Necessity of parametric conditions for monotonic TSK fuzzy systems

Input-output monotonicity is an important property found in many physical systems. This paper reviews parametric conditions of monotonic Takagi-Sugeno-Kang (TSK) fuzzy systems, and discusses its sufficiency and necessity. In author´s previous work, the sufficiency of the conditions was shown, but the necessity has remained an open question. This study analytically proves that the necessity holds when a single-input TSK fuzzy system employs two rules. For the cases of three rules, counter examples are provided to verify that the necessity of the conditions violates.