Abstract
We propose algorithms for the synthesis of decentralized state-feedback controllers with partial observation of infinite state systems, which are modeled by Symbolic Transition Systems. We first consider the computation of safe controllers ensuring the avoidance of a set of forbidden states and then extend this result to the deadlock free case. The termination of the algorithms solving these problems is ensured by the use of abstract interpretation techniques, but at the price of overapproximations, in particular, in the computation of the states which must be avoided. We then extend our algorithms to the case where the system to be controlled is given by a collection of subsystems (modules). This structure is exploited to locally compute a controller for each module. Our tool SMACS gives an empirical evaluation of our methods by showing their feasibility, usability and efficiency.
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Notes
Note that in Jeannet et al. (2005), this alphabet is assumed to be infinite.
For convenience, in the guards and update functions of the transitions of the system, we omit the conditions and assignments related to the locations. For example, the transition δ 9 is defined by \(\langle{Choice\_X, \mathsf{T}, \mathsf{Id}}\rangle\), whereas it should be defined by \(\langle{Choice\_X, l = \mathsf{Choice}; l := PX}\rangle\).
To remain coherent with the formalization of the state space \(\mathcal{D}_{{V}}\), we have chosen to define the observation space \(\mathcal{D}_{Obs}\) by means of a variable Obs whose domain is \(\mathcal{D}_{Obs}\). In particular, it allows us to use predicate transformers w.r.t. this variable.
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.
Making a parallel with the classical language-based approach, the language L Bad generated by the system from which the set of states Bad has been removed is not controllable w.r.t. the language L of the system, whereas the one generated by the system to which I(Bad) has been removed is actually the largest controllable sub-language of L Bad w.r.t. L. Note that none of these languages is regular.
Roughly, a widening operator tries to guess the limit of an ascending sequence of elements of the abstract domain in a finite number of steps (see Cousot and Cousot 1977).
Recall that \(\widehat{\mathcal F}^{\mathcal T}_{i}(\sigma, B)\) gives the set of states for which σ must be forbidden by the controller \(\mathcal C_i\) to prevent B to be reached. We use \(\widehat{\mathcal F}^{\mathcal T}_{i}\) instead of \(\mathcal F^{\mathcal T}_{i}\), because the first function can be computed locally unlike the second one.
Note that in this situation, the control architecture (i.e the fusion rule) that we considered in this paper is no longer valid.
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Gabriel 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. Decentralized Control of Infinite Systems. Discrete Event Dyn Syst 21, 359–393 (2011). https://doi.org/10.1007/s10626-011-0106-y
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DOI: https://doi.org/10.1007/s10626-011-0106-y