^{1}

^{2}

^{1}

^{1}

^{2}

This paper is concerned with the problem of robust reliable control for a class of uncertain discrete impulsive switched systems with state delays, where the actuators are subjected to failures. The parameter uncertainties are assumed to be norm-bounded, and the average dwell time approach is utilized for the stability analysis and controller design. Firstly, an exponential stability criterion is established in terms of linear matrix inequalities (LMIs). Then, a state feedback controller is constructed for the underlying system such that the resulting closed-loop system is exponentially stable. A numerical example is given to illustrate the effectiveness of the proposed method.

Switched systems are a class of dynamical systems comprised of several continuous-time or discrete-time subsystems and a rule that orchestrates the switching among different subsystems. These systems have attracted considerable attention because of their applicability and significance in various areas, such as power electronics, embedded systems, chemical processes, and computer-controlled systems [

Impulsive switched systems have received a considerable research attention for more than one decade. The problems of stability, controllability, and observability for impulsive switched systems have been successfully investigated, and a rich body of the literature has been available [

On the other hand, it is inevitable that the actuators will be subjected to failures in a real environment. A control system is said to be reliable if it retains certain properties when there exist failures. When failure occurs, the conventional controller will become conservative and may not satisfy certain control performance indexes. In this case, reliable control is a kind of effective control approach to improve system reliability. Recently, several approaches for designing reliable controllers have been proposed, and some of them have been used to research the problem of reliable control for switched systems [

In this paper, we will focus our interest on robust reliable control problem for a class of uncertain discrete impulsive switched systems with state delays. The dwell time approach is utilized for the stability analysis and controller design. The main contributions of this paper can be summarized as follows: (i) stability and reliability of discrete impulsive switched systems in the presence of actuators failures are first considered; (ii) a state feedback design methodology is proposed to achieve the exponential stability and reliability for the underlying systems.

The remainder of the paper is organized as follows. In Section

Consider the following uncertain discrete impulsive switched systems with state delays:

The impulsive jump at the switching instant

For each

For simplicity, we define

Before ending this section, we introduce the following definitions and lemmas.

Let

In this paper, the average dwell time method is used to restrict the switching number during a time interval such that the stability of system (

The system (

For a given matrix

Let

For matrices

In this subsection, we consider the exponential stability of the following uncertain discrete impulsive switched systems with state delays:

Consider system (

Choose the following piecewise Lyapunov function candidate for system (

Let

When

When

This completes the proof.

When

It should be noted that some stability results of discrete delayed systems withandwithoutimpulsive jumps have been obtained by using standard Lyapunov-Krasovskii function approach (see [

In this subsection, we are interested in designing a state feedback controller such that the resulting closed-loop system is exponentially stable.

For system (

Consider the system (

Then, under the reliable controller

From Theorem

Denote that

This completes the proof.

In Theorem

It should be noted that

The detailed procedure of controller design can be given in the following algorithm.

We have the following.

In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider system (

Given

According to conditions (

Switching signal.

State trajectories of the closed-loop system.

From Figures

This paper has investigated the problem of robust reliable control for a class of uncertain discrete impulsive switched systems with state delays. By employing the average dwell time approach, an exponential stability criterion has been proposed in terms of a set of LMIs. On the basis of the obtained stability criterion, the robust reliable controller has been designed. An illustrative example has also been given to illustrate the applicability of the proposed approach.

This work was supported by the National Natural Science Foundation of China under Grant Nos. 60974027 and 61273120.