Robust finite-horizon filtering for nonlinear timedelay Markovian jump systems with weighted tryonce-discard

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1 Systems Scence & Control Engneerng An Open Access Journal ISSN: (Prnt) (Onlne) Journal homepage: Robust fnte-horzon flterng for nonlnear tmedelay Markovan jump systems wth weghted tryonce-dscard protocol Le Lu Ywen Wang Lfeng Ma Je Zhang & Yumng Bo To cte ths artcle: Le Lu Ywen Wang Lfeng Ma Je Zhang & Yumng Bo (218) Robust fnte-horzon flterng for nonlnear tme-delay Markovan jump systems wth weghted try-once-dscard protocol Systems Scence & Control Engneerng 6: DOI: 1.18/ To lnk to ths artcle: The Author(s). Publshed by Informa UK Lmted tradng as Taylor & Francs Group. Publshed onlne: 14 May 218. Submt your artcle to ths journal Artcle vews: 221 Vew Crossmark data Full Terms & Condtons of access and use can be found at

2 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 218 VOL. 6 NO Robust fnte-horzon flterng for nonlnear tme-delay Markovan jump systems wth weghted try-once-dscard protocol Le Lu Ywen Wang Lfeng Ma Je Zhang and Yumng Bo School of Automaton Nanjng Unversty of Scence and Technology Nanjng Chna ABSTRACT In ths paper the network-based robust H flterng problem s nvestgated for a class of nonlnear delayed Markovan jump systems subject to packet dropouts. To regulate the transmsson order of multple sensor nodes the Weghted Try-Once-Dscard (WTOD) protocol s adopted to avod data collson and mprove transmsson accuracy and effcency. Va the schedulng of WTOD protocol only one sensor node s allowed to get access to the shared communcaton network at each step to send data. The purpose of the addressed problem s to desgn a network-based flter such that the prespecfed H dsturbance attenuaton level s guaranteed wth the help of schedulng of WTOD protocol. Suffcent condtons are establshed for the exstence of desred flter the flter parameters are obtaned by solvng the proposed seres of recursve lnear matrx nequaltes (RLMIs). Fnally an llustratve example s gven to demonstrate the effectveness of the presented flter desgn method. ARTICLE HISTORY Receved 22 March 218 Accepted 5 May 218 KEYWORDS Weghted try-once-dscard protocol; Markovan jump systems; H flterng; packet dropouts; networked systems 1. Introducton As a class of stochastc hybrd systems Markovan jump systems (MJSs) have been extensvely utlzed to model the dynamc systems wth random abrupt varatons of system structure (Boukas 26; Sh Mahmoud Nguang &Ismal26; Wu Sh & Gao 21). Consequently the control and flterng problems for Markovan jump systems have attracted persstent research attentons and frutful results have been so far avalable n the lterature see e.g. Sh Boukas and Agarwal (1999) Zhang and Boukas (29) Chen Nu and Zou (213) Chen Nu and Zou (214) Xu Lam and Mao (27)andMaWang Han and Lu (217). Among vares of flter desgn algorthms the H flterng approach has been exploted to handle systems subject to external nose sgnals wth bounded energy but unknown statstcs (Ma Wang Han &Lam218; Wang Wang Han & We 218; Zhang et al. 217; Zhang Wang Dng & Lu 215). For nstance n Chen et al. (213) the adaptve sldng mode control problem has been nvestgated for a class of Markovan jump systems wth actuator degradaton. The modedependent H flter desgn problem has been solved n Zhang and Boukas (29) for Markovan jump lnear systems wth partly unknown transton probabltes. In recent years networked systems have strred much attenton from both theoretcal research and ndustral applcaton due to the advantages of low cost wreless transmsson and hgh relablty and accordngly the relevant control and flterng ssues have become hot research drectons (Chen 199; Dng Wang Ho & We 217; Dng Wang Shen & Dong 215b; Dng Wang Shen & We 215; Ma Wang & Lam 217; Ma Wang Lam & Kyrakouls 217; Shen Wang & Qao 217; Sun Xe & Xao 28; Yuan Wang & Guo 217; YuanYuan Wang Guo & Yang 217; ZhuHua&Wang28). In a networked system the components (e.g. sensors flter and controller) are connected through a shared network transmsson channel. In comparson to the tradtonal pont-to-pont connecton systems the ntroducton of the transmsson network nevtably brngs a lot of network-nduced phenomena such as packet dropouts (Dong Lam & Gao 211; Sun Xe Xao & Soh 28; Wen Wang Hu Lu & Alsaad 218) sgnal quantzaton (Chen Wang Qan & Alsaad 218; Dong Wang Dng &Gao215b; Gao & Chen 27; Geng Wang Lang Cheng & Alsaad 217b; Tan Yue & Peng 28; Zhang Ma & Lu 216; Zhang Wang & Ma 216) communcaton delay (Dng et al. 215b; Zhu et al. 28) fadng channels (Dng Wang Shen & Dong 215a; Dong Wang Dng & Gao 215a; Geng Wang Lang Cheng & Alsaad 217a; Zhang et al. 218; Zhang Wang Zou &Fang217; Zhang Wang Zou & Lu 216) to name CONTACT Lfeng Ma malfeng@njust.edu.cn 218 The Author(s). Publshed by Informa UK Lmted tradng as Taylor & Francs Group. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgnal work s properly cted.

