Finite-time control of switching linear systems: The uncertain resetting times case

In recent years, a number of papers have treated the problem of the finite-time stability and stabilization of impulsive (or, more in general, switching) dynamical linear systems. Generally, these works assume that the sequence of switching (in the following resetting) times is a priori known. In this paper, we remove such (strong) assumption, so making the technique more appealing from the practical control engineering point of view. A first result provided in this work is a sufficient condition for finite-time stability when the resetting times are known with a certain degree of uncertainty. Such condition requires the solution of a suitable feasibility problem based on coupled difference/differential LMIs. We show that as the uncertainty intervals reduce in size, our condition becomes less conservative, becoming necessary and sufficient in the certain case (i.e., the resetting instants are perfectly known). Eventually, we consider the conceptually different situation in which the resetting times are totally unknown, namely, the arbitrary switching case. The analysis results are then used to derive sufficient conditions for the existence of state-feedback controllers that finite time stabilizes the closed-loop system in the three cases mentioned earlier. A nontrivial example, considering the finite-time control of the liquid levels into three interconnected reservoirs, shows the effectiveness of the proposed approach.