Fuzzy granulation and modular tiling: a useful duality

Fuzzy granulation refers to a family of very powerful techniques that allow computers to represent and manipulate the kind of knowledge that humans tend to actually possess regarding system capabilities, desirable system performance, and control policies to use the former to approach the latter. One of the most well established of this family of techniques may be referred to as Mamdani Granulation. This is usually viewed as a collection of fuzzy linguistic if-then rules, where each rule specifies the pairing of a fuzzy value of the dependent variable (for example, the control signal) with fuzzy value(s) for the independent variable(s) such as system state variables. Imposing the further restriction that the fuzzy values for a given variable all have identically-shaped membership functions that are evenly spaced over that variable's domain and have domain widths equal to twice the step size still captures a very large number of systems in both applied and theoretical settings. The term "Cartesian Mamdani Granulations" (CMG) refers to systems that satisfy this restriction. One additional restriction is convenient for the analysis without seriously misrepresenting the majority of CMG's in the literature. This is the restriction of fuzzy continuity, which says if two granules differ by one step on one independent variable and are identical on the other independent variable(s), then they cannot differ by more than one step on the dependent variable. Thus, the class of systems to be considered in the remainder of the paper consists of Continuous Cartesian Mamdani Granulations, or CCMGs.