set of allpossible states xes)at timesforthe system(4.2.1)withuncertainty inputsandinitialconditionssatisfying theconstraint(4.2.2). Remarks Inthesequel.wewillshowthatthesolutiontothisset-valuedstate esti-mationproblem leadsto asetXs[xo, d]which isan ellipsoid in Rn. If apoint-valuedstate estimate isrequiredrather than aset-valued stateestimate,thenthecenterof thisellipsoidcanbeusedtoprovidea point-valuedstate estimate. Theconnectionbetweentheaboveset-valuedstate estimationproblem and the model yalidation problem canbe seen fromhe followingsimple ob-servation.If foragiveninput-outputpair[uo('),Yo(')]'thecorresponding setof possiblestates isempty,the givenmeasure-mentsmustbe incompatiblewith theuncertain systemmodel.Hence,the uncertain systemmodelhasbeen invalidated. 4.3Designof Set-ValuedState Estimator The solution to the abovemodel validationand set-valued state estimation problemsinvolvethe followingRiccatidifferentialequation: Pet) B1(t)Q(t)-1 Bl(t)' + A(t)P(t) + P(t)A(t)' +P(t)[K(t)' K(t) - G(t)' R(t)G(t)]P(t).(4.3.1) 4.3Designof Set-Valued StateEstimator63 Also,weconsiderthefilterstate equations 5:(t)=[A(t)+ P(t)[K(t)' K(t) - G(t)' R(t)G(t)]] x(t) +P(t)G(t)' R(t)Yo(t)+ [P(t)K(t)'G(t) + B2(t)]UO(t). (4.3.2) The followingtheorem givesasolution toboth the set-valued state esti-mationproblemandthe modelvalidationproblem. Theorem4.3.1Let N=N'> 0beagwenmatnx,and QU =QU'and RU=R(o)'begwenmatnx functwnssuchthatthecondztwnQ(t)::::8I andR(t)::::8Iholdsonthetzmewterval[O,Tj.Cons2dertheuncertaw system(4.2.1),(4.2.2).Thenthefollowwgstatementshold: M Thesystem(4.2.1),(4.2.2)2Sstrzctly venfiableon[0. T]2f and only 2f thesolutwnP()totheR2ccatzequatwn(43.1)wzthzmtzalcondztwn P(O)=N-1 2Sdefinedandposdwe-defimteonthewterval[0. T]. (n)Supposethesystem (4.2.1),(4.2.2)zsstnctly venfiableon [0, T].Also, let8E(0, TjbegwenandletXoERn beagwenvector,d>0 beagwenconstant,anduo(t)andyo(t)begwenvectorfunctwns definedon[0,8].Then,thepmr[uo(),Yo(')]28reahzablezf andonly zf Ps[uoU. Yo(')]::::-d where Ps[uo(),yoU] 61' [[[CK(t)x(t)+ G(t)uo(t))[[2] =0-(C(t)x(t) - yo(t))'R(t)(G(t)x(t)- yo(t))dt ( 4.3.3) and x()2Sdefined by theequatwn(4.3.2)wzth wztwl cond2twn X(O)= Xo (iii)If thesystem(4.2.1),(4.2.2)zsstnctlyvenfiable,then (4.3.4) forallsE [0, T]. Proof (z)NecessztyInthiscase.wemustestablishtheexistenceofapositive-definite solution to Riccati equation(4.3.1).This willbe achievedby show-ing that the cost function in acorresponding linear quadratic optimal con-trolproblem isboundedfrombelow. 644.Continuous-Time Set-ValuedState Estimation FirstletsE(0, T]begivenandconsider uncertainsystem(4.2.1), (4.2.2)defined on[0,s].Givena pairluoU. YoU]'wehaveby the definition of Xs[xo,d].that XsEXs[xo, uoC)YOnd] if and only if there exist vector functionsx(), w()andv()satisfying equa-tion(4.2.1)andsuchthatx(s)=xs,thecomltraint(4.2.2)holds.and yo(t)= C(t)x(t)+ 1!(t) ( 4.3.5) foralltE[0,s].Substitutionof(4.3.5)into(J.2.2)impliesthat XsEXs[xo, d] if andonlyif thereexistsanuncertaintyinputwe)EL2 [0,s]suchthat J[xs, wC)]::;d whereJ[xs,w(-)]isdefinedby J[xs, wU] (x(O)- xo)' N(x(O)- xo) ( 4.3.6) +t( w(t)'Q(t)w(t)- 11(j{(t)x(t) + G(t)uo(t))112)dt Jo +(Yo(t)- C(t)x(t))' R(t)(Yo(t)- C(t)x(t)) (4.3.7) andxC)isthe solution to (4.2.1)with uncertainty input we) andboundary conditionx( s)= Xs' Nowconsider the functional(4.3.7)with Xo= 0,uo(-)==and Yo(-)==0. In this case,Jisahomogeneousquadratic functionalwithaterminalcost term.Also,consider the set Xs [0, 0. 0,1]corre@ponding to Xo= 0,uo(-)==0, Yo e)==and d=1.SinceXs [0,0,0. 1]isbounded.there existsaconstant hs> suchthatallvectorsXsERn withIlxsll=hsdonotbelongto the setX.[O,0. 0.1].Hence, ( 4.3.8) forallXsERnsuchthatIlxs II=hsandforallwe)EL2 [O,sJ.Since.Jis ahomogeneousquadraticfunctional,wehave J[axs, aw(.)]=a2 J[xs' we)] and(4.3.8)impliesthat infJ[xs, w()]> w()EL2 [O,sJ ( 4.3.9) forallsE[0, T]and allXs"I- 0.Note that in the aboveinfimum.the vector Xs"I- isfixedandusing(4.3.8).thisenablesustoconcludethestrict inequality. 