2015 IEEE 54th Annual Conference on Decision and Control (CDC) December 15-18, 2015. Osaka, Japan Stability Analysis of Discrete-time Systems with Poisson-distributed Delays Kun Liu, Karl Henrik Johansson, Emilia Fridman and Yuanqing Xia Abstract— This paper is concerned with the stability analysis robust H∞ control problem was discussed for discrete-time of linear discrete-time systems with poisson-distributed delays. fuzzy systems with infinite distributed delays. It should be Firstly, the exponential stability condition of system with pointed out that all the results reported in [7], [16], [17], poisson-distributed delays is derived when the corresponding system with the zero-delay or the system without the delayed [18] are concerned with constant kernel function. Moreover, term is asymptotically stable. Then, an augmented Lyapunov when the corresponding system without the delay as well as functional is suggested to handle the case that the corresponding the system without the delayed term are not asymptotically system without the delay as well as the system without the stable, the proposed methods in [7], [16], [17], [18] are delayed term are not necessary to be asymptotically stable. not applicable. It is well-known that poisson distribution is Furthermore, we show that the results can be further improved by formulating the system as a higher-order augmented one and widespread in queuing theory [4]. In [11], the experimental applying the corresponding augmented Lyapunov functional. data on the arrivals of pulses in indoor environments revealed Finally, the efficiency of the proposed results is illustrated by that each cluster’s time-delay is poisson-distributed, see also some numerical examples. [5] for more explanations. Keywords: Infinite delays, poisson-distributed delays, sta- In the present paper, we consider linear discrete-time bilizing delays, Lyapunov method. systems with poisson-distributed delays. The objective is to derive sufficient exponential stability conditions for the I. INTRODUCTION system via appropriate Lyapunov functionals. It is allowed Systems with distributed delays are frequently encountered that the corresponding system without the delay as well as in modeling the physiological behavior, the traffic flow, the the system without the delayed term are not asymptotically population dynamics, and the control over networks [9], [10]. stable. Thus, the considered infinite distributed delays with In general, there are two main classes of distributed de- a gap in the paper have stabilizing effects. We derive the lays, namely: finite distributed delays and infinite distributed results by transforming the system to an augmented one and delays. A great number of results have been reported for applying augmented Lyapunov functionals [12], [14]. Due the stability and control of systems with finite distributed to the effect of poisson-distributed delays, the augmented delays, e.g., [1], [2], [3], [12] and the reference therein. system contains not only distributed but also discrete delays. For the case of infinite distributed delays, we refer to [8], This is different from the continuous-time counterpart in [14] [9], [13], where necessary and sufficient conditions for the for the general case of gamma-distributed delays, where only stability of continuous-time systems with gamma-distributed distributed delays were included in the resulting augmented delays were derived in the frequency domain. In the time system. Furthermore, we show that the results can be fur- domain, sufficient conditions for the stability of continuous- ther improved by formulating the system as a higher-order time systems with gamma-distributed delays were derived augmented one and applying the corresponding augmented in [14] via appropriate Lyapunov functionals. Recently, the Lyapunov functional. Lyapunov-based stability and passivity analysis for diffusion The structure of the paper is organized as follows. Section partial differential equations with infinite distributed delays II presents the systems with poisson-distributed delays and were presented in [15]. the summation inequalities that will be employed. The ex- Discrete-time systems with infinite distributed delays have ponential stability of systems with poisson-distributed delays been analyzed in the literature. For example, the synchroniza- is studied in Section III when the corresponding system tion problem for an array of coupled complex discrete-time without the delay or the system without the delayed term networks with infinite distributed delays