Abstract

In the context of control and estimation under information constraints, restoration entropy measures the minimal required data rate above which the state of a system can be estimated so that the estimation quality does not degrade over time and, conversely, can be improved. The remote observer here is assumed to receive its data through a communication channel of finite bit-rate capacity. In this paper, we provide a new characterization of the restoration entropy which does not require to compute any temporal limit, i.e., an asymptotic quantity. Our new formula is based on the idea of finding a specific Riemannian metric on the state space which makes the metric-dependent upper estimate of the restoration entropy as tight as one wishes.

Highlights

  • Recent decades have witnessed a substantially growing attention to networked control systems [16, 15] and related problems of control and/or state estimation via communication channels with constrained bit-rates; for extended surveys of this area, we refer the reader to [28, 44, 1] and references therein

  • A minimal threshold of this rate, which still ensures that the remote observer is able to keep track of the system, is the quantity one likes to evaluate in a constructive manner

  • The communication rate between the system and the observer has to exceed the rate at which the system “generates information”, while the latter concept is classically formalized in a form of entropy-like characteristic of the dynamical system at hands

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Summary

Introduction

Recent decades have witnessed a substantially growing attention to networked control systems [16, 15] and related problems of control and/or state estimation via communication channels with constrained bit-rates; for extended surveys of this area, we refer the reader to [28, 44, 1] and references therein. [33, 28, 30] and references therein) - their various versions coexist to handle various kinds of observability and models of both the plant and the constrained communication channel Those results deliver a consistent message that the concept of the topological entropy (TE) of the system and its recent offshoots provide the figure-of-merit needed to evaluate the channel capacity for control applications; the mentioned modifications of TE are partly aimed to properly respond to miscellaneous phenomena crucial for control problems, like uncertainties in the observed system [38, 22, 23], implications of control actions [9, 13, 8, 37], the decay rate of the estimation error [26], or Lipschitz-like relations between the exactness of estimation and the initial state uncertainty [30].

State estimation via limited bit-rate communication and restoration entropy
Main ideas and results
Discrete-time systems
Continuous-time systems
Example
Proposition The following statements hold
Proposition The following relations hold:
Derivatives of singular values and other matrix functionals
Linear cocycles
D Technical conclusions from the proofs
Full Text
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