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This paper deals with the finitetime stability and H∞ control of linear discretetime delay systems. The system under consideration is subject to interval timevarying delay and normbounded disturbances. Linear matrix inequality approach is used to solve the finitetime stability problem. First, new sufficient conditions are established for robust finitetime stability of the linear discretetime delay system subjected to normbounded disturbances, then the state feedback controller is designed to robustly finitetime stabilize the system and guarantee an adequate level of system performance. The delaydependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed results.
Finite-time stability and H∞ control of linear discrete-time delay systems with norm-bounded disturbances Le A. Tuana and Vu N. Phatb,∗ a Department of Mathematics College of Sciences, Hue University, Hue, Vietnam b Institute of Mathematics, VAST 18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam ∗ Corresponding author: vnphat@math.ac.vn Abstract This paper deals with the finite-time stability and H∞ control of linear discretetime delay systems. The system under consideration is subject to interval time-varying delay and norm-bounded disturbances. Linear matrix inequality approach is used to solve the finite-time stability problem. First, new sufficient conditions are established for robust finite-time stability of the linear discrete-time delay system subjected to norm-bounded disturbances, then the state feedback controller is designed to robustly finite-time stabilize the system and guarantee an adequate level of system performance. The delay-dependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed results. Key words. Finite-time stability, H∞ control, time-varying delay, disturbances, linear matrix inequalities. 2000 Mathematics Subject Classifications: 34D10, 34K20, 49M7 1 Introduction Finite-time stability (FTS) introduced by Dorato in [1] involves dynamical systems whose solutions converge to an equilibrium state in finite time. Compared with the Lyapunov stability, FTS is a more practical property, useful to study the behavior of the system within a finite interval time, and therefore it finds many applications. A lot of interesting results on finite-time stability and stabilization in the context of linear discrete-time delay systems have been obtained (see, e.g. [2-5] and the references therein). 1 On the other hand, problem of finite-time H∞ control has attracted much attention due to its both practical and theoretical importance. Various approaches have been developed and a great number of results for linear continuous and discrete-time systems have been reported in the literatures (see, e.g. [6-10] and the references therein). Note that these papers were limited either to Lyapunov stability or to the system with constant delays. Based on the solution to some LMIs, the finite-time H∞ control for discrete-time systems without delay was proposed in [11, 12]. Recently, the authors in [13] have considered the finite-time H∞ control for discrete-time system with constant delays. To the best of our knowledge, finite-time H∞ control problem for linear discrete-time systems with interval time-varying delay has not fully investigated. The problem is important and challenging in many practice