Abstract
In this paper, the first of two companion works, we deal with linear systems and extend to the finite-time setting the concepts of stabilizability and detectability. After stating necessary and sufficient conditions for finite-time stabilizability (FT-stab) and detectability (FT-det), in terms of feasibility conditions constrained by differential linear matrix inequalities (DLMIs), we investigate the relationships between the existence of dynamical output feedback finite-time stabilizing controllers and the concepts of FT-stab and FT-det. In the classical Lya-punov framework, a consequence of the well known Separation Principle is that stabilizability via dynamic output feedback controllers is equivalent to stabilizability (via state feedback) plus detectablity. In this paper, we show that, even in the finite-time context, stabilizability and detectability play a role into the existence of stabilizing dynamical controllers, although with some differences. Indeed we prove that a dynamic output feedback controller which finite-time stabilizes the overall closed loop system exists if and only if the open loop system is finite-time detectable and stabilizable plus a further LMI condition. Based on this result, a procedure for the design of a suitable output feedback 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.
Published Version
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