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A polynomial algorithm for checking diagnosability of Petri nets
Abstract
Diagnosability of discrete event systems was previously defined in terms of finite state machines by Sampath et al. Two algorithms of polynomial complexity in the number of states were proposed later for checking their diagnosability. In this paper, we present an algorithm of polynomial complexity in the number of nodes for computing a sufficient condition of diagnosability of discrete event systems modeled by Petri nets. In other words, our algorithm is more efficient than previous ones since no state enumeration is necessary. This gives us an advantage to solve large real-world problems. Our algorithm is formulated as a linear programming problem, which is well-known to be of polynomial complexity in the worst case. Examples are given in the paper to illustrate our approach.
... The early works that addressed DES diagnosability issues mostly considered FSA models [10,23]. Then, diagnosability analysis was extended to the Petri Net (PN) formalism [1,2,5,7,8,14,19,28,29], taking advantage of its mathematical and graphical representations. On the one hand, the idea behind the works that investigate the graphical representation of PN state set consists in extending the FSA based techniques (i.e., diagnoserbased and verifier-based techniques) by considering the ★ This work is part of the ELSAT2020 program co-funded by the European Union with the European Regional Development Fund, the French state and the Hauts-de-France. ...
... Consider an LPN under hypothesis H0 and fault class . Given ∈ N * , is -diagnosable if either of the following two conditions is fulfilled: -i- (29) has no solution, or -ii-(29) has a solution and max( . ) < . ...
... Proof. i-If (29) has no solution, then (23) has no solution either. Thus, according to Theorem 9 and Corollary 2, is -diagnosable and in particular 1-diagnosable. ...
We propose an algebraic approach to investigate K-diagnosability of partially observed labeled Petri nets which can be either bounded or unbounded. Namely, a necessary and sufficient condition for K-diagnosability is established based on the resolution of an Integer Linear Programming (ILP) problem. When the system is K-diagnosable, our approach also yields the minimal value Kmin≤K that ensures Kmin-diagnosability. The value of Kmin is calculated directly, using the same ILP formulation, i.e, without testing 1,…,(Kmin−1)-diagnosability. A second K-diagnosability approach, which is derived from the first one, is also developed on a compacted horizon providing a sufficient condition for K-diagnosability. This second technique allows for reducing the system dimensionality yielding a higher computational efficiency and allowing the characterization of the length of the sequences that lead to the fault occurrence, which is necessary to perform the K-diagnosability test of the first approach.
... Moreover, the intrinsically distributed nature of PNs where the notion of state (i.e., marking) and action (i.e., transition) is local reduces the computational complexity involved in solving a diagnosis problem. Several adaptation of automata-based techniques and original theoretical approaches have been proposed to deal with diagnosis in Petri nets (PNs) framework [Wen et al., 2005, Ramírez-Treviño et al., 2007, Basile et al., 2008, Basile et al., 2009, Basile et al., 2010, Basile et al., 2012a Chapter 2. Model-Based Diagnosis et al., 2009, Basile, 2014, Jiroveanu and Boel, 2010, Cabasino et al., 2009a, Germanos et al., 2015, Madalinski and Khomenko, 2010, Ushio et al., 1998, Chung, 2005, Jiroveanu and Boel, 2004, Cabasino et al., 2010, Cabasino et al., 2009b, Liu et al., 2014b, Li et al., 2015c, Li et al., 2015b. Other formalisms have been used to deal with fault diagnosis of DESs. ...
... The early works that addressed fault diagnosis issues mostly considered nite state automaton models [Lin, 1994, Sampath et al., 1995, Zad et al., 2003, Jiang and Huang, 2001, Yoo and Lafortune, 2002b. Afterwards, fault diagnosis issues have also been dealt with within the Petri nets (PNs) framework [Lefebvre and Leclercq, 2015, Lefebvre et al., 2013, Wen et al., 2005, Ramírez-Treviño et al., 2007, Basile et al., 2008, Basile et al., 2009, Basile et al., 2010, Lefebvre, 2014, Lefebvre and Delherm, 2007, Basile, 2014, Jiroveanu and Boel, 2010, Cabasino et al., 2009a, Germanos et al., 2015, Madalinski and Khomenko, 2010, Ushio et al., 1998, Chung, 2005, Jiroveanu and Boel, 2004, Cabasino et al., 2010, Cabasino et al., 2009b, Liu et al., 2014b, Li et al., 2015c. ...
... The diagnosis approaches based on PN models can be classied into two main classes: [Wen et al., 2005] for computing a sucient condition of diagnosability. In a series of works [Basile et al., 2008, Basile et al., 2009, Basile et al., 2010, Basile et al. have developed two conditions regarding diagnosability: the rst is a necessary condition, while the second is a sucient one. ...
This PhD thesis deals with fault diagnosis of discrete-event systems modeled as finite state automata with some extensions to bounded Petri net models. The developed contributions can be classified regarding two pioneering approaches from the literature: the diagnoser-based technique and the twin-plant based technique.
