33 9 2016 9 DOI: 10.7641/CTA.2016.60012 Control Theory & Applcatons Vol. 33 No. 9 Sep. 2016 1 1 2 (1. 163318; 2. 163318) (RONs)... H. Lyapunov H.. ; ; ; TP273 A Research on fault detecton for Markovan jump systems wth tme-varyng delays and randomly occurrng nonlneartes REN We-jan 1 WANG Chun-le 1 LU Yang 2 (1. College of Electrc and Informaton Engneerng Northeast Petroleum Unversty Daqng Helongjang 163318 Chna; 2. College of Informaton Technology Helongjang Bay Agrcultural Unversty Daqng Helongjang 163318 Chna) Abstract: Ths paper s concerned wth the fault detecton problem for a class of dscrete-tme Markovan jump systems wth multple tme-varyng stochastc delays and randomly occurrng nonlneartes. Accordng to the stochastc characterstc of multple tme-varyng delays an ndvdual Bernoull dstrbuton s used to descrbe ths phenomenon. Furthermore a knd of stochastc nonlnear dsturbance s also consdered n the fault detecton research by means of a Bernoull dstrbuted whte sequence. The man purpose of ths paper s to desgn a fault detecton flter such that the resdual system s mean square exponental stable and the relevant H performance s satsfed. Through Lyapunov stablty theory and stochastc analyss technology suffcent condtons are set up for the desred exponental stablty and H dsturbance attenuaton a convex optmzaton problem s solved by the semdefnte programmng technque the gan characterstc of the fault detecton flter s obtaned. Fnally a smulaton example s provded to llustrate the usefulness and effectveness of the proposed desgn approach. Key words: fault detecton; Markovan jump system; stochastc tme-varyng delays; randomly occurrng nonlneartes 1 (Introducton)... [1. (Markovan jump systems MJSs) [2. MJSs. MJSs [3 4 [5 11 [12. MJSs. 2016 01 07; 2016 06 08.. E-mal: wangcl001@163.com; Tel.: +86 15845859180.. (61374127) (LBH Q12143) (QC2013C066). Supported by Natonal Natural Scence Foundaton of Chna (61374127) Helongjang Postdoctoral Research Foundaton (LBH Q12143) Helongjang Youth Foundaton (QC2013C066).
