Global exponential stability of switched neutral systems with time-varying delays

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 3-11 This paper is available online at GLOBAL EXPONENTIAL STABILITY OF SWITCHED NEUTRAL SYSTEMS WITH TIME-VARYING DELAYS Le Van Hien Faculty of Mathematics, Hanoi National University of Education Abstract. This paper deals with the problem of global exponential stability for a class of switched neutral systems with time-varying delays. Based on improved Lyapunov-Krasovskii functional method combined with the Leibniz-Newton formula, linear matrix inequalities (LMIs) conditions are proposed for the exponential stability of the considered systems under arbitrary switching signal. A numerical example showing the effectiveness of our conditions is given. Keywords: Switched systems, time-varying delay, exponential stability, linear matrix inequalities. 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. Switching systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models, such as manufacturing, communication networks, automotive engineering control and chemical processes [7, 8, 10]. Recently, the problem of stability analysis and control design for switched systems as well as the stability problem of uncertain linear time-delay systems and applications to control theory has attracted a lot of attention (see [1-7] and the references therein). The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functionals and linear matrix inequalities (LMIs) for constructing suitable Lyapunov-Krasovskii functionals. Although some important results have been obtained for linear switched systems, there are few results concerning the exponential stability of switched neutral systems with time-varying delays. In [12], the problem of globally quadratic stabilization for a class of un-delayed switched cascade systems composed of two subsystems was considered. Received September 5, 2012. Accepted October 1, 2012. Mathematics Subject Classification: 34D20; 37C75; 93D20. Contact Le Van Hien, e-mail address: hienlv@hnue.edu.vn 3 Le Van Hien Based on a single Lyapunov function, the feedback control law and the switching law are designed respectively when the first part is stabilized under some switching law and when both parts can be stabilized under some switching laws. Exponential stability of linear uncertain switched neutral systems with interval state delay is consider in [6]. By representing the considered switched system as a convex combination of all subsystems, LMIs conditions are proposed for the exponential stability of the system under arbitrary switching signal with restriction on the neutral matrix and exponential convergence rate. In this paper, we study the problem of exponential stability for a class of switched neutral systems with time-varying delays. The novel feature of the results obtained in this paper is twofold. First, the system considered in this paper is switched neutral systems with non-differentiable state delay. This allows the time-delay to be a fast time-varying function. Second, by employing an improved Lyapunov-Krasovskii functional, delay-dependent sufficient conditions for the exponential stability of the system under arbitrary switching signal are obtained in terms of LMI conditions, which can be solved effectively by various computation tools. 2. Preliminaries The following notations will be used throughout this paper. R+ denotes the set of all nonnegative real numbers; Rn denotes the n−dimensional Euclidean space with the norm ‖.