# Exponential stability and stabilization for uncertain discrete-time periodic systems with time-varying delay

Exponential stability and stabilization for uncertain discrete-time periodic systems with... Abstract This article addresses the problems of robustly exponential stability and exponential stabilization for uncertain linear discrete-time periodic systems with time delay in the state variables and polytopic-type parameter uncertainty. By constructing the novel uncertainty-dependent Lyapunov–Krasovskii functionals, we establish some sufficient conditions in forms of linear matrix inequalities, which guarantee the uncertain linear discrete-time periodic system with time delay is robustly exponentially stable. Then, by utilizing static periodic state feedback and free weighting matrix technique, we give some sufficient conditions to ensure the robustly exponential stabilization of uncertain linear discrete-time periodic systems with time delay. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed method. 1. Introduction In recent years, the control community has been attracted by the periodic systems because this class of systems represents an important subclass of time-varying systems. Indeed, many dynamic systems show periodic behaviour: networked control systems with communication constraints, celestial mechanics, sampled-data control, signal processing in the presence of cyclostationary noise, control of multirate plants and multiplexed systems etc., see Bittanti (1986), Bittanti & Colaneri (2009) and the references therein. In practice, time delays are often encountered in many real-world problems such as automatic control engineering, quadratic optimization methods and economic classification and biological reproduction. It is widely known that time delay arises pervasively in dynamic systems and causes instability and poor performance of control systems. Over the last two decades, many researches have focused on stability analysis and control design for discrete-time systems with time delays (Gonzlez, 2013; Phat & Ratchagit, 2011; Lee et al., 2015; Zhang & Yu, 2009; Kwon & Park, 2006; Boukas, 2007). The robust stabilization for linear discrete-time systems with time-varying input delay was studied in Gonzlez (2013). Some methods for dealing with constant delays in discrete-time systems have been given (Lee et al., 2015; Zhang & Yu, 2009; Kwon & Park, 2006). State feedback stabilization of non-linear discrete-time systems with time-varying delays was considered in Boukas (2007). The Lyapunov–Krasovskii method has been useful to investigate the stability and stabilization of linear discrete-time systems with time-varying delays (Gao & Cui, 2009; He et al., 2008; Tang et al., 2010). Dong et al. (2014) proposed new sufficient conditions for observer-based stabilization of a class of discrete-time non-linear systems with time delay. Exponential stability, as an important index to get the convergence rates of systems, has also been studied by many researchers (Liu, 2009; Botmart et al., 2011; Wang & Fei, 2014; Dong et al., 2016; Dong et al., 2012). Liu (2009) given the delay-dependent stability criteria by the Lyapunov–Krasovskii functional method. Botmart et al. (2011) proposed exponential stabilization delay-dependent sufficient conditions for linear systems with time-varying delay by choosing improved Lyapunov–Krasovskii functionals and using Leibniz–Newton’s formula. Dong et al. (2016) investigated the exponential stabilization and $$L_{2}$$-gain for a class of uncertain switched non-linear systems with interval time-varying delay and established novel delay-dependent sufficient conditions of exponential stabilization for a class of uncertain switched non-linear delay systems. The problem of robust exponential stabilization for dynamical non-linear systems with uncertainties and time-varying delay was investigated in Dong et al. (2012). Dong et al. (2012) presented the robust exponential stabilization criterion based on the Razumikhin theorem and the Lyapunov function method. Periodic linear systems have the potential to model many practical control systems including multirate sample data systems (Chen & Francis, 1995) and systems that operate periodically such as satellites. Therefore, there has been a persistent interest with the control community in linear periodic systems over the past two decades. Significant advances have been made in a variety of topics about the theories of control and state estimation (Letyagina & Zhabko, 2009; Grasselli et al., 1996). Robust stability analysis of linear periodic systems with time delay was presented in Letyagina & Zhabko (2009). Robust output regulation and tracking for linear periodic systems under structured uncertainties were addressed in Grasselli et al. (1996). Qiu et al. (2008) considered $$H_{\infty}$$ filtering design for discrete-time polytopic linear delay systems. However, among the developments in the literature, little effort has been devoted to the problems of robust stability and stabilization for linear periodic systems with a time delayed state. Our article investigates the problems of robustly exponential stability and robustly exponential stabilization for uncertain linear discrete-time periodic systems with time-varying delay in the state variables and polytopic-type parameter uncertainty. To the best of our knowledge, results about the robustly exponential stability and robustly exponential stabilization of uncertain linear discrete-time periodic systems are rarely reported in the literature. Our research is conducted in the hope to fill this gap and here we listed our contributions. (1) We construct an appropriate uncertainty-dependent Lyapunov–Krasovskii functional to exhibit the delay-dependent discrete-time periodic systems and establish the criteria of robustly exponential stability for uncertain linear discrete-time periodic systems without control. (2) Based on periodic state feedback control, we provide robustly exponential stabilization methods. The closed-loop discrete-time periodic system is robustly exponential stable under our established new sufficient conditions. (3) The calculation method of the periodic control gain matrix is presented. (4) Two simulation examples are given to show the performances of our method. (5) Compared with the existing ones, our method leads to less conservatism because of the introduced free weight matrices $$N_{1}$$, $$Z_{1}, Z_{2}\,and\,Z_{3}$$. The rest of this article is organized as follows. Section 2 gives the problem formulation and preliminary results. Then, the robust exponential stability problem is solved in Section 3. In Section 4, the robustly exponential stabilizing periodic state feedback controller is designed, and novel criteria of exponential stabilization for the uncertain linear discrete-time periodic systems are established. In Section 5, the performances of our method are illustrated by two examples. Finally, Section 6 concluded the paper with some further comments. Notation. Throughout this article, $${R^n}$$ denotes the $$n$$-dimensional real Euclidean space. $${R^{m \times n}}$$ represents the set of all $$m \times n$$ real matrices. $$I$$ and 0 represent the identity matrix and null matrix of appropriate dimensions. $$Z$$ represents the set of integers, $${Z^ + }$$ represents the set of non-negative integers. In symmetric block matrices, we use an asterisk ‘$$\ast$$’ to represent a term induced by symmetry. diag $$\{ \cdots \}$$ denotes a block-diagonal matrix. A matrix $$M(k)$$ is denoted N-periodic, where $$0 < N \in {Z^ + },$$ if $$M(k + N) = M(k),$$ for all $$k \in Z$$. $${A^T}$$ stands for the transpose of $$A$$. For a square matrix $$P,$$$$P > 0(< 0,\,\, \le 0,\,\, \ge 0)$$ means that this matrix is positive (negative, semi-negative and semi-positive) definite. $$\|\cdot\|$$ is the Euclidean vector norm. 2. System description and preliminaries Consider the following uncertain discrete-time periodic system: x(k+1)=A(k)x(k)+Ad(k)x(k−h(k))+B(k)u(k),x(k)=φ(k),k=−d2,−d2+1,⋯,0, (2.1) where $$x(k) \in {R^n}$$ is the state vector, $$u(k) \in {R^m}$$ is the control, $$A(k),{A_d}(k)$$ and $$B(k)$$ are uncertain $$N$$-periodic real matrices with appropriate dimensions that are assumed to be confined to the following polytope: Ω(k)={Υ(k):Υ(k)=∑i=1vλiΥi(k),λi≥0,∑i=1vλi=1}, (2.2) where Υ(k)=[A(k),Ad(k),B(k)], (2.3) Υi(k)=[Ai(k),Adi(k),Bi(k)] (2.4) with $${A_i}(k),{A_{di}}(k)$$ and $${B_i}(k),i = 1,2, \cdots,v,$$ being given $$N$$-periodic real matrices. The sequence $$\varphi (k)$$ is the initial condition; the time-varying delay $$h(k)$$ is a $$N$$-periodic function and satisfies the following condition: 0<d1≤h(k)≤d2, where $${d_1}$$ and $${d_2}$$ are non-negative integers representing the lower and upper bounds of the interval time-delay. Definition 2.1 The delayed discrete-time system (2.1) is said to be robustly exponentially stable with a convergence rate $$\alpha$$, if there exist scalars $$\alpha > 0,\beta > 0$$ such that ‖x(k)‖≤βe−αk‖φ‖∀k∈Z+, where $$\left\| \varphi \right\| = \mathop {\sup }\limits_{k \in Z \cap [- {d_2},0]} \left\| {x(k)} \right\|.$$ The following lemmas will play important roles in this article. Lemma 2.1 (Ramakrishnan & Ray, 2013) For any constant matrix $$W \in {R^{n \times n}}$$ with $$W = {W^T} > 0$$, integers $${n_1} < {n_2},$$ vector function $$\omega :\;\left\{ {{n_1},{n_1} + 1, \cdots,{n_2}} \right\} \to {R^q}$$ such that the sums concerned are well defined, then (n2−n1+1)∑i=n1n2ωT(i)Wω(i)≥(∑i=n1n2ω(i))TW(∑i=n1n2ωT(i)). Lemma 2.2 (Wang et al., 1992) For any $$x,y \in {R^n}$$ and any positive-definite matrix $$P \in {R^{n \times n}}$$, we have 2xTy≤xTPx+yTP−1y. Lemma 2.3 (Boyd et al., 1994) Given constant symmetric matrices $${S_1},{S_2},{S_3},$$ and $${S_1} = S_1^T < 0,$$$${S_3} = S_3^T > 0,$$ then $${S_1} + {S_2}S_3^{-1}S_2^T < 0$$ if and only if [S1S2S2T−S3]<0. This article aims at designing a stabilizing static state feedback for the system (2.1) as below u(k)=K(k)x(k), where $$K(k)$$ is an $$N$$-periodic matrix to be found such that the closed-loop system x(k+1)=(A(k)+B(k)K(k))x(k)+Ad(k)x(k−h(k)) (2.5) is exponentially stable for all system matrices belonging to the uncertainty polytope $${\it{\Omega}} (k)$$. 