State Feedback Control via Positive Invariance for Max-plus Linear Systems using Γ-algorithm

It is now almost classical that max-plus linear systems modelize discrete events systems (DES) of practical interest. The control of such systems is a recent topic of research which has not yet been so developed than control theory in classical algebra. In this paper, we identify and characterize the positive invariance of a max-plus polyhedron set by max-plus linear dynamical systems. We show that positive invariance of max-plus polyhedron set can be expressed under the form of polyhedron sets inclusion. Using Haar´s Lemma (1918) which provides the algebraic characterization of the inclusion of two polyhedral sets, Necessary and Sufficient Conditions (NSC) are provided under which the positive invariance of a max-plus polyhedron set hold. As an application we propose a method using Γ-algorithm to compute a static state feedback control. Finally, the proposed methods are illustrated by an example.