Robust Finite-Time H Filtering for Itô Stochastic Systems

Transcription

1 Journal o Applied Mahemaics and Physics, 26, 4, 75-7 hp:// ISSN Online: ISSN Prin: Robus Finie-ime H Filering or Iô Sochasic Sysems Aiqing Zhang College o Mahemaics and Compuer Science, Jianghan Universiy, Wuhan, China How o cie his paper: Zhang, A.Q. (26) Robus Finie-ime H Filering or Iô Sochasic Sysems. Journal o Applied Mahemaics and Physics, 4, hp://dx.doi.org/.426/jamp Received: Augus 7, 26 Acceped: Sepember 7, 26 Published: Sepember 2, 26 Copyrigh 26 by auhor and Scieniic Research Publishing Inc. his work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY 4.). hp://creaivecommons.org/licenses/by/4./ Open Access Absrac his paper invesigaes he problem o robus inie-ime H iler design or Iô sochasic sysems. Based on linear marix inequaliies (LMIS) echniques and sabiliy heory o sochasic dierenial equaions, sochasic Lyapunov uncion mehod is adoped o design a inie-ime H iler such ha, or all admissible uncerainies, he ilering error sysem is sochasic inie-ime sable (SFS). A suicien condiion or he exisence o a inie-ime H iler or he sochasic sysem under consideraion is achieved in erms o LMIS. Moreover, he explici expression o he desired iler parameers is given. A numerical example is provided o illusrae he eeciveness o he proposed mehod. Keywords Sochasic Sysems, H Filer, Finie-ime Sabiliy, Linear Marix Inequaliies (LMIS). Inroducion Since sochasic sysems play an imporan role in many branches o science and engineering applicaions, here has been a rapidly growing ineres in sochasic sysems. In he pas ew years, much aenion has been ocused on he robus H ilering problems o sochasic sysems; many conribuions have been repored in he lieraure []-[6]. In [], a H iler was designed or nonlinear sochasic sysems. From he dissipaion poin o view, a H ilering heory and a H -ype heory or a class o sochasic nonlinear sysems were esablished in [2] []. H ilering problems or discree-ime nonlinear sochasic sysems were addressed in [4]. he H ilering problems or uncerain sochasic sysems wih delays were sudied in [5]. A robus uzzy iler or a class o nonlinear sochasic sysems was designed in [6]. DOI:.426/jamp Sepember 2, 26

2 he previously menioned lieraure was based on Lyapunov asympoic sabiliy which ocuses on he seady-sae behavior o plans over an ininie-ime inerval. However, in many pracical applicaions, he goal is o keep he sae rajecories wihin some prescribed bounds during a ixed ime inerval. In hese cases, we need o guaranee ha he sysem saes remain wihin he given bounds, which is called inie-ime sabiliy. Recenly, inie-ime sabiliy or shor ime sabiliy and conrol problems or many ypes o dynamic sysems were sudied widely in [7]-[2]. he problem o inie-ime sabiliy and sabilizaion or a class o linear sysems wih ime delay was addressed in [7]. In [8], he suicien condiions were achieved or he inie-ime sabiliy o linear ime-varying sysems wih jumps. he auhors provided he suicien condiions o inie-ime sabiliy or sochasic nonlinear sysems in [9]. he problem o robus inie-ime sabilizaion or impulsive dynamical linear sysems was invesigaed in []. In [] uzzy conrol mehod was adoped o solve inie-ime sabilizaion o a class o sochasic sysem. A robus inie-ime iler was esablished or singular discree-ime sochasic sysem in [2]. I can be poined ou ha all he FS-relaed works or inie-ime problems menioned above were discussed or sochasic sysems. o he bes o he auhor s knowledge, he problem o robus inie-ime ilering or sochasic sysems has no been ully invesigaed. his moivaes us o invesigae he presen sudy. One applicaion o hese new resuls could be used o deec generaion o residuals or aul diagnosis problems. his paper is organized as ollows. Some preliminaries and he problem ormulaion are inroduced in Secion 2. In Secion, a suicien condiion or SFS o he corresponding ilering error sysem is esablished and he mehod o design a inie-ime iler is presened. Secion 4 presens a numerical example o demonsrae he aeciviy o he menioned mehodology. Some conclusions are drawn in Secion 5. We use (respecively, X, where X and Y are real symmeric marices), means ha he marix X R n o denoe he n-dimensional Euclidean space. he noaion X Y > Y Y is posiive deinie (respecively, posiive semi-deinie). I and denoe he ideniy and zero marices wih appropriae dimensions. λ ( Q) and ( Q) max λ denoe he maximum and he minimum o he eigenvalues o a real symmeric marix Q. he superscrip denoes he ranspose or vecors or marices. he symbol * in a marix denoes a erm ha is deined by symmery o he marix. 2. Sysems Descripions and Problem Formulaion Consider an uncerain Iô sochasic sysem, which can be described as ollows: { } { } dx = A+ Ax+ Av d+ D+ D x+ Dv dw ; () { } { } dy = C+ Cx+ Cv d+ E+ E x+ Ev dw ; (2), z = Lx x = x. () n m p q where x R, y( ) R, v R, z( ) R disurbance inpu, and conrolled oupu respecively, where v L 2 [, ) min are sae vecor, measuremen,, and 76

