Exponential stability of nonlinear neutral systems with time-varying delay

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2013, Vol. 00, No. 0, pp. 1-11 Exponential stability of nonlinear neutral systems with time-varying delay Le Van Hien† and Hoang Van Thi †† † Hanoi National University of Education †† Hong Duc University, Thanh Hoa E-mail: Hienlv@hnue.edu.vn Abstract. In this paper, the problem of exponential stability for a class of nonlin- ear neutral systems with interval time-varying delay is studied. Based on im- proved Lyapunov-Krasovskii functionals combine with Leibniz-Newton’s for- mula, new delay-dependent sufficient conditions for the exponential stability of the systems are established in terms of linear matrix inequalities (LMIs), which allows to compute the maximal bound of the exponential stability rate of the solution. Numerical examples are also given to show the effectiveness of the obtained results. Keywords: Neutral systems; interval time-varying delay; nonlinear uncer- tainty; exponential stability; linear matrix inequality 1. Introduction Time-delay occurs in most of practical models, such as, aircraft stabilization, chemi- cal engineering systems, inferred grindingmodel, manual control, neural network, nuclear reactor, population dynamic model, ship stabilization, and systems with lossless transmis- sion lines. The existence of this time-delay may be the source for instability and bad per- formance of the system. Hence, the problem of stability analysis for time-delay systems has received much attention of many researchers in recent years, see [4, 5, 7, 11, 12, 14, 17] and references therein. In many practical systems, the system models can be described by functional differ- ential equations of neutral type, which depend on both state and state derivatives. Neutral system examples include distributed networks, heat exchanges, and processes involving steam. Recently, the stability analysis of neutral systems has been widely investigated by many researchers, see [3, 7] for time-varying delay, and [8, 10-12] for interval time- varying delay. The main approach is Lyapunov-Krasovskii functional method and linear matrix inequality technique. However, in most of this results, the time-varying delay is assumed to be differentiable, which makes stability conditions more conservatism. 1 L.V. Hien & H.V. Thi In this paper, we consider exponential stability problem for a class of nonlinear neutral systems with interval time-varying delay. By using improved Lyapunov-Krasovskii functionals combined with LMIs technique, we propose new criteria for the exponential stability of the system. The delay-dependent conditions are formulated in terms of LMIs, being thus solvable by utilizing Matlab’s LMI Control Toolbox available in the literature to date. Compared to the existing results, our result has its own advantages. First, it deals with the neutral system considered in this paper is subjected to nonlinear uncertainties. Second, the time delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not necessary to be differentiable. This allows the time-delay to be a fast time-varying function and the lower bound is not restricted to being