3 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 181 but a few. These network-nduced phenomena lead to addtonal dffcultes n analyss/synthess problems and therefore have attracted consderable research nterest. Based on the event-trggered communcaton mechansm n Dong et al. (215a) the H flterng problem s researched for a class of nonlnear tme-varyng systems wth fadng channels and multplcatve noses. It should be ponted out that most of exstng results concernng the flterng problems of networked systems are based on the assumpton that all sensor nodes could smultaneously send measurement sgnals va the shared network channel. However n real-world communcaton network due to the lmted network bandwdth the smultaneous transmsson of measurement sgnals from multple sensor nodes to the shared transmsson network wll nevtably cause data collsons (Zhang Yu & Feng 211; Zou Wang Gao & Alsaad 217; Zou Wang Hu & Gao 217). Therefore the aforementoned assumpton s unreasonable and communcaton protocols are needed to determne the transmsson order of multple nodes wth whch only one node s permtted to send sgnal to the shared network at each transmsson nstant. So far a lot of communcaton protocols have been proposed and wdely used n practcal engneerng such as the Round-Robn protocol (Ugrnovsk & Frdman 214) the Weghted Try-Once-Dscard (WTOD) protocol (Donkers Heemels van de Wouw & Hetel 211; Walsh Ye & Bushnell 22; Zou Wang & Gao 216a) and stochastc communcaton protocol (Tabbara & Nesc 28; Zou Wang & Gao 216b). Compared wth the other two transmsson protocols WTOD protocol s relatvely complex and dffcult to be analyzed and desgned. Nevertheless as a dynamc protocol t could choose the most valuable measurement sgnal from multple sensor nodes. Recently the control and flterng problems of networked systems under the schedulng of WTOD protocol have appeared n some ntal results (Donkers et al. 211; Zou et al. 216a). For nstance n Donkers et al. (211) under the schedulng of WTOD protocol a swtched lnear system approach has been proposed to analyze the stablty of networked control systems. The set-membershp flter has been desgned n Zou et al. (216a) for lnear mxed tme-delay system under WTOD protocol and Round-Robn protocol. Unfortunately when WTOD protocol s consdered the network-based H flterng problem for Markovan jump systems s stll an open and challengng ssue. As such to shorten such a gap the WTOD protocol s utlzed n ths research for the networked flterng problem of Markovan jump systems. On the other hand as s well known packet dropout s a common network-nduced phenomenon (Dong et al. 211) whch may degrade the flterng performance of networked systems. As such takng the phenomenon of packet dropouts nto consderaton wll make the research result more practcal. However the couplng between WTOD protocol and packet dropout s a key ssue whch wll make the networked flterng problem become complex. Ths motvates our research nterest and the couplng problem wll be solved n the followng work. It s well known that tme-delays and nonlneartes wdely exst n actual system and have great nfluence on system performance (Shen Wang & Tan 218). The control and flterng problems for networked systems wth tme-delays and nonlneartes have been studed n a large amount see e.g. Ca Wang Xu Lu and Alsaad (215) Ma Wang and Lam (217) Dong Wang Shen and Dng (216) Ma Wang Lu and Alsaad (217) and Lyu and Bo (217) and the references theren. Summarzng the above dscussons an nterestng and challengng research problem s clear that s nvestgatng the H flterng problem for nonlnear Markovan jump systems wth packet dropout under the schedulng of WTOD protocol. Motvated by the above dscussons the networkbased H fnte-horzon flterng problem s nvestgated for a class of dscrete tme-varyng nonlnear Markovan jump systems wth packet dropouts and tme-delay under the schedulng of WTOD protocol. Suffcent condtons are establshed for the exstence of the desgned flter n terms of the feasblty of a seres of recursve lnear matrx nequaltes (RLMIs). A smulaton example s presented to show the effectveness of the proposed method. The man contrbutons of ths paper are hghlghted as follows: (1) The H flterng problem s for the frst tme researched for dscrete-tme nonlnear Markovan jump systems under the schedulng of WTOD protocol. (2) For the purpose of descrbng the transmsson of measurement sgnals from multple sensor nodes to the remote flter a unfed network transmsson framework combnng the WTOD protocol and packet dropouts s constructed and descrbed n a mathematcal way. (3) The mpact from the WTOD protocol on the flter parameters s consdered durng the process of flter desgn. (4) The flter desgn algorthm s addressed and the flter gan matrces can be obtaned by solvng the correspondng RLMIs. The rest of ths paper s organzed as follows. In Secton 2 the networked H flterng problem for a class

