This double seminar will present part of the results of three months of research stay at our Department:

a) José Luis García Malacara: “Duality of Controllability-Observability in a class of Hybrid Systems”.

We are concerned with controllability and observability of timed continuous Petri nets (TCPN) under the infinite server semantics.  In particular, we deal with structural and generic controllability (i.e., controllability for all values of the firing rates, and for almost all values of those parameters, respectively). Using ideas developed for linear structured systems theory, the concept of generic controllability is employed to characterize controllability for TCPN in terms of the structure of the Petri net graph. Joins, conflicts, forks and attributions are used to explain how control signals are transmitted or blocked through the net.

Based on the net graph, the obtained results are viewed as the dual of the previous obtained for structural and generic observability in TCPN. Clear relationships are stated using the duality concept on Petri net graphs.

b) Manuel Navarro: “Bifurcations in timed continuous Petri nets”.

The dynamic behavior of TCPN under the infinite server semantics will be considered. In particular, we study the change of the throughput in steady state due to the variation of a firing rate. Jumps in this value (equivalently in the steady state marking) are considered as bifurcations. They occur in nets in which the equilibrium marking of a region goes from being stable to unstable. In equal-conflict nets the bifurcation may occur depending on the firing rates of transitions in conflicts; if transitions are also joins, the steady state marking lies in the boundary between regions.

We show that in consistent and conservative choice-free Petri nets the change in the steady state throughput is continuous. This kind of result can be extended to equal-conflict nets when we vary the firing rates of transitions which are persistent. So, the throughput in steady state is continuous, moreover also monotone for these classes of nets.

Similarly, the variation of the initial marking is considered. There are relationships between the observed behavior of the throughput due to the change of a firing rate and the variation of components of the initial marking. Bifurcations are critical for the performance of the net system, thus very interesting in order to control them.