zero. Third, our approach allows us to obtain novel exponential stability conditions established in terms of LMIs, which allows to compute the maximal bound of the exponential stability rate of the solution. Therefore, our results are more general than the related previous results. The paper is organized as follows: Section 2 presents definitions and some well- known technical propositions needed for the proof of the main results. Delay-dependent exponential stability conditions of the system is presented in Section 3. Numerical exam- ples are given in Section 4. The paper ends with conclusions and cited references. Notations. 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 and I denote the identity matrix. A matrix Q ≥ 0 (Q > 0, resp.) means that Q is semi-positive definite (positive definite, resp.) i.e. 〈Qx, x〉 ≥ 0 for all x ∈ Rn (resp. 〈Qx, x〉 > 0 for all x 6= 0); A ≥ B means A − B ≥ 0; C1([a, b], Rn) denotes the set of all continuously differentiable functions on [a, b]. The segment of the trajectory x(t) is denoted by xt = {x(t+ s) : s ∈ [−h¯, 0]}. 2. Preliminaries Consider a nonlinear neutral system with interval time-varying delay of the form { x˙(t)−Dx˙(t− τ ) = A0x(t) + A1x(t− h(t)) + f (t, x(t), x(t− h(t)), x˙(t− τ )) , t ≥ 0, x(t) = φ(t), t ∈ [−h¯, 0], (2.1) where x(t) ∈ Rn is the system state; A0, A1, D are given real matrices; time varying delay h(t) satisfies 0 ≤ hm ≤ h(t) ≤ hM , constant τ ≥ 0 and h¯ = max{τ, hM}; nonlinear uncertainty function f : R+ ×Rn ×Rn × Rn → Rn satisfies ‖f(t, x, y, z)‖2 ≤ a20‖x‖ 2 + a21‖y‖ 2 + a22‖z‖ 2, ∀(x, y, z), t ≥ 0, (2.2) 2 Exponential stability of nonlinear neutral systems with time-varying delay where, a0, a1, a2 are given nonnegative constants. The initial function φ ∈ C 1([−h¯, 0], Rn) with its norm ‖φ‖s = sup−h¯≤t≤0 √ ‖φ(t)‖2 + ‖φ˙(t)‖2. Definition 2.1. System (2.1) is said to be globally exponentially stable if there exist con- stants α > 0, γ ≥ 1 such that all solution x(t, φ) of the system satisfies the following condition ‖x(t, φ)‖ ≤ γ‖φ‖se −αt, ∀t ≥ 0. We introduce the following technical well-known propositions, which will be used in the proof of our results. Proposition 2.1. (Schur Complement, see Boyd et. al. [1]) For given matrices X, Y, Z with appropriate dimensions satisfying X = XT, Y T = Y > 0. Then X + ZTY −1Z < 0 if and only if [ X ZT Z −Y ] < 0 or [ −Y Z ZT X ] < 0. Proposition 2.2. (Completing square) Let S be a symmetric positive definite matrix. Then for any x, y ∈ Rn and matrix F , we have 2〈Fy, x〉 − 〈Sy, y〉 ≤ 〈FS−1F Tx, x〉. The proof of the above proposition is easily derived from completing square: 〈S(y − S−1F Tx), y − S−1F Tx〉 ≥ 0. Proposition 2.3. (See, Gu [2]) For any symmetric positive definite matrix W , scalar ν > 0 and vector function w : [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 Consider system (2.1), where the delay function h(t) satisfies 0 ≤ hm ≤ h(t) ≤ hM , constant τ ≥ 0 and h¯ = max{τ, hM} and the nonlinear perturbation function f(.) sat- isfies the condition (2.2). For given symmetric positive definitematricesP,Q,R, S, T, Z,W we set ρ(α) = 2αλmax(P ) + ( 1− e−2ατ ) (λmax(R) + λmax(S)) + ( 1− e−2αhm ) λmax(Q) + h 2 M ( e2αhM − 1 ) λmax(T ) + ( hM − hm )2( e2αhM − 1 ) λmax(Z) + τ 2 ( e2ατ − 1 ) λmax(W ). 