Stabilization of LPV systems: state feedback, state estimation and duality

In this paper, the problem of the stabilization of linear parameter varying systems by means of gain scheduling control is considered. This technique consists on designing a controller which is able to update its parameters on-line according to the variations of the plant parameters. We first consider the state feedback case and we show a design procedure based on the construction of a Lyapunov function for discrete-time LPV systems in which the parameter variations are affine and occur in the state matrix only. This procedure produces a nonlinear static controller. We show that, differently from the robust stabilization case, we can always derive a linear controller, that is nonlinear controllers cannot outperform linear ones for the gain scheduling problem. Then we show that this procedure has a dual version, which leads to the construction of a linear gain scheduling observer. The two procedures may be combined to derive an observer-based gain scheduling compensator.