1148 33 3 [13 [14 ; [15. [16 17. [18 20.... [21 23... [24.. 1). 2). 3). 4) Lyapunov H. 2 ; 3 Lyapunov LMI ; 4 ; 5. 1. M T M ; R n n R n m n m ; Z ; I 0 ; l 2 ([0 ); R n ) n [0 ) ; P>0 P ; E{x} x ; x x ; dag{a 1 A 2 A n } A 1 A 2 A n *. 2 (Problem formulaton) x(k +1)=A(θ k )x(k)+γ(k)g(θ k x(k))+ A d1 (θ k ) q α r (k)x (k τ r (k)) + D 1 (θ k )ω(k)+g(θ k )f(k) y(k) =C(θ k )x(k)+d 2 (θ k )ω(k)+ E(θ k )f(k) x(k) =ψ(k) k Z (1) x(k) R n ; y(k) R m ; ω(k) R p l 2 [0 ) ; ( f(k) l ) 2 [0 );R l. A(θ k ) A d1 (θ k ) G(θ k ) C(θ k ) D 1 (θ k ) D 2 (θ k ) E(θ k ). τ r (k)(r =1 2 q) ; ψ(k) ; θ k S = {1 2 N} p j =P{θ k+1 = j θ k = } N θ j =1. j=1 g(θ k x(k)) g(θ k 0) = 0 g(θ k x(k)+δ(k)) g(θ k x(k)) B 1 (θ k )δ(k) (2) B 1 (θ k ) δ(k). α r (k)(r =1 2 q) γ(k) 0 1 (RONs) P{α r (k) =1} =ᾱ r P{α r (k) =0} =1 ᾱ r (3) P{γ(k) =1} = γ P{γ(k) =0} =1 γ ᾱ r [0 1 γ [0 1. 2 g(θ k x(k)). γ(k) g(θ k x(k)) (RONs) RONs. τ r (k)(r =1 2 q)
9 1149 d m τ r (k) d M d m d M. 3 1) 3 ; 2).. A(θ k )=A A d1 (θ k )=A d1 g(θ k x(k)) = g (x(k)) G(θ k )=G C(θ k )=C D 1 (θ k ) = D 1 D 2 (θ k )=D 2 E(θ k )=E B 1 (θ k )=B 1. ˆx(k +1)=A F (θ k )ˆx(k)+B F (θ k )y(k) (4) r(k) =C F (θ k )ˆx(k)+D F (θ k )y(k) ˆx k R n r(k) R l A F (θ k ) B F (θ k ) C F (θ k ) D F (θ k ). A F (θ k )= A F B F (θ k )=B F C F (θ k )=C F D F (θ k )=D F. (1) (4) x(k +1)=Ā(θ k) x(k)+ q (Ādr(θ k )+ à dr (θ k )) x(k τ r (k))+ ( γ + γ(k))zg(θ k x(k))+ (5) D(θ k )v(k) r(k) = C(θ k ) x(k)+ D F (θ k )v(k) x(k) = [ x T (k) ˆx T (k) T r(k) =r(k) f(k) v(k)= [ ω T (k) f T (k) [ T A 0 Ā(θk )= B F C A F [ᾱr A d1 0 [ αr (k)a d1 0 Ā dr (θ k )= à dr (θ k )= D1 G D(θ k )=[ Z=[I 0 T B F D 2 B F E C(θ k )=[D F C C F D F (θ k )=[D F D 2 D F E I α r (k) =α r (k) ᾱ r γ(k)=γ(k) γ. E{ α r (k)} =0 E{ α r(k)} 2 =ᾱ r (1 ᾱ r ) E{ γ(k)} =0 E{ γ 2 (k)} = γ(1 γ). Ā(θ k)=ā Ā dr (θ k )=Ādr à dr (θ k ) = Ãdr D(θk )= D C(θ k )= C D F (θ k )= D F. 1 v(k) =0 δ >0 0 <κ<1 E{ x(k) 2 } δ k κ sup Z E{ ψ() 2 } k 0 (5).. (4) A F B F C F D F a) (5). b) γ > 0 v(k) r(k) E{ r(k) 2 } γ 2 E{ v(k) 2 } (6) γ >0 (6). J(k) J th : J(k) ={ L r T (h)r(h)} 1 2 h=0 (7) J th = sup E{J(k)} ω k l 2f k =0 L. (7). J(k) >J th J(k) J th. 3 (Man results) Lyapunov (1).. 1 ( Schur [25.) S 1 S 2 S 3 S 1 = S1 T 0 <S 2 = S2 T S 1 + S3 T S 1 2 S 3 < 0 [ [ S1 S3 T S2 S 3 < 0 < 0. (8) S 3 S 2 S3 T S 1 2 x R n y R n Q >0 x T Qy + y T Qx x T Qx + y T Qy. (6). 1 (5) H γ >0 P > 0( N) Q r > 0(r =1 2 3 q) ρ >0
