Time delays are frequently encountered in many practical systems. They are common in many industrial and engineering systems such as communication networks, manufacturing devices, and biology. Since the system stability is an essential requirement in many applications, much effort has been devoted to investigate the stability criteria for various time-delay systems during the last few decades.^[1–4] It is well recognized that the presence of time delay in a dynamical system is often a primary source of instability and performance degradation. Moreover, the delay-dependent^[5] results are generally less conservative than the delay-independent ones,^[6] especially when the delay is small. Recently, much attention has been paid to the robust stability problem for uncertain systems with time delays.^[7,8]

Markovian jump systems are a special sort of hybrid system, which are driven by a Markov chain, where they may undergo unexpected changes in their structures and parameters, e.g., aerospace systems, power systems, etc.^[9–11] Such systems can be described by a set of time-delayed linear systems with the transitions between models determined by a Markov chain in a finite mode set. Applications of this class of systems may be found in processes including those in production systems and economic problems, etc. In the last few years, some quite significant results on the stability analysis and controller synthesis for Markovian jumping systems with time-delay have been reported.^[11,12]

The control design for uncertain systems with time-varying delays has been a difficult problem of control theory.^[13,14] Theoretically, this issue could be solved by the Lyapunov method. The H_∞ performance is usually analyzed in the control theory to synthesize controllers achieving stabilization with guaranteed performance. Stability is one of the most important problems in the synthesis of control systems. Therefore, the problems of delay-dependent stability analysis and H_∞ control for delayed systems have received substantial attention among the control community for the last few decades.^[15]

The H_∞ control concept was proposed to reduce the effect of the disturbance input on the regulated output within a prescribed level and guarantee that the closed-loop system is stable. In Refs. [16] and [17], the H_∞ technology was proposed to solve the disturbance attenuation problem in various difficult systems. In this paper, we provide a delay-dependent H_∞ control for Markovian jump uncertain systems with interval time-varying and distributed delays. A linear matrix inequality (LMI) optimization approach is used to achieve the best effect of disturbance attenuation. The obtained results in this paper are generally less conservative than those of some existing methods.

The following notations are used. Throughout this paper, ℝⁿ and ℝ^n×n denote the n-dimensional Euclidean space and the set of all n × n real matrices, respectively. For a matrix B and two symmetric matrices A and C, denotes a symmetric matrix, where the notation * represents the entries implied by symmetry. A^T and A^–1 are the matrix transpose and inverse of A, respectively. X > 0 means that matrix X is a symmetric positive definite matrix with appropriate dimensions. I denotes the identity matrix with appropriate dimensions. ε denotes the expectation operator with respect to some probability measure . Let be a complete probability space which relates to an increasing family of σ algebras where Ω is the sample space, is the σ algebra of subsets of the sample space, and is the probability measure on . Let f(t) ∈ L₂[0,∞), ||f|| refers to the Euclidean norm of function f(t) at time t. is the space of square integrable vectors on [0,∞).

Let {r_t,t ≥ 0} be a right-continuous Markov chain on a probability space taking values in a finite state space with generator Π = {π_pq} given by

Consider the following uncertain Markovian jump system with interval time-varying and distributed delays:

Lemma 1 (Schur complement^[18]) Let M, P, Q be given matrices and Q > 0, then

Lemma 2^[18] For any constant matrix M ∈ ℝ^{n × n}, M = M^T > 0, scalar η > 0, vector function w: [0, η] → ℝⁿ such that the integrations concerned are well defined,

Lemma 3^[19] Let D, E, and F(t) be real matrices of appropriate dimensions, and F(t) satisfies F^T(t) F(t) ≤ I, then the following inequality holds for any constant ε > 0:

For the sake of notation simplification, in the following, for each possible r_t = p (p = 1,2,…,m), we simply write L(r_t) as L_i; for instance, A(r_t,t) is denoted by A_p(t), and A_d(r_t,t) by A_dp, and so on. On the other hand, we have u(r_t) = u_p(t). We now rewrite system (1) by using state feedback control u_p(t) = K_px(t) as follows:

