Analysis On Discrete-Time Fuzzy Stochastic Control Systems

The observability, controllability and filtering of stochastic control systems are the fundamental theoretical problems of stochastic control systems. In modern control theory, the frameworks of observability, controllability and filtering have already been systematic. However, due to the uncertainties existing in the real world, the frameworks still have some limitations, particularly when the observations of experiments or the states of systems include fuzzy data. How to analyze the observability, controllability and filtering of fuzzy-state stochastic control systems is the focus we discuss in the thesis.Some achievements have been made in the theories of fuzzy number space and fuzzy random variable with the past 20 years’ development, and have provided foundation for the research on fuzzy stochastic control systems. Recently, there are some advances on continuous-time fuzzy-state control systems. However, there is little discussion on the discrete-time fuzzy-state control systems which are widely involved in engineering. With the development of theories on fuzzy stochastic processes, the study on fuzzy stochastic dynamical systems appeared correspondingly. However, the research of fuzzy stochastic systems with a fuzzy-state initial state and fuzzy stochastic noises remains rare. Therefore discrete-time fuzzy-state stochastic control systems are studied in the thesis. By applying the representation theorem for fuzzy numbers, we transform the discrete-time fuzzy-state control systems into two families of discrete-time crisp control systems. And according to modern control theories, we obtain the conditions for fuzzy control systems to be observable, controllable as well as the filter of fuzzy stochastic control systems.This paper studies discrete-time fuzzy stochastic control systems and mainly includes two parts: Analysis on the observability and quasi-controllability of discrete-time fuzzy-state control systems and the Kalman filter of fuzzy stochastic systems. In the first part, we study the linear discrete-time fuzzy-state control system in the form of difference equations. We transform the linear fuzzy-state control system into two families of crisp linear control systems, the observability is then investigated with sufficient and necessary conditions according to classical control theories. Based on our study, the sufficient and necessary conditions of the observability of discrete-time fuzzy-state control systems are stronger than the counterparts of crisp control systems. In accordance with the characteristics of fuzzy-state systems, the concept of strong observability is further addressed and the sufficient condition of strong observability is given afterwards. Meanwhile some examples are shown to explain the theorems. In the analysis of quasi-controllability of discrete-time fuzzy-state control systems, according to the fact that fuzzy-state control systems can not be controllable on the whole fuzzy number space, we give the new concept of "quasi-controllability". And the sufficient and necessary conditions for the discrete-time fuzzy-state control systems to be quasi-controllable are further addressed.In the second part, we study discrete-time fuzzy stochastic control systems in the form of difference equations. Different from previous references, the noises in our study are ordinary fuzzy random variables without subject to Gaussian distribution. We use the same idea as before to transform the linear discrete-time dynamic fuzzy stochastic system into two families of crisp linear discrete-time dynamic systems. The fuzzy stochastic Kalman filter is gained according to classical filtering theories. Meanwhile we can prove the estimations obtained by the two crisp filters can constitute a triangle fuzzy number under certain restrictions, which is the optimal estimation of the fuzzy stochastic system. The sufficient conditions for the fuzzy stochastic Kalman filter to be stable are further addressed. We find the conditions of stability of the fuzzy stochastic Kalman filter are stronger than the counterparts of crisp Kalman filter. Meanwhile examples are given to explain the algorithm and conditions.The approach to transform a fuzzy model into two families of real-valued models and to apply classical theories to discuss the models afterwards is of universal significance. It provides a good idea for us to study other issues on fuzzy stochastic systems.