4.3Designof Set-ValuedStateEstimator65 The optimization problem (4.3.9)with anunconstrained terminal condi-tionxes)iandsubjecttotheconstraintdefinedbythesystem(4.2.1) isalinearquadratic optimal controlprobleminwhichtimeisreversed.In thislinearquadraticoptimalcontrolproblem,asignindefinitequadratic costfunctionisbeingconsidered.Wenowuseaknownresultfromlin-earquadraticoptimalcontroltheorywhichstatesthatiftheinfimumof thecostfunctionisstrictlypositiveforallterminalconditions,then there existsasolutiontothecorrespondingRiccatiequation.Furthermore,the terminalvalueof theRiccatisolutionispositive-definite;e.g.,seepage23 of[30].Thus,weconcludethatcondition(4.3.9)impliesthatthereexists asolutionX(.)to theRiccati equation - Xes)K(s)' K(s)- C(s)' R(s)C(s) +X(s)A(s) + A(s)' Xes)+ X(s)B(s)Q(s)-l B(s)' Xes) (4.3.10) withinitialconditionX(O)=N.Furthermore,since(4.3.9)holdsforany terminaltimesE[0, TJ,thissolutionispositive-definiteon[0, T].From this,itfollowsthattherequiredsolutiontoRiccatiequation(4.3.1)is givenbyP (.) XU -1.Notethatinordertoestablishtheexistenceof asuitablesolutiontoRiccatiequation(4.3.10),itwasonlynecessaryto considerthecaseXo=0,uoU==0,yoU==andd=1.Thiscompletes theproof of thispartof thetheorem. (i)Sufficiency Foragiven time interval[0,s],wehave shownabovethat ifandonlyifthereexistsanuncertaintyinputwUEL2[0,s]suchthat condition (4.3.6)holds for the functional(4.3.7).Now consider the following minimizationproblem infJ[xs, w( .)] w(-)EL2[O,s] (4.3.11) wheretheinfimumistakenoverallxC)andwC)connectedby(4.2.1) with the boundary condition xes)=Xs'Thisproblemisalinearquadratic optimal trackingprobleminwhich the systemoperates inreversetime. \Vewishtoconvertthe abovetrackingproblemintoatrackingproblem of the formconsideredin[76]and[16].First defineXl(t)to be the solution to the state equations (4.3.12) Nowlet x(t) x(t) - Xl(t). 664.Continuous-TimeSet-ValuedStateEstimation Then,it followsfrom(4.2.1)and(4.3.12)that x(t)satisfies the state equa-tions xCt)=ACt)x(t) + Bl(t)W(t)( 4.3.13) where X(O)=x(O).Furthermore, the cost function(4.3.7)can be re-written as J[xs, w()]i[xs, w(-)] (x(O)- xo)' N(x(O)- xo) t( w(t)'Q(t)w(t) + v(t)' R(t)v(t)) + Jo-11(K(t)[x(t) + Xl(t)]+ G(t)uO(t))1I2dt (4.3.14) where xes)=Xs=Xs- Xl(S)and vet) Yo(t)- C(t)[x(t)+ Xl(t)].(4.3.15) Equations(4.3.13)and(4.3.14)nowdefineatrackingproblemof the form consideredin[76]whereYo (.),uo (-)andXl (.)arealltreatedasreference inputs.In fact,theonlydifferencebetweenthistrackingproblemandthe tracking problem consideredin the proof of the result of [16]isthat in this theorem,wehaveasignindefinite quadraticcostfunction. Thesolutiontothistrackingproblemiswellknown(e.g.,see[76]).In-deed,if the Riccati equation(4.3.1)hasapositive-definite solutiondefined on[0, T]withinitialcondition P(O)=Xo\ then thematrix function XC)=p(-)-l> 0 isthe solution to Riccati equation (4.3.10)with initial condition X(O)= N. Fromthis,itfollowsthat theinfimumin infi[xs, w(-)] w(-)EL2[O,S] willbe achievedforanyXo,uoC)and Yo(-).Furthermore as in[16]'wecan write mini[xs, w(-)] w(')EL 2[O,S] = (xs- :Tl(S))'X(s)(xs - :Tl(S))- Ps[uo(')' Yo()](4.3.16) where [(.)(.)] t(II(K(t)[Xl(t)+ :TI(t)]+ G(t)uo(t)) 112)dt PsUo,Yo- Jo -v(t)' R(t)v(t) 4 3DeSIgnof Set-Valued State Estimator67 andXl (s)ISthe solutIOntostate equatIOns Xl(S)=[A(s)+ P(s)[K(s)' K(s)- C(s)' R(s)C(s)]] Xl(S) +P(s)[K(s)' K(s)- C(s)' R(s)C(S)]Xl(S) +P(s)C(s)' R(s)yo(s) + P(s)K(s)'G(s)uo(s)(4317) withinitial conditionXl (0)=XoNowlet Using the fact that Xs=Xs-Xl (s).It followsthat (4.3.16) can be re-written as mfJ[xs,w()]=(xs- x(s))'X(s)(xs- xes))- Ps[uo(),yo(-)J w()EL2[Os] where t>t[ II(K(t)x(t) + G(t)uo(t))112] Ps[uo(), Yo()]=Jo-(C(t)x(t) - yo(t))' R(t)(C(t)x(t) _yo(t))dt andx( s)ISthesolutIOntostateequatIOns(43 2)WIthmltIalcondItIOn X(O)=XoFrom thISwecan concludethat the setXs[xo, uo(Yo(d] ISgIvenby Xs[xo, uo( Yo(d] {xsERnmmJ[xs, w()]::;d} w()EL2[Os] { XsERn} =(xs- x(s)),p(s)-I(xs - xes)) ::;d + Ps [uo ( ), Yo ( )] (4.318) ThIScompletestheproof of thISpart of the theorem (n)It ISclearthatthepair[uo(),Yo()]definedonanmterval[O,s] IS realIzableIfandonlyIf the setXs[xo,uo( not emptySmce thesystem(42 1),(42 2)ISstnctly venfiable,Itnowfollowsthat theset ISdefinedby(4318)Hence,the setXs[xo, uo(Yo(d]ls notempty If and onlyIfPs[uo(), Yo()]2-dThIScompletes the proof of thISpart of the theorem (m)In the proof of thepart(I)of the theorem,wehavealreadyproved thatthe set Xs[xo,uo(ISdescnbedby(43 18)ThIScompletes the proof of the theorem0 Remarks Notethat theonlydifferencebetweenthetrackingproblemconsideredin the proof of the above theorem andthe tracking problem consideredin the proof of theresultof [16]isthatinthischapter,wehaveasignindefinite quadraticcostfunction. 4.4TimeInvariantSet-ValuedStateEstimation We nowconsidera time-invariant version of the set-valued state estimation problem.Inthisproblem,weconsiderthefollowingtime-invariantuncer-tain system definedontheinfinitetimeinterval[0,00): i( t) z(t) y(t) Ax(t) + BIW(t) + B2u(t); Kx(t); Gx(t) + v(t)(4.4.1) wherex(t)ERnisthestate.w(t)ERPandv(t)ERlaretheuncertainty inputs,u(t)ERhisthecontrolinput,z(t)ERqistheuncertaintyoutput andy(t)ERlisthemeasuredoutput. LetQ=Q'> 0andR=R'> 0begivenmatricesassociatedwiththe system(4.4.1).Then theuncertainty inputsandinitialconditionsforthis uncertainsystemarerequiredtosatisfythefollowingintegralquadratic constraint: (x(O)- xo)' N(x(O)- xo)+ 18 (w(t)'Qw(t) + v(t)' Rv(t))dt ::;d + is Ilz(t)112dt.