Integrated Fault Diagnosis Based on Petri Net Models Manuel Manyari-Rivera, João Carlos Basilio, Amit Bhaya Abstract— This paper extends an existing sensor mapping procedure, defines compatibility of models and proposes an integrated methodology based on existing methodologies for the construction of diagnosers for discrete event systems modeled by Petri Nets. An industrial application is used as a case study to illustrate the theoretical results of the paper. I. I NTRODUCTION Modern industrial production systems and process control possess significant complexity in modeling analysis, reliabil- ity and planning, so that it is important to take appropriate decisions in order to maintain them in safe operation. On- line fault diagnosis and isolation systems aim at determining the fault type, size, location and time of occurrence. Recent research in this area includes the study of quantitative and qualitative methods using analytical redundancy, fault tree methods, expert systems, methods based on statisti- cal hypothesis testing and signature analysis, and discrete event systems (DES) approaches (see [1] and the references therein). This last approach makes possible to represent a wide variety of industrial applications, since, to some level of abstraction, any continuous-variable dynamical system can be viewed as a DES. In [2], [3], Sampath et. al. introduce the definition of lan- guage diagnosability and present a necessary and sufficient condition for diagnosability, namely that, a language L is diagnosable if and only if its diagnoser has no indeterminate cycles (cycles with states labeled with both faulty and non- faulty events). It is assumed in [2], [3] that the DES is modeled as a finite state automaton. Another approach is to model DES using Petri nets. A Petri net is a tool that can be used both to describe and study systems modeled as concurrent, asynchronous, distributed, parallel and stochastic [5], [6]. Ushio et. al. [7] consider some extensions of Sampath’s work for systems modeled by Petri nets with an infinite number of reachable markings. Wen and Jeng [8] continue the study presented in [7] and consider an approach to verify diagnosability based on the structural properties of the diagnoser; in [7], [8] it was assumed that some places were observable and that all tran- sitions were unobservable. More recently [11], an algebraic approach has been developed to build an automaton to be used as a diagnoser of Petri nets, without considering the construction of models, and, in [10], a distributed algorithm The authors gratefully acknowledge the support of the Brazilian Research Councils (CNPq, CAPES and FAPERJ) M. Manyari-Rivera, J. C. Basilio and A. Bhaya are with the Dept. of Electrical Engineering, Federal University of Rio de Janeiro, PEE/COPPE/UFRJ. PO Box 68504, Rio de Janeiro, 21945-970, Brazil. manuel@vishnu.coep.ufrj.br, basilio@dee.ufrj.br, amit@nacad.ufrj.br was presented for fault detection of DES modeled by Petri nets, without studying its diagnosability properties. The focus of this paper is to extend the theory of sensor mapping given in [2], [3], to systems modeled by Petri nets, by extracting qualitative characteristics from quanti- tative measures, and to define the notion of compatibility of models, with the view to integrating the diagnosability study and techniques given in [2], [3], [7] for the construction of diagnosers, as well as to propose a systematic procedure for the design of automatic fault diagnosis systems for real DES, using Petri net models. It is important to remark that, in contrast to the assumptions of [7] and [8], in this paper, places and transitions can be either observable or unobservable. This assumption is more realistic, since events are associated with transitions, and it is the latest that change the place markings (states) of the Petri net. II. MODELING OF SYSTEMS WITH PETRI NETS In this section, a few concepts on Petri nets, essential to the development of the paper, are reviewed. A. Basic definitions and notation A Petri net N is defined by the four-tuple N =(P,T,Pre,Post), where P is the set of places, T is the set of transitions, T = T o ∪ T u , with T o and T u denoting, respectively, the set of observable and unobservable transitions, |P | = m, |T | = n, with |.| denoting cardinality, Pre : P × T → N is the input weighting function, and P ost : T × P → N is the output weighting function. Throughout this paper I (t j ) and O(t j ) denote, respectively, the sets of input and output places of transition t j , and M (p), the number of markings in the place p (tokens in p). Therefore the marking vector M (state vector) is of the following form: M =[M (p 1 ) M (p 2 ) ... M (p m )] T ,M ∈ N m . A Petri net N with initial marking (state) M 0 will be denoted by 〈N,M 0 〉. A place p s in a Petri net 〈N,M 0 〉 with no input transition (I (p s )= ∅) and such that the initial state M 0 has one token in p s and no tokens elsewhere is called a starting place; the corresponding Petri net 〈N,M 0 〉 will be referred to as a Petri net with starting place [5]. A transition t j ∈ T is said to be enabled if and only if M (p) ≥ Pre(p), ∀p ∈ I (t j ). Assume that t j is enabled in M and let M ′ be the marking defined as: M ′ (p)= M (p) − Pre(p, t j )+ P ost(t j ,p). (1) Therefore, according to equation (1), the firing of t j takes M to M ′ , being denoted as M [t j >M ′ . Let T ⋆ denote the Kleene closure of T . 16th IEEE International Conference on Control Applications Part of IEEE Multi-conference on Systems and Control Singapore, 1-3 October 2007 TuC05.3 1-4244-0443-6/07/$20.00 ©2007 IEEE. 958