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This paper focuses on the problem of robust

Switched systems have constituted a very active field of current scientific research; many real world processes and systems can be modeled as switched systems, such as chemical processes and computer controlled systems. Besides, switched systems are extensively applied in many domains, including mechanical systems, automation, aircraft and air traffic control, and many other fields. And stability analysis and control, as the most important topics for the study of switched systems, have been studied widely [

On the other hand, time-delay phenomenon is very common in many kinds of engineering systems, for instance, long-distance transportation systems, hydraulic pressure systems, network control systems, and so on. It is regularly a source of instability and often causes undesirable performance and even makes the system out of control. So time-delay systems have also drawn more and more attention [

The contribution of this paper lies in the following aspects. First, we address the problem of

The remainder of this paper is organized as follows. Firstly, problem formulation and preliminaries are stated in Section

Consider a class of uncertain switched delay systems of the form

Consider

With regard to the switching signal

In general, the nonlinear uncertainties

Consider

Compared with the switched system in [

In this paper, we are interested in designing a state-feedback controller which is described by

System (

For any

The concept of average dwell time, which was an effective tool for the stability analysis of switched systems, was put forward for continuous switched systems firstly by Hespanha and Morse (see [

For

system (

under zero initial conditions system (

Now the following lemma which will be used to draw the main results in this paper is presented.

For any function vector

In this paper, we focus on the switched delay system with time-varying delays and constant delays, respectively. And we will tackle the robust

Consider the nonswitched system (

Then, along the trajectory of the system, when

The following proof is motivated by the method in [

For the function

Considering (

Hence

When the delays in the nonswitched system (

For given constants

The proof is similar to that of Lemma

For

Assumption (

Now, we will design a controller

For a given matrix

Define a set of Lyapunov-Razumikhin function candidates as follows:

Then when

Letting

Now we will show the weighted

Inequality (

When the delays in system (

For given constants

Define the Lyapunov-Razumikhin function candidate in Theorem

Inequality (

In [

To illustrate the main results, we consider the following examples.

Consider the time-varying delay switched system

Therefore according to Theorem

The state response of the closed-loop system (

Consider the constant time-delay switched system with

It is obvious that

The state response of the closed-loop system (

In the figures, horizontal axis stands for time and vertical axis stands for the states

It is easy to verify that both subsystems are unstable. And using the method given in [

In this paper, we considered the problem of robust

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by the National Natural Science Foundation of China (11171131) and Jilin Provincial Natural Science Foundation of China (201115043). The authors are very much thankful to the Associate Editor and anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions to improve the quality of this paper.

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_{2}-gain analysis for switched symmetric systems with time delay