3. Robust exponential stability analysis In this section, we shall develop linear matrix inequality (LMI)-based conditions for exponential stability of uncertain discrete-time periodic systems with time-delay. When the time-delay $$h(k)$$ is a constant $$d$$, i.e., $$h(k)=d$$, the system (2.1) can be written as x(k+1)=A(k)x(k)+Ad(k)x(k−d)+B(k)u(k),x(k)=φ(k),k=−d,−d+1,⋯,0. (3.1) Theorem 3.1 For a given scalar $$\alpha > 0,$$ the system (3.1) with $$u(k) = 0$$ is robustly exponentially stable with a convergence rate $$\alpha$$, if there exist $$N$$-periodic matrices $$P(k) > 0, {Q_i}(k) > 0, {R_i}(k) > 0,i = 1,2, \cdots,v,$$ and a matrix $${Z_1} > 0$$ satisfying the following LMIs: (N⌣i(k)Z1eαAiT(k)P(k+1)0Γ1∗M¯i(k)eα(d+1)AdiT(k)P(k+1)Γ2Γ3∗∗−P(k+1)00∗∗∗−P(k+1)0∗∗∗∗−I)<0,k=1,2,⋯,N,i=1,2,⋯,v, (3.2) where N⌣i(k)=−P(k)+dRi(k)+Q(k)−Z1,M¯i(k)=−Qi(k−d)−Z1−Ri(k−d),Γ1=d(eαAiT(k)−I)Z1,Γ2=eα(d+1)AdiT(k)P(k+1),Γ3=deα(d+1)AdiT(k)Z1. Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (3.1) with $$u(k) = 0$$ is reduced to z(k+1))=eαA(k)z(k)+eα(d+1)Ad(k)z(k−d). (3.3) Consider the Lyapunov–Krasovskii functional: V(k)=V1(k)+V2(k)+V3(k)+V4(k), (3.4) where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−dk−1zT(i)Q(i)z(i),V3(k)=∑i=−d−1∑j=k+ik−1zT(j)R(j)z(j),V4(k)=d∑i=−d−1∑j=k+ik−1ηT(j)Z1η(j),η(j)=z(j+1)−z(j),R(k)=∑i=1vλiRi(k),Q(k)=∑i=1vλiQi(k) (3.5) and $${\lambda _i},\;i = 1,2, \cdots,v,$$ are the uncertain non-negative scalars characterizing the polytope $${\it{\Omega}} (k)$$ as defined in (2.2). We can verify that λ¯1‖z(k)‖2≤V(k)≤λ¯2‖zk‖2, (3.6) where λ¯1=mink∈{1,2,⋯,N}⁡λmin(P(k)),‖zk‖2=supj∈Z∩[−d,0]⁡‖z(k+j)‖2,λ¯2=maxi∈{1,2,⋯,N}⁡{λmax(P(i))+dλmax(Q(i))+12d(d+1)λmax(R(i))}+2d2(d+1)λmax(Z1). Define $${\it{\Delta}} V(k) = V(k + 1) - V(k)$$. Then along the solution of (3.3), we have ΔV1(k)=V1(k+1)−V1(k)=zT(k)[e2αAT(k)P(k+1)A(k)−P(k)]z(k)+2eα(d+2)zT(k)AT(k)P(k+1)Ad(k)z(k−d)   +e2α(d+1)zT(k−d)AdT(k)P(k+1)Ad(k)z(k−d), (3.7) ΔV2(k)=V2(k+1)−V2(k)=zT(k)Q(k)z(k)−zT(k−d)Q(k−d)z(k−d), (3.8) ΔV3(k)=V3(k+1)−V3(k)=∑i=−d−1∑j=k+1+ikzT(j)R(j)z(j)−∑i=−d−1∑j=k+ik−1zT(j)R(j)z(j)=dzT(k)R(k)z(k)−∑j=k−dk−1zT(j)R(j)z(j). (3.9) Using Lemma 2.1, we have ΔV4(k)=d2ηT(k)Z1η(k)−d∑j=k−dk−1ηT(j)Z1η(j)≤d2ηT(k)Z1η(k)−(z(k)−z(k−d))TZ1(z(k)−z(k−d)). (3.10) From (3.7)–(3.10), we have ΔV(k)≤zT(k)[e2αAT(k)P(k+1)A(k)−P(k)+dR(k)+Q(k)−Z1]z(k)+zT(k−d)Z1z(k)+2eα(d+2)zT(k)AT(k)P(k+1)Ad(k)z(k−d)+zT(k)Z1z(k−d)+d2ηT(k)Z1η(k)+zT(k−d)[e2α(d+1)AdT(k)P(k+1)Ad(k)−Q(k−d)−R(k−d)−Z1]z(k−d)=(zT(k)zT(k−d))N(k)(z(k)z(k−d)), (3.11) where N(k)=(N1(k)N2(k)N2T(k)N3(k)),N1(k)=e2αAT(k)P(k+1)A(k)−P(k)+dR(k)+Q(k)−Z1+d2(eαAT(k)−I)Z1(eαA(k)−I),N2(k)=eα(d+2)AT(k)P(k+1)Ad(k)+Z1+d2(eαAT(k)−I)Z1eα(d+1)Ad(k),N3(k)=e2α(d+1)AdT(k)P(k+1)Ad(k)−Q(k−d)−Z1−R(k−d)+d2e2α(d+1)AdT(k)Z1Ad(k). By Lemma 2.3, we have that $${\rm N}(k) < 0$$ is equivalent to: (N¯1(k)Z1eαAT(k)0N¯3(k)∗N¯2(k)eα(d+1)AdT(k)eα(d+1)AdT(k)N¯4(k)∗∗−P−1(k+1)00∗∗∗−P−1(k+1)0∗∗∗∗−Z1−1)<0, (3.12) where N¯1(k)=−P(k)+dR(k)+Q(k)−Z1,N¯2(k)=−Q(k−d)−Z1−R(k−d).N¯3(k)=d(eαAT(k)−I),N¯4(k)=deα(d+1)AdT(k). Let $$M(k) = diag\{ I,I,P(k{\rm{ + }}1),P(k{\rm{ + }}1),{Z_1}\}.$$ Pre- and post-multiplying (3.12) by $${M^T}(k)$$ and $$M(k)$$, respectively, (3.12) can be cast as below: (N¯1(k)Z1eαAT(k)P(k+1)0N¯3(k)Z1∗N¯2(k)eα(d+1)AdT(k)P(k+1)N¯5(k)N¯4(k)Z1∗∗−P(k+1)00∗∗∗−P(k+1)0∗∗∗∗−Z1)<0, (3.13) where $${\bar {\rm N}_5}(k) = {e^{\alpha (d + 1)}}A_d^T(k)P(k{\rm{ + }}1).$$ From (3.2), we get that (3.13) holds. Therefore, from (3.11) it follows that $${\it{\Delta}} V(k) \le 0,$$ which implies that the function $$V(k)$$ is decreasing and V(k)≤V(0),∀k∈Z+. Hence, from (3.6) it follows that ‖z(k)‖≤λ¯2λ¯1‖φ¯‖,∀k∈Z+. Returning to the variable $$x(k) = {e^{-\alpha k}}z(k)$$, we have ‖x(k)‖≤λ¯2λ¯1e−αk‖φ‖,∀k∈Z+, which implies that the system (3.1) is exponentially stable. This completes the proof of the Theorem 3.1. □ Theorem 3.2 For given scalars $$\alpha > 0,$$$${d_1}$$ and $${d_2}$$ with $${d_2} > {d_1} > 0$$, the system (2.1) with $$u(k) = 0$$ is robustly exponentially stable with a convergence rate $$\alpha$$, if there exist $$N$$-periodic matrices $$P(k) > 0,$$$${Q_{ij}}(k) > 0,$$$$j = 1,2,i = 1,2, \cdots v,$$ and matrices $${Z_1} > 0,{Z_2} > 0,{Z_3} > 0, Q > 0,\;{N_1} > 0,$$ any matrices $${T_i},\;\,i = 1,2,3,$$ with appropriate dimensions satisfying the following LMIs: (Ψ1Ψ2∗Ψ3)<0,k=1,2,⋯,N,i=1,2,⋯,v, (3.14) where Ψ1=(Π¯11(k)0Z1Z2−G∗−QT1T−T1TΠ^25(k)∗∗Π¯33(k)−T2T+T30∗∗∗Π¯44(k)0∗∗∗∗Π¯55(k)),Ψ2=(Π^16(k)0Π^18(k)00Π^27(k)Π^28(k)T1T000T2T000T3T0000),Ψ3=diag{−P(k+1),−P(k+1),−N1,−Z3},Π¯11(k)=−P(k)+Qi1(k)+Qi2(k)+(1+d12)Q−Z1−Z2+G,Π^16(k)=eα(1+eαd2)AiT(k)P(k+1),d12=d2−d1,Π^18(k)=e0.5(−αd1+α)AiT(k)N1,Π^28(k)=0.5e0.5α(1+d1)AdiT(k)N1,Π^25(k)=0.5AdiT(k)N1,Π^27(k)=eαeαd2+e2αd2AdiT(k)P(k+1),Π¯33(k)=T2+T2T−Qi1(k−d1)−Z1,G=d12Z1+d22Z2+d122Z3,Π¯44(k)=−Z2−T3T−T3−Qi2(k−d2),Π¯55(k)=G−N1e−α(d2+1). Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (2.1) with $$u(k) = 0$$ is reduced to z(k+1)=eαA(k)z(k)+eα(h(k)+1)Ad(k)z(k−h(k)). (3.15) Consider the Lyapunov–Krasovskii functional: V(k)=∑i=16Vi(k), (3.16) where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−d1k−1zT(i)Q1(i)z(i)+∑i=k−d2k−1zT(i)Q2(i)z(i)+∑i=k−h(k)k−1zT(i)Qz(i)V3(k)=∑i=−d2+1−d1∑j=k+ik−1zT(j)Qz(j),V4(k)=d1∑i=−d1−1∑j=k+ik−1ηT(j)Z1η(j),V5(k)=d2∑i=−d2−1∑j=k+ik−1ηT(j)Z2η(j),V6(k)=d12∑i=−d2−d1−1∑j=k+ik−1ηT(j)Z3η(j),η(j)=z(j+1)−z(j),Qj(k)=∑i=1vλiQij(k),j=1,2 (3.17) and $${\lambda _i},\;i = 1,2, \cdots,v,$$ are the uncertain non-negative scalars characterizing the polytope $${\it{\Omega}} (k)$$ as defined in (2.2). Then along the solution of (3.3), we have ΔV1(k)=V1(k+1)−V1(k)=zT(k)[e2αAT(k)P(k+1)A(k)−P(k)]z(k)+2eα(h(k)+2)zT(k)AT(k)P(k+1)Ad(k)z(k−h(k))+e2α(h(k)+1)zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k)). Using Lemma 2.2, we get ΔV1(k)≤eα(d2+2)[zT(k)AT(k)P(k+1)A(k)z(k)+zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))]+zT(k)[e2αAT(k)P(k+1)A(k)−P(k)]z(k)+e2α(d2+1)zT(k−h(k))×AdT(k)P(k+1)Ad(k)z(k−h(k)), (3.18) ΔV2(k)=V2(k+1)−V2(k)≤zT(k)(Q1(k)+Q2(k)+Q)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)−zT(k−d2)×Q2(k−d2)z(k−d2)−zT(k−h(k))Qz(k−h(k))+∑j=k+1−d2k−d1zT(j)Qz(j), (3.19) ΔV3(k)=d12zT(k)Qz(k)−∑j=k+1−d2k−d1zT(j)Qz(j), (3.20) ΔV4(k)=d12ηT(k)Z1η(k)−d1∑j=k−d1k−1ηT(j)Z1η(j)≤d12ηT(k)Z1η(k)−(z(k)−z(k−d1))TZ1(z(k)−z(k−d1)), (3.21) ΔV5(k)=d22ηT(k)Z2η(k)−d2∑j=k−d1k−1ηT(j)Z2η(j)≤d22ηT(k)Z2η(k)−(z(k)−z(k−d2))TZ2(z(k)−z(k−d2)), (3.22) ΔV6(k)=d122ηT(k)Z3η(k)−d12∑j=k−d2k−d1−1ηT(j)Z3η(j)}≤d122ηT(k)Z3η(k)−(∑j=k−d2k−d1−1η(j))TZ3(∑j=k−d2k−d1−1η(j)). (3.23) Define $\gamma (k) = {\left(\begin{array}{ccc} {{z^T}(k - h(k))} & {{z^T}(k - {d_1})} & {{z^T}(k - {d_2})} \end{array} \right)^T},$ ${T^T} = \begin{array}{ccc} ({{T_1}} & {{T_2}} & {{T_3}} \end{array}).$ Using Lemma 2.2, we get −(∑j=k−d2k−d1−1η(k))TZ3(∑j=k−d2k−d1−1η(k))≤γT(k)TZ3−1TTγ(k)−2γT(k)T∑j=k−d2k−d1−1η(k)=γT(k)TZ3−1TTγ(k)−γT(k)[(0T−T)+(0T−T)T]γ(k). (3.24) From (3.18)–(3.24), it follows that ΔV(k)≤zT(k)[(e2α+eα(d2+2))AT(k)P(k+1)A(k)−P(k)]z(k)+zT(k)(Q1(k)+Q2(k)+Q)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)×z(k−h(k))−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Qz(k−h(k))+d12zT(k)×Qz(k)+d12ηT(k)Z1η(k)−(z(k)−z(k−d1))TZ1(z(k)−z(k−d1))+d22ηT(k)Z2η(k)−(z(k)−z(k−d2))TZ2(z(k)−z(k−d2))+d122ηT(k)Z3η(k)+γT(k)TZ3−1TTγ(k)−γT(k)[(0T−T)+(0T−TT]γ(k). (3.25) From (3.15), we have e−α(h(k)+1)z(k+1)−(e−αh(k)A(k)z(k)+Ad(k)z(k−h(k)))=0. For any $$N_{1}$$ with appropriate dimensions, it follows that zT(k+1)N1e−α(h(k)+1)z(k+1)−zT(k+1)N1e−αh(k)A(k)z(k)−zT(k+1)N1Ad(k)z(k−h(k))=0,zT(k)AT(k)N1T[e−αh(k)z(k+1)−e−α(h(k))+αA(k)z(k)−eαAd(k)z(k−h(k))]=0. (3.26) From (3.26), we get zT(k+1)N1e−α(h(k)+1)z(k+1)−zT(k+1)N1Ad(k)z(k−h(k))+zT(k)AT(k)N1T[−e−α(h(k))+αA(k)z(k)−eαAd(k)z(k−h(k))]=0. (3.27) From (3.25) and (3.27), we have ΔV(k)≤zT(k)[(e2α+eα(d2+2))AT(k)P(k+1)A(k)−P(k)]z(k)+zT(k)(Q1(k)+Q2(k)+Q)z(k)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))−zT(k−d1)Q1(k−d1)×z(k−d1)−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Qz(k−h(k))+d12zT(k)Qz(k)−(z(k)−z(k−d1))TZ1(z(k)−z(k−d1))+(z(k+1)−z(k))T[d12Z1+d22Z2+d122Z3]×(z(k+1)−z(k))−(z(k)−z(k−d2))TZ2(z(k)−z(k−d2))+(zT(k−h(k)),zT(k−d1),zT(k−d2))(T1TT2TT3T)Z3−1(T1T2T3)γ(k)−γT(k)[(0T−T+(0T−T)T]γ(k)−zT(k+1)N1e−α(h(k)+1)z(k+1)+zT(k+1)×N1Ad(k)z(k−h(k))+zT(k)AT(k)N1T[e−α(h(k))+αA(k)z(k)+eαAd(k)z(k−h(k))]}≤ξT(k)Π(k)ξ(k), (3.28) where ξ(k)=(zT(k),zT(k−h(k)),zT(k−d1),zT(k−d2),zT(k+1))T,G=d12Z1+d22Z2+d122Z3,Π(k)=ΠT(k)=(Πij(k)),i,j=1,2,⋯5,Π11(k)=(e2α+eα(d2+2))AT(k)P(k+1)A(k)−P(k)+Q1(k)+Q2(k)+(1+d12)Q  −Z1−Z2+G+AT(k)N1Te−αd1+αA(k),Π12(k)=0.5eαAT(k)N1TAd(k),Π13(k)=Z1,Π14(k)=Z2, Π15(k)=−G,Π22(k)=(eα(d2+2)+e2α(d2+1))AdT(k)P(k+1)Ad(k)−Q+T1TZ3−1T1,Π23(k)=T1TZ3−1T2+T1T,Π24(k)=T1TZ3−1T3−T1T,Π25(k)=0.5AdT(k)N1T,Π33(k)=T2+T2T−Q1(k−d1)−Z1+T2TZ3−1T2,Π34(k)=−T2T+T3+T2TZ3−1T3,Π44(k)=T3TZ3−1T3−Z2−T3T−T3−Q2(k−d2),Π55(k)=G−N1e−α(d2+1). From Lemma 2.3, we get that $${\it{\Pi}} (k) < 0,$$ if Π¯=(X1X2∗X3)<0, (3.29) where X1=(Π¯11(k)0Z1Z2−G∗−QT1T−T1TΠ25(k)∗∗Π¯33(k)Π¯34(k)0∗∗∗Π¯44(k)0∗∗∗∗Π55(k)), X2=(Π¯16(k)0Π18(k)00βAdT(k)Π28(k)T1T000T2T000T3T0000),X3=diag{−P−1(k+1),−P−1(k+1),−N1−T,−Z3},Π¯11(k)=−P(k)+Q1(k)+Q2(k)+(1+d12)Q−Z1−Z2+G,Π¯16(k)=eα(1+eαd2)AT(k),Π18(k)=e0.5(−αd1+α)AT(k),Π28(k)=0.5e0.5α(1+d1)AdT(k),  β=eαeαd2+e2αd2,  Π¯34(k)=−T2T+T3,Π¯33(k)=T2+T2T−Q1(k−d1)−Z1,Π¯44(k)=−Z2−T3T−T3−Q2(k−d2). Let $$\bar M(k) = diag\{ I,I,I,I,I,P(k{\rm{ + }}1),P(k{\rm{ + }}1),{N_1}^T,I\}.