Invariant subspaces and sensor placement for observability in Continuous Timed Petri Nets
Enrique Aguayo-Lara, Antonio Ramírez‐Treviño, Javier Ruiz‐León · 2011
This work deals with the problem of sensor placement for observability in Continuous Timed Petri Nets (ContPN) with infinite server semantics. ContPN are represented by a family of linear systems F = Σ1= (A1, B1, S1), ..., Σn= (An, Bn, Sn) switching between them. The observability in ContPN requires the observability of each linear system Σkand the distinguishability of each pair of systems Σk, Σj. Thus this work uses the well known result that a system Σkis observable iff the maximum Ak-invariant subspace contained in the kernel of Skis the null space to build output functions Skin such a way that the ContPN becomes observable. This work characterizes the Ak-invariant subspaces associated to nonzero eigenvalues from the ContPN incidence matrix and the others Ak-invariant subspaces from the ContPN structure. Using this information an output matrix S = S1= ... = Snis computed in such a way that no Ak-invariant subspace is contained in the kernel of S. From the construction of S we can guarantee that every pair of systems is distinguishable from each other. Thus the ContPN becomes observable.