Invariant subspaces and sensor placement for observability in Continuous Timed Petri Nets

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), ..., Σ_n = (A_n, B_n, S_n) switching between them. The observability in ContPN requires the observability of each linear system Σ_k and the distinguishability of each pair of systems Σ_k, Σ_j. Thus this work uses the well known result that a system Σ_k is observable iff the maximum A_k-invariant subspace contained in the kernel of S_k is the null space to build output functions S_k in such a way that the ContPN becomes observable. This work characterizes the A_k-invariant subspaces associated to nonzero eigenvalues from the ContPN incidence matrix and the others A_k-invariant subspaces from the ContPN structure. Using this information an output matrix S = S₁ = ... = S_n is computed in such a way that no A_k-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.