Offline analysis of the relaxed upper boundedness for online estimation ofoptimal event sequences in Partially Observable Petri Nets

Abstract

The aim of this paper is the analysis of the property of the relaxed structurally boundedness of the unobservable subnet of the Petri net which brings a condition guaranteeing the finitude of all possible sequence lengths in the context of an on-line estimation in Partially Observable Petri Nets relevant to a sliding horizon or a receding horizon starting from the initial marking. Based on specific invariants defined over the real numbers, the approach focuses on an offline structural analysis, that is, the determination of the parts of the unobservable subnet where an online estimation for any criterion can be made. The decomposition-composition technique is based on a block triangular form obtained with any technique. The composition of the substructures leads to a propagation of the relaxed structurally boundedness property through the structure. The study of a large-scale manufacturing system shows that the direct treatment of the large system system can be avoided and that the triangular form brings a sequential treatment allowing a computation based on smaller systems independently of the resolution of the complete system.

Appendix

This appendix presents the main lines of the Dulmage-Mendelsohn decomposition Dulmage and Mendelsohn (1958) Dulmage and Mendelsohn (1959) which brings a partition of the transitions and places in notable sub-structures. Generalizing the classical triangular form where the substructures are square, this structural analysis is based on a canonical decomposition of any bipartite graph (and its relevant table which is rectangular).

The DM decomposition presents a large scope of applications as it has been exploited in many fields such that: the resolution of large scale systems Pothen and Fan (December 1990); Murota (1987); Cane et al. (2014); the simulation of continuous systems where the debugging of software based on modeling languages is necessary Bunus and Fritzson (2002); the fault detection in continuous systems Declerck and Staroswiecki (1991a, 1991b). Similar to bond graphs, the study Frisk et al. (2012) improves this structural approach by taking into account integral and differential causal interpretations for differential constraints.

The classical algorithms of maximum matching as the classical "Hungarian method" (developed by H. Kuhn) and the algorithm of permutation of the rows and columns of a matrix Murota (1987); Pothen and Fan (December 1990); Declerck and Staroswiecki (1991a, 1991b) leading to its DM decomposition are out the scope of the appendix.

Let us define the initial table and adapt the DM decomposition to the Petri nets.

As the resolution focuses on the unknown variables, we separate the transitions of $TR\$and the relevant columns in two sets:

• The set of observable transitions $TR_{obs}$ which correspond to the known variables $\overline{y}$.

• The set of unobservable transitions $TR_{un}$ which are relevant to unknown variables $\overline{x}$.

Therefore, a permutation of the columns allows to establish the incidence matrix $W=\left( \begin{array}{ll} W_{obs}&W_{un} \end{array} \right) .$ Moreover, a structural point of view of the unobservable induced subnet is taken. Each (oriented) arc of the Petri net is replaced by a (non-oriented) edge which is non-valuated, that is, the valuations and the orientations of the arcs of the Petri net are neglected: only the existence of a connection (which is often represented by the symbol $\times$ or 1 in the literature) is considered in this appendix. We now show that the analysis of this non-oriented bipartite graph of the unobservable induced subnet allows to determine notable sub-systems which are the support of a possible resolution and an algebraic interpretation.

The matching between the relations P and the unobservable transitions $TR_{un}$ is defined as follows:

Definition 2

A matching C is a set of pairs $(p_{i},x_{j})$ where:

• Each place $p_{i}$ is associated with a transition $x_{j}$ at the most.

• Each transition $x_{j}\in TR_{un}$ is associated with a place $p_{i}$ at the most.

So, in a matching C, a unique transition is associated to each place and a unique place is associated to each transition at the most. In the bipartite graph, the matching is represented by a set of edges without common vertices.

As the non-oriented bipartite graph of the unobservable induced subnet is considered, a possible pair can be composed of a place and one of its input or output transitions. Moreover, we focus on maximum matchings where the number of its pairs is maximum. Different maximum matchings can be obtained but all of them have the same cardinality. In the Tables 2 and 4, each pair of the matching is expressed by a symbol in bold.

In this context, the maximum matching is the support of the canonical decomposition developed by A. L. Dulmage and N. S. Mendelsohn Dulmage and Mendelsohn (1958, 1959) in graph theory where a structural decomposition of the table leads to a diagonal of specific block substructures. Three distinct canonical structures named Just-, Over- and Under-structures are highlighted and the block substructures of the Just-structure are square. If the matching C is maximum, there is a unique partition of rows of P and columns of $TR_{un}$ denoted as X such that: $P=P^{>}\cup P^{=}\cup P^{<}$ and $X=X^{>}\cup X^{=}\cup X^{<}$ with empty intersections. This partition highlights three important sub-structures: the Over-structure $S^{>}=(P^{>},X^{>})$, the Just-structure $S^{=}=(P^{=},X^{=})$ and the Under-structure $S^{<}=(P^{<},X^{<}).$ Moreover, we have $|C|=|C^{>}|+\left| C^{=}\right| +|C^{<}|$ (expression $\left| .\right|$ denotes the number of pairs in the matching) where $C^{>},C^{=}$ and $C^{<}$ satisfy the following points.

