STATE FEEDBACK CONTROLLER DESIGN OF NETWORKED CONTROL SYSTEMS WITH PARAMETER UNCERTAINTY AND STATE-DELAY

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1 Asian Journal of Conrol, Vol. 8, No. 4, pp , December SAE FEEDBACK CONROLLER DESIGN OF NEWORKED CONROL SYSEMS WIH PARAMEER UNCERAINY AND SAE-DELAY Chen Peng and Dong Yue ABSRAC his paper is concerned wih he conroller design of newored conrol sysems. he coninuous ime plan wih parameer uncerainy and sae delay is sudies. A new model of he newored conrol sysem is provided under consideraion of he nonideal newor condiions. In erms of he given model, a conroller design mehod is proposed based on a delay dependen approach. he maximum allowable synheical bounds relaed wih he diarded daa pace and newor-included delay and he feedbac gain of a memoryless conroller can be derived by solving a se of linear marix inequaliies for he sabilizabliy of he newored conrol sysem based on Lyapunov funcional mehod. An example is given o show he effeciveness of our mehod. KeyWords: Newored conrol sysems, linear marix inequaliies, maximum allowable synheical bounds, delay sysem. I. INRODUCION A newored conrol sysem (NCS) involves communicaion paerns in which boh informaional and physical conrol loops are closed hrough a real-ime newor. Recenly, much aenion has been paid o he sudy of sabiliy analysis and conrol design of NCS [-4], since heir low cos, reduced weigh and power requiremens, simple insallaion and mainenance, and high reliabiliy. In an NCS, one of he imporan issues o rea is he effec of he newor-induced delay on he sysem performance. Performance of he feedbac conrol in he NCS is direcly dependen upon he newor-induced delay. ime-varying characerisics of he newor-induced delay no only degrade conrol performance bu also inroduce disorion of he conroller signal [5,6]. Manurip received June 22, 2004; revised June 9, 2005; acceped Sepember 8, he auhors are wih he Insiue of Informaion & Conrol Engineering echnology, Nanjing Normal Universiy, 78 Banchang Sree Nanjing, Jiangsu, China ( pc@ . njnu.edu.cn). his wor was parially suppored by Naional Naural Foundaion of China ( ), Minisry of Educaion of Jiangshu (330B3). In feedbac conrol sysems, i is imporan ha sampled daa should be ransmied wihin a sampling period and ha sabiliy of conrol sysems should be guaraneed. While a shorer sampling period is preferable in mos conrol sysems, for some purposes i can be lenghened up o a cerain bound wihin which sabiliy of he sysem is guaraneed in spie of he performance degradaion. his cerain bound is called a maximum allowable delay bound (MADB) [7]. An MADB has been obained from sabiliy condiions of conrol sysems. here have been some resuls on he MADB for sabiliy in non-newored conrol sysems [8,9]. In hese papers, he MADB is obained using he Ricai equaion approach, which yields conservaive delay bounds. here have been also some resuls on he MADB for sabiliy in newored conrol sysems, bu hese were concerned wih sabiliy or heduling of he NCS wih an allowable delay [,4,0]. Walsh e al. inroduce he noion of maximum allowable ransfer inerval (MAI)[], denoed by τ, which supposes ha successive sensor messages are separaed by a mos τ seconds. Less conservaive resuls on he MADB in newored conrol sysems are repored in [7]. However, hese resuls sill remain o be improved. he MADB hus obained can be exended as a maximum bound of a sampling period in he NCS. ha is, he sampling periods deermined by he pro-

