Abstract—The paper proves some properties of a previously proposed identification algorithm that builds on line the Petri net model of Discrete Event Systems (DES). The procedure uses the real time observation of the DES events and the corresponding available output vectors that partially provide the place markings. The paper shows how the considered identification method allows us to define a supervisory controller via monitor places enforcing generalized mutual exclusion constraints. To show the efficiency of the proposed approach, a communication gateway case study is presented. I. INTRODUCTION YSTEM identification deals with choosing mathematical models from a known model set to characterize the input-output behavior of an unknown system from finite data. Modern man made systems are modeled as Discrete Event Systems (DES) and the DES behavior can be described by a language that specifies all the admissible sequences of events that the DES is capable of “processing” or “generating” [4]. The problem of identifying a language and an automaton from finite data has been explored in [1, 12]. Moreover, paper [13] presents an algorithm for the construction of a free-labeled PN, i.e., a net where each transition is associated to a unique label, from the knowledge of a finite set of its firing sequences. An analogous problem is solved in [2] that proposes a method to synthesize a PN on the basis of a set of structural constraints. In two recent papers [3,10] the authors propose a linear algebraic approach for the identification of a PN from the knowledge of the DES language, i.e., the finite set of strings generated by the set of transitions. The approach solves the identification problem by Integer Linear Programming (ILP) that is applied to λ-free labeled nets [3], i.e., nets where two transitions may share the same event label but no transition is labeled with the empty string. The identification problem of DES on the basis of an on line identification approach has been applied by Chung et al. [5] that propose an on line modeling refinement technique able to incrementally update the observed sample path. Moreover, Meda et al. [14] use an incremental synthesis approach to identify an interpreted PN (i.e., an extension of Manuscript received April 29, 2007. M.P. Fanti is with the Department of Elettrotecnica ed Elettronica, Politecnico di Bari, Bari, Italy. (+30-080-5963643; fax: +39-080-5963410; e-mail: fanti@ deemail.poliba.it). M. Dotoli is with the Department of Elettrotecnica ed Elettronica, Politecnico di Bari, Bari, Italy. (e-mail: dotoli@ deemail.poliba.it). A.M. Mangini is with the Department of Elettrotecnica ed Elettronica, Politecnico di Bari, Bari, Italy. (e-mail: mangini@ deemail.poliba.it). PN) modeling a DES on the basis of the partial knowledge of input and output symbols. An on line identification approach is proposed by Dotoli et al. [6,7] assuming that the DES evolution and the initial state are not perfectly known and only an upper bound of the cardinality of the place set is given. Moreover, an Identification Algorithm (IA) waits until a new event occurs, updates the transition set and the labeling function, defines and solves an ILP problem. Hence, the set of places, of transitions and the λ-free labeling function are recursively determined. This paper proves some important properties of the PNs provided by the IA presented in [6,7]. It shows that even if it is possible to obtain an infinite number of PN systems modeling the DES, under suitable properties, each PN system provided by the IA is characterized by the same incidence matrix. Consequently, we prove that the proposed method can be used to specify a supervisory controller via monitor places enforcing generalized mutual exclusion constraints [9]. To this aim we consider a communication gateway case study that is modeled by an identified λ-free labeled PN and is controlled by monitors enforcing mutual exclusions constraints. II. BACKGROUND ON PETRI NETS A. Petri nets A PN is a bipartite graph [15] described by the four-tuple PN=(P, T, Pre, Post) where P is a set of places with cardinality m, T is a set of transitions with cardinality n, Pre:P×T→ mn × ’ and Post:P×T→ mn × ’ are the pre- and post-incidence matrices respectively, which specify the arcs connecting places and transitions. More precisely, for each p∈P and t∈T element Pre(p,t) (Post(p,t)) is equal to a natural number indicating the arc multiplicity if an arc going from p to t (from t to p) exists, and it equals 0 otherwise. Note that ’ is the set of non-negative integers. The m×n incidence matrix of the net is defined as C=Post-Pre. For the pre- and post-set we use the dot notation, e.g., •t={p∈P: Pre(p,t)>0}. A marking is a mapping M: P→ m ’ , assigning to each place of the net a nonnegative number of tokens. A PN system 0 , PN M is a net PN with an initial marking M 0 . Given a PN system 0 , PN M and a set of places P v ⊆P, we denote by M v the restriction of M to P v . Here, we consider the set of places partitioned into two disjoint subsets: P k , the set of measurable places, whose marking M k is known, and P uk , the set of non measurable places, whose marking M uk is On Line Identification of Discrete Event Systems via Petri Nets: an Application to Monitor Specification Mariagrazia Dotoli, Member, IEEE, Maria Pia Fanti, Senior, IEEE, and Agostino M. Mangini S