A new class of universal Lyapunov functions for the control of uncertain linear systems

We analyze the problem of synthesizing a state feedback control for the class of uncertain continuous-time linear systems affected by time-varying memoryless parametric uncertainties. We consider as candidate Lyapunov functions the elements of the class Σ_p ^z which is formed by special homogeneous positive definite functions. We show that this class is universal, in the sense that a Lyapunov function exists if and only if there exists a Lyapunov function in Σ_p^z. We prove this result in a constructive way, showing that such Lyapunov function can always be obtained by “smoothing” a polyhedral function for which construction algorithms are available. We show how we can associate to the considered functions controllers in a global explicit form