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

In this paper, 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 Sigma(p)(z), which is formed by special homogeneous positive definite functions. show that this class is universal in the sense that a Lyapunov function exists if and only if there exists a Lyapunov function in Sigma(p)(z). They 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. The authors show that unlike the polyhedral Lyapunov functions, these functions allow us to derive explicit formulas for the stabilizing controller.