applications, which motivates the main purpose of our research. In this paper, we extend further the results of finite-time stability and H∞ control for linear discrete-time delay systems with norm-bounded disturbances. Our main propose is to design a state feedback controller which guarantees FTS of the closed-loop system and reduces the effect of the disturbance input on the controlled output to a prescribed level. The novel features of this paper are: (i) The system under consideration subjected to interval time-varying delays in both the state input and observation output; (ii) Using new bounding LMI estimation technique, a set of improved Lyapunov-like functionals is constructed to design the H∞ feedback controller in terms of LMIs, which can be determined by utilizing MATLAB’s LMI Control Toolbox [14]. The paper is organized as follows. In Section 2 some preliminary definitions are provided and the problem we deal with is precisely stated. Section 3 presents the main results of the paper: sufficient conditions for robust finite-time H∞ boundedness and control in terms of LMIs. Numerical examples showing the effectiveness of the proposed method are given. 2 Preliminaries The following notations will be used throughout this paper. Z+ denotes the set of all nonnegative integers; Rn denotes the n−dimensional space with the scalar product x⊤ y; Rn×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 of appropriate dimension. Matrix A is called semi-positive definite (A 0) if x⊤ 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. The symmetric term in a matrix is denoted by ∗. Consider the following linear discrete-time systems with time-varying delay x(k + 1) = Ax(k) + Ad x(k − d(k)) + Bu(k) + Gω(k), z(k) = Cx(k) + Cd x(k − d(k)), k ∈ Z+ , x(k) = φ(k), k ∈ {−d2 , −d2 + 1, . . . , 0}, (1) where x(k) ∈ Rn is the state; u(k) ∈ Rm is the control input; z(k) ∈ Rp is the observation output; A, Ad ∈ Rn×n , B ∈ Rn×m , G ∈ Rn×q , C, Cd ∈ Rp×n are given real constant matrices; 2 d(k) is delay function satisfying the condition 0 < d1 d(k) d2 ∀k ∈ Z+ , (2) where d1 , d2 are known positive integers; φ(k) is the initial function; ω(k) ∈ Rq satisfying the condition N ∑ ∃d > 0 : w⊤ (k)w(k) < d. (3) k=0 Definition 2.1. (Finite-time stability) Given positive numbers N, c1 , c2 , c1 < c2 , and a symmetric positive-definite matrix R, the discrete-time delay system (1) with u(k) = 0 is said to be robustly finite-time stable w.r.t. (c1 , c2 , R, N ) if max x⊤ (k)Rx(k) k∈{−d2 ,−d2 +1,...,0} c1 =⇒ x⊤ (k)Rx(k) < c2 , ∀k = 1, 2, . . . , N, for all disturbances ω(k) satisfying (3). Definition 2.2. (Finite-time H∞ boundedness) Given positive numbers γ, N, c1 , c2 , c1 < c2 , and a symmetric positive-definite matrix R, the system (1) with u(k) = 0 is said to be robustly finite-time H∞ bounded w.r.t. (c1 , c2 , R, N ) if the following two conditions hold: (i) The system (1) with u(k) = 0 is robustly finite-time stable w.r.t. (c1 , c2 , R, N ). (ii) Under the zero initial condition (i.e., φ(k) = 0 ∀k ∈ {−d2 , −d2 + 1, . . . , 0}), the output z(k) satisfies N ∑ ⊤ z (k)z(k) γ N ∑ w⊤ (k)w(k), (4) k=0 k=0 for all disturbances ω(k) satisfying (3). Definition 2.3. (Finite-time H∞ control) Given positive numbers γ, N, c1 , c2 , c1 < c2 , and a symmetric positive-definite matrix R, the finite-time H∞ control problem for the system (1) has a solution if there exists a state