Duality of controllability and observability in proportional equal conflict timed continuous Petri Nets

Abstract

Controllability and observability properties have been widely studied in Timed Continuous Petri Nets (TCPNs), a class of piecewise affine systems, in order to analyze and control crowded discrete event systems. This work studies the concept of duality applied to TCPNs as a vehicle to establish links between controllability and observability, i.e., a synergy to improve the understanding of these properties and to enlarge the class of nets that can be analyzed. To achieve this, we study the concepts of rank-controllability and rank-observability. They capture structural conditions for controllability and observability. Afterwards, the computation of dual nets for Fork-Attribution (FA), Choice-Free (CF), Join-Free (JF), and Proportional Equal Conflict (PEQ) TCPNs subclasses are presented. By using the dual definition, several relations between the primal’s controllability and its dual’s observability are stated. Particularly, in FA rank-controllability and rank-observability are dual properties. In consistent and strongly connected CF, JF, and PEQ nets, the rank-observability of the dual is sufficient for the rank-controllability of the primal. The opposite implication holds for CF and PEQ if the self-loop places, added by the dual construction methodology, are measurable.