Real Time Identification of Time Petri Net models

The identification problem is formally stated as follows: Given a set Lobs of observed timed sequences that belongs to the timed language L of the unknown system T ′ and a set of places P of cardinality m, the problem consists in the identification of the net structure N , the timing structure I , and the initial marking m0 such that the timed language generated by T = 〈N,m0, I〉, named L(T ), contains Lobs, i.e. L(T ) ⊇ Lobs. The unknowns are the matrices Pre, Post, the vector m0 and the lower and upper bounds of each timed transition tj ∈ T . The proposed approach is based on the formulation of a MIPP (Mixed Integer Programming Problem) where the constraints are obtained by the set of logical conditions, named G(σT ), representing the enabling and disabling conditions for the firing of transitions included in each observed timed sequence σT . Let Lobs be the set of observed timed sequences and P the place set of the system to identify. A solution to the identification problem can be computed solving the system of equations