Stability Analysis of Discrete-time Systems with Poisson-distributed Delays

05 IEEE 54th Annual Conference on Decision Control (CDC) December 5-8, 05. Osaka, Japan Stability Analysis of Discrete-time Systems with Poisson-distributed Delays Kun Liu, Karl Henrik Johansson, Emilia Fridman Yuanqing Xia Abstract This paper is concerned with the stability analysis of linear discrete-time systems with poisson-distributed delays. Firstly, the exponential stability condition of system with poisson-distributed delays is derived when the corresponding system with the zero-delay or the system without the delayed term is asymptotically stable. Then, an augmented Lyapunov functional is suggested to hle the case that the corresponding system without the delay as well as the system without the delayed term are not necessary to be asymptotically stable. Furthermore, we show that the results can be further improved by formulating the system as a higher-order augmented one applying the corresponding augmented Lyapunov functional. Finally, the efficiency of the proposed results is illustrated by some numerical examples. Keywords: Infinite delays, poisson-distributed delays, stabilizing delays, Lyapunov method. I. INTRODUCTION Systems with distributed delays are frequently encountered in modeling the physiological behavior, the traffic flow, the population dynamics, the control over networks [9, [0. In general, there are two main classes of distributed delays, namely: finite distributed delays infinite distributed delays. A great number of results have been reported for the stability control of systems with finite distributed delays, e.g., [, [, [3, [ the reference therein. For the case of infinite distributed delays, we refer to [8, [9, [3, where necessary sufficient conditions for the stability of continuous-time systems with gamma-distributed delays were derived in the frequency domain. In the time domain, sufficient conditions for the stability of continuoustime systems with gamma-distributed delays were derived in [4 via appropriate Lyapunov functionals. Recently, the Lyapunov-based stability passivity analysis for diffusion partial differential equations with infinite distributed delays were presented in [5. Discrete-time systems with infinite distributed delays have been analyzed in the literature. For example, the synchronization problem for an array of coupled complex discrete-time networks with infinite distributed delays was investigated in [7. The state feedback control was considered in [6 for discrete-time stochastic systems with infinite distributed delays nonlinear disturbances. In [7 [8, the K. Liu is with the School of Automation, Beijing Institute of Technology, Beijing 0008, China. E-mail: kunliu@kth.se. His work was done when he was with KTH Royal Institute of Technology. K. H. Johansson is with the ACCESS Linnaeus Centre School of Electrical Engineering, KTH Royal Institute of Technology, SE-00 44, Stockholm, Sweden. E-mail: kallej@kth.se. E. Fridman is with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail: emilia@eng.tau.ac.il. Y. Xia is with the School of Automation, Beijing Institute of Technology, Beijing 0008, China. E-mail: xia yuanqing@bit.edu.cn. robust H control problem was discussed for discrete-time fuzzy systems with infinite distributed delays. It should be pointed out that all the results reported in [7, [6, [7, [8 are concerned with constant kernel function. Moreover, when the corresponding system without the delay as well as the system without the delayed term are not asymptotically stable, the proposed methods in [7, [6, [7, [8 are not applicable. It is well-known that poisson distribution is widespread in queuing theory [4. In [, the experimental data on the arrivals of pulses in indoor environments revealed that each cluster s time-delay is poisson-distributed, see also [5 for more