Fault-Tolerant Guaranteed Cost Control of Uncertain Networked Control Systems with Time-varying Delay

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1 IJCSNS Inernionl Journl of Compuer Science nd Nework Securiy VOL.9 No.6 June 9 3 Ful-olern Gurneed Cos Conrol of Uncerin Neworked Conrol Sysems wih ime-vrying Dely Guo Yi-nn Zhng Qin-ying Chin Universiy of ining nd echnology School of Informion nd Elecronic Engineering Xuzhou 64-6 Chin Summry In prcice here inevibly exiss he offse or disurbnces induced by cuor filures. his is n ignored problem in inegriy ful-olern conrol heory. In order o sisfy he robusness of sysem he closed-loop model of uncerin neworked conrol sysem wih ime-vrying dely nd cuor filures is esblished iming clss of conrolled objecive wih uncerin prmeers. he enuion performnce index of sysem for ful is defined. Combing wih ful-olern conrol nd gurneed cos conrol ful-olern gurneed cos conroller is designed doping Lypunov sbiliy nlysis mehod. Simulion resuls indice he conroller cn no only gurnee he sympoic sbiliy bu lso ensure he robusness nd ni-disurbnce performnce. Key words: Neworked conrol sysems Ful-olern Gurneed cos ime-vrying dely Uncerin. Inroducion Neworked conrol sysem (NCS) is ho reserch re. A presenmny workes hve been done bou sysem modelingsbiliy nlysisgurneed cos conroller design nd so on. Reserches on robus gurneed cos conroller ensure he sbiliy nd robusness of neworked conrol sysem. his mees he prcicl needs nd hs he imporn mening. Gurneed cos conrol is o design conroller which no only mkes uncerin closed-loop sysem sble bu lso limis he bound of cerin performnce index. In Ref[] gurneed cos conrol of discree neworked conrol sysem is sudied iming ime-vrying dely. Gurneed cos conrol of neworked conrol sysem is given bsed on discree jump sysem iming rndom dely[]. Yue[3] proposes gurneed cos conrol bsed on model given in Ref[4] iming qudric performnces. In Ref[5] H gurneed cos conrol of neworked conrol sysem doping proporion-inegrl oupu feedbck conroller iming qudric cos funcion. For neworked conrol sysems wih ime-vrying dely less hn one smple period nd d-pcke dropou compensor is inroduced o compense he effec of d dropou[6]. And NCS is modeled s discree swiched sysem wih prmericl uncerinies. Bsed on his model cooperive design pproch of conroller nd he compensor re given in erms of group of liner mrix inequliy. Gurneed cos conrol of neworked conrol sysem wih uncerin ime dely doping oupu feedbck conroller is sudied in Ref[7]. In exising chievemensgurneed cos conrol of neworked conrol sysem wih fuls is no ken ino ccoun. Oherwise mos of corol mehods re given bsed on discree neworked conrol sysem wih consn nework-induced ime dely. However he robusness of coninuous neworked conrol sysem wih unceriny nd ime-vrying dely is seldom considered. In he pper gurneed cos conroller of uncerin neworked conrol sysem wih ime-vrying dely nd cuor filures is designed. Firsly neworked conrol sysem wih ime-vrying dely nd cuor filures is modeled. Secondly he sbiliy of closed-loop ful sysem is nlyzed considering zero disurbnce cused by cuor fuls. When disurbnce cused by cuor fuls is no zero performnce index reflecing disurbnce degrdion is defined. And gurneed cos