Metrics Over the Class of Polyhedra and Several Correspondences in Constrained Control

The present paper deals with the computation of distances between polyhedral sets. The classical reference in this sense is the Hausdorff formulation, but several alternative distance functions can be cited. Their geometrical properties are revisited here together with the induced metrics over this specific class of convex bodies. The genuine goal of the analysis is to clarify which are the reasonable formulations to be employed in the framework of control laws for systems under constraints. The specific assumptions in the definition of distance functions may lead to particular considerations when operating with polyhedral feasible domains. Not to be neglected is the real time aspect of the control laws and consequently the need for computational efficiency. The complexity of the evaluation is examined with respect to the complexity of the optimisation problem which cast the chosen metric.