Exact Analysis of a Nonlinear Feedback Control System subjected to Sinusoidal Input

The frequency response of a feedback control system containing a nonlinear element of piecewise-linear type subjected to sinusoial input is exactly obtained by a new method utilizing appropriate Fourier series expansion. This method is in principle the same as that introduced by the author to solve the steady vibrations in an unsymmetrical piecewise-linear system subjected to a harmonic exciting force. It has the following main features : 1) Linealizing the nonlinear differential equation governing the dynamic behavior of the system by expanding the nonlinear part of the output of the piecewise-linear element into a Fourier series with the same period as of the given sinusoidal input. 2) Obtaining the formal solution of thus linealized equation by regarding the above-mentioned nonlinear part as if it were an exciting force from without. This solution contains unknown coefficients of the Fourier expansion above assumed. 3) Determining these unknown coefficients from the conditions that the above obtained solution satisfies the given piecewise-linear characteristics of the system. Certain process of convergency improvement by means of series transformation may prove highly effective in this step. As an example of the piecewise-linear element, we treat an element with a dead zone, and give the procedure of analysis for a feedback control system with such an element. Some numerical results and diagrams for the frequency response of the system are added. Also a few examples of wave forms for certain amplitudes and frequencies of the sinusoidal input are shown. These procedures of analysis and their results may contribute to the removal of uncertainties involved in the describing function analysis usually applied to obtain the frequency response of the nonlinear control system.