• For the Over-structure, the maximum matching $C^{>}$ satisfies $|C^{>}|=\left| X^{\prime }\right| <\left| P^{\prime }\right| .$ All elements of $X^{\prime }$ are matched but there is at least a non-matched element in $P^{\prime }.$

• For the Just-structure, the maximum matching $C^{=}$ satisfies $\left| C^{=}\right| =\left| P^{\prime }\right| =\left| X^{\prime }\right| .$ All elements of $P^{\prime }$ and $X^{\prime }$ are matched in the case of a Just-structure which can be decomposed in square blocks.

• For the Under-structure, the maximum matching $C^{<}$ satisfies $|C^{<}|=\left| P^{\prime }\right| <\left| X^{\prime }\right| .$ All elements of $P^{\prime }$ are matched but there is at least a non-matched element in $X^{\prime }.$

Given a matching, an alternating path is a path whose edges belong alternatively to the matching and not to the matching. Using these notions, the following theorem transposed from Dulmage and Mendelsohn (1958, 1959); Declerck and Staroswiecki (1991a, 1991b) allows to determine different notable substructures. To facilitate the presentation of the results, a direction is added to the edges of the non-oriented bipartite graph. Each pair $(p_{i},x_{j})$ of the maximum matching C is oriented from $p_{i}$ to $x_{j}$ (graphically, $p_{i} \overset{C}{\longrightarrow }x_{j})$ and in the opposite direction when $(p_{i},x_{j})\notin C$ (graphically, $p_{i}\leftarrow x_{j}$ ).

Theorem 9

Let us assume that the matching is maximum.

• The places and transitions of an alternating path belong to the Over-structure $S^{>}=(P^{>},X^{>})$ when this path starts from a matched place and finishes in a non-matched place.

• The places and transitions of an alternating path belong to the Under-structure $S^{<}=(P^{<},X^{<})$ when this path starts from a non-matched transition and finishes in a matched transition.

• The Just-structure is defined by $S^{=}=(P^{=},X^{=})$ with $P^{=}=P\backslash (P^{>}\cup P^{<})$ and $X^{=}=TR_{un}\backslash (X^{>}\cup X^{<}).$ $\blacksquare$

Example 4

In the Petri net of example 4, the sets of observable and unobservable transitions are {$y_{1},y_{2}$} and{$x_{1},x_{2},\ldots ,x_{8}$} respectively (Fig. 3). Remember that the firing number of $y_{1},y_{2}$ is finite (Assumption $\mathcal {AS-}3$). Table 4 presents the non-oriented form of the incidence matrix $W=\left( \begin{array}{ll} W_{obs}&W_{un} \end{array} \right)$ for the Petri net in Fig. 3: The values 1 and -1 are replaced by 1 which represents the presence of a connection between the relevant place and the unobservable transition. For clarity, the labels of the places and transitions in Petri nets Fig. 3 have been chosen such that they illustrates the DM decomposition without making a reorganization of the columns and rows of the table. A possible maximum matching C whose cardinality is 8 is represented in bold (this matching is suggested by the orientation of the Petri net): $C= \{(p_{1},x_{1}),(p_{2},x_{2}),(p_{4},x_{3}),(p_{5},x_{4}),(p_{6},x_{5}),(p_{7},x_{6}),(p_{8},x_{8}),(p_{9},x_{9})\}.$ Other maximum matchings are available as $\{(p_{2},x_{1}),(p_{3},x_{2}),(p_{4},x_{3}),\!(p_{5},x_{5}),\!(p_{6},x_{4}),\!(p_{7},x_{6}),$ $(p_{8},x_{7}),(p_{9},x_{8})\}$.

Fig. 3

Petri net of example 4: the matched pairs (place and its output transition) and the relevant outgoing arcs are in bold; Over-, Just- and Under-substructures are represented with bold and thin lines

Table 4 Canonical decomposition of the structure of the incidence matrix of the unobservable induced Petri net in Fig. 3 (example 4). Each value 1 represents a connection for a pair $(p_{i},x_{j})$ while 0 expresses an absence of link.