explanations. In the present paper, we consider linear discrete-time systems with poisson-distributed delays. The objective is to derive sufficient exponential stability conditions for the system via appropriate Lyapunov functionals. It is allowed that the corresponding system without the delay as well as the system without the delayed term are not asymptotically stable. Thus, the considered infinite distributed delays with a gap in the paper have stabilizing effects. We derive the results by transforming the system to an augmented one applying augmented Lyapunov functionals [, [4. Due to the effect of poisson-distributed delays, the augmented system contains not only distributed but also discrete delays. This is different from the continuous-time counterpart in [4 for the general case of gamma-distributed delays, where only distributed delays were included in the resulting augmented system. Furthermore, we show that the results can be further improved by formulating the system as a higher-order augmented one applying the corresponding augmented Lyapunov functional. The structure of the paper is organized as follows. Section II presents the systems with poisson-distributed delays the summation inequalities that will be employed. The exponential stability of systems with poisson-distributed delays is studied in Section III when the corresponding system without the delay or the system without the delayed term is asymptotically stable. Section IV shows the exponential stability conditions of systems with poisson-distributed delays when the corresponding system with the zero-delay as well as the system without the delayed term are allowed to be not asymptotically stable. Section V illustrates the efficiency of the presented approach with some examples. Finally, the conclusions the further work are stated in Section VI. Notations: The notations used throughout the paper are stard. The superscript T sts for matrix transposition, R n denotes the n dimensional Euclidean space with vector norm, R n m is the set of all n m real matrices. P 0 (P 0) means that P is positive definite (positive semi- 978--4799-7886-/5/$3.00 05 IEEE 7736

definite). denotes the term that is induced by symmetry I represents the unit matrix of appropriate dimensions. The symbol Z + denotes the set of non-negative integers. II. SYSTEM DESCRIPTION Consider the following linear discrete-time system with poisson-distributed delays: x(k +) = Ax(k)+A + τ=0 p(τ)x(k τ), k Z+, () where x(k) R n is the state vector, the system matrices A A are constant with appropriate dimensions. The initial condition is given as col{x(0),x( ),x( ),...} = col{φ(0),φ( ),φ( ),...}. The functionp(θ) is a poisson distribution with a gap h Z + : p(θ) = { e θ h (θ h)! θ h, 0 θ < h. The gap h can be interpreted e.g., in the network as the minimal propagation delay, which is always strictly positive. The mean value of p is +h. Due to the fact that τ=0 p(τ)x(k τ) = τ=h = θ=0 p(τ)x(k τ) p(θ +h)x(k θ h), we arrive at the equivalent system to () as follows: x(k +)=Ax(k)+A + τ=0 P(τ)x(k τ h), k Z+, () where P(τ) = e τ τ!. Moreover, some elementary calculus shows that for scalar 0 < δ δ i h P(i) = δ h e (δ ) = p 0δ, δ i h (i+h)p(i) = δ h e (δ ) (δ +h) = p δ, p = pδ δ= = +h. The derivation of stability conditions for system () is based on the summation inequalities with infinite sequences formulated in the following lemma. Lemma Given an n n matrix R 0, an integer h 0, scalar functions M(i) R, α(i) R + \{0}, a vector function x(i) R n such that the series concerned are convergent. Then the inequality α(i) M(i) xt (i)rx(i) TR [ M0 M(i)x(i) +, M(i)x(i) (3) its double summation extension hold, where h j=k i h α(i) M(i) xt (j)rx(j) TR M j=k i h M(i)x(j) k j=k i h, M(i)x(j) M 0 = α (i) M(i), M h = α (i)(i+h) M(i). (4) (5) Proof: The proofs of (3) (4) follow from those in [4 by involving sums