Analytical Analysis and Feedback Linearization Tracking Control of the General Takagi-Sugeno Fuzzy D - Systems, Man, and Cybernetics, Part C, IEEE Transactions on

Takagi–Sugeno (TS) fuzzy modeling technique, a black-box discrete-time approach for system identification, has widely been used to model behaviors of complex dynamic systems. Analytical structure of TS fuzzy models, however, is presently unknown, nor is its possible connection with the traditional models, causing at least two major problems. First, the fuzzy models can hardly be utilized to design controllers for control of the physical systems modeled. Second, there lacks a systematic technique for designing a controller capable of controlling any given TS fuzzy model to achieve desired tracking or setpoint control performance. In this paper, we provide solutions to these problems. First of all, we have proved that a general class of TS fuzzy models is nonlinear timevarying Auto-Regressive with the eXtra input (ARX) model. The fuzzy models in this study are general because they use arbitrary continuous input fuzzy sets, any types of fuzzy logic AND operators, TS fuzzy rules with linear consequent and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case. Furthermore, we have established a simple necessary and sufficient condition for analytically determining local stability of the general TS fuzzy dynamic models. The condition can also be used to analytically check quality of a TS fuzzy model and invalidate the model if the condition warrants. More importantly, we have developed a feedback linearization technique for systematically designing an output tracking controller so that output of a controlled TS fuzzy system, which may or may not be stable, of the general class achieves perfect tracking of any bounded time-varying trajectory. We have investigated stability of the tracking controller and established a necessary and sufficient condition, in relation to stability of nonminimum phase systems, for analytically deciding whether a stable tracking controller can be designed using our method for any given TS fuzzy system. Three numerical examples are provided to illustrate the effectiveness and utility of our results and techniques.