Some mathematical aspects of fuzzy systems

Abstract

In this work, three topics which are important for the further development of fuzzy systems are chosen to be investigated. First, the mathematical aspects of fuzzy relational equations (FREs) are explored. Solving FREs is one of the most important problems in fuzzy systems. In order to identify the algebraic information of the fuzzy space, two new tools, called fuzzy multiplicative inversion and additive inversion, are proposed. Based on these tools, the relationship among fuzzy vectors in fuzzy space is studied. Analytical expressions of maximum and mean solutions for FREs, and an optimal algorithm for calculating minimum solutions are developed. Second, the possibility of applying functional analysis theory to Takagi-Sugeno (T-S) fuzzy systems design is investigated. Fuzzy transforms, which are based on the generalised Fourier transform in functional analysis, are proposed. It is demonstrated that, mathematically, a T-S fuzzy model is equivalent to a fuzzy transform. Hence the parameters of a T-S fuzzy system can be identified by solving equations constructed using the inner product between membership functions and a given target function. The functional point of view leads to an insight into the behaviour of a fuzzy system. It provides a theoretical basis for exploring improvements to the efficiency of T-S fuzzy modelling. Third, the mathematical aspects of model-based fuzzy control (MBFC) are investigated. MBFC theory is not suitable for general nonlinear systems, due to an implicit linearity assumption. This assumption limits fuzzy controller design to a special case of linear time-varying systems control. To apply MBFC in general nonlinear control, a new stability criterion for general nonlinear fuzzy system is proposed. The mathematical aspects investigated in this research, provide a systematic guidance on issues such as efficient fuzzy systems modelling, balanced 'soft' and 'hard' computing in fuzzy system design, and applicability of fuzzy control to general nonlinear systems. They serve as a theoretical basis for further development of fuzzy systems