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- 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 grinding model, 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 k.k and scalar product xT y 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. hQx, xi ≥ 0 for all x ∈ Rn (resp. hQx, xi > 0 for all x 6= 0); A ≥ B means A − B ≥ 0; C 1 ([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 ( ˙ − Dx(t x(t) ˙ − τ ) = A0x(t) + A1 x(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) n where x(t) ∈ R 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 + n n n n uncertainty function f : R × R × R × R → R satisfies kf(t, x, y, z)k2 ≤ a20kxk2 + a21kyk2 + a22kzk2 , ∀(x, y, z), t ≥ 0, (2.2) 2
- Exponential stability of nonlinear neutral systems with time-varying delay 1 ¯ n q constants. The initial function φ ∈ C ([−h, 0], R ) where, a0, a1, a2 are given nonnegative with its norm kφks = sup ¯ ˙ kφ(t)k2 + kφ(t)k 2. −h≤t≤0 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 kx(t, φ)k ≤ γkφks e−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 = X T , Y T = Y > 0. Then X + Z T Y −1 Z < 0 if and only if X ZT −Y Z < 0 or < 0. Z −Y ZT X Proposition 2.2. (Completing square) Let S be a symmetric positive definite matrix. Then for any x, y ∈ Rn and matrix F , we have 2hF y, xi − hSy, yi ≤ hF S −1F T x, xi. The proof of the above proposition is easily derived from completing square: hS(y − S −1 F T x), y − S −1 F Txi ≥ 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 Z ν T Z ν Z ν w(s)ds W w(s)ds ≤ ν wT (s)W w(s)ds. 0 0 0 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 definite matrices P, Q, R, S, T, Z, W we set −2ατ ρ(α) = 2αλmax (P ) + 1 − e (λmax (R) + λmax (S)) + 1 − e−2αhm λmax (Q) + h2M e2αhM − 1 λmax (T ) 2 + hM − hm 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 1 λ2 = λmax (P ) + hm λmax (Q) + τ λmax (R) + λmax (S) + h3M e2α∗hM λmax (T ) 2 1 1 + (hM − hm )2 (hM + hm )e2α∗hM λmax (Z) + τ 3e2α∗τ λmax (W ). 2 2 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: Ξ11 AT0 U2 + W Ξ13 AT0 U4 −U1T + AT0 U5 Ξ16 Ξ17 ∗ −R − W U2T A1 0 −U2T U2T D U2T ∗ T T T T T T T ∗ Ξ 33 Z + A 1 U4 −U 3 + A 1 U5 U3 D + A 1 U 6 U3 + A 1 U 7 T T T Ξ= ∗ ∗ ∗ −Q − Z −U4 U4 D U4 < 0, ∗ T T ∗ ∗ ∗ Ξ 55 U 5 D − U 6 U 5 − U 7 ∗ T T ∗ ∗ ∗ ∗ Ξ66 U6 + D U7 ∗ ∗ ∗ ∗ ∗ ∗ Ξ77 (3.1) where Ξ11 = AT0 (P + U1 ) + (P + U1T )A0 + a20I + Q + R − T − W ; Ξ13 = P A1 + U1T A1 + AT0 U3 + T ; Ξ16 = P D + U1T D + AT0 U6 ; Ξ17 = AT0 U7 + P + U1T ; Ξ33 = −T − Z + AT1 U3 + U3T A1 + a21I; Ξ55 = −U5 − U5T + S + h2M T + (hM − hm )2Z + τ 2 W ; Ξ66 = −S + DT U6 + U6T D + a22I; Ξ77 = −I + U7T + U7 . Then the system (2.1) is globally exponentially stable. Moreover, every solution x(t, φ) of the system satisfies r λ2 kx(t, φ)k ≤ kφks e−αt , ∀α ∈ (0, α∗ ], ∀t ≥ 0. λ1 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) 7 X V (t, xt) = Vk , (3.2) i=1 4
