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 semiglobally 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 present a converse theorem for USPAS and (2) we establish USPAS of cascaded systems without the requirement of a Lyapunov function. Compared to other converse theorems in the literature, ours has the advantage of guaranteeing a specific relationship between the upper and lower bounds on the generated Lyapunov function VV and of providing a time-invariant bound on the gradient of VV, which is fundamental to establish theorems for cascades.