Robust exponential stability of uncertain switching time-delay systems with nonlinear perturbations

ROBUST EXPONENTIAL STABILITY OF UNCERTAIN SWITCHING TIME - DELAY SYSTEMS WITH NONLINEAR PERTURBATIONS Le Van Hien Hanoi National University of Education Abstract. In this paper, a class of uncertain switching time-delay systems with nonlinear perturbations is considered. The system parameter uncertainties are time-varying and unknown with norm-bounded. The delay in the system states is also time-varying. By using an improved Lyapunov-Krasovskii functional, a state dependent switching rule for robust exponential stability is designed in terms of solution of Lyapunov-type equations and growth bound of perturbations. The approach allows for computation of the bounds that characterize the exponential stability rate of the solution. Numerical examples are given to illustrate the results. KEY WORDS: Switching system, time-delay system, uncertainty, nonlinear perturba- tion, exponential stability, Lyapunov equation. 1 Introduction Switching systems belong to an important class of hybrid systems, which are described by a family of differential equations together with specified rules to switch between them. A switching system can be represented by a differential equation of the form x˙(t) = fσ(t)(t, x), t ≥ 0, where {fσ(., .) : σ ∈ I} is a family of functions parameterized by some index set I, which is typically a finite set, and σ(.), which depends on system state at each time, is the switching rule/signal determining a switching sequence for the given system. Switching systems arise in many practical processes that cannot be described by ex- clusively continuous or exclusively discrete models, such as manufacturing, communication networks, automotive engineering control, chemical processes (e.g. see [2], [4], [9], [10], [11] and the references therein). Although many important results have been obtained for linear switched systems, there are few results concerning the stability of switched linear systems with time delay. Under the assumption on commutative system matrices, it was shown in [5] that when all subsystems are asymptotically stable, the switching system is asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear systems with time delay has been studied in [15], but the result was limited to symmetric systems. In [7], [8], [13], delay-dependent asymptotic stability conditions are extended to discrete-time linear switching systems with time delay. There are some other results concerning asymptotic stability for switching linear systems with time delay, but most of them provide conditions for asymptotic stability for arbitrary switching signal without focusing on exponential stability. 1 The exponential stability problem was considered in [16] for switching linear systems with impulsive effects by using the matrix measure concept, and in [14] for nonholonomic chained systems with strongly nonlinear input/state driven disturbances and drifts. On the other hand, it is worth noting that the existing stability conditions for time-delay systems must be solved upon a grid of the parameter space, which results in testing a nonlinear Riccati-type equation or a finite number of LMIs. In this case, the results using finite gridding points are unreliable and the numerical complexity of the tests grows rapidly. Therefore, finding new conditions for the exponential stability of uncertain linear switching time-delay systems is of interest. In this paper, the problem of exponential stability for a class of uncertain hybrid time varying-delay systems with nonlinear perturbations is addressed. Our objective