2 386 Asian Journal of Conrol, Vol. 8, No. 4, December 2006 posed sampling period decision algorihm can be se o values less han he MADB. Furhermore, hey can no deal wih diarding daa pace and error order ec. here have many sudies in he sabiliy of he sysem in an NCS. Under an assumpion ha he newor-induced delay is less han he sampling period (τ < h), sabiliy of he NCS has been invesigaed in [4]. ime-invarian delay case was considered in [7,2]. A perurbaion mehod of sabiliy analysis of he NCS was given in [,3]. However, he conrol law used was designed in advance wihou considering he presence of he newor in all above references. he sochasic opimal conroller were proposed based on diree-ime model for he cases when he newor-induced delay is shorer [4] or larger [5] han he sampling period. Bu i need a large requiremen of conroller memory o sore a large amoun of pas informaion from he iniial poin. Moreover, o implemen he conroller, he informaion of all pas delays mus be nown as a priori. And in hese mehods, he effec of conrollero-acuaor delays was negleced. Moreover, no mehod was given in he above references how o esimae he noindeal newor saus ha guaranees he sabilizabiliy of he NCS. In his paper, we will be concerned wih maximum allowable synheical bound (MASB) relaed wih he daa pace dropou, newor-included delay and he conroller design of newored conrol sysems. Under consideraion of newor-induced delay, a model of he NCS is presened based on wor of [3,6], where he sensor is cloc-driven and he conroller and acuaor are even-driven and he daa is ransmied wih a single-pace. In conras wih he conroller design mehod based on coninuous linear ime-invarying model, our mehod considers he NCS is a piecewise coninuous uncerain parameers sysem wih sae-delay. Moreover, he newor-induced delay considered in his paper can be slowly or fas ime-varying. hen, a conroller design mehod is proposed based on a delay dependen approach. From he derived crieria, he memoryless conroller can be designed and he MASB can be deermined by solving a se of linear marix inequaliies (LMIs). One example is finally given o show he effeciveness of our mehod. Noaion. R n denoes he n-dimensional Euclidean space, R n m is he se of n m real marices, I is he ideniy marix of appropriae dimensions, sands for he Euclidean vecor norm or he induced marix 2-norm as appropriae. he noaion X > 0 (respecively, X 0), for X R n n means ha he marix X is real symmeric posiive definie (respecively, posiive semi-definie). λ max (P)(λ min (P)) denoes he maximum (minimum) of eigenvalue of real symmeric marix P. For an arbirarily marix B and wo sym- meric marices A and C, A B C denoes a symmeric marix, where * denoes a bloc marix enry implied by symmery.. Modeling of NCS In his paper we consider he following sysem wih sae-delay and parameer uncerainy: x& () = ( A+Δ A) x() + ( A+ΔA) x( d) + ( B+ΔB) u() () x () =ϕ (), [ d, 0] (2) where x() R n and u() R m are he sae vecor and he conrol inpu vecor respecively. A, A, and B are consan marices wih appropriae dimensions. ΔA, ΔA, and ΔB are uncerain marices which can be ime-varying. d is a alar represening he delay in he sysem. Suppose he parameer uncerainies ΔA, ΔA, and ΔB of sysem model are norm-bounded and saisfies [ ΔA ΔA Δ B] = DF( )[ E E E ] (3) 2 3 where F() R i j is an uncerain marix which saisfies F () F() I. D, E, E 2, and E 3 are consan marices of appropriae dimensions. Suppose he sensor are cloc-driven and conroller and acuaor are even-driven, he daa is ransmied wih a single-pace and he full sae variables are available for measuremens, he real inpu u() realize hrough zero-order hold is piecewise consan funcion. Assume ha here are no error codes in he ransmission and ha he sae variables are available for sae feedbac conrol. Apparenly, he daa