Stability Analysis of Discrete-Time Systems with Poisson-Distributed Delays

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

0, 0 <δ < 1 and an A. Condition I: an augmented system integer h 0, assume that there exist n n positive definite ≥ × From (10), it follows that system (2) can be transformed matrices W, G1 and H1, such that the following LMI holds: into the following augmented one: T −1 T T Ξ=Σ+ F0 W F0 p1δ F12H1F12 + p11F01H1F01 0, x(k +1)= Ax(k)+ A1f(k), − ≺ −λ τ (9) +∞ e λ f(k +1)= τ=0 τ! x(k +1 τ h) where ∞ −−λ τ − = e−λx(k +1 h)+ + e λ x(k +1 τ h) −1 τ=1 τ! Σ = diag G δW, p G , − P − +∞ −λ τ+1 − − 1 0δ 1 = e λx(k +1 h)+ e λ x(k τ h) { − − } Pτ=0 (τ+1)! F0 = [A A1], F01 = [A I A1], F12 = [I I]. −λ − −λ − − − − = e Ax(k h)+ e A1f(k h) Then the system (2) is exponentially stable with the decay +∞ − P − + τ=0 Q(τ)x(k τ h), rate √δ. − − (14) −λ τ+1 where Q(Pτ)= e λ . Proof: Denote (τ+1)! +∞ f(k)= τ=0 P(τ)x(k τ h), Remark 3 The stability of system (14) implies the stability − − Z+ (10) ξ(k) = col x(k),f(k) , k . of system (2), but not vice versa. For a positive integer h, P{ } ∈ the effect of poisson-distributed delays leads to the aug- Taking difference of V (k) along (2) leads to ∞ mented system (14) that contains one term + Q(τ)x(k ∆V (k)= V (k + 1) δV (k) τ=0 − T T − T τ h) corresponding to distributed delays, and two terms ξ (k)F0 W F0ξ(k)+ x (k)(G1 δW )x(k) −−λ −λ P ≤ T − e Ax(k h), e A1f(k h) with discrete delays. This +p11η1 (k)H1η(k) is different− from the continuous-time− counterpart in [14] for +∞ i+h T i=0 δ P(i)x (k i h)G1x(k i h) the general case of gamma-distributed delays, where only − +∞ k−1 i+−h − T − − − − δ P(i)η1 (s)H1η1(s). distributed delays were included in the resulting augmented − Pi=0 s=k i h (11) system. ApplyingP furtherP the inequalities (3) and (4), we obtain +∞ i+h T Simple computation shows that i=0 δ P(i)x (k i h)G1x(k i h) − − − − − − (12) ∞ p 1f T (k)G f(k) + −i−h 1−h −λ λ/δ ∆ 0δ 1 i=0 δ Q(i)= δ e (e 1) =q0δ, ≤−P +∞ −i−h − and i=0 δ (i + h)Q(i) P ∆ +∞ k−1 −h −λ λ/δ λ/δ δi+h i ηT s H η s = δ e [λe + δ(h 1)(e 1)] =q1δ, (15) i=0 s=k−i−h P( ) 1 ( ) 1 1( ) P − − − T ∆ −λ −1 +∞ k−1 q01 =q0δ δ=1 =1 e , Pp P − − P(i)η1(s) H1 | − ≤− 1δ i=0 s=k i h ∆ −λ +∞ k−1 (13) q11 =q1δ δ=1 = λ + (h 1)(1 e ). hP P − − P(i)η1(s) i | − − × i=0 s=k i h −1 T Consider system (14) with both distributed and discrete = ph [x(k) f(k)] H1[x(k) if(k)] − −1δP P− − delays. We suggest the following discrete-time Lyapunov = p 1ξT (k)F T H F ξ(k). − 1δ 12 1 12 functional: Then (11)–(13) yield ∆V (k) = V (k + 1) δV (k) ˆ T ˆ 2 V (k)=ξ (k)W ξ(k)+ V i (k)+ V i (k)+ V i (k) , ξT (k)Ξξ(k) 0 if LMI (9) holds. Due to the fact− that ≤ i=1 G H S ≤ +∞ k−1 k−s−1 T 2 k 2 VG2 (k)= i=0 s=kP−i−h δh Q(i)x (s)G2x(s), i λ (W ) x(k) V (k) δ V (0), V (0) β φ , ∞ − min c V (k)= + i+h k 1 δk−s−1Q(i)ηT (s)H η (s), | | ≤ ≤ ≤ k k H2 Pi=0 Pj=1 s=k−j 1 2 1 k−1 k−s−1 T where β > 0 is a scalar and φ c = sups=0,−1,−2,... φ(s) , V (k)= δ x (s)S x(s) k k | | S1 Ps=k−Ph P 1 the system (2) is exponentially stable with the decay rate √δ −1 k−1 k−s−1 T +h j=−h s=k+j δ η1 (s)R1η1(s), for given scalars λ> 0 and h 0. Pk−1 − − V (k)= δk s 1f T (s)S f(s) ≥ S2 s=Pk−h P 2 −1 k−1 k−s−1 T Remark 2 If A is Schur stable, Lyapunov functional (7) with +h j=−h s=k+j δ η2 (s)R2η2(s), P (16) H = 0 can be applied to yield the following simple but 1 where the termsPV (k)Pand V (k) are defined in (7), ξ(k) conservative LMI condition: G1 H1 and η k are given in (10) and (8), respectively, <δ< , T T 1( ) 0 1 G1 + A W A δW A W A1 G 0, H 0, S 0, R 0, i =1, 2, and, − −1 T 0. 2 2 i i p G1 + A1 W A1 ≺ ≻ ≻ ≻ ≻ ∗ − 0δ PQ Wˆ = 0, (17) IV. STABILITY IN THE CASE THAT A AND A + A1 ARE Z ≻ ALLOWEDTOBENOT SCHUR STABLE ∗ and In this section, we will derive LMI conditions for the η2(k)= f(k + 1) f(k). (18) exponential stability of system (2), where A and A + A1 − may not be Schur stable. This means that the considered Here the last two terms VS1 (k) and VS2 (k) are added to infinite distributed delays with a gap have stabilizing effects. “compensate”, respectively, the delayed terms x(k h) and We provide two sufficient stability conditions, the efficiency f(k h) of (14). Therefore, for system (14) with−h = 0, − of which is illustrated by the given examples below. VS1 (k) and VS2 (k) are not necessary. We derive conditions