3 L.V. Hien & H.V. Thi Note that, the scalar function ρ(α) is continuous and strictly increasing function in α ∈ [0,∞), ρ(0) = 0, ρ(α) → ∞ as α → ∞. Hence, for any λ0 > 0, there is a unique positive solution α∗ of the equation ρ(α) = λ0, and ρ(α) < λ0 for all α ∈ (0, α∗). Let us set λ1 = λmin(P ), and λ2 = λmax(P ) + hmλmax(Q) + τ ( λmax(R) + λmax(S) ) + 1 2 h3 M e2α∗hMλmax(T ) + 1 2 (hM − hm) 2(hM + hm)e 2α∗hMλmax(Z) + 1 2 τ 3e2α∗τλmax(W ). The exponential stability of system (2.1) is summarized in the following theorem. Theorem 3.1. Assume that, for system (2.1), there exist matrices Uk, (k = 1, . . . , 7), symmetric positive definite matrices P,Q,R, S, T, Z,W, and positive number , such that the following linear matrix inequality hold: Ξ =   Ξ11 A T 0U2 +W Ξ13 A T 0U4 −U T 1 + A T 0U5 Ξ16 Ξ17 ∗ −R−W UT2 A1 0 −U T 2 U T 2 D U T 2 ∗ ∗ Ξ33 Z + A T 1U4 −U T 3 + A T 1U5 U T 3 D + A T 1U6 U T 3 + A T 1U7 ∗ ∗ ∗ −Q− Z −UT4 U T 4 D U T 4 ∗ ∗ ∗ ∗ Ξ55 U T 5 D − U6 U T 5 − U7 ∗ ∗ ∗ ∗ ∗ Ξ66 U T 6 +D TU7 ∗ ∗ ∗ ∗ ∗ ∗ Ξ77   < 0, (3.1) where Ξ11 = A T 0 (P + U1) + (P + U T 1 )A0 + a 2 0I +Q+R− T −W ; Ξ13 = PA1 + U T 1 A1 + A T 0U3 + T ; Ξ16 = PD + U T 1 D + A T 0U6; Ξ17 = A T 0U7 + P + U T 1 ; Ξ33 = −T − Z + A T 1U3 + U T 3 A1 + a 2 1I ; Ξ55 = −U5 − U T 5 + S + h 2 M T + (hM − hm) 2Z + τ 2W ; Ξ66 = −S +D TU6 + U T 6 D + a 2 2I ; Ξ77 = −I + U T 7 + U7. Then the system (2.1) is globally exponentially stable. Moreover, every solution x(t, φ) of the system satisfies ‖x(t, φ)‖ ≤ √ λ2 λ1 ‖φ‖se −αt, ∀α ∈ (0, α∗], ∀t ≥ 0. Chứng minh. Let λ0 = λmin ( −Ξ ) > 0 (due to (3.1)). Taking any α > 0 from the interval (0, α∗], we consider the following Lyapunov-Krasovskii functional for the system (2.1) V (t, xt) = 7∑ i=1 Vk, (3.2) 4 Exponential stability of nonlinear neutral systems with time-varying delay where, V1 = x T(t)Px(t), V2 = ∫ t t−hm e2α(s−t)xT(s)Qx(s)ds V3 = ∫ t t−τ e2α(s−t)xT(s)Rx(s)ds, V4 = ∫ t t−τ e2α(s−t)x˙T(s)Sx˙(s)ds, V5 = hM ∫ t t−hM ∫ t s e2α(θ−t+hM )x˙T(θ)T x˙(θ)dθds, V6 = (hM − hm) ∫ t−hm t−hM ∫ t s e2α(θ−t+hM )x˙T(θ)Zx˙(θ)dθds, V7 = τ ∫ t t−τ ∫ t s e2α(θ+τ−t)x˙T(θ)Wx˙(θ)dθds. Taking the derivative of V1 along the solution of system (2.1) we have V˙1 = 2x T(t)Px˙(t) = xT(t) [ PA0 + A T 0P ] x(t) + 2xT(t)P [ A1x(t− h(t)) +Dx˙(t− τ ) + f(t) ] , where, for convenient, we denote f(t) =: f(t, x(t), x(t− h(t)), x˙(t− τ )). From (2.2) we obtain  [ a20x T(t)x(t) + a21x T(t− h(t))x(t− h(t)) + a22x˙ T(t− τ )x˙(t− τ )− fT(t)f(t) ] ≥ 0, for any  > 0. Therefore, the derivative of V1 satisfies V˙1 ≤ x T(t) [ PA0 + A T 0P + a 2 0I ] x(t) + 2xT(t)P [ A1x(t− h(t)) +Dx˙(t− τ ) + f(t) ] +  [ a21x T(t− h(t))x(t− h(t)) + a22x˙ T(t− τ )x˙(t− τ )− fT(t)f(t) ] . (3.3) 5 L.V. Hien & H.V. Thi Next, the derivatives of Vk, k = 2, . . . , 7 give V˙2 = x T(t)Qx(t)− e−2αhmxT(t− hm)Qx(t− hm)− 2αV2; V˙3 = x T(t)Rx(t)− e−2ατxT(t− τ )Rx(t− τ )− 2αV3; V˙4 = x˙ T(t)Sx˙(t)− e−2ατ x˙T(t− τ )Sx˙(t− τ )− 2αV4; V˙5 = h 2 M e2αhM x˙T(t)T x˙(t)− hM ∫ t t−hM e2α(s−t+hM )x˙T(s)T x˙(s)ds− 2αV5 ≤ h2 M e2αhM x˙T(t)T x˙(t)− hM ∫ t t−hM x˙T(s)T x˙(s)ds− 2αV5; (3.4) and V˙6 = (hM − hm) 2e2αhmx˙T(t)Zx˙(t) − (hM − hm) ∫ t−hm t−hM e2α(s−t+hM)x˙T(s)Zx˙(s)ds− 