Design And Stability Analysis Of Fuzzy Control System Based On Analytical Structure

Since American professor Zadeh firstly proposed the concept of fuzzy set in 1965 and English engineer Mamdani applied fuzzy theory in steamer control successfully in 1975, fuzzy control has been widely used in many fields such as chemical engineering, metallurgy, aerospace engineering, communication and transportation etc. Compared with the traditional control, it is proved that fuzzy control has a better control effect on those systems which has the features of nonlinear, big time delay, and non-mathematical model etc. It has become an important branch of intelligence control.Compared with the traditional control, fuzzy control lacks enough research on the systematic analysis and design, due to its special control theory and complicated structure, for example, the dividing of fuzzy universe, the design of membership function, and acquisition of fuzzy rules etc., specially, the stability analysis of fuzzy system which is relative to the feasibility of system. These problems make fuzzy controller be a"black-box". So it is hard to mathematically analyze the characters of fuzzy systems and to explain the better nature of fuzzy control. As a result, these above shortages make the development of fuzzy control theory fall behind its application. Thus, how to make a research on fuzzy theory and systematic design method of fuzzy controller, has become an important research issue in fuzzy control field. Specially, how to establish a kind of design method for the general, applicable and engineering application of fuzzy controller, is an challengable job for designers.The analytical structure of fuzzy controller builds the explicit expression between input and output of fuzzy controller, and establishes the relationship between fuzzy controller and traditional PID controller. So with the analytical structure of fuzzy controller, it is possible to reveal the essence of fuzzy control. The analytical structure of fuzzy controller provides a useful theoretical tool to design and analyze fuzzy controller. At present, many researchers have proposed many kinds of analytical structures of fuzzy controllers at different perspectives and attain a lot of important results. It is valuable to deeply study on the analytical structures of the different fuzzy controllers, and specially, on the systematic analysis and design of fuzzy controllers with their analytical structures.In order to develop analytical theory, simplify design process and make full use of engineering application of fuzzy controller, some key issues on the design and stability analysis of fuzzy controller are researched in detail in the paper, based on the derivation of analytical structure of fuzzy controllers. Firstly, the fuzzy Lyapunov function is used in stability analysis of T-S fuzzy control systems and H∞stability of Mamdani fuzzy control systems by the analytical structure of a typical fuzzy controller. Secondly, with the discussion of input and output membership function, a kind of general linear membership function is put forward and its universal approximation is discussed. Finally, a new method for the design of fuzzy controller is presented, whose parameters are defined according to the traditional PID control technology. Simulation and experiment results manifest the effect of the researches in the paper. As a conclusion, the main contents of the paper are as follows:(1) Apply fuzzy Lyapunov function in the stability analysis and design of T-S and Mamdani fuzzy system. For T-S fuzzy control system, the sufficient conditions and parameters design method, for the state observer, optimal controller and a parallel distributed compensation controller with a compensator, are derived by formulating the fuzzy Lyapunov function of the fuzzy system, the parallel distributed compensation strategy and linear matrix inequality. Based on the above researches, the fuzzy Lyapunov function is adopted in the H∞stability analysis of Mamdani fuzzy system, and a new sufficient condition is derived on the basis of the analytical structure of the fuzzy controller.(2) A kind of systematic design method is put forward about the input and output membership function. The design factors of membership function are analyzed in detail on the basis of analytical structure of a typical fuzzy controller. And the research provides theoretical foundation for the design of fuzzy controller.(3) Based on the summary of the characteristic of triangle and trapezoid membership function, a kind of general linear membership function is put forward, and triangle and trapezoid membership function can be look as its special cases. It is proved that the controller with general linear membership function is the sum of a global multi-delay controller with an offset and a local PD(or PI) controller. Meanwhile, the limiting structure and nonlinear character are derived.(4) The universal approximation and sufficient condition of Mamdani fuzzy controller with general linear membership function is derived and proved that the controller can approximate any continuous real function with arbitrary precision in discourse. The effect of the designed membership function is verified by the research of universal approximation of fuzzy controller. (5) A kind of PID fuzzy controller with a weighted factor is designed and a systematic design method is proposed to set parameters of fuzzy controller according to the ordinary PID controller. On the basis of analytical structure of fuzzy controller, the control essence of PID fuzzy controller is revealed and the stability of fuzzy system is researched with the fuzzy Lyapunov method. These results manifest that the performance of the designed fuzzy controller is better than that of ordinary PID controller. Secondly, a kind of PID fuzzy controller with a changeable integral parameter is designed to control these systems with big inertial and delay, etc. Finally, the designed fuzzy controllers are used in temperature and level control of the process control experiment system. Simulation and experiment results show that the designed fuzzy controllers are simple and easy to be realized, and meanwhile, they have some advantages of robustness and stability.