7738 to guarantee exponential stability of system (14) in the Therefore, (21)–(24) yield ∆Vˆ (k) = Vˆ (k + 1) δVˆ (k) following proposition: ξˆT (k)Ξˆξˆ(k). Then if LMI (19) holds for given scalars− λ>≤0 and h 0, the system (14) is exponentially stable with the Proposition 2 Given scalars λ > 0, 0 <δ < 1 and an decay rate≥ √δ. integer h 0, let there exist n n positive definite matrices P,Z, G ,≥ H , S , R , i =1, 2,×and an n n matrix Q such B. Condition II: a higher-order augmented system i i i i × that (17) and the following LMI are satisfied: For the case that A and A + A1 are not necessary to − be Schur stable, alternatively, we analyze the stability of Ξ=ˆ Σ+ˆ FˆT Wˆ Fˆ δFˆT Wˆ Fˆ p 1FˆT H Fˆ 0 0 1 1 1δ 12 1 12 system (2) by transforming (2) into the following higher- ˆT − 2 −ˆ 2 ˆT ˆ +F01[p11H1 +q11H2 + h R1]F01 + h F02R2F02 order augmented one: −1 ˆT ˆ h ˆT ˆ h ˆT ˆ q1δ F15H2F15 δ F13R1F13 δ F24R2F24 0, − − − ≺ (19) x(k +1)= Ax(k)+ A1f(k), ˆ −1 f(k +1)= e−λAx(k h)+ e−λA f(k h)+ υ(k), where Σ = diag S1 + G1 + q01G2, p0δ G1 + 1 h h −{1 − −λ − −λ − S2, δ S1, δ S2, q G2 , υ(k +1)= λe Ax(k h)+ λe A1f(k h)+ g(k), − − − 0δ } − − (25) A A 0 0 0 Fˆ = 1 , where f(k) and υ(k) are defined in (10) and (20), respec- 0 0 0 e−λA e−λA I 1 tively, and ˆ I 0 0 00 ˆ ∞ F1 = , F01 = [A I A1 0 0 0], g(k)= + U(τ)x(k τ h), 0 I 0 0 0 − −τλ=0τ − − e λ +2 − − Fˆ = [0 I e λA e λA I], Fˆ = [I I 0 0 0], U(τ)= . 02 − 1 12 − P(τ+2)! Fˆ = [I 0 I 0 0], Fˆ = [q I 000 I], 13 − 15 01 − Similar to (15), we have Fˆ24 = [0 I 0 I 0]. − +∞ −i−h 2−h −λ λ/δ −1 ∆ i=0 δ U(i)= δ e (e δ λ 1) = u0δ, Then the system (14) is exponentially stable with the decay ∞ − − − − + δ i h(i + h)U(i) rate √δ. Pi=0 2−h −λ −2 2 −1 λ/δ −1 ∆ =δ e [δ λ +(h 2+δ λ)(e δ λ 1)] = u1δ, Proof: Define P − − − ∞ and υ(k)= + Q(τ)x(k τ h) (20) τ=0 − − u =∆ u =1 e−λ λe−λ, 01 0δ|δ=1 − − and ξˆ(k) = col x(k)P,f(k), x(k h),f(k h),υ(k) . By ∆ −λ −λ −λ u11 = u1δ δ=1 = λ λe + (h 2)(1 e λe ). taking difference{ of Vˆ (k) along− (14) and− applying Jensen} | − − − − inequality with finite sequences (see e.g., [6]), we have For system (25), we introduce the following augmented Lyapunov functional: ξT (k + 1)Wˆ ξ(k + 1) δξT (k)Wˆ ξ(k) T T −T (21) ¯ T ¯ 2 = ξˆ (k)[Fˆ Wˆ Fˆ δFˆ Wˆ Fˆ ]ξˆ(k) V (k)=¯x (k)W x¯(k)+ i=1 VGi (k)+ VHi (k)+ VSi (k) 0 0 − 1 1 +V (k)+ V (k) and G3 H3P h i 2 with i=1 VGi (k +1)+ VHi (k +1)+ VSi (k + 1) +∞ k−1 k−s−1 T VG3 (k)= i=0 s=k−i−h δ U(i)x (s)G3x(s), h δV i (k) δV i (k) δV i (k) P G H S +∞ i+h k−1 k−s−1 T − − − VH (k)= − δ U(i)η (s)H3η1(s), ˆT ˆ ˆT 2 i ˆ 3 Pi=0 Pj=1 s=k j 1 ξ (k) Σ+ F01[p11H1 +q11H2 + h R1]F01 ≤ where the terms V i (k), V i (k) and V i (k), i = 1, 2, are −1 ˆT ˆ 2 ˆT ˆ P PG PH S h p F12H1F12 + h F02R2F02 − 1δ defined in (16), 0 <δ < 1, G3 0, H3 0, x¯(k) = h ˆT ˆ h ˆT ˆ ˆ ≻ ≻ δ F R1F13 δ F R2F24 ξ(k) col x(k), f(k), υ(k) , η (k) is given by (8), and, − 13 − 24 1 −1 T { } +q υ (k)G2υ(k) i 0δ ∞ PQ1 Q2 + δi+h i xT k i h G x k i h i=0 Q( ) ( ) 2 ( ) W¯ = ZQ3 0. (26) − +∞ k−1 i+h− − T − −  ∗  ≻ − − δ Q(i)η (s)H2η1(s). R − Pi=0 s=k i h 1 ∗ ∗ (22)   ApplyingP furtherP the inequality (3), we obtain Remark 4 Differently from the continuous-time counterpart +∞ i+h T in [14] for the general case of gamma-distributed delays, to i=0 δ Q(i)x (k i h)G2x(k i h) − −1 T − − − − (23) analyze stability of the higher-order augmented system (25) q0δ υ (k)G2υ(k). ≤−P for poisson-distributed delays, two additional terms VG3 (k) Furthermore, the application of (4) leads to and VH3 (k) are necessary to “compensate” the term g(k)= +∞ +∞ k−1 i+h T U(τ)x(k τ h) of (25). i=0 s=k−i−h δ Q(i)η1 (s)H2η1(s) τ=0 − − − T −1 +∞ k−1 Pq P − − Q(i)η1(s) H2 FollowingP the proof of Proposition 2, we derive the following ≤− 1δ i=0 s=k i h +∞ k−1 (24) condition for exponential stability of system (25): hP P − − Q(i)η1(s) i × i=0 s=k i h −1 T = qh1δ [q01x(k) υ(k)] H2[q01xi(k) υ(k)] Proposition 3 Given scalars λ > 0, 0 <δ < 1 and an − −P P − − = q 1ξˆT (k)FˆT H Fˆ ξˆ(k). integer h 0, let there exist n n positive definite matrices − 1δ 15 2 15 ≥ × 7739 TABLE I TABLE II COMPLEXITYOFDIFFERENTSTABILITYCONDITIONS EXAMPLE 1: MAXIMUM ALLOWABLE VALUES OF λ FOR DIFFERENT h Method Decision variables Number and order of LMIs [max λ] \ h 0 1 2 8 Proposition 1 1.5n2 + 1.5n one of 3n × 3n Proposition 1 8.25 7.25 6.25 0.25 Proposition 2 6n2 + 5n one of 7n × 7n Proposition 2 8.41 7.41 6.41 0.41 one of 2n × 2n Proposition 3 8.55 7.55 6.55 0.55 Proposition 3 9.5n2 + 6.5n one of 8n × 8n one of 3n × 3n