instead of integral. Since R 0, application of Schur complements implies that the following holds [ α(i) M(i) x T (i)rx(i) x T (i)m(i) α (i) M(i) R 0 (6) for any i [0,+, i Z +. Summation of (6) from 0 to + leads to [ α(i) M(i) xt (i)rx(i) xt (i)m(i) M 0 R 0. By Schur complements, the above matrix inequality yields (3). Furthermore, double summation of [ α(i) M(i) x T (j)rx(j) x T (j)m(i) α (i) M(i) R 0 from k i h to k in j from 0 to + in i, where j=k i h α (i) M(i) = M h Schur complements ensure that the inequality (4) holds. In the sequel, the summation inequalities (3) (4) with infinite sequences play an important role in the stability problem of discrete-time systems with poisson-distributed delays. III. STABILITY IN THE CASE THAT A OR A+A IS SCHUR STABLE Consider system (). It is assumed that A or A + A is Schur stable. Our stability analysis will be based on the following discrete-time Lyapunov functional: V(k) = x T (k)wx(k)+v G (k)+v H (k), V G (k) = s=k i h δk s P(i)x T (s)g x(s), V H (k)= i+h j= s=k j δk s P(i)η T (s)h η (s), (7) where 0 < δ <, W 0, G 0, H 0, η (k) = x(k +) x(k). (8) Remark The term V G (k) compensates the delayed term in () when A is Schur stable. The term V H (k) extends the triple integrals of [4 to discrete case. It compensates the summation term in x(k +) = (A+A )x(k) +A τ=0 P(τ)[x(k τ h) x(k), k Z+, allows us to derive stability conditions of system () when A + A is Schur stable whereas A is not. Moreover, the V H (k) term also improves the results provided that A is Schur stable. By the stard arguments for discrete-time systems, we arrive at the following linear matrix inequality (LMI) condition for exponential stability of (): 7737

Proposition Given scalars > 0, 0 < δ < an integer h 0, assume that there exist n n positive definite matrices W, G H, such that the following LMI holds: Ξ = Σ+F0 T WF 0 p δ FT H F +p F0H T F 0 0, (9) where Σ = diag{g δw, p 0δ G }, F 0 = [A A, F 0 = [A I A, F = [I I. Then the system () is exponentially stable with the decay rate δ. Proof: Denote f(k) = τ=0p(τ)x(k τ h), ξ(k) = col{x(k),f(k)}, k Z +. Taking difference of V(k) along () leads to V(k) = V(k +) δv(k) ξ T (k)f0 TWF 0ξ(k)+x T (k)(g δw)x(k) +p η T(k)H η(k) δi+h P(i)x T (k i h)g x(k i h) (0) s=k i h δi+h P(i)η T (s)h η (s). () Applying further the inequalities (3) (4), we obtain δi+h P(i)x T (k i h)g x(k i h) p 0δ ft (k)g f(k) p δ s=k i h δi+h P(i)η T (s)h η (s) TH s=k i h (s) P(i)η k s=k i h P(i)η (s) = p δ [x(k) f(k)t H [x(k) f(k) = p δ ξt (k)f TH F ξ(k). () (3) Then () (3) yield V(k) = V(k + ) δv(k) ξ T (k)ξξ(k) 0 if LMI (9) holds. Due to the fact that min (W) x(k) V(k) δ k V(0), V(0) β φ c, where β > 0 is a scalar φ c = sup s=0,,,... φ(s), the system () is exponentially stable with the decay rate δ for given scalars > 0 h 0. Remark IfAis Schur stable, Lyapunov functional (7) with H = 0 can be applied to yield the following simple but conservative LMI condition: [ G +A T WA δw A T WA p 0δ G +A T WA 0. IV. STABILITY IN THE CASE THAT A AND A+A ARE ALLOWED TO BE NOT SCHUR STABLE In this section, we will derive LMI conditions for the exponential stability of system (), where A A + A may not be Schur stable. This means that the considered infinite distributed delays with a gap have stabilizing effects. We provide two sufficient stability conditions, the efficiency of which is illustrated by the given examples below. A. Condition I: an augmented system From (0), it follows that system () can be transformed into the following augmented one: x(k +) = Ax(k)+A f(k), f(k +) = τ=0 e τ τ! x(k + τ h) = e x(k + h)+ τ= e τ τ! x(k + τ h) = e x(k + h)+ τ=0 e τ+ (τ+)! x(k τ h) = e Ax(k h)+e A f(k h) + τ=0 Q(τ)x(k τ h), (4) where Q(τ) = e τ+ (τ+)!. Remark 3 The stability of system (4) implies the stability of system (), but not vice versa. For a positive integer h, the effect of poisson-distributed delays leads