conroller is designed in erms of Lypunov sbiliy nlysis mehod. A ls he vlidiy of proposed mehod is vlided by wo exmples.. odeling of he Closed-loop Ful Sysems Consider he coninuous-ime liner pln described by se-spce equions of he form &x = Ax + Bu y = Cx n m x R is he se nd u R is he conrol p inpu. y R nn is he oupu of pln. A R is he se n m mrix nd B R p n is he inpu mrix. C R is he oupu mrix. And A B C re consns mrices. In his pper i is ssumed h: Coninuous pln wihou nework nd se feedbck re sbiliy or mee cerin needs of conrol. Conroller is ime-vrying nd coninuous. (3)Noise of sysem is no ken ino ccoun. And no error exis in communicion. nuscrip received June 5 9 nuscrip revised June 9

2 4 IJCSNS Inernionl Journl of Compuer Science nd Nework Securiy VOL.9 No.6 June 9 (4)Suppose d sc is ime delys cused by nework d from sensor o conroller. sc denoes ime delys cused by nework from conroller o cuor. Above ime delys re regrded s d = dsc + dc. Under bove ssumpion sysem model considering ime-vrying nework-induced dely is obined: x& = Ax + Bu( d ) d = d sc + dc is ime-vrying dely sisfied d ( ) τ. Here τ is upper limi of ime-vrying dely. Consider cuor filure ful model of neworked conrol sysem is formed: x& = Ax + BLu( d) + B( I L) f L = dig { l l L l } m is cuor filure mrix sisfied L nd L ψ. Here ψ indices n ggrege conining ll possible cuor filure mrixes. (3) i-h cuor is norml li = i = L m i-h cuor is filure (4) I is obvious h sysem is norml when L= I. If L I here exiss cuor fuls nd f is he disurbnce or offse cused by cuor fuls. Consindering unceriny of sysem ΔA nd ΔB re unknown limied coefficien mrixes. h is ΔA ΔA A ΔB ΔB B n n. Here A R nd n n B R re known consn mrixes. If se feedbck conroller is doped shown s u = Kx bove model of sysem shown in formul(3) is rnsformed s follows. &x( ) = ( A+Δ A( )) x( ) + ( B+ΔB( )) LKx( d( )) + B( I L) f ( ) (5) Besd on bove-menioned model reserches re done iming differen disurbnce: f mens h when cuor fuls hppen oupu of conroller is zero. he gol of he conrol sysem is o obin sufficiency condiions ensuring he sympoic sbiliy of closed-loop ful sysem nd deermine proporion consn mrix of conroller doping Lypunov sbiliy nlysis mehod. If f he offse of fuls is regrded s unknown disurbnce of sysem. In order o decrese he influence cused by disurbnce degrdion performnce y < ρ f shll sisfy. Here << ρ. Corresponding performnce index is defined s J = y y ρ f f d. he gol of he conrol sysem is o deermine proporion consn mrix of conroller which ensures he sympoic sbiliy of sysem wheher cuor ful hppens nd mke J <. 3 Design of Ful-olern Gurneed Cos Conroller 3. Sbiliy nlysis of closed-loop ful sysems heorem : Considering he sysem shown in formul(5) for given posiive consne κ nd τ if here exis posiive definie symmeric mrix X symmery mrix Y nd posiive sclrs ζ η δ δ he following liner mrix inequliy is sisfied: = < > I A > s δ δ hen he sysem is exponenilly sympoiclly sble. From group of fesible soluion (XY) conroller wih K = YX is obined = XA + AX + Y L B + BLY + ζ A + η B = X Y τκλ BL τκλ B τ XA τ X ( ) τy L B τy L B B τy τσ mx B Y = dig { ζ I ηi τκ( λ δ) I τ( I δ A ) I τδ I τi τi τi τi} Proof: Bsed on Newon-Leibniz heorem { } Q ( ) = [ A+Δ A( + ] x ( + + B( I L) f ( )) d d + θ θ + [ B+Δ B( + ] LKx ( d( + + he closed-loop model of ful sysem is rewrien: &x = [( A+Δ A) + ( B+ΔB) LK] x + BI ( Lf ) ( ) ( B+ΔB( )) LKQ ( ) V x = V Consruc Lypunov funcion s x + V x. ( ) V ( ) ( ) x = x Px V ( x ) = V ( x ) + V ( x ) + V ( x ) 3 (6) (7) (8)

