Analytical resolution and numerical identification of fuzzy relational systems

Since Sanchez's seminal paper on fuzzy relational equations, both their theories and applications have been continuously exploited by researchers. However, the solvable conditions of a system of fuzzy relational equations, also known as a fuzzy relational system (FRS), are still poorly established and their relationship with the methods for obtaining approximate solutions are unclear. When the FRS is adopted to model a fuzzy system, most of the existing identification algorithms focus on parameter estimation and less on the structure identification. In this paper, these two issues are addressed. New theoretical understandings on solving a system of fuzzy relational equations exactly and approximately are presented and their implications on the use of FRS to encode fuzzy rulebases are highlighted. Based upon the guided evolutionary simulated annealing (GESA) algorithm [11], an evolutionary identification formulation called EVIDENT capable for both parameter and structure identifications in fuzzy system models is proposed. As demonstrated by the simulation results, the new algorithm not only is effective in determining the structure of the systems, but also identifies a better parametric solution, as compared with that of the existing FRS identification algorithms.