Stabilization of LPV systems

We study in this paper the static state-feedback stabilization of linear finite dimensional systems depending polynomially upon a finite set of real, bounded, parameters. These parameters are a priori unknown, but available in real-time for control. We state two main results. First, we show that stabilizability of the class of systems obtained for frozen values of the parameters may be expressed equivalently by some LMI conditions, linked to certain class of parameter-dependent Lyapunov functions. Second, we show that existence of such a Lyapunov function for the LPV systems subject to bounded rate of variation of the parameters with respect to time, may be in the same manner expressed equivalently by some LMI conditions. In both cases, the method provides explicitly parameter-dependent stabilizing gain.