On the Monotonicity of Fuzzy-Inference Methods Related to T–S Inference Method

Yubazaki et al. have proposed a ¿single-input rule modules connected-type fuzzy-inference method¿ (SIRMs method) whose final output is obtained by combining the products of the importance degrees and the inference results from single-input fuzzy-rule modules. Moreover, Seki et al. have proposed a ¿functional-type SIRMs method¿ whose consequent parts are generalized to functions from real numbers. It is expected that inference results from the functional-type SIRMs method are monotone, if the antecedent parts and the consequent parts of fuzzy rules in the functional-type SIRMs rule modules are monotone. However, this paper points out that even if consequent parts in the functional-type SIRMs rule modules are monotone, the inference results are not necessarily monotone when the antecedent parts are noncomparable fuzzy sets, and it clarifies the conditions for the monotonicity of inference results from the functional-type SIRMs method. Moreover, for the Takagi-Sugeno (T-S) inference method, the monotonicity condition is clarified in the case of two inputs by using the equivalence relation of fuzzy inference.

I. Introduction

Fuzzy inference plays a significant role in fuzzy applications [1]–[5]. However, in the traditional fuzzy-inference methods, the number of fuzzy rules tends to become large, and hence, the setup and tuning of fuzzy rules turn out to be difficult. This is due to the fact that all the input items of the system are specified in the antecedent parts of the fuzzy rules, while all the output items must be specified in the consequent parts. On the other hand, the “single-input rule modules connected-type fuzzy-inference method” (SIRMs method) [7]–[12], which unifies the inference outputs from fuzzy-rule modules consisting of just one input type “if–then” rule, is able to reduce the number of fuzzy rules drastically. This method has been applied to nonlinear function identification, control of a first-order lag system with dead time, orbital pursuit control of a nonrestrained object, stabilization control of a handstand system [8], etc., and good results were obtained. However, since the number of rules of the SIRMs method is limited as compared with the traditional inference methods, inference results obtained by the SIRMs method are rather simple, in general.