pace ransmied delay τ i of sensor o conroller do no change he value of x(i h), so x(i h) = xi ( h+τ i ). Similarly, he daa ca pace ransmied delay τ i of conroller o acuaor do ca no change he value of Kx( i h +τ i ), so ui ( ) h+τ i +τ i = Kx( i h +τ i ), where K is he sae feedbac gain. From he above analysis, we can obain u(i h + τ i ) = Kx(i h) (see he parallel dashed line in Fig. ) and hence + u ( ) = Kx ( τ ), { i h+τ, =, 2, L } (4) i i + where u ( ) = lim ˆ ˆ + 0 u ( ), h be he sampling period, i ( =, 2, 3, ) be some inegers such ha {i, i 2, i 3, } {0,, 2, 3, }. he newor-induced delay τ i is he ime from he insan i h when sensors sample from he plan o he insan when acuaor ransmi daa o he plan ca ca ( τ i = τ i +τ i, where τ i is he sensor-o-conroller delay, τ i is conroller-o-acuaor delay and compue and overhead delay is included in τ i ). he ime inerval beween he insan ih+τ i of a pace arriving a he acuaor and he nex arrival insan i+ h +τ i is he effecive + duraion of he hold operaion. Obviously, U [ ih +τ = i, i + h+τ ) = [ 0, ), 0 0. i +

3 C. Peng and D. Yue: Sae Feedbac Conroller Design of Newored Conrol Sysems 387 According o (4), he inpu u() realized hrough a zero-order hold is a piecewise consan funcion and he effecive conrol sysem can be modelled as x& () = ( A+Δ A) x() + ( A+ΔA) x( d) + ( B+Δ B) u(), [ i h+τ i, i ) + h +τ i+ + u ( ) = Kx ( τ ), { i h+τ, =, 2, K } (5) i i Noice ha i is no required o have i + > i. If i + i =, i means ha here is no daa pace dropou in he ransmission. If i + > i +, here are some daa pace dropou and bu he daa are ordered correcly. If i + < i, i means unordered daa arrival sequence occurs, which includes τ = τ 0 and τ < h as he special cases. hese possible nonideal newor condiions are aen in accoun in (5) which are illusraed in Fig.. From Fig., i can be observed ha: h 2h daa pace dropou may occur beween sensor and conroller or conroller and acuaor 2h 3h daa from sensor o conroller are ordered, bu daa from conroller o acuaor are unordered 4h 5h daa from sensor o conroller are unordered and conroller o acuaor are also unordered 6h 7h daa from sensor o conroller are unordered, bu conroller o acuaor are ordered In (5), when he daa arrival sequence is unordered, i + < i. For example, in Fig. when 2h 3h, i h + τ i = 3h + τ 3, i + h + τ i + = 2h + τ 2 and 6h 7h, i h + τ i = 7h + τ 7, i + h + τ i + = 6h + τ 6. he conrol u() mainains a a consan value of u(i h + τ i ) by he zero-order hold when [i h + τ i, i + h + τ i ) + as observed when [τ 0, 3h + τ 3) or [3h + τ 3, 2h + τ 2 ). In his paper, we assume ha u() = 0 before he firs conrol signal reaches he plan. he sysem (5) can be rewrien as x& () = ( A+Δ A) x() + ( A+ΔA) x( d) + ( B+Δ B) Kx( i h), [ i h+τ i, i ) + h +τ i (6) + I is easy o see ha he soluions of (6) are coninuous on [ 0 ). o faciliae developmen, we firs inroduce he following definiion. Definiion. A maximum allowable synheical bound (MASB) of NCS, denoed by η, which saisfies (i + i )h τ η, =, 2, 3, + i + Remar. MASB is relaed wih he number of daa pace dropou ( i + i ), newor-induced delay τi + and sampling period h. herefore, when he MASB is obained, we can use i as a beer heduling mehod for NCS. II. CONROLLER DESIGNS OF NCS In his secion, we assume ha he full sae variables are available for measuremens. We presen a MASB calculaion mehod and relevan conroller design mehod for sysem (6) based on a Lyapunov funcional mehod. Firs, le s consider he simple case wihou he parameer uncerainies ΔA, ΔA, and ΔB. We presen a conroller design mehod for sysem (6) based on an LMI approach. Fig.. Nonideal ransmission process of daa in NCS.

