Stability and Persistent Excitation in Signal Sets

Persistent excitation (PE) conditions have been widely used to analyze stability properties of various parameter identification algorithms and to establish uniform global asymptotic stability (UGAS) for a large class of nonlinear time-varying systems. In order to generalize such conditions to a more general setting, a new PE condition is proposed with three basic ingredients: a signal set to represent a family of time functions (e.g., trajectories); a pseudo distance measure to describe the convergence; and some binary relations (e.g., state-to-output relations). Closely related to detectability, this PE condition is a necessary condition to guarantee UGAS. Under uniform global stability and an integral inequality, it becomes a sufficient condition of UGAS. A novel concept: M-pair, which aims at simplifying the checking of the PE condition, is introduced. By using M-pair, it is possible to simplify the structure of the referred signal set (in the spirit of the classic Krasovskii-LaSalle theorem) and to extend the dimension of the reference signal set (similar to the Matrosov theorem). Thus, the framework of M-pair not only unifies these well-known results, but also generates more flexibility in checking the PE conditions. When applied to nonlinear switched systems, three new tools to verify the PE condition are obtained. Finally, an example illustrates that a nonlinear time-varying switched system with arbitrary switching can be shown to be UGAS without using a common Lyapunov function.