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For the first time, the finitetime stabilization with guaranteed cost control for linear autonomous timevarying delay systems with bounded controls is studied in this paper. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel sufficient conditions for designing nonlinear feedback controllers that guarantee the robust finitetime stabilization of the closedloop system are established. The obtained stabilization condition is then adapted to solve the problem of guaranteed cost control. A numerical example is given to show the effectiveness of the proposed results
Finite-time stabilization and guaranteed cost control of linear autonomous delay systems with bounded controls1 P. NIAMSUPa , V. N. PHATb,∗ a Department of Mathematics Chiang Mai University, Chiang Mai 50200, Thailand b Institute of Mathematics, VAST 18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam ∗ Corresponding author: vnphat@math.ac.vn Abstract For the first time, the finite-time stabilization with guaranteed cost control for linear autonomous time-varying delay systems with bounded controls is studied in this paper. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel sufficient conditions for designing nonlinear feedback controllers that guarantee the robust finite-time stabilization of the closed-loop system are established. The obtained stabilization condition is then adapted to solve the problem of guaranteed cost control. A numerical example is given to show the effectiveness of the proposed results. Key words. Finite-time stabilization, Bounded control, Guaranteed cost control, Timevarying delay, Lyapunov function, Linear matrix inequality. 1 Introduction In the last decade, we have witnessed an increasing interest to the problem of finite-time stability and control for linear time-delay systems [1-4]. The finite-time stability (FTS) introduced in [5] means that the state of a system does not exceed some bound during a fixed interval time. The finite-time stabilization concerns with the design of a feedback controller which ensures the FTS of the closed-loop system and the problem of guaranteed cost control (GCC) is to find a feedback controller to finite-time stabilize the system guaranteeing an adequate cost level of performance. Based on linear matrix inequality techniques, some results have been obtained for FTS and GCC for a class of linear time-delay systems, for instance [6-10]. However, according to the author’s knowledge, there is no result available yet on FTS and GCC for linear time-delay systems with bounded controls. ————————————————– 1 This work was completed when the second author was visiting the Vietnam Institute for Advance Study in Mathematics (VIASM). He would like to thank the VIASM for financial support and hospitality. 1 The study of stabilizations of systems with control constraints is not only a natural mathematical problem, but also arises often in many applied areas [11-14]. It is clear that control constraints on the structure of the feedbacks and the neglect of geometric constraints on the control are hardly in accord with present-day requirements for control systems. Moreover, finite-time stability analysis for linear time-varying delay systems is more difficult, because time-varying delay systems have more complicated dynamic behaviors than the systems without delay or with constant delays. Furthermore, it