feedback controller u(k) = Kx(k) such that the resulting closed-loop system is robustly finite-time H∞ bounded w.r.t. (c1 , c2 , R, N ). Proposition 2.1. (Schur Complement Lemma, [15]). Given constant matrices X, Y, Z with appropriate dimensions satisfying X = X ⊤ , Y = Y ⊤ > 0, then ] [ X Z⊤ ⊤ −1 < 0. X + Z Y Z < 0 ⇐⇒ Z −Y 3 3 Main results This section provides sufficient conditions for finite-time H∞ boundedness and control for the system (1) with interval time-varying delay and norm-bounded disturbances. Theorem 3.1. Given positive constants γ, N, c1 , c2 , c1 < c2 , a symmetric positive-definite matrix R, the system (1) with u(k) = 0 is robustly finite-time H∞ bounded w.r.t. (c1 , c2 , R, N ) if for scalar δ > 1, there exist symmetric positive-definite matrices P, Q, positive scalars λ1 , λ2 , λ3 such that the following LMIs hold: λ1 R < P < λ2 R, Q < λ3 R, −δP + (d2 − d1 + 1)Q 0 0 A⊤ P C ⊤ ⊤ ∗ −δ d1 Q 0 A⊤ d P Cd γ ⊤ ∗ ∗ − δN I G P 0 < 0, ∗ ∗ ∗ −P 0 ∗ ∗ ∗ ∗ −I N +1 γd − c2 δλ1 c1 δ λ2 ρλ3 N +1 ∗ −c1 δ λ2 0 < 0, ∗ ∗ −ρλ3 ] [ 1 (d1 −1) . where ρ := c1 δ N +d2 −1 d2 δ + d2 (d2 −1)−d 2 (5) (6) (7) Proof. Consider the following non-negative quadratic functions: V (k) = V1 (k) + V2 (k) + V3 (k), where V1 (k) = x⊤ (k)P x(k), k−1 ∑ V2 (k) = δ k−1−s x⊤ (s)Qx(s) s=k−d(k) V3 (k) = −d 1 +1 ∑ k−1 ∑ δ k−1−t x⊤ (t)Qx(t). s=−d2 +2 t=k−1+s Taking the difference variation of Vi (k), i = 1, 2, 3, we have V1 (k + 1) − δV1 (k) = x⊤ (k + 1)P x(k + 1) − δx⊤ (k)P x(k) ⊤ ⊤ x(k) x(k) A [ ] P A Ad G x(k − d(k)) − δx⊤ (k)P x(k), = x(k − d(k)) A⊤ d ω(k) G⊤ ω(k) (V2 + V3 )(k + 1) − δ(V2 + V3 )(k) = k ∑ s=k+1−d(k+1) + −d 1 +1 ∑ k ∑ δ k−t x⊤ (t)Qx(t) − s=−d2 +2 t=k+s k−1 ∑ δ k−s x⊤ (s)Qx(s) − s=k−d(k) −d 1 +1 ∑ k−1 ∑ s=−d2 +2 t=k−1+s 4 δ k−s x⊤ (s)Qx(s) δ k−t x⊤ (t)Qx(t) k−1 ∑ ⊤ = x (k)Qx(k) + δ k−d ∑1 k−s ⊤ x (s)Qx(s) + s=k−d1 +1 k−1 ∑ − δ k−s x⊤ (s)Qx(s) s=k+1−d(k+1) δ k−s x⊤ (s)Qx(s) − δ d(k) x⊤ (k − d(k))Qx(k − d(k)) s=k−d(k)+1 + −d 1 +1 ∑ [ k−1 ∑ x⊤ (k)Qx(k) + s=−d2 +2 δ k−t x⊤ (t)Qx(t) − t=k+s k−1 ∑ = x (k)Qx(k) + δ + ] 1−s ⊤ x (k − 1 + s)Qx(k − 1 + s) k−1 ∑ k−s ⊤ x (s)Qx(s) − s=k−d1 +1 k−d ∑1 δ k−t x⊤ (t)Qx(t) t=k+s −δ ⊤ k−1 ∑ δ k−s x⊤ (s)Qx(s) s=k−d(k)+1 δ k−s x⊤ (s)Qx(s) − δ d(k) x⊤ (k − d(k))Qx(k − d(k)) s=k+1−d(k+1) + −d 1 +1 ∑ [ ⊤ ] x (k)Qx(k) − δ 1−s x⊤ (k − 1 + s)Qx(k − 1 + s) s=−d2 +2 k−d ∑1 ⊤ x (k)Qx(k) + δ k−s x⊤ (s)Qx(s) s=k+1−d(k+1) −δ − d(k) ⊤ x (k − d(k))Qx(k − d(k)) + (d2 − d1 )x⊤ (k)Qx(k) −d 1 +1 ∑ δ 1−s x⊤ (k − 1 + s)Qx(k − 1 + s) s=−d2 +2 (d2 − d1 + 1)x⊤ (k)Qx(k) − δ d1 x⊤ (k − d(k))Qx(k − d(k)) + k−d ∑1 δ k−s ⊤ x (s)Qx(s) − k−d ∑1 δ k−s x⊤ (s)Qx(s) s=k+1−d2 s=k+1−d(k+1) ⊤ d1 ⊤ (d2 − d1 + 1)x (k)Qx(k) − δ x (k − d(k))Qx(k − d(k)). Thus we have V (k + 1) − δV (k) ⊤ ⊤ x(k) A x(k) [ ] x(k − d(k)) A⊤ P A Ad G x(k − d(k)) d ⊤ ω(k) G ω(k) [ ] + x⊤ (k) −δP + (d2 − d1 + 1)Q x(k) − δ d1 x⊤ (k − d(k))Qx(k − d(k)) γ γ + z ⊤ (k)z(k) − N ω ⊤ (k)ω(k) + N ω ⊤ (k)ω(k) − z ⊤ (k)z(k). δ δ Note that by setting [ ]⊤ ξ(k) := x⊤ (k) x⊤ (k − d(k)) ω ⊤ (k) , [ ] Υ := P A P Ad P G , 5 −δP + (d2 − d1 + 1)Q + C ⊤ C C ⊤ Cd 0 ∗ −δ d1 Q + Cd⊤ Cd 0 , Φ := ∗ ∗ − δγN I we can see that ⊤ ⊤ x(k) x(k) A [ ] x(k − d(k)) A⊤ P A Ad G x(k − d(k)) = ξ ⊤ (k)Υ⊤ P −1 Υξ(k) d G⊤ ω(k) ω(k) and [ ] x⊤ (k) −δP +(d2 − d1 + 1)Q x(k) − δ d1 x⊤ (k − d(k))Qx(k − d(k)) γ + z ⊤ (k)z(k) − N ω ⊤ (k)ω(k) δ [ ] ⊤ = x (k) −δP + (d2 − d1 + 1)Q + C ⊤ C x(k) + 2x⊤ (k)C ⊤ Cd x(k − d(k)) [ ] γ + x⊤ (k − d(k)) −δ d1 Q + Cd⊤ Cd x(k − d(k)) − N ω ⊤ (k)ω(k) δ ⊤ = ξ (k)Φξ(k). Therefore, we get [ ] γ ξ ⊤ (k) Φ + Υ⊤ P −1 Υ ξ(k) + N ω ⊤ (k)ω(k) − z ⊤ (k)z(k). (8) δ Next, by applying the Proposition 2.1, we have −δP + (d2 − d1 + 1)Q + C ⊤ C C ⊤ Cd 0 A⊤ P ∗ −δ d1 Q + Cd⊤ Cd 0 A⊤ d P Φ+Υ⊤ P −1 Υ < 0 ⇐⇒ λ1 x⊤ (k)Rx(k), V (k) ∀k ∈ Z+ . (13) Note that by the Proposition 2.1, the LMI (7) is equivalent to [ ] [ ] [ N +1 ] c1 δ N +1 λ2 0 −1 c1 δ N +1 λ2 λ2 ρλ3 γd − c2 δλ1 + c1 δ 1, there exists symmetric positive definite matrices P, Q, positive scalars λ1 , λ2 , λ3 such that the LMIs (5)-(7) hold, therein matrix A + BK will in place of the matrix A. In other words, in proportion to the LMI (6), we have −δP + (d2 − d1 + 1)Q 0 0 (A + BK)⊤ P C ⊤ ∗ −δ d1 Q 0 A⊤ Cd⊤ dP γ ⊤ < 0. I G P 0 ∗ ∗ − (22) N δ ∗ ∗ ∗ −P 0 ∗ ∗ ∗ ∗ −I Pre- and post-multipling (22) by matrix diag{P −1 , P −1 , I, P −1 , I} > 0 yields −δP −1 + (d2 − d1 + 1)P −1 QP −1 0 0 P −1 (A + BK)⊤ P −1 C ⊤ ∗ −δ d1 P −1 QP −1 0 P −1 A⊤ P −1 Cd⊤ d γ ⊤ < 0. ∗ ∗ − I G 0 N δ −1 ∗ ∗ ∗ −P 0 ∗ ∗ ∗ ∗ −I (23) Let’s define new matrix variables as follows: U = P −1 , V = P −1 QP −1 . Then, (23) becomes −δU + (d2 − d1 + 1)V 0 0 U (A + BK)⊤ U C ⊤ ∗ −δ d1 V 0 U A⊤ U Cd⊤ d γ ⊤ < 0. I G 0 ∗ ∗ − N δ ∗ ∗ ∗ −U 0 ∗ ∗ ∗ ∗ −I Letting Y ⊤ = U K ⊤ , K = Y U −1 , we get the LMI (19). For getting LMI (20), we note that the inequality (7) can be regarded as (γd − c2 δλ1 )I c1 δ N +1 λ2 I ρλ3 I ∗ −c1 δ N +1 λ2 I 0 < 0. (24) ∗ ∗ −ρλ3 I Post-multipling (24) by matrix diag{R, R, R} > 0 gives γdR − c2 δλ1 R c1 δ N +1 λ2 R ρλ3 R ∗ −c1 δ N +1 λ2 R 0 < 0. ∗ ∗ −ρλ3 R (25) Again pre- and post-multipling (25) by matrix diag{P −1 , P −1 , P −1 } > 0, we get γdP −1 RP −1 − c2 δP −1 (λ1 R)P −1 c1 δ N +1 P −1 (λ2 R)P −1 ρP −1 (λ3 R)P −1 < 0. (26) ∗ −c1 δ N +1 P −1 (λ2 R)P −1 0 −1 −1 ∗ ∗ −ρP (λ3 R)P Setting new variables W1 = −γdP −1 RP −1 + c2 δP −1 (λ1 R)P −1 , W2 = P −1 (λ2 R)P −1 , 9 W3 = P −1 (λ3 R)P −1 , the LMI (26) reduces to the LMI (20) as desired. To obtain the LMI (18), we just pre- and post-multipling (5) by the matrix P −1 . Finally, note that W1 = −γdP −1 RP −1 + c2 δP −1 (λ1 R)P −1 < −γdU RU + c2 δU, we obtain W1 − c2 δU + γdU R[γdR]−1 γdRU < 0, which is obviously equivalent to the LMI (21) by the Schur Complement Lemma. The proof of the theorem is completed. Remark 3.1. Different from the previous results [8, 11-13], the Lyapunov function method is not used for the proof of Theorem 3.1. All the sufficient conditions of Theorem 3.1, Theorem 3.2 are given in terms of LMIs, which can be easily calculated by the LMI Toolbox in MATLAB. Remark 3.2. In the papers [6, 8, 11-13], additional unknowns and free-weighting matrices are introduced to make the flexibility to solve the resulting LMIs. However, too many unknowns and free-weighting matrices employed in the existing methods complicate the system analysis and significantly increase the computational demand. Compared with the free matrix method used in these papers, our simpler uncorrelated augmented matrix method uses fewer variables, e.g., the LMI (6) has no free-weighting matrices, the LMI (19) has one free-weighting matrix. Consequently, our criterions are less conservative in comparison with others. This effectiveness of the results will be illustrated by the following examples. Example 3.1. Consider the linear discrete-time delay system (1) with u(k) = 