4 388 Asian Journal of Conrol, Vol. 8, No. 4, December 2006 heorem. For given alars η > 0 and λ i (i = 2, 3, 4), if here exis marices P %, S %, and R % > 0, a nonsingualr X and marices Y and M % i (i =, 2, 3, 4) wih appropriae dimensions such ha % % 2 % 3 % 4 % 22 % 23 % 24 2 % 33 % 34 η M% 3 < 0 % 44 4 ηr% ( i + ) (7) ( i + i ) h+τ η, =, 2,K (8) are saisfied, where % % % % % % = S+ AX + XA + M+ M 2 = AX +λ 2 XA + M2, % 3 = BY +λ3 XA M% + M% 3 % 4 = P% +λ4xa X + M% 4 % 22 = S% +λ 2 A X +λ 2 XA % =λ 23 2BY+λ3XA M % 2 % 24 = λ 2 X +λ 4 XA % =λ 33 3BY+λ3Y B M % 3 M % 3 % 34 = λ 3 +λ4 % % 4 44 =ηr% λ 4 X λ4 X X Y B M hen he sysem (6) is asympoically sable wih he feedbac gain K = YX. Proof. We se y() = x& () and consruc a Lyapunov funcional candidae as 2 3 V () = V() + V() + V() d where V () = x () Px(), V2 () = x ( v) Sx( v) dv, V () y () v Ry () v dvds 3 η s =, and P > 0, S > 0 and R > 0. aing he ime derivaive of V() for [i h + τ i, i + h + τ i ), + we have & () 2 x () Py() (9) V = and [ x ( ) N x ( d) N x ( i h) N + y () N4][ Ax () + Ax ( d) + BKxi ( h) y ()] = 0 (3) hen using (2) and (3) and he whole ime derivaive of V() for [i h + τ, i + h + τ yields: i V &() = V& () + V& 2() + V& 3() i ) [ x ( ) M + x ( d) M + x ( i h) M + y () M4 ] x() x( i h) Y( s) ds ih [ x ( ) N + x ( d) N + x ( i h) N + y () N ][ Ax() A x( d) BKx( i h) + y()] 4 = x ()[ S + N A+ A N + M + M ] x() x ( )[ N A + A N + M ] x( d) x ( )[ N BK + A N M + M ] x( τ ) x ( )[ P+ A N N + M ] y( ) + x ( d)[ S + N A + A N ] x( d) x ( d)[ N BK + A N M ] x( i h) + 2 x ( d)[ N + A N ] y( ) x ( i h)[ N BK + K B N M M ] x( i h) x ( ih)[ N + K B N M ] y ( ) + η 4 4 η y ()[ R N N ] y () y ( v) Ryv ( ) dv 2 2[ x ( ) M + x ( d) M x ( i h) M + y ( ) M ] y( s) ds (4) ih From (8), we can obain ha, when [i h + τ i + ), τ i, i + h + & () x () Sx () x ( d) Sx ( d) (0) V 2 = V& 3 () =η y () Ry () y ( s ) Ry ( s ) ds () η and y () s Ry() s ds y () v Ry() v dv (5) η i h Using he Newon-Leibniz formula x() x(i h) ys () ds= 0 and (6), we can show ha, for arbirary ih marices N i and M i (i =, 2, 3, 4) of appropriae dimensions, [ x ( ) M x ( d) M x ( i h) M y ( ) M ] x () xi ( h) ys ( ) ds = 0 (2) ih 2 3 2[ x ( ) M + x ( d) M + x ( i h) M 4 ih + y () M ] y( s) ds ηξ () M R Mξ() y ih + () v Ry() v dv (6) where ξ () = [x () x ( d) x (i h) y ()], M = [ M M M M ]. Combining (4)-(6), we obain

5 C. Peng and D. Yue: Sae Feedbac Conroller Design of Newored Conrol Sysems V &() ξ () ξ () +ηξ () M R Mξ() (7) where S N A A N M M 2 N A A N2 M2 = = + + = NBK+ AN M + M = P+ AN N+ M = S + N A + A N = N BK + A N M = N + A N = N BK + K B N M M = N + K B N M =ηr N N (8) Suppose (7) is saisfied, i is apparen ha % 44 =ηr% λ 4 X λ 4 X < 0, According o he heorem, η > 0, R % > 0, assume λ 4 > 0, we can obain X + X < 0 and X is nonsingual. Defining N 2 = λ 2 N, N 3 = λ 3 N, N 4 = λ 4 N, and Y = KX, defining N = X, P % = XPX, R % = XRX, S % = XSX, and M % i = XM i X (i =, 2, 3, 4) hen pre, posmuliplying boh sides of (9) wih diag (X X X X) and is ranspose, we can ge (7), herefore, (9) is equivalen o (7) ηm ηm η M 3 < 0 44 ηm4 ηr (9) According o (9), by Suchr complemens, we can obain ha (7) is smaller han zero ( V &( ) < 0) for [i h + τ i, i + h + τ i ) +. Since [ i h+τ i, i ) + h +τ = i+ = [ 0, ), 0 0 and V() is coninuous in [ 0, ) since x() is coninuous in [4]. We can deduce V & () < 0 for [ 0, ). herefore, by using he Lyapunov-Krasovsii heorem, he closed-loop sysem (6) is asympoically sable. his complees he proof. Remar 2. From he proof, we can obain ha X is nonsingual. So in he proof, we can se Y = KX and obain K = YX. Oherwise, In his paper, he sysem is asympoically sable means ha an equilibrium is asympoically sable. Considering he effec of he parameer uncerainies ΔA, ΔA, and ΔB, we conclude he following resul for he uncerain ime-delay sysems (6). heorem 2. For given alars η and λ i (i = 2, 3, 4), if here exis alars ε i > 0 (i =, 2, 3), marices P %, S %, and R % > 0, a nonsingualr X and marices Y and M % i (, 2, 3, 4), wih appropriae dimension, if U % +Φ % 2 % 3 % 4 λxe 0 0 % 22 +λ2φ % 23 % λxe M Y E % +λ Φ % η % λ 3 % 44 +λ4φ < 0 ηr% λε I 0 0 λε2 I 0 λε3 I (20)

6 390 Asian Journal of Conrol, Vol. 8, No. 4, December 2006 (( i i) h i + ) + +τ η (2) +λ 2 +λ 3 +λ 4 > 0 (22) where Φ = (ε + ε 2 + ε 3 ) DD, λ = + λ 2 + λ 3 + λ 4, hen he sysem (6) wih he feedbac gain K = YX is asympoically sable. Proof. Replace A, A, and B wih A + DF()E, A + DF()E 2, and B + DF()E 3 in (4). hen, following he similar procedure as in he proof of heorem, we can obain Θ % 2 % 3 % 4 Ξ % 23 % 24 2 Φ % 34 η M% 3 < 0 % 44 +λ4φ 4 ηr% (23) where Θ= % +Φ+λε XE EX + XQX, Ξ= % 22 +λ2φ+λε2 XE2E2 X, Φ= % 33 +λ3φ+λε3 Y E3 E3Y + Y RY. Using Schur complemen, we can obain (20) from (23) and 22 from λ > 0. his complees he proof. From he heorem and heorem 2, i is apparen ha differen λ i (i = 2, 3, 4) have differen value of η, so we purpose such search algorihm o find he opimal value of λ i (i = 2, 3, 4) o obain he sub-maximum η s max. o obain he η s max, we presen following search algorihm: Algorihm: Given he α i, β i as upper and lower bound of λ i, ζ i (i = 2, 3, 4) as sep incremen. Se η 0 = 0 and K = 0. According o heorem, under he consrain of ζ i, α i, and β i, group differen λ i o obain corresponding η and K = YX based on LMI, subjec o (??) and (8). If η > η 0, se η = η 0 and K = K 0. Oupu η 0 = η s max and K = K 0. he feedbac gain of newored conroller is designed as K = YX, he η s max ha guaranees he sabilizabiliy of he sysem (6), we apply he heorem 2 and he search algorihm o derive he η s max and he corresponding feedbac gain K wih λ 2 =., λ 3 = 0.5, λ 4 = 8.4, i has been found ha he maximum value of η s max is 2.95 and he corresponding feedbac gain is K = [ ]. In oher words, as long as (( i+ i) h +τi ) 2. 