3 ADCC EE L are known consan ma- are unknown marices ha represen he ime-varying parameer uncerainies and are assumed o be o he orm w R is a sandard Wiener process.,,,,,, rices o appropriae dimensions and A( ), D( ), C( ), E( ) is an unknown marix wih Lebesgue measurable elemens sais-. HG,, G2, G, G 4 are known consan marices wih appropriae dimensions. We now consider he ollowing iler or sysem ()-(): k k where F( ) R φ ϕ ying F ( ) F( ) I [ ] A D C E = HF G G2 G G4. where x ˆ A B C are he iler parameers wih compai- R ble dimensions o be deermined. Deine ξ ( ) x ( ) xˆ ( ) lowing ilering error sysem: x ˆ = Ax ˆ + B y = ˆ ˆ = d d d ; z ˆ Cx, x x n is he iler sae,,, = and e( ) z( ) zˆ ( ) (4) =, hen we can obain he ol- ( ) = A ( ) + Bv + D ( ) + Dv w dξ ξ d ξ d ; (5) = ξ, ξ ξ e L = (6) where A+ A A A =, B = B C C A BC + D D = D = L = L C D+ D,, B E+ E BE We inroduce he ollowing deiniions and lemmas, which will be useul in he succeeding discussion. Deiniion []: he ilering error sysem (5) (6) is said o be sochasic inie-ime sable (SSFS) wih respec o ( c, c2,, R ), where R >, < c < c2 i or a given ime-consan >, he ollowing relaion holds: { Ξ xrx} < c Ξ { x Rx} < c2 [, ]. Deiniion 2: Given a disurbance aenuaion level γ >, he ilering error sysem (5) (6) is said o be robusly sochasic inie-ime sable (SFS) wih respec o c, c,, R wih a prescribed H disurbance aenuaion level γ, i i is robusly ( 2 ) sochasic inie-ime sable in he sense o Deiniion and e 2 γ v all nonzero 2 [ ) and all admissible uncerainies. 2 2 Ξ or 2 2 Lemma [4]: Le M, N and F be marices o appropriae dimensions, and hen or any scalar ε >, MFN N F M εmm ε N N F F I, + +. (7) Lemma 2 [5]: Le ABCF,,, and S be marices o appropriae dimensions such 77