(4.4.2) Theorem4.4.1Consider theuncertain system {4.4.1}with weighting ma-tricesQ=Q'>0andR=R'>0andsupposethatthepair(A, B)is stabilizable.If thealgebraicRiccatiequatwn AP + PA' + P[K'K - G'RG]P += 0 ( 4.4.3) hasasolutwnP>0suchthatthematrix[A'- [G'RC - K'K]P]is stable.thenforanymatrixN=N'>0suchthatN-1 ::;P,theset of allpossiblestatesx(s)attimescorresponding toa input-outputpair[uo(-,yoU]isdescribedby { XsERn :} Xs[xo, d]=(xs- x(s))' P(s)-l(xs - x(s)) ::;d + Ps[uoO,yoU] 4.4TimeInvariantSet-ValuedState Estimation69 whereP()zsthesolutwnof theRzccatzdzjjerentwlequatwn p(t) =AP(t) + P(t)A' + P(t)[K' K- C'RClP(t) + B1Q-l~with zmtial condztwn P(O)=N-l, x()zsthesolutwn tothestateequations .i:( t)[A+ P(t)[K' K- C'RCll x(t) +P(t)C'RYo(t) + [P(t)KG + B2l uo(t) wzthzmtzalcondztwnx(O)=Xoand Ps [uoO Yo 0] ~ t[ II(Kx(t)+ Guo(t))112- ]dt io(Cx(t)- yo(t))'R(Cx(t)- yo(t)). ( 4.4.4) Moreover,PC)zsdefinedon[0,00)andhasthepropertyP(s)-+Pas s-+ 00. Proof Thefirstpartof thetheoremfollowsdirectlyfromTheorem4.3.1.Hence, tocomplete the proof of thistheorem,it remains only to show thatP()is definedon[0.00)andhastheproperty P (s)-+Pass --->00. UsingthefactthatP >0isasolutiontoRiccatiequation(4.4.3),it followsthatP alsosatisfiestheRiccatiequation AP + P A' + P K' K P+ Q =0 whereA=A- PC'RCandQ =PC' RC P+ BQ-l B'~ O.Furthermore. sincethematrix[A'- [C'RC - K'K]P]isstable.itfollowsthat thema-trixA + PK' Kisstable.Fromthis,wecanusetheStrictBoundedReal Lemma(Lemma2.3.4)toconcludethatthematrixA=A- PC'RCis stable.Hence,thepair(C, A)mustbe detectable. Nowusing the factthat the pair (A, B) isstabilizable and the pair(C, A) isdetectable, it followsfrom Theorem 4.1of [159]and the remarks following that theorem, that PC) isdefined on [0,00)and has the property P( s)--->P ass--->00.D Remark Theabovetheoremshowsthatforatime-invariantuncertainsystemde-finedoveraninfinitetimeinterval,thestateestimator(4.4.4)converges asymptotically to atime-invariant state estimator.Furthermore, this time-invariant state estimator can be constructed via the solution to an algebraic Riccatiequation. 5 Discrete-Time Set-ValuedState Estimation 5.1Introduction Inthischapter,weconsider theproblem of set-valuedstate estimation for uncertain discrete-time systems.Asin Chapter 4.the starting point forour approachisthedeterministicinterpretationof thediscrete-timeKalman Filtergivenin[16].In[16].theKalmanFilterisshowntogiveastate estimateintheformofanellipsoidalsetofallpossiblestatesconsistent with the given process measurements andadeterministic description of the noise.Weextendtheapproachof[16]toconsiderdiscrete-timeuncertain process models whichhaveadeterministicdescription of the noiseandun-certainty.Thisuncertainty descriptionisreferredto astheSumQuadratic Constraint(SQC)uncertaintydescriptionandisthediscrete-timeversion oftheIQCuncertaintydescriptionconsideredinChapter4.Asforthe IQCuncertaintydescription,theSQCuncertaintydescriptionallowsfor alargeclassof nonlinear,dynamicuncertainties.Furthermore,ourmain resultshowsthatforthisuncertainty description,ourrobustfiltercanbe usedto determine if the assumed model isconsistent with the given output measurements.Suchmodelvalidationresultscannotbeobtainedwitha stochasticdescriptionof noiseanduncertainty. Oneofthemainresultsof thischapterisaset-valuedstateestimator whichgivesastate estimate in the formof an ellipsoidalsetof allpossible states consistent with the givenprocessmeasurementsandour determinis-ticdescriptionof thenoiseanduncertainty.Inparticular,thisresultgives an exact characterization of this set.Asin the standard Kalman Filter, our 725.Discrete-TimeSet-ValuedStateEstimation stateestimateisconstructedrecursivelyfromtheoutputmeasurements. However,theformof ourstateestimatorequationsissomewhatdifferent fromthestandardKalmanFilterequations.Theresultondiscrete-time set-valuedstateestimationpresentedinthischapterwasoriginallypre-sentedinthepapers[122,139].Someotherresultsonset-valuedrobust discrete-timefilteringcanbe foundin[86]. Aswellasconsidering the standard set-valuedstate estimationproblem fordiscrete-time uncertain systems, this chapter alsoconsiders aset-valued stateestimationproblemforuncertaindiscrete-timesystemsforthecase in whichsomeof themeasureddata pointsaremissing.Inmanypractical filteringproblems,thereisapossibility that someof the observationdata maybemissing.Thisproblemof missingdata mayarisefromtemporary sensorfailureor congestionof the communicationsnetwork connecting the sensorstotheprocessor.ThestandardKalmanFilteringproblemwith missingmeasurementdatahasbeenconsideredbyanumberofprevious authors:e.g.,see[27.91].Inthischapter,wealsoconsideraproblemof robustKalmanFiltering whichallowsformissingmeasurementdata.This result on discrete-time set-valued state estimation with missing data which ispresentedinthischapter originallyappearedinthepapers[130,137]. 