A linear matrix inequality approach for the control of uncertain fuzzy systems

This article proposes a linear controller to tackle nonlinear plants represented by a fuzzy plant model whose membership functions depend on some unknown parameters of the nonlinear plant. The unknown parameters are within known bounds. The stability of the closed-loop system is analyzed based on the Lyapunov stability theory. We show that the stability condition derived is the same as that of the relaxed stability condition given by Wang et al. (1996), in which the fuzzy controller depends on membership functions of the fuzzy plant model. However, the structure of the proposed linear controller is much simpler than that of the nonlinear fuzzy controller. The derived stability condition is formulated into a linear matrix inequality (LMI) problem. By solving the LMIs, the parameters of the linear controller can be obtained. The LMIs can be solved readily by using software tools such as MATLAB.

Fuzzy Plant Model with Unknown Parameters and Linear Controller

An uncertain multivariable nonlinear system is to be controlled. In this article, we consider multivariable nonlinear systems of the following form:

$$\dot{{\bf x}}(t)={\bf F}({\bf x}(t)){\bf x}(t)+{\bf B}({\bf x}(t)){\bf u}(t), \eqno{\hbox{(1)}}$$ where and are the system matrix and input matrix, respectively, both of which have known structure but are subject to unknown parameters, is the system state vector, and is the input vector. The system of (1) is represented by a fuzzy plant model that expresses the multivariable nonlinear system as a weighted sum of linear systems. A linear controller will be designed to close the feedback loop. A block diagram of a closed-loop control system is shown in Fig. 1. Figure 1.

Block diagram of the closed-loop control system.