3 IJCSNS Inernionl Journl of Compuer Science nd Nework Securiy VOL.9 No.6 June 9 5 = ( ϑ)[ ( ϑ)] [ ( ϑ)] ( ϑ) dϑ x A+Δ A A+ΔA x τ + θ + ( ϑ) [ ( ϑ d ( ) ] x K BL+Δ B + + L τ d( + + θ [ B+Δ B( ϑ + d ( + )] LKx( ϑ) dϑ + f ( ϑ)( I L) B B( I L) f ( ϑ) dϑ τ + θ Along wih rndom rjecory of sysem he derivive of V( x ) is V & ( x ) = V & ( x ) + V & ( x ). ( ) ( ) ( ) V& ( ) ( ) x = x& Px + x Px& (9) = x {[( A+Δ A + ( B+ΔB) LK] P + x PB( I L) f x P( B+ΔB) LK Q x ( ) P( B + ΔB( )) LK Q ( ) = x PB ( +Δ B) LK( A+Δ A( + ) x ( + d d d x PB ( +Δ B) LK( B+Δ B( + ) x ( d ( + + x PB ( +ΔB) LKBI ( L) f ( + Lemm [8]: For ny vecors or mrices XYZ nd ny posiive consns > > he following inequliies re sisfied: X Y + Y X X X + Y Y ± Z Y Z Z + Y Y Bsed on Lemm fomul is rewrien s: x ( ) P( B + ΔB( )) LKB( I L) f ( + θ ) d( ) τ x P( B + Δ B) LKK ( BL+ ΔB L) Px + f ( + θ )( I L) B B( I L) f ( + d( ) Now consider V & ( x ) = V & ( x ) + V & ( x ) + V & 3( x ). V& ( x ) = τ x ( )( A+Δ A( )) ( A+ΔA( )) x( ) x ( + ( A +Δ A ( + ) ( A +Δ A ( + ) x ( + τ τ x ( )( A+Δ A( )) ( A+ΔA( )) x( ) ( ( ( ) ( ( ) ( x + A+Δ A + A+Δ A + x + d V& τ ( x ) = x K ( BL+Δ B( + d( + L)) ( BL+Δ B( + d( + L) Kx ( d( ( ( ) x + + K BL+Δ B + L τ ( BL+Δ B( + L) Kx ( d( + + τ x K ( BL+Δ B( + d( + L)) ( BL+Δ B( + d( + L) Kx x ( ( ) ) ( ( ) ) d+ θ + θ K BL+Δ B+ θ L d ( BL+Δ B( + L) Kx ( d( + + (3) V& 3( x ) = τ f ( )( I L) B B( I L) f( ) f ( + ( I L) B B( I L) f( + τ τ f ( )( I L) B B( I L) f( ) f ( + θ )( I L) B B( I L) f ( + d (4) Wih (9)-(4) he following inequliies re obined: V& ( x ) x ( ){[( A +Δ A ( ) + ( B +Δ B ( )) LK ] P + PA [( +Δ A( ) + ( B+Δ B( )) LKx ]} ( ) + x( ) PBI ( Lf ) ( ) + τ ( + + ) x( ) PB ( +Δ B( )) LKK( BL+ΔB( ) L) Px( ) τ + x( A+Δ A)( A+Δ A) x + τf ( I L) BBI ( Lf ) τ + x K ( BL+Δ B( + d ( + L)) ( BL+Δ B( + d ( + L) Kx (5) According o Lemm [8] he following inequliies re obined: P Δ A ( ) + Δ A ( ) P ζ P A P + I ζ PΔ B LK + K L ΔB P η PB P + KK η τ K ( BL +Δ B( + d( + L)) ( BL +Δ B( + d( + L) K τ = K [ L B BL+ L B Δ B( + d( + L) + L Δ B ( + d( + ) Δ B( + d( + L)] K τ K[ LBBL+ LBBBL+ ( + σ mx ( B)) IK ] (6) Lemm [9]: Le A nd ΔA be n n rel mrices nd ssume inequliy ΔA ΔA A is sisfied A is symmeric mrix. hen for ny < ε< we hve ( A+Δ A)( A+ΔA) AA + A ε ε δ ( + + ) = δ Le hen τ( + + ) P ( B +Δ B( )) LKK ( BL+ΔB( ) L) P ( + + ) ( + + ) τκ P ( BLL B + B ) P + + δ δ (7)

4 6 IJCSNS Inernionl Journl of Compuer Science nd Nework Securiy VOL.9 No.6 June δ >. κ is given consn sisfying κ > nd KK κ I. Lemm 3[9]: Le A nd ΔA be n n rel mrices nd ssume inequliy ΔA ΔA A is sisfied A is symmeric mrix. hen for ny << ε nd εa > we hve ( A+Δ A) ( A+ΔA) A ( εa) A+ σi ε ΔA σ = ' oherwise δ = δ. Le he following inequliies re sisfied: τ x ( A+Δ A) ( A+ΔA) x τ[ A ( I δa ) A+ I] δ Bsed on bove inequliies formul (5) becomes: V& ( x ) x ( )( A P + AP + ( BLK) P + PBLK) x( ) + x {( ζpa P+ I+ ηpb P+ KK ) ζ η ( + + ) ( + + ) + τκp BLLB+ B P δ δ ( ) + + δ σmx + τ[ A ( I δ A ) A+ I] τ + K[ LBBL+ LBBBL+ ( + ( B)) IKx ] } ( ) + x ( ) PB( I L) f( ) + τ f ( )( I L ) B B ( I L ) f ( ) (8) If f he sbiliy of sysem is only considered hen x PBI ( Lf ) ( ) + τ f I LBBI ( Lf ) = ˆ So formul(8) is simplified sv &( x ) x ( ) x( ) ˆ = A P + AP + ( BLK) P + PBLK + ζ PA P + I ζ ( + + ) ( + + ) + τκ P BLL B+ B P ( ) + + δ δ σ mx τ + K ( LB BL+ LB BBL+ ( + ( B)) I) K + + τ δ + + η η KK ( A ( I A ) A I) PB P δ (9) herefore if ˆ < V& ( x ) <. h mens he closed-loop sysem express by (5) is sble. According o Shur Lemm ˆ < is equl o mrix inequliy: ˆ ˆ ˆ = < ˆ ˆ Where ˆ = A P + AP + ( BLK) P + PBLK + ζpa P + ηpb P ˆ = I K τκλ PBL τκλ PB τ A τ I τk L B τk L B B τk τσmx ( B) K ] ˆ = dig { ζ I ηi τκ( λ δ) I τ( I δ A ) I τδi τi τi τi τi} Becuse mrix inequliies is nonlier mrix inequliy bou P nd K i cn no compue using Liner rix Inequliy oolbox. So dig{ P IIIIIIIII } is muliplied wih boh sides of inequliy(9). Le X = P Y = KX hen bove mrix inequliy becomes inequliy (6). From group of fesible soluion s (XY) conroller wih K = YX is obined. 3. Design of ful-olern gurneed cos conroller heorem :Consider he sysem (5) for given posiive consne k τ ρ if here exis posiive definie symmery mrix X symmery mrix Y nd posiive sclr ζ η δ δhe following liner mrix inequliy is sisfied: < f + + δ s.. > I δ A > hen he sysem shown in formul(5) wih K = YX is sble nd J <. When sysem is norml is rewrien s. XC B( I L) f = I ( ρ τ( ) ( ) I I L B B I L Proof: Considering he performnce index: J = y y ρ f f d ˆ x + C C PBI ( L) x d ( ) ( ρ τ( ) ( )) f I L B P I I L B BI L f V( ) ˆ + CC PBI ( L) < If J< ( I L) BP ( ρ I τ( I L) BBI ( L)) ( ρi τ( I-L) B B( I-L ) >. According o Shur Lemm formul is equl o mrix inequliy: ˆ C PB( I L) I < ( ρ τ( ) ( )) I I L B B I L Becuse fomul is nonlier mrix inequliy i cn no be compued using LI oolbox. So

5 IJCSNS Inernionl Journl of Compuer Science nd Nework Securiy VOL.9 No.6 June 9 7 dig{ P IIIIIIIII } is muliplied wih boh sides of inequliy we hve ˆ ( ) I < ( ρi τ( I L) B B( I L) P P P C B I L (3) Proporion consn of ful-olern gurneed cos K = conroller is Besd on bove prmeers response curves of ses re shown in Fig.. Le X= P Y = KX hen bove mrix inequliy becomes: XC B( I L) f : = I < ( ρ τ( ) ( ) I I L B B I L (4) If inequliy hs soluion closed-loop sysem shown in formul (5) cn gurnee he sympoic sbiliy nd sisfy J<. 4 Simulions Smples nd Anlysis Consider nework conrol sysem wih uncerin prmeers he closed-loop model is x& ( ) = ( A+Δ A( )) x( ) + ( B+ΔB( )) LKx( d( )) + B( I L) f( ).3.5 A.sin =.7.8 A.5 Δ =.cos B=.sin. Δ B=.cos C = A= B=.. In generl iniil ses re rndomly chosen s [ ] x x = [ ]. Suppose dsc = dc =.+.sin is nework-induced ime-dely. So d =. +.sin. Le τ =.4. Impulse signl is doped s disurbnce cused by fuls. If no cuor ful hppens in conrol sysem L = dig { } L =dig. { } L =dig or { } respecively indice -h cuor ful or -h cuor ful. A. Exmple: Ful-olern conroller Suppose k =.5. Globl opiml soluion of inequliy(6) is obined s min =.55 doping LI oolbox. Becuse of min < LI is fesible. And group of fesible soluions re obined. h is ζ = η=94.99 λ = δ = δ = =.793 =59.5. Vrible mrixes of X = he sysem is Y = x ( ) x ( ) Fig. Response curves of ses I is obvious h wheher cuor ful