4 182 L. LIU ET AL. of nonlnear Markovan jump systems wth tme-delay under the schedulng of WTOD protocol s ntroduced and formulated. In Secton 3 several useful lemmas are lsted and suffcent condtons guaranteeng the exstence of the desred H flter are derved. The flter desgn algorthm under the schedulng of WTOD protocol s solved n Secton 4 and a numercal example s gven n Secton 5 to demonstrate the proposed method. Fnally we conclude n Secton 6. Notaton. The notaton used here s farly standard except otherwse stated. R n denotes the n- dmensonal Eucldean space. l 2 N 1 s the space of square-summable vector functons over N 1. E{x} and E{x y} respectvely stand for the expectaton of the stochastc varable x and expectaton of x condtonal on y. x descrbes the Eucldean norm of a vector x. A T represents the transpose of A. I n denotes the dentty matrx of n dmensons. The notaton X Y (respectvely X > Y ) X and Y are symmetrc matrces means that X Y s postve sem-defnte (respectvely postve defnte). dag{f 1 F 2...} stands for a block-dagonal matrx whose dagonal blocks are gven by F 1 F 2... The Kronecker delta functon δ(l) s a bnary functon that equals 1 f l = and otherwse. The symbol * n a matrx means that the correspondng term of the matrx can be obtaned by symmetrc property. 2. Problem formulaton and prelmnares 2.1. The weghted try-once-dscard (WTOD) communcaton protocol The communcaton network under the schedulng of WTOD protocol s descrbed as follows. Consder a networked system wth L nodes and the set of nodes s labeled as P ={ L}. In ths system the shared communcaton network s utlzed to transmt sgnals n whch only one node s allowed to get access to the network at each transmsson nstant. Let ζ P be the selected node whch could send sgnals to the communcaton network at tme nstant k. Under the schedulng of WTOD protocol the value of ζ s determned as follows: ζ := arg max (y l yl 1 l L )T W l (y l yl ) (1) yl represents the last successfully transmtted sgnal before tme nstant k of node l and W l (l P) s a known postve defnte weght matrx of the lth node. In practcal engneerng W l (l P) can be determned on bass of the physcal meanng of measurement output. Defne y y T 1 yt 2 yt L T and y (y 1 )T (y 2 )T (y L )T T.Werewrte(1)as ζ = arg max 1 l L (y y ) T W l (y y ) (2) W = dag{w 1 W 2... W L } W l = W l l = dag {δ(l 1)I δ(l 2)I... δ(l L)I}(1 l L) and δ( ) { 1} s the Kronecker delta functon. Remark 2.1: Consderng the exstence of the networknduced packet dropouts the WTOD protocol s modfed accordngly. It s assumed that the nformaton of packet dropouts could be transmtted from the flter end to the protocol end. Therefore n the schedulng rule (2) of WTOD protocol y s not the last transmtted sgnal before tme nstantk but the last successfully transmtted sgnal before tme nstant k whch s dfferent from Zou et al. (216a) Problem formulaton Let r (k N) be a Markov chan takng values n a fnte state space S ={ M} wth transton probablty matrx ˆ = λ j gvenby Prob{r(k + 1) = j r = } =λ j j S λ j ( j S) s the transton probablty from to j and M j=1 λ j = 1 ( S). In ths paper we consder the followng class of stochastc nonlnear tme-delay Markovan jump system defned on k N: x(k + 1) = (A(k r) + A(k r))x + (A d (k r) + A d (k r)) x(k d) + αg(k x) + B(k r)w y = C(k r)x + D(k r)v z = L(k r)x x R n x represents the state vector y R n y s the process output z R n z s the sgnal to be estmated w R n w and v R n v are the external dsturbance sgnals that belong to l 2 N 1 and g( ) : R + R n R n s nonlnear vector functon. d > sa known postve scalar. For fxed system mode A(k r) A d (k r) C(k r) B(k r) D(k r) and L(k r) are known real-valued tme-varyng matrces wth approprate dmensons. For smplcty for fxed system mode r = amatrxu(k ) wll be denoted by U U = {A A d C B D L}. (3)