5.2SumQuadraticConstraintUncertainty Description LetTE{1. 2. 3,... }.Considerthetime-varyinguncertaindiscrete-time systemdefinedfork= 0.1. .... T: x(k + 1) z(k) y(k) A(k)x(k) + Bdk)w(k); K(k)x(k); C(k)x(k) + v(k)(5.2.1) where x(k)ERn isthestate,w(k)ERPand v(k)E Rl are the uncertainty inputs,z(k)ERqistheuncertaintyoutput,y(k)ERlisthemeasured output,andA(k). Bl (k), K(k)andC(k)are givenmatricessuch that A(k) isnon-singularfork= 0.1, .... T. SystemUncertainty Theuncertaintyintheabovesystemisdescribedbyanequationofthe form: [ w(k)]_ v(k)- q;(k.x()) wherethe followingSumQuadraticConstraintissatisfied.Let N=N'>O 5.2SumQuadraticConstraintUncertaintyDescription73 beagivenmatrix,XoERnbeagivenvector,d> 0beagivenconstant, andQ(k)andR(k)begivenpositive-definitesymmetricmatricesdefined fork=0,L..., T.Then wewillconsidertheullcertaintyinputsw(-)and v()andinitialconditionsx(O)suchthatthefollowingSumQuadratic Constraintissatisfied: (x(O)- xoY N(x(O)- xo) T-l + L (w(k)'Q(k)w(k) + tI(k + l}'R(k + l)t1(k + 1)) k=O T-l d + L Ilz(k + 1)112. (5.2.2) k=O AsinChapter4,thisuncertainty descriptioncanberepresentedby the blockdiagramshowninFigure4.2.1.Also,asinChapter4,theuncertain system(5.2.1),(5.2.2)allowsforuncertaintysatisfyingastandardnorm boundconstraint.Forexample,consideranuncertain system describedby the state equations x(k + 1) y(k) [A(k) + Bll (k)6.(k)K(k)]x(k) + Bdk)nl (k); C(k)x(k) + n2(k); II::;1(5.2.3) where6.(k)isthe uncertainty matrix, nl(k) and n2(k)are noise sequences, B 1 (k)=[B 11 (k)B 12 (k)],andII. IIdenotesthestandardinducedmatrix norm.Alsosuppose,initialconditionsandnoisesequencessatisfythein-equality (x(O)- xo)' N(x(O)- xo)+ x(O)' K(O)'Q(O)K(O)x(O) T-lT + L nl (k)'Q(k)nl (k)+ L n2(k)' R(k)n2(k)::;d. k=Ok=1 Toverifythatsuchuncertaintyisadmissiblefortheuncertainsystem (5.2.1),(5.2.2),let w(k)=[ ] fork =0,1, ...,T andv(k)=n2(k)fork =1,2,. ,.,Twhere II::;1 forallk.Thencondition(5.2.2)issatisfied. It isclearthat theSumQuadraticConstraint(5.2.2)isadiscrete--time version of the IntegralQuadraticConstraint(4.2.2). 745.Discrete-Time Set-ValuedState Estimation Set- ValuedStateEst2matwnProblem Thefirstresultofthischapterconcernsthefollowingstateestimation problem.Lety(k)=yo(k)beafixedmeasuredoutputof theuncertain system(5.2.1),(52.2)fork=1,2,..,T.Then,findthecorresponding setXT[xo,Yo(-)Ii, d]ofallpossiblestatesx(T)attimeTforthesystem (5.2.1)withuncertaintyinputsandinitialconditionssatisfyingthecon-straint(5.2.2). Definition5.2.1Thesystem(5.2.1),(5.2.2)2Ssmdtobestrictlyveri-T fiable2j theset XT[xo, Yo(-)11 ,d]2Sbounded jor any Xo,any Yo(')'andany d.Letyo(k),k=1,2, ...,T beagwenoutputsequence.Theoutput Yo(') 2Ssmdtoberealizable2jthereexzstsequences[xO, w(), vC)]satzsjymg conditions(5.2.1),(5.2.2)wzthy(k)= yo(k). Inadditiontosolvingthestateestimationproblemmentionedabove. themainresultofthischapteralsosolvesthefollowingproblem:Given anoutputsequenceYo('),determineifthisoutputisrealizableforthe uncertainsystem(5.2.1),(5.2.2).If agivenmeasuredoutputsequenceis notrealizableforthegivenuncertainsystemmodel,wecansaythat this model isinvalidated by the measured data.Thus, the results of this chapter areusefulinthequestionof model validation. 5.3Designof Discrete-TimeSet-ValuedState Estimator Oursolutiontotheabovestate estimationprobleminvolvesthefollowing Riccatidifferenceequation: F(k + 1) S(k + 1) S(O)=N where [B1(k)'S(k)B1(k)+ Q(k)] #ih(k)'S(k)A.(k), A.(k)'S(k)[A.(k)- iJr(k)F(k + 1)] +C(k + 1)' R(k + l)C(k + 1)- K(k + I)' K(k + 1), (5.3.1) A.(k) A(k)-l,ih(k) A(k)-lB1(k) and 0# denotes the Moore - Penrose pseudo-inverse; e.g., see[2].Solutions to thisRiccatiequationwillberequiredtosatisfy the followingcondition: Bl(k)'S(k)Bl(k) + Q(k)>0& N(ih(k)'S(k)ih(k) + Q(k))cN(A.(k)'S(k)ih(k))(5.3.2) fork=1,2, ...,T.HereN()denotestheoperationoftakingthenull space of amatrix. 