Theorem 1 For a prescribed scalar γ > 0 and some given positive scalars τ_m, τ_M, h, μ and matrix K, system (4) with condition (2) is asymptotically stable and satisfies ||z(t)||₂ ≤ γ||w(t)||₂ for all nonzero if there exist matrices P_p > 0, Q_i > 0 (i = 1,2,3), R > 0, S_j > 0 (j = 1,2), and appropriate dimension matrices N_l, O_l (l = 1,2,3,4,5) such that the following LMI is satisfied:

Proof Define the Lyapunov functional candidate as

Substituting Eqs. (8)–(11) into Eq. (7), we obtain

Theorem 2 For a prescribed scalar γ > 0 and some given positive scalars τ_m, τ_M, u, h and matrix K, system (4) with condition (2) is robustly asymptotically stable and satisfies ||z(t)||₂ ≤ γ||w(t)||₂ for all nonzero if there are matrices P_p > 0, Q_i > 0 (i = 1,2,3), R > 0, S_j > 0 (j = 1,2), and appropriate dimension matrices N_l, O_l (l = 1,2,3,4,5), and a scalar ε_p > 0 such that the following LMI is satisfied:

Proof Replace A_p, A_dp, A_fp, and B_p in LMI (5) by A_p + ΔA_p(t), A_dp + ΔA_dp(t), A_fp + ΔA_fp(t), and B_p + ΔB_p(t), respectively, we obtain

Theorem 3 For a prescribed scalar γ > 0 and some given positive scalars τ_m,τ_M,u,h, system (4) with condition (2) is robustly asymptotically stable and satisfies ||z(t)||₂ ≤ γ||w(t)||₂ for all nonzero if there are matrices Q̄_i > 0 (i = 1,2,3), R > 0, S_j > 0 (j = 1,2), X_p > 0, Y_p, appropriate dimension matrices N_l, O_l (l = 1,2,3,4,5), and a scalar ε_p > 0 such that the following LMI is satisfied:

Proof We define

In this section, some numerical examples are provided to demonstrate that the method proposed in this paper is effective and less conservative than some existing methods.

Example 1 Consider system (1) with the following matrix parameters.

Mode 1:

Mode 2:

Example 2 Consider the following time delay system with system mode p = 1:

Taking parameters h = 0 and unknown μ, and using LMI in Theorem 1, we obtain the minimum allowable H_∞ performance γ for different values of τ_M and τ_m as shown in Table 1, which clearly shows that the results of the proposed method in this paper are significantly better than those in Refs. [7] and [8].

Table 1.

Table 1.

Table 1. Example 4: minimized H∞ performance for different τM and τm. . Method τM = 0.8695 τM = 1 τm = 0 τm = 0.5695 τm = 0 τm = 0.5 Ref. [7] 0.87 0.81 4.05 3.27 Ref. [8] 0.81 0.72 2.95 2.13 Thm 3 0.5614 0.5305 1.7458 1.1254

Table 1. Example 4: minimized H∞ performance for different τM and τm. .

Example 3 Consider system (18) with the following matrix parameters in Ref. [9]:

Table 2.

Table 2.

Table 2. Example 4: τM and γ. . Method μ 0.5 1.0 1.5 2.0 Ref. [9] 0 0.47 0.52 0.55 0.56 Thm 3 0.60 0.64 0.67 0.69 Ref. [10] 0.5 0.3666 0.4293 0.4450 0.4524 Thm 3 0.5355 0.5754 0.5863 0.5898

Table 2. Example 4: τM and γ. .

We have studied the H_∞ control problem for systems with interval time-varying and distributed delays. On the basis of the Lyapunov functional, a delay-dependent H_∞ control scheme is presented in terms of LMIs. It is shown that a desired state feedback controller can be constructed when the given LMIs are feasible, and it leads to less conservative results than the existing methods. Finally, numerical examples are given to illustrate the effectiveness of the proposed method. We are interested to extend this work to the discrete-time uncertain stochastic systems with discrete interval and distributed time-varying delays. The results will be obtained in the future.