Distributed State Estimation for Hybrid and Discrete Event Systems Using l-Complete Approximations

Abstract

The topic of this paper is distributed state estimation for time-invariant systems with finite input and output spaces. We assume that the system under investigation can be realised by a hybrid I/S/O-machine, where some of the discrete states may also represent failure modes. Our approach is based on previous work, e.g., Moor and Raisch (1999); Moor et al. (2002), where l-complete approximations were proposed as discrete event abstractions for hybrid dynamical systems. In particular, it has been shown that l-complete approximations can be used to provide set-valued estimates for the unknown system state. Estimates are conservative in the sense that the true state can be guaranteed to be contained in the set-valued estimate. In this contribution, we show that for a class of hybrid systems the same estimate can be obtained via a distributed, or decentralised, approach involving several less complex approximations, which are run in parallel. For a larger class of systems, it can be shown that this approach provides an outer approximation of the estimate provided by a monolithic l-complete estimator. The proposed procedure implies significant computational savings during estimator synthesis, with an only modest increase in on-line effort. The latter is a result of "assembling" the global estimate from the available local estimates. The resulting computational trade-off is explicitly discussed.

Comparing asynchronous l-complete approximations and quotient based abstractions

“…We now investigate how to use this set as a state space in the construction of different state machine realizations of the SAlCA of a system Σ. This is be done on the basis of a state machine realization Q of Σ satisfying (6). For this, we first investigate how a string ζ ∈ Π l (B(Q)) can correspond to a state x ∈ X of Q.…”

“…Definition 6. Given (6) and (11), let Σ S = (N 0 , W × X, B S (Q)) be a dynamical system, where B S (Q) is the extension of π W ×X (B f (Q)) to Z as discussed in Sec. II.…”

Decentralized set-valued state estimation based on non-deterministic chains

“…-Complete approximation . We consider here the realization algorithm for the strongest -complete abstraction Σ as follows (see also [4]).…”

“…The model Σ = (N 0 , W, B ), ∈ N, is a strongest -complete approximation of Σ if (i) B is -complete; (ii) B ⊇ B, and (iii) B ⊇ B, B being -complete ⇒ B ⊇ B . We consider here the realization algorithm for the strongest -complete abstraction Σ as follows (see also [4]).…”

“…A simple algorithm is developed based on the concept of so-called "non-deterministic chains", which represent a special class of transition relations featuring inherent injective transition functions. Our computational framework relates to the work [4] which focuses on distributed setvalued state estimation for I/S/O machines. Yet it includes additional perspectives of decentralized computation.…”