Regarding the diagnoser-based technique, we propose a new diagnoser variant with some interesting features that allow us to reformulate a necessary and sufficient condition for diagnosability of permanent faults and propose a systematic procedure for checking such a condition without building any intermediate model. An on-the-fly algorithm, for simultaneously constructing the diagnoser and verifying diagnosability is then developed. The established diagnoser is then extended to deal with fault diagnosis of intermittent faults. A Hybrid version (in the sense of combining enumerative and symbolic representations) of our diagnoser is also established in order to deal with fault diagnosis of labeled bounded Petri nets. The developed approaches are implemented in dedicated tools and evaluated through benchmarks with respect to the reference approaches in the domain.
Regarding twin-plant based technique, our contribution consists in elaborating a model-checking framework that extends the Cimatti’s work for the actual verification of various diagnosability concepts pertaining to permanent and intermittent failures based on the twin-plant structure. The main idea is to reformulate and express the diagnosability issues as temporal logics and then to tackle them using the model-checking engines.
... In (Chung et al., 2003) some transitions are assumed to be observable in order to increase the database used by the diagnoser. An algorithm, based on linear programming, of polynomial complexity in the worst case for computing a sufficient condition of diagnosability has been also proposed (Wen et al., 2005). Let us consider the Petri net named PN2 in figure 2 as an example. ...
... State space method have been used to state the diagnosability of an automatic temperature control system (ATC) for automobile applications (Wen et al., 2005). The PN models of ATC has 3 components (figure 4a-b-c): ...
... c) The controller has four observable states and four events. (Wen et al., 2005) There are two failure types. Failure types F1 and F2 stand for pump fails off and pump fails on respectively. ...
The investigation of diagnosis methods for discrete event systems shows that Petri nets is efficient not only to model the considered systems but also to support the diagnosis methods. Several approaches can be used in order to check diagnosability, to select sensors and to work out diagnosers. The table 5 sums up the main characteristics of these method. As a conclusion it is important to notice the great effort, observed this last years to develop and improve diagnosis methods for DES. The strong connection with observation properties in automata and the use of advances in computer science like the coding theory have played an important role in that development. Now, the challenges are, from our
... As a mathematical model abstracted from an AMS, Petri nets [18]- [21] describe a system with simple and intuitive graphics, and are widely used in modeling, analysis and control of AMSs. Petri nets can be used to investigate a deadlock control policy of AMSs in a mathematical way [22], [23]. At present, the deadlock analysis and control of AMSs based on Petri net modeling mainly have the following two ways: (1) obtain a deadlock avoidance policy, based on the reachability graph analysis and the theory of regions [24]- [28]; (2) add control places to make a system deadlock-free [29]- [31], based on the structural analysis VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. ...
... 3. Taking the dead marking set of the simplest model as the initial data, the algorithm is called to calculate all the dead zone markings (DZ) of the simplest model. There are four pseudo safe markings to be obtained from DZ: M 1 = p 5 + p 6 + 2p 8 + p 10 + 2p 12 + p 13 + p 19 + p 23 , M 2 = p 5 + p 6 + 2p 8 + p 10 + p 11 + p 12 + p 13 + p 18 + p 23 , M 3 = p 5 + p 6 + 2p 8 + p 10 + p 11 + 2p 12 + p 17 + p 23 , M 4 = p 5 + p 6 + p 7 + p 8 + p 10 + 2p 12 + p 13 + p 20 + p 23 4. Obtain the pseudo safe markings of the simplest model from the DZ. These markings and the reachable markings of the subnet structure where p 2 and p 3 are located are calculated by synchronous product operations to calculate all pseudo safe markings of the model in Figure 16. ...
Automated manufacturing systems (AMSs) are prone to be in deadlock states when resource allocation is unreasonable. Reasonably allocating system resources to achieve deadlock control is a primary task of the design of an AMS. This paper proposes a multi-step look-ahead deadlock prediction method to obtain an optimal deadlock avoidance policy for a class Petri nets, without calculating a complete reachability graph. As the reachability graph of a large-scale Petri net model is usually large, the analysis and calculation process to obtain the optimal deadlock avoidance policy is complicated. We first simplify a Petri net model by using the existing Petri net reduction methods or removing non-shared resources to simplify the model structure. Then we calculate the dead zone markings of the simplified model through a reverse generation method, and finally develop a multi-step look-ahead deadlock prediction method to obtain an optimal deadlock avoidance policy for a class of AMSs.
... In the framework of PN, an automaton called ROFautomaton (ROF stands for ''the reduced automaton whose event set comprises the observable events and fault events only'') whose state space is significantly smaller than the reachability graph of a PN, is proposed to check the diagnosability of a bounded PN without unobservable cycles [33]. In particular, under the assumption that the markings and the transitions are partially observable and by exploring the relation between diagnosability and the properties of the T-invariants of a net, a sufficient condition for diagnosability is proposed based on linear programming in [34]. In [35] a sufficient condition is also presented for safe and strongly connected PN with an output function that associates an output vector with each net marking (interpreted PN). ...
... We consider the case that the states and events of a VDES are partially observable. In other words, the sensors of a VDES are placed in some components of states and some events, which corresponds to the formalism of partially observed PN [34]. ...