to the augmented system (4) that contains one term τ=0 Q(τ)x(k τ h) corresponding to distributed delays, two terms e Ax(k h), e A f(k h) with discrete delays. This is different from the continuous-time counterpart in [4 for the general case of gamma-distributed delays, where only distributed delays were included in the resulting augmented system. Simple computation shows that δ i h Q(i) = δ h e (e /δ ) = q 0δ, δ i h (i+h)q(i) = δ h e [e /δ +δ(h )(e /δ ) = q δ, q 0 = q0δ δ= = e, q = qδ δ= = +(h )( e ). (5) Consider system (4) with both distributed discrete delays. We suggest the following discrete-time Lyapunov functional: [ ˆV(k)=ξ T (k)ŵξ(k)+ V Gi (k)+v Hi (k)+v Si (k), V G (k) = i= i+h j= s=k i h δk s Q(i)x T (s)g x(s), V H (k) = s=k j δk s Q(i)η T (s)h η (s), V S (k) = s=k h δk s x T (s)s x(s) +h j= h s=k+j δk s η T(s)R η (s), V S (k) = s=k h δk s f T (s)s f(s) +h j= h s=k+j δk s η T(s)R η (s), (6) where the terms V G (k) V H (k) are defined in (7), ξ(k) η (k) are given in (0) (8), respectively, 0 < δ <, G 0, H 0, S i 0, R i 0, i =,,, Ŵ = [ P Q Z 0, (7) η (k) = f(k +) f(k). (8) Here the last two terms V S (k) V S (k) are added to compensate, respectively, the delayed terms x(k h) f(k h) of (4). Therefore, for system (4) with h = 0, V S (k) V S (k) are not necessary. We derive conditions 7738

to guarantee exponential stability of system (4) in the following proposition: Proposition Given scalars > 0, 0 < δ < an integer h 0, let there exist n n positive definite matrices P,Z, G i, H i, S i, R i, i =,, an n n matrix Q such that (7) the following LMI are satisfied: ˆΞ = ˆΣ+ ˆF 0 TŴ ˆF 0 δ ˆF TŴ ˆF p ˆF δ TH ˆF +ˆF 0 T[p H +q H +h R ˆF 0 +h ˆFT 0 R ˆF0 q ˆF δ 5 TH ˆF 5 δ h ˆFT 3 R ˆF3 δ h ˆFT 4 R ˆF4 0, (9) where ˆΣ = diag{s + G + q 0 G, p 0δ G + S, δ h S, δ h S, q 0δ G }, [ A A 0 0 0 ˆF 0 = 0 0 e A e, [ A I I 0 0 0 0 ˆF =, ˆF0 = [A I A 0 I 0 0 0 0 0 0, ˆF 0 = [0 I e A e A I, ˆF = [I I 0 0 0, ˆF 3 = [I 0 I 0 0, ˆF5 = [q 0 I 0 0 0 I, ˆF 4 = [0 I 0 I 0. Then the system (4) is exponentially stable with the decay rate δ. Proof: Define υ(k) = τ=0q(τ)x(k τ h) (0) ˆξ(k) = col{x(k),f(k),x(k h),f(k h),υ(k)}. By taking difference of ˆV(k) along (4) applying Jensen inequality with finite sequences (see e.g., [6), we have i= ξ T (k +)Ŵξ(k +) δξt (k)ŵξ(k) = ˆξ T (k)[ˆf 0 TŴ ˆF 0 δ ˆF TŴ ˆF ˆξ(k) [ V Gi (k +)+V Hi (k +)+V Si (k +) δv Gi (k) δv Hi (k) δv Si (k) ˆξ T (k)[ˆσ+ ˆFT 0 [p H +q H +h R ˆF 0 p δ ˆF T H ˆF +h ˆFT 0 R ˆF0 δ h ˆFT 3 R ˆF3 δ h ˆFT 4 R ˆF4 ˆξ(k) () +q 0δ υt (k)g υ(k) δi+h Q(i)x T (k i h)g x(k i h) s=k i h δi+h Q(i)η T(s)H η (s). () Applying further the inequality (3), we obtain δi+h Q(i)x T (k i h)g x(k i h) q 0δ υt (k)g υ(k). Furthermore, the application of (4) leads to q δ s=k i h δi+h Q(i)η T (s)h η (s) TH s=k i h (s) Q(i)η k s=k i h Q(i)η (s) = q δ [q 0x(k) υ(k) T H [q 0 x(k) υ(k) = q ˆξ δ T (k)ˆf 5 TH ˆF 5ˆξ(k). (3) (4) Therefore, () (4) yield ˆV(k) = ˆV(k +) δˆv(k) ˆξ T (k)ˆξˆξ(k). Then if LMI (9) holds for given scalars > 0 h 0, the system (4) is exponentially stable with the decay rate δ. B. Condition II: a higher-order augmented system For the case that A A + A are not necessary to be Schur stable, alternatively, we analyze the stability of system () by transforming () into the following higherorder augmented one: x(k +) = Ax(k)+A f(k), f(k +) = e Ax(k h)+e A f(k h)+υ(k), υ(k +) = e Ax(k h)+e A f(k h)+g(k), (5) where f(k) υ(k) are defined in (0) (0), respectively, Similar to (5), we have g(k) = τ=0u(τ)x(k τ h), U(τ) = e τ+ (τ+)!. δ i h U(i) = δ h e (e /δ δ ) = u 0δ, δ i h (i+h)u(i) =δ h e [δ +(h +δ )(e /δ δ ) = u δ, u 0 = u0δ δ= = e e, u = uδ δ= = e +(h )( e e ). For system (5), we introduce the following augmented Lyapunov functional: V(k)= x T (k) W x(k)+ [ i= V Gi (k)+v Hi (k)+v Si (k) +V G3 (k)+v H3 (k) with V G3 (k) = V H3 (k) = i+h j= s=k i h δk s U(i)x T (s)g 3 x(s), s=k j δk s U(i)η T (s)h 