On the Response of Nonlinear Control Systems to Periodic Input Signals

The describing-function technique is often used to determine the response of nonlinear control systems to sinusoidal input signals. In this approach,* which is applicable to systems of any order but which is ordinarily restricted to systems containing only one nonlinear element, it is assumed that the response is periodic, with only the component at the input frequency significant. Although the describing-function technique is of considerable practical value and indeed is one of the most powerful analytical tools available to the control system synthesist, it appears that, except with regard to predicting the existence of self-sustained oscillations, 6 there has been no rigorous discussion of its validity.! In this paper we study a broad class of nonlinear control systems containing a single memoryless nonlinear element. We present conditions under which there exists a unique periodic response, with a given period, to an arbitrary periodic input with the same period, and we * The describing-function technique was discovered independently by engineers in at least five different countries. 1 " 6 f However, some interesting relevant ideas have been presented by Johnson. 7 911