Jump Criteria of Nonlinear Control Systems and the Validity of Statistical Linearization Approximation

It is well known that jump resonance can occur in nonlinear control systems with attendant worsening of the control performance. In the case of periodic input signals, the rigorous conditions for the unique response,* or equivalently, for the absence of jump resonance, are available. 1 In addition, various authors have studied the conditions for the absence of jump resonance using the describing function method (see Refs. 2 and 3); the describing function method criteria * Although the present terminology is widely used, a more precise term will be "unique solution to the equations arising from the steady state situation for a given input realization." 2529 2530 T H E BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1969 for jump resonance have been found for many common nonlinearities. For systems with random inputs, the exact condition for the unique response is not known, although a rigorous condition for the convergence of a successive approximation is available. 4 A useful approximate technique for studying the performance of nonlinear feedback systems subject to random inputs is Booton's method of statistical linearization. 5 Although the method of statistical linearization has been widely used, the conditions for its validity are not fully known. The first part of this paper concerns the determination of the criteria for unique response, in a class of nonlinear control systems subject to random inputs, using statistical linearization approximation. We present the statistical linearization criteria for unique response for several common nonlinearities.