4 ha S >, and F F ollowing inequaliy holds I. hen or any scalar ε >, such ha ( ε ) S ε BB >, he A + BFC S A + BFC A S BB A + ε C C. (8) Lemma [6] (Gronwall inequaliy): Le v be a nonnegaive uncion such ha or some consans, v a+ b vsd s,< < ab, hen we have v a exp b, < <. Lemma 4 [7] [8] (Schur complemen): Given a symmeric marix he ollowing hree condiions are equivalen o each oher: ) φ < ; 2) ) φ < and φ φ φ φ < ; φ < and φ φ φ φ <.. Robus Finie-ime H Filer Design φ φ2 φ = φ2 φ 22 heorem : Suppose ha he iler parameers A, B, C in (4) are given. he iler- c, c,, R, i here exis ing error sysem (5) (6) is robusly SSFS wih respec o scalars ε, ε 2, ε, ε 4, α > and symmeric posiive deinie marix P saisying P p = = R QR P P ε HH >, P ε B HH B >, such ha he ollowing LMIs hold Π<, (9) and Π C B P2 D B E G G G2 G 4 * Π22 * * Π * * * Π44 Π= * * * * εi * * * * * ε 2I * * * * * * ε I * * * * * * * ε 4I λ λ max min ( Q) Q c α e c 2 () () Π = P A + A P + ε P HH P αp, Π = P A + A P + ε P B HH B P αp, ε HH P Π =, Π = ε B HH B P

5 where * denoes he ransposed elemens in he symmeric posiions. Proo: Consider a sochasic Lyapunov uncion candidae deined as ollows: ( ξ) ξ ξ By Iô ormula, we have he sochasic dierenial dv ( ( ) ) o sysem (5) (6) wih v = as ollows: dv ( ξ( ) ) LV ( ξ( ) ) 2ξ ( ) P D( ) ξ( ) dw( ) where V = P, (2) = + ξ along he rajecories ( ξ) ξ ξ ξ ξ ξ ξ LV = P A + A P + D PD PA + AP CBP 2 P H = ξ ( ) F + ( )[ G ] PBC 2 P2 A + AP 2 P H F G F G F G PBH 2 PBH 2 + [ ] + [ ] [ + ] + ([ D ] + HF ( )[ G2 ] ) P( [ D ] + HF ( )[ G2 ] ) + ( BE + BHF [ G4 ] ) P 2 BE + BHF [ G4 ]} ξ ( ) We prove ( ξ) α ( ξ) LV < V () By Lemma and Lemma 2, we have PA + AP α P CBP 2 LV ( ξ( ) ) αv ( ξ( ) ) ξ ( ) PBC 2 P2 A + AP 2 α P2 P H G + ε [ H P ξ + ε G ] G + ε2 H B P2 ε2 PBH + 2 D G + + [ G ] 2 ( P εhh ) [ D ] ε [ G2 ] B E G 4 P2 ε4bhh B BE + ε4 G4 ξ + } [ ] By Lemma 4 and (7) (8), i ollows ha ( ξ) α ( ξ) ξ ξ LV V Π <, Inegraing boh sides o () rom o wih [, ] have By Lemma, i ollows ( ξ) { ( ξ)} ( ξ) α ( ξ) { } Ξ V < V + Ξ V s ds { V } V ( ξ) Ξ < α e, and aking expecaion, we 79

6 { } min 2 2 { ( )} { } Ξ V ξ =Ξ ξ R QR ξ λ Q Ξ ξ Rξ (4) From (2) and (), we obain V { } α ( ) 2 2 α ξ e = ξ R QR ξ e λmax ( Q) Rξ ξ α λ ( Q) ce max λ Ξ { ξ ξ} I implies Ξ ξ ξ, [, ] R c 2 e ( Q) ( Q) max α R ce λmin. heorem 2: For given γ >. he ilering error sysem (5) (6) is robusly SSFS wih respec o (,,, ) c c2 R, wih a prescribed H disurbance aenuaion level γ, i here exis scalars ε5, ε 6, α > and marices Θ, Θ2, Θ and he same marix P as heorem, saisying such ha () and he ollowing LMIs hold P >, ε5hh α Ω<. (5) AP + P A BP HP P G P P L P P D P P G P α P 2 * γ I * * ε 6 I Ω= * * * ε 6I * * * * I * * * * * ( ε5hh P ) * * * * * * ε 5I In his case, he suiable iler parameers A, B, C in sysem (4) can be given by A = P Θ, B = P Θ, C = P Θ P Proo: I ollows rom heorem and Schur complemens lemma ha he ilering error sysem (5) (6) is robusly SFS wih respec o (,,, ) Nex, we shall show ha he sysem (5) (6)) saisies 2 γ v where he Lyapunov uncion candidae V (, ( ) ) By Iô ormula, we have he sochasic dierenial as where c c R and () is ollowed. 2 Ξ e. (6) ξ is given in (2) ( ξ) = ( ξ) + ξ ξ + d V, LV, 2 P D D dw, (, ξ) = ξ ξ + + ξ + ξ LV P A Bv A Bv P ξ ξ + D + Dv P D + Dv 7

