Research Article H Estimates for Discrete-Time Markovian Jump Linear Systems

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1 Mathematca Probems n Engneerng Voume 213 Artce ID pages Research Artce H Estmates for Dscrete-Tme Markovan Jump Lnear Systems Marco H Terra 1 Gdson Jesus 2 and João Y Ishhara 3 1 Department of Eectrca Engneerng Unversty of São Pauo at São Caros CP São Caros SP Braz 2 DepartmentofScencesandTechnoogyUnverstyofSantaCruz Ihéus BA Braz 3 Department of Eectrca Engneerng Unversty of BrasíaCP Brasía DF Braz Correspondence shoud be addressed to Marco H Terra; terra@scuspbr Receved 1 February 213; Accepted 8 June 213 Academc Edtor: Weha Zhang Copyrght 213 Marco H Terra et a Ths s an open access artce dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgna work s propery cted Ths paper deas wth the probem of H fterng for dscrete-tme Markovan jump near systems Predcted and ftered estmates are obtaned based on the game theory Both fters are soved through recursve agorthms The Markovan system consdered assumes that the jump parameters are not accessbe ecessary and suffcent condtons are provded to the exstence of the fters A numerca exampe s provded n order to show the effectveness of the approach proposed 1 Introducton Fterng of Markovan jump near systems (MJLSs) has been subject of ntensve study n the ast years Dfferent and creatve approaches to dea wth ths cass of fter have been consdered n the terature One of the man aternatves to sove ths knd of probem s based on Kaman fter agorthms see for nstance 1 2 Fters proposed n these references are based on near mnmum mean square error estmates of dscrete-tme MJLS They present an nterestng feature reated wth recursveness; however they are not robust n nature When the robustness s reevant to the fterng process and demands an extra performance of the fterng approach H technques are aways consdered as one of the best soutons to be adopted In ths way a cosed-oop transfer functon from the unknown dsturbances to the estmaton error s desgned n order to satsfy a prescrbed H -norm constrant In genera agorthms deveoped to deduce H Markovanftersarebasedonnearmatrxnequates (LMIs) see for nstance 3 12 In partcuar 4 6 assume that the jump parameter of the Markov chan s not avaabe In ths paper we propose H fters for dscrete-tme MJLS whch are cacuated n terms of recursve agorthms based on Rccat equatons Foowng the approach consdered n 13 we deveop predcted and ftered estmates based onthetwo-payersgametheorythedeaofgametheorywas asoadoptednsthochastch 2 /H contro see for nstance 14 We defne the status of the payers n the fterng probems consdered n ths paper n order to reach an equbrum between two contradctory objectves The frst payer can be nterpreted as the maxmzer of the estmaton costwhereasthesecondpayertrestofndanestmatethat brngs the quadratc cost to a mnmum A souton exsts for a specfed γ-eve f the resutng cost s postve We assume n these Markovan fterng probems that the jump parameter s not accessbe The recursveness of the approach we are proposng s the man advantage of these Markovan fters f compared wth the fters aforementoned As a byproduct of ths approach we provde necessary condtons for the exstence of them based on ony known parameters of the Markovan system Thspapersorganzedasfoows:nSecton 2wepresent the probem statement; n Secton 3 H fters for dscretetmemjlsarepresented;andnsecton 4 acomparatve study based on numerca exampe between the approach we are proposng and the fter deveoped n 4s performed

