International Journal of Control, Automation, and Systems (2013) 11(6):1095-1105 DOI 10.1007/s12555-012-0366-9 ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555 Fault Diagnosis of Timed Discrete Event Systems using Dioid Algebra Sobhi Baniardalani* and Javad Askari Abstract: This paper deals with the fault diagnosis problem in a concurrent Timed Discrete Event Sys- tem (TDES). In a TDES, concurrency leads to more complexity in the diagnoser and appears where, at a certain time, some user must choose among several resources. To cope with this problem, a new model-based diagnoser is proposed in this paper. This diagnoser uses Durational Graph (DG), a main subclass of timed automata for representing the time evolution of the TDES. The proposed diagnoser predicts all possible timed–event trajectories that may be generated by the DG. This prediction proce- dure is complicated for nondeterministic DG’s that are obtained for concurrent TDES’s. To solve this problem, a new Dioid Algebra, Union-Plus Algebra is introduced in this paper. Based on this Algebra, a reachability matrix is defined for a DG that plays an essential role in predicting the time behavior of TDES. By using reachability matrix, a prediction procedure is carried on via an effective equation set that is similar to linear system state equations in ordinary algebra. These results provide a suitable framework for designing an observer-based diagnoser that is illustrated by an example. Keywords: Dioid algebra, durational graph, event scheduling table, fault diagnosis, timed discrete event system. 1. INTRODUCTION In recent years, process diagnosis methods in dynami- cal systems, have increasingly been developed. Based on the model and the measurement information, most of the cited works can be categorized into two main categories: Diagnosis based on quantitative models: These me- thods use quantitative models such as differential equa- tions describing the dynamical system, and faults are interpreted as external input signals or parameter devia- tions. In these diagnosers detection of faults is carried out by algorithms that use the measured input and output variables u(t) and y(t) [1-7]. These methods need quantit- ative models and numerically measured signals that are not thoroughly and easily available in many real applica- tions. Diagnosis based on qualitative models: In these approaches the values of signals are qualitative; like intervals or symbols, and qualitative models describe a relation among these qualitative signals. In these methods the diagnostic task is solved by means of symbolic knowledge processing as elaborated in the fields of artificial intelligence, fuzzy logic or Discrete Event System (DES) theory [8-14]. These approaches are suitable for systems with qualitative faults [1]. The present work goes into the latter category and aims at detecting the faults of a DES with unknown quantitative model. It is assumed that only some basic events and their occurrence time instances are observable in the underlying system. In fact, we are faced with a Timed Discrete Event System (TDES), a DES with some timing information of events (e.g., the time of occurred events) [15]. Using time characteristics of the events can enhance the capability of the diagnostic tasks, as well as the diagnoser complexity [10,16]. Thus, our presented diagnostic method lies in the TDES fault diagnosis methods. From a formal perspective, a DES can be thought of as a dynamic system, namely an entity equipped with a state space and a state-transition structure. In particular, a DES is discrete in time and (usually) in state space; it is asynchronous or event- driven: that is, driven by events other than or in addition to, the tick of a clock; and it may be nondeterministic: that is, capable of transitional “choices” by internal chance or other mechanisms not necessarily modeled by the system analyst [17]. A suitable model that has recently been proposed for fault diagnosis in TDES’s is Durational Graph (DG) [11- 13]. In a DG which is a special subclass of Timed Automata (TA) [14], only the sojourn time of each event is defined. Sojourn time indicates the time duration which the automaton remains in each state before it makes a transition to the next possible state(s). Obtaining the DG model of a system is usually easy and straightforward. Moreover, DG is a good choice for modeling a system the dynamics of which is not known completely [18,19]. 1.1. Problem definition and motivation To the best of the author knowledge, against the advantages of DG, it has not been widely used in TDES © ICROS, KIEE and Springer 2013 __________ Manuscript received August 26, 2012; revised March 25, 2013; accepted June 19, 2013. Recommended by Editorial Board mem- ber Bin Jiang under the direction of Editor Myotaeg Lim. Sobhi Baniardalani is with Kermanshah Power and Water Insti- tute of Technology, Kermanshah, Iran (e-mail: ardalani@ec.iut.ac. ir). Javad Askari is with the Department of Electrical and Comput- er Engineering, Isfahan University of Technology, Isfahan, Iran (e-mail: j-askari@cc.iut.ac.ir). * Corresponding author.