is to de- rive delay-dependent conditions for the robust exponential stability by using an improved Lyapunov-Krasovskii functional. The conditions will be presented in terms of the solution of Lyapunov-type equations and growth bounds of perturbations. Comparing with the previous results, a simple geometric design is employed to find the switching rule and our approach allows to compute simultaneously the two bounds that characterize the exponen- tial stability rate of the solution. The results obtained can be considered as an extension of existing results for linear switching time-delay systems. The paper is organized as follows: Section 2 presents notations, definitions and a technical lemma required for the proof of the main results. Sufficient conditions for the robust exponential stability and illustrative examples are presented in Section 3. The paper ends with a conclusion followed by cited references. 2 Problem formulation The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product 〈., .〉 and the vector norm ‖.‖; Rn×r denotes the space of all matrices of (n×r)− dimension. AT denotes the transpose of A; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax(A) = max{Reλ : λ ∈ λ(A)}; λmin(A) = min{Reλ : λ ∈ λ(A)}; A matrix A is semi-positive definite (A ≥ 0) if 〈Ax, x〉 ≥ 0, for all x ∈ Rn;A is positive definite (A > 0) if 〈Ax, x〉 > 0 for all x 6= 0; A ≥ B means A−B ≥ 0. Consider a class of uncertain hybrid time-delay systems of the form{ x˙(t) = Aσ(t)x(t) +Dσ(t)(t)x(t− h(t)) + fσ(t)(t, x(t), x(t − h(t))), t ∈ R +, x(t) = φ(t), t ∈ [−h, 0], (2.1) where x(t) ∈ Rn is the system state; σ(.) : R+ → I := {1, 2, . . . ,m} is the switching func- tion, which will be designed. Aσ,Dσ ∈ {[Ai,Di], i = 1, 2, . . . ,m}, Ai,Di are given matrices and φ(t) ∈ C([−h, 0], Rn) is the initial function with the norm ‖φ‖ = sups∈[−h,0] ‖φ(s)‖; the parameter perturbations fi(t, y, z) are unknown and assumed to satisfy the following condition: ‖fi(t, y, z)‖ ≤ βi‖y‖+ γi‖z‖, ∀t ∈ R +, y, z ∈ Rn, where βi, γi, i = 1, 2, . . . ,m are given non-negative constants. The time-varying delay func- tion h(t) is assumed to satisfy the following well-studied condition 0 ≤ h(t) ≤ h, h˙(t) ≤ µ < 1, 2 where h and µ are given non-negative constants. This assumption means that the time delay may change from time to time but the rate of changing is bounded, i.e. the delay cannot increase as fast as the time itself. Definition 2.1. Given α > 0. The system (2.1) is robustly α−exponentially stable if there exists a switching function σ(.) and positive number N such that any solution x(t, φ) of the system satisfies ‖x(t, φ)‖ ≤ Ne−αt‖φ‖, ∀t ∈ R+, (2.2) for all the admissible uncertainties. Definition 2.2. (see [12]) The system of matrices {Li}, i = 1, 2, . . . ,m, is said to be strictly complete if for any x ∈ Rn\{0} there is i ∈ {1, 2, . . . ,m} such that xTLix < 0. Let us define Ωi = {x ∈ R n : xTLix < 0}, i = 1, 2, . . . ,m. It's easy to show that the system {Li}, i = 1, 2, . . . ,m, is strictly complete if and only if m⋃ i=1 Ωi = R n\{0}. (2.3) Remark 2.3. A sufficient condition for the strict completeness of the system of ma- trices {Li} is that there exists λi ≥ 0, i = 1, 2, . . . ,m such that ∑m i=1 λi > 0 and m∑ i=1 λiLi < 0. As shown in [12], if m = 2 then the above condition is also necessary for the strict completeness. Next, we introduce the following well known lemma, which will be used in the proof of our results. Lemma 2.4.