Fuzzy-Logic Controller Synthesis Based on Sinusoidal-Input Describing Functions & Optimization

We present a new method for the synthesis of fuzzylogic controllers (flcs) for amplitude-sensitive nonlinear plants based on sinusoidal-input describing-function (sidf) methods plus step-response optimization. This technique exploits the fact that two traditional classes of flcs are, in functional terms, of the proportionalplus-derivative (pd) and proportional-plus-integral (pi) types. This method involves a two-step process wherein an initial controller is obtained via the direct generation of the membership functions and output levels based on the “frequency response” of the nonlinear plant in the describing-function sense (G(jω, a) where a is the sinusoidal input amplitude), then the flc is perfected via optimization of the step responses for a specified set of input amplitudes. The resulting fuzzy-logic controller obtained in this paper includes derivative action in an inner-loop feedback path (nonlinear rate feedback) and nonlinear pi compensation in the forward path; the performance of the closed-loop system is, by design, quite insensitive to reference-input amplitude. An illustration of the method and its effectiveness is provided, based on a prototypical position control problem where a servo motor plus mechanical load are characterized by torque saturation and nonlinear friction (stiction). We emphasize, however, that this approach is capable of treating nonlinear systems of a very general nature, with no restrictions as to system order, number of nonlinearities, configuration, or nonlinearity type.