0 and its parameters are described by ] −0.25 0.1 , A= 0.2 0.3 [ [ ] 0.1 −0.2 , C= −0.15 0.15 h(k) = 2 + 8 sin2 kπ , 2 ] −0.12 0.1 , Ad = 0.15 0.1 [ ] 0.2 0.1 , G= 0.2 0.25 ] −0.1 0.25 Cd = , 0.2 −0.15 [ [ R = I, k ∈ Z+ . Note that the delay function h(k) is interval time-varying and d1 = 2, d2 = 10. For given N = 200, d = 1, c1 = 1, c2 = 7, and γ = 1, by using LMI Control Toolbox in MATLAB, the LMIs (5)-(7) are feasible with δ = 1.0001 and [ ] [ ] 2.1225 0.0261 0.1901 −0.0194 P = , Q= , 0.0261 2.0246 −0.0194 0.1548 λ1 = 2.0180, λ2 = 2.1292, λ3 = 0.1987. By the Theorem 3.1, the system is robustly finite-time H∞ bounded w.r.t. (1, 7, I, 200). Example 3.2. Consider the system (1) where: 10 Example 3.2. Consider the system (1) where: [ ] 0.4 0.1 A= , 0.3 0.5 [ ] 0.2 −0.15 Ad = , 0.15 0.1 [ ] C = 0.2 0.3 , [ ] Cd = 0.2 0.15 , h(k) = 2 + 10 cos2 kπ , 2 [ ] 0.1 B= , 0.2 [ ] 0.25 G= , 0.3 [ ] 1.2 0 R= , 0 1.3 k ∈ Z+ . For given d1 = 2, d2 = 12, N = 140, d = 1, c1 = 2, c2 = 16 and γ = 1, the LMIs (18)-(21) are feasible with δ = 1.00027 and [ ] 0.2836 0.1250 U= , 0.1250 0.3413 ] 0.3417 0.0824 , W2 = 0.0824 0.3739 [ [ ] 0.0217 0.0118 V = , 0.0118 0.0274 ] 0.0222 0.0114 , W3 = 0.0114 0.0276 [ [ ] 4.3514 1.9538 W1 = , 1.9538 5.2522 [ ] Y = −0.9638 −1.2080 . By the Theorem 3.2, the finite-time H∞ control problem of the system (1) has a solution and the state feedback controller is given by [ ] u(k) = −2.1922 −2.7365 x(k), k ∈ Z+ . 4 Conclusion In this paper, finite-time stability and H∞ control problems have been investigated for a class of linear discrete-time systems subjected to interval time-varying delay and norm-bounded disturbances. By constructing a set of improved Lyapunov-like functionals, sufficient conditions for robust finite-time stability of the system are proposed. Starting from these results, we have provided sufficient conditions for the solution of the H∞ control of the system. The proposed conditions are expressed in terms of LMIs. Numerical examples are presented to illustrate the effectiveness of the proposed results. Acknowledgments. This work was completed when the second author was visiting the Vietnam Institute for Advance Study in Mathematics (VIASM). 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D.L., Antic D.S., The application of different Lyapunov-like functionals and some aggregate norm approximations of the delayed states for finitetime stability analysis of linear discrete time -delay systems, Journal of the Franklin Institute, 351(2014), 3914-3931 [6] He Y., Wu M., Han Q-L., She J -H. , Delay- dependent H control of linear discretetime systems with an interval-like time-varying delay, International... systems with norm-bounded disturbance, Journal of the Franklin Institute, 348(2011), 331-352 [12] Wang K., Shen Y., Liu W., Jiang Q., Finite-time H control for a class of discretetime linear system, Chinese Control and Decision Conference (CCDC), 2010 Chinese, 546-549 [13] Song H. , Yu L., Zhang D., Zhang W-A., Finite-time H control for a class of discretetime switched time -delay systems with quantized... Analysis and design conditions, Automatica, 46(2010), 919-924 [3] Zuo Z., Li H. , Wang Y., New criterion for finite-time stability of linear discrete-time systems with time-varying delay, Journal of the Franklin Institute, 350(2013), 27452756 [4] Zhang Z., Zhang H. , Zheng B., Kamiri H. R., Finite-time