3 η (s), where the terms V Gi (k), V Hi (k) V Si (k), i =,, are defined in (6), 0 < δ <, G 3 0, H 3 0, x(k) = col{x(k), f(k), υ(k)}, η (k) is given by (8),, W = P Q Q Z Q 3 0. (6) R Remark 4 Differently from the continuous-time counterpart in [4 for the general case of gamma-distributed delays, to analyze stability of the higher-order augmented system (5) for poisson-distributed delays, two additional terms V G3 (k) V H3 (k) are necessary to compensate the term g(k) = τ=0 U(τ)x(k τ h) of (5). Following the proof of Proposition, we derive the following condition for exponential stability of system (5): Proposition 3 Given scalars > 0, 0 < δ < an integer h 0, let there exist n n positive definite matrices 7739

TABLE I COMPLEXITY OF DIFFERENT STABILITY CONDITIONS Method Decision variables Number order of LMIs Proposition.5n +.5n one of 3n 3n Proposition 6n +5n one of 7n 7n one of n n Proposition 3 9.5n +6.5n one of 8n 8n one of 3n 3n P, Z, R, S i, R i, i =,, G j, H j, j =,,3, n n matrices Q j, j =,,3, such that (6) the following LMI are feasible: Ξ = Σ+ F 0 T W F 0 δ F T W F p F δ H T F + F 0 T[p H +q H +u H 3 +h R F 0 +h FT 0 R F0 q F δ 5 TH F 5 u F δ 6 TH 3 F 6 δ h FT 3 R F3 δ h FT 4 R F4 0, (7) where Σ = diag{s + G + q 0 G + u 0 G 3, p 0δ G + S, δ h S, δ h S, q 0δ G, u 0δ G 3}, A A 0 0 0 0 F 0 = 0 0 e A e A I 0, 0 0 e A e A 0 I F = I 0 0 0 0 0 0 I 0 0 0 0, F0 = [A I A 0 0 0 0, 0 0 0 0 I 0 F 0 = [0 I e A e A I 0, F = [I I 0 0 0 0, F 3 = [I 0 I 0 0 0, F5 = [q 0 I 0 0 0 I 0, F 6 = [u 0 I 0 0 0 0 I, F4 = [0 I 0 I 0 0. Then the system (5) is exponentially stable with the decay rate δ. Remark 5 Compare the number of scalar decision variables the resulting LMIs. See Table I for the complexity of different stability conditions. Note that Proposition 3 achieves less conservative results than Propositions on account of computational complexity (see examples below). Remark 6 The system () with poisson-distributed delays can be also transformed into an augmented system with respect to the state col{x(k),f(k),υ(k),g(k)}. By virtue of corresponding augmented Lyapunov functional, the achieved less conservative results suffer from more decision variables higher order LMIs. V. ILLUSTRATIVE EXAMPLES In this section, we provide two examples to demonstrate the efficiency of stability conditions. A. Example Consider the linear discrete-time system () with [ [.05 0.0 0.5 0 A = A 0..05 = 0. 0. Here A + A is Schur stable whereas A is not. For the values of h given in Table II, by applying Propositions,. TABLE II EXAMPLE : MAXIMUM ALLOWABLE VALUES OF FOR DIFFERENT h [max \ h 0 8 Proposition 8.5 7.5 6.5 0.5 Proposition 8.4 7.4 6.4 0.4 Proposition 3 8.55 7.55 6.55 0.55 3 with δ =, we obtain the maximum allowable values of that guarantee the asymptotic stability (see Table II). For δ = the stability region in the (, h) plane that preserves the asymptotic stability is depicted in Figs. 3 by using Propositions, 3, respectively. The simulation results in Figs. 3 verify that Proposition with the number {6n + 5n} n= = 34 of variables induces more dense stability region than Proposition with {.5n +.5n} n= = 9 variables, but guarantees sparser stability region than Proposition 3 with the number{9.5n + 6.5n} n= = 5 of variables. h 9 8 7 6 5 4 3 Proposition 0 0 3 4 5 6 7 8 9 Fig.. Example : stability region by Proposition with δ = h 9 8 7 6 5 4 3 Proposition 0 0 3 4 5 6 7 8 9 Fig.. Example : stability region by Proposition with δ = B. Example Consider the linear discrete-time system () with the coefficient matrices: [ [ 0.5 0 0.5 0.8 A = A 0 =. 0.5 0. 7740

