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Failure detection and localization for timed event graphs in \((\max \limits ,+)\)-algebra

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Abstract

In this paper, we address the problem of failure detection and localization in a Timed Discrete Event System (TDES) such \((\max \limits ,+)\)-linear system graphically modeled by a Timed Event Graph (TEG). The considered failures are changes on holding times or tokens of the TEG places that can provoke shifts between an observed outcoming timed flow and an expected outcoming timed flow (for a given incoming timed flow). Indicators are built to first detect such shifts relying on the \((\max \limits ,+)\) algebraic framework and the residuation theory. An analysis of the indicators’ values provides information about time or event failure that could have happen. Then, thanks to the knowledge of the behavior of the system through its corresponding TEG, sets of failures that could explain the detected shifts are obtained. It comes from matrices of signatures for each indicator built on each observable output of the system. An example of application is proposed to experiment exhaustively failures of type time and event on each place of the TEG.

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Notes

  1. As in usual algebra, ⊗ will be omitted when no confusion is possible.

  2. Notation a without the bracket will be adopted in the sequel.

  3. A simplification of writing gives \(\varepsilon = \gamma ^{+\infty }\delta ^{-\infty }\) and \(\top = \gamma ^{-\infty }\delta ^{+\infty }\).

  4. We use the same notation for a transition of the TEG and its corresponding series in the \((\max \limits ,+)\)-linear system.

  5. But this is not because of the equivalence relation in \({{\mathscr{M}}}^{ax}_{in}\llbracket \gamma ,\delta \rrbracket \) (see Remark 3).

  6. When there is only one output, such as in SISO (Single Input - Single Output) or MISO systems, notation I(u,y) with \(\tau (y, \tilde {y})\) and \(\nu (y, \tilde {y})\) holds.

  7. \(\tilde {y}\) and y are respectively equal to series a and b from Example 11.

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Authors and Affiliations

  1. LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France

    Euriell Le Corronc, Yannick Pencolé, Alexandre Sahuguède & Claire Paya

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  1. Euriell Le Corronc
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  2. Yannick Pencolé
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  3. Alexandre Sahuguède
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  4. Claire Paya
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Correspondence to Euriell Le Corronc.

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This article belongs to the Topical Collection: Topical Collection on Smart Manufacturing -A New DES Frontier Guest Editors: Rong Su and Bengt Lennartson

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Le Corronc, E., Pencolé, Y., Sahuguède, A. et al. Failure detection and localization for timed event graphs in \((\max \limits ,+)\)-algebra. Discrete Event Dyn Syst 31, 513–552 (2021). https://doi.org/10.1007/s10626-020-00329-7

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