2αV6 ≤ (hM − hm) 2e2αhmx˙T(t)Zx˙(t) − (hM − hm) ∫ t−hm t−hM x˙T(s)Zx˙(s)ds− 2αV6; V˙7 = τ 2e2ατ x˙T(t)Wx˙(t)− τ ∫ t t−τ e2α(s+τ−t)x˙T(s)Wx˙(s)ds− 2αV7 ≤ τ 2e2ατ x˙T(t)Wx˙(t)− τ ∫ t t−τ x˙T(s)Wx˙(s)ds− 2αV7; (3.5) Applying Proposition 3 and the Leibniz-Newton formula, we have −hM ∫ t t−hM 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)) ] ; (3.6) −(hM − hm) ∫ t−hm t−hM x˙T(s)Zx˙(s)ds ≤ −(h(t)− hm) ∫ t−hm t−h(t) x˙T(s)Zx˙(s)ds ≤ − [∫ t−hm t−h(t) x˙(s)ds ]T Z [∫ t−hm t−h(t) x˙(s)ds ] ≤ − [ x(t− hm)− x(t− h(t)) ]T Z [ x(t− hm)− x(t− h(t)) ] ; (3.7) 6 Exponential stability of nonlinear neutral systems with time-varying delay and −τ ∫ t t−τ x˙T(s)Wx˙(s)ds ≤ − [∫ t t−τ x˙(s)ds ]T W [∫ t t−τ x˙(s)ds ] ≤ − [ x(t)− x(t− τ ) ] T W [ x(t)− x(t− τ ) ] . (3.8) By using the following identity relation −x˙(t) +Dx˙(t− τ ) + A0x(t) + A1x(t− h(t)) + f(t) = 0, we obtain 2 [ xT(t)UT1 + x T(t− τ )UT2 + x T(t− h(t))UT3 + xT(t− hm)U T 4 + x˙ T(t)UT5 + x˙ T(t− τ )UT6 + f T(t)UT7 ] × [ −x˙(t) +Dx˙(t− τ ) + A0x(t) + A1x(t− h(t)) + f(t) ] = 0. (3.9) Therefore, from (3.3)-(3.9) we have V˙ (t, xt) + 2αV (t, xt) ≤ η T(t)Φη(t), (3.10) where, ηT(t) = [ xT(t) xT(t− τ ) xT(t− h(t)) xT(t− hm) x˙ T(t) x˙T(t− τ ) fT(t) ] , Φ =   Φ11 A T 0U2 +W Φ13 A T 0U4 −U T 1 + A T 0U5 Φ16 Φ17 ∗ Φ22 U T 2 A1 0 −U T 2 U T 2 D U T 2 ∗ ∗ Φ33 Z + A T 1U4 −U T 3 + A T 1U5 U T 3 D + A T 1U6 U T 3 + A T 1U7 ∗ ∗ ∗ Φ44 −U T 4 U T 4 D U T 4 ∗ ∗ ∗ ∗ Φ55 U T 5 D − U6 U T 5 − U7 ∗ ∗ ∗ ∗ ∗ Φ66 U T 6 +D TU7 ∗ ∗ ∗ ∗ ∗ ∗ Φ77   , and Φ11 = (A0 + αI) TP + P (A0 + αI) + A T 0U1 + U T 1 A0 + a 2 0I +Q+R −W − T ; Φ13 = PA1 + U T 1 A1 + A T 0U3 + T ; Φ16 = PD + U T 1 D + A T 0U6; Φ17 = P + U T 1 + A T 0U7; Φ22 = −e −2ατR−W ; Φ33 = a 2 1I − T − Z + A T 1U3 + U T 3 A1; Φ44 = −e −2αhmQ− Z; Φ55 = S + h 2 M e2αhMT + (hM − hm) 2e2αhMZ + τ 2e2ατW − U5 − U T 5 ; Φ66 = −e −2ατS + UT6 D +D TU6 + a 2 2I ; Φ77 = −I + U7 + U T 7 . 7 L.V. Hien & H.V. Thi Observe that Φ = Ξ +Ψ, where, Ψ = diag { 2αP, (1− e−2ατ)R, 0, (1− e−2αhm)Q, h2 M (e2αhM − 1)T + (hM − hm) 2(e2αhM − 1)Z + τ 2(e2ατ − 1)W, (1− e−2ατ)S, 0 } . hence V˙ (t, xt) + 2αV (t, xt) ≤ η T(t)(Ξ + Ψ)η(t). (3.11) Taking (3.11) into account, we finally obtain V˙ (t, xt) + 2αV (t, xt) ≤ [ ρ(α)− λ0 ] ‖η(t)‖2 ≤ 0, (3.12) which implies V (t, xt) ≤ V (0, x0)e −2αt, t ≥ 0. To estimate the exponential stability rate of the solution, we use (3.2) that λ1‖x(t)‖ 2 ≤ V (t, xt) ≤ λ2‖xt‖ 2 s , t ∈ R+. and from the differential inequality (3.12), we obtain ‖x(t, φ)‖ ≤ √ λ2 λ1 ‖φ‖se −αt, t ≥ 0 which completes the proof of the theorem. Remark 3.1. The exponential convergence rate α in Theorem 1 can be obtained by solv- ing a nonlinear scalar equation ρ(α) = λ0. For this equation, many algorithms and com- putational methods can be used, e.g., iterative or Newton’s method [9]. However, for a more explicit condition, we estimate the exponential rate α as follow: From the fact that, e2αh¯ − 1 ≥ 2αh¯, we have ρ(α) ≤ γ ( e2αh¯ − 1 ) , where, γ = λmax(P ) h¯ + [ λmax(Q) + λmax(R) + λmax(S) ] + h2 M λmax(T ) + (hM − hm) 2λmax(Z). Therefore, system (2.1) is exponentially stable with the exponential rate 0 < α ≤ 1 2h¯ ln ( 1 + λ0 γ ) . Remark 3.2. Theorem 1 gives conditions for the exponential stability