$$ Pre- and post-multiplying (3.29) by $${\bar M^T}(k)$$ and $${\bar M}(k)$$, respectively, (3.29) can be cast as below: Π~=(X1X~2∗X~3)<0, (3.30) where X~2=(Π~16(k)0Π~18(k)00Π~27(k)Π~28(k)T1T000T2T000T3T0000),X~3=diag{−P(k+1),−P(k+1),−N1T,−Z3},Π~16(k)=eα(1+eαd2)AT(k)P(k+1),Π~18(k)=e0.5(−αd1+α)AT(k)N1T,Π~27(k)=βAdT(k)P(k+1),Π~28(k)=0.5e0.5α(1+d1)AdT(k)N1T. From (3.14), we get that (3.30) holds. Therefore, from (3.11) it follows that $${\it{\Delta}} V(k) \le 0$$ which implies that the function $$V(k)$$ is decreasing and V(k)≤V(0),∀k∈Z+. We can verify that δ1‖z(k)‖2≤V(k)≤V(0)≤δ2‖φ¯‖2, (3.31) where δ1=mink∈{1,2,⋯,N}⁡λmin(P(k)),δ2=maxk∈{1,2,⋯,N}⁡λmax(P(k))+eα(d2+2){maxk∈{1,2,⋯,N}⁡[d1λmax(Q1(k))+d2λmax(Q2(k))]+d2λmax(Q)+12d12(d1+d2−1)λmax(Q)+2d12(d1+1)λmax(Z1)+2d22(d2+1)λmax(Z2)+2d122(d2+d1+1)λmax(Z3)}. (3.32) Hence from (3.31) it follows that ‖z(k)‖≤δ2δ1‖φ¯‖,∀k∈Z+. Returning to the variable $$x(k) = {e^{-\alpha k}}z(k)$$ we have ‖x(k)‖≤δ2δ1e−αk‖φ‖,∀k∈Z+. This completes the proof of the Theorem 3.2. □ Corollary 3.1 For given scalars $$\alpha > 0,$$$${d_1}$$ and $${d_2}$$ with $${d_2} > {d_1} > 0$$, the system (2.1) with $$u(k) = 0$$ is robustly exponentially stable with a convergence rate $$\alpha$$, if there exist matrices $$P > 0,$$$${Q_{ij}} > 0,$$$$j = 1,2,i = 1,2, \cdots v,$$ and matrices $${Z_1} > 0,{Z_2} > 0,{Z_3} > 0,$$$$Q > 0,$$$${N_1} > 0,$$ any matrices $${T_i},\;\,i = 1,2,3,$$ with appropriate dimensions satisfying the following LMIs: (Ψ1Ψ2∗Ψ3)<0,k=1,2,⋯,N,i=1,2,⋯,v, where Ψ1=(Π¯110Z1Z2−G∗−QT1T−T1TΠ^25(k)∗∗Π¯33−T2T+T30∗∗∗Π¯440∗∗∗∗Π¯55),Ψ2=(Π^16(k)0Π^18(k)00Π^27(k)Π^28(k)T1T000T2T000T3T0000),Ψ3=diag{−P,−P,−N1,−Z3},Π¯11=−P+Qi1+Qi2+(1+d12)Q−Z1−Z2+G,Π^16(k)=eα(1+eαd2)AiT(k)P,Π^18(k)=e0.5(−αd1+α)AiT(k)N1,Π^28(k)=0.5e0.5α(1+d1)AdiT(k)N1,Π^25(k)=0.5AdiT(k)N1,Π^27(k)=eαeαd2+e2αd2AdiT(k)P,Π¯33=T2+T2T−Qi1−Z1,G=d12Z1+d22Z2+d122Z3,Π¯44=−Z2−T3T−T3−Qi2,Π¯55=G−N1e−α(d2+1),d12=d2−d1. Remark 3.1 The problem of robust stability analysis of linear discrete time periodic systems was discussed in Souza & Coutinho (2014). But, the problem of exponential stability of linear discrete time periodic system was not involved. We know that the exponential stability implied the stability. In this article, we deal with the problems of robustly exponential stability and robustly exponential stabilization for uncertain linear discrete-time periodic systems with time-delay. Compared with Souza & Coutinho (2014), obtained results in our article have a greater range of applications. 4. Robust exponential stabilization This section deals with the problem of robustly exponential stabilization for the uncertain periodic time-delay system (2.1)–(2.4). Theorem 4.1 For a given scalar $$\alpha > 0,$$ the system (3.1) with $$u(k) = {W^T}(k){L^{ - 1}}(k)x(k)$$ is exponentially stable with a convergence rate $$\alpha$$, if there exist $$N$$-periodic matrices $$S(k) > 0,$$$$L(k) > 0,$$$$W(k)$$ satisfying the following LMIs: (−0.5L(k)0Λ130L(k)∗Λ22Λ23Λ230∗∗−L(k+1)00∗∗∗−L(k+1)0∗∗∗∗−1dS(k))<0,k=1,2,⋯,N, i=1,2,⋯,v, (4.1) where Λ13=eα(L(k)AiT(k)+W(k)BiT(k)),Λ22=−0.5L(k−d),Λ23=eα(d+1)L(k−d)AdT(k). Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (3.1) is reduced to z(k+1)=eα(A(k)+B(k)K(k))z(k)+eα(d+1)Ad(k)z(k−d). (4.2) Consider the Lyapunov–Krasovskii functional: V(k)=V1(k)+V2(k)+V3(k), where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−dk−1zT(i)Q(i)z(i),V3(k)=∑i=−d+1−1∑j=k+ik−1zT(j)R(j)z(j),P(k)=L−1(k),R(k)=S−1(k). Define $${\it{\Delta}} V(k) = V(k + 1) - V(k)$$. Then along the solution of (4.2), we have ΔV(k)≤(zT(k)zT(k−d))N^(k)(z(k)z(k−d)), (4.3) where N^(k)=(N^1(k)N^2(k)N^2T(k)N^3(k)),N^1(k)=e2α(A(k)+B(k)K(k))TP(k+1)(A(k)+B(k)K(k))−P(k)+dR(k)+Q(k),N^2(k)=eα(d+2)(A(k)+B(k)K(k))TP(k+1)Ad(k),N^3(k)=e2α(d+1)AdT(k)P(k+1)Ad(k)−Q(k−d). By Lemma 2.3, we have that $$\hat {\rm N}(k) < 0$$ is equivalent to: (N~1(k)0N~2(k)0∗−Q(k−d)eα(d+1)AdT(k)eα(d+1)AdT(k)∗∗−P−1(k+1)0∗∗∗−P−1(k+1))<0, (4.4) where $${\tilde {\rm N}_1}(k) = - P(k) + dR(k) + Q(k),$$$${\tilde {\rm N}_2}(k) = {e^\alpha }{(A(k) + B(k)K(k))^T}.$$ Let $${M_1}(k) = diag\{ {P^{ - 1}}(k),{P^{ - 1}}(k - d),I,I\},$$$$Q(k) = 0.5P(k).$$ Pre- and post-multiplying (4.4) by $${M_1}^T(k)$$ and $$M_1(k)$$, respectively, (4.4) can be cast as below: N⌢(k)=(N⌢1(k)0N⌢2(k)0∗−0.5P−1(k−d)N⌢3(k)N⌢3(k)∗∗−P−1(k+1)0∗∗∗−P−1(k+1))<0, (4.5) where N⌢1(k)=−0.5P−1(k)+P−1(k)(dR(k))P−1(k),N⌢2(k)=eαP−1(k)(A(k)+B(k)K(k))T,N⌢3(k)=eα(d+1)P−1(k−d)AdT(k). By Lemma 2.3, we have that $$\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over {\rm N}} (k) < 0$$ is equivalent to: (−0.5P−1(k))0N⌢2(k)0P−1(k)∗−0.5P−1(k−d)N⌢3(k)N⌢3(k)0∗∗−P−1(k+1)00∗∗∗−P−1(k+1)0∗∗∗∗−1d(R(k))−1)<0. (4.6) Let $$L(k) = {P^{-1}}(k),\quad W(k) = L(k){K^T}(k),\quad S(k) = {R^{-1}}(k).$$ From (4.1), we get $${\it{\Delta}} V(k) \le 0$$ which implies that the function $$V(k)$$ is decreasing and V(k)≤V(0)∀k∈Z+. Hence from (3.6) it follows that ‖z(k)‖≤λ~2λ~1‖φ¯‖∀k∈Z+, where λ~1=mink∈{1,2,⋯,N}⁡λmin(P(k)),λ~2=maxi∈{1,2,⋯,N}⁡{λmax(P(i))+dλmax(Q(i))+12d(d+1)λmax(R(i))}. Returning to the variable $$x(k) = {e^{-\alpha k}}z(k)$$ we have ‖x(k)‖≤λ~2λ~1e−αk‖φ‖,∀k∈Z+. This completes the proof of the Theorem 4.1. □ Theorem 4.2 For given scalars $$\alpha > 0,$$$$d_{1}$$ and $$d_{1}$$ with $${d_2} > {d_1} > 0,$$ the system (2.1) with $$u(k) = {W^T}(k){L^{ - 1}}(k)x(k)$$ is exponentially stable with a convergence rate $$\alpha$$, if there exist matrices $$U > 0,$$$${F_1} > 0,$$ and $$N$$ -periodic matrices $$L(k) > 0,$$$$W(k)$$ satisfying the following LMIs: (Π⌢11(k)00Π⌢14(k)0Π⌢16(k)L(k)∗−UΠ⌢23(k)0Π⌢25(k)Π⌢26(k)0∗∗Π⌢330000∗∗∗−L(k+1)000∗∗∗∗−L(k+1)00∗∗∗∗∗−F10∗∗∗∗∗∗−Π⌢77)<0,k=1,2,⋯,N,i=1,2,⋯,v, (4.7) where Π⌢11(k)=−0.5L(k),Π⌢14(k)=e2α+eα(d2+2)(L(k)AiT(k)+W(k)BiT(k)),Π⌢23(k)=0.5UAdiT(k),Π⌢16(k)=e0.5(−αd1+α)(L(k)AiT(k)+W(k)BiT(k)),Π⌢26(k)=0.5e0.5(αd1+α)UAdT(k),Π⌢25(k)=β1UAdT(k),Π¯33=−e−α(d2+1)F1T,Π⌢77=(1/(1+d12))U,β1=eα(d2+2)+e2α(d2+1). Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (2.1) is reduced to z(k+1)=A¯(k)z(k)+A¯d(k)z(k−h(k)), (4.8) where $$\bar A(k) = {e^\alpha }(A(k) + B(k)K(k)),\quad {\bar A_d}(k) = {e^{\alpha (h(k) + 1)}}{A_d}(k).$$ Consider the Lyapunov–Krasovskii functional V(k)=V1(k)+V2(k)+V3(k), (4.9) where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−d1k−1zT(i)Q1(i)z(i)+∑i=k−d2k−1zT(i)Q2(i)z(i)+∑i=k−h(k)k−1zT(i)Q3z(i)),V3(k)=∑i=−d2+1−d1∑j=k+ik−1zT(j)Q3z(j). Then along the solution of (4.8), we have ΔV(k)≤zT(k)[(e2α+eα(d2+2))A¯T(k)P(k+1)A¯(k)−P(k)]z(k)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))+zT(k)(Q1(k)+Q2(k)+Q3)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Q3z(k−d(k))+d12zT(k)Q3z(k). (4.10) From (4.8), we can obtain e−α(h(k)+1)z(k+1)−(e−αh(k)A¯(k)z(k)+Ad(k)z(k−h(k)))=0. (4.11) For any $${N_1}$$, it follows that zT(k+1)N1e−α(h(k)+1)z(k+1)−zT(k+1)N1Ad(k)z(k−h(k))+zT(k)A¯T(k)N1T[−eα(h(k))+αA¯(k)z(k)−eαAd(k)z(k−h(k))]=0. (4.12) From (4.10) and (4.12), we have ΔV(k)≤zT(k)[(e2α+eα(d2+2))A¯T(k)P(k+1)A¯(k)−P(k)]z(k)+d12zT(k)Q3z(k)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))+zT(k)(Q1(k)+Q2(k)+Q3)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Q3z(k−h(k))−zT(k+1)N1e−α(d2+1)z(k+1)+zT(k+1)N1Ad(k)z(k−h(k))+zT(k)A¯T(k)N1T[e−αd1+αA¯(k)z(k)+eαAd(k)z(k−h(k))]≤ξT(k)Ξ(k)ξ(k), (4.13) where ξ(k)=(zT(k)zT(k−h(k))zT(k+1))T,Ξ(k)=ΞT(k)=(Ξij(k)),i,j=1,2,3,Ξ11(k)=(e2α+eα(d2+2))A¯T(k)P(k+1)A¯(k)−P(k)+Q1(k)+Q2(k)+(1+d12)Q3+A¯T(k)N1Te−αd1+αA¯(k),Ξ12(k)=0.5eαA¯T(k)N1TAd(k),Π13(k)=0,Π23(k)=0.5AdT(k)N1T,Ξ22(k)=(eα(d2+2)+e2α(d2+1))AdT(k)P(k+1)Ad(k)−Q3,Π33=−N1e−α(d2+1). From Lemma 2.3, we get that $${\it{\Xi}} (k) < 0,$$ if (Π¯11(k)00Π¯14(k)0Π16(k)∗−Q3Π23(k)0β1AdT(k)Π26(k)∗∗Π¯33000∗∗∗−P−1(k+1)00∗∗∗∗−P−1(k+1)0∗∗∗∗∗−N1−T)<0, (4.14) where Π¯11(k)=−P(k)+Q1(k)+Q2(k)+(1+d12)Q3,β1=eα(d2+2)+e2α(d2+1),Π26(k)=0.5e0.5(αd1+α)AdT(k),Π¯14(k)=β2A¯T(k),β2=e2α+eα(d2+2),Π16(k)=e0.5(−αd1+α)A¯T(k),Π¯23(k)=0.5AdT(k)N1T,Π¯33=−N1e−α(d2+1). Let $${M_2}(k) = diag\{ {P^{ - 1}}(k),Q_3^{ - 1},N_1^{ - 1},I,I,I\}.$$ Pre- and post-multiplying (4.14) by $${M_2}^T(k)$$ and $${M_2}(k)$$, respectively, (4.14) can be cast as below: (Π^11(k)00Π^14(k)0Π^16(k)∗−Q3−1Π^23(k)0β1Q3−1AdT(k)Π^26(k)∗∗Π^33(k)000∗∗∗−P−1(k+1)00∗∗∗∗−P−1(k+1)0∗∗∗∗∗−N1−T)<0, (4.15) where Π^11(k)=−P−1(k)+P−1(k)[Q1(k)+Q2(k)+(1+d12)Q3]P−1(k),Π^14(k)=β2P−1(k)A¯T(k),Π^16(k)=e0.5(−αd1+α)P−1(k)A¯T(k),Π^23(k)=0.5Q3−1AdT(k),Π^26(k)=0.5e0.5(αd1+α)Q3−1AdT(k),Π^33=−e−α(d2+1)N1−T. Set $${Q_1}(k) = {Q_2}(k) = \frac{1}{4}P(k).$$ From Lemma 2.3, we get that (4.15) is equality to (Φ1Φ2∗Φ3)<0, (4.16) where Φ1=(Π~11(k)00Π^14(k)∗−Q3−1Π^23(k)0∗∗Π^330∗∗∗−P−1(k+1)),Φ2=(0Π^16(k)P−1(k)β1Q3−1AdT(k)Π^26(k)0000000),Φ3=diag{−P−1(k+1)−N1−TΠ^77},Π~11(k)=−0.5P−1(k),Π^14(k)=β2P−1(k)A¯T(k),Π^26(k)=0.5e0.5(αd1+α)Q3−1AdT(k),Π^16(k)=e0.5(−αd1+α)P−1(k)(A(k)+B(k)K(k))T,Π^23(k)=0.5Q3−1AdT(k),Π¯33=−e−α(d2+1)N1−T,Π^77=−1(1+d12)Q3−1.  Let $$L(k) = {P^{ - 1}}(k),\quad W(k) = L(k){K^T}(k),\quad U = Q_3^{-1},\quad {F_1} = N_1^{ - 1}.$$ (4.7) implies that (4.16) holds. Hence, we get $${\it{\Delta}} V(k) \le 0.$$ The rest proofs are similar to the proof of Theorem 3.2, which are omitted here. □ Corollary 4.1 For given scalars $$\alpha > 0,$$$$d_{1}$$ and $$d_{1}$$ with $${d_2} > {d_1} > 0,$$ the system (2.1) with $$u(k) = {W^T}{L^{ - 1}}x(k)$$ is exponentially stable with a convergence rate $$\alpha$$, if there exist matrices $$U > 0,$$$${F_1} > 0,\;$$$$L > 0,$$ and $$W$$ satisfying the following LMIs: (Π⌢1100Π⌢14(k)0Π⌢16(k)L∗−UΠ⌢23(k)0Π⌢25(k)Π⌢26(k)0∗∗Π⌢330000∗∗∗−L000∗∗∗∗−L00∗∗∗∗∗−F10∗∗∗∗∗∗−Π⌢77)<0,k=1,2,⋯,N,i=1,2,⋯,v, (4.17) where Π⌢11(k)=−0.5L,Π⌢23(k)=0.5UAdiT(k),Π⌢14(k)=e2α+eα(d2+2)(LAiT(k)+WBiT(k)),Π⌢16(k)=e0.5(−αd1+α)(LAiT(k)+WBiT(k)),Π⌢26(k)=0.5e0.5(αd1+α)UAdT(k),Π⌢25(k)=β1UAdT(k),Π¯33=−e−α(d2+1)F1T,Π⌢77=(1/(1+d12))U,β1=eα(d2+2)+e2α(d2+1). Remark 4.1 The problems of stability and stabilization for discrete-time periodic linear systems are concerned in Zhou et al. (2011). But, the robust stability and stabilization for discrete-time periodic linear systems did not considered. Compared with Zhou et al. (2011), obtained results in our article have a greater range of applications. 