1150 33 Φ 1 + C T C 2 γ x T Φ = Ẑ T P Zg (x(k))+ P Ā Φ 2 < 0 q D T P Ā + D F T C x DT P Ẑ Φ (k τ r (k))ãt P dr à dr x(k τ r (k))+ 3 (9) 2( q Ā dr x(k τ r (k))) T P D v(k)+ Z T P Z ρi (10) v T (k) P D v(k)+ (6) 2 γ( q Ā dr x T (k τ r (k))) T P Zg (x(k))+ P = N p j P j γg T (x(k))z T P Zg (x(k)) j=1 (12) Φ 1 =2ĀT P Ā + q (d M d m +1)Q r P + ρ B dr à dr } = Φ 2 =dag{ Q 1 + Ã1 Q 2 + Ã2 Q q + [ T [ αr (k)a d1 0 αr (k)a d1 0 à q } +2ẐT P Ẑ E{ P } = Φ 3 =2 D T P D + D T D F F γ 2 I [ T [ à r =ᾱ r (1 ᾱ r )ÂT P Ad1 0 Ad1 0 d1  d1 2 q ᾱ r (1 ᾱ r ) P = Ẑ =[Ād1 Ā d2 Ā dq ᾱ B =dag{(3 γ 2 + γ)b T r (1 ᾱ r B 0} d1  d1. (13) 2  d1 =dag{a d1 0}. 2 γ x T P Zg (x(k)) η(k) := [ x T (k) x T (k τ 1 (k)) x T (k τ q (k)) T x T. P Ā x(k)+ γ 2 g T (x(k))z T P Zg (x(k)) (14) θ k = N 2 γg T (x(k))z T P D v(k) Lyapunov-Krasovsk γ 2 g T (x(k))z T P Zg (x(k)) + v T (k) D T P D v(k) V ( x(k)θ k )= (15) V 1 ( x(k)θ k )+V 2 ( x(k)θ k )+V 3 ( x(k)θ k ) (11) 2 γ( q Ā dr x(k τ r (k))) T P Zg (x(k)) ( q Ā dr x(k τ r (k))) T P ( q Ā dr x(k τ r (k)))+ V 1 ( x(k)θ k )= x T (k)p (θ k ) x(k) V 2 ( x(k)θ k )= q k 1 x T (l)q r x(l) V 3 ( x(k)θ k )= q l=k τ r (k) d m k 1 m= d M+1 l=k+m x T (l)q r x(l). ΔV 1 ( x(k)θ k )= E {V 1 ( x(k +1)θ k+1 ) x(k)θ k = } V 1 ( x(k)θ k )= N p j x T (k +1)P j x(k +1) x T (k)p x(k) = j=1 x T (k)(āt P Ā P ) x(k)+ 2 x T (k)āt P ( q Ā dr x(k τ r (k)))+ 2 x T (k)āt P D v(k)+2 γg T (x(k))z T P D v(k)+ ( q Ā dr x(k τ r (k))) T P ( q Ā dr x(k τ r (k)))+ γ 2 g T (x(k))z T P Zg (x(k)). (16) (2) g T (x(k))(3 γ 2 + γ)z T P Zg (x(k)) x T (k)(3 γ 2 + γ)ρb T B x(k) = x T (k)ρ B x(k) (12) (17) ΔV 1 ( x(k)θ k )= E {V 1 ( x(k +1)θ k+1 ) x(k)θ k = } V 1 ( x(k)θ k ) x T (k)(2āt P Ā P + ρ B ) x(k)+ 2 x T (k)āt P ( q Ā dr x(k τ r (k)))+ 2 x T (k)āt (17) P D v(k)+2( q Ā dr x(k τ r (k))) T
9 1151 P ( q Ā dr x(k τ r (k)))+ k d m l=k d M+1 x T (l)q r x(l)) (21) q x T (k τ r (k))ãt P dr à dr x(k τ r (k))+ v(k) =0 2( q E{ΔV ( x(k)θ k )} η T (k)φ η(k) (22) Ā dr x(k τ r (k))) T P D v(k)+ 2v T (k) D [ T P D v(k) (18) Φ1 Φ =. ΔV 2 ( x(k)θ k )= Ẑ T P Ā Φ 2 E {V 2 ( x(k +1)θ k+1 ) x(k)θ k = } V 2 ( x(k)θ k )= 1 Φ < 0. (5). N p j ( q k x T (5) H (l)q r x(l) j=1 l=k τ r (k)+1 q k 1 x T (l)q r x(l)) J N =E [ r T (k) r(k) β 2 v T (k)v(k) = l=k τ r(k) q ( x T E [ r (k)q r x(k) (k) r(k)+δv( x(k)θ k ) x T β 2 v T (k)v(k) E {V (k τ r (k))q r x(k τ r (k))+ k+1 ( x(k+1)θ k+1 )} k d m E [ r T (k) r(k) β 2 v T (k)v(k)+ x T (l)q r x(l)) (19) l=k d M +1 ΔV ( x(k)θ k ) ΔV 3 ( x(k)θ k )= E {V 3 ( x(k +1)θ k+1 ) x(k)θ k = } η T (k)φη(k) < 0. V 3 ( x(k)θ k )= q ((d M d m ) x T (k)q r x(k) E{ r(k) 2 } γ 2 E{ v(k) 2 }. k d m x T (l)q r x(l)). (20). l=k d M +1. (18) (20) 2 ΔV ( x(k)θ k ) x T (k)(2āt P Ā P + ρ B ) x(k)+. H γ >0. P > 0( N) Q r > 0 2 x T (k)āt P ( q (r =1 2 3 q) ρ>0 X K Ā dr x(k τ r (k)))+ 2 x T (k)āt P D v(k)+2( q Ā dr x(k τ r (k))) T Y = ˆΓ P ( q 11 Ā dr x(k τ r (k)))+ 0 γ 2 I q x T (k τ r (k))ãt P ˆΓ dr à dr x(k τ r (k))+ 31 P ˆD0 + X ˆR2 P < 0 ˆΓ 41 P 2( q ˆΓ Ā dr x(k τ r (k))) T 51 ˆΓ52 ˆP P D v(k)+ (23) 2v T (k) D T P D v(k)+ q ( x T (k)q r x(k) Z T P Z ρi (24) x T (4) (k τ r (k))q r x(k τ r (k))+. k d m l=k d M +1 x T (l)q r x(l))+ q ((d M d m ) x T (k)q r x(k) ˆΓ 11 =dag{γ 11 Γ 22 } ˆΓ 31 =[ P  0 + X ˆR1 P Ẑ ˆΓ 41 =dag{ P  0 + X ˆR1 P Ẑ }
1152 33 Γ 11 = ρ B + q (d M d m +1)Q r P. Γ 22 =dag{ Q 1 + Ã1 Q 2 + Ã2 [ Q q + [ Ãq} 0.2 0.8 ˆΨ =. 0.3 0.7 ˆΓ 51 = K ˆR1 0 [ [ P ˆD0 + X ˆR2 0n n ˆΓ 52 = Ê = [ [ K ˆR2 ÊT 1 I n n 0.6 0.2 0.2 0.5 P =dag { P P } ˆP =dag { P I } A 1 = A 2 =.7.5 [ Ê 1 =[0 l p I l p T Ê 2 =[0 m n I m n T 0.03 0 A [ [ d11 = 0.02 0.03 A 0 D1 G [ Â 0 = ˆD0 = 0.05 0 A [ [ d12 = 0.04 0.05 0 I [ [ ˆR 1 = ˆR2 =. C 0 D 2 E 0.8 0 0.4 0 D 11 = D 12 =.3.4 (P Q r X K ρ) (23) (24) (4) C 1 = C 2 =[0.2 0.1 H 1 = H 2 =0.9 D 21 =[0.6 0.1 D 22 =[0.5 0.1 [A F B F =[ÊT P Ê 1 Ê T X G (25) 1 =[ 0.1 0.2 T G 2 =[ 0.1 0.2 T. [C F D F =K. 2 τ r (k) 6(r =1 2) P Q r 1. ᾱ 1 =E{α 1 (k)} =0.9 Ā = Â0 + ÊK ˆR 1 1 ᾱ 2 =E{α 2 (k)} =0.7 D = ˆD 0 + ÊK ˆR 1 2 (26) γ =E{γ(k)} =0.8. C = K ˆR1 D F = K ˆR2 ÊT 1 K 1 =[A F B F. Shur (9) ˆΓ 11 0 γ 2 I ˆΓ 31 ˆD0 +ÊK ˆR 1 2 P 1 1 ˆΓ 41 P ˆΓ 51 ˆΓ52 ˆP 1 ˆΓ 31 =[Â0 + ÊK ˆR 1 1 Ẑ ˆΓ 41 =dag{â0 + ÊK ˆR 1 1 [ Ẑ} ˆD0 + ˆΓ 52 = ÊK ˆR 1 2. K ˆR2 ÊT 1 <0 (27) 1 X = P ÊK 1 (23). 4 (Numercal example) g 1 (x(k)) = g 2 (x(k)) = 0.5x 1 (k)sn(x 2 (k)) ε(k) =1 B(k) =dag{0.2 0.15} (2). 2 (25) [ 0.0025 0.0410 A F1 = B F1 = 0.0237 0.0021 [ C F1 = 2.255.1310 D F1 =0.5039 [ 0.0019 0.0162 A F2 = B F2 = 0.0022 0.0106 [ 0.0005 0.0288 [ 0.0015 0.0065 C F2 =[ 0.0013 0.0033 D F2 =0.7791. k =0 1 150 f(k) { 1 40 k 80 f(k) = 0. ˆΨ θ(0) = 1 θ(k) 1.
9 1153 { [rand[0 1 1.2rand[0 1 T 0 k 50 ω(k) = 0 1 Fg. 1 Modes evoluton ω(k) =0 r(k) J(k) 2 3. rand [0 1. r(k) J(k) 4 5. J th =sup f=0 r T (s)r(s)} 1/2 E{ 150 s=0 200 J th 1.1129. 5 0.9469=J(40)<J th <J(41) = 1.3016 1.. 4 Fg. 4 Resdual sgnal wth dsturbance 2 Fg. 2 Resdual sgnal wthout dsturbance 5 Fg. 5 Resdual evaluaton functon wth dsturbance 3 Fg. 3 Resdual evaluaton functon wthout dsturbance 5 (Conclusons)..