2 2 Mathematca Probems n Engneerng 2 Probem Defnton The H recursve fters deveoped n ths paper are based on the foowng dscrete-tme MJLS: x +1 =F Θ x +G Θ u y =H Θ x +D Θ w s =L Θ x +R Θ V =1 where x R n sthevauedstatey R m s the vaued output sequence s R p s the vaued sgna to be estmated u R p w R q andv R q are random dsturbances; Θ s a dscrete-tme Markov chan wth fnte state space {1} and transton probabty matrx P = p jk Wesetπ j := P(Θ =j) F k R n n G k R n p H k R m n D k R m q L k R p n andr k R p q k {1}for The random dsturbances {u } {w }and{v } are assumed to be nu mean wth fnte second order moments ndependent wde sense statonary sequences mutuay ndependent wth covarance matrces equa to the U W andv respectvey x 1 {Θ =k} are random vectors wth E{x 1 {Θ =k}} =μ k (where 1 { } denotes Drac measure) and E{x x T 1 {Θ =k}} =V k ; x and {Θ } are ndependent of {u } {w } and {V }Ascaarγ>and asequenceofsetsofobservatons (1) {y } {y y 1 } {y y } (2) are defned to fnd at each nstant a ftered estmate s of s such that x x F x 2 P s satsfed for =and {x } = x F x 2 P s 2 L Θ x V + = = + y H Θ x s 2 L Θ x V y 2 H Θ x W + = 2 W <γ 2 (3) <γ 2 x 2 +1 F Θ x U (4) s satsfed for > For the predcted estmate gven γ> and a sequence of observatons (2) the probem s to fnd at each nstant apredctons +1 of s +1 such that x s satsfed for =and {x } +1 = x F x 2 P s 2 L Θ x V + = x F x 2 P +1 = s 2 L Θ x V y 2 H Θ x W + = <γ 2 (5) <γ 2 x 2 +1 F Θ x U (6) s satsfed for Thesequenceofsoutonss and s +1 s outputs of the respectve H fters Due to the hybrd nature of ths cass of system at each nstant of tme a new mode of a countabe set of modes s used to cacuate the functonas (4) and(6) In genera to synthesze recursve fterng agorthms for ths cass of probems s not an easy task Thanks to the augmented mode of (1) proposedn1 we can redefne these functonas n order to deveop H recursve fters smar to those we fnd n the terature for standard state-space systems wthout jumps The augmented mode of (1) can be wrtten as foows: z +1 = F z +ψ y = H z +φ s = L z +σ =1 where the parameter matrces are gven by F := p 11 F 1 p 1 F d p 1 F 1 p F H := H 1 H L := L 1 L whose dmensons are defned by F R n n H R m n andl R p n ; and the state varabe s defned as z := z T 1 zt T R n z k := x 1 {Θ =k} R n We defne for and k {1} Z k := E {z k z T k } Rn n Z := E {z z T }=dag Z k R n n where Z k s gven by the foowng recursve equaton: Z +1k = p jk F j Z j F T j + p jk π j G j U G T j Z k =V k (7) (8) (9) (1) (11) We defne aso n the next emma weghtng matrces Λ Π andγ as varances of random dsturbances gven by ψ φ andσ respectvey Lemma 1 Let ψ φ andσ defned by ψ := M +1 z +θ φ := D Θ w σ := R Θ V (12)

3 Mathematca Probems n Engneerng 3 for and jk {1}where M T +1 := MT +11 MT +1 M +1k := (1 {Θ+1 =k} p 1k )F 1 1 {Θ =1} (1 {Θ+1 =k} p k )F 1 {Θ =} 1 {θ(+1)=1} G θ := Θ u 1 {θ(+1)=} G Θ u (13) The varances of ψ φ andσ can be obtaned by the foowng equatons: respectvey where Γ := dag p jk F j Z j F T j F Z F T + dag p jk π j G j U G T j Π := D D T Λ := R R T (14) D := D 1 π 1/2 1 W1/2 D π 1/2 W1/2 (15) R := R 1 π 1/2 1 V1/2 R π 1/2 V1/2 (16) Proof The proof foows the arguments proposed n 15 Chapter 3 and n 1 3 H Estmates for DMJLS In ths secton we propose recursve H estmates for dscrete-tme MJLS (DMJLS) It s known that for standard state-space systems ths knd of fterng approach s dffcut to be mpemented onne due to the fact that we do not know the mnmum γ for each step of the recurson In genera t depends on the estmate error varance matrx whch shoud be cacuated at the same nstant of tme In ths sense the exstence condton of ths fter s not known apror To ameorate ths mtaton we provde for ths Markovan probem necessary condtons to tune ths parameter dependng on the varances of the random dsturbances ψ φ andσ and on the known parameter matrces of the augmented mode of (1)gvenn(7) 31 H Fter Consder (3)and(4) for the augmented mode (7): z s L z 2 Λ z F z 2 Z + y H z 2 Π <γ 2 a (17) for =and {z } = = s 2 L z Λ z F z 2 Z + = y H z 2 Π + = z +1 F z 2 Γ <γ 2 a (18) for >WecanrewrtethsH fterng probem n terms of the foowng optmzaton probem: where mn {z } = {s } = max J f ({y } = {s } = {z } = )> (19) J f := z F z + 2 Z y H z 2 Π J f a s L z 2 = Λ := z F z 2 Z + = a = + = z +1 F z 2 Γ y H z 2 Π s L z 2 > Λ (2) (21) otce that t s not necessary to maxmze J f over {s } n the optmzaton probem (19) The souton of = the H fter s guaranteed f and ony f there exsts a { s } = for whch Jf ({y } = { s } = {z } = ) > has a mnmum { z } Therefore n order to dea wth ony the = mnmzaton probem (19) for each t s easy to show that (21)canberewrttenas where J f := (U X B ) T R (U X B ) (22) X := z z 1 z B := R R := d R Z Z 1 Z E A E d U := d A 1 E 1 A E

4 4 Mathematca Probems n Engneerng I E := H R := L A := F Γ Π a Λ Z := y 1 s Z := F z y s Γ := Z (23) For a wecanconcudethatthefoowngrecurrent reatons are vad: R := R B R := Z B z X := U X := E α U α := A (24) Accordng to 16 there exsts a mnmum for (22)fand ony f the postveness of U T R U s guaranteed Lemma 2 Consder matrces U and R and coumn vectors B and X of approprate dmensons wth R symmetrc For any B we have nf x Jf > (25) f and ony f U T R U and Ker(U T R U ) Ker(R U ) If the mnmum s attaned t s unque f and ony f U T R U > and the optma souton s gven by X = (U T R U ) U T R B Wth ths fundamenta emma n mnd we can obtan an exstence condton for the fter we are proposng n the foowng based on the known parameter matrces of the Markovan mode For >thetermu T R U of X can be wrtten as ET R E E T R α α T R E U T R U +α T R α (26) and a necessary condton for U T R U >s the postveness of E T R E whch n terms of the augmented Markovan mode of (1) can be estabshed as foows: Γ + HT Π H a LT Λ L > (27) Remark 3 Aowerboundforγ a canbecacuatedthrough (27) based on the augmented mode (7)andLemma 1otce that (27) s a natura and nterestng extenson of the fterng of standard state-space systems 13 wthout jumps to the DMJLSwearedeangwthnthspaper ow we are n a poston to deduce the H recursve ftered estmate for DMJLS The next theorem provdes the souton for ths probem based on the augmented mode (7) and on the sequence of measurements (2) of the orgna Markovan system (1) Theorem 4 Consder the augmented Markovan mode (7) There exsts a recursve H fter for ths system defned by the foowng recursve equatons: Z := Z + H T Π H a LT Λ L Z := (Γ + F Z F T ) + H T Π H a LT Λ L z := (P +γ 2 a LT Λ L ) ( Z FT z + H T Π y ) z := ( Z +γ 2 a LT Λ L ) (X F z + H T Π y ) X := Γ + F Z F T (28) (29) s := L z (3) fandonyf Z > =1 and the attenuaton eve γ a s postve Proof Ths proof foows standard arguments used to deduce Kaman fters and bascay conssts n checkng the postveness of (26) and n fndng the equaton of the fter through the Lemma 2 In order to have U T R U >wecanrewrte ths term as (26) The (2 2) sub bock of (26)mustbepostve defnte Assumng that the postveness of U T R U s guaranteed and α T R α = FT Γ F (31) the (2 2) term of (26) s postve defnte and U T R U >f and ony f the Schur compement of the (2 2) bock M := E T (R +α (U T R U ) α T ) E (32) s postve defnte For M s the (1 1) bock of (U T R U ) Wthα defned n (24) we obtan M := E T (R + A M A T ) E (33) Consderng Z := M n terms of the orgna data (23) we obtan (28) From Lemma 2 we have the souton for the mnmzaton of (22)Basedontherecurrentreatons(23)

