With the development of modern industrial production, more and more systems encountered in practice can no longer be completely described by a single continuous-time dynamic process or discrete event process. Therefore, hybrid systems such as switched systems and impulsive systems have aroused great interest among researchers. Switched systems are a class of dynamical systems that consist of a finite number of subsystems and a logical rule that orchestrates switching between these subsystems. It is important to study switched systems because of the fact that they provide a natural and convenient unified framework for mathematical modeling of many physical phenomena [1][2] and practical applications [3]. An impulsive system is a kind of hybrid system that shows the continuous dynamics on the continuous-time interval and the discrete dynamics at some moments which impulses occur. Due to characters of lower cost and higher confidentiality, there are many applications of impulsive systems, such as economics [4], communication security [5], and biomedical engineering [6].

It is worth noting that the subsystems in switched systems are easily affected by impulses when switching happens, a more comprehensive model, i.e., impulsive switched systems are naturally formed, which have extensive applications in many fields such as biological population management [7], network control [8], flying object motions [9], frequency-modulated signal processing systems [10]. Impulsive switched systems, as is known to all, do not retain the property of constituent modes. Moreover, impulsive effects that the state of a system may undergo abrupt change at some moments can be classified into two types: stabilizing impulses and destabilizing impulses, which may affect the system dynamics in a positive and a negative way respectively. To be specific, the stabilizing impulses can promote stability and enhance prescribed system performance, while destabilizing impulses undermine system stability and lead to poor performance. Therefore, the study of impulsive switched systems is more comprehensive and challenging. In [11], the authors studied exponential stability and L 2 -gain for nonlinear impulsive switched systems by the multiple Lyapunov function method. In [12], based on multiple Lyapunov functions, some conditions of finite-time stability have been derived for a class of nonlinear impulsive switched systems. By using the Lyapunov function and the average impulsive switched internal approach, the input-to-state stability of nonlinear impulsive switched systems has been investigated in [13]. In [14], under the novel Lyapunov-like function, the authors discussed the globally uniformly exponentially stable of impulsive switched linear systems.

External disturbance is inevitable in actual systems [15][16]. The existence of external disturbance seriously affects the performance of systems and even leads to instability. H ∞ control is an effective and important approach to dealing with external disturbance. It can be successfully used to design a controller such that the output is upper bounded by limiting the external disturbance while the internal stability is ensured. Many interesting results on H ∞ control for switched systems have been reported in recent years. By utilizing Riccati-Metzler inequalities, the dynamic output feedback H ∞ control of switched linear systems has been investigated in [17]. Ref. [18] considered H ∞ control of nonlinear switched systems via introducing average dwell time (ADT) and piecewise Lyapunov functions approaches. Based on multiple Lyapunov functions and mode-dependent average dwell time, [19] studied the finite-time H ∞ control problem of switched linear systems. [20] developed the multiple Lyapunov functions to H ∞ control nonlinear switched systems. Some other interesting results on H ∞ control for switched systems refer to [21]-[23]. It is worth noting that the impulse effect has not been considered in the above contributions. In [24], based on the ADT method and Lyapunov function, the authors studied the exponential stability and L 2 -gain analysis of impulsive switched linear systems, where both stabilizing impulses and destabilizing impulses were considered. In [25], from the point of impulsive control, finite-time H ∞ control for nonlinear impulsive switched systems with a dynamic output feedback controller was investigated via multiple Lyapunov functions and a mode-dependent average dwell time condition. Although the above papers take the impulse effect into account when switching happens, time-delays are excluded in their studied switched systems. Nevertheless, in practical systems, time delays are unavoidable, which are the main cause of poor performance or even instability. Therefore, it is necessary to study the H ∞ control for impulsive switched systems with time-delays. On the other hand, it is well known that the L-K functional and the L-R technique are two main distinct approaches to dealing with the stability problems of various delayed systems. Especially when the time-delays are unknown or unmeasurable, the L-R technique is more efficient and powerful. In [26], based on Razumikhin technique and Lyapunov functions, authors attempt to derive robust exponential stability of nonlinear impulsive switched time-delays systems when the impulses are stabilizing. By employing L-K functional method, [27] focused on the finite-time control for nonlinear impulsive switched with time-varying delays under destabilizing impulses. However, under the effect of the external disturbance, a basic and important issue on how to develop L-K functional method and L-R technique for H ∞ control of impulsive switched time-delays systems remains unresolved. Especially when both stabilizing impulses and destabilizing impulses are considered, it is worth deep study, which motivates this work.

In this paper, we investigate the H ∞ control problem of impulsive switched linear systems with time-varying delays. Some sufficient conditions, which rely on a relation among switchings, impulses, and external disturbance, are presented. The main contributions are threefold: 1) Compared with [24][25], the effect of time-varying delays on the systems are taken into account in this paper; 2) Two types of impulses, i.e. the destabilizing impulses and stabilizing impulses, are regarded respectively 3) In the case of destabilizing impulses, utilizing L-K functional and ADT condition, the H ∞ control performance of the delayed impulsive switched systems is guaranteed. The L-R technique and reverse ADT condition is used to obtain the H ∞ control for the delayed impulsive switched systems under the stabilizing impulses.

