Parameter-dependent Lyapunov functions for time varying polytopic systems

This paper provides global asymptotic stability conditions for continuous-time varying polytopic systems, using a parameter dependent Lyapunov function. The time varying parameter uncertainty as well as its time derivative are modelled as belonging to polytopic convex sets and their dependence is made explicit to get less conservative results. A particular case, characterized by the parameter uncertainty satisfying a linear differential equation is analyzed and a simpler version of the aforementioned stability conditions is presented. The results are expressed in terms of linear matrix inequalities being thus numerically solvable with no big difficulty. The theory is illustrated by the determination of the asymptotic stability region of a Mathieu's type equation with an uncertain time varying parameter.