5 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 183 The real-valued matrces A and A d represent the norm-bounded parameter uncertantes of the followng structure A A d = H FE A E d (4) H E A and E d are known real tme-varyng matrces and F s an unknown matrx satsfyng the followng condton F T F I (5) The stochastc varable α n (3) s ntroduced to account for the random nature of the occurrence of the nonlnearty whch s a Bernoull dstrbuted whte sequence takng values on or 1 wth The measurement output after transmsson through the network s denoted as ỹ = ỹ T 1 ỹt 2 ỹt L T The ntal state of ỹ s assumed to be ỹ(j) = ỹ for j < ỹ s a known vector. When WTOD protocol and the phenomenon of packet dropouts are consdered the updatng rule of ỹ l (k N + l P) s set to be (1 β)ỹ l (k 1) f l = ζ ỹ l = +βy l ỹ l (k 1) otherwse (9) Prob{α = 1} =E{α} =ᾱ Prob{α = } =1 ᾱ (6) Accordng to (9) we have ᾱ 1 s a known constant. Smlar to Dong Wang and Gao (212)andZhangMa et al. (217) the nonlnear functon g(k x) s assumed to satsfy g = and for arbtrary σ R n x g(k x + σ) g(k x) Gσ (7) G s a known matrx. Next the sgnal transmsson of the measurement output y va the protocol-based network wll be ntroduced. To ths end we dvde the sensors of the system nto L sensor nodes accordng to ther spatal dstrbuton. Therefore the measurement output y can be rewrtten as y = y T 1 yt 2 yt L T y l (l P) s the measurement of the lth sensor node before transmsson. The communcaton network s scheduled by WTOD protocol ζ P s the selected sensor node obtanng access to the communcaton network at tme nstant k. The value of ζ s determned by the schedulng of WTOD protocol. As s well known owng to varous reasons the broadcast nnovaton could face packet dropouts. In vew of ths the stochastc varable β s ntroduced to cater for the phenomenon of packet dropouts. β s a Bernoull dstrbuted whte sequence whch s uncorrelated to α and takes values on or 1 wth Prob{β = 1} = E{β} = β Prob{β = } =1 β β 1 s a known constant. (8) ỹ = β ζ y + (I ny β ζ )ỹ(k 1) ỹ(j) = ỹ j < (1) Remark 2.2: For tme-varyng Markovan jump system the dynamc schedulng characterstcs of WTOD protocol may brng effectve flterng performance. The WTOD protocol wll be executed before the data transmsson and the phenomenon of packet dropouts s occurred durng the process of measurement transmsson. In (1) the unfed measurement transmsson model for a networked system wth WTOD protocol and packet dropouts s establshed the phenomenon of packet dropouts s descrbed by the commonly used Bernoull sequence β and the schedulng rule of WTOD protocol s explaned n (2). Based on ths measurement transmsson model the H flterng problem under the schedulng of WTOD protocol wll be nvestgated n the followng part. Lettng η = x T ỹ T (k 1) T and w = w T v T T system (3) wth WTOD protocol schedulng can be reformulated as follows: η(k + 1) = (Ā + βã )η + Ā d η(k d) + α g(k η) + ( B + β B ) w ỹ = ( C + β C )η + ( D + β D ) w z = L η (11)

6 184 L. LIU ET AL. A + A Ā = β ζ C I ny β ζ Ã = ζ C ζ Ad + A Ā d = d B B = β ζ D B = ζ D g(k x) g(k η) = β = β β C = β ζ C I ny β ζ C = ζ C ζ D = β ζ D D = ζ D L = L 2.3. Tme-varyng flter In ths paper a mode-dependent flter based on the sgnal ỹ s consdered for the augmented system (11) whch s of the form ˆx(k + 1) = A f ˆx + B f ỹ ẑ = L f ˆx (12) ˆx R n x represents the estmate of x ẑ R n z s the estmate of the output z. For fxed system mode r = the tme-varyng matrces A f B f and L f are flter gan matrces wth approprate dmensons to be determned. By lettng η = η T ˆx T T z = z ẑwe subsequently obtan the followng augmented system to be nvestgated: η(k + 1) = (A + β A ) η + αḡ(k η) + A d E η(k d) + (B + β B w z = L η (13) Ā A = B f C A f A Ã = B f C g(k η) ḡ(k η) = Ād A d = E = I B B = B f D B B = B f D L = L L f Our am n ths paper s to desgn a tme-varyng flter of the form (12) such that under the schedulng of WTOD protocol for the gven postve scalar γ the matrces U () > ( S) U(j) > (j = d) and the ntal state x and ỹ system (13) satsfes the followng H flterng performance requrement: J : = E{ z 2 2 γ 2 w 2 2 } γ 2 η T ()U () η() γ 2 1 j= d η T (j)e T U(j)E η(j) ({ w} x ) (14) for all stochastc nonlneartes N 1 z 2 2 = z 2 k= 3. Analyss of H performances N 1 w 2 2 = w 2 k= Frst of all we ntroduce the followng lemmas whch wll be used n ths paper. Lemma 3.1: (Schur complement Boyd Ghaou Feron & Balakrshnan 1994): Gven constant matrces S 1 S 2 and S 3 S 1 = S T 1 and < S 2 = S T 2 then S 1 + S T 3 S 1 2 S 3 < f and only f S1 S T 3 S2 S < or 3 S 3 S 2 S T < (15) 3 S 1 Lemma 3.2 (S-procedure Boyd et al. 1994): Let N = N T H and E be real matrces of approprate dmensons and F T F I. Then nequalty N + HFE + (HFE) T < f and