5. Numerical examples In this section numerical examples are provided to show the high performance of the proposed approach. Example 5.1 Consider the uncertain 3-periodic system (3.1) with the following parameters: A1(1)=(0.3500−1.9),A1(2)=(−0.200−0.7),A1(3)=(0.100−0.4),Ad1(1)=(−0.10.2−0.1−0.1),Ad1(2)=(−0.10.0−0.1−0.1),Ad1(3)=(−0.10.3−0.1−0.7),A2(1)=(0.480.0100.68),  A2(2)=(−0.180.10−0.78),A2(3)=(0.28000.38),Ad2(1)=(−0.140.18−0.11−0.11),  Ad2(2)=(−0.120.0−0.14−0.14),  Ad2(3)=(−0.320.28−0.12−0.82),Bi(k)=(01)T,i=1,2,k=1,2,3,d=2. Applying Theorem 4.1 with $$\alpha = 0.5,$$ and using the Matlab LMI control toolbox, we solve (4.1) and obtain a set of feasible solutions as follows: W(1)=(0.5861−0.0597),W(2)=(0.39050.0452),  S(1)=108(3.50980.00000.00003.5098),L(1)=(0.35090.03420.03420.0723), L(2)=(1.30020.26820.26820.8442),L(3)=(0.6950−0.1389−0.13890.0782),S(2)=108(3.50980.00000.00003.5098),S(3)=108(3.51250.00200.00203.5115),  W(3)=(0.1843−0.3443). We obtain the following state feedback gains: K(1)=(1.8354   −1.6924),K(2)=(0.3096   −0.0448),K(3)=(−0.9519   −6.0907). Therefore, according to Theorem 4.1, the system (3.1) with the above parameters and $$u(k) = {W^T}(k) \times{L^{ - 1}}(k)x(k)$$ is exponentially stable, which is further verified by the simulation results given in Fig. 1. Fig. 1. View largeDownload slide State trajectories of the closed-loop system in Example 5.1. Fig. 1. View largeDownload slide State trajectories of the closed-loop system in Example 5.1. Figure 1 represents the trajectories of $${x_1}(k)$$ and $${x_2}(k)$$ of the closed-loop system. It is obvious that the considered system in Example 5.1 is exponentially stabilizable through the obtained feedback controller gain. Example 5.2 Consider the uncertain 2-periodic time-delay system (2.1) with the following parameters: A1(1)=(−0.1−0.0100.01−0.15000−0.1),A1(2)=(−0.1270.01−0.010.01−0.14000−0.2),Ad1(1)=(−0.1−0.0100−10.100.01−0.3),Ad1(2)=(−0.1000.01−0.1000.02−0.36),A2(1)=(−0.120.10−0.01−1000−1.1),A2(2)=(−0.20.1−0.1−0.01−0.2000−0.18),Ad2(1)=(−0.160.0100.1−0.100−0.01−0.27),Ad2(2)=(−0.200.01−0.01−0.30−0.020−0.18),h(k)=2+sin⁡(kπ2),Bi(1)=(0.01−0.200.03−0.10.02),Bi(2)=(0.020.10.20.1−0.150.03),i=1,2,d1=1,d2=3. Applying Theorem 4.2 with $$\alpha = 0.5,$$ and using the Matlab LMI control toolbox, we solve (4.7) and obtain a set of feasible solutions as follows: L(1)=(1.36990.1009−0.02200.10090.66720.0186−0.02200.01860.8388),L(2)=(1.14750.22280.05720.22280.67840.01940.05720.01941.4694),F1=108(1.3057−0.0000−0.0000−0.00001.30730.0000−0.00000.00001.3057),U=(0.8381−0.0631−0.1710−0.06310.31570.0950−0.17100.09500.9170),W(1)=(0.71421.38580.00290.8250−1.72050.7779),W(2)=(0.9485−2.5252−1.4823−1.80280.8355−0.9316). We obtain the following state feedback gains: K(1)=(0.4896−0.0129−2.03810.94791.06730.9286),K(2)=(1.3098−2.63070.5524−1.7741−2.0594−0.5378). Figure 2 shows the simulation results for states $${x_1}(k),$$$${x_2}(k)$$ and $${x_3}(k)$$ of the closed-loop system. Fig. 2. View largeDownload slide State trajectories of the closed-loop system in Example 5.2. Fig. 2. View largeDownload slide State trajectories of the closed-loop system in Example 5.2. The simulation results reveal that the considered system is exponentially stabilizable through the obtained feedback controller gains. Simulation results demonstrate that our proposed design is very effective. 6. Conclusion In this article, we have investigated the problems of robustly exponential stability and exponential stabilization for uncertain linear discrete-time periodic systems with time-delay in the state variables and polytopic-type parameter uncertainty. Using the uncertainty-dependent Lyapunov–Krasovskii functionals, we have provided some novel sufficient conditions in terms of LMIs. These conditions guarantee that the discrete-time periodic system with time-delay is robustly exponentially stable. Furthermore, employing a static periodic state feedback control law, we have given robust exponential stabilization criteria for uncertain linear discrete-time periodic systems. Besides, we have presented the calculation method of the periodic control gain matrix. Finally, effectiveness of our proposed approach has been illustrated by simulation results. We would like to take the mixed time-varying delays and $$H_{\infty}$$ control into consideration for the further research. Funding This work is supported by the National Nature Science Foundation of China under Grants 61603272 and 11526149. References Bittanti S. ( 1986 ) Deterministic and stochastic linear periodic systems . Time Series and Linear Systems ( Bittanti S. ed.). Berlin : Springer , pp. 141 – 182 . Bittanti S. & Colaneri P. ( 2009 ) Period Systems: Filtering and Control . London : Springer . Botmart T., Niamsup P. & Phat V. N. ( 2011 ) Delay-dependent exponential stabilization for uncertain linear systems with interval non-differentiable time-varying delays . Appl. Math. Comput. , 217 , 8236 – 8247 . Boukas E. K. ( 2007 ) State feedback stabilization of nonlinear discrete-time systems with time-varying delays . Nonlinear Anal. , 66 , 1341 – 1350 . Google Scholar CrossRef Search ADS Boyd S., Ghaoui L. E., Feron E. & Balakrishnan V. ( 1994 ) Linear Matrix Inequalities in System and Control Theory . Philadelphia : SIAM . Chen T. & Francis B.A. ( 1995 ) Optimal Sampled-Data Control Systems . New York: Springer . de Souza C. E. & Coutinho D. ( 2014 ) Robust stability and control of uncertain linear discrete-time periodic systems with time-delay . Automatica , 50 , 431 – 441 . Google Scholar CrossRef Search ADS Dong Y., Li T. & Mei S. ( 2016 ) Exponential stabilization and $$L_2$$-gain for uncertain switched nonlinear systems with interval time-varying delay . Math. Meth. Appl. Sci. , 39 , 3836 – 3854 . Google Scholar CrossRef Search ADS Dong Y., Wang X., Mei S. & Li W. ( 2012 ) Exponential stabilization of nonlinear uncertain systems with time-varying delay . J. Eng. Math. , 77 , 225 – 237 . Google Scholar CrossRef Search ADS Dong Y., Zhang Y. & Zhang X. ( 2014 ) Design of observer-based feedback control for a class of discrete-time nonlinear systems with time-delay . Appl. Math. Comput. , 13 , 107 – 121 . Gao M. & Cui B. ( 2009 ) Global robust exponential stability of discrete-time interval BAM neural networks with time-varying delays . Appl. Math. Model. , 33 , 1270 – 1284 . Google Scholar CrossRef Search ADS Gonzlez A. ( 2013 ) Robust stabilization of linear discrete-time systems with time-varying input delay . Automatica , 49 , 2919 – 2922 . Google Scholar CrossRef Search ADS Grasselli O. M., Longhi S., Tornamb A. & Valigi P. ( 1996 ) Robust output regulation and tracking for linear periodic systems under structured uncertainties . Automatica , 32 , 1015 – 1019 . Google Scholar CrossRef Search ADS He Y., Wu M., Liu G. P. & She J. H. ( 2008 ) Output feedback stabilization for a discrete system with a time-varying delay . IEEE Trans. Automat. Control , 53 , 2372 – 2378 . Google Scholar CrossRef Search ADS Kwon O. M. & Park Ju H. ( 2006 ) Exponential stability of uncertain dynamic systems including state delay . Appl. Math. Lett. , 19 , 901 – 907 . Google Scholar CrossRef Search ADS Lee D. H., Joo Y. H. & Tak M. H. ( 2015 ) Periodically time-varying memory static output feedback control design for discrete-time LTI systems . Automatica , 52 , 47 – 54 . Google Scholar CrossRef Search ADS Letyagina O. N. & Zhabko A. P. ( 2009 ) Robust stability analysis of linear periodic systems with time-delay . Int. J. Mod. Phys. A , 24 , 893 – 907 . Google Scholar CrossRef Search ADS Liu P.-L. ( 2009 ) Robust exponential stability for uncertain time-varying delay systems with delay dependence . J. Franklin Inst. , 346 , 958 – 968 . Google Scholar CrossRef Search ADS Phat V. N. & Ratchagit K. ( 2011 ) Stability and stabilization of switched linear discrete-time systems with interval time-varying delay . Nonlinear Anal. Hybrid Syst. , 5 , 605 – 612 . Google Scholar CrossRef Search ADS Qiu J., Feng G. & Yang J. ( 2008 ) Improved delay-dependent $$H_\infty$$ filtering design for discrete-time polytopic linear delay systems . IEEE Trans. Circuits Syst. II Express Briefs , 55 , 178 – 182 . Google Scholar CrossRef Search ADS Ramakrishnan K. & Ray G. ( 2013 ) Robust stability criteria for a class of uncertain discrete-time systems with time-varying delay . Appl. Math. Model. , 37 , 1468 – 1479 . Google Scholar CrossRef Search ADS Souza C. E. D. & Coutinho D. ( 2014 ) Robust stability and control of uncertain linear discrete-time periodic systems with time-delay , Automatica , 50 , 431 – 441 . Google Scholar CrossRef Search ADS Tang Y., Fang J., Xia M. & Gu X. ( 2010 ) Synchronization of Takagi Sugeno fuzzy stochastic discrete-time complex networks with mixed time-varying delays . Appl. Math. Model. , 34 , 843 – 855 . Google Scholar CrossRef Search ADS Wang R. & Fei S. ( 2014 ) New stability and stabilization results for discrete-time switched systems . Appl. Math. Comput. , 238 , 358 – 369 . Wang Y., Xie L. & Souza C. E. D. ( 1992 ) Robust control of a class of uncertain nonlinear systems . Syst. Control Lett. , 19 , 139 – 149 . Google Scholar CrossRef Search ADS Zhang W. A. & Yu L. ( 2009 ) Stability analysis for discrete-time switched time delay systems . Automatica , 45 , 2265 – 2271 . Google Scholar CrossRef Search ADS Zhou B., Zheng W. X. & Duan G.-R. ( 2011 ) Stability and stabilization of discrete-time periodic linear systems with actuator saturation . Automatica , 47 , 1813 – 1820 . Google Scholar CrossRef Search ADS © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

# Exponential stability and stabilization for uncertain discrete-time periodic systems with time-varying delay

, Volume Advance Article – Mar 12, 2017
24 pages

/lp/ou_press/exponential-stability-and-stabilization-for-uncertain-discrete-time-vq9mNn1Wpw
Publisher
Oxford University Press
ISSN
0265-0754
eISSN
1471-6887
D.O.I.