‖ and scalar product xTy of two vectors x, y; λmax(A)(λmin(A), resp.) denotes the maximal (the minimal, resp.) number of the real part of eigenvalues of A; AT denotes the transpose of the matrix A; Rn×m denotes the set of all (n×m)-matrices; Q ≥ 0 (Q > 0, resp.) meansQ is semi-positive definite (positive definite, resp.), A ≥ B means A−B ≥ 0; C1([a, b], Rn) denotes the set of all continuously differentiable functions on [a, b] with the norm ‖φ‖ = supa≤t≤b √ ‖φ(t)‖2 + ‖φ˙(t)‖2. Consider a linear switched neutral system with time-varying delays of the form (Σα) : { x˙(t)− Cαx˙(t− τ(t)) = Aαx(t) +Dαx(t− h(t)), t ≥ 0, x(t) = φ(t), t ∈ [−h¯, 0], (2.1) where x(t) ∈ Rn is the state; Ai, Ci, Di ∈ Rn×n are given real matrices; φ ∈ C1([−h¯, 0], Rn) is the initial function and α ∈ {1, 2, . . . , N} is a piecewise constant switching signal depending on the system state. A switching rule is a rule which determines a switching sequence for a given switching system. Moreover, α(x) = i implies that the system realization is chosen as i = 1, 2, . . . , N . It is seen that the system (Σα) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(t) hits predefined boundaries, i.e., the switching rule is dependent on the system trajectory. Time-varying delay functions h(t), τ(t) satisfy 0 ≤ h(t) ≤ h, 0 ≤ τ(t) ≤ τ, τ˙(t) ≤ µ < 1 (2.2) and h¯ = max{h, τ}. 4 Global exponential stability of switched neutral systems with time-varying delays Definition 2.1. For given β > 0, system (2.1) is said to be β-exponentially stable under an arbitrary switching signal if there exists a number γ ≥ 1 such that every solution x(t, φ) of the system satisfies the following inequality ‖x(t, φ)‖ ≤ γ‖φ‖e−βt, ∀t ≥ 0. The following well-known proposition will be used in the proof of our results. Proposition 2.1. For any symmetric positive definite matrixW , scalar ν > 0 and vector function ω : [0, ν] −→ Rn such that the concerned integrals are well defined, then[∫ ν 0 w(s)ds ]T W [∫ ν 0 w(s)ds ] ≤ ν ∫ ν 0 wT(s)Ww(s)ds. 3. Main results The exponential stability of system (2.1) under arbitrary switching signal is presented in the following theorem. Theorem 3.1. For given β > 0, system (2.1) is β-exponentially stable if there exist matrices U1i, U2i, U3i, U4i, U5i, i = 1, 2, . . . , N and symmetric positive definite matrices P,Q,R, T,W, such that the following linear matrix inequalities hold for all i = 1, 2, . . . , N Φ(i) =  Φ (i) 11 Φ (i) 12 A T i U3i +W A T i U4i − UT1i PCi + UT1iCi + ATi U5i ∗ Φ(i)22 DTi U3i DTi U4i − UT2i UT2iCi +DTi U5i ∗ ∗ Φ(i)33 −UT3i UT3iCi ∗ ∗ ∗ Φ(i)44 UT4iCi − U5i ∗ ∗ ∗ ∗ Φ(i)55  ≤ 0, (3.1) where ∗ represents the symmetric matrix form and Φ (i) 11 = A T i (P + U1i) + (P + U T 1i)Ai + 2βP +Q +R− T −W ; Φ (i) 12 = PDi + U T 1iDi + A T i U2i + T ; Φ (i) 22 = −T + UT2iDi +DTi U2i; Φ (i) 33 = −(1− µ)e−2βτQ−W ; Φ (i) 44 = R + h 2e2βhT + τ 2e2βτ 1− µ W − U4i − U T 4i; Φ (i) 55 = −(1− µ)e−2βτR + UT5iCi + CTi U5i. Moreover, every solution x(t, φ) of the system satisfies the following inequality ‖x(t, φ)‖ ≤ √ λ2 λ1 ‖φ‖e−βt, t ≥ 0, 5 Le Van Hien where, λ1 = λmin(P ), and λ2 =λmax(P ) + τ ( λmax(Q) + λmax(R) ) + 1 2 h3e2βhλmax(T ) + τ 3 2(1− µ)e 2βτλmax(W ). Proof. Consider the following Lyapunov-Krasovskii functional V (t, xt) = 5∑ k=1 Vk, (3.2) where, V1 = x T(t)Px(t), V2 = ∫ t t−τ(t) e2β(s−t)xT(s)Qx(s)ds, V3 = ∫ t t−τ(t) e2β(s−t)x˙T(s)Rx˙(s)ds, V4 = h ∫ t t−h ∫ t s e2β(θ−t+h)x˙T(θ)T x˙(θ)dθds, V5 = τ 1− µ ∫ t t−τ(t) ∫ t s e2β(θ+τ−t)x˙T(θ)Wx˙(θ)dθds. It is easy to verify from (3.2) that λ1‖x(t)‖2 ≤ V (t, xt) ≤ λ2‖xt‖2, t ∈ R+. (3.3) Taking derivative of Vk, k = 1, 2, . . . , 5, along trajectories of i th subsystem of (2.1), we