Abstract
In this chapter, we extend the decentralized state estimation and event inference techniques discussed in Chap. 9 to distributed settings. We again consider an underlying monolithic system that is modeled as a labeled nondeterministic finite automaton and is observed by multiple observation sites with different observation capabilities (i.e., each site has its own set of observable events). The difference is that we focus on distributed observation architectures, i.e., settings in which each observation site cannot only send its observations, estimates, or decisions to other sites or to a coordinator (if one is present), but it is also able to receive and process similar information from other observation sites (or the coordinator). This should be contrasted with the decentralized protocols of Chap. 9 where the only entity that could receive information was the coordinator. We consider synchronization-based protocols, generalized to apply to such distributed settings, and describe the information exchange strategies and the corresponding run-time executions of the resulting distributed state estimation and event inference algorithms. The chapter also discusses the implications of these distributed protocols on the verification of properties of interest, such as detectability, opacity, and fault diagnosis. In particular, under some choices, certain properties of interest, such as synchronization-based distributed diagnosability which is discussed explicitly, 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.
We will see (also via Example 10.2) later in this chapter that this renders the distributed approach more powerful than the decentralized one.
- 2.
In a directed communication topology, a node cannot necessarily communicate with its in-neighbors (not unless they are also out-neighbors); however, there are many applications where a node might be able to send some simple signal to in-neighboring nodes (to request information from them). We also discuss synchronization steps in the case when it is not possible for a node to request information from in-neighboring nodes.
- 3.
This information could be, for instance, the state estimates obtained by the coordinator or the totally ordered sequences that match the partially ordered sequences that have been reported at the coordinator by the various observation sites.
- 4.
In fault diagnosis applications, we may have situations where one observation site is certain about the “fault” (or “no fault”) condition; in such scenarios, a synchronization in Case III distributed state estimation (but not in Case III decentralized state estimation) will enable other observation sites that may be “uncertain” to realize the presence (or absence) of a fault. Clearly, this would influence their subsequent operation (e.g., they only need to retain all estimates associated with the given condition), but this is irrelevant in the sense that the decision (“fault” or “no fault”) can already be made.
- 5.
Technically, in order to reach this conclusion, we need to also verify that the fault will be detected for all other executions that contain a fault (i.e., any execution of the form \(edabfd(b+c)^*\)).
- 6.
In other words, there might not necessarily exist a sequence of events s that generates a projection \(P_i(s)\) which drives the local diagnoser \(D_i\) to this set of state estimates.
- 7.
In the set Sync, event \(sync_i\) is the same as event \(sync_{\{ i \}}\).
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Hadjicostis, C.N. (2020). Distributed 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_10
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