10.1093/imamci/dnx003
Publisher site
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### Abstract

Abstract This article addresses the problems of robustly exponential stability and exponential stabilization for uncertain linear discrete-time periodic systems with time delay in the state variables and polytopic-type parameter uncertainty. By constructing the novel uncertainty-dependent Lyapunov–Krasovskii functionals, we establish some sufficient conditions in forms of linear matrix inequalities, which guarantee the uncertain linear discrete-time periodic system with time delay is robustly exponentially stable. Then, by utilizing static periodic state feedback and free weighting matrix technique, we give some sufficient conditions to ensure the robustly exponential stabilization of uncertain linear discrete-time periodic systems with time delay. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed method. 1. Introduction In recent years, the control community has been attracted by the periodic systems because this class of systems represents an important subclass of time-varying systems. Indeed, many dynamic systems show periodic behaviour: networked control systems with communication constraints, celestial mechanics, sampled-data control, signal processing in the presence of cyclostationary noise, control of multirate plants and multiplexed systems etc., see Bittanti (1986), Bittanti & Colaneri (2009) and the references therein. In practice, time delays are often encountered in many real-world problems such as automatic control engineering, quadratic optimization methods and economic classification and biological reproduction. It is widely known that time delay arises pervasively in dynamic systems and causes instability and poor performance of control systems. Over the last two decades, many researches have focused on stability analysis and control design for discrete-time systems with time delays (Gonzlez, 2013; Phat & Ratchagit, 2011; Lee et al., 2015; Zhang & Yu, 2009; Kwon & Park, 2006; Boukas, 2007). The robust stabilization for linear discrete-time systems with time-varying input delay was studied in Gonzlez (2013). Some methods for dealing with constant delays in discrete-time systems have been given (Lee et al., 2015; Zhang & Yu, 2009; Kwon & Park, 2006). State feedback stabilization of non-linear discrete-time systems with time-varying delays was considered in Boukas (2007). The Lyapunov–Krasovskii method has been useful to investigate the stability and stabilization of linear discrete-time systems with time-varying delays (Gao & Cui, 2009; He et al., 2008; Tang et al., 2010). Dong et al. (2014) proposed new sufficient conditions for observer-based stabilization of a class of discrete-time non-linear systems with time delay. Exponential stability, as an important index to get the convergence rates of systems, has also been studied by many researchers (Liu, 2009; Botmart et al., 2011; Wang & Fei, 2014; Dong et al., 2016; Dong et al., 2012). Liu (2009) given the delay-dependent stability criteria by the Lyapunov–Krasovskii functional method. Botmart et al. (2011) proposed exponential stabilization delay-dependent sufficient conditions for linear systems with time-varying delay by choosing improved Lyapunov–Krasovskii functionals and using Leibniz–Newton’s formula. Dong et al. (2016) investigated the exponential stabilization and $$L_{2}$$-gain for a class of uncertain switched non-linear systems with interval time-varying delay and established novel delay-dependent sufficient conditions of exponential stabilization for a class of uncertain switched non-linear delay systems. The problem of robust exponential stabilization for dynamical non-linear systems with uncertainties and time-varying delay was investigated in Dong et al. (2012). Dong et al. (2012) presented the robust exponential stabilization criterion based on the Razumikhin theorem and the Lyapunov function method. Periodic linear systems have the potential to model many practical control systems including multirate sample data systems (Chen & Francis, 1995) and systems that operate periodically such as satellites. Therefore, there has been a persistent interest with the control community in linear periodic systems over the past two decades. Significant advances have been made in a variety of topics about the theories of control and state estimation (Letyagina & Zhabko, 2009; Grasselli et al., 1996). Robust stability analysis of linear periodic systems with time delay was presented in Letyagina & Zhabko (2009). Robust output regulation and tracking for linear periodic systems under structured uncertainties were addressed in Grasselli et al. (1996). Qiu et al. (2008) considered $$H_{\infty}$$ filtering design for discrete-time polytopic linear delay systems. However, among the developments in the literature, little effort has been devoted to the problems of robust stability and stabilization for linear periodic systems with a time delayed state. Our article investigates the problems of robustly exponential stability and robustly exponential stabilization for uncertain linear discrete-time periodic systems with time-varying delay in the state variables and polytopic-type parameter uncertainty. To the best of our knowledge, results about the robustly exponential stability and robustly exponential stabilization of uncertain linear discrete-time periodic systems are rarely reported in the literature. Our research is conducted in the hope to fill this gap and here we listed our contributions. (1) We construct an appropriate uncertainty-dependent Lyapunov–Krasovskii functional to exhibit the delay-dependent discrete-time periodic systems and establish the criteria of robustly exponential stability for uncertain linear discrete-time periodic systems without control. (2) Based on periodic state feedback control, we provide robustly exponential stabilization methods. The closed-loop discrete-time periodic system is robustly exponential stable under our established new sufficient conditions. (3) The calculation method of the periodic control gain matrix is presented. (4) Two simulation examples are given to show the performances of our method. (5) Compared with the existing ones, our method leads to less conservatism because of the introduced free weight matrices $$N_{1}$$, $$Z_{1}, Z_{2}\,and\,Z_{3}$$. The rest of this article is organized as follows. Section 2 gives the problem formulation and preliminary results. Then, the robust exponential stability problem is solved in Section 3. In Section 4, the robustly exponential stabilizing periodic state feedback controller is designed, and novel criteria of exponential stabilization for the uncertain linear discrete-time periodic systems are established. In Section 5, the performances of our method are illustrated by two examples. Finally, Section 6 concluded the paper with some further comments. Notation. Throughout this article, $${R^n}$$ denotes the $$n$$-dimensional real Euclidean space. $${R^{m \times n}}$$ represents the set of all $$m \times n$$ real matrices. $$I$$ and 0 represent the identity matrix and null matrix of appropriate dimensions. $$Z$$ represents the set of integers, $${Z^ + }$$ represents the set of non-negative integers. In symmetric block matrices, we use an asterisk ‘$$\ast$$’ to represent a term induced by symmetry. diag $$\{ \cdots \}$$ denotes a block-diagonal matrix. A matrix $$M(k)$$ is denoted N-periodic, where $$0 < N \in {Z^ + },$$ if $$M(k + N) = M(k),$$ for all $$k \in Z$$. $${A^T}$$ stands for the transpose of $$A$$. For a square matrix $$P,$$$$P > 0(< 0,\,\, \le 0,\,\, \ge 0)$$ means that this matrix is positive (negative, semi-negative and semi-positive) definite. $$\|\cdot\|$$ is the Euclidean vector norm. 2. System description and preliminaries Consider the following uncertain discrete-time periodic system: x(k+1)=A(k)x(k)+Ad(k)x(k−h(k))+B(k)u(k),x(k)=φ(k),k=−d2,−d2+1,⋯,0, (2.1) where $$x(k) \in {R^n}$$ is the state vector, $$u(k) \in {R^m}$$ is the control, $$A(k),{A_d}(k)$$ and $$B(k)$$ are uncertain $$N$$-periodic real matrices with appropriate dimensions that are assumed to be confined to the following polytope: Ω(k)={Υ(k):Υ(k)=∑i=1vλiΥi(k),λi≥0,∑i=1vλi=1}, (2.2) where Υ(k)=[A(k),Ad(k),B(k)], (2.3) Υi(k)=[Ai(k),Adi(k),Bi(k)] (2.4) with $${A_i}(k),{A_{di}}(k)$$ and $${B_i}(k),i = 1,2, \cdots,v,$$ being given $$N$$-periodic real matrices. The sequence $$\varphi (k)$$ is the initial condition; the time-varying delay $$h(k)$$ is a $$N$$-periodic function and satisfies the following condition: 0<d1≤h(k)≤d2, where $${d_1}$$ and $${d_2}$$ are non-negative integers representing the lower and upper bounds of the interval time-delay. Definition 2.1 The delayed discrete-time system (2.1) is said to be robustly exponentially stable with a convergence rate $$\alpha$$, if there exist scalars $$\alpha > 0,\beta > 0$$ such that ‖x(k)‖≤βe−αk‖φ‖∀k∈Z+, where $$\left\| \varphi \right\| = \mathop {\sup }\limits_{k \in Z \cap [- {d_2},0]} \left\| {x(k)} \right\|.