was investigated is asymptotically stable. Section IV shows the exponential in [7]. The state feedback control was considered in [16] stability conditions of systems with poisson-distributed de- for discrete-time stochastic systems with infinite distributed lays when the corresponding system with the zero-delay as delays and nonlinear disturbances. In [17] and [18], the well as the system without the delayed term are allowed to be K. Liu is with the School of Automation, Beijing Institute of Technology, not asymptotically stable. Section V illustrates the efficiency Beijing 100081, China. E-mail: [email protected]. His work was done when of the presented approach with some examples. Finally, the he was with KTH Royal Institute of Technology. conclusions and the further work are stated in Section VI. K. H. Johansson is with the ACCESS Linnaeus Centre and School of Electrical Engineering, KTH Royal Institute of Technology, SE-100 44, Notations: The notations used throughout the paper are Stockholm, Sweden. E-mail: [email protected]. standard. The superscript ‘T ’ stands for matrix transposition, E. Fridman is with the School of Electrical Engineering, Tel-Aviv Rn denotes the n dimensional Euclidean space with vector University, Tel-Aviv 69978, Israel. E-mail: [email protected]. norm , Rn×m is the set of all n m real matrices. P 0 Y. Xia is with the School of Automation, Beijing Institute of Technology, | · | × ≻ Beijing 100081, China. E-mail: xia [email protected]. (P 0) means that P is positive definite (positive semi- 978-1-4799-7886-1/15/$31.00 ©2015 IEEE 7736 definite). denotes the term that is induced by symmetry Proof: The proofs of (3) and (4) follow from those in [14] by and I represents∗ the unit matrix of appropriate dimensions. involving sums instead of integral. Since R 0, application The symbol Z+ denotes the set of non-negative integers. of Schur complements implies that the following≻ holds T T II. SYSTEM DESCRIPTION α(i) M(i) x (i)Rx(i) x (i)M(i) | | − − 0 α 1(i) M(i) R 1 Consider the following linear discrete-time system with ∗ | | (6) poisson-distributed delays: for any i [0, + ], i Z+. Summation of (6) from 0 to ∞ ∈ ∞ ∈ x k Ax k A + p τ x k τ , k Z+, + leads to ( +1)= ( )+ 1 τ=0 ( ) ( ) ∞ − ∈ ∞ ∞ (1) + α(i) M(i) xT (i)Rx(i) + xT (i)M(i) n P i=0 i=0 where x(k) R is the state vector, the system matrices | | −1 0. ∈ M0R A and A1 are constant with appropriate dimensions. The P ∗ P initial condition is given as col x(0), x( 1), x( 2),... = By Schur complements, the above matrix inequality yields col φ(0), φ( 1), φ( 2),... The{ function− p(θ)−is a poisson} (3). Furthermore, double summation of { − − }+ distribution with a gap h Z : T T ∈ α(i) M(i) x (j)Rx(j) x (j)M(i) −λ θ−h | | −1 −1 0 e λ θ h, α (i) M(i) R p(θ)= (θ−h)! ≥ ∗ | | ( 0 θ < h. from k i h to k 1 in j and from 0 to + in i, where − − − ∞ The gap h can be interpreted e.g., in the network as the +∞ k−1 −1 i=0 j=k−i−h α (i) M(i) = M1h minimal propagation delay, which is always strictly positive. | | The mean value of p is λ + h. Due to the fact that and SchurP complementsP ensure that the inequality (4) holds. +∞ +∞ τ=0 p(τ)x(k τ) = τ=h p(τ)x(k τ) In the sequel, the summation inequalities (3) and (4) with − +∞ − = p(θ + h)x(k θ h), infinite sequences play an important role in the stability P Pθ=0 − − problem of discrete-time systems with poisson-distributed we arrive at the equivalent systemP to (1) as follows: delays. ∞ x(k +1)=Ax(k)+A + P(τ)x(k τ h), k Z+, 1 τ=0 − − ∈ (2) III. STABILITY IN THE CASE THAT A OR A + A1 IS −λ τ e λ P SCHUR STABLE where P(τ) = τ! . Moreover, some elementary calculus shows that for scalar 0 <δ 1 ≤ Consider system (2). It is assumed that A or A + A1 − +∞ −i−h −h (δ 1−1)λ ∆ is Schur stable. Our stability analysis will be based on the i=0 δ P(i)= δ e = p0δ, −1 following discrete-time Lyapunov functional: +∞ −i−h −h (δ −1)λ −1 ∆ i=0 δ (i + h)P(i)= δ e (λδ +h) = p1δ, P T ∆ V (k)= x (k)W x(k)+ VG (k)+ VH (k), p11 = p1δ δ=1 = λ + h. 1 1 P | +∞ k−1 k−s−1 T VG1 (k)= i=0 s=k−i−h δ P(i)x (s)G1x(s), The derivation of stability conditions for system (2) is +∞ i+h k−1 k−s−1 T VH1 (k)= i=0 j=1 s=k−j δ P(i)η1 (s)H1η1(s), based on the summation inequalities with infinite sequences P P (7) formulated in the following lemma. where 0 <δ<P 1,P W P0, G 0, H 0, and ≻ 1 ≻ 1 ≻ Lemma 1 Given an n n matrix R 0, an integer h 0, η1(k)= x(k + 1) x(k). (8) − scalar functions M(i) × R, α(i) R≻+ 0 , and a vector≥ ∈ ∈ \{ } function x(i) Rn such that the series concerned are Remark 1 The term V (k) “compensates” the delayed ∈ G1 convergent.