Piecewise quadratic functions for finite-time stability analysis

In this paper we consider the finite-time stability (FTS) problem for linear time varying systems. In most of the previous literature, the definition of FTS exploits the standard weighted quadratic norm to define the initial and trajectory domains, which, therefore, turn out to be ellipsoidal; this is consistent with the fact that quadratic Lyapunov functions are used to derive the FTS conditions. Conversely, the recent paper [1], considers the case where the above domains are polytopic and, consequently, the analysis is performed with the aid of polyhedral Lyapunov functions. In the current work, the class of Piecewise Quadratic Lyapunov functions is considered. First, it is shown that such class of functions recovers as particular cases both quadratic and polyhedral Lyapunov functions; then a novel sufficient condition for FTS of linear time-varying systems is provided. A procedure is proposed to convert such condition into a computationally tractable problem. The examples illustrated at the end of the paper show the benefits of the proposed technique with respect to the methodologies available in the literature.