Counter approach for the estimation of optimal sequences in Partially Observable Untimed Petri Nets

Abstract

In this paper, we consider the on-line estimation of optimal current subsequences in Partially Observable Untimed Petri Nets. Applying the counter approach classically used in max-plus algebra for Timed Petri nets, the idea is to exploit the assumption of a non immediate consumption of the tokens for each place which introduces an order of precedence between events. The approach can estimate a global price depending on the costs and gains provided by the tasks. The estimation of optimal sequences is based on the determination of a time horizon necessary to describe the sequences. The estimation is relevant to a step defined by two successive occurrences of observable transition firings. We show that the approach can consider any optimization problem if the dates of the observations are known or, if a guaranteed horizon can be computed which is always possible when the unobservable subnet satisfies a weak assumption close to the structural boundedness (relaxed structurally boundedness). As the technique avoids the generation of sets, the approach does not depend on their cardinalities and is numerically efficient.

Two-staged approach for estimation of sequences in partially observable P-time Petri nets on a sliding horizon with schedulability analysis

“…3) The general linear criterion is not restrictive and can be adapted to many fields. This criterion can be a global price which is expressed with a function depending of the costs and the gains of the process [12] or expresses the presence and absence of faults in fault detection [5] [6].…”

“…In this paper, the aim is to treat P-time PNs by a simplification under the form of a P-timed PN while the paper [11] considers P-timed PNs by building an untimed PN and analyzing the candidate count vectors. The objective of [12] is also different as this study focuses on the estimation of sequences in untimed PNs. The proposed technique differs from [2] which develops a state observer of the form of an automaton whose nodes can be viewed as macro-markings and from [3] where the state observer for P-time labeled PNs with some indistinguishable events is built online in a decentralized context.…”

“…After presenting the aim of the two-phased strategy, the following section IV describes the Phase 1 where we present the procedure of the sequence estimation. Contrary to [12], we consider time models and focus on the adaptation of the estimation to a horizon composed of two successive steps. The generation of the relevant untimed sequences and the starting marking is discussed.…”

“…denoted h , the dimension of vector κ <k−1>→<k> is denoted by n = (h + 1).|T R un | while the dimension of vector γ <k−1>→<k> is (h + 1).|T R obs |. Below the form generalizes the polyhedron exploited in [12]. Moreover, the time durations can be unitary but also null.…”

“…Some criteria can be used in diagnosis [5]. Generalizing a criterion presented in [12], we can focus on the minimum or maximum number of events of a sequence inside the step < k >, that is, more formally, the sum of the event numbers with c 1x|T Run| .x <k> where c 1x|T Run| is unitary. As The incidence matrices W un , W obs , W − obs and the initial marking are as follows:…”