5.3Designof Discrete-TimeSet-ValuedState Estimator75 Also.weconsiderasetof state equations of the form T)(k+1) v(O) g(k + 1) g(O)= [A(k)- Bl(k)F(k + 1)], T)(k) +G(k + I)'R(k + l)yo(k + 1), Nxo, g(k) + yo(k + 1)'R(k + I)Yo(k + 1) -7)(k)' ih(k)[B1(k)' S(k)Bl(k) -+- Q(k)] #Bl(k)'1](k), (5.3.3) Notethatif thematrixih(k)'S(k)B1(k) + Q(k)ispositive-definite,then condition(5.3.2)holdsautomatically and the pseudo-inverse in(5.3.1)and (5.3.3)can be replaced by anormal matrix inverse.This situation willhold inalmostallcasesforwhichasuitable solutionexiststoRiccatiequation (53.1). Theorem5.3.1theuncertamsystem(5.2.1),(5.2.2).Thenthe followmgstatementshold: (z)Theuncertam system (5.2.1),(5.2.2) strzctly verzfiableif and onlytherea solutwntoRiccahequatwn(5.3.1)(5.3.2)andS(T)>O. (n)Let yo(k),k=1,2, ...,T beagwenoutput sequenceandsupposethe system(5.2.1),(5.2.2)strzctlyverzfiable.Then,Yo (-)realzzable zf andonlyPT(YO(-));::- -d where PT(YO(-)) 7)(T)'S(T)-l7)(T)- g(T) and 7)(T)and g(T)aredefinedbytheequatwns(5.3.3). (iii)If theuncertamsystem(5.2.1),(5.2.2)isstnctlyvenfiable,then { XTERn:} XT[XO,yo(-)IJ, d]=II(S(T)hT - . :::;p(Yo())+ d(5.3.4) Proof Given an output sequence Yo (-) , we have by the definition of XT[xo, yo( )IJ, d], that XTEXT[xo, Yo(-)IJ, d] ifandonlyifthereexistsequencesxU, w()andv()satisfyingequation (5.2.1)and such thatx(T) = XT,the constraint(5.2.2)holds,and yo(k)= G(k)x(k) + v(k). 765.Discrete-Time Set-ValuedState Estimation' Substitution of thisinto(5.2.2)impliesthat if andonly if thereexistsaninputsequence w(-)such thatJ[XT' w()J::;d whereJ[XT' we)Jisdefinedby J[XT' w(-)J (X(O)- xo)' N(x(O)- xo) +L-x(k + 1)' K(k + I)' K(k + l)x(k +1) T-l(w(k)'Q(k)w(k)) k=O+v(k + 1)' R(k + l)v(k + 1) (5.3.5) where v(k + 1) yo(k+ 1)- C(k + l)x(k + 1) andx()isthe solutionto(5.2.1)withinputwe-)andboundary condition x(T)=XT. Nowsupposetheuncertainsystem(5.2.1),(5.2.2)isstrictlyverifiable andconsider the functional(5.3.5)withXo= andYo(-)==0.In this case, Jisahomogeneousquadraticfunctionalwithaterminalcostterm.Also, considerthesetXT[O,O,1]correspondingtoXo=0,Yo(-)==andd=1. SinceXT[O, 0,1Jisbounded,thereexistsaconstanthT> suchthatall vectorsXTERnwithIlxT11=hTdonotbelongtothesetXT[O,O, 1J. Hence,J[XT,W()J>1forallXTERnsuchthatIlxrII=hTandforall wO.Since.Jisahomogeneousquadraticfunctional,wehave J[aXT, awOJ= a2 J[XT, wO] andthecondition,J[XT' w()]> 1forIlxT11=hT,impliesthat m(xT)> 0 forallXTi0where m(XT) inf J[XT'W(')]' w() The optilnization problem infJ[XT' w()] w() subject to the constraint definedby the system(5.2.1)isalinear quadratic optimal controlprobleminwhichtimeisreversed.Inthislinearquadratic optimalcontrolproblem,asignindefinitequadraticcostfunctionisbeing considered.Usingaknownresultfromlinearquadraticoptimalcontrol theory, we conclude that the condition m(xT)> implies that there exists a solution to Riccati equation(5.3.1)satisfying condition(5.3.2)andSeT)> 0;e.g.,see[30,76]. 5.3DesignofDiscrete-TimeSet-ValuedStateEstimator77 Wehaveshownabovethatanoutput sequenceYoUisrealizableif and onlyif thereexistsavectorXTERn andanuncertaintyinputu{)such thattheconditionJ[XT. w()]:::;dholds.NowWithJdefinedasin(5.3.5), considertheoptimizationproblem infJ[XT, wO] w(-) Thisproblemisalinearquadraticoptimal trackingproblem inwhichthe systemoperatesinreversetime.Infact,thedifferencebetweenthis trackingproblemandthetrackingproblemin[76]ishere.we haveasignindefinitequadraticcostfunctionandtimeisreversed.The solutiontothistrackingproblemiswellknow!),(e.g.,see[76]).Indeed,if thereexistsasolutiontoRiccatiequation(5.3.1)satisfying(5.3.2)and S(T)> 0,then theinfimum infJ[XT'W(')] m(/ willbeachievedforanyXoandYo(');e.g., [30,76].Furthermore,as in[76],wecanwrite m(XT)=min J[XT' w()]= + g(T) w() (5.3.6) where[7)(-),g(')]isthesolutiontostate equaticms(5.3.3).However,since S(T)> 0,itfollowsthat the set isbounded.Since,xo,Yo(-),andd werearbitrar.'y,it followsthat the uncer-tainsystemmustbestrictly verifiable.Conver:,\ely,wehaveprovedabove that if the uncertain system(5.2.1),(5.2.2)isstrictly verifiable,then there exists asolution to Riccati equation (5.3.1)satiSfying (5.3.2)andS(T)>O. Thus,wehaveestablishedstatement(i). Nowsupposethesystemisstrictlyverifiable.Asabove,itfollowsthat there willexistasolutiontoRiccati equation(5.3.1)satisfying(5.3.2)and S(T)> O.Also,itisclearthattheoutputyoUisrealizableifandonly if there exists avector XTERn such that m(xT):s:d.This and(5.3.6)imply that the realizabilityof Yo (.)isequivalenttothecondition PT(YO('))2:-d. Thuswehaveestablishedstatement(ii).Also,wehaveshownabovethat XT[xo,Yo(-)li',d]isthesetofallXTERn SUththatm(xT):::;d.Then from(5.3.6),condition(5.3.4)followsimmedial;ely.Thus,wehaveproved statement(iii).0 785.Discrete-Time Set-ValuedState Estimation 5.4Discrete-TimeUncertainSystemswithMissing Data Wenowconsideraproblemofset-valuedstateestimationforuncertain discrete time systems whichhave missingmeasurementdata.Considerthe time-varyinguncertaindiscrete-timesystem(5.2.1),(5.2.2).Wesuppose thatthe measurementsequencey(-)isincomplete.That is,let beagivenvectorfork=1,2, .... TsuchthatM'(k)E{O,1}forany i=1. ...,landanyk=1, ...,T.Then,theithcomponenty'(k)of theoutputvectory(k)isknownifM'(k)=1andy'(k)isunknownif M'(k)= O.Thematrix M [M(l)M(2)M(T)] isreferredtoastheincompletenessmatrix forthesystem(5.2.1). AssociatedwiththeincompletenessmatrixM,wealsodefinetwose-quences of matricesE(k)andE(k)asfollows:For eachk,the matrix E(k) isdefinedtobethediagonalmatrixwhosediagonalelementsare givenby the elements of the vector M (k).To define the matrix E(k ),let Sbe the set of standardunitvectorse,inRlsuchthat!vItek)=O.Then thecolumns ofthematrixE(k)aretheunitvectorsinthesetSinnumericalorder. If ]\I(k)isavectorofones,thenE(k)isthezerovector.It followsfrom thisdefinitionthat the rangespaceof the matrixE(k)correspondsto the unknownelementsof the output vectory(k). Set- ValuedStateEstimatwnwithMissingData Theset-valuedstateestimationproblemwithmissingdataisdefinedas follows:Let!IIbeagivenincompletenessmatrixandlety(k)=yo(k)be the output of the uncertain system(5.2.1),(5.2.2)fork=1,2, .... T.Also, define a corresponding knownoutput sequence Yo (k)such that Yo (k)=yO( k) ifM'(k)=1andy'6(k)=0ifM'(k)=O.Thatis.theknownoutput sequenceisobtained by settingunknownelementsin the outputvector to zero.It followsfromthisdefinitionthatwecanwriteyo(k)=E(k)yo(k) wherethematrixE(k)isdefinedasabove.Weconsidertheproblemof finding the corresponding set XT[M, xo, Yo(') Ii, d]of all possible states x(T) attimeTforthesystem(5.2.1)withtheincompletenessmatrixMand uncertainty inputsandinitialconditionssatisfyingtheconstraint(5.2.2). Remark Theaboverobuststate estimationproblem withmissingdata includesthe problemofrobustpredictionasaspecialcase;seealso[87].Thisfollows ifweconsidertheincompletenessmatrixtobesuchthatM(k)=0for 5.5Designof aSet-ValuedStateEstimator withMissingData79 k=To+ LTo+ 2,...,T.Then,thesetXT[M,xo,yoOli.d]givesthe predictedsetof possiblestatesattimek=Tgivenmeasurementsupto timek=To. Definition5.4.1Thesystem(5.2.1).(5.2.2)issaid tobestrictly verifi-able with the incompleteness matrix Mif theset XT[M,xo,yoOli,d] isboundedforanyxo,anyknownoutputsequenceYoOandanyd. 5.5Designof aSet-ValuedStateEstimatorwith MissingData Our solution to the above set-valued state estimation problem with missing data involvesthefoUowingRiccati differenceequation: where F(k + 1) S(k + 1) [B1(k)'5(k)B1(k)+ Q(k))", Bl(k)'5(k)A(k), A(k)'S(k)[A(k)- B1(k)F(k + 1)] +C(k + 1)' E(k + I)R(k + l)E(k + I)C(k + 1) -K(k + 1)' K(k + 1),5(0)=N(5.5.1) R(k+l) R(k+l)-R(k+l)E(k+l) x(E(k + 1)'R(k + I)E(k + 1)) -1 E(k + 1)'R(k + 1) >O. If E(k+ 1)= 0,welet R(k + 1)=R(k + 1).Note,(.)#denotes the Moore-Penrosepseudo-inverse.Solutionsto thisRiccatiequation willbe required to satisfy the followingcondition: Bl(k)'5(k)B1(k) + Q(k)>0& N(Bl(k)'5(k)B1(k) + Q(k))cN(4(k)'5(k)B1(k))(5.5.2) fork=1. 2, . ... T.HereN (.)denotesthe oftakingthenull spaceof amatrix. 805.Discrete-TimeSet-ValuedState Estimation Also,weconsiderasetof state equations of the form 7](k+I)= v(O) g(k + 1) g(O)= [liCk)- Bl(k)F(k + 1)]' 7](k) +C(k +I)' E(k +I)R(k +1)yo(k +1), Nxo, g(k)+yo(k+1)'R(k +I)Yo(k+1) -7](k)'Bl(k)[B1(k)'S(k)B1(k)+ Q(k)t Bl(k)'7](k), x ~ N x o (5.5.3) NotethatifthematrixBl(k)'S(k)Bdk) + Q(k)ispositive-definite,then condition(5.5.2)holdsautomatically andthe pseudo-inverse in(5.5.1)and (5.5.3)can be replacedby anormal matrix inverse.This situation willhold inalmostallcasesforwhichasuitablesolution existstoRiccatiequation (5.5.1). Theorem5.5.1Cons2dertheuncertam system(5.2.1),(5.2.2)wdh m2SS-mgdataandletIv!beagwenmcompletenessmatT'tX.ThpTlthefollowmg statementshold: (z)Theuncertam system(5.2.1),(5.2.2)zsstrzctlyverzjiablewzththem-completenessmatrzx Mzf and only 2f thereexzstsa solutzontoR2ccati dziJerenceequatwn(5.5.1)satzsfymgcondztzon(5.5.2)and SeT)>O. (n)If theunceriamsystem(5.2.1),(5.2.2)2Sstrzctlyverzjiablewzththe mcompletenessmatrix M,then whereT)(T)and geT)aredefinedbytheequatzons(5.5.3). Proof Givenaknownoutput sequenceyo(),wehavebythe definition of the set XT[M, Xo,yoOli, d], thatXTEXT[M.xo,YoOli,djifandonlyifthereexistsequencesYO('), xC,),w(-)andv(-)satisfyingequation(5.2.1)andsuchthatx(T)=XT, theconstraint(5.2.2)and Yo(k)=E(k )yo( k). Nowsince yo(k)=E(k)[C(k)x(k) + v(k)], 5.5Designof aSet-Valued State Estua.tor"with MissingData81 it followsthat wecanwrite v(k)= Ya(k)- E(k)C(k)x(k) + E(k)J-l(k) wherethevectorJ-l(k)isarbitrary.Substitutionof thisinto(5.2.2)implies that XTEXT[M, Xa,Ya(')li, d] if and onlyif there existsaninputsequencew()andasequenceJ-lC)such that (x(O)- xa)' N(x(O)- xa) T-l(w(k)'Q(k)w(k)) + L-x(k + 1)' K(k + 1)' K(I + l)x(k + 1):::;d(5.5.5) k=av(k)' R(k + l)v(k) where v(k) Ya(k+ 1)- E(k + I)C(k + l)x(k + 1)+ E(k + 1)J-l(k + 1). Minimizingthelefthandsideof thisinequalitywithrespecttoJ-l(k+ 1), itthenfollowsthat XTEXT[M, Xa,YaOli, d]if and only if there existsan