is difficult to find an suitable Lyapunov-Krasovskii functional satisfying the derivative conditions of the FTS and GCC as shown in [15]. Traditionally, Lyapunov function theory has served as a powerful tool for stability and control analysis. The idea of a Lyapunov function was extended in [16] in the context of control design to yield control Lyapunov functions (CLFs). For continuous linear control systems, there exist a well known method to construct CLFs, which essentially involves finding a positive definite solution of Riccati equations or linear matrix inequalities (LMIs) [14, 17-19]. However, the procedures are derived under the assumption of unconstrained control action. For linear bounded control systems without delays, the system matrix satisfies some appropriate spectral and controllability properties, papers [20-22] proposed a nonlinear feedback control to Lyapunov stabilizes the system without delays. It is worth noting that the approach in these works cannot be applied readily to the systems with time-varying delays. The main difficulty is that the investigation of the spectrum of the time-varying delay system matrices is still complicated and there are no appropriate properties available as in the un-delayed case. Consequently, the problem of the finite-time control of linear time-delay systems with bounded controls is of interest in its own right. Motivated by the above discussions, we study the problem of finite-time control for linear autonomous time-varying delay systems with bounded controls. Our main propose is to design a nonlinear feedback controller which guarantees the closed-loop system finite-time stable and guarantees an upper bound on the quadratic cost performance. First, we show how to obtain sufficient conditions for robust finite-time stabilization of linear autonomous delay systems with bounded control by using Lyapunov function method and LMI techniques. Then, we will demonstrate how the obtained stabilization result can be applied to solve the GCC problem for the system. The conditions are obtained in terms of LMIs, which can be determined by utilizing MATLABs LMI Control Toolbox [23]. Finally, an example is given to show the effectiveness of the proposed results. The structure of the paper is as follows. Section 2 gives the necessary background on linear time-varying delay systems with bounded controls and some technical lemmas. In Section 3, the state feedback controller design for robust finite-time stabilization and GCC is presented together with an illustrative example. Section 4 gives some conclusions. 2 Preliminaries In this section, we introduce some notations and lemmas. R+ denotes the set of all real non-negative numbers; Rn denotes the n−dimensional space with the scalar product x⊤ y; M n×r denotes the space of all matrices of (n × r)−dimensions. A⊤ denotes the transpose of matrix A; A is symmetric if A = A⊤ ; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax (A) = max{Reλ; λ ∈ λ(A)}. xt := {x(t + s) : s ∈ [−h, 0]}, ∥ xt ∥= sups∈[−h,0] ∥ x(t + s) ∥; C 1 ([0, t], Rn ) denotes the set of all Rn −valued continuously 2 ˙ C } . L2 ([0, t], Rm ) differentiable functions on [0, t] with the norm: ∥φ∥C 1 = max {∥φ∥C , ∥φ∥ m denotes the set of all the R −valued square integrable functions on [0, t]; Matrix A is called semi-positive definite (A ≥ 0) if xT Ax ≥ 0, for all x ∈ Rn ; A is positive definite (A > 0) if x⊤ Ax > 0 for all x ̸= 0; A > B means A − B > 0. The notation diag{. . .