On Controllability of Timed Continuous Petri Net Systems: the Join Free Case

Abstract

Timed continuous PNs have been used as relaxed models to "approximately" evaluate the performance of the underlying discrete systems. Moreover, the control of the continuized systems can approximate the scheduling of the (discrete) PNs. This paper analyses controllability of conservative and consistent join free net systems under infinite server semantics. They are positive systems in which classic control theory is not directly applicable: in this domain input actions are non-negative and dynamically bounded, leading to polytope constrained state space instead of a vectorial space. Thus a new concept of controllability is proposed. The "controllability space" (CS), included in this polytope, is studied depending on the set of controlled transitions. The full state space is "controllable" iff all the transitions are controlled. On the other hand, a given state can always be "controlled" (reached and maintained) without using at least one transition. The CS obtained by controlling just one transition is a straight segment, and the CS obtained with several transitions includes the convex of the CS obtained independently with every transition. If additionally the system is choice-free the state space is a partition of the CS obtained with the entire set of transitions except one. Nevertheless borders belong to all neighbour regions.