Abstract
In this chapter, we extend the state estimation and event inference techniques discussed in earlier chapters to settings where the underlying system might be observed by multiple observation sites with distinct observation capabilities. We focus on the case of a single (monolithic) system that is modeled as a labeled nondeterministic finite automaton and assume that each observation site has its own natural projection observation mapping (i.e., each site has its own set of observable events, some of which may also be observable to other sites). We discuss decentralized observation architectures, i.e., settings in which each observation site operates in isolation but is able to communicate (perhaps periodically or when otherwise prompted) its observations, estimates, or decisions to a coordinator. The coordinator is in charge of forming the final state estimate or making the ultimate decision, and may or may not have knowledge of the system model. We study synchronization-based decentralized estimation/inference protocols, in which the decision to send information to the coordinator is determined by strategies that can be described by finite automata. Apart from describing the information exchange strategies and the run-time executions of the resulting synchronization-based decentralized algorithms, the chapter also discusses their implications on the verification of properties of interest, such as detectability, fault diagnosis, and opacity. In particular, we establish that, under certain choices for these protocols, certain properties of interest (including synchronization-based decentralized diagnosability) can be verified with complexity that is polynomial in the size of the state space of the given system and exponential in the number of observation sites.
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Notes
- 1.
Clearly, no decentralized protocol for state estimation or event inference can exceed the performance of the centralized observation site. In fact, due to loss of information (e.g., absence of timing information between events or the limited information communicated from the observation sites), a decentralized/distributed scheme will likely have inferior performance compared to the performance of centralized state estimation or event inference at the fictitious \(O_c\).
- 2.
We assume that \(R_{\mathbb {M}}(Q', \sigma )\) for any \(\sigma \in (\varSigma _{o_1} \cup \varSigma _{o_2})\) can be obtained with complexity \(|Q|^2\) by precomputing all \(R_{\mathbb {M}}(\{q\}, \sigma )\) for each \(q \in Q\), and then simply taking unions of sets.
- 3.
Note that, under the assumptions in the lemma, it is not possible for \(\alpha _{L_1} \in \varSigma _{o_1} \cap \varSigma _{o_2}\) and \(\beta _{L_2} \in \varSigma _{o_1} \cap \varSigma _{o_2}\), unless \(\alpha _{L_1} = \beta _{L_2}\).
- 4.
Earlier synchronizations might result in all observation sites reporting “U” (in which case no decision can be made at the coordinator) but eventually, at least if the system is co-diagnosable, one diagnoser will report “F” (which will imply that a definite decision “F” can be taken at the coordinator).
- 5.
As mentioned earlier, in Case III decentralized detectability, the decision at each observation site is “detectable” (“D”) or “not detectable” (“ND”), depending on whether that local observation site is able, at that particular instant, to determine the state of the system exactly or not. When a synchronization occurs, the coordinator decides “D” if at least one observation site is reporting a “D”; otherwise, the coordinator decides “ND”. Note that unlike fault diagnosis (where an “F” decision remains an “F” decision due to the absorbing property of the F label—refer to Remark 7.3 of Chap. 7) the decision about the set of state estimates being a singleton set is not absorbing, i.e., it can change from “D” to “ND” and vice versa. This also implies that at different synchronization points, we may have different observation sites that are aware of the exact state of the system; furthermore, in-between synchronization points it is possible that no observation site is aware of the exact state of the system.
- 6.
Recall that we have adopted the usual assumptions that (i) G is live, and (ii) G possesses no unobservable cycles (see, for example, the discussions in Chap. 6).
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Hadjicostis, C.N. (2020). Decentralized State Estimation. In: Estimation and Inference in Discrete Event Systems. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-30821-6_9
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