- Exponential stability of nonlinear neutral systems with time-varying delay where, V1 = xT (t)P x(t), Z t V2 = e2α(s−t)xT (s)Qx(s)ds t−h Z t m V3 = e2α(s−t)xT (s)Rx(s)ds, t−τ Z t V4 = e2α(s−t)x˙ T (s)S x(s)ds, ˙ t−τ Z t Z t V5 = hM e2α(θ−t+hM ) x˙ T (θ)T x(θ)dθds, ˙ t−hM s Z t−hm Z t V6 = (hM − hm ) e2α(θ−t+hM ) x˙ T (θ)Z x(θ)dθds, ˙ t−hM s Z t Z t V7 = τ e2α(θ+τ −t)x˙ T (θ)W x(θ)dθds. ˙ t−τ s Taking the derivative of V1 along the solution of system (2.1) we have V˙1 = 2xT (t)P x(t) ˙ h i = xT (t) P A0 + AT0 P x(t) h i T + 2x (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 h i a20xT (t)x(t) + a21xT (t − h(t))x(t − h(t)) + a22x˙ T (t − τ )x(t ˙ − τ ) − f T (t)f(t) ≥ 0, for any > 0. Therefore, the derivative of V1 satisfies h i ˙ T T 2 V1 ≤ x (t) P A0 + A0 P + a0I x(t) h i + 2xT (t)P A1x(t − h(t)) + Dx(t ˙ − τ ) + f(t) (3.3) h i + a21xT (t − h(t))x(t − h(t)) + a22x˙ T (t − τ )x(t ˙ − τ ) − f T (t)f(t) . 5
- L.V. Hien & H.V. Thi Next, the derivatives of Vk , k = 2, . . . , 7 give V˙2 = xT (t)Qx(t) − e−2αhm xT (t − hm )Qx(t − hm ) − 2αV2 ; V˙3 = xT (t)Rx(t) − e−2ατ xT (t − τ )Rx(t − τ ) − 2αV3 ; V˙4 = x˙ T (t)S x(t) ˙ − e−2ατ x˙ T (t − τ )S x(t ˙ − τ ) − 2αV4 ; Z t (3.4) V˙5 = h2M e2αhM x˙ T (t)T x(t) ˙ − hM e2α(s−t+hM ) x˙ T (s)T x(s)ds ˙ − 2αV5 t−hM Z t ≤ h2M e2αhM x˙ T (t)T x(t) ˙ − hM x˙ T (s)T x(s)ds ˙ − 2αV5 ; t−hM and V˙6 = (hM − hm )2e2αhm x˙ T (t)Z x(t) ˙ Z t−hm − (hM − hm ) e2α(s−t+hM ) x˙ T (s)Z x(s)ds ˙ − 2αV6 t−hM ≤ (hM − hm )2 e2αhm x˙ T (t)Z x(t) ˙ Z t−hm − (hM − hm ) x˙ T (s)Z x(s)ds ˙ − 2αV6 ; (3.5) t−hM Z t ˙ 2 2ατ T V7 = τ e x˙ (t)W x(t) ˙ −τ e2α(s+τ −t)x˙ T (s)W x(s)ds ˙ − 2αV7 t−τ Z t 2 2ατ T ≤ τ e x˙ (t)W x(t) ˙ −τ x˙ T (s)W x(s)ds ˙ − 2αV7 ; t−τ Applying Proposition 3 and the Leibniz-Newton formula, we have Z t Z t T −hM x˙ (s)T x(s)ds ˙ ≤ −h(t) x˙ T (s)T x(s)ds ˙ t−hM t−h(t) Z t T Z t ≤− ˙ x(s)ds T ˙ x(s)ds (3.6) t−h(t) t−h(t) h iT h i ≤ − x(t) − x(t − h(t)) T x(t) − x(t − h(t)) ; Z t−hm Z t−hm T −(hM − hm ) x˙ (s)Z x(s)ds ˙ ≤ −(h(t) − hm ) x˙ T (s)Z x(s)ds ˙ t−hM t−h(t) Z t−hm T Z t−hm ≤− ˙ x(s)ds Z ˙ x(s)ds (3.7) t−h(t) t−h(t) h iT h i ≤ − x(t − hm ) − x(t − h(t)) Z x(t − hm ) − x(t − h(t)) ; 6
- Exponential stability of nonlinear neutral systems with time-varying delay and Z t Z t T Z t T −τ x˙ (s)W x(s)ds ˙ ≤− ˙ x(s)ds W ˙ x(s)ds t−τ h t−τ iT i t−τ h (3.8) ≤ − x(t) − x(t − τ ) W x(t) − x(t − τ ) . By using the following identity relation −x(t) ˙ + Dx(t ˙ − τ ) + A0x(t) + A1x(t − h(t)) + f(t) = 0, we obtain h 2 xT (t)U1T + xT (t − τ )U2T + xT (t − h(t))U3T i T + x (t − hm )U4T + x˙ T (t)U5T + x˙ (t − +fT τ )U6T T (t)U7T (3.9) h i × −x(t) ˙ + Dx(t ˙ − τ ) + A0x(t) + A1x(t − h(t)) + f(t) = 0. Therefore, from (3.3)-(3.9) we have V˙ (t, xt) + 2αV (t, xt) ≤ η T (t)Φη(t), (3.10) where, η T (t) = xT (t) xT (t − τ ) xT (t − h(t)) xT (t − hm ) x˙ T (t) x˙ T (t − τ ) f T (t) , Φ11 AT0 U2 + W Φ13 AT0 U4 −U1T + AT0 U5 Φ16 Φ17 ∗ Φ22 U2T A1 0 −U2T U2T D U2T ∗ T T T T T T T ∗ Φ 33 Z + A 1 U4 −U3 + A 1 U5 U3 D + A 1 U 6 U 3 + A 1 U 7 T T T Φ= ∗ ∗ ∗ Φ44 −U4 U4 D U4 , ∗ T T ∗ ∗ ∗ Φ55 U 5 D − U 6 U 5 − U 7 ∗ T T ∗ ∗ ∗ ∗ Φ66 U6 + D U7 ∗ ∗ ∗ ∗ ∗ ∗ Φ77 and Φ11 = (A0 + αI)T P + P (A0 + αI) + AT0 U1 + U1T A0 + a20I + Q + R − W − T ; Φ13 = P A1 + U1T A1 + AT0 U3 + T ; Φ16 = P D + U1T D + AT0 U6 ; Φ17 = P + U1T + AT0 U7 ; Φ22 = −e−2ατ R − W ; Φ33 = a21I − T − Z + AT1 U3 + U3T A1; Φ44 = −e−2αhm Q − Z; Φ55 = S + h2M e2αhM T + (hM − hm )2e2αhM Z + τ 2 e2ατ W − U5 − U5T ; Φ66 = −e−2ατ S + U6T D + DT U6 + a22I; Φ77 = −I + U7 + U7T . 7