} stands for a block-diagonal matrix. Matrix (Mij )n×m denotes the matrix of Mi,j , i = 1, 2, ..., n, j = 1, 2, ..., m, Mij = Mji , i ̸= j. The symmetric term in a matrix is denoted by ∗. Consider the following linear autonomous time-varying delay systems with bounded control { x(t) ˙ = Ax(t) + Dx(t − h(t)) + Bu(t) + B1 w(t)) t ≥ 0, (2.1) x(t) = φ(t), t ∈ [−h2 , 0], where x(t) ∈ Rn , u(t) ∈ Rm , w ∈ Rr are, respectively, the state, the control, the disturbance vector, A, D ∈ Rn×n , B ∈ Rn×m , B1 ∈ Rn×p are given constant matrices. The initial function φ(t) ∈ C 1 ([−h2 , 0], Rn ) . The delay function h(t) is continuous and satisfying 0 ≤ h1 ≤ h(t) ≤ h2 , ∀t ≥ 0. (2.2) The control u ∈ L2 ([0, T ], Rm ) satisfies ∃r > 0 : ||u(t)|| ≤ r, ∀t ≥ 0. (2.3) The disturbance w(t) ∈ L2 ([0, T ], Rp ) satisfies ∫ T ∃d > 0 : w⊤ (t)w(t)dt ≤ d. (2.4) 0 The performance index associated with the system (2.1) is the following function ∫ T [x⊤ (t)Q1 x(t) + x⊤ (t − h(t))Q2 x(t − h(t)) + u⊤ (t)Q3 u(t)]dt, J(u) = (2.5) 0 where Q1 , Q2 ∈ Rn×n , Q3 ∈ Rm×m , are given symmetric positive definite matrices. The objective of this paper is to design a feedback controller u(t) = Kx(t) satisfying (2.3) and a positive number J ∗ such that the resulting closed-loop system x(t) ˙ = (A + BK)x(t) + Dx(t − h(t)) + B1 w(t), (2.6) is finite-time stable for all disturbance w(t) satisfying (2.2) and the value of the cost function (2.4) is bounded by J ∗ . Definition 2.1. For given positive numbers T, c1 , c2 , c2 > c1 , and symmetric positive definite matrix R, the unforced control system (2.1) is robustly finite-time stable w.r.t (c1 , c2 , T, R) if { } sup φ⊤ (s)Rφ(s) ≤ c1 =⇒ x⊤ (t)Rx(t) < c2 , ∀t ∈ [0, T ], −h2 ≤s≤0 for all disturbance w(t) satisfying (2.4). Definition 2.2. If there exist a feedback control law u∗ (t) = Kx(t) satisfying (2.3) and a positive number J ∗ such that the closed-loop system (2.6) is robustly finite-time stable and the cost function (2.5) satisfies J(u∗ ) ≤ J ∗ , then the value J ∗ is a guaranteed cost value and the designed control u∗ (t) is said to be a guaranteed cost controller. 3 We introduce the following technical propositions, which will be used in the proof of our results. Proposition 2.1. (Schur complement lemma [24]). Given constant matrices X, Y, Z with appropriate dimensions satisfying Y = Y ⊤ > 0, X = X ⊤ . Then X + Z ⊤ Y −1 Z < 0 if and only if ( ) X Z⊤ < 0. Z −Y Proposition 2.2. (Generalized Jensen inequality [25]) For a given symmetric matrix R > 0, any differentiable function φ : [a, b] → Rn , then the following inequality holds ∫ b 1 12 ⊤ (φ(b) − φ(a))⊤ R(φ(b) − φ(a)) + Ω RΩ, b − a b −a a ∫ b φ(b) + φ(a) 1 φ(u)du. where Ω = − 2 b−a a 3 φ˙ ⊤ (u)Rφ(u)du ˙ ≥ Main result In this section, Lyapunov function approach is applied in order to design guaranteed cost controllers for the time-delay system (2.1). The following lemma is necessary for the proof of main theorem. ⊤ BB P x Lemma. Let f (x) = −r 1+∥B T P x∥ , b = ||B||. Then (i) f (x) is global Lipschitz in Rn . (ii) ||f (x)|| ≤ 3rb2 ||P x||, ∀x ∈ Rn . Proof. Let x1 , x2 ∈ Rn and y1 = B ⊤ P x1 , y2 = B ⊤ P x2 . We have [ y ] y1 2 ∥f (x1 ) − f (x2 )∥ = r B − 1 + ∥y2 ∥ 1 + ∥y1 ∥ y1 y2 ≤ rb − 1 + ∥y2 ∥ 1 + ∥y1 ∥ ∥y2 − y1 ∥ + ∥y1 ∥y2 − ∥y2 ∥y1 ≤ rb . (1 + ∥y1 ∥)(1 + ∥y2 ∥) Since y2 ∥y1 ∥ − y1 ∥y2 ∥ = y2 (∥y1 ∥ − ∥y2 ∥) + ∥y2 ∥(y2 − y1 ) ≤ ∥y2 ∥(∥y1 − y2 ∥) + ∥y2 ∥(∥y1 − y2 ∥) = 2∥y2 ∥(∥y1 − y2 ∥). and ∥y1 − y2 ∥ ≤ ∥y1 − y2 ∥, (1 + ∥y1 ∥)(1 + ∥y2 ∥) ∥y2 ∥ ≤ 1, (1 + ∥y1 ∥)(1 + ∥y2 ∥) we have ||f (x1 ) − f (x2 )|| ≤ 3rb||y1 − y2 || ≤ 3rpb2 ||x1 − x2 ||, 4 (3.1) where p = ||P ||. (ii) From (3.1) , we have ||f (x1 ) − f (x2 )|| ≤ 3rb||y1 − y2 || = 3rb||B ⊤ P x1 − B ⊤ P x2 ||. Taking x2 = 0, we have ||f (x)|| ≤ 3rb||B ⊤ P x|| ≤ 3rb2 ||P x||, ∀x ∈ Rn . Theorem 3.1. If there exist symmetric positive definite matrices P, Si , i = 1, 2, 3, X1 , X2 , a positive number η > 0 satisfying the following conditions (Mij )12×12 < 0, (3.2) α2 c1 + 2βd ≤ c2 e−ηT , α1 (3.3) then r B ⊤ P x(t), t ≥ 0, (3.4) 1+ is a guaranteed cost controller for the system (2.1) and the guaranteed cost value is given by u(t) = − ||B ⊤ P x(t)|| J ∗ = α3 ∥φ∥2 + 2βd, where P¯ = R−1/2 P R−1/2 , S¯i = R−1/2 Si R−1/2 , i = 1, 2, 3, γ = r2 λmax (Q3 ), α1 = λmin (P¯ ), { } β = λmax (B1⊤ P B1 ) + 2λmax (B1⊤ S4 B1 ), ∥φ∥ = sup φ⊤ (s)R(s)φ(s), φ˙ ⊤ (s)Rφ(s) ˙ , −h2 ≤s≤0 α2 = λmax (P¯ ) + α3 = λmax (P ) + 2 ∑ hi λmax (S¯i ) + 0.5.(h2 − h1 )2 (h2 + h1 )λmax (S¯3 ) + 2 ∑ i=1 i=1 2 ∑ 2 ∑ hi λmax (Si ) + 0.5.(h2 − h1 )2 (h2 + h1 )λmax (S3 ) + i=1 M11 = AT P + P A − 4 ¯ i ), 0.5h3i λmax (X 0.5h3i λmax (Xi ), i=1 2 ∑ i=1 Xi + 2 ∑ Si + P + Q1 , M22 = −S1 − 4S3 − 4X1 , i=1 M33 = −S2 − 4S3 − 4X2 , M44 = −8S3 , M55 = (h2 − h1 )2 S3 + h21 X1 + h22 X2 − 2S4 , 1 M66 = −12X1 , M77 = −12X2 , M88 = M99 = −12S3 , M10,10 = − I, 1 + 27r2 b4 1 M11,11 = − , M12,12 = −0.5I, M12 = −2X1 , M13 = −2X2 , M14 = P D, M15 = A⊤ S4 , γ M16 = 6X1 , M17 = 6X2 , M18 = M19 = 0, M1,10 = P, M1,11 = P B, M1,12 = 0 M23 = 0, M24 = −2S3 , M25 = 0, M26 = 6X1 , M27 = 0, M28 = 6S3 , M29 = M2,10 = M2,11 = M2,12 = 0, M34 = −2S3 , M35 = M36 = 0, M37 = 6X2 , M38 = 0, M39 = 6S3 , M3,10 = M3,11 = M3,12 = 0, M45 = DT S4 , M46 = M47 = 0, M48 = M49 = 6S3 , M4,10 = M4,11 = M4,12 = 0, M56 = M57 = M58 = M59 = M5,10 = M5,11 = 0, M5,12 = S4 , M67 = M68 = M69 = M6,10 = M6,11 = M6,12 = 0, M78 = M79 = M7,10 = M7,11 = M7,12 = 0, M89 = M8,10 = M8,11 = M8,12 = 0, M9,10 = M9,11 = M9,12 = M10,11 = M10,12 = 0, M11,12 = 0. 5 Proof. Let us consider the bounded feedback control (3.4). By Lemma (i), the function BB T P x f (x) = −r 1+∥B T P x∥ is global Lipschitz, hence, the closed-loop system x(t) ˙ = Ax(t) + A1 x(t − h(t)) + B1 w(t) + f (x(t)), t ∈ R+ , (3.5) has an unique solution. Consider the following Lyapunov-Krasovskii functionals for system (3.5): 4 ∑ Vi (t, xt ), V (t, xt ) = i=1 where ηt ⊤ V1 (t, xt ) =e x (t)P x(t), ∫ V3 (t, xt ) =(h2 − h1 )eηt V4 (t, xt ) = 2 ∑ ∫ hi eηt i=1 We prove that Since V2 (t, xt ) = −h1 ∫ −h2 ∫ 0 −hi t 2 ∑ ∫ e i=1 t t ηt x⊤ (s)Si x(s)ds, t−hi x˙ ⊤ (τ )S3 x(τ ˙ )dτ ds t+s x(τ ˙ )⊤ Xi x(τ ˙ )dτ ds, t+s α1 x⊤ (t)Rx(t) ≤ V (t, xt ), ∀t : 0 ≤ t ≤ T. (3.6) V1 (.) = eηt x⊤ P x = eηt x⊤ R1/2 R−1/2 P R−1/2 R1/2 x = eηt xT R1/2 P¯ R1/2 x, where P¯ = R−1/2 P R−1/2 , we have V1 (.) = eηt x⊤ R1/2 P¯ R1/2 x ≥ x⊤ R1/2 P¯ R1/2 x ≥ λmin (P¯ )x⊤ Rx, and α1 = λmin (P¯ ). Similarly, we can verify the following estimations { } V (0, x0 ) ≤ α2 sup φ⊤ (s)R(s)φ(s), φ˙ ⊤ (s)Rφ(s) ˙ ≤ α2 c1 , −h2 ≤s≤0 V (0, x0 ) ≤ α3 ∥φ∥2 . (3.7) (3.8) Taking the derivative of V1 (.) we have V˙1 = ηeηt 2y ⊤ (t)P x(t) + 2eηt x⊤ (t)P x(t) ˙ [ ] = eηt y T (t)[A⊤ P + P A]x(t) + 2x⊤ (t)P Dx(t − h(t)) + 2x⊤ (t)P f (x(t)) + 2xT (t)P B1 w(t) + ηV1 (.) 