The diagnosability problem of faults is studied in the framework of vector discrete-event system (VDES). A VDES is a discrete-event system model in which a system state is represented by a vector with integer components, and state transitions are represented by integer vector addition. Predicates are employed to verify the fault diagnosability of VDES, since, defined as functions, predicates can conveniently identify particular state sets of interests. Specifically, system states are partitioned into different subsets by predicates, and the fault diagnosability of a system is verified by checking a subset of states. A sufficient condition for fault diagnosability of VDES is presented first. A necessary and sufficient condition is then developed. According to the two conditions, two types of predicates are given to partition the states in a VDES. In this work, a diagnoser or a full state enumeration is not constructed, whose complexity is exponential with respect to the system state size. In order to verify whether a system satisfies the proposed conditions, several polynomial algorithms and an algorithm by constructing a tree automaton are developed. Several examples are provided to illustrate the results obtained in this paper.
... In order to overcome this problem, Some subsequent automatabased approaches Yoo and Lafortune, 2002b] have been proposed for reducing the computational complexity, without the construction of a diagnoser automaton. Besides, a series of works [Aghasaryan and Fabre, 1998;Basile et al., 2010;Bouyer et al., 2005;Brandin and Wonham, 1994;Cabasino et al., 2010;Genc and Lafortune, 2007;Ghazel et al., 2009;Haar, 2009;Ru and Hadjicostis, 2009;Ushio et al., 1998;Wen et al., 2005] concerning the diagnosis and diagnosability of DESs turned to PN modeling, thus benefiting from the expressiveness and the well-developed theory of PNs. ...
... The classic diagnosability can be analyzed based on incrementally investigating K-diagnosability with increasing the value of K, and the K min value which ensures the diagnosability will be eventually found (for diagnosable systems), as will be discussed in Section 4.4.1. Note that some ILP-based approaches [Basile et al., 2012;Wen et al., 2005] have to rebuild and solve the equation system (or inequalities) when seeking out K min , without using the previous search results. ...
This PhD thesis deals with fault diagnosis of DES in both untimed and timed contexts using Petri net models. Some on-the-fly and incremental techniques are developed to reduce the state explosion problem while analyzing diagnosability. In the untimed context, an algebraic representation for labeled Petri net (LPN) is developed to feature the system behavior. The diagnosability of LPN models is tackled by analyzing a series of K-diagnosability problems, where K is increased progressively. Two models called respectively FM-graph and FM-set tree are developed and built on the fly to record the necessary information for diagnosability analysis and online diagnosis. Finally, a diagnoser is derived from the FM-set tree for online diagnosis. In the timed context, time interval splitting techniques are developed in order to generate a state representation of labeled time Petri net models, for which techniques from the untimed context can be used to analyze diagnosability and perform online diagnosis. Based on this, necessary and sufficient conditions for the diagnosability of LTPN models are determined. Moreover, we provide the solution for the minimum delay that ensures diagnosability. From a practical point of view, diagnosability analysis is performed on the basis of on-the-fly building of a structure that we call ASG and which holds fault information about the LTPN states. Generally, using on-the-fly analysis and incremental techniques makes it possible to build and investigate only a part of the state space. Analysis results obtained on some chosen benchmarks show the efficiency in terms of time and memory compared with the traditional approaches based on state enumeration.
... In this case the mathematical representation of PNs is exploited. A classification with respect to the adopted technique is the following: • algebraic techniques [17], [18]; • T-invariant approaches [19], [20]; • compact representation of reachability set under partial observation [21], [22]; • verifier net [23]; • net unfolding approach [24]; • integer linear programming approaches [25], [26]; • compact representation of net marking under partial observation [27], [28]. ...
... Using the assumption that the net marking and the transitions set are partially observable, and investigating the relation between diagnosability and the properties of the Tinvariants of the net, a sufficient condition for diagnosability based on linear programming is proposed in [19]. In [20] an interpreted diagnoser has been devised for a safe and strongly connected PN with an input function that associates an event to each transition ( if the transition is associated with an internal event) and an output function that associates an output vector to each net marking -i.e., an interpreted PN (IPN). ...
Fault diagnosis of discrete-event systems has received a lot of attention in the literature since the early 90s. At the beginning, the problem has been approached using the framework of finite state automata and regular languages. Recently, the problem has been tackled within the Petri nets (PNs) framework. This paper overviews the main ideas behind the fault diagnosis approaches based on PNs.
... In [7] Wen and Jeng propose an approach to test diagnosability by checking the structural property of the T-invariants of the nets. In [8] In this paper, we consider the diagnosability problem of unbounded Petri nets. The model we consider is a labeled Petri net where some transitions are indistinguishable. ...
... Note that our problem statement is related to prior works on diagnosability analysis of regular languages represented by finite-state automata and is different from prior work on diagnosability analysis of Petri nets. Specifically, we consider that only a subset of transitions are unobservable, while in [5], [7], [8] the authors consider that some places are unobservable as well. Moreover, we consider labeled Petri nets where two or more transitions can share the same label, rather than free-labeled Petri nets. ...
In this paper we consider the property of diagnosability for labeled unbounded Petri nets, namely Petri nets where the number of tokens in one or more places can grow indefinitely. We give necessary and sufficient conditions for diagnosability and we present a test to study diagnosability based on the analysis of the coverability graph of a particular net, called verifier net, that is built starting from the initial system. To the best of our knowledge, this is the first available test for diagnosability analysis of labeled unbounded Petri nets. We also discuss existing methods to perform diagnosis of unbounded Petri nets.
