Detection of Temporal Anomalies for Partially Observed Timed PNs

Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 2821078, 10 pages https://doi.org/10.1155/2017/2821078 Research Article Detection of Temporal Anomalies for Partially Observed Timed PNs Dimitri Lefebvre Normandie Université, UNIHAVRE, GREAH, 76600 Le Havre, France Correspondence should be addressed to Dimitri Lefebvre; Received 13 October 2016; Revised 1 March 2017; Accepted 16 March 2017; Published 12 April 2017 Academic Editor: Fazal M. Mahomed Copyright © 2017 Dimitri Lefebvre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article concerns faults detection and isolation for timed stochastic discrete event systems modeled with partially observed timed Petri nets. Events occur according to arbitrary probability density functions. The models include the sensors used to measure events and markings and also the temporal constraints to be satisfied by the system operations. These temporal constraints are defined according to tolerance intervals specified for each transition. A fault is an operation that ends too early or too late. The set of trajectories consistent with a given timed measured trajectory is first computed. Then, the probability that the temporal specifications are unsatisfied is estimated for any sequence of measurements and the probability that a temporal fault has occurred is obtained as a consequence. 1. Introduction The prevention of faults is a critical issue in numerous systems to preserve the safety of both equipment and human operators. These issues have been addressed in numerous studies with fault detection and diagnosis (FDD) methods. The aim of fault detection is to create an alarm each time a fault occurs, and the aim of diagnosis is to isolate the fault within a group of candidates [1]. In the domain of discrete event systems (DESs), FDD has been often formulated with automata, Petri nets (PNs) [2], in particular labeled PNs (LPNs) [3] or partially observed Petri nets (POPNs) [4]. The main reason for developing FDD tests with PN extensions is that such models include graphical representations that can be disseminated widely in numerous application domains. They also offer mathematical supports that are consistent with standard tools. The proposed methods are useful for a large variety of technological systems, ranging from computer or chemical engineering to manufacturing and intelligent transportation systems. In numerous contributions, the faults that are considered are unexpected events that may occur in event sequences and that cannot be directly measured. Various approaches have been proposed with PN extensions to detect and isolate such unexpected events. These approaches are based either on the analysis of the PN reachability graph [5–9], on the direct properties of the PNs [10, 11], or on PN unfolding [12, 13]. A few results also concern the introduction of temporal information in the diagnosis process. At first, dates of events have been introduced in usual extensions of untimed PNs. Such dates lead to a more accurate estimation of the past and future fault occurrence probabilities [14] and are also useful to propose an evaluation of the unknown fault dates [15]. The design and identification of models that include temporal faults have been also considered [16, 17]. Then, fuzzy Petri nets have been used to model and check temporal constraints between event occurrences [18]. Partial orders with unfolding and (max, +)-linear inequalities have been used with timed PN models [19, 20]. Monotonic monitoring and stratification have been introduced, when the monitoring is fragmented because of the uncertain temporal observation [21]. Finally, indirect monitoring has been used by comparing the actual cycle periods with the expected one in order to detect faults [22]. This paper takes place in the context where both transitions and places are assumed to be partially observed and 2 Mathematical Problems in Engineering f(d) f(d) 2/(b − a) a
훿 Δ b d a
훿 (a + b)/2 Δ (a) b d (b) Figure 1: Probability density functions of the transition firing durations: bounded uniform (a); symmetrical triangular (b). consider only temporal faults. For that purpose, temporal constraints are defined by tolerance intervals that are associated with the transitions and that represent the normal durations of the system operations. The aim of the diagnosis system is to generate alarms when the temporal constraints are no longer satisfied. For that purpose, timed POPNs (POTPNs) are introduced. POTPNs take into consideration some measurable events that correspond to dated and labeled transition firings and also to partial measurements of the marking vector that is dated. This formalism, fully described in [23], is useful to represent incomplete history of dated measurements collected by SCADA systems. In the present work, this model is extended by adding temporal constraints that give upper and lower bounds for each transition duration. The paper is organized as follows. In Section 2, temporal constraints and POTPNs are introduced. In Section 3, the main results are detailed. Examples are detailed throughout the paper. Section 4 concludes the paper. 2. Context and Notations 2.1. PNs with Temporal Specifications. A PN structure is defined as 𝐺 = ⟨P, T, 𝑊𝑃𝑅 , 𝑊𝑃𝑂⟩, where P = {𝑃1 , . . . , 𝑃𝑛 } is a set of 𝑛 places and T = {𝑇1 , . . . , 𝑇𝑞 } is a set of 𝑞 transitions, 𝑊𝑃𝑂 ∈ (N)𝑛×𝑞 and 𝑊𝑃𝑅 ∈ (N)𝑛×𝑞 are the post- and preincidence matrices (N is the set of nonnegative integer numbers), and 𝑊 = 𝑊𝑃𝑂 − 𝑊𝑃𝑅 is the incidence matrix. A PN is choicefree if |(𝑃𝑖 )∘ | ≤ 1 (the postset of 𝑃𝑖 contains at most a single transition). ⟨𝐺, 𝑀𝐼 ⟩ is a PN system with initial marking 𝑀𝐼 and 𝑀 ∈ (N)𝑛 represents the PN marking vector. A PN system is 1-bounded if and only if (iff) 𝑀 ≤ 1𝑛 where 1𝑛 = (1 ⋅ ⋅ ⋅ 1)𝑇 (inequality 𝑀 ≤ 1𝑛 is considered component wise). A transition 𝑇𝑗 is enabled at marking 𝑀 iff 𝑀 ≥ 𝑊𝑃𝑅 (:, 𝑗), where 𝑊𝑃𝑅 (:, 𝑗) is the column 𝑗 of preincidence matrix; this is denoted as 𝑀[𝑇𝑗 ⟩. When 𝑇𝑗 is enabled, it may fire, and when 𝑇𝑗 fires once, the marking varies according to Δ𝑀 = 𝑀󸀠 − 𝑀 = 𝑊(:, 𝑗). This is denoted as 𝑀[𝑇𝑗 ⟩𝑀󸀠 . A sequence of size 𝐻 = |𝜎| fired at marking 𝑀 is a sequence of 𝐻 transitions 𝜎 = 𝑇(1)𝑇(2) ⋅ ⋅ ⋅ 𝑇(𝐻), with 𝑇(𝑗) ∈ T, 𝑗 = 1, . . . , 𝐻 that consecutively fire from marking 𝑀 to marking 𝑀󸀠 . This is denoted as 𝑀[𝜎⟩𝑀󸀠 . The integer 𝑥𝑗 (𝜎) is the number of occurrences of transition 𝑇𝑗 in 𝜎, and 𝑋(𝜎) = (𝑥𝑗 (𝜎)) ∈ (N)𝑞 is the firing count vector for 𝜎. A sequence 𝜎 fired at 𝑀 leads to an untimed trajectory (𝜎, 𝑀) detailed in (𝜎, 𝑀) = 𝑀 (0) [ 𝑇 (1)⟩ 𝑀 (1) ⋅ ⋅ ⋅ [ 𝑇 (𝐻)⟩ 𝑀 (𝐻) , (1) with 𝑀(0) = 𝑀. A marking 𝑀 is said to be reachable from initial marking 𝑀𝐼 if there exists a firing sequence 𝜎 such that 𝑀𝐼 [𝜎⟩𝑀. The set of all reachable markings from initia (...truncated)