Almost Sure Control Of Stochastic Fuzzy Systems

In recent years, the control of nonlinear stochastic systems has been investigated intensively. Meanwhile, the technique of fuzzy control has attracted much attention in both theory and practice. How to apply fuzzy control theory into nonlinear stochastic systems and solve the related practical problems is a major concern in the control and stochastic fields. In the first chapter of the thesis, we give a brief outline of the recent advance in stochastic and fuzzy systems and the structure of this paper.There are several different concepts of stability of stochastic systems, such as asymptotic stability in probability, almost sure exponential stability and mean square exponential stability etc. The stochastic fuzzy system is a class of nonlinear stochastic system in terms of Takagi-Sugeno(T-S) fuzzy model. In the second chapter, this paper considers a general class of the T-S stochastic fuzzy systems involving multi-noises, and noises in controller structure, and proposes a new relaxed design method of fuzzy controllers. By the example, we illustrate the relaxed method can design fuzzy controller to stabilize the stochastic systems, which can not be stabilized by the existing methods in the literature.Generally speaking, both mean square exponential stability and almost sure exponential stability imply asymptotic stability in probability, but they do not imply each other. However, in many situations, e.g. linear systems, mean square exponential stability implies almost sure exponential stability. So far, stability issues of the stochastic fuzzy systems in almost sure sense are not addressed in the literature yet. In the third chapter, this paper studies the almost sure exponential stabilization of the T-S stochastic fuzzy systems, by combining the method of mean square exponential stabilization of the T-S stochastic fuzzy systems and the almost sure exponential stability theory of stochastic system. We propose two different design methods of fuzzy controller. By Schur complement etc, the results are expressed in terms of linear matrix inequalities(LMIs), which take quite simple forms easy to solve, and we give several illustration examples by using Matlab software.In practice, a complicated dynamic system is usually described by a relatively simplified model, but the difference between the original system and the simplified model causes the uncertainty of the model.In the fourth chapter of the thesis, we discuss the T-S stochastic fuzzy system with the uncertainty of system parameters, and propose a design method of the robust controller.By the example, we illustrate this method can design robust controller to stabilize the T-S stochastic fuzzy systems almost surely.However, in practice, the design methods fore-mentioned can only stabilize the systems, but they can not stabilize the systems at a given velocity. For the stabilization of stochastic systems, Damm and Hinrichsen proposed an algorithm of rational matrix inequality by introducing Newton iteration, which provided a new method for the design of the controller. In the fifth chapter of the thesis, we transform the problems into a class of generalized Riccati equation. We propose a new design method of the fuzzy controller, based on the Newton method and the solution of stochastic Riccati equations. As a kind of numerical method, the iteration method is more complicated to solve than the method of linear matrix inequalities, but this method is less conservative. The example illustrates the feasibility of this iteration method. Finally, in the sixth chapter of the thesis, we make a conclusion of this paper, and propose two issues which deserve further research.