Stability analysis for a general class of discrete-time polynomial fuzzy dynamic systems

The polynomial fuzzy models are capable of modeling complex dynamic systems. They have attracted increasing attention in the recent years as the models of choice for the development of more advanced fuzzy controllers. Little effort has been made to study the models themselves though. Like many other types of models, a polynomial fuzzy model aims at describing the physical system´s dynamics based on the measured input-output data of the system. Importantly, a polynomial fuzzy model that appears to mimic the measured data reasonably well does not guarantee its validity. One way to assess model´s quality is to check whether its stability is consistent with that of the physical system, which is the theme of our investigation. In this paper, we first propose a type of discrete-time polynomial fuzzy dynamic models, which comprises the general Takagi-Sugeno (T-S) fuzzy model as a special case. Then, based on the Lyapunov´s linearization method, a necessary and sufficient condition is established for analytically determining the local asymptotic stability of the proposed models. A numerical example is given to illustrate the effectiveness and utility of our method.