Hysteretic Control Lyapunov Functions with Application to Global Asymptotic Tracking for Underwater Vehicles

Abstract

This paper introduces hysteretic control Lyapunov functions (HCLFs) for hybrid feedback control of a class of continuous-time systems. A family of HCLFs consists of local control Lyapunov functions defined on open domains, and include finite collections of open and closed sets that cover the state-space, implicitly defining a hysteresis-based switching mechanism. Given an HCLF family, we derive sufficient conditions for the existence of globally asymptotically stabilizing hybrid control laws. Moreover, we provide a constructive design procedure for synthesis of optimization-based feedback laws under mild conditions on the objective functions. We construct HCLFs for an underwater vehicle and demonstrate their applicability to hybrid control design for global asymptotic trajectory tracking for underwater vehicles.