1154 33 H. Lyapunov H.... (References): [1 WANG Yongqang YE Hao WANG Guzeng. Recent development of fault detecton technques for networked control systems [J. Control Theory and Applcatons 2009 26(4): 400 409. (. [J. 2009 26(4): 400 409.) [2 DONG H L WANG Z D HO DWCetal.RobustH flterng for Markovan jump systems wth randomly occurrng nonlneartes and sensor saturaton: the fnte-horzon case [J. IEEE Transactons on Sgnal Processng 2011 59(7): 3048 3057. [3 WANG G L ZHANG Q L YANG C Y. Stablzaton of sngular Markovan jump systems wth tme-varyng swtchngs [J. Informaton Scences 2015 297(1): 254 270. [4 LIU X H XI H S. Stochastc stablty for uncertan neutral Markovan jump systems wth nonlnear perturbatons [J. Journal of Dynamcal and Control Systems 2015 21(2): 285 305. [5 ZHANG Y Q LIU C X SONG Y D. Fnte-tme H flterng for dscrete-tme Markovan jump systems [J. Journal of the Frankln Insttute 2013 350(6): 1579 1595. [6 ZHANG B Y LI Y M. Exponental flterng for dstrbuted delay systems wth Markovan jumpng parameters [J. Sgnal Processng 2013 93(1): 206 216. [7 ZHUANG G M SONG G F XU S Y. H flterng for Markovan jump delay systems wth parameter uncertantes and lmted communcaton capacty [J. Control Theory and Applcatons 2014 8(14): 1337 1353. [8 TERRA M H ISHIHARA J Y JUNIOR A P. Array algorthm for flterng of dscrete-tme Markovan jump lnear systems [J. IEEE Transactons on Automatc Control 2007 52(7): 1293 1296. [9 LI Z C YU Z D. Further result on H flter desgn for contnuoustme Markovan jump systems wth tme-varyng delay [J. Journal of the Frankln Insttute 2014 351(9): 4619 4635. [10 ZHUANG G M LU J W ZHANG M S. Robust H flter desgn for uncertan stochastc Markovan jump Hopfeld neural networks wth mode-dependent tme-varyng delays [J. Neurocomputng 2014 127(1): 181 189. [11 LONG S H ZHONG S M LIU Z J. H flterng for a class of sngular Markovan jump systems wth tme-varyng delay [J. Sgnal Processng 2012 92(11): 2759 2768. [12 LAM J. Robust H control of descrptor dscrete-tme Markovan jump systems [J. Internatonal Journal of Control 2007 80(3): 374 385. [13 LIU M HO DWC NIUYG. RobustStablzaton of Markovan jump lnear system over networks wth random communcaton delay [J. Automatca 2009 45(2): 416 421. [14 WANG Hongru WANG Changhong GAO Hujun. Robust fault detecton for dscrete-tme Markovan jump systems wth tmedelays [J. Control and Decson 2006 21(7): 796 800. (. [J. 2006 21(7): 796 800.) [15 WANG H R WANG C H MOU S S et al. Robust fault detecton for dscrete tme Markovan jump systems wth mode-dependent tmedelays [J. Journal of Control Theory and Applcatons 2007 5(2): 139 144. [16 OUALI A MAKHLOUF A B HAMMAMI M A et al. State feedback control law for a class of nonlnear tme-varyng system under unknown tme-varyng delay [J. Nonlnear Dynamcs 2015 82(1/2): 349 355. [17 PHAT V N NIAMSUP P. Global stablzaton of lnear tme-varyng delay systems wth bounded controls [J. Appled Mathematcs Letters 2015 46(1): 11 16. [18 HU J WANG Z D GAO H J et al. Extended Kalman flterng wth stochastc nonlneartes and multple mssng measurements [J. Automatca 2012 48(9): 2007 2015. [19 SHEN B WANG Z D HUNG Y S. Dstrbuted consensus H flterng n sensor networks wth multple mssng measurements: the fnte-horzon case [J. Automatca 2010 46(10): 1682 1688. [20 YANG F W WANG Z D HUNG Y S et al. H control for networked systems wth random communcaton delays [J. IEEE Transacton on Automatc Control 2006 51(3): 511 518. [21 DING D R WANG Z D SHEN B et al. Envelope-constraned H flterng wth fadng measurements and randomly occurrng nonlneartes: the fnte horzon case [J. Automatca 2015 55(1): 37 45. [22 DONG H L WANG Z D DING S X et al. Event-based H flter desgn for a class of nonlnear tme-varyng systems wth fadng channels and multplcatve noses [J. IEEE Transactons on Sgnal Processng 2015 63(14): 3387 3395. [23 DONG H L WANG Z D GAO H J. On desgn of quantzed fault detecton flters wth randomly occurrng nonlneartes and mxed tme-delays [J. Sgnal Processng 2015 92(4): 1117 1125. [24 ZHANG J H SHI P QIU J Q. Non-fragle guaranteed cost control for uncertan stochastc nonlnear tme-delay systems [J. Journal of the Frankln Insttute 2009 346(7): 676 690. [25 BOYD S GHAOUI L E FERON E et al. Lnear Matrx Inequaltes n System and Control Theory [M. Phladelpha: SIAM 1994. (1963 ) E-mal: renwj@126.com; (1991 ) E-mal: wangcl001@163.com; (1976 ) E-mal: luyanga@sna.com.