1. Introduction

The detection of the trajectories in a system is important for several purposes. For example, in Discrete-Event Systems (DES) some control schemes and observer designs use the sequence of events as feedback, while several techniques for the failure detection employ the system trajectories to establish a suitable recovery point [13], [16], [18], [19], [22].

Several approaches reported in the literature dealing with the formal analysis of DES have studied the system trajectories as part of other problems. For instance, the design of controllers, observers, trackers and detectors for DES have all been studied within the automaton field [10], [15], [19], [20], [21].

Petri nets (PN’s) have also been used for the analysis of almost the same problems as in the automaton framework. For example, the modeling of manufacturing systems, the supervisory control, state feedback schemes, sequence detection and the observability of PN’s and the design of controllers and observers, have been reported in [1], [5], [6], [7], [9], [11], [16].

Previous related work of the authors of this paper is also reported in the literature. In [2], a technique that allows controlling a DES using a supervisor with the feedback provided by an observer of sequences is developed. In [3], preliminary results for addressing the sequence detection of a subclass of PN, called Free-Choice nets, are provided.

This paper extends the previous research of the authors and provides novel results and efficient algorithms addressing the problem of the sequence detection in DES that are modelled by S-Systems, a subclass of PN with well-defined structure, where efficient solutions are derived. Firstly, safe nets are considered. For this case, the construction of the Sequence-Detectability table provides a necessary and sufficient condition for the testing of the sequence-detectability. Secondly, the safeness requirement is relaxed and its implication on the Sequence-Detectability is analyzed. For the non-safe case, the construction of the Event-Detectability table provides a necessary and sufficient condition for the verification of the property. Finally, the relationship among the sequence-detectability and the observability of S-Systems is outlined.

The rest of this paper is organized as follows. Section 2 introduces some basic PN’s notions used in this work. Section 3 presents the Sequence-Detectability property and its characterization in well-formed S-Systems. Section 4 outlines how the sequence-detectability supports solving the observability of the net. Section 5 provides the conclusions of this work and a final section shows the bibliographical references.

REFERENCES

1. CAMPOS-RODRIGUEZ, R., M. ALCARAZ-MEJIA, A Matlab/Simulink Framework for the Design of Controllers and Observers for Discrete-Event Systems, Journal of Electronics and Electrical Engineering, vol. 99(3), 2010, pp. 63-68.
2. CAMPOS-RODRIGUEZ, R., M. ALCARAZ-MEJIA, J. MIRELES-GARCIA, Supervisory Control of Discrete Event Systems Using Observers, Proc. of the IEEE Mediterranean Conference on Control and Automation, vol. 1, 2007, pp. 1-7.
3. CAMPOS-RODRIGUEZ, R., A. RAMIREZ-TREVINO, E. LOPEZ-MELLADO, Observability Analysis of Free-Choice Petri Net Models, Proceedings of IEEE/SMC Conference on Systems of Systems Engineering 6, 2006,
   pp. 77-82.