2 [ ] ∑ ηt ⊤ ˙ V2 = e x (t)(S1 + S2 )x(t) − x⊤ (t − hi )Si x(t − hi ) + ηV2 (.) i=1 [ ∫ V˙ 3 = eηt (h2 − h1 )2 x⊤(t)S ˙ ˙ − (h2 − h1 ) 3 x(t) [ = eηt (h2 − h1 )2 x˙ ⊤ (t)S3 x(t) ˙ − (h2 − h1 ) ∫ − (h2 − h1 ) ∫ t−h1 t−h2 t−h(t) t−h2 t−h1 ] x˙ (s)S3 x(s)ds ˙ + ηV3 (.) ] x˙ ⊤ (s)S3 x(s)ds ˙ + ηV3 (.) t−h(t) 6 ⊤ x˙ ⊤ (s)S3 x(s)ds ˙ Applying Proposition 2.2 gives ∫ −(h2 − h1 ) t−h(t) ∫ ⊤ x˙ (s)S3 x(s)ds ˙ ≤ −(h2 − h(t)) t−h2 t−h(t) x˙ ⊤ (s)S3 x(s)ds ˙ t−h2 ≤ − (x(t − h(t)) − x(t − h2 ))⊤ S3 (x(t − h(t) − x(t − h2 ))) ( )⊤ ∫ t−h(t) x(t − h(t)) + x(t − h2 ) 1 − 12 − x(s)ds S3 2 h2 − h(t) t−h2 ) ( ∫ t−h(t) 1 x(t − h(t)) + x(t − h2 ) − x(s)ds 2 h2 − h(t) t−h2 ≤ − 4xT (t − h(t))S3 x(t − h(t)) − 4x⊤ (t − h2 )S3 x(t − h2 ) − 4x⊤ (t − h(t))U x(t − h2 ) ∫ t−h(t) ∫ t−h(t) 12 12 ⊤ ⊤ + x (t − h(t))S3 x (t − h2 )S3 x(s)ds + x(s)ds h2 − h(t) h2 − h(t) t−h2 t−h2 ∫ t−h(t) ∫ t−h(t) 12 ⊤ x (s)dsS x(s)ds. − 3 (h2 − h(t))2 t−h2 t−h2 and similarly we have ∫ −(h2 − h1 ) t−h1 x˙ T (s)S3 x(s)ds ˙ ≤ −4x⊤ (t − h(t))U x(t − h(t)) − 4y ⊤ (t − h1 )S3 x(t − h1 ) t−h(t) ∫ t−h1 12 ⊤ x (t − h1 )S3 x(s)ds − 4x (t − h(t))U x(t − h1 ) + h(t) − h1 t−h(t) ∫ t−h1 ∫ t−h1 ∫ t−h1 12 12 ⊤ x⊤ (t − h(t))S3 x(s)ds − + x (s)dsS x(s)ds. 3 h(t) − h1 (h(t) − h1 )2 t−h(t) t−h(t) t−h(t) ⊤ Then, we have [ V˙3 ≤ eηt (h2 − h1 )2 x˙ ⊤ (t)S3 x(t) ˙ − 8y ⊤ (t − h(t))S3 x(t − h(t)) − 4x⊤ (t − h2 )S3 x(t − h2 ) − 4x⊤ (t − h1 )S3 x(t − h1 ) − 4y ⊤ (t − h(t))U x(t − h2 ) ∫ t−h(t) ∫ t−h(t) 12 ⊤ ⊤ x(s)ds − 4x (t − h(t))S3 x(t − h1 ) − x (s)dsS3 (h2 − h(t))2 t−h2 t−h2 ∫ t−h1 ∫ t−h1 12 T − x (s)dsS3 x(s)ds + ηV3 (.) (h(t) − h1 )2 t−h(t) t−h(t) ∫ t−h(t) ∫ t−h(t) 12 12 T ⊤ x (t − h(t))S3 x(s)ds + x (t − h2 )S3 y(s)ds + h2 − h(t) h2 − h(t) t−h2 t−h2 ∫ t−h1 ∫ t−h1 ] 12 12 ⊤ ⊤ + x (t − h1 )S3 x(s)ds + x (t − h(t))S3 x(s)ds . h(t) − h1 h(t) − h1 t−h(t) t−h(t) 7 Using the same calculation as in V˙3 (t, xt ), we get ( V˙4 (t, xt ) ≤ηV4 + eηt x(t) ˙ ⊤ (h21 X1 + h22 X2 )x(t) ˙ + x(t)⊤ (−4X1 − 4X2 )x(t) − 4x(t − h1 )⊤ X1 x(t − h1 ) − 4x(t − h2 )⊤ X2 x(t − h2 ) − 4x(t)⊤ X1 x(t − h1 ) ∫ t 2 ∑ 12 ⊤ ⊤ x(t) Xi − 4x(t) X2 x(t − h2 ) + x(s)ds h t−hi i=1 i ∫ t ∫ ∫ t 2 2 ) ∑ ∑ 12 t 12 ⊤ x(s)ds − x(t − hi )⊤ Xi + x(s) dsX x(s)ds . i 2 hi h t−h t−h t−h i i i i i=1 i=1 Thus, we obtain that [ V˙ (.) ≤eηt x⊤ (t)[AT P + P A − 4(X1 + X2 )]x(t) + 2x⊤ (t)P Dx(t − h(t)) + 2x⊤ (t)P f (x(t)) + 2x⊤ (t)P B1 w(t) + x⊤ (t)(S1 + S2 )x(t) − 2 ∑ x⊤ (t − hi )Si x(t − hi ) i=1 + x˙ ⊤ (t)[(h2 − h1 )2 S3 + h21 X1 + h22 X2 )]x(t) ˙ − 8x⊤ (t − h(t))S3 x(t − h(t)) − 2 ∑ 4x⊤ (t − hi )(S3 + Xi )x(t − hi ) − i=1 2 ∑ 4x⊤ (t − h(t))S3 y(t − hi ) i=1 ∫ t−h(t) ∫ t−h(t) 12 ⊤ x(s)ds − 4x (t)Xi x(t − hi ) − x (s)dsS3 (h2 − h(t))2 t−h2 t−h2 i=1 ∫ t ∫ t−h1 ∫ t−h1 2 ∑ 12 12 T + x(t)⊤ Xi x(s)ds − x (s)dsS x(s)ds 3 hi (h(t) − h1 )2 t−h(t) t−h t−h(t) i i=1 ∫ t ∫ t ∫ 2 2 ) ∑ ∑ 12 t 12 ⊤ ⊤ x(s)ds x(s)ds − x(s) dsX x(t − hi ) Xi + i h h2 t−hi t−hi i=1 i t−hi i=1 i 2 ∑ ⊤ ∫ t−h(t) ∫ t−h(t) 12 12 ⊤ ⊤ + x (t − h(t))S3 x(s)ds + y (t − h2 )S3 x(s)ds h2 − h(t) h2 − h(t) t−h2 t−h2 ∫ t−h1 ∫ t−h1 ] 12 12 ⊤ ⊤ + x (t − h1 )S3 x(s)ds + x (t − h(t))S3 x(s)ds h(t) − h1 h(t) − h1 t−h(t) t−h(t) (3.9) + ηV (t, xt ) Multiplying both sides of equation (3.5) with 4eηt x˙ ⊤ (t)S4 , we obtain eηt [−4x˙ ⊤ (t)S4 x(t) ˙ + 4x˙ ⊤ (t)S4 Ax(t) + 4x˙ ⊤ (t)S4 Dx(t − h(t)) + 4x˙ ⊤ (t)S4 B1 w(t) + 4x˙ ⊤ (t)S4 f (x(t))] = 0. (3.10) Adding all the zero items of (3.10) and zero term eηt f 0 (t, x(t), x(t−h(t)), u(t))− eηt f 0 (t, x(t), x(t − h(t)), u(t)) = 0 into (3.9), respectively and using Lemma (ii) and the Cauchy