7 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 185 only f there exsts a postve scalar ε such that N + εhh T + ε 1 E T E < or equvalently N εh E T εh εi < (16) E εi Lemma 3.3 (Boyd et al. 1994): Let V (χ) V 1 (χ)... V p (χ) be quadratc functons of χ R n V (χ) = χ T T χ ( = 1... p) wth T T = T. Then the followng s true V 1 (χ)... V p (χ) V (χ) f there exst γ 1... γ p > such that p T γ T (17) =1 Theorem 3.1: Consder system (3) the WTOD protocol gven by (1) and the flter (12). Let the dsturbance attenuaton level γ> the flter gan matrces {A f } k N 1 {B f } k N 1 and {L f } k N 1 be gven. For gven U () > ( S) and U(j) > (j = d) the H performance requrement defned n (14) s acheved for all nonzero w f there exst sequences of postve defnte matrces {P } k N {Q} k dn 1 and postve scalars {ε 1 } k N 1 and {ρ l } k N 1l P satsfyng the followng recursve matrx nequaltes: ˆƔ 11 ˆƔ 12 ˆƔ 13 ˆƔ = ˆƔ 22 (18) I (3ᾱ 2 +ᾱ) P (k + 1) ε 1 I ( S) (19) wth the ntal condton ˆƔ 11 = η T ()(P () γ 2 U ()) η() + 1 l= d η T (l)e T (Q(l) γ 2 U(l))E η(l) (2) P + ε 1 Ĝ T Ĝ + E T QE L ρ l V T W lζ V l=1 L ρ l V T W lζ ˆD l=1 Q(k d) γ 2 I L ρ l ˆD T W lζ ˆD l=1 ˆƔ 13 = L T ˆƔ 22 = dag{ P (k + 1) P (k + 1) P (k + 1) P (k + 1) P (k + 1)} Ḡ = G Ĝ = Ḡ V = C I ˆD = D W lζ = W l W ζ P (k + 1) = M λ j P j (k + 1) j=1 ˆƔ 12 A T P (k + 1) A T P (k + 1) = A T d P (k + 1) A T d P (k + 1) B T P (k + 1) β(1 β) A T P (k + 1) B T P (k + 1) β(1 β) B T P (k + 1) Proof: Defne the followng functon for system (13) V(k r) := V 1 (k r) + V 2 (k r) (21) V 1 (k r) = η T P r η V 2 (k r) = k 1 j=k d η T (j)e T Q(j)E η(j) wth P r and Q satsfyng (18) and (19). Then along the trajectory of system (13) we have E{ V r = } = E{V(k + 1 r(k + 1)) r = } V(k r) = 2 E{ V r = } (22) =1 Calculatng E{ V 1 r = } along the trajectory of system(13)wehave E{ V 1 r = } = E{ η T (k + 1) P (k + 1) η(k + 1) η T P η r = } = E{ η T (A T P (k + 1)A + β(1 β) A T P (k + 1) A P ) η + 2 η T A T P (k + 1)A d η(k d) + 2ᾱ η T A T P (k + 1)ḡ(k η) + 2 η T (A T P (k + 1)B + β(1 β) A T P (k + 1) B ) w

8 186 L. LIU ET AL. + η T (k d)a T d P (k + 1)A d η(k d) + 2ᾱη T (k d)a T d P (k + 1)ḡ(k η) + 2η T (k d)a T d P (k + 1) w +ᾱḡ T (k η) P (k + 1)ḡ(k η) + 2ᾱḡ T (k η) P (k + 1)B w + w T (B T P (k + 1)B + β(1 β) B T P (k + 1) B ) w} E{ η T (2A T P (k + 1)A + β(1 β) A T P (k + 1) A P ) η + 2 η T A T P (k + 1)A d η(k d) + 2 η T (A T P (k + 1)B + β(1 β) A T P (k + 1) B ) w + 2η T (k d)a T d P (k + 1)A d η(k d) + 2η T (k d)a T d P (k + 1) w + (3ᾱ 2 +ᾱ)ḡ T (k η) P (k + 1)ḡ(k η) + w T (2B T P (k + 1)B + β(1 β) B T P (k + 1) B ) w} (23) Consderng nequalty (19) and the nonlnear constrant condton (7) we have (3ᾱ 2 +ᾱ)ḡ T (k η) P (k + 1)ḡ(k η) ε 1 ḡ T (k η)ḡ(k η) ε 1 η T Ĝ T Ĝ η (24) Next t can be derved that E{ V 2 r = } = E{V 2 (k + 1 r(k + 1)) r = } V 2 (k r) = E{η T Qη η T (k d)q(k d)η(k d)} (25) Lettng ξ = η T η T (k d) w T T and combnng (23) (24) and (25) result n E{ V r = } E{ξ T ˆ ξ}. (26) ˆ 11 ˆ = ˆ 13 A T d P (k + 1)B ˆ 33 A T P (k + 1)A d A T d P (k + 1)A d Q(k d) ˆ 11 = 2A T P (k + 1)A + E T QE + β(1 β) A T P (k + 1) A P + ε 1 Ĝ T Ĝ ˆ 13 = A T P (k + 1)B + β(1 β) A T P (k + 1) B ˆ 33 = B T P (k + 1)B + β(1 β) B T P (k + 1) B Now n order to obtan the H performance defned n (14) we defne: J = E{ z T z γ 2 w T w} (27) Addng zero term E{ V r = } E{ V r = } to the rght sde of equaton (27) and summng up (27) on both sdes from to N 1 wth respect to kweobtan N 1 k= = N 1 J E{ ξ T ξ}+v( r()) (28) k= ˆ 11 + L T L A T P (k + 1)A d ˆ 13 A T d P (k + 1)A d A T d P (k + 1) Q(k d) B ˆ 33 γ 2 I Moreover t follows from the above nequalty (28) that J = E{ z 2 2 γ 2 w 2 2 } γ 2 η T ()U () η() γ 2 1 j= d η T (j)e T U(j)E η(j)

