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 ($TCPN$s), 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 $TCPN$s 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 ($PEQ$) $TCPN$s 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.