hppens he conroller cn gurnee he sympoic sbiliy. B. Exmple:Ful-olern gurneed cos conroller Suppose k =.5. ρ =.9 is degrdion degree of disurbnce. Globl opiml soluion of inequliy (4) is obined s min =.3 doping LI oolbox. Becuse of min < LI is fesible which ensure J <. And group of fesible soluions re obined. h is ζ =.745 η=.797 λ =.46 δ =.46 δ =.4999 =.358 =.33. Vrible mrixes of he X = sysem is.88.3 Y = Proporion consn of ful-olern gurneed cos.9.43 K = conroller is Besd on bove prmeers response curves of ses re shown in Fig.. x ( ) x ( ) Fig. Response curves of ses

6 8 IJCSNS Inernionl Journl of Compuer Science nd Nework Securiy VOL.9 No.6 June 9 I is obvious h wheher cuor ful hppens he conroller cn no only gurnee he sympoic sbiliy bu lso ensure he robusness nd ni-disurbnce performnce. 4. Conclusion Aiming clss of conrolled pln wih uncerin prmeers he closed-loop model of uncerin neworked conrol sysem wih ime-vrying dely is esblished considering disurbnce cused by cuor ful. he degrdion performnce index of sysems for ful is defined. Combing wih ful-olern conrol nd gurneed cos conrol ful-olern gurneed cos conroller is designed doping Lypunov sbiliy nlysis mehod. Simulion resuls indice he conroller cn no only gurnee he sympoic sbiliy bu lso ensure he robusness nd ni-disurbnce performnce. [8] Zhou k.khrgonekr P.P. Robus sbilizion of liner sysems wih norm-bounded ime-vrying unceriny Sysems nd Conrol Leers vol. pp [9] C. Cheng Q. Wu Decenrlized robus conroller for uncerin lrge-scle sysems wih conrol dely Inernionl Journl of Sysems Science vol.3 pp.33-4 Guo Yi-nn received he Ph.D. degrees from Chin Univ. of ining nd echnology in 3. Afer working s lecure (from 999) Posdocor (from 4) in he School of Informion nd Elecronic Engineering Chin Univ. of ining nd echnology she hs been n ssocie professor Chin Univ. of ining nd echnology since 5. Her reserch ineress include knowledge-induced evoluionry lgorihms culurl lgorihms nd heir pplicions. She is member of SCAI CSC. Acknowledgmen his work ws suppored by Nionl Nurl Science Foundion of Chin under Grn 6855 he 863 Projec of Chin under Grn 7AAZ6 nd Qingln Projecive of Jingsu. References [] Alexopoulos C Griffin P Ph plnning for mobile robo IEEE rnscions on Sysems n nd Cyberneics vol.pp [] Shnbin Li Zhi Wng Youxin Sun Gurneed cos conrol nd is pplicion o neworked conrol sysems IEEE Inernionl Symposium on Indusril Elecronics pp [] Wng Yufeng Wng Chnghong Hung Xu Gurneed cos conrol wih rndom communicion delys vi jump liner sysem pproch Inernionl Conference on Conrol Auomion Roboics nd Vision pp [3] Chen Peng Dong Yue Subopiml gurneed cos conroller design for neworked conrol sysems he 6h World Congress on Inelligen Conrol nd Auomion pp [4] Dong YueQing-long HnJmes Lsm Neworked-bsed robus H conrol of sysems wih unceriny Auomic vol.4pp [5] Zhng inging Zhng Qinglin Bo Gng H Gurneed Cos Conrol of he Neworked Conrol Sysem Informion nd Conrol vol.36 pp.38-34: 7 [6] Wng Yn Chen Qingwei Fn Weihue l Gurneed cos conrol of neworked conrol sysems wih d-pcke dropou Conrol heory & Applicions vol.4(7)pp [7] Qiu Zhnzhi Zhng Qinlin Liu ing Gurneed performnce conrol for oupu feedbck neworked conrol sysems wih uncerin ime-dely Conrol heory & Applicions vol.4(7)pp