A Linear Matrix Inequality Approach for the Control of Uncertain Fuzzy Systems

The control of nonlinear systems is difficult because no systematic mathematical tools exist to help find necessary and sufficient conditions to guarantee their stability and performance. The problem becomes yet more complex if some of the plant parameters are unknown. By using a Takagi-Sugeno-Kang (TSK) fuzzy plant model [1]-[4], a nonlinear system can be expressed as a weighted sum of some simple subsystems. This model provides a fixed structure to some nonlinear systems and facilitates the analysis of the systems. There are two ways to obtain the fuzzy plant model: 1) by performing system identification methods based on the input-output data of the plant [1]-[4] or 2) directly from the mathematical model of the nonlinear plant [5]. The stability of fuzzy systems formed by a fuzzy plant model and a fuzzy controller has been investigated recently. Various stability conditions have been obtained by employing Lyapunov stability theory [3], [6], [11], passivity theory [12], and other methods [5], [13]-[17]. A linear controller [18] was also proposed to control the plant represented by the fuzzy plant model. Most of the fuzzy controllers proposed are functions of the grades of membership of the fuzzy plant model. Hence, the membership functions of this model must be known. In other words, the parameters of the nonlinear plant must be known or be constant when the identification method is used to derive the fuzzy plant model. Practically, the parameters of many nonlinear plants will change during their operation (e.g., the load of a dc-dc converter or the number of passengers on board a train). In these cases, the robustness property of the fuzzy controller is an important concern. 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 will be analyzed based on the Lyapunov stability theory. We will show that the stability condition derived will be the same as that of the relaxed stability condition in [7], 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 will be formulated into a linear matrix inequality (LMI) problem [19]. 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.