The use of complex equivalent gain of a memory-type nonlinearity for the analysis of feedback control systems with random inputs

The paper presents a technique of linearization of a memory-type nonlinearity with a Gaussian input by applying the concept of "complex describing function." The quasi-linearized transfer function for the nonlinearity with the statistical input is termed "complex equivalent gain." The magnitude of the complex equivalent gain is defined as the ratio of the rms value of the output correlated component to the rms value of the input of the nonlinearity while the phase is assumed to be frequency independent. Furthermore, it is pointed out that the complex describing function for a memory-type nonlinearity is a limiting case of complex equivalent gain where the input to the nonlinearity is a simple sine wave. Finally, a closed-loop control system incorporating a memory-type nonlinearity and impressed by a Gaussian input is analyzed by using the proposed quasi-linearization technique. The results obtained from an analog computer study of the system which agree closely with the results of analysis are also presented.