A necessary condition for local asymptotic stability of nonlinear systems with exogenous disturbance

Local asymptotic stability of nonlinear systems with real-parametric uncertainty or disturbance is one of the important problems in control systems literature. In this paper, we investigate this problem for nonlinear systems with time-varying disturbance. We assume that the disturbance vector is generated by an exosystem, which is neutrally stable. Thus, the disturbances that we consider include both constant and periodic signals. For this class of nonlinear systems with time-varying disturbance, we derive a necessary condition for local asymptotic stability of equilibria. As corollaries of our general result, we deduce the necessary condition obtained by Byrnes and Sundarapandian [1] for nonlinear systems with constant real parametric uncertainty, and the necessary condition obtained by Brockett [2] for nonlinear autonomous systems. We illustrate our result with several examples.