We face the problem of stabilizing Linear Parameter Varying systems by means of Gain Scheduling Control. 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 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.