As C is maximum, we can apply the DM decomposition (Theorem 9). The edges of the non-oriented bipartite graph are now oriented with the direction presented above and the oriented path can be interpreted. In this simple example, the orientation deduced from the chosen maximum matching corresponds to the orientation of the Petri net. For $S^{>}$, an alternating path starts from $p_{1}$ (matched) and finishes in $p_{3}$ (non-matched): $p_{1}\overset{C}{\longrightarrow } x_{1}\longrightarrow p_{2}\overset{C}{\longrightarrow }x_{2}\longrightarrow p_{3}.$ For $S^{<}$, an alternating path starts from $x_{7}$ (non-matched) and finishes in $x_{9}$ (matched): $x_{7}\longrightarrow p_{8}\overset{C}{ \longrightarrow }x_{8}\longrightarrow p_{9}\overset{C}{\longrightarrow } x_{9}$. For $S^{=}$, an alternating path starts from $p_{4}$ (matched) and finishes in $x_{5}$ (matched): $p_{4}\overset{C}{\longrightarrow } x_{3}\longrightarrow p_{5}\overset{C}{\longrightarrow }x_{4}\longrightarrow p_{6}\overset{C}{\longrightarrow }x_{5}$. Another one starts from $p_{7}$ (matched) and finishes in $x_{6}$ (matched): $p_{7}\overset{C}{ \longrightarrow }x_{6}$. $\blacksquare$

Let us consider the Just-structure $S^{=}.$ Let us assume that a fictitious self-loop of null length connecting $x_{i}$ to $x_{i}$ is added to each vertex $x_{i}\in X^{=}.$ For any pair of matched vertexes ($x_{i},$ $x_{j})$ with $x_{j},x_{i}\in X^{=},$ we can focus on the case where there is a path from $x_{i}$ to $x_{j}$ and a path from $x_{j}$ to $x_{i}$ (mutual dependence of $x_{i}$ and $x_{j}$). This case defines a pair of dependent transitions corresponding to a circuit and we can define a substructure composed of transitions where each transition is connected to any transition of the substructure with a circuit composed of places. Formally, the substructure contains a directed path from to $x_{i}$ to $x_{j}$ and a directed path from $x_{j}$ to $x_{i}$ for every pair of vertices $x_{i},$ $x_{j}.$ These substructures are usually named strongly connected substructure and, irreducible substructure for the corresponding representation in the table. As this type of substructure is remarkable, the approach is based on the determination of all these substructures which leads to a partition of the Just-structure $S^{=}.$

Example 4 continued.

Represented in grey in the Just-structure $S^{=}$ of Table 4, substructures $(\{p_{4}\},\{x_{3}\})$, $(\{p_{5},p_{6}\},\{x_{4},x_{5}\})$ and $(\{p_{7}\},\{x_{6}\})$ are irreducible. $\blacksquare$

Note that each DM decomposition is common to a set of Petri nets. In fact, each filling of the lower-left corner of $W_{un}$ defines a new Petri net with the same DM decomposition.

Example 4 continued.

The Petri net in Fig. 4 is a variant of the Petri net in Fig. 3 which presents the same DM decomposition. The difference is the addition of components in the lower-left corner of $W_{un}$: $\left( W_{un}\right) _{4,1}=1,$ $\left( W_{un}\right) _{4,2}=1,$ $\left( W_{un}\right) _{7,3}=1,$ $\left( W_{un}\right) _{7,4}=1,$ $\left( W_{un}\right) _{8,1}=1,$ $\left( W_{un}\right) _{8,3}=1,$ $\left( W_{un}\right) _{8,6}=-1.$ $\blacksquare$

Fig. 4

Variant of Petri net Fig. 3 with the same DM decomposition

Example 3 continued (case study).

The determination of a maximum matching gives a matching where the non-matched places are $p_{33},p_{36},p_{38},p_{26},p_{11},p_{33},p_{7},p_{8},\!p_{17},\!p_{4},\!p_{29},$ $p_{25},$ $p_{34},$ $p_{30},$ $p_{13},p_{35}$ (in Table 2, the matched components are in bold). The application of the DM decomposition of $W_{un}$ is a priori unsuccessful as it leads to a unique under-determined structure which does not generate smaller subsystems. Keeping only the rows of the matched places as $p_{9},$ $p_{10},$ $p_{5},$..., a second DM decomposition leads to a just-structure which is a triangular form (shaded in darker grey) suggesting an order of resolution. The size of all the blocks in the main diagonal except the substructure $\{p_{5},p_{24}\}\times \{x_{15},x_{17}\}$ is $1\times 1$. The addition of the rows of the non-matched places proposes the subsystems in darker grey and light grey. Note that, as the DM decomposition has not been applied in the classical form, this decomposition of subsystems is not unique a priori. $\blacksquare$