Constructing nonlinear variable gain controllers via the Takagi-Sugeno fuzzy control

We investigated analytical structure of the Takagi-Sugeno (TS) type of fuzzy controllers, which was unavailable in the Literature. The TS fuzzy controllers we studied employ a new and simplified TS control rule scheme in which all the rule consequent use a common function and are proportional to one another, greatly reducing the number of parameters needed in the rules. Other components of the fuzzy controllers are general: arbitrary input fuzzy sets, any type of fuzzy logic, and the generalized defuzzifer, which contains the popular centroid defuzzifier as a special case. We proved that all these TS fuzzy controllers were nonlinear variable gain controllers and characteristics of the gain variation were parametrized and governed by the rule proportionality. We conducted an in-depth analysis on a class of nonlinear variable gain proportional-derivative (PD) controllers. We present the results to show: 1) how to analyze the characteristics of the variable gains in the contest of control; 2) why the nonlinear variable gain PD controllers can outperform their linear counterpart; and 3) how to generate various gain variation characteristics through the manipulation of the rule proportionality.