"This work has been submitted to the "IEEE Transactions on Control Systems Technology" for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible." This report deals with the adaptation of a real-time controller's sampling period to account for the available computing resource variations. The design of such controllers requires a parameter-dependent discrete-time model of the plant, where the parameter is the sampling period. A polytopic approach for LPV (Linear Parameter Varying) systems is then developed to get an $H_{\infty}$ sampling period dependent controller. A reduction of the polytope size is here performed which drastically reduces the conservatism of the approach and makes easier the controller implementation. Some experimental results on a T inverted pendulum are provided to show the efficiency of the approach.