inputsequencew()suchthatJ[XT, wO]:::;dwhereJ[XT' wO]isdefined by il. J[XT' w()]= (x(O)- xa)' N(x(O)- xa) T-l(w(k)'Q(k)w(k) - x(k + 1)'K(k + 1)'K(k + l)x(k + 1)) + L+ [jJo(k+ 1)- E(k + I)C(k + l)x(k + 1)]' k=axR(k + 1) [Yo(k+ 1)- E(k + I)C(k + l)x(k + 1)] andx()isthe solution to(5.2.1)with inputw()andboundary condition x(T)= XT. Theremainderoftheproofofthistheoremissimilartotheproofof Theorem5.3.1inSection5.3.0 Remark It should be noted that the Riccati difference equation (5.5.1)is such that to determineS(k),the matricesE(k)andE(k)arerequired.Thesematrices willdepend on which data are missing at timek.Thus.in order to operate ourrobuststateestimatorinrealtime,theRiccatidifferenceequation (5.5.1)togetherwiththestateequations(5.5.3)mustbesolvedon-line. Thecoefficientsinequations(5.5.1)and(5.5.3)wouldthenbeadjusted depending on whichmeasurementswereavailableat time stepk. 825.Discrete-Time Set-ValuedStateEstimation 5.6ARobustDeconvolutionProblem Toillustratetheresultsof thischapteronset-valuedstateestimationfor discretetimeuncertainsystems.wenowconsiderarobustdeconvolution problem similar to those considered in[28].However,weuse adifferentun-certaintydescriptionthanwasconsideredin[28].Ablockdiagramof the system under consideration is shown if Figure 5.6.1.In this robust deconvo-+ SignalModel 0.707 Channel Model u(k) 0.0127 0.4 z-0.2 + y(k) State Estimator u(k) FIGURE5.6.1.Robustdeconvolution system. lutionproblem,theuncertainparameter~ k ) representstheuncertainty inthesignalmodelnaturalfrequency.Combiningthesignalmodeland thechannelmodel,weobtainthefollowinguncertainsystemof theform (5.2.3) : (5.6.1) where T-lT (10 + 0.01272)llx(0)112 + L nl(k)2 + L n2(k)2~ 1(5.6.2) k=Ok=l andtheuncertainparameter~ k ) satisfies 1 1 ~ k ) 1 1 :::;1. (5.6.3) 5.6ARobustDetonvolutionProblem83 Inthisstatespacedescription,xI(k)andx2(k)arethestatevariables of thesignalmodel.Therequiredsignalu(k)cotrespondstoxI(k).Also, x3(k)isthestate variableof thechannelmodel.Weconsiderthissystem overafinitetimeintervalofT=100samples.Toapplyourresultsto thisdeconvolutionproblem,weconsideracorrespondinguncertain system oftheform(5.2.1)inwhichtheuncertaintysatisfiesthesumquadratic constraint.Inthiscase.thematricesA,N,BI,XO,K.C,Q,andRare givenby A [ 1.98-1 o ][10 0 ~ o ] 10o: N=010 0.400.200 [ 0.707 0707][ 0] 0o: Xo=0 0o0 c [ a aI ] :K=[0.0127a a 1: Q=IandR=I. (5.6.4) Also,theconstantdisgivenbyd=1. Toillustratetheperformanceofourstateestimator,weconsiderthe uncertaintiesandnoisesignalstobesuchthatD,.(k)==1,nI (0)=0.5. nICk)=0fork =1,2, ... ,100,and 1. n2(k)=105sm(k/10) fork=1, 2.... 100.Withtheinitialcondition:teO)= 0,it is straightfor-wardto verifythatthe uncertainty inputsequences w(k)=[x(k)'K'D,.'nl(k)']' andv(k)=n2(k)satisfythesumquadraticconstraint(5.2.2).Wenow apply our state estimator to the linearsystem corresponding to thisuncer-taintyrealization.Figure5.6.2showstheresultingestimateof thesignal u(k).upperandlowerboundsonu(k),andthetruevalueofu(k)for k=1, 2,.... 100.Theestimatedvalueof thest''1tevectorcorrespondsto thecenterof the ellipsoidof possiblestatesdesctibedbyequation(5.3.4). Indeed,referringtoequation(5.3.4),thestate estimateattimekisgiven by i(k) =S(k)-IT}(k).In this example,the requiredestimate of the signal u( k)corresponds to the firstcomponent of this estimated value of the state vector.Also,theupperandlowerboundsonthe signalu(k)areobtained byprojecting theellipsoidalsetof possiblevaluesof the state vectoronto itsfirstcomponent. In addition to the uncertainty realization described above wealso consid-eredanotheruncertaintyrealizationinwhicht ~ ~ valueof D,.wasreplaced byD,.=-1. Apart fromthis change.whichcorre:spondstoachangeinthe 815.Discrete-Time Set-ValuedState Estimation 10 o -5 o , \ 1020 - true value ofu(k) - - estimated value ofu(k) - upper bound onu(k) , , 30 lower bound on u(k) 405060 timestep , I 7080 \ \ 90100 FIGURE:5.6.2.Estimatedvalueof u(k)withD.=1. naturalfrequencyofthesignalmodel ,theuncertaintyrealizationisthe sameasabove.Figure5.6.3showsthecorrespondingsimulationsforthis case.Notethatinbothcases,agoodestimateisobtainedfortheactual signalu(k)inspite of largeuncertamty inthe signalmodel. 5.7ARobustDeconvolutionProblem withMissing Data Toillustrate theresults of thischapter on set-valuedstate estimation with missingdata,weconsiderarobustdeconvolutionproblemsimilartothat consideredin Section5.6.However,unlikeSection 5.6,wewillassume that overthe timeinterv;:tlof interest,thereexistperiodsof missingdata.The blockdiagram of the systemunder considerationremainsasshowninFig-ure5.6.l.Inthisrobustdeconvolutionproblem,theuncertainparame-ter 6.