State estimation via limited capacity noisy communication channels

Abstract

The paper addresses a state estimation problem involving communication errors and capacity constraints. Discrete-time partially observed linear systems perturbed by stochastic unbounded additive disturbances are studied. Unlike the classic theory, the sensor signals are communicated to the estimator over a limited capacity noisy digital link modeled as a stochastic discrete memoryless channel. It is shown that the capability of the noisy channel to ensure state estimation with a bounded in probability error is identical to its capability to transmit information with as small probability of error as desired. In other words, the classic Shannon capacity of the channel constitutes the boundary of the observability domain. It is shown that whenever the Shannon capacity bound is met, a reliable observation can be ensured by means of a state estimator consuming a bounded (as time progresses) computational complexity and memory per unit time. The corresponding state estimator is constructed explicitly and is based on the classic block coding approach, so that traditional block encoding-decoding procedures can be employed for its implementation.

Design of Information Channels for Optimization and Stabilization in Networked Control

“…Particularly related references include [11, 28-30, 32, 37, 42, 43, 45, 46, 54, 55, 59]. In the context of discrete channels, many of these papers considered a bounded noise assumption, except notably [30,36,37,55,64], and [56]. We refer the reader to [32] and [60] for a detailed literature review.…”

Stochastic stability and ergodicity of linear Gaussian systems controlled over discrete channels

“…[25] considered systems driven by bounded noise and considered a number of stability criteria: Almost sure stability for noise-free systems, moment stability for systems with bounded noise (lim sup t→∞ E[|x t | p ] < ∞,) as well as stability in probability (defined in [18]) for systems with bounded noise. Stability in probability is defined as follows: For every p > 0, there exists a ζ such that P (|x t | > ζ) < p for all t ∈ N. [25] also offered a novel and insightful characterization for reliability for controlling unstable processes, named, any-time capacity, defined for the following criterion: lim sup t→∞ E[|x t | p ] < ∞, for positive moments p. In a related context, [15], [25], [18] and [17] considered the relevance to Shannon capacity. [15] observed that when the moment coefficient goes to zero, Shannon capacity is the right measure for a channel when noise is bounded.…”

“…A parallel argument is provided by [25] also for bounded noise signals. With a departure from the bounded noise assumption, [17], made the discussion in [25] more explicit and considered a more general model of multi-dimensional systems driven by an unbounded noise process considering again stability in probability. [17] also showed that when the discrete noisy channel has capacity less than log 2 (|a|), there exists no stabilizing scheme, and if the capacity is strictly greater than this number, there exists a stabilizing scheme.…”

“…⋄ For the proof, we consider the following update algorithm. The algorithm and its variations have been used in source coding and networked control literature: See for example the earlier papers [8], [14], [4] (zooming algorithms); and more recent ones [23], [19], [18], [17]. Let n be a given block length.…”

Exponential state estimation, entropy and Lyapunov exponents

“…The possibility of such an estimation is crucial for many control tasks. This prob-lem was studied in [27,30] for linear systems in a stochastic framework with the objective to bound the estimation error in probability. Here the well-known criterion (known as the datarate theorem) was obtained, which states that the critical channel capacity is given by the sum of the unstable eigenvalues of the dynamical matrix.…”