... [16] proposes the definitions of weak and strong diagnosability and solve the problem by using net unfolding techniques. [17] considers a structural approach to give a sufficient condition for diagnosability relying on T-invariants and formulates the problem as a linear programming problem. [18] gives sufficient conditions for diagnosability and undiagnosability of a fault transition based on the notion of the so called g-markings [19], representing a set of inequalities to be solved. ...
Checking the diagnosability of a timed discrete-event system usually consists in determining whether a single fault event can always be identified with certainty after a finite amount of time. The aim of this article is to extend this type of analysis to more complex behaviors, called event patterns, and to propose an effective method to check diagnosability with the use of model-checking techniques. To do so, we propose to convert the pattern diagnosability problem into checking a linear-time property over a specific time Petri net.
... In (Cabasino et al. 2009a), the authors introduce verifier nets suitable for unbounded systems (the diagnosability test searches for certain types of cycles of states in the coverability graph of the verifier net). Note that other approaches analyze directly the PN model with dedicated algorithms that are based on the investigation of the T-invariants of the net and linear programming (Wen et al. 2005;Ramírez-Treviño et al. 2007;Basile et al. 2008;Basile et al. 2009). Diagnosis issues have also been addressed when the net marking is partially observable (Ru and Hadjicostis 2009;Lefebvre and Delherm 2007), however, they do not explicitly concern the problem of diagnosability. ...
Various aspects of privacy and safety in many application domains can be assessed based on proper analysis of successive measurements that are collected about a given system. This work is devoted to such issues in the context of timed stochastic discrete event systems (DES) that are modeled with partially observed timed stochastic Petri net models. The first contribution is to introduce a k-step trajectory-observer, which is a construction that captures all possible k-suffixes of the trajectories that are consistent with a given sequence of measurements that has been recorded. When the system behaves according to Markovian dynamics (i.e., all event occurrences are distributed in time with exponential probability density functions), a parallel-like composition of the timed system with the resulting observer is proposed that leads to a Markovian process. The second contribution is to take advantage of the Markovian analysis to compute certain important characteristic times during which the underlying system should satisfy a given property (based on the suffixes of length k of a given trajectory). To illustrate the approach, we consider two particular properties, namely k-suffix language opacity and k-diagnosability, which can be studied in a stochastic timed context using the Markovian trajectory observer.
... Une méthode limitée à l'identification des erreurs de capteur ou de processus a été proposée suite à une variation du nombre de jetons dans un P-invariant [Prock 1991]. Les auteurs de [Wen et al. 2005] Plusieurs travaux ont été proposés pour répondre à cette question [Jiang et Kumar 2004, Jéron et al. 2006]. ...
Cette thèse porte sur le diagnostic des défauts dans des Systèmes à Evènements Discrets (SED) pour lesquels l’occurrence des événements est soumise à des contraintes temporelles. Les domaines d’application potentiels sont nombreux et variés allant de la production manufacturière aux systèmes de transport en passant par les réseaux de communication et les systèmes d’information. La complexité croissante de ces systèmes nécessite l’élaboration de méthodes de surveillance de plus en plus efficaces et performantes pour garantir leur sécurité, leur disponibilité ainsi que le maintien de leurs propriétés dans le temps. Les réseaux de Petri Stochastiques Temporisés Partiellement Observés (RdPSTPO) sont utilisés pour modéliser le système ainsi que les défauts qui peuvent l’affecter. L’étude concerne particulièrement les défauts qui se traduisent par la violation des contraintes temporelles. Une exploitation pertinente des informations temporelles, en vue du diagnostic, constitue la contribution majeure de la thèse. En effet, une meilleure utilisation de ces informations permet de discerner avec précision les différents comportements qui expliquent les mesures. De plus, la probabilité d’occurrence des défauts est calculée en fonction des dates des mesures collectées. Deux approches sont développées : la première est dédiée aux défauts ponctuels et la seconde aux dérives lentes dans les systèmes cycliques.
... Des RdP représentant le comportement normal du système sont utilisés et les fautes sont détectées suite à une variation du nombre de jetons dans un P-invariant. Les auteurs de (Wen et al. 2005) supposent, quant à eux, une mesure partielle des événements et du marquage. En étudiant la relation entre la diagnosticabilité et les invariants du RdP, ils établissent une condition suffisante pour la diagnosticabilité. ...
La complexification des systèmes et la réduction du nombre de capteurs nécessitent l’élaboration de méthodes de surveillance de plus en plus efficaces. Le travail de cette thèse s’inscrit dans ce contexte et porte sur le diagnostic et le pronostic des Systèmes à Événements Discrets (SED) temporisés. Les réseaux de Petri stochastiques partiellement mesurés sont utilisés pour modéliser le système. Le modèle représente à la fois le comportement nominal et le comportement dysfonctionnel du système. Il permet aussi de représenter ses capteurs à travers une mesure partielle des transitions et des places. Notre contribution porte sur l’exploitation de l’information temporelle pour le diagnostic et le pronostic des SED. À partir d’une suite de mesures datées, les comportements du système qui expliqueraient ces mesures sont d’abord déterminés. La probabilité de ces comportements est ensuite évaluée pour fournir un diagnostic du système en termes de probabilité d’occurrence d’un défaut. Dans le cas où une faute est diagnostiquée, une approche permettant d’estimer la distribution de sa date d’occurrence est proposée. L’objectif est de donner plus de détails sur cette faute afin de mieux la caractériser. Par ailleurs, la probabilité des comportements compatibles est exploitée pour estimer l’état actuel du système. Il s’agit de déterminer les marquages compatibles avec les mesures ainsi que leurs probabilités associées. À partir de cette estimation d’état, la prise en considération des évolutions possibles du système permet d’envisager la prédiction de la faute avant son occurrence. Une estimation de la probabilité d’occurrence de la faute sur un horizon de temps futur est ainsi obtenue. Celle-ci est ensuite étendue à l’évaluation de la durée de vie résiduelle du système. Enfin, une application des différentes approches développées sur un cas d’un système de tri est proposée.
