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

The authors 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. They consider as candidate Lyapunov functions the elements of the class Σ_p^z which is formed by special homogeneous positive definite functions. They 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. They prove this result in a constructive way, showing that such a Lyapunov function can always be obtained by “smoothing” a polyhedral function for which construction algorithms are available. The authors show that unlike the polyhedral Lyapunov functions, these functions allow us to derive explicit formulas for the stabilizing controller