have V˙1 = x T(t)[PAi + A T i P ]x(t) + 2x T(t)P [Dix(t− h(t)) + Cix˙(t− τ(t))]; V˙2 = x T(t)Qx(t)− (1− τ˙ (t))e−2βτ(t)xT(t− τ(t))Qx(t − τ(t))− 2βV2 ≤ xT(t)Qx(t)− (1− µ)e−2βτxT(t− τ(t))Qx(t− τ(t))− 2βV2; V˙3 = x˙ T(t)Rx˙(t)− (1− τ˙(t))e−2βτ(t)x˙T(t− τ(t))Rx˙(t− τ(t))− 2βV3 ≤ x˙T(t)Rx˙(t)− (1− µ)e−2βτ x˙T(t− τ(t))Rx˙(t− τ(t))− 2βV3; V˙4 = h 2e2βhx˙T(t)T x˙(t)− h ∫ t t−h e2β(s−t+h)x˙T(s)T x˙(s)ds− 2βV4 ≤ h2e2βhx˙T(t)T x˙(t)− h ∫ t t−h x˙T(s)T x˙(s)ds− 2βV4; V˙5 ≤ τ 2 1− µe 2βτ x˙T(t)Wx˙(t) − τ 1− µ(1− τ˙(t)) ∫ t t−τ(t) e2β(s+τ−t)x˙T(s)Wx˙(s)ds− 2βV5 ≤ τ 2 1− µe 2βτ x˙T(t)Wx˙(t)− τ ∫ t t−τ(t) x˙T(s)Wx˙(s)ds− 2βV5. (3.4) 6 Global exponential stability of switched neutral systems with time-varying delays Next, by applying proposition 2.1 and Leibniz-Newton formula, we have −h ∫ t t−h x˙T(s)T x˙(s)ds ≤ −h(t) ∫ t t−h(t) x˙T(s)T x˙(s)ds ≤ − [∫ t t−h(t) x˙(s)ds ]T T [∫ t t−h(t) x˙(s)ds ] ≤ − [ x(t)− x(t− h(t)) ]T T [ x(t)− x(t− h(t)) ] ; (3.5) and −τ ∫ t t−τ(t) x˙T(s)Wx˙(s)ds ≤ − [∫ t t−τ(t) x˙(s)ds ]T W [∫ t t−τ(t) x˙(s)ds ] ≤ − [ x(t)− x(t− τ(t)) ]T W [ x(t)− x(t− τ(t)) ] . (3.6) By using the following identity −x˙(t) + Cix˙(t− τ(t)) + Aix(t) +Dix(t− h(t)) = 0, we have 2 [ xT(t)UT1i + x T(t− h(t))UT2i + xT(t− τ(t))UT3i + x˙T(t)UT4i + x˙T(t− τ(t))UT5i ] × [ −x˙(t) + Cix˙(t− τ(t)) + Aix(t) +Dix(t− h(t)) ] = 0. (3.7) Denote ηT(t) = [ xT(t) xT(t− h(t)) xT(t− τ(t)) x˙T(t) x˙T(t− τ(t))] . Combining (3.4) - (3.7) we have V˙ (t, xt) + 2βV (t, xt) ≤ ηT(t)Φ(i)η(t) ≤ 0, ∀t ≥ 0. (3.8) From (3.8) it follows that V (t, xt) ≤ V (0, x0)e−2βt, t ∈ R+. Taking (3.3) into account we obtain λ1‖x(t, φ)‖2 ≤ V (t, xt) ≤ λ2‖φ‖2e−2βt, t ∈ R+. Finally, we get ‖x(t, φ)‖ ≤ √ λ2 λ1 ‖φ‖e−βt, ∀t ≥ 0, which concludes the proof of the theorem. 7 Le Van Hien Remark 3.1. In this paper, both constrains on the differentiability and slow variation h˙(t) ≤ hD < 1 of the state delay are not imposed. However, in case the state delay h(t) is differentiable and h˙(t) ≤ hD < ∞, the above criterion is still available since it does not depend on hD. Remark 3.2. In [6], the neutral term matrix Cα is assumed to be a common matrix for all subsystems. Based on the assumption ‖C‖ < 1, the exponential convergence rate β satisfies 0 < β < − ln(‖C‖) τ . Differ from [6], we do not require these assumptions and the exponential rate is determined by a set of LMIs. By solving iteratively a set of LMIs we can find the maximum allowable bound (MAB) for convergence rate β. Remark 3.3. By applying the result of theorem 3.1 for the case of N = 1, we obtain the exponential stability conditions for neutral systems with time-varying delays which are stated in the following corollary. Consider a neutral system with time-varying delays of the form{ x˙(t)− Cx˙(t− τ(t)) = Ax(t) +Dx(t− h(t)), t ≥ 0, x(t) = φ(t), t ∈ [−h, 0], (3.9) where, A,D,C are given real matrices and time delays h(t), τ(t) satisfy conditions (2.2). Corollary 3.1. For given β > 0, the system (3.9) is β-exponentially stable if there exist matrices U1, U2, U3, U4, U5 and symmetric positive definite matrices P,Q,R, T,W, such that the following linear matrix inequality hold: Ξ11 Ξ12 A TU3 +W A TU4 − UT1 PC + UT1 C + ATU5 ∗ Ξ22 DTU3 DTU4 − UT2 UT2 C +DTU5 ∗ ∗ Ξ33 −UT3 UT3 C ∗ ∗ ∗ Ξ44 UT4 C − U5 ∗ ∗ ∗ ∗ Ξ55  ≤ 0, where, Ξ11 = A T(P + U1) + (P + U T 1 )A+ 2βP +Q +R− T −W ; Ξ12 = PD + U T 1 D + A TU2 + T ; Ξ22 = −T + UT2 D +DTU2; Ξ33 = −(1− µ)e−2βτQ−W ; Ξ44 = R + h 2e2βhT + τ 2e2βτ 1− µ W − U4 − U T 4 ; Ξ55 = −(1− µ)e−2βτR + UT5 C + CTU5. 