# POSITIVE INVARIANCE OF POLYHEDRAL SETS AND LINEAR CONSTRAINED REGULATION PROBLEM IN THE CONTEXT OF THE DELTA OPERATOR

### Abstract

In the literature, methods are available to obtain positively invariant sets for continuous-time and discrete-time systems using the shift operator. However, there are not references showing how to develop methods using the delta operator. In this work, positive invariance relations of polyhedral sets are proposed in the context of the delta operator model for linear discrete-time systems. The delta operator approach is known to be of interest when using high sample rates and it also allows to unify discrete-time and ontinuous-time concepts and results. In this context, the proposed delta operator positive invariance relations which are obtained from the classical shift operator results, are also shown to recover the continuous-time invariance relations when the sample period tends to zero. Due to the interest of using the positive invariance property and polyhedral sets in constrained control, a linear programming optimization approach is also proposed in the context of the delta operator to solve a discrete-time linear constrained regulation problem. A numerical example is exploited to show that the proposed delta operator solution closely follows the continuous-time one.