7 and ABDD,,, are deined in par 2. Ξ LV x s d s. ( ) Assume ( ) By (), we have ha Observe ha { } { LV s e s e s v s v s ds} 2 =Ξ γ d 2 Ξ ( ξ) γ + αv ( ξ( ) ) J e s e s v s v s s <Ξ 2 ( ξ) + γ α ( ξ) ξ( ) PA + AP+ LLαP PB 2 v( ) B P γ I LV e e v v V ( ) D ξ = + P D D D v By Lemma and Lemma 2, we have where, G = [ G G ] D P D D = D + HF G P D + HF G D 2 4, D D BE D ( ε ) D P HH D + G G 5 ε5 =, H = H BH. + + α B P + B P γ I PA AP LL P PB 2 γ I P A+ A P+ L L α P+ P B ε6p HH P ε6 G G 2 A where A =, H = H H B,. G = G G BC A hereore, using Lemma 4, i ollows ha 2 { ( ) ( ) LV ξ s e s e s γ v s v s αv ξ s ds} ξ ( s) ξ ( s) v ( s) ds Θ v( s) Ξ + Ξ where P A + A P+ L L αp+ ε6p HH P+ ε6 G G P B Θ= 2 B P γ I ( ε ) + D P 5HH D + ε6 G G (7) On he oher hand, le Θ = PA, Θ = PB, Θ = PCP,

8 pre- and pos-muliply (5) by {,,,,,,, } diag P I I I I I I I respecively. By Surch complemen, (5) implies Θ<. I ollows rom (7) ha diag P I I I I I I I, {,,,,,,, } J( ) (8) or all >. hen (6) ollows immediaely rom () and (8). 4. Numerical Example We now give a numerical example o illusrae he proposed approach. Suppose ha we have a Iô sochasic sysem in he orm o ()-() wih coeiciens A, A, D, D = = = = C =, C =, E =, E = H =, G =..2, G2 =.2..2 [ ] [ ] [.. ], [.. ], [.2 ] G = G = L =. 4 In his example by seing γ = 2, α =., c =.6, c 2 =.2, =, ε =., 4. ε =, R = diag { 2, 2} and applying heorem 2, we ind ha LMIs (5) is easible. hus he sysem is sochasic inie-ime sable wih respec o (.6,.2,, R ). Moreover, applying heorem 2, we can obain he corresponding iler parameers as ollows: 5. Conclusion A C =, B = = [ ] In his paper, he robus inie-ime H ilering problem has been sudied or Iô sochasic sysems. Based on LMI echnique, sochasic Lyapunov uncion mehod is adoped o obain a suicien condiion or he exisence o a inie-ime H iler. he resuling iler saisies prescribed perormance consrain. Reerences [] Berman, N. and Shaked, U. (25) H Filering or Nonlinear Sochasic Sysems. Proceedings o he h Medierranean Conerence on Conrol and Auomaion Limassol, Cyprus, June 25, [2] Wei, G.-L. and Shu, H.-S. (27) H Filering on Nonlinear Sochasic Sysems wih Delay. Chaos, Solions and Fracals,, hp://dx.doi.org/.6/j.chaos [] Zhang, W.-H., Chen, B. and seng, C.S. (25) Robus H Filering or Nonlinear Sochasic Sysems. IEEE racions on Signal Processing, 5,