- L.V. Hien & H.V. Thi Observe that Φ = Ξ + Ψ, where, n Ψ = diag 2αP, (1 − e−2ατ )R, 0, (1 − e−2αhm )Q, h2M (e2αhM − 1)T o 2 2αhM 2 2ατ −2ατ + (hM − hm ) (e − 1)Z + τ (e − 1)W, (1 − e )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 h i V˙ (t, xt) + 2αV (t, xt) ≤ ρ(α) − λ0 kη(t)k2 ≤ 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 kx(t)k2 ≤ V (t, xt) ≤ λ2 kxtk2s , t ∈ R+ . and from the differential inequality (3.12), we obtain r λ2 kx(t, φ)k ≤ kφks e−αt , t≥0 λ1 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, 2α¯ h ¯ 2α¯h λmax (P ) h e − 1 ≥ 2αh, we have ρ(α) ≤ γ e − 1 , where, γ = ¯ + λmax (Q) + i h λmax (R) + λmax (S) + h2M λmax (T ) + (hM − hm )2 λmax (Z). Therefore, system (2.1) is 1 λ0 exponentially stable with the exponential rate 0 < α ≤ ¯ ln 1 + . 2h 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 −2 0 0.1 −1 0.1 0 A0 = , A1 = , D= , 1 −4 0 −0.1 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 [2kπ, (2k + 1)π] 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 11.1343 −8.6199 6.4391 −2.7561 5.9522 −1.4762 P = , Q= , R= , −8.6199 33.9554 −2.7561 23.0890 −1.4762 10.8871 1.1647 −0.1417 0.4364 −0.0872 0.5411 −0.0846 S= , T = , W = , −0.1417 2.0581 −0.0872 0.8021 −0.0846 1.1756 2.6249 −1.2851 −14.3260 41.8769 0.1456 0.0452 Z= , U1 = , U2 = , −1.2851 12.4711 −7.9793 −28.4459 −0.0317 0.2562 0.3302 −0.7932 −0.2285 1.8629 3.3512 7.8831 U3 = , U4 = , U5 = , 0.7994 −8.1438 −0.0296 0.2517 −10.1070 7.1532 −0.0126 −0.8276 1.1847 −8.1379 U6 = , U7 = . 0.7984 0.0933 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 kx(t, φ)k ≤ 3.1803kφks e−0.0022t, ∀t ≥ 0. Example 4..2. Consider the system studied in ([15, 20]): d [x(t) − Dx(t − τ )] = A0 x(t) + A1x(t − τ ) + f(t, x(t), x(t − τ )), (4.1) dt where, −2 0.5 1 0.4 0.2 1 A0 = , A1 = , D= , a0 = 0.2, a1 = 0.1. 0 −1 0.4 −1 0 0.2 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. This work was partially supported by Hanoi National University of Education and the Min- istry of Education and Training, Vietnam. REFERENCES [1] S. Boyd, L.E. Ghaoui, E. Feron, & V. Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM. [2] K. Gu (2000). An integral inequality in the stability problem of time delay systems Proc. IEEE Conf. on Decision and Control, New York. [3] Q.L. Han & L. Yu (2004). Robust stability of of linear neutral systems with nonlinear parameter perturbations. IEE Proceeding, Control Theory and Application. 151(5), 539-546. [4] Y. He, Q. Wang, C. Lin & M. Wu (2007). Delay-range-dependent stability for systems with time-varying delay. Automatica, 43, 371-376. [5] L.V. Hien & V.N. Phat (2009). Exponential stability and stabilization of a class of uncertain linear time-delay systems. Journal of the Franklin Institute, 346, 611-625. [6] L.V. Hien & V.N. Phat (2009). Stability and stabilization of switched linear dynamic systems with time delay and uncertainties. Applied Mathematics and Computation, 210, 223-231. [7] X. Jiang & Q.L. Han (2006). Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica, 42, 1059-1065. [8] X. Jiang & Q. Han (2008). New stability criteria uncertain linear systems with interval time-varying delay. Automatica, 44, 2680-2685. [9] C.T. Kelley (2003). Solving nonlinear equations with Newton’s method, Philadelphia: SIAM. [10] O.M. Kwon, J.H. Park & S.M. Lee (2009). Augmented Lyapunov functional ap- proach to stability of uncertain neutral systems with time-varying delays. Applied Mathematics and Computation, 207, 202-212. 10
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