9 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 187 N 1 E{ ξ T ξ}+v( r()) k= γ 2 η T ()U () η() γ 2 1 j= d η T (j)e T U(j)E η(j) (29) Accordng to the ntal condton (2) t s easy to obtan that V( r()) γ 2 η T ()U () η() γ 2 1 j= d η T (j)e T U(j)E η(j) ( S) (3) Now to prove J we are n the poston to derve E{ N 1 k= ξ T ξ}. By analyzng the schedulng mechansm of packet dropouts-constraned WTOD protocol we obtan that for any l P (y y ) T ( W l W ζ )(y y ) (31) whch can be wrtten n terms of η as η T Ū T ( W l W ζ )Ū η (32) Ū = V ˆD Accordng to Lemma 3.3 f there exst ρ 1 ρ 2... ρ L >suchthat = L ρ l Ū T ( W l W ζ )Ū l=1 = (33) ˆ 11 + L T L L l=1 ρ l V T W lζ V A T P (k + 1)A d A T d P (k + 1)A d Q(k d) ˆ 13 L l=1 ρ l V T W lζ ˆD A T d P (k + 1)B ˆ 33 γ 2 I L l=1 ρ l ˆD T W lζ ˆD then nequalty (32) mples E{ N 1 k= ξ T ξ}. Furthermore accordng to Lemma 3.1 t s easy to see that nequalty (33) s equvalent to (18). Summarzng the above dervaton we obtan J. Therefore the H ndex defned n (14) s guaranteed. Ths proof s completed. Remark 3.1: In Theorem 3.1 the mpact from the WTOD protocol on the H flterng problem for the addressed dscrete tme-varyng Markovan systems has been analyzed. The suffcent condtons have been derved that guarantee the exstence of desred flter under the schedulng of WTOD protocol. However due to the cross couplng of matrces n (18) t s dffcult to drectly desgn the desred flter by Theorem 3.1. To elmnate the couplng terms smlar to Zhang and Boukas (29) the slack matrx varables wll be ntroduced and the correspondng result s shown n Theorem 3.2. Theorem 3.2: Consder system (3). Let the dsturbance attenuaton level γ> the flter gan matrces {A f } k N 1 {B f } k N 1 and {L f } k N 1 be gven. For gven U () > ( S) U(j) > (j = d) the H performance requrement defned n (14) s acheved for all nonzero w f there exst sequencesofpostvedefntematrces{p } k N {Q} k dn 1 sequences of postve scalars {ε 1 } k N 1 and {ρ l } k N 1l P and a sequence of real-valued matrces {R } k N 1 satsfyng the followng recursve matrx nequaltes: Ɣ = ˆƔ 11 Ɣ 12 ˆƔ 13 Ɣ 22 I (34) (3ᾱ 2 +ᾱ) P (k + 1) ε 1 I ( S) (35) wth the ntal condton η T ()(P () γ 2 U ()) η() + 1 l= d Ɣ 12 = η T (l)e T (Q(l) γ 2 U(l))E η(l) (36) A T R A T R A T d R A T d R B TR β(1 β) A T R B TR β(1 β) B TR