5 Mathematca Probems n Engneerng 5 and (24) consderng α X j = A x j for j and ntroducng W e := Π γ 2 a Λ Z P 12 P 21 P 22 := ET R E E T R α α T R E U T R U +α T R α (34) z +1 := F z + H L Z H T L T + F M H T (Π + H M H T ) (38) hods (29) takng nto account that s = s := L z (y H z ) 32 H Predctor In ths subsecton we present the recursve H predctor fter for DMJLS The orgna H probem (5)-(6) can be rewrtten n terms of the foowng optmzaton probem: where mn max J p {z } +1 = {s } +1 ({y } = {s } +1 = {z } +1 = )> (35) = J p := z F z 2 Z J p a s L z 2 = Λ := z F z 2 Z + = a + = z +1 F z 2 Γ +1 = y H z 2 Π s L z 2 Λ (36) The H predctor fter exsts at the nstant f and ony f there exsts a sequence { s } +1 = such that Jp ({y } = { s } +1 = {z } +1 = ) has a mnmum { z } +1 for whch = J p ({y } = { s } +1 = { z } +1 = )> (37) The predcton optmzaton probem s equvaent n nature to the fter probem aforementoned In the next theorem we present the H predctor fter whose proof foows the ne of the ftered verson Theorem 5 The Markovan H predcton probem (35) s sovabe f and ony f Z +1 γ 2 a LT +1 Λ +1 L +1 >where the sequence { Z +1 } and the predctor fter are cacuated by the recursons Z := Z Z +1 := Γ + F Z F T W e H L Z F F Z H T L M := Z γ 2 a LT Λ L s +1 := L +1 z +1 In order to recover the orgna predcted and ftered state estmates of System (1) we cacuate the foowng summatons: x = z j x = z j (39) where z := z T 1 zt T R n and z k := x 1 {Θ =k} R n x s defned n the same way Remark 6 To estabsh the stabty of the statonary H predcton fter t s assumed that a matrces of (1) and the transton probabtes p jk are tme nvarant; System (1) s mean square stabe (MSS) and ts Markov chan {Θ } s ergodc There exsts a unque postve-defnte souton Z (wth ) to the agebrac Rccat equaton Z :=Γ+F ZF T F Z H L TW e H L ZF W e := Π γ 2 a Λ+H L ZH T L T (4) provdng that Z a LT Λ L >foranyγ a fxed whch guarantees that W e >and thus the above nverse s we defned Foowng the gudenes defned n 1 the stabty of the predctor fter s assured wth r σ (F FMH T (Π + HMH T ) H)<1 (41) where r σ ( ) denotes spectra radus of the dynamc matrx ofthefterwthstatonarygantheasymptotcstabtyof the ftered verson can aso be assured foowng ths ne of argumentaton Remark 7 Smar to the ftered case we can aso fnd a necessary exstence condton to the Markovan H predcton fter gven by Z a LT Λ L > = Γ a LT +1 Λ +1 L +1 > (42)

6 6 Mathematca Probems n Engneerng Remark 8 If we consder γ a theh fter proposed n ths paper reduces to the Kaman fter for MJLS proposed n 1 For the case wth no jumps ( = 1)thepredctedand the ftered H estmates reduce to the H fters for statespacesystemsgvenn umerca Exampe rms 4 In ths secton we compare the fterng approach proposed wth the fter deveoped n 4 Wth two Markovan states the mode (1) s defned based on the foowng parameters: 9 1 P= 9 1 F 1 = F 2 = G 1 =G 2 = H 1 =H 2 =1 D 1 =D 2 =8 L 1 =L 2 =5 R 1 =R 2 =1 3 (43) Wecomparebothftersnthepredctedformwrttennthe foowng way: where z +1 = A z + B y s +1 = L +1 z +1 A = F F M H T (Π + H M H T ) H B = F M H T (Π + H M H T ) The fter of 4 s consdered n the strcty proper form as x f (k+1) =A f x f (k) +B f y (k) z f (k) =C f x f (k) (44) (45) (46) whose souton s gven n terms of near matrx nequates We show n Fgure 1 the root-mean-square errors (rms) of both fters We performed 1 Monte Caro smuatons from = 6wth the vaues of Θ generated randomy The nta condton x s consdered Gaussan wth mean T and varance ; Θ {1 2} u w and V are ndependent sequences of noses wth U W andv dentty matrces wth approprate dmensons π 1 () = 5 and π 2 () = 95 Weobtanedγ a = for the fter Fter (LMI) Fter (Rccat) Fgure 1: Root-mean-square errors for H ftersbasedonrccat recursve equaton and near matrx nequates proposed n ths paper and γ = 2886 forthefterproposed n 4 The parameter matrces of our fter were computed as A = B = L +1 = andforthefterof4werecomputedas A f = B f = C f = (47) (48) In spte of the smaer γ provded by the fter of 4 the rms of both fters are equvaent Ths dfference n γ s due to the fact that the functonas of both approaches are dfferent n nature However f we consder