Steady-State Control Reference and Token Conservation Laws in Continuous Petri Net Systems

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper addresses several questions related to the control of timed continuous Petri nets under infinite server semantics. First, some results regarding equilibrium states and control actions are given. In particular, it is shown that the considered systems are piecewise linear, and for every linear subsystem the possible steady states are characterized. Second, optimal steady-state control is studied, a problem that surprisingly can be computed in polynomial time, when all transitions are controllable and the objective function is linear. Third, an interpretation of some controllability aspects in the framework of linear dynamic systems is presented. An interesting finding is that noncontrollable poles are zero valued. Note to Practitioners-Petri nets are a well-known formalism for the analysis and design of discrete event systems. Due to the state- explosion problem some kind of relaxation is frequently used. In particular, fluidification is a classical approximation technique that it is usually employed in the analysis of manufacturing or logistic systems, specially when heavily loaded. This work focuses on timed continuous Petri nets which are a fluidified version of discrete Petri nets with timing associated to transitions (representing stations with servers that perform activities). Furthermore, transitions have associated control actions which can slow down the corresponding activities from an initial maximum speed, that depends on the marking, and possibly halt them completely. The steady-state control problem of this kind of system and some invariant-dynamical properties are addressed. </para>