Sequence Estimation and Schedulability Analysis for Partially Observable Petri Nets (Part I) Ph. Declerck Sequence Estimation and Schedulability Aim Analysis for Partially Observable Petri Nets Preliminaries (Part I) Estimation in P-timed Petri nets Diagnosis in P-timed Petri nets Ph. Declerck Conclusion References University of Angers 15 November 2018
Plan Sequence - Introduction Estimation and Schedulability - Preliminaries Analysis for Partially - Estimation/Diagnostic in P-timed Petri nets Observable Petri Nets (Part I) - Schedulability Analysis : Generation and checking of count Ph. Declerck vectors (next talk !) Aim - Fault detection for non-modelled faults (next talk !) Preliminaries Estimation in P-timed Petri nets Diagnosis in P-timed Petri nets Conclusion References
Aim Sequence - To make estimation in Petri nets with application to fault Estimation and Schedulability detection Analysis for Partially - Let TR = TR obs ∪ TR un where TR obs is the set of Observable Petri Nets (Part I) observable transitions while TR un is the set of unobservable Ph. Declerck ones. Aim - For a sequence (word) ω (or a subsequence) observed, the Preliminaries aim is to compute a firing sequence (or some firing Estimation in sequences) of unobservable transitions necessary to complete P-timed Petri nets ω into a fireable sequence of the Petri net consistent with its Diagnosis in P-timed Petri nets evolution. Conclusion References
A difficulty Sequence Ru, Y., and Hadjicostis, C. N. (2009). Bounds on the Estimation and Schedulability number of markings consistent with label observations in Analysis for Partially Petri nets. IEEE Transactions on Automation Science and Observable Petri Nets (Part I) Engineering, 6(2), 334-344. Ph. Declerck - The number of consistent markings in a Petri net with Aim nondeterministic transitions (unobservable transitions and/or Preliminaries transitions that share the same label) is at most polynomial Estimation in in the length of the observation sequence. → Increasing with P-timed Petri nets the new observations in the worst case... Diagnosis in P-timed Petri nets - The number of firing sequences can be exponential in the Conclusion length of the observation sequence. References → Guiding thread Compromise between the accuracy of the interpretation and the numerical efficiency of the approach.
� � � � � � � � � � � � � � � � � � � �� ����� ������ ������ �� �� � � Principe of estimation Sequence Estimation and Schedulability Analysis for Partially Observable Petri Nets (Part I) Ph. Declerck Figure : Example : Timed Event Graph. Aim Preliminaries Estimation in T = T 1 = T 2 is the minimum travel time from London to P-timed Petri nets Paris and conversely. Diagnosis in P-timed Petri nets - Let us assume that a person in Paris observes that 10 Conclusion planes coming from London have landed at time 100 References minutes or before. We can conclude that at least ten planes have taken off from London at time 100 − T minutes. So, the least number of plane take-offs from London is 10 at time 100 − T minutes. → If each plane has been checked before its take-off, we can estimate the minimum activity of the maintenance department at London.