(see [6]) For any x, y ∈ Rn and positive definite matrix X one has 2xT y ≤ xTX−1x+ yTXy. 3 Main results In the sequel, for the sake of brevity, we will denote σ for the switching signal σ(.). For given non-negative numbers α, h, µ and symmetric positive definite matrices P,R, S we set τ = (1− µ)−1, η = (2α+ e2αh), β = max 1≤i≤m βi, γ = max 1≤i≤m γi; Li = A T i P + PAi + ηP + τQ+ hR+ S;Q = m∑ i=1 DTi PDi (3.1) Theorem 3.1. The system (2.1) is robustly α-exponentially stable if there exists symmetric positive definite matrices P,R, S and a positive scalar ε such that the following conditions hold 3 i) The system of matrices {Li}, i = 1, 2, . . . ,m is strictly complete. ii) β2 + γ2 ≤ λmin [ εS − ε2(1 + τe2αh)PP ] . Moreover, the solution x(t, φ) of the system satisfies ‖x(t, φ)‖ ≤ √ N2 N1 e−αt‖φ‖, t ∈ R+, where N1 = λmin(P ); N2 = λmax(P ) + h[τ m∑ i=1 λmax(D T i PDi) + λmin(S − ε(1 + τe 2αh)PP ) + 1 2 hλmax(R)] Proof. Consider the following Lyapunov-Krasovskii functional V (xt) = V1(x(t)) + V2(xt) + V3(xt) + V4(xt) where xt(s) = x(t+ s), s ∈ [−h, 0] and V1(x(t)) = x T (t)Px(t), (3.2) V2(xt) = τ t∫ t−h(t) e2α(s−t)xT (s)Qx(s)ds, (3.3) V3(xt) = ε −1γ2 t∫ t−h(t) e2α(s−t)‖x(s)‖2ds, (3.4) V4(xt) = 0∫ −h t∫ t+s e2α(s−t)xT (θ)Rx(θ)dθds. (3.5) From (3.2) - (3.5) and ii) of Theorem 3.1, it's easy to verify that N1‖x(t)‖ 2 ≤ V (xt) ≤ N2‖xt‖ 2, t ≥ 0. (3.6) Taking derivative of V1(x(t)) = x T (t)Px(t) along trajectories of any subsystem ith we have V˙1(x(t)) = x T (t)[ATi P + PAi]x(t) + 2x T (t)PDix(t− h(t)) + 2xT (t)Pfi(t, x(t), x(t− h(t))). Applying Lemma 2.4 gives 2xT (t)PDix(t− h(t)) ≤ e 2αhxT (t)Px(t) + e−2αhxT (t− h(t))DTi PDix(t− h(t)) ≤ e2αhxT (t)Px(t) + e−2αhxT (t− h(t))Qx(t − h(t)). (3.7) 2xT (t)Pfi(t, x(t),x(t− h(t))) ≤ 2‖x T (t)P‖(βi‖x(t)‖ + γi‖x(t− h(t))‖) ≤ εxT (t)PPx(t) + ε−1β2i ‖x(t)‖ 2 + ετe2αhxT (t)PPx(t) + ε−1(1− µ)e−2αhγ2i ‖x(t− h(t))‖ 2. (3.8) 4 Next, taking derivative of V2(xt) and V3(xt), respectively, along trajectories of subsys- tem ith yields V˙2(xt) =− 2αV2(xt) + τx T (t)Qx(t)− τ(1− h˙(t))xT (t− h(t))e−2βh(t)Qx(t− h(t)) ≤− 2αV2(xt) + τx T (t)Qx(t)− e−2αhxT (t− h(t))Qx(t− h(t)), (3.9) V˙3(xt) =− 2αV3(xt) + ε −1γ2‖x(t)‖2 − ε−1γ2(1− h˙(t))e−2αh(t)‖x(t− h(t))‖2 ≤− 2αV3(xt) + ε −1γ2‖x(t)‖2 − e−1γ2(1− µ)e−2αh‖x(t− h(t))‖2. (3.10) Finally, taking derivative of V4(xt) we get V˙4(xt) =− 2αV4(xt) + hx T (t)Rx(t)− 0∫ −h e2αsxT (t+ s)Rx(t+ s)ds ≤− 2αV4(xt) + hx T (t)Rx(t). (3.11) From (3.1), (3.6) - (3.11) we get V˙ (xt) + 2αV (xt) ≤ x T (t)[ATi P + PAi + ηP + τQ+ hR]x(t) + ε(1 + τe2αh)xT (t)PPx(t) + β2 + γ2 ε ‖x(t)‖2 = xT (t)Lix(t)− 1 ε xT (t) [ εS − ε2(1 + τe2αh)PP − (β2 + γ2)I ] x(t). (3.12) From condition ii) of the theorem 3.1 we have xT (t) [ εS − ε2(1 + τe2αh)PP ] x(t)− (β2 + γ2)‖x(t)‖2 ≥ 0. Therefore, V˙ (xt) + 2αV (xt) ≤ x T (t)Lix(t). (3.13) Let us set Ωi = {x ∈ R n : xTLix < 0}, i = 1, 2, . . . ,m. Then by the strict completeness of the system of matrices {Li}, and from (2.3) it follows that m⋃ i=1 Ωi = R n\{0}. Defining the sets Ω˜1 = Ω1, Ω˜i = Ωi\ i−1⋃ j=1 Ω˜j, i = 2, 3, . . . ,m. we see that m⋃ i=1 Ω˜i = R n\{0}, Ω˜i ∩ Ω˜j = ∅, i 6= j. Therefore, for any x(t) ∈ Rn, t ≥ 0, there exists i ∈ {1, 2, . . . ,m} such that x(t) ∈ Ω˜i. By choosing switching rule as σ(x(t)) = i whenever x(t) ∈ Ω˜i, then from (3.13) we have V˙ (xt) + 2αV (xt) ≤ x T (t)Lix(t) ≤ 0, t ≥ 0, i = 1, 2, . . . ,m. 5 This implies that V (xt) ≤ V (φ)e −2αt, t ≥ 0. Taking (3.2) into account, we get N1‖x(t, φ)‖ 2 ≤ V (xt) ≤ V (φ)e −2αt ≤ N2e −2αt‖φ‖2, t ≥ 0, and then ‖x(t, φ)‖ ≤ √ N2 N1 e−αt‖φ‖, t ≥ 0, which concludes the proof of the theorem 3.1. Remark 3.2. Note that by Remark 2.2, the system {Li} is strictly complete if there exists λi ≥ 0, i = 1, 2, . . . ,m, ∑m i=1 λi > 0 such that m∑ i=1 λiLi < 0. (3.14) In this case, the switching rule can be chosen as σ(t) = argmin{xT (t)Lix(t)}, t ≥ 0. Indeed, as