A generalized method for determining the closed-loop frequency response of nonlinear systems

One of the formidable problems in the area of feedback control concerns the analysis and synthesis of nonlinear systems. Of the several methods available for treating nonlinear systems, one of the more widely used is the sinusoidal response method. Valuable information about the characteristics of nonlinear systems can be obtained from the open-loop frequency response, but the full potential of this method has not been exploited because apparently there is no widely accepted procedure for determining the closed-loop frequency response. This paper describes a method whereby the closed-loop frequency response of an important class of nonlinear systems can be determined easily. Development of this method is dependent upon representing the nonlinear element as a so-called equivalent gain. The method is described and illustrated by analytically determining the closed-loop frequency response of an acceleration-limited system and an error-limited system. The results of the use of this method compare favorably with solutions of the two problems as obtained from an analogue computer. A few of the nonlinear phenomena which can be reasonably approximated by this method are saturation, limiting, backlash, and dead space.