This paper investigates the problem of fuzzy controller design for nonaffine-in-control singularly perturbed switched systems (NCSPSSs). First, the NCSPSS is approximated by Takagi-Sugeno (T-S) models which include not only state but also control variables in the premise part of the rules. Then, a dynamic state feedback controller design method is proposed in terms of linear matrix inequalities. Under the controller, stability bound estimation problem of the closed-loop system is solved. Finally, an example is given to show the feasibility and effectiveness of the obtained methods.

Switched systems, which consist of a finite number of subsystems and a logical rule governing the switching among the subsystems, are widely encountered in mechanical systems, power systems, and aircraft [

On the other hand, many practical systems exhibit multiple time scale behavior, which can lead to high dimensionality and ill-conditioned numerical issues in the analysis and design problems [

Singularly perturbed switched systems (SPSSs) whose subsystems are SPSs are of practical interest in many industry processes [

Fuzzy control has found a great variety of applications in control engineering [

Motivated by [

The rest of this paper is organized as follows. In Section

Consider the following nonlinear singularly perturbed switched system:

By using the algorithm in [

The

Denote

The scalars

The T-S models used in [

The design of state feedback stabilizing fuzzy controllers for the fuzzy system (

The

Because the controller rules are the same as the plant rules, the controller is given as follows:

Then we have the closed-loop system:

Upon introducing the indicator function

Assume that

This section will present a controller design method.

If there exist matrices

Let

It follows from (

Using the Schur complement, inequality (

System (

Define

By simple calculation, we have

LMI conditions (

On the other hand, it is easy to see that

Then it follows from (

Let

Thus, for sufficiently small

Then, by (

Define Lyapunov function

Computing the derivative of

LMIs (

In [

LMIs (

T-S models, which use a set of fuzzy rules to describe a nonlinear system in terms of a set of local linear models, offer an efficient approach to stability analysis and controller design of complex nonlinear systems. In this framework, most of the stability analysis and controller design problems can be reduced to solve LMI problems. A larger number of individual subsystems or fuzzy rules will lead to larger computational burden. Fortunately, there have been some efficient algorithms to deal with LMI problems with reasonable large dimensions [

Stability bound is a key stability index of SPSs. Theorem

If there exists matrices

Let

It follows from (

We can rewrite (

Define

From LMIs (

Since

Define Lyapunov function

Computing the derivative of

Stability bound problem of SPSS is challenging. The existing method in [

It is known that LMI-based stability conditions for T-S fuzzy systems are usually sufficient conditions [

To illustrate the proposed results, we consider the following singularly perturbed switched system composed of two modes.

Mode 1:

Mode 2:

For the individual systems, we choose the interpolation points set as follows.

Mode 1:

Mode 2:

Using Algorithm 2.1 in [

Mode 1:

Mode 2:

Triangle-type membership functions are employed as the fuzzy basis functions (see Figures

Membership function of

Membership function of

Solving the LMIs in Theorem

Under this controller, the stability bound of the closed-loop system is

For simulation, we choose

Switching rule

State trajectories and control input.

In this paper, we considered the problem of fuzzy controller design for nonaffine-in-control singularly perturbed switched systems (NCSPSSs). A general class of Takagi-Sugeno (T-S) models including both state and control variables in the premise part of the rules were established to approximate the NCSPSS. By LMI technique, the controller design and stability bound estimation problems were solved. The presented example demonstrated the feasibility and effectiveness of the obtained methods. It can be seen that the proposed results are limited to standard NCSPSS. Thus, one of our future works is to investigate control and analysis problems of nonstandard NCSPSS.

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

This work was supported by the National Natural Science Foundation of China (61374043, 60904009, and 61020106003), the Jiangsu Provincial Natural Science Foundation of China (BK20130205), the China Postdoctoral Science Foundation funded Project (2013M530278, 2014T70558), the Fundamental Research Funds for the Central Universities (2013QNA50, 2013RC10, 2013RC12, and 2013XK09), and the Natural Science Foundation of Liaoning Province (201202201).

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