... However they do not actually construct a diagnoser to test diagnosability. Wen and Jeng [9] then proposed a diagnosability test algorithm by using a linear programming. It is of polynomial complexity in the number of net nodes and provides a sufficient condition for diagnosability of the nets. ...
For bounded Petri nets, Cabasino et al. propose a diagnosability test method that is based on the analysis of a modified basis reachability graph and a basis reachability diagnoser. However, its complexity is exponential in the number of nodes of the basis reachability diagnoser. In order to reduce the complexity of their method, this paper presents a new diagnosability test approach for bounded Petri nets. We present the concept of an extended basis reachability graph and prove that our approach is of polynomial complexity in the number of nodes of extended basis reachability graphs. An example is given to illustrate the application of the presented approach.
... In order to overcome this problem, some works turn to using Petri nets (PNs) [4][5][6][7][8][9][10][11][12][13][14][15][16][17], which offer powerful expressiveness and compact representation. In [10], the authors develop the basis reachability graph (BRG) to discuss diagnosability of bounded LPNs. ...
In this paper, we develop an on-the-fly and incremental technique for fault diagnosis of discrete event systems modeled by labeled Petri nets, in order to tackle the combinatorial explosion problem. K-diagnosability, diagnosability, Kmin (the minimum K ensuring diagnosability) and on-line diagnosis are solved on the basis of the on-the-fly and incremental building of two structures, called respectively fault marking graph and fault marking set graph, in parallel. We build on existing results, namely those establishing necessary and sufficient conditions for diagnosability, but we bring mechanisms to make the checking of such conditions potentially more efficient. We show that, in general, analyzing or even building the whole reachability graph is unnecessary to analyze diagnosability and build an on-line diagnoser. Our technique was implemented in a prototype tool called OF-PENDA, and a railway level crossing benchmark is used to make a comparative discussion pertaining to efficiency in terms of time and memory relative to some existing approaches.
... A diagnoser is constructed from the reachability graph. In (Wen and Li 2005), the authors proposed a sufficient condition for testing diagnosability by checking the structure of T -invariants of a PN. In (Cabasino et al. 2009) the modified basis reachability graph (MBRG) and basis reachability diagnoser (BRD), which provide a compact representation of the reachability graph, were developed. ...
In this paper, an efficient approach to verify diagnosability of discrete-event systems is proposed. The
approach consists in constructing a hybrid diagnoser based on the symbolic observation graph (SOG), which is a technique that combines symbolic and enumerative representations in order to build a deterministic observer from a partially observed model. The construction of the diagnoser as well as the verification of diagnosability are performed simultaneously on-the-fly, which can considerably reduce the generated state space of the diagnoser and thus the overall running time. Furthermore, the proposed approach provides a heuristic strategy in order to converge fast into the necessary part, of the diagnoser, for analysing diagnosability.
... Different papers deal also with approach 2). In particular, using the assumption that the net marking and the transitions set are partially observable, and investigating the relation between the diagnosability definition and the property of the T-invariants of the net, a sufficient condition for diagnosability based on linear programming is proposed in Wen et al. (2005). In Trevino et al. (2007) a sufficient condition has also been presented for safe and strongly connected PNs with an output function which associates an output vector to each net marking (interpreted PNs). ...
The problem of diagnosability of a fault after the firing of a finite number of observable events (i.e., K-diagnosability) is tackled in this paper. This problem corresponds to diagnosability of a fault within a finite delay in the context of discrete event systems (DESs). Two results for DESs modeled as labeled Petri nets are given: the first is a sufficient condition for K - undiagnosability of fault, while the second is a necessary and sufficient condition for K-diagnosability. The proposed results exploit the mathematical representation of Petri nets and the Integer Linear Programming standard optimization tool.
... Testing diagnosability was proved NLOGSPACE-hard for enumerative representations, and PSPACEhard for symbolic representations in Rintanen (2007). Some researchers tried to use other types of symbolic representations, such as Petri nets (see Wen et al., 2005). There are, however, Petri nets has its own advantages and disadvantages and the choice between them depends on the system we have to model and on what properties we need to study. ...
Model-based diagnosis in discrete event systems (DESs) is a major research topic in failure diagnosis, where diagnosability plays an important role in the construction of the diagnosis engine. To improve the solution efficiency for diagnosability, this paper proposes novel techniques to solve the problems of testing and optimizing for diagnosability. We propose a new concept, reverse twin plant, which is generated backwards from the final states of the DESs so there is no need to generate a complete copy of the DES model to determine the diagnosability. Such a design makes our testing algorithm much faster than existing methods. An efficient optimizing algorithm, which makes a non-diagnosable system diagnosable, is also proposed in the paper by expanding the minimal observable space with operation on just a part of the DES model. Examples and theoretical studies demonstrate the performance of the proposed designs.