$$ The following lemmas will play important roles in this article. Lemma 2.1 (Ramakrishnan & Ray, 2013) For any constant matrix $$W \in {R^{n \times n}}$$ with $$W = {W^T} > 0$$, integers $${n_1} < {n_2},$$ vector function $$\omega :\;\left\{ {{n_1},{n_1} + 1, \cdots,{n_2}} \right\} \to {R^q}$$ such that the sums concerned are well defined, then (n2−n1+1)∑i=n1n2ωT(i)Wω(i)≥(∑i=n1n2ω(i))TW(∑i=n1n2ωT(i)). Lemma 2.2 (Wang et al., 1992) For any $$x,y \in {R^n}$$ and any positive-definite matrix $$P \in {R^{n \times n}}$$, we have 2xTy≤xTPx+yTP−1y. Lemma 2.3 (Boyd et al., 1994) Given constant symmetric matrices $${S_1},{S_2},{S_3},$$ and $${S_1} = S_1^T < 0,$$$${S_3} = S_3^T > 0,$$ then $${S_1} + {S_2}S_3^{-1}S_2^T < 0$$ if and only if [S1S2S2T−S3]<0. This article aims at designing a stabilizing static state feedback for the system (2.1) as below u(k)=K(k)x(k), where $$K(k)$$ is an $$N$$-periodic matrix to be found such that the closed-loop system x(k+1)=(A(k)+B(k)K(k))x(k)+Ad(k)x(k−h(k)) (2.5) is exponentially stable for all system matrices belonging to the uncertainty polytope $${\it{\Omega}} (k)$$. 3. Robust exponential stability analysis In this section, we shall develop linear matrix inequality (LMI)-based conditions for exponential stability of uncertain discrete-time periodic systems with time-delay. When the time-delay $$h(k)$$ is a constant $$d$$, i.e., $$h(k)=d$$, the system (2.1) can be written as x(k+1)=A(k)x(k)+Ad(k)x(k−d)+B(k)u(k),x(k)=φ(k),k=−d,−d+1,⋯,0. (3.1) Theorem 3.1 For a given scalar $$\alpha > 0,$$ the system (3.1) with $$u(k) = 0$$ is robustly exponentially stable with a convergence rate $$\alpha$$, if there exist $$N$$-periodic matrices $$P(k) > 0, {Q_i}(k) > 0, {R_i}(k) > 0,i = 1,2, \cdots,v,$$ and a matrix $${Z_1} > 0$$ satisfying the following LMIs: (N⌣i(k)Z1eαAiT(k)P(k+1)0Γ1∗M¯i(k)eα(d+1)AdiT(k)P(k+1)Γ2Γ3∗∗−P(k+1)00∗∗∗−P(k+1)0∗∗∗∗−I)<0,k=1,2,⋯,N,i=1,2,⋯,v, (3.2) where N⌣i(k)=−P(k)+dRi(k)+Q(k)−Z1,M¯i(k)=−Qi(k−d)−Z1−Ri(k−d),Γ1=d(eαAiT(k)−I)Z1,Γ2=eα(d+1)AdiT(k)P(k+1),Γ3=deα(d+1)AdiT(k)Z1. Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (3.1) with $$u(k) = 0$$ is reduced to z(k+1))=eαA(k)z(k)+eα(d+1)Ad(k)z(k−d). (3.3) Consider the Lyapunov–Krasovskii functional: V(k)=V1(k)+V2(k)+V3(k)+V4(k), (3.4) where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−dk−1zT(i)Q(i)z(i),V3(k)=∑i=−d−1∑j=k+ik−1zT(j)R(j)z(j),V4(k)=d∑i=−d−1∑j=k+ik−1ηT(j)Z1η(j),η(j)=z(j+1)−z(j),R(k)=∑i=1vλiRi(k),Q(k)=∑i=1vλiQi(k) (3.5) and $${\lambda _i},\;i = 1,2, \cdots,v,$$ are the uncertain non-negative scalars characterizing the polytope $${\it{\Omega}} (k)$$ as defined in (2.2). We can verify that λ¯1‖z(k)‖2≤V(k)≤λ¯2‖zk‖2, (3.6) where λ¯1=mink∈{1,2,⋯,N}⁡λmin(P(k)),‖zk‖2=supj∈Z∩[−d,0]⁡‖z(k+j)‖2,λ¯2=maxi∈{1,2,⋯,N}⁡{λmax(P(i))+dλmax(Q(i))+12d(d+1)λmax(R(i))}+2d2(d+1)λmax(Z1). Define $${\it{\Delta}} V(k) = V(k + 1) - V(k)$$. Then along the solution of (3.3), we have ΔV1(k)=V1(k+1)−V1(k)=zT(k)[e2αAT(k)P(k+1)A(k)−P(k)]z(k)+2eα(d+2)zT(k)AT(k)P(k+1)Ad(k)z(k−d)   +e2α(d+1)zT(k−d)AdT(k)P(k+1)Ad(k)z(k−d), (3.7) ΔV2(k)=V2(k+1)−V2(k)=zT(k)Q(k)z(k)−zT(k−d)Q(k−d)z(k−d), (3.8) ΔV3(k)=V3(k+1)−V3(k)=∑i=−d−1∑j=k+1+ikzT(j)R(j)z(j)−∑i=−d−1∑j=k+ik−1zT(j)R(j)z(j)=dzT(k)R(k)z(k)−∑j=k−dk−1zT(j)R(j)z(j). (3.9) Using Lemma 2.1, we have ΔV4(k)=d2ηT(k)Z1η(k)−d∑j=k−dk−1ηT(j)Z1η(j)≤d2ηT(k)Z1η(k)−(z(k)−z(k−d))TZ1(z(k)−z(k−d)). (3.10) From (3.7)–(3.10), we have ΔV(k)≤zT(k)[e2αAT(k)P(k+1)A(k)−P(k)+dR(k)+Q(k)−Z1]z(k)+zT(k−d)Z1z(k)+2eα(d+2)zT(k)AT(k)P(k+1)Ad(k)z(k−d)+zT(k)Z1z(k−d)+d2ηT(k)Z1η(k)+zT(k−d)[e2α(d+1)AdT(k)P(k+1)Ad(k)−Q(k−d)−R(k−d)−Z1]z(k−d)=(zT(k)zT(k−d))N(k)(z(k)z(k−d)), (3.11) where N(k)=(N1(k)N2(k)N2T(k)N3(k)),N1(k)=e2αAT(k)P(k+1)A(k)−P(k)+dR(k)+Q(k)−Z1+d2(eαAT(k)−I)Z1(eαA(k)−I),N2(k)=eα(d+2)AT(k)P(k+1)Ad(k)+Z1+d2(eαAT(k)−I)Z1eα(d+1)Ad(k),N3(k)=e2α(d+1)AdT(k)P(k+1)Ad(k)−Q(k−d)−Z1−R(k−d)+d2e2α(d+1)AdT(k)Z1Ad(k). By Lemma 2.3, we have that $${\rm N}(k) < 0$$ is equivalent to: (N¯1(k)Z1eαAT(k)0N¯3(k)∗N¯2(k)eα(d+1)AdT(k)eα(d+1)AdT(k)N¯4(k)∗∗−P−1(k+1)00∗∗∗−P−1(k+1)0∗∗∗∗−Z1−1)<0, (3.12) where N¯1(k)=−P(k)+dR(k)+Q(k)−Z1,N¯2(k)=−Q(k−d)−Z1−R(k−d).N¯3(k)=d(eαAT(k)−I),N¯4(k)=deα(d+1)AdT(k). Let $$M(k) = diag\{ I,I,P(k{\rm{ + }}1),P(k{\rm{ + }}1),{Z_1}\}.$$ Pre- and post-multiplying (3.12) by $${M^T}(k)$$ and $$M(k)$$, respectively, (3.12) can be cast as below: (N¯1(k)Z1eαAT(k)P(k+1)0N¯3(k)Z1∗N¯2(k)eα(d+1)AdT(k)P(k+1)N¯5(k)N¯4(k)Z1∗∗−P(k+1)00∗∗∗−P(k+1)0∗∗∗∗−Z1)<0, (3.13) where $${\bar {\rm N}_5}(k) = {e^{\alpha (d + 1)}}A_d^T(k)P(k{\rm{ + }}1).$$ From (3.2), we get that (3.13) holds. Therefore, from (3.11) it follows that $${\it{\Delta}} V(k) \le 0,$$ which implies that the function $$V(k)$$ is decreasing and V(k)≤V(0),∀k∈Z+. Hence, from (3.6) it follows that ‖z(k)‖≤λ¯2λ¯1‖φ¯‖,∀k∈Z+. Returning to the variable $$x(k) = {e^{-\alpha k}}z(k)$$, we have ‖x(k)‖≤λ¯2λ¯1e−αk‖φ‖,∀k∈Z+, which implies that the system (3.1) is exponentially stable. This completes the proof of the Theorem 3.1. □ Theorem 3.2 For given scalars $$\alpha > 0,$$$${d_1}$$ and $${d_2}$$ with $${d_2} > {d_1} > 0$$, the system (2.1) with $$u(k) = 0$$ is robustly exponentially stable with a convergence rate $$\alpha$$, if there exist $$N$$-periodic matrices $$P(k) > 0,$$$${Q_{ij}}(k) > 0,$$$$j = 1,2,i = 1,2, \cdots v,$$ and matrices $${Z_1} > 0,{Z_2} > 0,{Z_3} > 0, Q > 0,\;{N_1} > 0,$$ any matrices $${T_i},\;\,i = 1,2,3,$$ with appropriate dimensions satisfying the following LMIs: (Ψ1Ψ2∗Ψ3)<0,k=1,2,⋯,N,i=1,2,⋯,v, (3.14) where Ψ1=(Π¯11(k)0Z1Z2−G∗−QT1T−T1TΠ^25(k)∗∗Π¯33(k)−T2T+T30∗∗∗Π¯44(k)0∗∗∗∗Π¯55(k)),Ψ2=(Π^16(k)0Π^18(k)00Π^27(k)Π^28(k)T1T000T2T000T3T0000),Ψ3=diag{−P(k+1),−P(k+1),−N1,−Z3},Π¯11(k)=−P(k)+Qi1(k)+Qi2(k)+(1+d12)Q−Z1−Z2+G,Π^16(k)=eα(1+eαd2)AiT(k)P(k+1),d12=d2−d1,Π^18(k)=e0.5(−αd1+α)AiT(k)N1,Π^28(k)=0.5e0.5α(1+d1)AdiT(k)N1,Π^25(k)=0.5AdiT(k)N1,Π^27(k)=eαeαd2+e2αd2AdiT(k)P(k+1),Π¯33(k)=T2+T2T−Qi1(k−d1)−Z1,G=d12Z1+d22Z2+d122Z3,Π¯44(k)=−Z2−T3T−T3−Qi2(k−d2),Π¯55(k)=G−N1e−α(d2+1). Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (2.1) with $$u(k) = 0$$ is reduced to z(k+1)=eαA(k)z(k)+eα(h(k)+1)Ad(k)z(k−h(k)). (3.15) Consider the Lyapunov–Krasovskii functional: V(k)=∑i=16Vi(k), (3.16) where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−d1k−1zT(i)Q1(i)z(i)+∑i=k−d2k−1zT(i)Q2(i)z(i)+∑i=k−h(k)k−1zT(i)Qz(i)V3(k)=∑i=−d2+1−d1∑j=k+ik−1zT(j)Qz(j),V4(k)=d1∑i=−d1−1∑j=k+ik−1ηT(j)Z1η(j),V5(k)=d2∑i=−d2−1∑j=k+ik−1ηT(j)Z2η(j),V6(k)=d12∑i=−d2−d1−1∑j=k+ik−1ηT(j)Z3η(j),η(j)=z(j+1)−z(j),Qj(k)=∑i=1vλiQij(k),j=1,2 (3.17) and $${\lambda _i},\;i = 1,2, \cdots,v,$$ are the uncertain non-negative scalars characterizing the polytope $${\it{\Omega}} (k)$$ as defined in (2.2). Then along the solution of (3.3), we have ΔV1(k)=V1(k+1)−V1(k)=zT(k)[e2αAT(k)P(k+1)A(k)−P(k)]z(k)+2eα(h(k)+2)zT(k)AT(k)P(k+1)Ad(k)z(k−h(k))+e2α(h(k)+1)zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k)). Using Lemma 2.2, we get ΔV1(k)≤eα(d2+2)[zT(k)AT(k)P(k+1)A(k)z(k)+zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))]+zT(k)[e2αAT(k)P(k+1)A(k)−P(k)]z(k)+e2α(d2+1)zT(k−h(k))×AdT(k)P(k+1)Ad(k)z(k−h(k)), (3.18) ΔV2(k)=V2(k+1)−V2(k)≤zT(k)(Q1(k)+Q2(k)+Q)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)−zT(k−d2)×Q2(k−d2)z(k−d2)−zT(k−h(k))Qz(k−h(k))+∑j=k+1−d2k−d1zT(j)Qz(j), (3.19) ΔV3(k)=d12zT(k)Qz(k)−∑j=k+1−d2k−d1zT(j)Qz(j), (3.20) ΔV4(k)=d12ηT(k)Z1η(k)−d1∑j=k−d1k−1ηT(j)Z1η(j)≤d12ηT(k)Z1η(k)−(z(k)−z(k−d1))TZ1(z(k)−z(k−d1)), (3.21) ΔV5(k)=d22ηT(k)Z2η(k)−d2∑j=k−d1k−1ηT(j)Z2η(j)≤d22ηT(k)Z2η(k)−(z(k)−z(k−d2))TZ2(z(k)−z(k−d2)), (3.22) ΔV6(k)=d122ηT(k)Z3η(k)−d12∑j=k−d2k−d1−1ηT(j)Z3η(j)}≤d122ηT(k)Z3η(k)−(∑j=k−d2k−d1−1η(j))TZ3(∑j=k−d2k−d1−1η(j)). (3.23) Define $\gamma (k) = {\left(\begin{array}{ccc} {{z^T}(k - h(k))} & {{z^T}(k - {d_1})} & {{z^T}(k - {d_2})} \end{array} \right)^T},$ ${T^T} = \begin{array}{ccc} ({{T_1}} & {{T_2}} & {{T_3}} \end{array}).