Notations. Let ℝ denote the set of real numbers, ℝ + the set of non-negative real numbers, ℤ + the set of positive integers and ℝ n the n-dimensional real space equipped with the Euclidean norm ‖   ⋅   ‖ . L 2 [ t 0 , ∞ ) denotes the space of square integrable vector functions on [ t 0 , + ∞ ) . A > 0 ( ≥ , < , ≤ 0 ) denotes that the matrix A is a symmetric positive definite (positive semi-definite, negative definite, negative semi-definite) matrix. Moreover, A T and A − 1 denote the transpose and the inverse of A , respectively. Let λ max ( A ) ( λ min ( A ) ) be the maximum (minimum) eigenvalue of A. I denotes the identity matrix with appropriate dimensions. ℳ = { 1 , 2 , ⋯ , m ¯ } , m ¯ ∈ ℤ + , is an index set. For any set U ⊆ ℝ k ( 1 ≤ k ≤ n ) , interval S ⊆ ℝ , P C ( S , U ) = { φ : S → U is continuous everywhere except for a finite number of points t, where φ ( t − ) , φ ( t + ) exist and φ ( t + ) = φ ( t ) } . Let ‖ φ ‖ τ = sup θ ∈ [ − τ , 0 ] ‖ φ ( t 0 + θ ) ‖ . In addition, the notation ⋆ always denotes the symmetric block in one symmetric matrix.

In this paper, we consider the following impulsive switched systems with time-varying delays:

where x ( t ) ∈ ℝ n is the system state, y ( t ) ∈ ℝ n is the output, ω ( t ) ∈ ℝ q is the exogenous disturbance which belongs to L 2 [ t 0 , ∞ ) , u ( t ) ∈ ℝ p is the control input; A σ ( t ) , B σ ( t ) , C σ ( t ) , D σ ( t ) , E σ ( t ) , F σ ( t ) , H σ ( t ) are constant matrices with appropriate dimensions; φ ( t ) ∈ P C ( [ t 0 − τ , t 0 ] , ℝ n ) is the initial condition; σ ( t ) : [ t 0 , ∞ ) → ℳ = { 1 , 2 , ⋯ , m ¯ } is the switching signal, which can be characterized by the switching sequence ( i 0 , t 0 ) , ( i 1 , t 1 ) , ⋯ , ( i j , t j ) , ⋯ and σ ( t ) = i k means that the i k th subsystem is activated when t ∈ [ t k , t k + 1 ) ; τ ( t ) : ℝ + → ℝ + denotes the time-varying delay; Δ x ( t ) = x ( t + ) − x ( t − ) , x ( t ) = x ( t + ) = lim h → 0 + x ( t + h ) , x ( t − ) = lim h → 0 − x ( t + h ) .

In this paper, we make the following assumptions.

Assumption 1. τ ( t ) is continuous and bounded

where τ is a positive constant.

Assumption 2. τ ( t ) is differentiable and its derivative is bounded

where d is a positive constant.

This paper, in order to achieve the H ∞ performance of the system, considers a class of state feedback controllers

where K σ ( t ) is the state feedback gain matrix. Substituting (2) into (1), one has

It is assumed that the switches and impulses occur at the same time. Let { t k } , k ∈ ℤ + be a sequence of discrete times representing the impulsive switching instants. To avoid Zeno behavior, the sequence t 0 < t 1 < t 2 < ⋯ < t k satisfies that t k → + ∞ as k → ∞ . Given an impulsive switching time sequence { t k } and a switching signal σ ( t ) , the set { t k , σ ( t ) } is called the impulsive switching signal which is denoted by ℱ .

Definition 1. ([7]) For a class of impulsive switching signal ℱ 1 [ τ α , N 1 ] satisfying the ADT condition given by

where N σ ( T , t ) is the number of impulses or switches occurring on the interval [ t , T ) , N 1 > 0 is called the chatter bounds and τ α is called ADT constant. Similarly, for a class of impulsive switching signal ℱ 2 [ τ α , N 2 ] satisfying the reverse ADT condition given by

where N σ ( T , t ) is the number of impulses or switches occurring over the interval [ t , T ) , N 2 > 0 is called the chatter bounds and τ α is called reverse ADT constant.

Definition 2. System (3) is said to be exponentially stable (ES) over the impulsive switching signal ℱ , if the solution x ( t ) of system (3) satisfies

Definition 3. ([28]) For a given constant γ > 0 , the system (1) is said to achieve H ∞ control performance under the impulsive switching signal ℱ if there exist control input u ( t ) and an impulsive switching signal ℱ satisfying the following conditions

1) The system (3) with ω ( t ) = 0 is ES;

2) Under the zero initial condition x ( t ) = 0 , t ∈ [ t 0 − τ , t 0 ] , the system (3) satisfies the H ∞ performance index for any ω ( t ) ≠ 0

Lemma 1. ([29]) Let S and T be real matrices of appropriate dimensions. Then, for any matrix Q > 0 of appropriate dimension and any scalar ε > 0 , the following inequality holds

In this section, some sufficient conditions are established to guarantee the H ∞ performance of delayed impulsive switched systems. Both destabilizing and stabilizing impulses are taken into account. To start with, the case of destabilizing impulses is considered.