Finite-time stabilizability and detectability of linear systems. Part II: Design of observer based output feedback finite-time stabilizing controllers

摘要

In this paper and the companion work, we deal with linear systems and investigate the concepts of stabilizability and detectability in the finite-time context. Here, our aim is to study the relationships between the existence of observer based output feedback finite-time stabilizing controllers and the concepts of finite-time stabilizability (FT-stab) and detectability (FT-det). In the classical Lyapunov context, the Separation Principle states that stabilizability and detectability of the given linear system are necessary and sufficient for the existence of an output feedback dynamical controller which, without loss of generality, can be given a controller-observer structure. We show that this is no longer true in the finite-time context; therefore, we provide a further sufficient condition for the existence of an observer-based output feedback controller which finite-time stabilizes the closed loop system. Based on this result, a procedure for the design of a suitable controller is provided; such procedure requires the solution of some optimization problems constrained by DLMIs. A numerical example illustrates the benefits of the proposed approach.