$ Using Lemma 2.2, we get −(∑j=k−d2k−d1−1η(k))TZ3(∑j=k−d2k−d1−1η(k))≤γT(k)TZ3−1TTγ(k)−2γT(k)T∑j=k−d2k−d1−1η(k)=γT(k)TZ3−1TTγ(k)−γT(k)[(0T−T)+(0T−T)T]γ(k). (3.24) From (3.18)–(3.24), it follows that ΔV(k)≤zT(k)[(e2α+eα(d2+2))AT(k)P(k+1)A(k)−P(k)]z(k)+zT(k)(Q1(k)+Q2(k)+Q)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)×z(k−h(k))−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Qz(k−h(k))+d12zT(k)×Qz(k)+d12ηT(k)Z1η(k)−(z(k)−z(k−d1))TZ1(z(k)−z(k−d1))+d22ηT(k)Z2η(k)−(z(k)−z(k−d2))TZ2(z(k)−z(k−d2))+d122ηT(k)Z3η(k)+γT(k)TZ3−1TTγ(k)−γT(k)[(0T−T)+(0T−TT]γ(k). (3.25) From (3.15), we have e−α(h(k)+1)z(k+1)−(e−αh(k)A(k)z(k)+Ad(k)z(k−h(k)))=0. For any $$N_{1}$$ with appropriate dimensions, it follows that zT(k+1)N1e−α(h(k)+1)z(k+1)−zT(k+1)N1e−αh(k)A(k)z(k)−zT(k+1)N1Ad(k)z(k−h(k))=0,zT(k)AT(k)N1T[e−αh(k)z(k+1)−e−α(h(k))+αA(k)z(k)−eαAd(k)z(k−h(k))]=0. (3.26) From (3.26), we get zT(k+1)N1e−α(h(k)+1)z(k+1)−zT(k+1)N1Ad(k)z(k−h(k))+zT(k)AT(k)N1T[−e−α(h(k))+αA(k)z(k)−eαAd(k)z(k−h(k))]=0. (3.27) From (3.25) and (3.27), we have ΔV(k)≤zT(k)[(e2α+eα(d2+2))AT(k)P(k+1)A(k)−P(k)]z(k)+zT(k)(Q1(k)+Q2(k)+Q)z(k)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))−zT(k−d1)Q1(k−d1)×z(k−d1)−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Qz(k−h(k))+d12zT(k)Qz(k)−(z(k)−z(k−d1))TZ1(z(k)−z(k−d1))+(z(k+1)−z(k))T[d12Z1+d22Z2+d122Z3]×(z(k+1)−z(k))−(z(k)−z(k−d2))TZ2(z(k)−z(k−d2))+(zT(k−h(k)),zT(k−d1),zT(k−d2))(T1TT2TT3T)Z3−1(T1T2T3)γ(k)−γT(k)[(0T−T+(0T−T)T]γ(k)−zT(k+1)N1e−α(h(k)+1)z(k+1)+zT(k+1)×N1Ad(k)z(k−h(k))+zT(k)AT(k)N1T[e−α(h(k))+αA(k)z(k)+eαAd(k)z(k−h(k))]}≤ξT(k)Π(k)ξ(k), (3.28) where ξ(k)=(zT(k),zT(k−h(k)),zT(k−d1),zT(k−d2),zT(k+1))T,G=d12Z1+d22Z2+d122Z3,Π(k)=ΠT(k)=(Πij(k)),i,j=1,2,⋯5,Π11(k)=(e2α+eα(d2+2))AT(k)P(k+1)A(k)−P(k)+Q1(k)+Q2(k)+(1+d12)Q  −Z1−Z2+G+AT(k)N1Te−αd1+αA(k),Π12(k)=0.5eαAT(k)N1TAd(k),Π13(k)=Z1,Π14(k)=Z2, Π15(k)=−G,Π22(k)=(eα(d2+2)+e2α(d2+1))AdT(k)P(k+1)Ad(k)−Q+T1TZ3−1T1,Π23(k)=T1TZ3−1T2+T1T,Π24(k)=T1TZ3−1T3−T1T,Π25(k)=0.5AdT(k)N1T,Π33(k)=T2+T2T−Q1(k−d1)−Z1+T2TZ3−1T2,Π34(k)=−T2T+T3+T2TZ3−1T3,Π44(k)=T3TZ3−1T3−Z2−T3T−T3−Q2(k−d2),Π55(k)=G−N1e−α(d2+1). From Lemma 2.3, we get that $${\it{\Pi}} (k) < 0,$$ if Π¯=(X1X2∗X3)<0, (3.29) where X1=(Π¯11(k)0Z1Z2−G∗−QT1T−T1TΠ25(k)∗∗Π¯33(k)Π¯34(k)0∗∗∗Π¯44(k)0∗∗∗∗Π55(k)), X2=(Π¯16(k)0Π18(k)00βAdT(k)Π28(k)T1T000T2T000T3T0000),X3=diag{−P−1(k+1),−P−1(k+1),−N1−T,−Z3},Π¯11(k)=−P(k)+Q1(k)+Q2(k)+(1+d12)Q−Z1−Z2+G,Π¯16(k)=eα(1+eαd2)AT(k),Π18(k)=e0.5(−αd1+α)AT(k),Π28(k)=0.5e0.5α(1+d1)AdT(k),  β=eαeαd2+e2αd2,  Π¯34(k)=−T2T+T3,Π¯33(k)=T2+T2T−Q1(k−d1)−Z1,Π¯44(k)=−Z2−T3T−T3−Q2(k−d2). Let $$\bar M(k) = diag\{ I,I,I,I,I,P(k{\rm{ + }}1),P(k{\rm{ + }}1),{N_1}^T,I\}.$$ Pre- and post-multiplying (3.29) by $${\bar M^T}(k)$$ and $${\bar M}(k)$$, respectively, (3.29) can be cast as below: Π~=(X1X~2∗X~3)<0, (3.30) where X~2=(Π~16(k)0Π~18(k)00Π~27(k)Π~28(k)T1T000T2T000T3T0000),X~3=diag{−P(k+1),−P(k+1),−N1T,−Z3},Π~16(k)=eα(1+eαd2)AT(k)P(k+1),Π~18(k)=e0.5(−αd1+α)AT(k)N1T,Π~27(k)=βAdT(k)P(k+1),Π~28(k)=0.5e0.5α(1+d1)AdT(k)N1T. From (3.14), we get that (3.30) holds. Therefore, from (3.11) it follows that $${\it{\Delta}} V(k) \le 0$$ which implies that the function $$V(k)$$ is decreasing and V(k)≤V(0),∀k∈Z+. We can verify that δ1‖z(k)‖2≤V(k)≤V(0)≤δ2‖φ¯‖2, (3.31) where δ1=mink∈{1,2,⋯,N}⁡λmin(P(k)),δ2=maxk∈{1,2,⋯,N}⁡λmax(P(k))+eα(d2+2){maxk∈{1,2,⋯,N}⁡[d1λmax(Q1(k))+d2λmax(Q2(k))]+d2λmax(Q)+12d12(d1+d2−1)λmax(Q)+2d12(d1+1)λmax(Z1)+2d22(d2+1)λmax(Z2)+2d122(d2+d1+1)λmax(Z3)}. (3.32) Hence from (3.31) it follows that ‖z(k)‖≤δ2δ1‖φ¯‖,∀k∈Z+. Returning to the variable $$x(k) = {e^{-\alpha k}}z(k)$$ we have ‖x(k)‖≤δ2δ1e−αk‖φ‖,∀k∈Z+. This completes the proof of the Theorem 3.2. □ Corollary 3.1 For given scalars $$\alpha > 0,$$$${d_1}$$ and $${d_2}$$ with $${d_2} > {d_1} > 0$$, the system (2.1) with $$u(k) = 0$$ is robustly exponentially stable with a convergence rate $$\alpha$$, if there exist matrices $$P > 0,$$$${Q_{ij}} > 0,$$$$j = 1,2,i = 1,2, \cdots v,$$ and matrices $${Z_1} > 0,{Z_2} > 0,{Z_3} > 0,$$$$Q > 0,$$$${N_1} > 0,$$ any matrices $${T_i},\;\,i = 1,2,3,$$ with appropriate dimensions satisfying the following LMIs: (Ψ1Ψ2∗Ψ3)<0,k=1,2,⋯,N,i=1,2,⋯,v, where Ψ1=(Π¯110Z1Z2−G∗−QT1T−T1TΠ^25(k)∗∗Π¯33−T2T+T30∗∗∗Π¯440∗∗∗∗Π¯55),Ψ2=(Π^16(k)0Π^18(k)00Π^27(k)Π^28(k)T1T000T2T000T3T0000),Ψ3=diag{−P,−P,−N1,−Z3},Π¯11=−P+Qi1+Qi2+(1+d12)Q−Z1−Z2+G,Π^16(k)=eα(1+eαd2)AiT(k)P,Π^18(k)=e0.5(−αd1+α)AiT(k)N1,Π^28(k)=0.5e0.5α(1+d1)AdiT(k)N1,Π^25(k)=0.5AdiT(k)N1,Π^27(k)=eαeαd2+e2αd2AdiT(k)P,Π¯33=T2+T2T−Qi1−Z1,G=d12Z1+d22Z2+d122Z3,Π¯44=−Z2−T3T−T3−Qi2,Π¯55=G−N1e−α(d2+1),d12=d2−d1. Remark 3.1 The problem of robust stability analysis of linear discrete time periodic systems was discussed in Souza & Coutinho (2014). But, the problem of exponential stability of linear discrete time periodic system was not involved. We know that the exponential stability implied the stability. In this article, we deal with the problems of robustly exponential stability and robustly exponential stabilization for uncertain linear discrete-time periodic systems with time-delay. Compared with Souza & Coutinho (2014), obtained results in our article have a greater range of applications. 4. Robust exponential stabilization This section deals with the problem of robustly exponential stabilization for the uncertain periodic time-delay system (2.1)–(2.4). Theorem 4.1 For a given scalar $$\alpha > 0,$$ the system (3.1) with $$u(k) = {W^T}(k){L^{ - 1}}(k)x(k)$$ is exponentially stable with a convergence rate $$\alpha$$, if there exist $$N$$-periodic matrices $$S(k) > 0,$$$$L(k) > 0,$$$$W(k)$$ satisfying the following LMIs: (−0.5L(k)0Λ130L(k)∗Λ22Λ23Λ230∗∗−L(k+1)00∗∗∗−L(k+1)0∗∗∗∗−1dS(k))<0,k=1,2,⋯,N, i=1,2,⋯,v, (4.1) where Λ13=eα(L(k)AiT(k)+W(k)BiT(k)),Λ22=−0.5L(k−d),Λ23=eα(d+1)L(k−d)AdT(k). Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (3.1) is reduced to z(k+1)=eα(A(k)+B(k)K(k))z(k)+eα(d+1)Ad(k)z(k−d). (4.2) Consider the Lyapunov–Krasovskii functional: V(k)=V1(k)+V2(k)+V3(k), where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−dk−1zT(i)Q(i)z(i),V3(k)=∑i=−d+1−1∑j=k+ik−1zT(j)R(j)z(j),P(k)=L−1(k),R(k)=S−1(k). Define $${\it{\Delta}} V(k) = V(k + 1) - V(k)$$. Then along the solution of (4.2), we have ΔV(k)≤(zT(k)zT(k−d))N^(k)(z(k)z(k−d)), (4.3) where N^(k)=(N^1(k)N^2(k)N^2T(k)N^3(k)),N^1(k)=e2α(A(k)+B(k)K(k))TP(k+1)(A(k)+B(k)K(k))−P(k)+dR(k)+Q(k),N^2(k)=eα(d+2)(A(k)+B(k)K(k))TP(k+1)Ad(k),N^3(k)=e2α(d+1)AdT(k)P(k+1)Ad(k)−Q(k−d). By Lemma 2.3, we have that $$\hat {\rm N}(k) < 0$$ is equivalent to: (N~1(k)0N~2(k)0∗−Q(k−d)eα(d+1)AdT(k)eα(d+1)AdT(k)∗∗−P−1(k+1)0∗∗∗−P−1(k+1))<0, (4.4) where $${\tilde {\rm N}_1}(k) = - P(k) + dR(k) + Q(k),$$$${\tilde {\rm N}_2}(k) = {e^\alpha }{(A(k) + B(k)K(k))^T}.$$ Let $${M_1}(k) = diag\{ {P^{ - 1}}(k),{P^{ - 1}}(k - d),I,I\},$$$$Q(k) = 0.5P(k).$$ Pre- and post-multiplying (4.4) by $${M_1}^T(k)$$ and $$M_1(k)$$, respectively, (4.4) can be cast as below: N⌢(k)=(N⌢1(k)0N⌢2(k)0∗−0.5P−1(k−d)N⌢3(k)N⌢3(k)∗∗−P−1(k+1)0∗∗∗−P−1(k+1))<0, (4.5) where N⌢1(k)=−0.5P−1(k)+P−1(k)(dR(k))P−1(k),N⌢2(k)=eαP−1(k)(A(k)+B(k)K(k))T,N⌢3(k)=eα(d+1)P−1(k−d)AdT(k). By Lemma 2.3, we have that $$\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over {\rm N}} (k) < 0$$ is equivalent to: (−0.5P−1(k))0N⌢2(k)0P−1(k)∗−0.5P−1(k−d)N⌢3(k)N⌢3(k)0∗∗−P−1(k+1)00∗∗∗−P−1(k+1)0∗∗∗∗−1d(R(k))−1)<0. (4.6) Let $$L(k) = {P^{-1}}(k),\quad W(k) = L(k){K^T}(k),\quad S(k) = {R^{-1}}(k).$$ From (4.1), we get $${\it{\Delta}} V(k) \le 0$$ which implies that the function $$V(k)$$ is decreasing and V(k)≤V(0)∀k∈Z+. Hence from (3.6) it follows that ‖z(k)‖≤λ~2λ~1‖φ¯‖∀k∈Z+, where λ~1=mink∈{1,2,⋯,N}⁡λmin(P(k)),λ~2=maxi∈{1,2,⋯,N}⁡{λmax(P(i))+dλmax(Q(i))+12d(d+1)λmax(R(i))}. Returning to the variable $$x(k) = {e^{-\alpha k}}z(k)$$ we have ‖x(k)‖≤λ~2λ~1e−αk‖φ‖,∀k∈Z+. This completes the proof of the Theorem 4.1. □ Theorem 4.2 For given scalars $$\alpha > 0,$$$$d_{1}$$ and $$d_{1}$$ with $${d_2} > {d_1} > 0,$$ the system (2.1) with $$u(k) = {W^T}(k){L^{ - 1}}(k)x(k)$$ is exponentially stable with a convergence rate $$\alpha$$, if there exist matrices $$U > 0,$$$${F_1} > 0,$$ and $$N$$ -periodic matrices $$L(k) > 0,$$$$W(k)$$ satisfying the following LMIs: (Π⌢11(k)00Π⌢14(k)0Π⌢16(k)L(k)∗−UΠ⌢23(k)0Π⌢25(k)Π⌢26(k)0∗∗Π⌢330000∗∗∗−L(k+1)000∗∗∗∗−L(k+1)00∗∗∗∗∗−F10∗∗∗∗∗∗−Π⌢77)<0,k=1,2,⋯,N,i=1,2,⋯,v, (4.7) where Π⌢11(k)=−0.5L(k),Π⌢14(k)=e2α+eα(d2+2)(L(k)AiT(k)+W(k)BiT(k)),Π⌢23(k)=0.5UAdiT(k),Π⌢16(k)=e0.5(−αd1+α)(L(k)AiT(k)+W(k)BiT(k)),Π⌢26(k)=0.5e0.5(αd1+α)UAdT(k),Π⌢25(k)=β1UAdT(k),Π¯33=−e−α(d2+1)F1T,Π⌢77=(1/(1+d12))U,β1=eα(d2+2)+e2α(d2+1). Proof. We introduce the new variable $$z(k) = {e^{\alpha k}}x(k)$$. The system (2.1) is reduced to z(k+1)=A¯(k)z(k)+A¯d(k)z(k−h(k)), (4.8) where $$\bar A(k) = {e^\alpha }(A(k) + B(k)K(k)),\quad {\bar A_d}(k) = {e^{\alpha (h(k) + 1)}}{A_d}(k).