General Takagi-Sugeno fuzzy systems with simplified linear rule consequent are universal controllers, models and filters

Abstract

Takagi-Sugeno (TS) fuzzy systems have successfully been employed, mainly in a trial-and-error manner, to solve many control and modeling problems; but their applications as signal filters remain to be fully explored. Compared to their nonfuzzy counterparts, TS fuzzy controllers and models are difficult to be efficiently constructed because there is a large number of design parameters in the rule consequent. The number grows dramatically with the increase of the number of input fuzzy sets and input variables. Furthermore, there exists little published result on relationship between TS fuzzy controllers/models/filters and their nonfuzzy counterparts. In this paper, we investigate, in relation to some popular nonfuzzy controllers, models and filters, analytical structure of a general class of multi-input single-output (MISO) TS fuzzy systems that use arbitrary fuzzy rules with our recently introduced simplified linear rule consequent. Other components of the fuzzy systems in this study are general: arbitrary continuous input fuzzy sets, any type of fuzzy logic AND and the generalized defuzzifier containing the widely used centroid defuzzifier as a special case. We prove that the general MISO TS fuzzy systems are: (1) nonlinear variable gain controllers when implemented as controllers, or (2) nonlinear time-varying auto-regressive with the extra input (ARX) models when implemented as models, or (3) nonlinear infinite impulse response (IIR) or finite impulse response (FIR) filters when implemented as filters. Furthermore, we constructively prove that the general TS fuzzy systems with the simplified linear rule consequent are universal approximators and can approximate any continuous function in closed domain arbitrarily well. The practical implication of these results is that these fuzzy systems, with much less design parameters, are always able to produce solutions to various control, modeling and filtering problems. We also establish sufficient conditions that can be used to calculate the number of input fuzzy sets and rules needed for achieving prespecified approximation accuracy.