Passive linear time-varying systems: State-space realizations, stability in feedback, and controller synthesis

Abstract

In this paper we consider linear time-varying passive systems. We state various theorems, which rely on the state-space matrices of the system, that identify when a linear-time varying system is purely passive, input strictly passive, output strictly passive, or input-state strictly passive which is a nonstandard notion of passivity defined in this paper. Two of our theorems resemble the Kalman-Yakubovich- Popov Lemma, one applicable to time-varying systems with a feedthrough matrix and the other for linear time-varying systems without one. The negative feedback interconnection of various systems is considered. We show that an output strictly passive system negatively interconnected with an input-state strictly passive system is globally asymptotically stable. We also show that both linear time-varying input-state and output strictly passive systems when connected in negative feedback with a sector bounded, memoryless nonlinearity are also globally asymptotically stable. The optimal design of a time-varying output strictly passive controller is also considered. We present an example: the position and velocity control of a time-varying mass controlled via a dynamic time-varying compensator and a sector bounded, memoryless nonlinearity.