TABLE OF CONTENTS The describing function method is useful in determining the stability of nonlinear systems. It is based on an analysis that neglects the effect of harmonics in the system. Therefore, it is classified as a frequency response method. This approximation is best made for systems exhibiting low-pass filtering. It usually improves in accuracy for increasing system order, as it represents an application of the method of harmonic linearization.22
First, we will examine the case of the single-input single-output (SISO) system of Fig. 2.1. The system is divided into a single nonlinearity followed by a linear element. This linear portion is designated by the transfer function G, a frequency dependent but amplitude-independent function. The nonlinear element N is amplitude dependent but usually frequency independent. Feeding the error e into the nonlinear element generates the control u. It consists of two parts, a quasilinear gain and a distortion term. Denoting this equivalent gain by K results in the control equation u = Ke + f( e) . K is a function of the input signal amplitude. This gain is the describing function K( g), where e = g sin ?t.
The proper choice for K( g) often allows the distortion term f( e) to be neglected. Minimizing the distortion f( e) in a mean-square sense is accomplished by choosing K(