95, he (6) + wih K = [ ] is asympoically sable. I means ha, if h = 0.2 ms and he nonideal ransmied daa pace can be negleced in he ransmission, he maximum allowable delay τi 275ms.. he designed conroller can sabilize he sysem (5) as long as he upper bound of he newor-induced delay is less han 2.75 ms. A plo of he saes of he above uncerain sysem wih differen conroller feedbac gain is shown in Fig. 3. I apparen ha he sysem is asympoically sable. According o he (( i+ i) h +τi ) η, if we have + nown he MADB of specified NCS, we can obain he relaionship of he sampling period h, number of daa pace dropou ( i + i ) and newor-induced delay τ i +, i s shown in able 2. From he able 2, if he sampling period is se o 0.4, η s max is 2.95, hen i allow daa pace dropou and nonordered daa arrival sequence happen. For example: No daa pace dropou i + i =, no unordered sequence i + > i, he MADB is 2.55 ( τi η h). Or + have an unordered sequence i + < i, he MADB is 3.35 ( τ η+ h). i + III. NUMERICAL EXAMPLE his secion presens an example o obain he η s max and he feedbac gain K. Consider an uncerain sae delay sysem x& () = ( A+Δ A) x() + ( A+ΔA) x( d) + ( B+ΔB) u() (24) where A=, A =, B = (25) D =, E = [0. 2 0], E2 = [0 0. ], E3 = [0. 4] 0 (26) Fig. 2. Sae variable versus sampling ime wih K = [ ].

7 C. Peng and D. Yue: Sae Feedbac Conroller Design of Newored Conrol Sysems 39 Have daa pace dropou i + i >, number of daa pace dropou ( i + i ) can be, 2, 3, under he consrain of (( i+ i) h +τi ) η. Which including no + unordered daa arrival sequence i + > i, and unordered daa arrival sequence i + < i. When have one daa pace dropou i + i = 2, he MADB is 2.35 ( τi η 2 h). + MASB provide a beer heduling mehod o NCS. Such as, o he specified newor (e.g. DeviceNe, ConrolNe ec.), we can esimae he newor-induced delay as a prior nowledge of newor heduling, hen adjus he sampling period (h) and he rae of acive daa pace dropou o ge opimal performance index. able 2. he relaionship of heduling parameer. MAEDB Sampling Period Daa Dropou Allowable Number Delay IV. CONCLUSIONS Performance of he feedbac conrol in he NCS is direcly dependen upon he newor- induced delay. imevarying characerisics of he newor-induced delay no only degrade conrol performance bu also inroduce disorion of he conroller signal. In his paper, he plan wih uncerain parameers and sae delay is sudies. We assure ha he full sae variables are available for measuremens, presen a calculaional mehod of a new rule and relevan conroller design mehod for sysem (5). he maximum allowable synheical bounds are obained for he sabilizabliy of he NCS based on Lyapunov funcional mehod and linear marix inequaliies (LMI) formulaion. As fuure wors, he daa dropou and he muli-loops newor heduling algorihm will be considered and he plan wih performance index such as guaraneed cos and H 2 /H are necessary o be sudied. REFERENCES. Aso, R. and H. Yoram, Inegraed Communicaion and Conrol Sysems: Par II-Design Consideraion, J. Dyn. Sys., Meas. Conr., Vol. 0, pp (988). 2. Peer, V.Z. and H.M. Richard, Newored Conrol Design for Linear Sysems, Auomaica, Vol. 39, pp (2003). 3. Yue, D., Q.L. Han, and C. Peng, Sae Feedbac Conroller Design of Newored Conrol Sysems, IEEE rans. Circuis Sys. II: Analog Digial Signal Process., Vol. 5, pp (2004). 4. Zhang, W., M.S. Branicy, and S.M. Phillips, Sabiliy of Newored Conrol Sysems, IEEE Conr. Sys. Mag., Vol. 2, pp (200). 5. Hong, S.H., Scheduling Algorihm of Daa Sampling imes in he Inegraed Communicaion and Conrol Sysems, IEEE rans. Conr. Sys. echnol., Vol. 3, pp (995). 