matrix 8 inequality for the following estimations 4x˙ ⊤ (t)S4 B1 w(t) ≤ 2x˙ ⊤ (t)S4 x(t) ˙ + 2w⊤ (t)B1T S4 B1 w(t); 4x˙ ⊤ (t)S4 f (x(t)) ≤ 2x˙ ⊤ (t)S4 S4 x(t) ˙ + 2∥f (x(t))∥2 ≤ 2x˙ ⊤ (t)S4 S4 x(t) ˙ + 18r2 b4 x⊤ (t)P P x(t); 2x⊤ (t)P f (x(t)) ≤ x⊤ (t)P P x(t) + ∥f (x(t))∥2 ≤ x⊤ (t)P P x(t) + 9r2 b4 x⊤ (t)P P x(t); 2xT (t)P B1 w(t) ≤ w⊤ (t)B1⊤ P B1 w(t) + x⊤ (t)P x(t), u⊤ Q3 u(t) ≤ r2 λmax (Q3 )x⊤ P BB T P x(t) we obtain V˙ (.) ≤ηV (t, xt ) + eηt ζ ⊤ (t)Vζ(t) − eηt f 0 (t, x(t), x(t − h(t)), u(t)) + eηt βw⊤ (t)w(t) (3.11) where β = λmax (B1⊤ P B1 ) + 2λmax (B1⊤ S4 B1 ) and ∫t ∫t 1 ⊤ (s)ds, 1 ⊤ ζ(t) = [x(t), x(t − h1 ), x(t − h2 ), x(t − h(t)), x(t), ˙ x h1 t−h1 h2 t−h2 x (s)ds ∫ t−h1 ∫ t−h(t) 1 1 ⊤ ⊤ (h(t)−h1 ) t−h(t) x(s) ds, (h2 −h(t)) t−h2 x(s) ds], V = (Nij )9×9 , γ = r2 λmax (Q3 ), N11 = A⊤ P + P A − 4 2 ∑ i=1 Xi + 2 ∑ Si + P + γP BB ⊤ P + (1 + 27r2 b4 )P P + Q1 , i=1 N22 = −S1 − 4S3 − 4X1 , N33 = −S2 − 4S3 − 4X2 , N44 = −8S3 , N55 = (h2 − h1 )2 S3 + h21 X1 + h22 X2 − 2S4 + 2S4 S4 , N66 = −12X1 , N77 = −12X2 , N88 = N99 = −12S3 , N12 = −2X1 , N13 = −2X2 , N14 = P D, N15 = A⊤ S4 , N16 = 6X1 , N17 = 6X2 , N18 = N19 = 0, N23 = 0, N24 = −2S3 , N25 = 0, N26 = 6X1 , N27 = 0, N28 = 6S3 , N29 = 0, N34 = −2S3 , N35 = N36 = 0, N37 = 6X2 , N38 = 0, N39 = 6S3 , N45 = DT S4 , N46 = N47 = 0, N48 = N49 = 6S3 , N56 = N57 = N58 = N59 = 0, N67 = N68 = N69 = 0, N78 = N79 = 0, N89 = 0. By Proposition 2.2, the conditions V < 0 is equivalent to the condition (2.2). Therefore, we obtain from (3.11) that V˙ (t, xt ) < ηV (t, xt ) + 2eηt βw⊤ (t)w(t), ∀t ∈ [0, T ]. (3.12) Multiplying both sides of (3.12) with e−ηt we have e−ηt V˙ (t, xt ) − ηe−ηt V (t, xt ) < 2βw⊤ (t)w(t), Integrating both sides of (3.13) from 0 to t, we obtain ∫ t e−ηt V (t, xt ) < V (0, x0 ) + 2β w⊤ (s)w(s)ds, ∀t ∈ [0, T ]. ∀t ∈ [0, T ]. 0 Therefore, from (3.6), (3.7) it follows that α1 e−ηt x(t)⊤ Rx(t) < e−ηt V (t, xt ) ≤ α2 c1 + 2βd, 9 (3.13) and hence α2 c1 + 2βd ηt e ≤ c2 , ∀t ∈ [0, T ]. α1 which implies that the closed-loop systems is robustly finite-time stable w.r.t. (c1 , c2 , T, R). x(t)T Rx(t) < To find the value of the cost function (2.4), we derive from (3.13) that e−ηt V˙ (t, xt ) − ηe−ηt V (t, xt ) ≤ −f 0 (t, x(t), x(t − h(t)), u(t)) + 2βw⊤ (t)w(t), Integrating both sides of (3.14) from 0 to T leads to ∫ T ∫ 0 −ηt f (t, x(t), x(t − h(t)), u(t))dt ≤V (0, x0 ) − e V (t, xt ) + 2β 0 T t ≥ 0. (3.14) w⊤ (t)w(t)dt 0 ≤V (0, x0 ) + 2βd, due to V (t, xt ) ≥ 0. Hence, from (3.8), (3.14) it follows that J ≤ V (0, x0 ) + 2βd ≤ α3 ∥ϕ∥2 + 2βd = J ∗ . This completes the proof of the theorem. Remark 3.1. We note that the condition (3.3) is not LMI with respect to η. Since η does not include in (3.2), we can first find the solutions P, Si , Xi from LMI (3.2) and then determine η from (3.3). Example 3.1. Consider the system (2.1) where [ ] [ ] [ ] [ ] −2 −0.2 −0.5 −0.5 −0.2 −1 A= ,D= ,B= , B1 = 0.5 −3 −0.2 0.4 −0.5 −3 { with h(t) = 0.1 + 0.4 cos t 0.1 if if t ∈ I = ∪k≥0 [2kπ, (2k + 1)π] t ∈ R+ \ I. Note that the functions h(t) is non-differentiable. Given [ ] [ ] [ ] 0.02 0 0.01 0 R = I, Q1 = , Q2 = , Q3 = 0.5 . 0 0.01 0 0.05 By using the LMI Toolbox in MATLAB [22], the conditions (3.2) and (3.3) are satisfied with h1 = 0.1, h2 = 0.5, r = 0.4, r = 0.4, d = 0.01, c1 = 1, c2 = 2.5, T = 10 and η = 0.01, [ ] [ ] [ ] 1.1372 −0.0259 0.4505 −0.0518 0.4013 −0.0512 P = , S1 = , S2 = −0.0259 1.5739 −0.0518 0.8791 −0.0512 0.9069 [ ] [ ] [ ] 0.5447 −0.0097 0.2029 −0.0011 1.6168 0.0173 S3 = , S4 = , X1 = , −0.0097 0.5691 −0.0011 0.1913 0.0173 1.4009 [ ] 0.2733 −0.0054 X2 = . −0.0054 0.2335 Thus the system is robustly finite-time stable w.r.t. (1, 2.5, 10, I) by feedback controller u(t) = 0.4(0.2145x1 + 0.7817x2 ) . 