10 188 L. LIU ET AL. R = P (k + 1) R R T Ɣ 22 = dag{r... R } }{{} 5 Proof: By Theorem 3.1 the augmented system (13) satsfes the H performance defned n (14) f nequaltes (18) and (19) hold wth the ntal condton (2). Compared wth Theorem 3.2 we just need to verfy that nequalty (18) can be derved from nequalty (34). In fact from nequalty (34) we know that P (k + 1) R R T. Accordng to P (k + 1) > we have R + R T > whch means that R are nonsngular matrces. On the other hand for arbtrary R the followng s true: ( P (k + 1) R ) T P 1 (k + 1)( P (k + 1) R ) (37) whch s equvalent to P (k + 1) R R T RT 1 (k + 1) P R (k + 1) (38) It follows from nequalty (34) that ˆƔ 11 Ɣ 12 ˆƔ 13 Ɣ = Ɣ 22 (39) I R = R T (k + 1) P 1 R (k + 1) Ɣ 22 = dag{ R... R } }{{} 5 Performng now a congruence transformaton on (39) usng dag{i I I R 1 P (k + 1) R 1 P (k + 1) R 1 P (k + 1) R 1 P (k + 1) R 1 P (k + 1) I}wecanconclude that nequalty (18) holds. Ths proof s completed. 4. Fnte horzon flter desgn under the schedulng of WTOD protocol The flter desgn algorthm for the Markovan systems under the schedulng of WTOD protocol s proposed n the followng theorem the desred flter matrces could be obtaned by solvng a set of recursve lnear matrx nequaltes (RLMIs). Theorem 4.1: Consder system (3). For the gven dsturbance attenuaton level γ> U () > ( S) U(j) > (j = d) the H performance requrement defned n (14) s acheved for all nonzero w f there exst a sequence of postve defnte matrces {Q} k dn 1 sequences of postve scalars {ε 1 } k N 1 {ε 2 } k N 1 and {ρ l } k N 1l P and famles of real-valued matrces {P 1 } k N {P 2 } k N {P 3 } k N {R 11 } k N 1 {R 12 } k N 1 {R 22 } k N 1 {X } k N 1 {Y } k N 1 and{l f } k N 1 satsfyngthefollowngrecursvematrx nequaltes: P1 P P = 3 > P 2 (4) = (41) (3ᾱ 2 +ᾱ) P 1 (k + 1) ε 1 I (3ᾱ 2 +ᾱ) P 3 (k + 1) (3ᾱ 2 +ᾱ) P 2 (k + 1) ε 1 I ( S) (42) wth the ntal condton η T ()(P () γ 2 U ()) η() + 1 l= d 11 = η T (l)e T (Q(l) γ 2 U(l))E η(l) (43) (11) 11 P 3 P 2 Q(k d) L ρ l Ĉ T W lζ ˆD l=1 γ 2 I L ρ l ˆD T W lζ ˆD l=1 (11) 12 = 12 (13) 12 (21) 12 (22) 12 (23) = ε 2 (11) = dag{p... P I} }{{} 5 23 = (12) = dag{ ε 2 I... ε 2 I} }{{} 4

11 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 189 (11) 11 = P 1 + ε 1 Ḡ T Ḡ + Q L ρ l Ĉ T W lζ Ĉ l=1  T (11) R 11  T 12 = + C T R 12 YT U + C TYT X T U X T  T R 11  T + C T R 12 YT U + C TYT X T U X T β(1 β)(ã T (13) R = + C TYT U) β(1 β)(ã T R 12 L + C T T YT ) (21) 12 = L T f  T d R 11 B T R 11 + D T YT U  T d R 12 B T R 12 + D T YT (22) ÂT 12 = d R 11  T d R 12 B T R 11 B T + D T R 12 YT U + D T YT (23) 12 = β(1 β)( B T R 11 + D T YT U) (11) 13 = (12) 23 = β(1 β)( B T R 12 + B T YT ) ÊA T Êd T R 1 R 2 R T 11 Ĥ R T 11 Ĥ R 1 = R T 12 Ĥ R T 12 Ĥ R T 11 Ĥ R 2 = R T 12 Ĥ R T 11 Ĥ R T 12 Ĥ P P = 1 R 11 R T 11 P 3 R 12 U T R T 22 P 2 R 22 R T 22 A  = β ζ C I ny β ζ Ad  d = H Ĥ = Ê d = E d Ê A = E A Ĉ = C Moreover f (41) and (42) have feasble solutons then the system matrx L f of the admssble flter n the form of (12) can be drectly obtaned and A f and B f can be obtaned by means of the matrces X Y and R 22 as follows: A f = R T 22 X B f = R T 22 Y (44) Proof: Frst we assume that P and R have the followng forms: P1 P P = 3 P 2 (45) R11 R R = 12 R 22 U R 22 U s a gven matrx wth approprate dmenson. Then by defnng new matrx varables X = R T 22 A f and Y = R T 22 B f (34) can be wrtten as follows: < (46) 22 (11) 12 = 12 (13) 12 (21) 12 (22) 12 (23) 12 Ā T (11) R 11 Ā T 12 = + C T R 12 YT U + C TYT X T U X T Ā T R 11 Ā T + C T R 12 YT U + C TYT X T U X T

12 19 L. LIU ET AL. (21) 12 = Ā T d R 11 B T R 11 + D T YT U Ā T d R 12 B T R 12 + D T YT (22) ĀT 12 = d R 11 Ā T d R 12 B T R 11 + D T YT U B T R 12 + D T YT In order to elmnate the parameter uncertanty A and A d n (46) we rewrte t n the followng form: N FE + ( N FE ) T < (47) N = E = ( (12) 23 ) T (11) 13 F = dag{f F} Accordng to Lemma 3.2 nequalty (47) holds f and only f (41) holds. On the other hand t s easy to see that (35) and (36) are equvalent to (42) and (43) respectvely. Thus accordng to Theorem 3.2 under the schedulng of WTOD protocol the H performance requrement of the flterng error system (13) s satsfed wth ntal condtons (43). Ths completes the proof. Remark 4.1: So far we have solved the flter desgn problem for a class of dscrete tme-varyng nonlnear Markovan systems under the schedulng of WTOD protocol. For the flterng error dynamcs (13) the suffcent condtons have been derved n Theorem 3.1 guaranteeng the exstence of the desred H flter. Furthermore to elmnate the matrx product the slack matrx varables have been ntroduced n Theorem 3.2. Then the protocolbased flter desgn method for a class of Markovan jump systems has been proposed n terms of a set of RLMIs n Theorem 4.1. By means of Theorem 4.1 we can summarze the Protocol-Based Robust H Flter Desgn Algorthm (PBRHFDA) as follows. 5. Numercal smulatons In ths secton a numercal smulaton example s presented to llustrate the effectveness of the proposed flter desgn method. The system parameters are gven as follows: Mode 1: A 1 = sn (2k) cos (2k) A d1 = B 1 = cos (2k) sn sn (3k) C 1 = D 1 = sn L 1 = sn (3k).1 H 1 = E A1 = E d1 = Mode 2:.5.1 A 2 = sn (2k) +.1 sn cos (2k) cos (2k).1.1 A d2 = cos (2k) B 2 =.1.3 sn (3k) D 2 = sn (3k) C 2 = L 2 = sn (3k).2.2 H 2 =.2 E A2 = E d2 =.2.1.1