8 Global exponential stability of switched neutral systems with time-varying delays Moreover, every solution x(t, φ) of the system satisfies the following inequality ‖x(t, φ)‖ ≤ √ λ2 λ1 ‖φ‖e−βt, t ≥ 0, where, λ1 = λmin(P ), and λ2 =λmax(P ) + τ ( λmax(Q) + λmax(R) ) + 1 2 h3e2βhλmax(T ) + τ 3 2(1− µ)e 2βτλmax(W ). 4. Numerical example In this section, we give a numerical example to illustrate the effectiveness and conservativness of our obtained results. Example 4.1. Consider the system (2.1) with N = 3, τ = 0.2, µ = 0.1, h(t) = { 0.5 sin(t) if t ∈ I = ∪k≥0[2kπ, (2k + 1)π] 0 if t ∈ R+ \ I , A1 = [−2 0 0 −1 ] , A2 = [−1.5 1 0 −1 ] , A3 = [−0.5 0 0 −3 ] ; D1 = [−1 −1 0 −0.5 ] , D2 = [ 0.6 −1 0 −0.4 ] , D3 = [−1 1 0 1 ] ; C1 = [ 0.1 0 0 0.2 ] , C2 = [ 0.2 0 0 0.1 ] , C3 = [ 0.1 0 0 0.1 ] . Note that, the state delay function h(t) is not differentiable on R+, therefore, the stability criteria proposed in [6, 7, 9, 11] are not applicable. For β = 0.25, LMIs (3.1) are feasible with P = [ 2.1098 −0.4847 −0.4847 8.2588 ] , Q = [ 0.5180 −0.1899 −0.1899 1.9153 ] , R = [ 0.2552 −0.1499 −0.1499 1.8616 ] , T = [ 1.7519 −0.9708 −0.9708 5.8449 ] , W = [ 0.9465 −0.0162 −0.0162 1.0031 ] , U11 = [ 0.2748 −0.0364 −0.0671 0.4460 ] , U12 = [ 0.2748 −0.0364 −0.0671 0.4460 ] , U13 = [ 0.2748 −0.0364 −0.0671 0.4460 ] , U21 = [ 0.5256 155.7949 −313.5386 −312.7399 ] , U22 = [ 0.6360 46.0251 71.3101 −118.6846 ] , 9 Le Van Hien U23 = [−0.1829 4.3068 3.7746 −2.0157 ] , U31 = [ 0.2748 −0.0364 −0.0671 0.4460 ] , U32 = [ 0.2610 −0.1368 0.0647 0.4245 ] , U33 = [ 0.1507 −0.0888 −0.0181 0.2814 ] , U41 = [ 0.8823 312.9300 −313.8181 3.3591 ] , U42 = [ 1.3745 118.2655 −119.0609 3.1961 ] , U43 = [ 0.9489 −4.3089 3.8504 2.6230 ] , U51 = [−0.0078 −62.7151 31.3122 0.5408 ] , U52 = [−0.1129 −11.9349 23.7641 0.2754 ] , U53 = [−0.0219 0.3186 −0.4102 0.0211 ] . By theorem 3.1, the system (2.1) is exponentially stable with the convergence rate β = 0.25 under arbitrary switching signal. Moreover, every solution x(t, φ) of the system satisfies the following estimate ‖x(t, φ)‖ ≤ 2.1627‖φ‖e−0.25t, t ∈ R+. For τ = 0.2 and µ = 0. Table 1 gives the upper bounds on state delay for some values of convergence rate β Table 1. Allowable upper bounds of time delay function h(t) β 0.1 0.2 0.3 0.4 0.5 h 0.768 0.707 0.654 0.556 0.409 This system with C1 = C2 = C3 = 0 and h˙(t) ≤ hD is considered in [6, 11]. For β = 0, hD ≥ 1 or unknown then the upper bound of the time delay is given in Table 2 below. Table 2. Allowable upper bounds of time delay function h(t) (with β = 0, hD ≥ 1 or unknown) Wang et al. [11] Lien et al. [6] Our result 0 ≤ h(t) ≤ 0.9339 0 ≤ h(t) ≤ 0.9428 0 ≤ h(t) ≤ 0.9428 5. Conclusion Based on Lyapunov-Krasovskii functional approach combined with Leibniz-Newton formula, new delay-dependent conditions for exponential stability of linear switched neutral systems with non-differentiable state delay are proposed. The conditions are presented in terms of linear matrix inequalities which allows us to compute simultaneously the two bounds that characterize the exponential stability rate of the solution. 10 Global exponential stability of switched neutral systems with time-varying delays Acknowledgments. This work was supported by Hanoi National University of Education, Grant HNUE-12-103. REFERENCES [1] K. Gu, 2000. An integral inequality in the stability problem of time delay systems, in: IEEE Control Systems Society and Proceedings of IEEE Conference on Decision and Control, IEEE Publisher, New York. [2] L.V. Hien, Q.P. Ha, V.N. Phat, 2009. Stability and stabilization of switched linear dynamic systems with time delay and uncertainties. Appl. Math. 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