of neutral systems with nonlinear uncertainties and interval-time varying state delay. These conditions are derived in terms of linear matrix inequalities which can be solved effectively by various computation tools [1]. Different from [5, 6, 12, 13], where the α-exponential stability problem is considered, the exponential rate α is given and enters as nonlinear terms in the stability conditions. In this paper, the exponential convergence rate is determined in terms of linear matrix inequalities. 8 Exponential stability of nonlinear neutral systems with time-varying delay 4. Numerical examples In this section, we give some numerical examples to illustrate the effectiveness of our obtained results in comparison with the existing results. Example 4..1. Consider neutral system (2.1), where A0 = [ −2 0 1 −4 ] , A1 = [ 0.1 −1 0 −0.1 ] , D = [ 0.1 0 0 0.1 ] , a0 = 0.2, a1 = 0.2, a2 = 0.1, τ = 1, and h(t) = 1 + ψ(t), where, ψ(t) = 0.5 sin(t) if t ∈ I = ∪k≥0[2kpi, (2k + 1)pi] and ψ(t) = 0 if t ∈ R+\I. Note that, the delay function h(t) is continuous, but non-differentiable on R+. Therefore, the stability results obtained in [3, 10-12, 16, 18-21] are not applicable. By using LMI toolbox of Matlab, we can verify that, the LMI (3.1) is satisfied with hm = 1, hM = 1.5,  = 10 and P = [ 11.1343 −8.6199 −8.6199 33.9554 ] , Q = [ 6.4391 −2.7561 −2.7561 23.0890 ] , R = [ 5.9522 −1.4762 −1.4762 10.8871 ] , S = [ 1.1647 −0.1417 −0.1417 2.0581 ] , T = [ 0.4364 −0.0872 −0.0872 0.8021 ] , W = [ 0.5411 −0.0846 −0.0846 1.1756 ] , Z = [ 2.6249 −1.2851 −1.2851 12.4711 ] , U1 = [ −14.3260 41.8769 −7.9793 −28.4459 ] , U2 = [ 0.1456 0.0452 −0.0317 0.2562 ] , U3 = [ 0.3302 −0.7932 0.7994 −8.1438 ] , U4 = [ −0.2285 1.8629 −0.0296 0.2517 ] , U5 = [ 3.3512 7.8831 −10.1070 7.1532 ] , U6 = [ −0.0126 −0.8276 0.7984 0.0933 ] , U7 = [ 1.1847 −8.1379 8.1228 1.3701 ] . We have λ0 = 0.3635 and ρ(α) = 73.6906α + 36.9084 ( 1− e−2α ) + 1.1867 ( e2α − 1 ) + 5.0081 ( e3α − 1 ) . The unique positive solution of equation ρ(α) = λ0 is α∗ = 0.0022057. Then all solution x(t, φ) of the system satisfies the following inequality ‖x(t, φ)‖ ≤ 3.1803‖φ‖se −0.0022t, ∀t ≥ 0. Example 4..2. Consider the system studied in ([15, 20]): d dt [x(t)−Dx(t − τ )] = A0x(t) + A1x(t− τ ) + f(t, x(t), x(t− τ )), (4.1) where, A0 = [ −2 0.5 0 −1 ] , A1 = [ 1 0.4 0.4 −1 ] , D = [ 0.2 1 0 0.2 ] , a0 = 0.2, a1 = 0.1. 9 L.V. Hien & H.V. Thi Applying Corollary 1 for hm = 0, hM = τ and a2 = 0 we obtain the allowable value of the delay for the asymptotic stability of system (4.1) is τ = 1.8106, while the upper bound of value τ given in [15] and [20] is 0.583 and 1.7043, respectively. 5. Conclusion In this paper, we have proposed new delay-dependent exponential stability condi- tions for a class of nonlinear neutral systems with non-differentiable interval time-varying delay. Based on the improved Lyapunov-Krasovskii functionals and linear matrix inequal- ity technique, new delay-dependent sufficient conditions for the exponential stability of the systems have been established in terms of LMIs. Numerical examples are given to show the effectiveness of our results. Acknowledgments. 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