2 and 3 with δ = 1, we obtain the maximum allowable values of λ that guarantee the asymptotic stability (see P, Z, R, S , R , i , , G , H , j , , , and n n i i = 1 2 j j = 1 2 3 Table II). For δ = 1 the stability region in the (λ, h) matrices Q , j , , , such that (26) and the following× j = 1 2 3 plane that preserves the asymptotic stability is depicted in LMI are feasible: Figs. 1–3 by using Propositions 1, 2 and 3, respectively. ¯ ¯ ¯T ¯ ¯ ¯T ¯ ¯ −1 ¯T ¯ The simulation results in Figs. 1–3 verify that Proposition Ξ= Σ+ F0 W F0 δF1 W F1 p1δ F12H1F12 ¯T − − 2 ¯ 2 +F01[p11H1 +q11H2 + u11H3 + h R1]F01 2 with the number 6n + 5n n=2 = 34 of variables − − (27) 2 ¯T ¯ 1 ¯T ¯ 1 ¯T ¯ induces more dense stability{ region} than Proposition 1 with +h F02R2F02 q1δ F15H2F15 u1δ F16H3F16 h ¯T ¯ − h ¯T ¯ − 2 δ F13R1F13 δ F24R2F24 0, 1.5n +1.5n n=2 = 9 variables, but guarantees sparser − − ≺ stability{ region} than Proposition 3 with the number 9.5n2 + −1 where Σ¯ = diag S1 + G1 +q01G2 + u01G3, p G1 + { − − 0δ 6.5n n=2 = 51 of variables. S , δhS , δhS{, q 1G , u 1G , − } 2 − 1 − 2 − 0δ 2 − 0δ 3} A A1 0 0 00 9 ¯ −λ −λ Proposition 1 F0 = 0 0 e A e A1 I 0 , 8  −λ −λ  0 0 λe A λe A1 0 I 7  I 00000  6 F¯1 = 0 I 00 0 0 , F¯01 = [A I A1 0 0 0 0],   − 5

