Analytical analysis and feedback linearization tracking control of the general Takagi-Sugeno fuzzy dynamic systems

The 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. The analytical structure of TS fuzzy models, however, is unknown, causing at two major problems. First, the fuzzy models cannot be utilized to design controllers of the physical systems modeled. Second, there is no systematic technique for designing a controller that is capable of controlling any given TS fuzzy model to achieve the desired tracking or setpoint control performance. In this paper, we provide solutions to these problems. We have proved that a general class of TS fuzzy models is a nonlinear time-varying ARX (Auto-Regressive with eXtra input) model. We have established a simple condition for analytically determining the local stability of the general TS fuzzy dynamic model. The condition can also be used to analytically check the quality of a TS fuzzy model and invalidate the model if the condition warrants. We have developed a feedback linearization technique for systematically designing an output tracking controller so that the output of a controlled TS fuzzy system of the general class achieves perfect tracking of any bounded time-varying trajectory. We have investigated the stability of the tracking controller and established a condition, in relation to the stability of non-minimum 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.