- Let us assume that a person in Paris observes that at the Sequence most 10 planes have taken off at time 100 minutes or before Estimation and Schedulability at Paris. We can conclude that the greatest number of 10 Analysis for Partially planes have landed in London at time 100+ T minutes. It Observable Petri Nets (Part I) could be lower : A pilot can decide to return or to land at Ph. Declerck another airport for technical reasons. The least number of Aim landings is zero in the worst case. Preliminaries → Maximum activity of the maintenance at London Estimation in P-timed Petri nets Diagnosis in P-timed Petri nets Conclusion References
Fault detection Sequence No-modelled faults Estimation and Schedulability The objective is the detection of variations of the model Analysis for Partially which are not described. Observable Petri Nets (Part I) On-line comparison of consistency of informations with Ph. Declerck models or submodels Aim Ex : connection of three pipes in continuous systems Preliminaries - Nominal model : Q 1 + Q 2 = Q 3 Estimation in - Real model (leak in a pipe) : Q 1 + Q 2 � = Q 3 P-timed Petri nets Faults or changes in the process in Petri nets Diagnosis in P-timed Petri nets - Variation of a temporisation (deterioration of a machine, Conclusion repairing) ; References - Loss or addition of a token (loss of a ressource, addition of a part) ; - Another graph (new schedule)
� � � � � � � � ����� ����� � � Principle Sequence Estimation and Schedulability Analysis for Partially Observable Petri Nets (Part I) Ph. Declerck Aim Figure : Example : P-Time Petri net. Preliminaries Estimation in P-timed Petri nets Journey of a vehicle from the town A to B between 2 and 3 Diagnosis in P-timed Petri nets hours). Journey from B to C between 5 and 6 hours. Conclusion Observable transitions : u and y. References Unobservable : Time x Time u known, x ∈ [ u + 2 , u + 3]. Time y known, x ∈ [ y − 6 , y − 5]. Therefore, x ∈ [max( u + 2 , y − 6), min( u + 3 , y − 5)] otherwise, model � =reality
Modelled faults Sequence Estimation and Schedulability TR = TR obs ∪ TR un Analysis for Partially Observable Petri We assume that the faults occurring in the process are Nets (Part I) modeled by unobservable transitions and the notation TR f Ph. Declerck represents the relevant set. The set of unobservable Aim transitions describing a normal behavior is denoted TR n . Preliminaries Therefore, TR f ⊂ TR un and Estimation in P-timed Petri nets Diagnosis in TR un = TR n ∪ TR f (1) P-timed Petri nets Conclusion References
� � � � � � � � � � � � � � � � � � � � � � � � � �� �� �� Determination of a fault state D Sequence If min Z ( c det . x ) ≥ 1 with c det ≥ 0 a row-vector, then at least a Estimation and Schedulability Analysis for fault is detected on the horizon (State D). Partially Observable Petri - It always exists as x ≥ 0 . Nets (Part I) - The minimum presents an interest if path from an Ph. Declerck unobservable transition x i to an observable transition y Aim Example on minimum Preliminaries Estimation in P-timed Petri nets Diagnosis in P-timed Petri nets Conclusion References Figure : Example : Timed Petri net - chain x 4 and x 5 . If y = 1 , then x − 1 = 1, x − 2 = 1 for the marking M = 0 → State D for x 1 , x 2 . Also x + 3 = 0 and x + 4 = 0 → Cannot lead to the state D . Now, if the marking in the place p 2 is M 2 = 3 , then x − 1 = 1 but x − 2 = 0
� � � � � � � � � � � � � � � � � � � � � � � � � �� �� �� Determination of a normal state N Sequence If max Z ( c det . x ) = 0, then no fault is detected (State N ). Estimation and Schedulability Analysis for The maximum presents an interest if path from an Partially Observable Petri observable transition y to an unobservable transition x i Nets (Part I) It can be equal to + ∞ : source transition which is a Ph. Declerck perturbation, circuit with no input place → infinite marking. Aim Example on maximum Preliminaries Estimation in P-timed Petri nets Diagnosis in P-timed Petri nets Conclusion References Figure : Example : Timed Petri net - chain. If y = 1 , then x + 2 = + ∞ (perturbation), x + 1 = + ∞ and also x + 3 = M 3 + 1 and x + 4 = M 4 + M 3 + 1 If y = 0 , then x + 2 = + ∞ (perturbation), x + 1 = + ∞ and also x + 3 = M 3 and x + 4 = M 4 + M 3 → State N for x 3 , x 4 if M 3 = M 4 = 0 .
Preliminaries Sequence Petri nets Estimation and Schedulability - A Place/Transition net (a P / TR net) is a structure Analysis for Partially N = ( P , TR , W + , W − ), where P is a set of | P | places and Observable Petri Nets (Part I) TR is a set of | TR | transitions which are denoted by x Ph. Declerck - Notation t corresponds to the current time, T l to the Aim temporization of place p l ∈ P , and T to the transposition of Preliminaries a matrix. - Matrices W + and W − are | P | × | TR | post- and Estimation in P-timed Petri nets pre-incidence matrices over N where each row Diagnosis in P-timed Petri nets l ∈ { 1 , . . . , | P |} specifies the weight of the incoming and Conclusion outgoing arcs of place p l ∈ P respectively. The incidence References matrix is W = W + − W − . - Vector M l is the marking of place p l with l ∈ { 1 , . . . , | P |} . A net system ( N , M init ) is a net N with an initial marking M init .
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