0 0 00 I 0 h −λ −λ 4 F¯02 = [0 I e A e A1 I 0], F¯12 = [I I 0 0 0 0], − − F¯ = [I 0 I 0 0 0], F¯ = [q I 000 I 0], 3 13 − 15 01 − F¯ = [u I 0000 I], F¯ = [0 I 0 I 0 0]. 2 16 01 − 24 − Then the system (25) is exponentially stable with the decay 1 0 rate √δ. 0 1 2 3 4 5 6 7 8 9 λ

Remark 5 Compare the number of scalar decision variables Fig. 1. Example 1: stability region by Proposition 1 with δ = 1 and the resulting LMIs. See Table I for the complexity of different stability conditions. Note that Proposition 3 achieves less conservative results than Propositions 1 and 9 2 on account of computational complexity (see examples Proposition 2 below). 8

Remark 6 The system (2) with poisson-distributed delays 6 can be also transformed into an augmented system with 5 respect to the state x k ,f k ,υ k ,g k . By virtue of h col ( ) ( ) ( ) ( ) 4 corresponding augmented{ Lyapunov functional,} the achieved 3 less conservative results suffer from more decision variables and higher order LMIs. 2 1

V. ILLUSTRATIVE EXAMPLES 0 0 1 2 3 4 5 6 7 8 9 In this section, we provide two examples to demonstrate λ the efﬁciency of stability conditions. Fig. 2. Example 1: stability region by Proposition 2 with δ = 1 A. Example 1 Consider the linear discrete-time system (1) with B. Example 2 1.05 0.01 0.15 0 Consider the linear discrete-time system (1) with the A = and A1 = − . 0.1 1.05 0.1 0.1 coefﬁcient matrices: − Here A + A is Schur stable whereas A is not. For the 0.5 0 0.5 0.8 1 A = and A = . values of h given in Table II, by applying Propositions 1, −0 1 1 −0.5 0.2 − 7740 9 National Natural Science Foundation of China (grant no. Proposition 3 8 61503026, 61440058).

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0 0.5 1 1.5 2 2.5 3 a class of discrete-time stochastic systems with distributed delays and λ nonlinear disturbances. Automatica, 46(3):543–548, 2010. [17] G.L. Wei, G. Feng, and Z.D Wang. Robust control for discrete-time fuzzy systems with inﬁnite-distributed delays. IEEE Transactions on Fig. 4. Example 2: allowable values of λ by Propositions 2 and 3 with h δ Fuzzy Systems, 17(1):224–232, 2009. = 0 and = 1 [18] Z.G. Wu, P. Shi, H.Y. Su, and J. Chu. Reliable control for discrete- time fuzzy systems with inﬁnite-distributed delay. IEEE Transactions on Fuzzy Systems, 20(1):22–31, 2012. 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 handled 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.

VII. ACKNOWLEDGEMENT This work was partially supported by the Israel Science Foundation (grant no. 1128/14), the Knut and Alice Wal- lenberg Foundation, the Swedish Research Council, and the

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