Robust Stability of linear uncertain systems through Piecewise Quadratic Lyapunov Functions defined over conical partitions

In this paper a class of Piecewise Quadratic Lyapunov Functions (PQLFs) for the analysis of the robust stability of linear systems subject to polytopic uncertainties is considered. These functions are obtained by partitioning the state space into polyhedric conical sets and by associating to each cone a quadratic form. This class of Lyapunov functions is not only a generalization of quadratic Lyapunov functions but it also (strictly) contains the class of polyhedral Lyapunov functions. The latter implication directly proves the universality of these functions for the robust stability problem and justifies the effort of finding a Lyapunov function belonging to this class. The technicalities for an effective use of this class of functions to test the robust stability of linear uncertain systems will be detailed. Numerical examples show that the proposed methodology allows one to obtain results that, compared with the actual state of the art, perform better under several viewpoints.