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Abstract. Feedback control with limited data rates is an emerging area which incorporates ideas from both control and information theory. A fundamental question it poses is how low the closed loop data rate can be made before a given dynamical system is impossible to stabilize by any coding and control law. Analagously to source coding, this defines the smallest error-free data rate sufficient to achieve "reliable" control, and explicit expressions for it have been derived for linear timeinvariant systems without disturbances. In this paper, the more general case of finite-dimensional linear systems with process and observation noise is considered, the object being mean square state stability. By inductive arguments employing the entropy power inequality of information theory, and a new quantizer error bound, an explicit expression for the infimum stabilizing data rate is derived, under very mild conditions on the initial state and noise probability distributions.Key words. stochastic control, communication theory, source coding, entropy AMS subject classifications. 93E15, 94A05, 94A17, 94A291. Introduction. Communications and control have traditionally been areas with little common ground. For the most part communications theory is concerned with the reliable transmission of information from one point to another, and is relatively indifferent to its specific purpose or whether it is eventually fed back to the source. Control theory, in contrast, is concerned mainly with using information in a feedback loop to achieve some performance objective, and usually assumes that limitations in the communications links are not significant enough to affect performance drastically.The reasons usually given for this mutual indifference are firstly, that a communications system is generally used for a broad range of purposes and can rarely be designed to match a particular objective and, secondly, that to explicitly model communication limitations would complicate controller synthesis. However, in recent years emerging applications such as micro-electromechanical systems, mobile telephone power control, and networked industrial control systems have begun to cross the boundary between these disciplines. In these applications, the aim is to control a dynamical system consisting of many separate components connected by a digital communication network. Although the total available capacity in bits per second may be large, each component is effectively allocated only a small portion. This can introduce significant quantization errors and delays, due to the low resolution and finite transmission time of each discrete-valued, digital symbol. Quantization resolution can be improved at the expense of delay and vice-versa, but nonetheless there remains an upper bound on the amount of information, in some sense, that may be exchanged per unit time. Clearly, by designing coders and decoders that are matched to the dynamical system and controllers, a more economical use of communication

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Abstract-A data rate theorem for stabilization of a linear, discrete-time, dynamical system with arbitrarily large disturbances, over a rate-limited, time-varying communication channel is presented. Necessary and sufficient conditions for stabilization are derived, their implications and relationships with related results in the literature are discussed. The proof techniques rely on both information-theoretic and control-theoretic tools.

In communications, unknown variables are usually modelled as random variables, and concepts such as independence, entropy and information are defined in terms of the underlying probability distributions. In contrast, control theory often treats uncertainties and disturbances as bounded unknowns having no statistical structure. The area of networked control combines both fields, raising the question of whether it is possible to construct meaningful analogues of stochastic concepts such as independence, Markovness, entropy and information without assuming a probability space. This paper introduces a framework for doing so, leading to the construction of a maximin information functional for nonstochastic variables. It is shown that the largest maximin information rate through a memoryless, error-prone channel in this framework coincides with the block-coding zeroerror capacity of the channel. Maximin information is then used to derive tight conditions for uniformly estimating the state of a linear time-invariant system over such a channel, paralleling recent results of Matveev and Savkin.

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It is well known in the field of dynamical systems that entropy can be defined rigorously for completely deterministic open-loop systems. However, such definitions have found limited application in engineering, unlike Shannon's statistical entropy. In this paper, it is shown that the problem of communication-limited stabilization is related to the concept of topological entropy, introduced by Adler et al. as a measure of the information rate of a continuous map on a compact topological space. Using similar open cover techniques, the notion of topological feedback entropy (TFE) is defined in this paper and proposed as a measure of the inherent rate at which a map on a noncompact topological space with inputs generates stability information. It is then proven that a topological dynamical plant can be stabilized into a compact set if and only if the data rate in the feedback loop exceeds the TFE of the plant on the set. By taking appropriate limits in a metric space, the concept of local TFE (LTFE) is defined at fixed points of the plant, and it is shown that the plant is locally uniformly asymptotically stabilizable to a fixed point if and only if the data rate exceeds the plant LTFE at the fixed point. For continuously differentiable plants inEuclidean space, real Jordan forms and volume partitioning arguments are then used to derive an expression for LTFE in terms of the unstable eigenvalues of the fixed point Jacobian.