Towards a paradigm for fuzzy logic control

The purpose of this paper is to lay a foundation for fuzzy logic control (FLC) design, constructing a class of fuzzy logic controllers that is suitable for a broad range of controls applications. Some scalar definitions in FLC are extended to the n-dimensional case, including vector fuzzy number and membership vector. A rigorous mathematical expression is given for the function g(x) (called the 'reasoning surface') manufactured by a fuzzy associative memory (FAM). For the existence and uniqueness of solutions in any closed-loop system with a fuzzy logic controller, it is shown that the FAM function g(x) must be Lipschitz; this places restrictions on the allowed rules in the FLC. It is argued that, for effective control design, the membership functions should not be uniformly spaced, the triangular membership functions generally suffice, and product inferencing should be used. Under these conditions, the FLC manufactures a piecewise-multilinear Lipschitz FAM function g(x) that is a generalized proportional-plus-derivative-plus-offset controller. The scalar and two-dimensional cases are studied in detail.