9 8 Proposition 3 National Natural Science Foundation of China (grant no. 650306, 6440058). h 7 6 5 4 3 0 0 3 4 5 6 7 8 9 Fig. 3. Example : stability region by Proposition 3 with δ = It is worth mentioning that in this system both A A+A are not Schur stable. Thus, Proposition is not applicable. For h = 0 the allowable values of that guarantee the asymptotic stability of the system by Propositions 3 with δ =, are shown in Fig. 4. Therefore, the delays have stabilizing effects. It is observed that in comparison with Proposition, Proposition 3 improves the results but at the price of {3.5n +.5n} n= = 7 additional decision variables. Proposition Proposition 3 0 0.5.5.5 3 Fig. 4. Example : allowable values of by Propositions 3 with h = 0 δ = VI. CONCLUSIONS In this paper, we have presented LMI conditions for exponential stability of linear discrete-time systems with poisson-distributed delays. Especially, by formulating the system as an augmented one we have hled the case of stabilizing delay, i.e., the corresponding system with the zero-delay as well as the system without the delayed term are not necessary to be asymptotically stable. Extensions of the proposed direct Lyapunov approach to networked control systems will be interesting topics for future research. REFERENCES [ W.H. Chen W.X. Zheng. Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica, 43():95 04, 007. [ E. Fridman. New Lyapunov-Krasovskii functionals for stability of linear retarded neutral type systems. Systems & Control Letters, 43(4):309 39, 00. [3 E. Fridman G. Tsodik. H control of distributed discrete delay systems via discretized Lyapunov functional. European Journal of Control, 5():84 94, 009. [4 D. Gross, J. Shortle, J. Thompson, C. Harris. Fundamentals of queueing theory. John Wiley & Sons, 008. [5 S.A. Kotsopoulos K. Loannou. Hbook of Research on Heterogeneous Next Generation Networking: Innovations Platforms: Innovations Platforms. IGI Global, 008. [6 K. Liu E. Fridman. Delay-dependent methods the first delay interval. Systems & Control Letters, 64():57 63, 04. [7 Y.R. Liu, Z.D. Wang, J.L. Liang, X.H. Liu. Synchronization state estimation for discrete-time complex networks with distributed delays. IEEE Transactions on Systems, Man, Cybernetics, Part B: Cybernetics, 38(5):34 35, 008. [8 W. Michiels, C. I. Morarescu, S. I. Niculescu. Consensus problems with distributed delays, with application to traffic flow models. SIAM Journal on Control Optimization, 48():77 0, 009. [9 C.I. Morarescu, S.I. Niculescu, K. Gu. Stability crossing curves of shifted gamma-distributed delay systems. SIAM Journal on Applied Dynamical Systems, 6():475 493, 007. [0 J.P. Richard. Time-delay systems: An overview of some recent advances open problems. Automatica, 39(0):667 694, 003. [ A. Saleh R. Valenzuela. A statistical model for indoor multipath propagation. IEEE Journal on Selected Areas in Communications, 5():8 37, 987. [ A. Seuret F. Gouaisbaut. Wirtinger-based integral inequality: application to time-delay systems. Automatica, 49(9):860 866, 03. [3 R. Sipahi, F. M. Atay, S. I. Niculescu. Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers. SIAM Journal on Applied Mathematics, 68(3):738 759, 007. [4 O. Solomon E. Fridman. New stability conditions for systems with distributed delays. Automatica, 49():3467 3475, 03. [5 O. Solomon E. Fridman. Stability passivity analysis of semilinear diffusion PDEs with time-delays. International Journal of Control, 88():80 9, 05. [6 Z.D. Wang, Y.R. Liu, G.L. Wei, X.H. Liu. A note on control of a class of discrete-time stochastic systems with distributed delays nonlinear disturbances. Automatica, 46(3):543 548, 00. [7 G.L. Wei, G. Feng, Z.D Wang. Robust control for discrete-time fuzzy systems with infinite-distributed delays. IEEE Transactions on Fuzzy Systems, 7():4 3, 009. [8 Z.G. Wu, P. Shi, H.Y. Su, J. Chu. Reliable control for discretetime fuzzy systems with infinite-distributed delay. IEEE Transactions on Fuzzy Systems, 0(): 3, 0. VII. ACKNOWLEDGEMENT This work was partially supported by the Israel Science Foundation (grant no. 8/4), the Knut Alice Wallenberg Foundation, the Swedish Research Council, the 774