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Symbolic Supervisory Control of Infinite Transition Systems Under Partial Observation Using Abstract Interpretation

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Abstract

We propose algorithms for the synthesis of state-feedback controllers with partial observation of infinite state discrete event systems modelled by Symbolic Transition Systems. We provide models of safe memoryless controllers both for potentially deadlocking and deadlock free controlled systems. The termination of the algorithms solving these problems is ensured using abstract interpretation techniques which provide an overapproximation of the transitions to disable. We then extend our algorithms to controllers with memory and to online controllers. We also propose improvements in the synthesis of controllers in the finite case which, to our knowledge, provide more permissive solutions than what was previously proposed in the literature. Our tool SMACS gives an empirical validation of our methods by showing their feasibility, usability and efficiency.

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Notes

  1. Miremadi et al. (2008b) present a similar model but the domain of the variables is assumed to be finite.

  2. The underlying structure used to encode the sets of states is mostly based on a data structure named Binary Decision diagram defined by Bryant (1986), that has been proved to be very efficient to encode boolean predicates.

  3. Note that Jeannet et al. (2005) assume this alphabet to be infinite.

  4. An STS is structurally deterministic if \(\forall \delta_1, \delta_2 \in \Delta: (\sigma_{\delta_1} = \sigma_{\delta_2})\implies (G_{\delta_1} \cap \, G_{\delta_2} = \emptyset)\). The structural determinism of an STS \({\cal T}\) implies that the corresponding LTS \({[\![{{\cal T}}]\!]}\) is deterministic.

  5. Where × denotes the classical synchronous product between STS such that \({\cal L}({\cal T}_1\times {\cal T}_2)={\cal L}({\cal T}_1)\cap {\cal L}({\cal T}_2) \) (see Jeannet et al. 2005 for details).

  6. We could have used an extended definition of permissiveness where if two controlled systems have equal reachable state space, inclusion of the transitions that can be fired from reachable states is also taken into account.

  7. In the sequel, for more clarity and conciseness in some examples, we sometimes directly define the system to be controlled by the LTS which corresponds to the STS modelling it.

  8. One can also consider interval (Cousot and Cousot 1977) or octagons (Miné 2001) abstract lattices. Note that Smacs (see Section 7) implements these three lattices

  9. In our algorithm, the S-observability condition holds trivially, because the supervisory function is defined on the observation states.

  10. We shall reuse this example with a different value for t further in the paper.

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Authors and Affiliations

  1. Université Libre de Bruxelles (U.L.B.), Campus Plaine CP 212, Boulevard du Triomphe, 1050, Bruxelles, Belgique

    Gabriel Kalyon, Tristan Le Gall & Thierry Massart

  2. INRIA, centre Rennes - Bretagne Atlantique, Campus Universitaire de Beaulieu, 35042, Rennes Cedex, France

    Hervé Marchand

Authors
  1. Gabriel Kalyon
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  2. Tristan Le Gall
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  3. Hervé Marchand
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  4. Thierry Massart
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Correspondence to Thierry Massart.

Additional information

G. Kalyon is supported by the Belgian National Science Foundation (FNRS) under a FRIA grant. This work has been done in the MoVES project (P6/39) which is part of the IAP-Phase VI Interuniversity Attraction Poles Programme funded by the Belgian State, Belgian Science Policy.

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Kalyon, G., Le Gall, T., Marchand, H. et al. Symbolic Supervisory Control of Infinite Transition Systems Under Partial Observation Using Abstract Interpretation. Discrete Event Dyn Syst 22, 121–161 (2012). https://doi.org/10.1007/s10626-011-0101-3

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