Toward Necessity of Parametric Conditions for Monotonic Fuzzy Systems

Input-output monotonicity is an important constraint found in many application domains. A monotonic fuzzy system (MFS) is defined as a Takagi-Sugeno-Kang (TSK) system whose output is monotonically increasing or decreasing with respect to one or more inputs. This paper reviews the authors' previous work, which derived the parametric conditions for the MFS, and discusses the rationale lying behind the conditions. An MFS is developed by creating a monotonic rule base while preserving the relative monotonicity among the membership functions corresponding to the fuzzy rules. This paper also proves that the parametric conditions are necessary and sufficient to build a single-input zeroth-order MFS with two rules. Only the sufficiency of the conditions holds for a multi-input first-order or higher order TSK fuzzy system with three or more rules.