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This paper addresses the problem of designing robust tracking controls for a class of switched fuzzy (SF) systems with time delay. A switched fuzzy system, which differs from existing ones, is firstly employed to describe a nonlinear system. Next, a fast switching controller consisting of a number of simple subcontrollers is proposed. The smooth transition is governed by using the fast switching controller. Tracking hybrid control schemes which are based upon a combination of the _{∞} tracking theory, fast switching control algorithm, and switching law design are developed such that the _{∞} model referent tracking performance is guaranteed. Since convex combination techniques are used to derive the delay independent criteria, some subsystems are allowed to be unstable. Finally, various comparisons of the elaborated examples are conducted to demonstrate the effectiveness of the proposed control design approach. All results illustrate good control performances as desired.

Recent years have witnessed rapidly growing interest in switched systems which are an important class of hybrid systems [

From the middle of the 1980s, there have appeared a number of analysis problems about T-S fuzzy systems [

To effectively achieve nonlinear control, we propose the SF model [

Comparing with the previous work [

On the other hand, traditional parallel distributed compensation (PDC) control [

In this paper, we investigate the problem of tracking control for SF systems with time delay by using a switching method. Sufficient conditions for the solvability of the tracking control problem are given. We use single Lyapunov function technique and the fast switching controller to design a tracking control law, such that the

This paper is organized as follows. Section

Consider a switching fuzzy model defined by Tanaka et al

region rule

local region rule

Where

The following final output of the switching fuzzy model (

The idea of Tanaka et al. is simple and natural. They explain the idea through the following example [

Consider a switching fuzzy model with the membership functions in Regions 1 and 2 as shown in Figure

Region rule 1:

then

Region rule 2: if

then

Membership functions of switching fuzzy systems in regions 1 and 2.

The membership functions of each local T-S fuzzy model are assigned as follows:

It can be seen from Figure

Differing from the existing switching fuzzy systems [

When subsystems of the switched systems are T-S fuzzy systems, the systems are switched fuzzy systems. Sketch map of the switched fuzzy systems is depicted in Figure

Sketch map of switched fuzzy system.

Now, we define SF model including

Comparing with the previous work [

So in here, we shall introduce an innovated representation modeling of SF time-delay systems. In this model, each subsystem is a fuzzy time-delay system, namely, sub fuzzy time-delay system.

Consider the SF time-delay model including

It is readily seen that the

Therefore the global model of the

Given a reference model

The system (

It should be noted, in this definition the existence of state is dependent on switching law

In here, for system (

Our purpose is to design a controller

The

Now a fast switching controller is employed to control the switched fuzzy time-delay model of (

The fast switching controller for the sub fuzzy time-delay system is described by

Combining (

Suppose there exist positive definite symmetric matrixes

And, the switching law is designed as

By Schur complement lemma, the condition (

Now define a quadratic Lyapunov-Krasovskii functional candidate

First, we will prove that the system (

Then from (

in which

Next, under the zero initial condition we prove

Differentiating the Lyapunov-Krasovskii functional candidate along the trajectories

That is (

It is noted that Theorem

For the convenience of the design, we assume

And

Hence, we can conclude that the closed-loop system of (

It can be seen from (

Suppose there exist positive definite matrixes

In order words, we formulate the finding of

Suppose there exist positive definite matrixes

Once we have

It can be seen that (

In order to demonstrate the efficiency and feasible performance of the proposed control synthesis, now we consider the model as follows:

: if

Also, this model is used for the extra design of the switching law (

region rule 1: if

Local Region Rule

Local Region Rule 2: if

region rule 2: if

Local Region Rule 1: if

Local Region Rule 2: if

We have

The fuzzy sets of “

Choosing

Taking the initial condition as

(a) Response evolution of system state variables with the fast switching controller. (b) Response evolution of system error variables with the fast switching controller. (c) Control signal of the fast switching controller.

To investigate the effectiveness of the proposed fast switching controller of the SF system here, we now compare the traditional PDC controller [

The state feedback gains of subsystems are obtained as

For the same data and the same initial condition

(a) Response evolution of system state variables with the traditional PDC controller. (b) Response evolution of system error variables with the traditional PDC controller. (c) Control signal of the traditional PDC fuzzy controller.

In this paper, tracking control for SF systems with time delay is investigated. We use single Lyapunov function technique and a switching law to design a tracking control law such that the

This work is supported by National Nature Science Foundation under Grant 61004039, the Science Foundation of Educational Department of Liaoning Province under Grant 2010375, and Excellent Talented Person Foundation of Educational Department of Liaoning Province under Grant LJQ2011127. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

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