9 hp://dx.doi.org/.9/sp [4] Wang, Z.-D., Lam, J. and Liu, X.-H. (27) H Filering or a Class o Nonlinear Discree-ime Sochasic Sysems wih Sae Delays. Journal o Compuaional and Applied Mahemaics, 2, 5-6. hp://dx.doi.org/.6/j.cam [5] Chen, Y., Wang, J.H., e al. (29) Dependen Filering o Uncerain Sochasic Sysems wih Delays. he 48h IEEE Conerence on Decision and Conrol, Shanghai, 6-8 December 29, [6] seng, C.S. (27) Fuzzy Filer Design or a Class o Nonlinear Sochasic Sysems. IEEE ransacions on Fuzzy Sysems, 5, hp://dx.doi.org/.9/fuzz [7] Moulay, E., Dambrine, M., Yeganear, N. and Perruquei, W. (28) Finie-ime Sabiliy and Sabilizaion o ime-delay Sysems. Sysems & Conrol Leers, 57, hp://dx.doi.org/.6/j.sysconle [8] Amao, F., Ambrosino, R., Ariola, M. and Cosenino, C. (29) Finie-ime Sabiliy o Linear ime-varying Sysems wih Jumps. Auomaica, 45, hp://dx.doi.org/.6/j.auomaica [9] Chen, W.S. and Jiao, L.C. (2) Io Finie-ime Sabiliy heorem o Sochasic Nonlinear Sysems. Auomaica, 46, hp://dx.doi.org/.6/j.auomaica [] Amao, F., Ambrosino, R., Ariola, M. and De ommasi, G. (2) Finie-ime Sabiliy o Impulsive Dynamical Linear Sysems Subjec o Norm-Bounded Uncerainies. Inernaional Journal o Robus and Nonlinear Conrol, 2, hp://dx.doi.org/.2/rnc.62 [] Xing, S.Y., Zhu, B.Y. and Zhang, Q.L. (2) Sochasic Finie-ime Sabilizaion o a Class o Sochasic -S Fuzzy Sysem wih he Io s-ype. Proceedings o he 2nd Chinese Conrol Conerence, Xi an, [2] Zhang, A.Q. and Campbell, S.L. (25), Robus Finie-ime Filering or Singular Discree-ime Sochasic Sysems. he 27h Chinese Conrol and Decision Conerence, Qingdao, [] Zhang, W.H. and An, X.Y. (28) Finie-ime Conrol o Linear Sochasic Sysems. Inernaional Journal o Innovaive Compuing, Inormaion and Conrol, 4, [4] Wang, Y. and DeSouza, C.E. (992) Robus Conrol o a Class o Uncerain Nonlinear Sysems. Sysem &. Conrol Leers, 9, hp://dx.doi.org/.6/67-69(92)997-c [5] Xu, S. and Chen,. (22) Robus H Conrol or Uncerain Sochasic Sysems wih Sae Delay. IEEE ransacions on Auomaic Conrol, 47, hp://dx.doi.org/.9/ac [6] Oksendal, B. (2) Sochasic Dierenial Equaions: An Inroducion wih Applicaions. 5h Ediion, Springer-Verlag, New York. [7] Boukas, E.K. (26) Saic Oupu Feedback Conrol or Sochasic Hybrid Sysems: LMI Approach. Auomaica, 42, hp://dx.doi.org/.6/j.auomaica [8] Boyd, S., Ghaoui, L.E., Feron, E. and Balakrishnan, V. (994) Linear Marix Inequaliy in Sysems and Conrol heory. SIAM Sudies in Applied Mahemaics. SIAM, Philadelphia. hp://dx.doi.org/.7/

10 Submi or recommend nex manuscrip o SCIRP and we will provide bes service or you: Acceping pre-submission inquiries hrough , Facebook, LinkedIn, wier, ec. A wide selecion o journals (inclusive o 9 subjecs, more han 2 journals) Providing 24-hour high-qualiy service User-riendly online submission sysem Fair and swi peer-review sysem Eicien ypeseing and prooreading procedure Display o he resul o downloads and visis, as well as he number o cied aricles Maximum disseminaion o your research work Submi your manuscrip a: hp://papersubmission.scirp.org/ Or conac jamp@scirp.org