Necessary and sufficient conditions for uniform semiglobal practical asymptotic stability: Application to cascaded systems

It is well established that for a cascade of two uniformly globally asymptotically stable (UGAS) systems, the origin remains UGAS provided that the solutions of the cascade are uniformly globally bounded. While this result has met considerable popularity in specific applications it remains restrictive since, in practice, it is often the case that the decoupled subsystems are only uniformly \emph{semiglobally} \emph{practically} asymptotically stable (USPAS). Recently, we established that the cascade of USPAS systems is USPAS under a local boundedness assumption and the hypothesis that one knows a Lyapunov function for the driven subsystem. The contribution of this paper is twofold: 1) we establish USPAS of cascaded systems without the requirement of a Lyapunov function and 2) we present a converse theorem for USPAS. While other converse theorems in the literature cover the case of USPAS ours has the advantage of providing a bound on the gradient of the Lyapunov function, which is fundamental to establish theorems for cascades.