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 = Σ<inf>1</inf> = (A<inf>1</inf>, B<inf>1</inf>, S<inf>1</inf>), …, Σ<inf>n</inf> = (A<inf>n</inf>, B<inf>n</inf>, S<inf>n</inf>) switching between them. The observability in ContPN requires the observability of each linear system Σ<inf>k</inf> and the distinguishability of each pair of systems Σ<inf>k</inf>, Σ<inf>j</inf>. Thus this work uses the well known result that a system Σ<inf>k</inf> is observable iff the maximum A<inf>k</inf>-invariant subspace contained in the kernel of S<inf>k</inf> is the null space to build output functions S<inf>k</inf> in such a way that the ContPN becomes observable. This work characterizes the A<inf>k</inf>-invariant subspaces associated to nonzero eigenvalues from the ContPN incidence matrix and the others A<inf>k</inf>-invariant subspaces from the ContPN structure. Using this information an output matrix S = S<inf>1</inf> = … = S<inf>n</inf> is computed in such a way that no A<inf>k</inf>-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.