The Center Of Gravity Fuzzy Systems And Probability Expression Based On CRI Method

During the recent years, many schoolers have been studying the universal approximation of the fuzzy system and fuzzy controllers, actually fuzzy systems when used as controllers. This paper studied majorly on a new feature of fuzzy systems:the probability nature, the universal approximation as well as the comparison between two different fuzzy systems. The major works are as follows.Firstly, the non-single point fuzzifier is proposed and a method of constructing single input and single output (SISO) fuzzy system is given by using CRI fuzzy inference. The fuzzy system is based on a set of input-output data, center-of-gravity defuzzifier and genuine many-valued implications. So this method solves a problem successfully that genuine many-valued implications can't be used to construct a fuzzy system with the CRI fuzzy inference. What' more, we got the expressions of the joint probability density function and numerical characteristics. Excitedly, it is pointed out that although the probability density functions based on different implications vary, the numerical characteristics are the same.Secondly, the concept of adaptive universe is proposed, which helps to construct the SISO fuzzy system based on functions and the probability density function. Upon that, we calculate the regressin function and prove that the center-of-gravity fuzzy system is actually the regression function. This coincidience reveals the interrelation between the fuzzy theory and probability theory.Thirdly, the universal approximation of SISO fuzzy system based on data is discussed and the sufficient condition is given, which indicates that the fuzzy system could approximate the real system in any degree of accuracy. It verifies the efficiency of the center of gravity fuzzy system as an almighty approximator.Finally, the comparison is discussed between the Mamdani fuzzy system and the Center of gravity fuzzy system. The first step is to build the Mamdani fuzzy system. Then the approximation error of the two fuzzy systems is calculated separately in the same number of inference rules. Sequentially, the majority of Center of gravity fuzzy systems (7/12) are better than Mamdani fuzzy systems.