stability analysis and stabilization for linear discrete-time system with time-varying delay, Journal of the... International Journal of Systems Science, 39(2008), 427-436 [7] Meng Q., Shen Y., Finite-time H control for linear continuous system with normbounded disturbance, Commun Nonlinear Sci Numer Simulat., 14(2009) 1043-1049 [8] Liu J., Zhang J., He M., Zhang H. , New results on robust H control for discrete-time systems with interval time-varying delays, J Control Theory Appl., 9(2011), 611-616 [9] Zhang Y., Liu... −0.9638 −1.2080 By the Theorem 3.2, the finite-time H control problem of the system (1) has a solution and the state feedback controller is given by [ ] u(k) = −2.1922 −2.7365 x(k), k ∈ Z+ 4 Conclusion In this paper, finite-time stability and H control problems have been investigated for a class of linear discrete-time systems subjected to interval time-varying delay and norm-bounded disturbances By constructing... finite-time H control of singular stochastic systems via static output feedback, Applied Mathematics and Computation, 218(2012) 56295640 [10] Tuan L.A., Nam P.T., Phat V.N., New H controller design for neural networks with interval time-varying delays in state and observation, Neural Process Lett., 37(2013), 235-249 [11] Xiang W., Xiao J., H finite-time control for switched nonlinear discrete-time systems with. .. set of improved Lyapunov-like functionals, sufficient conditions for robust finite-time stability of the system are proposed Starting from these results, we have provided sufficient conditions for the solution of the H control of the system The proposed conditions are expressed in terms of LMIs Numerical examples are presented to illustrate the effectiveness of the proposed results Acknowledgments This... 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 References [1] Dorato P., Short-time stability in linear time-varying systems, In: IRE International Convention Record, Part IV, 1961, 83-87 11 [2] Amato F., Ariola M., Cosentino C., Finite-time control of discrete-time linear systems: ... time -delay systems with quantized feedback, Commun Nonlinear Sci Numer Simulat., 17(2012), 4802-4814 [14] Gahinet P., Nemirovskii A., Laub A.J., Chilali M., LMI Control Toolbox For Use with MATLAB, Massachusetts, The MathWorks, Inc., 1995 [15] Boyd S., Ghaoui L El, Feron E., V Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994 12 ...Example 3.2 Consider the system (1) where: [ ] 0.4 0.1 A= , 0.3 0.5 [ ] 0.2 −0.15 Ad = , 0.15 0.1 [ ] C = 0.2 0.3 , [ ] Cd = 0.2 0.15 , h( k) = 2 + 10 cos2 kπ , 2 [ ] 0.1 B= , 0.2 [ ] 0.25 G= , 0.3 [ ] 1.2 0 R= , 0 1.3 k ∈ Z+ For given d1 = 2, d2 = 12, N = 140, d = 1, c1 = 2, c2 = 16 and γ = 1, the LMIs (18)-(21) are feasible with δ = 1.00027 and [ ] 0.2836 0.1250 U= , 0.1250 0.3413 ... the other hand, problem of finite-time H control has attracted much attention due to its both practical and theoretical importance Various approaches have been developed and a great number of. .. the system with constant delays Based on the solution to some LMIs, the finite-time H control for discrete-time systems without delay was proposed in [11, 12] Recently, the authors in [13] have... results of finite-time stability and H control for linear discrete-time delay systems with norm-bounded disturbances Our main propose is to design a state feedback controller which guarantees FTS of