6. Lian, F.L., J.R. Moyne, and D.M. ilbury, Performance Evaluaion of Conrol Newors: Eherne, Conrolne, and Devicene, IEEE Conr. Sys. Mag., Vol. 2, pp (200). 7. Kim, D.S., Y.S. Lee, W.H. Kwon, and H.S. Par, Maximum Allowable Delay Bounds of Newored Conrol Sysems, Conr. Eng. Prac., Vol., pp (2003). 8. Mori,., N. Fuuma, and M. Kuwahara, Simple Sabiliy Crieria for Single and Composie Linear Sysems wih ime Delays, In. J. Conr., Vol. 34, pp (98). 9. Su,.J. and C.G. Huang, Robus Sabiliy of Delay Dependence for Linear Uncerain Sysems, IEEE rans. Auoma. Conr., Vol. 37, pp (992). 0. Lian, F.L., J.R. Moyne, and D.M. ilbury, Newor Design Consideraion for Disribued Conrol Sysems, IEEE rans. Conr. Sys. echnol., Vol. 0, No. 2, pp (2002).. Walsh, G.C. and Y. Hong, Scheduling of Newored Conrol Sysems, IEEE Conr. Sys. Mag., Vol. 2, pp (200). 2. Par, H.S., Y.H. Kim, D.S. Kim, and W.H. Kwon, A Scheduling Mehod for Newor Based Conrol Sysems, IEEE rans. Conr. Sys. echnol., Vol. 0, pp (2002). 3. Walsh, G., O. Beldiman, and L. Bushnell, Asympoic Behavior of Nonlinear Newored Conrol Sysems, IEEE rans. Auoma. Conr., Vol. 46, pp (200). 4. Nilsson, J., B. Bernhardsson, and B. Wienmar, Sochasic Analysis and Conrol of Real-ime Sysems wih Random ime Delays, Auomaica, Vol. 34, pp (998). 5. Hu, S. and Q. Zhu, Sochasic Opimal Conrol and Analysis of Sabiliy of Newored Conrol Sysems wih Long Delay, Auomaica, Vol. 39, pp (2003). 6. Yue, D., Q.L. Han, and J. Lam, Newor-Based Robus Conrol of Sysems wih Uncerainy, Auomaica, Vol. 4, pp (2004).

8 392 Asian Journal of Conrol, Vol. 8, No. 4, December 2006 Chen Peng was born in China in 972. He received his B.Sc., M.Sc. and Ph.D. from Chinese Universiy of Mining echnology in 996, 999, and 2002 respecively. He was a Pos-docoral Research Fellow a Nanjing Normal Universiy. From November 2004 o January 2005, he was a Research Associae a Hong Kong Universiy. He is currenly a full associae Professor a Nanjing Normal Universiy. His research ineress include newored conrol sysems, Supply chain and nowledge diovery in daabase. Dong Yue was born in China in 964. He received his B.Sc. from Guilin Insiue of Elecrical Engineering in 985, M.Sc. from Anhui Universiy in 99 and Ph.D. from Souh China of echnology in 995, respecively. From 995 o 997, he was a Pos-docoral Research Fellow a China Universiy of Mining and echnology. He was a full Professor from 997 o 200 a China Universiy of Mining and echnology. From June 999 o Sep. 999, he was a Research Associae a Ciy Universiy of Hong Kong. From Aug., 2000 o Aug., 200, he wored a Pohang Universiy of Science and echnology as a Senior Scienis. From June 2002 o Sep. 2002, he was a Research Aassociae a Hong Kong Universiy. From Augus 2003 o Ocober 2003, he was a Visiing Professor a Cenral Queensland Universiy. From March 2004 o March 2005, he was a Research Fellow a Cenral Queensland Universiy. He is currenly a full Professor a Nanjing Normal Universiy and direcor of Research Cenre for Informaion and Conrol Engineering echnology. His research ineress include newored conrol sysems, ime delay sysems, robus conrol.