... More recently a series of interesting contributions have been proposed using Petri nets (PNs) [21,11,22,23,16,13,4,6,2]. ...
In this paper, we focus on labeled Petri nets with silent transitions that may either correspond to fault events or to regular unobservable events. We address the problem of deriving a procedure to determine if a given net system is diagnosable, i.e., the occurrence of a fault event may be detected for sure after a finite observation. The proposed procedure is based on our previous results on the diagnosis of discrete-event systems modeled with labeled Petri nets, whose key notions are those of basis markings and minimal explanations, and is inspired by the diagnosability approach for finite state automata proposed by Sampath in 1995. In particular, we first give necessary and sufficient conditions for diagnosability. Then, we present a method to test diagnosability that is based on the analysis of two graphs that depend on the structure of the net, including the faults model, and the initial marking.
... They use the diagnoser of [38] to prove that their method is correct, however they do not construct a diagnoser for the system to do online diagnosis. In [41], Wen et al. present a linear-programming-based algorithm of polynomial complexity in the number of nodes for computing a sucient condition of diagnosability of DES modeled by Petri nets. ...
In this paper, we analyze the diagnosability properties of labeled Petri nets. We consider the standard notion of diagnosability of languages, requiring that every occurrence of an unobservable fault event be eventually detected, as well as the stronger notion of diagnosability in K steps, where the detection must occur within a fixed bound of K event occurrences after the fault. We give necessary and sufficient conditions for these two notions of diagnosability for both bounded and unbounded Petri nets and then present an algorithmic technique for testing the conditions based on linear programming. Our approach is novel and based on the analysis of the reachability/coverability graph of a special Petri net, called Verifier Net, that is built from the Petri net model of the given system. In the case of systems that are diagnosable in K steps, we give a procedure to compute the bound K. To the best of our knowledge, this is the first time that necessary and sufficient conditions for diagnosability and diagnosability in K steps of labeled unbounded Petri nets are presented.
... They used Ushio's diagnoser to prove that their method is correct, however they do not construct a diagnoser for the system to do diagnosis. Wen et al. [46] also presented an algorithm, based on a linear programming problem, of polynomial complexity in the number of nodes, to compute a sufficient diagnosability condition for DES modeled by PNs. ...
This extended abstract serves as a support for the plenary address given by the author at the 32nd International Conference
on Application and Theory of Petri Nets and Concurrency.
This paper is about the diagnosability of fault patterns in timed stochastic discrete event systems. For this purpose, the diagnosability problem is formulated with labeled stochastic Petri net models and pure logical fault pattern nets. A particular composition of a labeled stochastic Petri net with a fault pattern net is proposed and is shown to characterize in an explicit way the fault patterns, including the timing and probabilistic aspects of the underlying system. Logical and probabilistic verifiers are derived, and used to establish a set of conditions to check not only the strong diagnosability property but also weaker notions of diagnosability.
In this paper, we present a diagnoser-based approach to deal with fault diagnosis of bounded labeled Petri nets. The approach consists in building a semi-symbolic diagnoser to analyze diagnosability and perform online diagnosis. The contribution of this paper is twofold: (i) from the theoretical point of view, we provide new conditions for checking diagnosability based on a novel diagnoser variant that explicitly separates the normal reachable markings from the faulty ones, in each diagnoser node. This allows us to independently keep tracking the normal and the faulty diagnoser paths more efficiently. (ii) From the practical point of view and in order to reduce the memory required to build the diagnoser efficiently, we establish a semi-symbolic encoding of the diagnoser state-space. Such a representation deploys a symbolic encoding of the diagnoser nodes content, using Binary Decision Diagrams, while it keeps an explicit encoding of the observable transitions between the nodes. In addition, we provide an on-the-fly algorithm to simultaneously construct the diagnoser and analyze diagnosability. The effectiveness of the approach is illustrated through some experimentation performed on a Petri net benchmark.
Fault diagnosis of Discrete Event Systems has become an active research area in recent years. The research activity in this area is driven by the needs of many different application domains such as manufacturing, process control, control systems, transportation, communication networks, software engineering, and others. The aim of this paper is to review the state-of the art of methods and techniques for fault diagnosis of Discrete Event Systems based on models that include faulty behaviour. Theoretical and practical issues related to model description tools, diagnosis processing structure, sensor selection, fault representation and inference are discussed.
This paper deals with the problem of diagnosability of a fault after the firing of a finite number events (i.e., KK-diagnosability). This problem corresponds to diagnosability of a fault within a finite delay in the context of discrete event systems. The main contribution of this paper is a necessary and sufficient condition for KK-diagnosability of bounded nets. The proposed approach exploits the mathematical representation of Petri nets and the Integer Linear Programming optimization tool. In particular no specific assumptions are made on the structure of the net induced by the unobservable transitions, since the proposed approach permits to detect also the undiagnosability due to the presence of unobservable cycles.
