Jump criteria of nonlinear control systems and the validity of statistical linearization approximation

We study the conditions for the unique response in a class of nonlinear control systems subject to random inputs using statistical linearization approximation. As in the case of sinusoidal inputs, we show that jump phenomena may occur if the inverse vector locus of the linear part passes through certain regions on the complex plane, where the regions are defined by the characteristics of nonlinear part. Such jump phenomena regions for several typical nonlinearities are given; we also show that, among a restricted class of nonlinearities, the saturation and dead zone produce the largest jump phenomena regions. A new result concerning the validity of statistical linearization approximation of nonlinear control systems is also presented. We show that the condition for the uniqueness of response to a given input in a nonlinear feedback system obtained through statistical linearization approximation is compatible with a related rigorous result, thus providing additional confidence in the applicability of statistical linearization.