The Stability Research Of Fuzzy Control Based On The Element Space Analysis

The stability analysis and design of fuzzy control systems have been the most important problems in fuzzy theory. The research of fuzzy control theory includes a series of main problems, such as the stability analysis, the system design approaches and the improvement of the system performance. It is difficult to analyse the stability for fuzzy control because of nonlinearity of fuzzy systems. Up to now, there have existed a lot of measures to ensure the stability of fuzzy control systems. However the perfect theory system has not been formed. It is necessary to study the theoretical problems more deeply.This paper aims at the thorough research on the stability of fuzzy systems, and some innovations in the theory are obtained, which is of benefit to providing new thoughts and new methods for the investigation into fuzzy systems.First, this paper summarizes the phylogeny and puzzle of automation and fuzzy control, and sums up the productions of the stability for fuzzy control. It also introduces the thoughtway and influence of the productions.Second, the basic kernel knowledge and methods on fuzzy sets are introduced.Third, for generalized fuzzy systems, it puts forward smart modelⅠ,and describes the concept of element space, and defines the equilibrium state based on imminent element transition matrix. For linguistic rule-based fuzzy systems, the stability can be determined through the transition matrix. As long as the control law can be showed in element space, transition matrix should be applied to analyse the stability of closed-loop fuzzy systems. Stable region are also presented when the system is not stable for all fuzzy states. The simulation result of the water level control of water container is presented to illustrate the methods above.Last, for stability analysis of T-S model, there is smart modelⅡwhich is based on the imminent element. By mapping the state-space to element space, we can judge system stability with the transition matrix. This method can be effective to divide the entire domain into stable region and instable region. If other nonlinear system model can be converted into this smart model, the system stability can also be judged. This is a new approach to the design of nonlinear controller.