1 + (0.2145x1 + 0.7817x2 ) The guaranteed cost valued of the closed-loop system is as follows: J ∗ = 2.1654∥ϕ∥2 + 0.3797. 10 4 Conclusions In this paper, the problem of robust finite-time stabilization and guaranteed cost control for linear time-varying delay systems with bounded controls has been studied. Based on the Lyapunov functional method and a generalized Jensen integral inequality, new delaydependent delay conditions for the existence of feedback controllers for robust finite-time stabilization are established. The proposed conditions have been applied to guaranteed cost control problem. A numerical example is given to illustrate the efficiency of the proposed method. Acknowledgments. This work was partially supported by the Chiang Mai University, Thailand and the the National Foundation for Science and Technology Development, Vietnam (grant 101.01-2014.35). References [1] Garcia G., Tarbouriech S. & Bernussou J. (2009), Finite-time stabilization of linear time-varying continuous systems. IEEE Trans. Auto. Contr., 54, pp. 364-369. [2] Amato F., Ambrosino R., Cosentino C. (2010), De Tommasi G., Input-output finite time stabilization of linear systems. Automatica, 46, 1558-1562. [3] Liu H., Shena Y. & Zhao X. 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(2006), Stabilization of linear nonautonomous systems with norm-bounded controls. J. Optim. Theory Appl., 131, 135-149. [22] Gahinet P., Nemirovskii A., Laub A.J. & Chilali M. (1995), LMI Control Toolbox For use with MATLAB, The MathWorks, Inc. [23] Boyd S., Ghaoui L.El, Feron E. & Balakrishnan V. (1994), Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM. [24] Seuret A. & Gouaisbaut F. (2013), Wirtinger-based integral inequality: Application to time-delay systems. Automatica, 49, 2860-2866. 12 [...]... criterion for finite-time stability of linear nonautonomous systems with delays Appl Math Lett., 30, 12-18 [8] Mai, T V & Nguyen, T H T (2011) Novel optimal guaranteed cost control of nonlinear systems with mixed multiple time-varying delays IMA J Math Control Inf., 28, 475-486 [9] Palarkci M.N (2009), Robust delay- dependent guaranteed cost controller design for uncertain neutral systems Appl Math Comput.,... Conclusions In this paper, the problem of robust finite-time stabilization and guaranteed cost control for linear time-varying delay systems with bounded controls has been studied Based on the Lyapunov functional method and a generalized Jensen integral inequality, new delaydependent delay conditions for the existence of feedback controllers for robust finite-time stabilization are established The proposed... non-fragile guaranteed cost control of uncertain large-scale systems with time-delays in subsystem interconnections Int J Sys Sci., 35, 233-241 [11] Thuan M.V., Phat V.N , Fernando T.L & H Trinh (2014), Exponential stabilization of time-varying delay systems with nonlinear perturbations, IMA J Contr Inform 31, 441-464 11 [12] Sussman H.J., Sontag E.D & Yang Y (1994), A General result on the stabilization of linear. .. linear systems using bounded controls IEEE Trans Aut Contr., 39, 2411-2425 [13] Zhou