Algebraic and geometric characterization of Petri net controllers using the theory of regions

This paper presents a formal treatment of Petri net controller design problems. Two supervisory control problems of plant Petri net models, forbidden state and forbidden state-transition problems, are defined. The theory of regions is used to provide algebraic characterizations of pure and impure control places for both problems. Thanks to Farkas-Minkowski's lemma, the algebraic characterizations lead to nice geometric characterization for the existence of control places for the two supervisory problems.