shown in the proof of Theorem 3.1, we have arrived at the estimation V˙ (xt) + 2αV (xt) ≤ x T (t)Lix(t), t ≥ 0. Since λi ≥ 0 and λ = ∑m i=1 λi > 0, so min i=1,2,...,m xT (t)Lix(t) ≤ λ −1 m∑ i=1 λix T (t)Lix(t). By choosing switching rule as σ(t) = argmin{xT (t)Lix(t)}, t ≥ 0, we have V˙ (xt) + 2αV (xt) ≤ x T (t)Lix(t) ≤ λ −1 m∑ i=1 λix T (t)Lix(t) ≤ 0. This leads to ‖x(t, φ)‖ ≤ √ N2 N1 e−αt‖φ‖, t ≥ 0, as desired. The following procedure can be applied to construct the switching rule. Step 1. Define the symmetric positive definite matrices P,Q,R, S (i.e. the solution of the matrix inequality (3.14)) such that the system {Li} is strictly complete. Step 2. Construct the sets Ωi, and then Ω˜i. Step 3. The switching rule is chosen as σ(x(t)) = i, whenever x(t) ∈ Ω˜i. Example 3.3. Consider the system (2.1), where m = 2, h(t) = 0.5sin2t and [A1,D1] = [( −41 1 −1 1 ) , ( 1 −1 1 −1 )] , [A2,D2] = [( 1 −1 1 −35 ) , ( 1 −1 3 −4 )] , 6 Note that, both matrices A1 and A2 are unstable. In this case, we have h = 0.5, µ = 0.5, τ = 2. For α = 0.5, we verify that the symmetric positive definite matrices P = ( 0.1402 −0.1445 −0.1445 0.2730 ) , R = ( 1.0416 0.0168 0.0168 1.0920 ) , S = I = ( 1 0 0 1 ) be solution of (3.14) and M = 0.5L1 + 0.5L2 < −0.04I, where L1 = ( −5.6592 0.2671 0.2671 9.3802 ) , L2 = ( 5.5418 −0.3343 −0.3343 −9.6992 ) . Therefore, the system {L1, L2} is strictly complete. The sets Ω1,Ω2 are defined as Ω1 = {(x, y) ∈ R 2 : −5.6592x2 + 0.5342xy + 9.3802y2 < 0}, Ω2 = {(x, y) ∈ R 2 : 5.5418x2 − 0.6686xy − 9.6992y2 < 0}, which can be represented by Figure 1. It can be seen that Ω1 ∪ Ω2 = R 2\{0}. Therefore, the switching regions are given as Ω˜1 = {(x, y) ∈ R 2 : −5.6592x2 + 0.5342xy + 9.3802y2 < 0}; Ω˜2 = {(x, y) ∈ R 2 : −5.6592x2 + 0.5342xy + 9.3802y2 ≥ 0, (x, y) 6= (0, 0)}. We have Ω˜1 ∪ Ω˜2 = R 2\{0}, Ω˜1 ∩ Ω˜2 = ∅. Choose ε = 0.89 then by Theorem 3.1, the system (2.1) is 0.5−exponentially stable with the switching rule σ(t) = { 1 if x(t) ∈ Ω˜1, 2 if x(t) ∈ Ω˜2 and the grown bounds of the perturbations satisfy β2 + γ2 ≤ 0.4349 Moreover, the solution of the system satisfies ‖x(t, φ)‖ ≤ 11.3e−0.5t‖φ‖, ∀t ∈ R+. Figure 1. 7 For the case when m = 1 (without switching) and α = 0, β = 0,D = 0, Theorem 3.1 gives a delay-dependent criteria for the robust stability of linear systems with delayed perturbations considered in [1]. Consider the linear system with delayed perturbations of the form [1] x˙(t) = Ax(t) + f(t, x(t− h(t))), t ∈ R+, (3.15) where ‖f(t, x(t− h(t)))‖ ≤ γ‖x(t− h(t))‖. Note that, the stability conditions in Theorem 1 in [1] is independent of the size of delay. If R is small, Theorem 1 in [1] can be derived from the following corollary. Corollary 3.4. The system (3.15) is robust asymptotically stable if there exist positive definite matrices P,R, S, a positive scalar ε such that the following conditions hold i) ATP + PA+ hR+ S < 0; ii) γ2 ≤ λmin [ εS − ε2 1− µ PP ] . Proof. Consider the following Lyapunov-Krasovski functional V (xt) =x T (t)Px(t) + ε−1γ2 t∫ t−h(t) ‖x(s)‖2ds + 0∫ −h t∫ t+s e2α(s−t)xT (θ)Rx(θ)dθds. By the same argument used in the proof of Theorem 3.1 we get V˙ (xt) ≤x T (t)[ATP + PA+ hR]x(t) + ε 1− µ xT (t)PPx(t) + ε−1γ2‖x(t)‖2 ≤xT (t)[ATP + PA+ hR+ S]x(t)− 1 ε xT (t) [ εS − ε2 1− µ PP − γ2I ] x(t) ≤xT (t)[ATP + PA+ hR+ S]x(t) < 0. Therefore, the system (3.15) is robust asymptotically stable [3, Theorem 2.1, p. 132], which conclude the proof. 4 Conclusion This paper has proposed a switching design for robust exponential stability of un- certain linear switching time-delay systems with nonlinear perturbations. 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