In this paper an approach to on-line diagnosis of discrete event systems based on labeled Petri nets is presented. The approach is based on the notion of basis markings and justifications and it can be applied both to bounded and unbounded Petri nets whose unobservable subnet is acyclic. Moreover it is shown that, in the case of bounded Petri nets, the most burdensome part of the procedure may be moved off-line, computing a particular graph called Basis Reachability Graph.Finally, the effectiveness of the proposed procedure is analyzed applying a MATLAB diagnosis toolbox we developed to a manufacturing example taken from the literature.Research highlights► We present an approach to on-line fault diagnosis based on labeled Petri nets. ► We consider both silent events and undistinguishable events. ► Basis markings and justifications avoid an exhaustive enumeration of the state space. ► In the case of bounded net systems, the procedure can be significantly simplified. ► A manufacturing system is considered to show the effectiveness of the procedure.
In this paper we present an approach to solve the problem of diagnosability of bounded Petri net systems. In particular, we first give necessary and sufficient conditions for diagnosability. Then, we present a method to test diagnosability that is based on the analysis of two graphs that depend on the structure of the net, including the faults model, and the initial marking. The first graph is called basis reachability diagnoser, the second one is called modified basis reachability graph.
This paper serves as a support for the plenary address given by the second author during the conference. In this paper we present an approach to on-line diagnosis of discrete event systems based on labeled Petri nets, that are a particular class of Petri nets where some events are undistinguishable, i.e., events that produce an output signal that is observable, but that is common to other events. Our approach is based on the notion of basis markings and justifications and it can be applied both to bounded and unbounded Petri nets whose unobservable subnet is acyclic. Moreover it is shown that, in the case of bounded Petri nets, the most burdensome part of the procedure may be moved off-line, computing a particular graph that we call Basis Reachability Graph. Finally we present a diagnosis MATLAB toolbox with some examples of application.
Reasoning about the knowledge of an agent is an important problem in many areas of AI. For exam- ple in diagnosis a basic question about a system is whether it is possible to diagnose it, that is, whether it is always possible to know whether a faulty be- havior has occurred. In this paper we investigate the complexity of this diagnosability problem and the size of automata that perform diagnosis. There are algorithms for testing diagnosability in polynomial time in the number of states in the sys- tem. For succinct system representations, which may be exponentially smaller than the state space of the system, the diagnosability problem is con- sequently in EXPTIME. We show that this upper bound is not tight and that the decision problem is in fact PSPACE-complete. On-line diagnosis can be carried out by diagnosers which are automata that recognize faulty behavior. We show that diagnosers in the worst case have a size that is exponential in the number of states, both for explicit and succinct system representa- tions. This is a consequence of the diagnoser hav- ing to maintain beliefs about the state of the system.
This paper presents an approach using Petri nets for designing diagnosable discrete event systems such as complex semiconductor manufacturing machines. The concept is based on diagnosability analysis and enhancement. In this paper, we interpret and formulate the diagnosability problem as a binary integer linear programming problem that may have a feasible solution. If the system is predicted to be non-diagnosable, the approach tries to add sensors to enhance its diagnosability, i.e., to make the system diagnosable. The idea is to separate any two undifferentiated event cycles by changing their labels as a result of adding sensors. Our approach is under the assumption that the costs of sensors are not considered. This assumption is well justified in semiconductor manufacturing. We use a real- world Metal-Organic Vapor Phase Epitaxy (MOVPE) system to illustrate that our proposed approach is practically useful.
This paper is about fault detection and identification of discrete event systems. The proposed approach is based on Petri nets (PNs) that are used to design reference and faulty models. The main contribution concerns the design and identification of these models according to the statistical analysis of the alarm sequences that are collected on the considered system. The model structure is described as a state graph, and the parameters of the probability density functions (pdfs) for transition firing periods are estimated. Normal and exponential pdfs are considered, and estimation is detailed in case of concurring behaviors. The reference models, described as timed PNs, are then used for fault detection and isolation issues. Finally, stochastic PNs with normal and exponen- tial pdfs are considered to include a representation of the faulty behaviors.
This work deals with model based fault diagnosis of discrete event systems. The model of the system, expressed as an interpreted Petri net (IPN) describes partially observed events and states, and includes all possible faulty states. Based on a modular modelling methodology, the input-output diagnosability property is introduced and structurally characterized. Then a diagnoser scheme is proposed allowing fault detection and location in polynomial time.
This paper studies the diagnosis of asynchronous discrete event systems. We follow a so-called true concurrency approach, in which neither the global state nor global time are available. Instead, we use only local states in combination with a partial order model of time; our basic mathematical tool is that of Petri net unfoldings. This study was motivated by the problem of event correlation in telecommunications network management.
This paper is concerned with the analysis of the observability of the discrete event systems (DES) modeled by interpreted Petri nets (IPN). This paper presents three major contributions on the field of the observability of DES. First, an observability definition for IPN is proposed. This definition is more precise than previous ones because it deals with the possibility of determining the system's initial state, using the knowledge of the system's inputs, outputs, and structure. Later, a novel characterization of the IPN exhibiting the observability property that is based on the IPN structure is presented. Finally, a method for designing asymptotic observers is discussed. The main advantage over other methods is that the observer presented herein is given as an IPN, allowing further analysis of the system-observer pair.