the same numerca vaue of a parameters but wth matrces R 1 and R 2 mutped by 1 6 weobtanfortheproposedrecursvefterγ a = 1 On the other hand we obtan an nfeasbe souton wth the LMIproposednSectonIVof4(computedthroughMatab 74) The mode-ndependent fter provded n 4 deas wth ony suffcent condtons Consderng offne computatons our approach provdes necessary and suffcent condtons for the exstence of the fter It s mportant to reca that the proposed approach enabes a parameter matrces of (1)tobetmevaryngItaso provdes a practca estmate for the mnmum admssbe γ through the necessary condton (27) whch depends on ony known parameters of the Markovan system

7 Mathematca Probems n Engneerng 7 5 Concuson Ths paper deveoped H predcted and ftered estmates for DMJLS They were deduced based on the assumpton that jump parameter s not accessbe The numerca exampeshowedtheeffectvenessofthsapproachthroughan augmented mode of the standard Markovan system these fters were deduced based on the game theory It aows us to dea wth recursve agorthms to sove ths knd of fterng probem where the parameter matrces can be tme varyng for each operaton mode of the Markovan system As future works we ntend to sove nonnear fterng and contro probems for DMJLS based on the probems rased by References 1 O L V Costa and S Guerra Statonary fter for near mnmum mean square error estmator of dscrete-tme Markovan jump systems IEEE Transactons on Automatc Contro vo 47 no 8 pp MHTerraJYIshharaandGJesus Informatonfterng and array agorthms for dscrete-tme Markovan jump near systems IEEE Transactons on Automatc Contro vo 54 no 1pp YDongJSunandQWu H fterng for a cass of stochastc Markovan jump systems wth mpusve effects Internatona Robust and onnear Contro vo18 no1pp A P C GonçavesARForavantandJCGerome H fterng of dscrete-tme Markov jump near systems through near matrx nequates IEEE Transactons on Automatc Controvo54no6pp H Lu F Sun K He and Z Sun Desgn of H reduced-order fter for Markovan jumpng systems wth tme deay IEEE Transactons Crcuts and Systems vo51no11pp C E Souza A mode-ndependent H fter desgn for dscrete-tme Markovan jump near systems n Proceedngs of the 42nd Conference on Decson and Contro Mau Hawa USA December 23 7 CEdeSouzaandMDFragoso H fterng for Markovan jump near systems Internatona Systems Scence vo33no11pp CEdeSouzaandMDFragoso H fterng for dscretetme near systems wth Markovan jumpng parameters Internatona Robust and onnear Contro vo13 no 14 pp C E de Souza A Trofno and K A Barbosa Modendependent H fters for Markovan jump near systems IEEE Transactons on Automatc Controvo51no11pp F Sun H Lu K He and Z Sun Reduced-order H fterng for near systems wth Markovan jump parameters Systems & Contro Letters vo 54 no 8 pp L Wu P Sh H Gao and C Wang H fterng for 2D Markovan jump systems Automatca vo 44 no 7 pp Z Ln Y Ln and W Zhang A unfed desgn for state and output feedback H contro of nonnear stochastc Markovan jump systems wth state and dsturbance-dependent nose Automatca vo 45 no 12 pp J Y Ishhara M H Terra and B M Espnoza H fterng for rectanguar dscrete-tme descrptor systems Automatca vo 45 no 7 pp B-S Chen and W Zhang Stochastc H 2 /H contro wth state-dependent nose IEEE Transactons on Automatc Controvo49no1pp OLVCostaMDFragosoandRPMarquesDscrete-Tme Markov Jump Lnear Systems: Probabty and Its Appcatons Sprnger London UK S Haykn Adaptve Fter Theory Prentce Ha Upper Sadde Rver J USA B Hassb A H Sayed and T Kaath Indefnte-Quadratc Estmaton and Contro: A Unfed Approach to H 2 and H Theores SIAM Studes n Apped Mathematcs SIAM Phadepha Pa USA W Zhang B-S Chen and C-S Tseng Robust H fterng for nonnear stochastc systems IEEE Transactons on Sgna Processng vo 53 no 2 part 1 pp

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