13 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 191 Algorthm PBRHFDA Step 1. Gven the dsturbance attenuaton level γ> the postve defnte matrces U ()( S) and U(l)(l = d d ). Choose ntal postve defnte matrces P ()( S) and Q(l)(l = d d ) to satsfy the condton (43) and set k =. Step 2. Obtan the values of matrces {P 1 (k + 1) P 2 (k + 1) P 3 (k + 1) Q R 11 (k + 1) R 12 (k + 1) R 22 (k + 1) X Y } and the desred flter parameters A f B f L f for the samplng nstant k by solvng (41) and (42). Step 3. Set k = k + 1 and then P 1 = P 1 (k + 1) P 2 = P 2 (k + 1) P 3 = P 3 (k + 1) Q(k d) = Q(k d + 1). Step 4. If k < NthengotoStep2elsegotoStep5. Step 5. Stop. Mode 3: A 3 = sn (2k) sn +.2 cos (2k) cos (2k).1.1 A d3 = B 3 = sn D 3 = sn (3k) C 3 = L 3 = sn (3k) H 3 = E d3 = E A3 = The nonlnear functon g(k x) and the external dsturbance sgnals w v are chosen as g(k x) =.2x 1 cos (x 3 ).5x 2 x x 3 sn (2x 2 ) w = exp (.5k) sn v = 2exp(.4k) cos (2k) process and matrx U n R are chosen as follows: ˆ = U = The sensors are dvded nto two sensor nodes and the weght matrces of WTOD protocol are set to be W 1 = W 2 = 1.2. The ntal value of state and ts estmaton are x() = T and ˆx() = T respectvely. For gven dsturbance attenuaton level γ =.95 and postve defnte matrces U () = 5I 8 ( S) U( 2) = U( 1) = 3I 5 choose P () = 3I 8 ( S) Q( 2) = Q( 1) = I 5 to satsfy the ntal condton (34). Accordng to PBRHFDA algorthm the RLMIs n Theorem (4.1) can be solved recursvely subject to gven ntal condtons. The smulaton results are shown n Fgures 1 5. Fgure 1 depcts the state varables x 2 and ts estmaton ˆx 2 and Fgure 2 plots the output z and ts estmaton ẑ as the estmaton error z ẑ s shown n Fgure 3. The mplementaton of WTOD protocol x ( = 1 2 3) denotes the th element of x. Then t s easy to see that the constrant (5) can be met wth G = The statstcal nformaton of stochastc varable α and β are taken as ᾱ =.5 and β =.9 respectvely. Set d = 2. The transton probablty matrx of the Markov Fgure 1. State x 2 and ts estmate.

14 192 L. LIU ET AL. Fgure 2. Output z and ts estmate. Fgure 4. The mplementaton of WTOD protocol schedulng. Fgure 3. Estmaton error z ẑ. schedulng s plotted n Fgure 4 whle 1 represents the measurement output y 1 of the frst sensor node s sent to the flter va the shared communcaton network and 2 represents the measurement output y 2 of the second sensor node s sent through the shared communcaton network. Fgure 5 plots the realzaton of the Markovan jump mode r.theh performance ndex s J 1 = ;. All the smulaton results confrm the effectveness of the proposed protocol-based flter desgn method that could well acheve the desred flterng requrement. 6. Conclusons In ths paper we have nvestgated the robust H fnte-horzon flterng problem for a class of dscrete Fgure 5. Realzaton of the Markovan jump mode r. tme-varyng nonlnear Markovan jump systems wth packet dropouts and tme-delay under the schedulng of WTOD protocol. Consderng lmted network bandwdth the WTOD protocol has been ntroduced wth whch only one node s permtted to send measurement sgnal to the shared communcaton network at each nstant n order to avod data collsons. Suffcent condtons for the exstence of the desred fnte-horzon flter satsfyng the H performance requrement have been presented n terms of the feasblty of a seres of RLMIs. Fnally an llustratve example has been set up to confrm the effectveness of

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