Fault detection and isolation is a crucial and challenging task in
the automatic control of large complex systems. We propose a
discrete-event system (DES) approach to the problem of failure
diagnosis. We introduce two related notions of diagnosability of DES's
in the framework of formal languages and compare diagnosability with the
related notions of observability and invertibility. We present a
systematic procedure for detection and isolation of failure events using
diagnosers and provide necessary and sufficient conditions for a
language to be diagnosable. The diagnoser performs diagnostics using
online observations of the system behavior; it is also used to state and
verify off-line the necessary and sufficient conditions for
diagnosability. These conditions are stated on the diagnoser or
variations thereof. The approach to failure diagnosis presented in this
paper is applicable to systems that fall naturally in the class of
DES's; moreover, for the purpose of diagnosis, most continuous variable
dynamic systems can be viewed as DES's at a higher level of abstraction
Failure diagnosis in large and complex systems is a critical task. In the realm of discrete event systems, Sampath et al. proposed a language based failure diagnosis approach. They introduced the diagnosability for discrete event systems and gave a method for testing the diagnosability by rst constructing a diagnoser for the system. The complexity of this method of testing diagnosability is exponential in the number of states of the system and doubly exponential in the number of failure types. In this paper, we give an algorithm for testing diagnosability that does not construct a diagnoser for the system, and its complexity is of 4th order in the number of states of the system and linear in the number of the failure types. Keywords: Discrete event system, failure diagnosis, diagnosability, complexity. 1
In this paper, we introduce an adaptive two‐mode diagnosis scheme for discrete event systems. When the adaptive diagnoser is initially in the passive mode, it keeps track of the latest state estimates based on the occurrence of observable events. If the system is diagnosable as defined by Sampath et al., (1995) the adaptive diagnoser will detect the occurrence of failures within a finite delay. However, when the adaptive diagnoser gets into an uncertain state, it is at the users’ discretion to switch the operational mode to the active mode. In the active mode, we propose an algorithm such that the active diagnoser has a testing mechanism to generate a resolution sequence in resolving the uncertain state, if the diagnosed system at the uncertain state is actively diagnosable.
Inside Synchrony Theory, this paper deals with a computation of structural synchronic invariants. It is based on the state equation of the place/transition (P/T) net and on Linear Programming theory. The approach is conceptually very simple, general and computationally very efficient. It is said to be a general approach in the sense that instead of single transitions, subsets of transitions are directly taken into account. Computationally it is very efficient because the linear programming problem is known to be of polynomial complexity. Two transition subsets are in a given synchronic relation if the corresponding synchronic invariant is bounded; for all conflict-resolution strategies, synchronic relations characterize the existence or non-existence of firing dependences. As a by-product of the proposed approach, a full algebraic characterization of structural synchronic relations (i.e. independent of the initial marking) is obtained.
Failure diagnosis in the context of DES was first formulated by Sampath et al. Where the notion of diagnosability and the associated diagnoser are proposed. Ushio et al. extended Sampath's study to systems modeled by Petri nets. This paper further assumes some of the transitions in a Petri net model are observable in the sense that its occurrence can be observed. The main contribution of this study shows how diagnosers and verifiers for systems modeled by Petri net are constructed accordingly. As shown by examples, the additional information from observed transitions adds diagnosability to the system.
We consider Petri net models of discrete event systems with faulty behaviors which are triggered by the firing of failure transitions. We extend a necessary and sufficient condition for diagnosability shown by Sampath et al. (1994, 1995) to unbounded Petri nets. But it is algorithmically impossible to check the condition in general. We introduce methods for modification of coverability trees in order to detect failure transitions. By using such approximated sets, we propose two kinds of diagnosers called a difference marking ω-diagnoser and an ω-refined diagnoser. For observable places whose token numbers are replaced by ω in the coverability trees, the former diagnoser calculates difference between token numbers before and after partially observed markings change, and detects failures. In the latter diagnoser, we refine the symbol ω, and modify the coverability tree in order to distinguish markings reachable by normal behaviors from those reachable by faulty behaviors.
We propose a discrete event systems (DES) approach to the failure
diagnosis problem. We present a methodology for modeling physical
systems in a DES framework. We discuss the notion of diagnosability and
present the construction procedure of the diagnoser. Finally, we
illustrate our approach using a heating, ventilation and air
conditioning (HVAC) system
In the above paper by Zuberek, conflict-free Petri nets with deterministic timing are used for modeling cluster tools. Performance analysis of the models is based on P-invariants. This comment tries to clarify some statements in the paper and points out a more efficient performance-analysis approach based on linear programming, which is of polynomial time complexity.
The problem of verifying the properties of diagnosability and I-diagnosability is considered. We present new polynomial-time algorithms for deciding diagnosability and I-diagnosability. These algorithms are based on the construction of a nondeterministic automaton called a verifier.
Starts with a brief review of the history and the application
areas considered in the literature. The author then proceeds with
introductory modeling examples, behavioral and structural properties,
three methods of analysis, subclasses of Petri nets and their analysis.
In particular, one section is devoted to marked graphs, the concurrent
system model most amenable to analysis. Introductory discussions on
stochastic nets with their application to performance modeling, and on
high-level nets with their application to logic programming, are
provided. Also included are recent results on reachability criteria.
Suggestions are provided for further reading on many subject areas of
Petri nets
Net Unfolding Approach
- A Benveniste
- E Fabre
- C Jard
- S Haar