$$ Consider the Lyapunov–Krasovskii functional V(k)=V1(k)+V2(k)+V3(k), (4.9) where V1(k)=zT(k)P(k)z(k),V2(k)=∑i=k−d1k−1zT(i)Q1(i)z(i)+∑i=k−d2k−1zT(i)Q2(i)z(i)+∑i=k−h(k)k−1zT(i)Q3z(i)),V3(k)=∑i=−d2+1−d1∑j=k+ik−1zT(j)Q3z(j). Then along the solution of (4.8), we have ΔV(k)≤zT(k)[(e2α+eα(d2+2))A¯T(k)P(k+1)A¯(k)−P(k)]z(k)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))+zT(k)(Q1(k)+Q2(k)+Q3)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Q3z(k−d(k))+d12zT(k)Q3z(k). (4.10) From (4.8), we can obtain e−α(h(k)+1)z(k+1)−(e−αh(k)A¯(k)z(k)+Ad(k)z(k−h(k)))=0. (4.11) For any $${N_1}$$, it follows that zT(k+1)N1e−α(h(k)+1)z(k+1)−zT(k+1)N1Ad(k)z(k−h(k))+zT(k)A¯T(k)N1T[−eα(h(k))+αA¯(k)z(k)−eαAd(k)z(k−h(k))]=0. (4.12) From (4.10) and (4.12), we have ΔV(k)≤zT(k)[(e2α+eα(d2+2))A¯T(k)P(k+1)A¯(k)−P(k)]z(k)+d12zT(k)Q3z(k)+(eα(d2+2)+e2α(d2+1))zT(k−h(k))AdT(k)P(k+1)Ad(k)z(k−h(k))+zT(k)(Q1(k)+Q2(k)+Q3)z(k)−zT(k−d1)Q1(k−d1)z(k−d1)−zT(k−d2)Q2(k−d2)z(k−d2)−zT(k−h(k))Q3z(k−h(k))−zT(k+1)N1e−α(d2+1)z(k+1)+zT(k+1)N1Ad(k)z(k−h(k))+zT(k)A¯T(k)N1T[e−αd1+αA¯(k)z(k)+eαAd(k)z(k−h(k))]≤ξT(k)Ξ(k)ξ(k), (4.13) where ξ(k)=(zT(k)zT(k−h(k))zT(k+1))T,Ξ(k)=ΞT(k)=(Ξij(k)),i,j=1,2,3,Ξ11(k)=(e2α+eα(d2+2))A¯T(k)P(k+1)A¯(k)−P(k)+Q1(k)+Q2(k)+(1+d12)Q3+A¯T(k)N1Te−αd1+αA¯(k),Ξ12(k)=0.5eαA¯T(k)N1TAd(k),Π13(k)=0,Π23(k)=0.5AdT(k)N1T,Ξ22(k)=(eα(d2+2)+e2α(d2+1))AdT(k)P(k+1)Ad(k)−Q3,Π33=−N1e−α(d2+1). From Lemma 2.3, we get that $${\it{\Xi}} (k) < 0,$$ if (Π¯11(k)00Π¯14(k)0Π16(k)∗−Q3Π23(k)0β1AdT(k)Π26(k)∗∗Π¯33000∗∗∗−P−1(k+1)00∗∗∗∗−P−1(k+1)0∗∗∗∗∗−N1−T)<0, (4.14) where Π¯11(k)=−P(k)+Q1(k)+Q2(k)+(1+d12)Q3,β1=eα(d2+2)+e2α(d2+1),Π26(k)=0.5e0.5(αd1+α)AdT(k),Π¯14(k)=β2A¯T(k),β2=e2α+eα(d2+2),Π16(k)=e0.5(−αd1+α)A¯T(k),Π¯23(k)=0.5AdT(k)N1T,Π¯33=−N1e−α(d2+1). Let $${M_2}(k) = diag\{ {P^{ - 1}}(k),Q_3^{ - 1},N_1^{ - 1},I,I,I\}.$$ Pre- and post-multiplying (4.14) by $${M_2}^T(k)$$ and $${M_2}(k)$$, respectively, (4.14) can be cast as below: (Π^11(k)00Π^14(k)0Π^16(k)∗−Q3−1Π^23(k)0β1Q3−1AdT(k)Π^26(k)∗∗Π^33(k)000∗∗∗−P−1(k+1)00∗∗∗∗−P−1(k+1)0∗∗∗∗∗−N1−T)<0, (4.15) where Π^11(k)=−P−1(k)+P−1(k)[Q1(k)+Q2(k)+(1+d12)Q3]P−1(k),Π^14(k)=β2P−1(k)A¯T(k),Π^16(k)=e0.5(−αd1+α)P−1(k)A¯T(k),Π^23(k)=0.5Q3−1AdT(k),Π^26(k)=0.5e0.5(αd1+α)Q3−1AdT(k),Π^33=−e−α(d2+1)N1−T. Set $${Q_1}(k) = {Q_2}(k) = \frac{1}{4}P(k).$$ From Lemma 2.3, we get that (4.15) is equality to (Φ1Φ2∗Φ3)<0, (4.16) where Φ1=(Π~11(k)00Π^14(k)∗−Q3−1Π^23(k)0∗∗Π^330∗∗∗−P−1(k+1)),Φ2=(0Π^16(k)P−1(k)β1Q3−1AdT(k)Π^26(k)0000000),Φ3=diag{−P−1(k+1)−N1−TΠ^77},Π~11(k)=−0.5P−1(k),Π^14(k)=β2P−1(k)A¯T(k),Π^26(k)=0.5e0.5(αd1+α)Q3−1AdT(k),Π^16(k)=e0.5(−αd1+α)P−1(k)(A(k)+B(k)K(k))T,Π^23(k)=0.5Q3−1AdT(k),Π¯33=−e−α(d2+1)N1−T,Π^77=−1(1+d12)Q3−1.  Let $$L(k) = {P^{ - 1}}(k),\quad W(k) = L(k){K^T}(k),\quad U = Q_3^{-1},\quad {F_1} = N_1^{ - 1}.$$ (4.7) implies that (4.16) holds. Hence, we get $${\it{\Delta}} V(k) \le 0.$$ The rest proofs are similar to the proof of Theorem 3.2, which are omitted here. □ Corollary 4.1 For given scalars $$\alpha > 0,$$$$d_{1}$$ and $$d_{1}$$ with $${d_2} > {d_1} > 0,$$ the system (2.1) with $$u(k) = {W^T}{L^{ - 1}}x(k)$$ is exponentially stable with a convergence rate $$\alpha$$, if there exist matrices $$U > 0,$$$${F_1} > 0,\;$$$$L > 0,$$ and $$W$$ satisfying the following LMIs: (Π⌢1100Π⌢14(k)0Π⌢16(k)L∗−UΠ⌢23(k)0Π⌢25(k)Π⌢26(k)0∗∗Π⌢330000∗∗∗−L000∗∗∗∗−L00∗∗∗∗∗−F10∗∗∗∗∗∗−Π⌢77)<0,k=1,2,⋯,N,i=1,2,⋯,v, (4.17) where Π⌢11(k)=−0.5L,Π⌢23(k)=0.5UAdiT(k),Π⌢14(k)=e2α+eα(d2+2)(LAiT(k)+WBiT(k)),Π⌢16(k)=e0.5(−αd1+α)(LAiT(k)+WBiT(k)),Π⌢26(k)=0.5e0.5(αd1+α)UAdT(k),Π⌢25(k)=β1UAdT(k),Π¯33=−e−α(d2+1)F1T,Π⌢77=(1/(1+d12))U,β1=eα(d2+2)+e2α(d2+1). Remark 4.1 The problems of stability and stabilization for discrete-time periodic linear systems are concerned in Zhou et al. (2011). But, the robust stability and stabilization for discrete-time periodic linear systems did not considered. Compared with Zhou et al. (2011), obtained results in our article have a greater range of applications. 5. Numerical examples In this section numerical examples are provided to show the high performance of the proposed approach. Example 5.1 Consider the uncertain 3-periodic system (3.1) with the following parameters: A1(1)=(0.3500−1.9),A1(2)=(−0.200−0.7),A1(3)=(0.100−0.4),Ad1(1)=(−0.10.2−0.1−0.1),Ad1(2)=(−0.10.0−0.1−0.1),Ad1(3)=(−0.10.3−0.1−0.7),A2(1)=(0.480.0100.68),  A2(2)=(−0.180.10−0.78),A2(3)=(0.28000.38),Ad2(1)=(−0.140.18−0.11−0.11),  Ad2(2)=(−0.120.0−0.14−0.14),  Ad2(3)=(−0.320.28−0.12−0.82),Bi(k)=(01)T,i=1,2,k=1,2,3,d=2. Applying Theorem 4.1 with $$\alpha = 0.5,$$ and using the Matlab LMI control toolbox, we solve (4.1) and obtain a set of feasible solutions as follows: W(1)=(0.5861−0.0597),W(2)=(0.39050.0452),  S(1)=108(3.50980.00000.00003.5098),L(1)=(0.35090.03420.03420.0723), L(2)=(1.30020.26820.26820.8442),L(3)=(0.6950−0.1389−0.13890.0782),S(2)=108(3.50980.00000.00003.5098),S(3)=108(3.51250.00200.00203.5115),  W(3)=(0.1843−0.3443). We obtain the following state feedback gains: K(1)=(1.8354   −1.6924),K(2)=(0.3096   −0.0448),K(3)=(−0.9519   −6.0907). Therefore, according to Theorem 4.1, the system (3.1) with the above parameters and $$u(k) = {W^T}(k) \times{L^{ - 1}}(k)x(k)$$ is exponentially stable, which is further verified by the simulation results given in Fig. 1. Fig. 1. View largeDownload slide State trajectories of the closed-loop system in Example 5.1. Fig. 1. View largeDownload slide State trajectories of the closed-loop system in Example 5.1. Figure 1 represents the trajectories of $${x_1}(k)$$ and $${x_2}(k)$$ of the closed-loop system. It is obvious that the considered system in Example 5.1 is exponentially stabilizable through the obtained feedback controller gain. Example 5.2 Consider the uncertain 2-periodic time-delay system (2.1) with the following parameters: A1(1)=(−0.1−0.0100.01−0.15000−0.1),A1(2)=(−0.1270.01−0.010.01−0.14000−0.2),Ad1(1)=(−0.1−0.0100−10.100.01−0.3),Ad1(2)=(−0.1000.01−0.1000.02−0.36),A2(1)=(−0.120.10−0.01−1000−1.1),A2(2)=(−0.20.1−0.1−0.01−0.2000−0.18),Ad2(1)=(−0.160.0100.1−0.100−0.01−0.27),Ad2(2)=(−0.200.01−0.01−0.30−0.020−0.18),h(k)=2+sin⁡(kπ2),Bi(1)=(0.01−0.200.03−0.10.02),Bi(2)=(0.020.10.20.1−0.150.03),i=1,2,d1=1,d2=3. Applying Theorem 4.2 with $$\alpha = 0.5,$$ and using the Matlab LMI control toolbox, we solve (4.7) and obtain a set of feasible solutions as follows: L(1)=(1.36990.1009−0.02200.10090.66720.0186−0.02200.01860.8388),L(2)=(1.14750.22280.05720.22280.67840.01940.05720.01941.4694),F1=108(1.3057−0.0000−0.0000−0.00001.30730.0000−0.00000.00001.3057),U=(0.8381−0.0631−0.1710−0.06310.31570.0950−0.17100.09500.9170),W(1)=(0.71421.38580.00290.8250−1.72050.7779),W(2)=(0.9485−2.5252−1.4823−1.80280.8355−0.9316). We obtain the following state feedback gains: K(1)=(0.4896−0.0129−2.03810.94791.06730.9286),K(2)=(1.3098−2.63070.5524−1.7741−2.0594−0.5378). Figure 2 shows the simulation results for states $${x_1}(k),$$$${x_2}(k)$$ and $${x_3}(k)$$ of the closed-loop system. Fig. 2. View largeDownload slide State trajectories of the closed-loop system in Example 5.2. Fig. 2. View largeDownload slide State trajectories of the closed-loop system in Example 5.2. The simulation results reveal that the considered system is exponentially stabilizable through the obtained feedback controller gains. Simulation results demonstrate that our proposed design is very effective. 6. Conclusion In this article, we have investigated the problems of robustly exponential stability and exponential stabilization for uncertain linear discrete-time periodic systems with time-delay in the state variables and polytopic-type parameter uncertainty. Using the uncertainty-dependent Lyapunov–Krasovskii functionals, we have provided some novel sufficient conditions in terms of LMIs. These conditions guarantee that the discrete-time periodic system with time-delay is robustly exponentially stable. Furthermore, employing a static periodic state feedback control law, we have given robust exponential stabilization criteria for uncertain linear discrete-time periodic systems. Besides, we have presented the calculation method of the periodic control gain matrix. 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### Journal

IMA Journal of Mathematical Control and InformationOxford University Press

Published: Mar 12, 2017

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