B & Duan G.R (2009), Global stabilization of linear systems via bounded controls Syst Contr Lett., 58, 54-61 [14] Moulay E., Dambrine M., Yeganefar N & Perruquetti W (2008), Finite-time stability and stabilization of time -delay systems Syst Contr Lett., 57, 561-566 [15] Gu K., Kharitonov V.L & Chen J (2003), Stability of. .. (2005), Control of linear systems subject to time-domain constraints with polynomial pole placement and LMIs, IEEE Trans Aut Contr., 50, 1360-1363 [19] Slemrod M (1989), Feedback stabilization of a linear control system in Hilbert space with a priori bounded control Math Contr Sig Sys., 22, 265-285 [20] Bounit H & Harmouri H (1997), Bounded feedback stabilization and global separation principle of distributed-parameter... Time Delay Systems Boston, Birkhauser [16] Gomes da Silva J.M & Tarbouriech S (2001), Local stabilization of discrete-time linear systems with saturating controls: an LMl-based approach IEEE Trans Aut Contr., 46, 119-124 [17] Cai X.S & Han Z.Z (2006), Universal construction of control Lyapunov functions for linear systems Lat Am Appl Res., 36, 15-22 [18] Henrion D., Tarbouriech S & Kucera V (2005), Control. .. criterion for finite-time stability of linear discrete-time systems with time-varying delay J Franklin Institute 350, 2745-2756 (2013) [5] Dorato P (1961), Short time stability in linear time-varying systems In: Proc IRE Int Convention Record, 4, 83-87 [6] Yanga R & Wang Y (2013), Finite-time stability analysis and H∞ control for a class of nonlinear time -delay Hamiltonian systems Automatica, 49, 390-401... distributed-parameter systems IEEE Trans Aut Contr., 42, 414-419 [21] Phat V.N & Niamsup P (2006), Stabilization of linear nonautonomous systems with norm -bounded controls J Optim Theory Appl., 131, 135-149 [22] Gahinet P., Nemirovskii A., Laub A.J & Chilali M (1995), LMI Control Toolbox For use with MATLAB, The MathWorks, Inc [23] Boyd S., Ghaoui L.El, Feron E & Balakrishnan V (1994), Linear Matrix Inequalities... linear time-varying continuous systems IEEE Trans Auto Contr., 54, pp 364-369 [2] Amato F., Ambrosino R., Cosentino C (2010), De Tommasi G., Input-output finite time stabilization of linear systems Automatica, 46, 1558-1562 [3] Liu H., Shena Y & Zhao X (2012), Delay- dependent observer-based H∞ finite-time control for switched systems with time-varying delay Nonl Anal.: Hybrid Systems, 6, 885-898 [4] Zuon... to guaranteed cost control problem A numerical example is given to illustrate the efficiency of the proposed method Acknowledgments This work was partially supported by the Chiang Mai University, Thailand and the the National Foundation for Science and Technology Development, Vietnam (grant 101.01-2014.35) References [1] Garcia G., Tarbouriech S & Bernussou J (2009), Finite-time stabilization of linear ... problem of robust finite-time stabilization and guaranteed cost control for linear time-varying delay systems with bounded controls has been studied Based on the Lyapunov functional method and a... result on the stabilization of linear systems using bounded controls IEEE Trans Aut Contr., 39, 2411-2425 [13] Zhou B & Duan G.R (2009), Global stabilization of linear systems via bounded controls. .. optimal guaranteed cost control of nonlinear systems with mixed multiple time-varying delays IMA J Math Control Inf., 28, 475-486 [9] Palarkci M.N (2009), Robust delay- dependent guaranteed cost controller