(k)representsthe uncertainty in the signalmodelnatural frequency. Combiningthesignalmodelandthechannelmodel.weobtainthe uncer-tain system(5.6.1) , (5.6.2) , (5.6.3).This isanuncertain system of theform (5.2.3).We consider this system overafinite time interval of T=100 sam-ples.Furthermore,weassume that the measuredsignaly( k)isunavailable tothe state estimatorduring the time periods: kE{5, 6,...,10,31,32, .. .,35,76,77, ... ,SO}. 5.7ARobustDeconvolutionProblemwithMissing DMa85 1 5 - - - ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - r - - - - - - ,10 ~ 5/ '5"I 0;, C 0> iii -5 o10203040 - true value of u(k) - - estimated value of u(k) - .upper boundonu(k) \ \ \ , 50 timestep lower bound onu(k) 60708090100 FIGURE5.6.3.Estimated valueof u(k)withDo=-1. Toapplyourresultstothisdeconvolutionproblem,weconsideracorre-spondinguncertainsystemoftheform(5.2.1)inwhichtheuncertainty satisfies the sum quadratic constraint.In this case.the matricesA,N,BI, Xc,K,C,Q,andRaregivenby(5.6.4).Also,theconstantdisgivenby d=1.Theincompletenessmatrixcorrespondingtothisproblemisarow vector M=[M(l)M(2) ... M(lOO)] whereM(k)= 0for kE{5, 6, ... ,10,31. 32, ... ,35,76.77, ... ,SO} andM(k)=1otherwise.CorrespondingtothismatrixM,thematrices E(k)andE(k)areasdefinedinSection5.4. Toillustratetheperformanceofourstateestimator,weconsiderthe uncertaintiesandnoisesignalstobesuchthat6;.(k)==1,nI(O)=0.5, nICk)=0fork =1,2, ... ,100,and 1. n2(k)=105sm(k/10) fork=1,2, ...,100.Withtheinitialconditionx(O)=0,itisstraight-forwardto verifythatthe uncertainty inputsequences w(k)=[x(k)'K'6;.'nI(k)']' 865Discrete-TimeSet-ValuedState Estimation andv(k)=n2(k)satisfythesumquadraticconstraint(5.2.2).Wenow apply ourstate estimator to the linear system corresponding to thisuncer-taintyrealization.Figure5.7.1showstheresultingestimateof thesignal u(k),upperandlowerboundsonu(k),andthetruevalueofu(k)for k=1. 2 .... ,100.Theestimatedvalueofthestatevectorcorrespondsto thecenterof theellipsoidof possiblestatesdescribedbyequation(5.5.4). Indeed.referringto equation(5.5.4),thestate estimate attimekisgiven by x(k)=S(k)-l7](k).Inthis example,the requiredestimate of the signal u(k)corresponds to the firstcomponent of this estimated value of the state vectorAlso,theupperandlowerboundsonthesignalu(k)areobtained byprojecting the ellipsoidalsetof possiblevaluesof thestatevectoronto itsfirstcomponent. In addition to the uncertainty realization described above. we also consid-eredanotheruncertaintyrealizationinwhichthe valueof ~ wasreplaced by,6.=-1.Apartfromthischange,whichcorresponds toachangeinthe naturalfrequencyofthesignalmodel.theuncertaintyrealizationisthe sameasabove.Figure5.7.2showsthecorrespondingsimulationsforthis case.Notethatinbothcases,agoodestimateisobtainedfortheactual signalu( k)inspite of largeuncertainty inthe signalmodel. 30 ~ 5- true valueofu(k) - - est, matedvalue of u(k) gO - - upper bound on u(k) lower boundonu(k) 15 L-________I, I , I, 10 I, 2 I :J I 'I Cii5 I I, , , /, I' I, c: Ol en Q -5 -10 -15 _ 2 0 L ~ ~ ~ L L ~ ~ ~ L ~orow~ ~ WW708090100 time step FIGURE57.1.Estimated valueof u(k)withll. =1. ? '5 Cii c en in 30 25 . ~1& 10 5 , , r , 0 -.$ -lit;) ~ I ~'I I I I, 5.7 A RobustDeconvolution Problem with MissingData87 - true value of u(k) - - estimated value of u(k) - - upper bound on u(k) lower bound on u(k) ,I ,I / I, _20L-__-L____L-__-L____L-__-L____L-__-L____ __ __ o102030405060708090100 timestep FIGURE5.7.2.EstImatedvalue of u(k)with.6.= -1. RobustStateEstimationwithDiscrete andContinuousMeasurements 6.1Introduction Inmanyfilteringandstateestimationproblems,theprocessbeingcon-sideredhasacontinuous-timemodelbutthemeasuredoutputdataare availableonlyatdiscretesamplinginstants.Hence,wearemotivatedto consider"hybrid"problems of problems of state estimation andmodel val-idation inwhich the underlying processmodeliscontinuousbut theavail-ablemeasureddata areavailableat discretesampling times.Furthermore, weextendthisproblemtoallowforthecasewhensomeof theoutputs canbemeasuredcontinuouslyand other outputs can onlybemeasuredat discretesamplinginstants.Suchasituationmayariseincomplexhybrid processesinvolvingbothdigitalandanalogblocks.Also,theremayarise situationsinwhichsomeofthesensorssupplydataataveryfastrate (whichcanbeapproximatedasacontinuous-timesignal)whereasother sensorssupplydataataslowersamplerate.Inthiscase,itisimportant that the signal processing algorithm be able to simultaneously handle both typesofmeasureddata.Themainresultsofthischapterextendtheset-valuedstateestimationandmodelvalidationresultsof Chapters4and5 toallowforhybriddiscrete-continuousdata.Theresultsofthischapter originallyappearedinthepapers[125,128,138]. Anotherissue whicharisesinmany realtime signal processing problems istheissueofmi