This paper studies discrete-time control systems subject to average data-rate limits. We focus on a situation where a noisy linear system has been designed assuming transparent feedback and, due to implementation constraints, a source-coding scheme (with unity signal transfer function) has to be deployed in the feedback path. For this situation, and by focusing on a class of source-coding schemes built around entropy coded dithered quantizers, we develop a framework to deal with average data-rate constraints in a tractable manner that combines ideas from both information and control theories. As an illustration of the uses of our framework, we apply it to study the interplay between stability and average data-rates in the considered architecture. It is shown that the proposed class of coding schemes can achieve mean square stability at average data-rates that are, at most, 1.254 bits per sample away from the absolute minimum rate for stability established by Nair and Evans. This rate penalty is compensated by the simplicity of our approach.

We improve the existing achievable rate regions for causal and for zero-delay source coding of stationary Gaussian sources under an average mean squared error (MSE) distortion measure. To begin with, we find a closed-form expression for the information-theoretic causal rate-distortion function (RDF) under such distortion measure, denoted by R , where R(D) denotes Shannon's RDF. Two of these bounds are strictly smaller than 0.5 bits/sample at all rates. These bounds differ from one another in their tightness and ease of evaluation; the tighter the bound, the more involved its evaluation.We then show that, for any source spectral density and any positive distortion D ≤ σ 2 x , R it c (D) can be realized by an AWGN channel surrounded by a unique set of causal pre-, post-, and feedback filters. We show that finding such filters constitutes a convex optimization problem. In order to solve the latter, we propose an iterative optimization procedure that yields the optimal filters and is guaranteed to converge to R it c (D). Finally, by establishing a connection to feedback quantization we design a causal and a zerodelay coding scheme which, for Gaussian sources, achieves an operational rate lower than R it c (D)+0.254

This paper studies networked control systems closed over noiseless digital channels. By focusing on noisy LTI plants with scalar-valued control inputs and sensor outputs, we derive an absolute lower bound on the minimal average data rate that allows one to achieve a prescribed level of stationary performance under Gaussianity assumptions. We also present a simple coding scheme that allows one to achieve average data rates that are at most 1.254 bits away from the derived lower bound, while satisfying the performance constraint. Our results are given in terms of the solution to a stationary signal-to-noise ratio minimization problem and builds upon a recently proposed framework to deal with average data rate constraints in feedback systems. A numerical example is presented to illustrate our findings. Index TermsNetworked control systems; optimal control; average data rate; signal-to-noise ratio.

We improve the existing achievable rate regions for causal and for zero-delay source coding of stationary Gaussian sources under an average mean squared error (MSE) distortion measure. To begin with, we find a closed-form expression for the information-theoretic causal rate-distortion function ( , where R(D) denotes Shannon's RDF. Two of these bounds are strictly smaller than 0.5 bits/sample at all rates. These bounds differ from one another in their tightness and ease of evaluation; the tighter the bound, the more involved its evaluation.We then show that, for any source spectral density and any positive distortion D ≤ σ 2 x , R it c (D) can be realized by an AWGN channel surrounded by a unique set of causal pre-, post-, and feedback filters. We show that finding such filters constitutes a convex optimization problem. In order to solve the latter, we propose an iterative optimization procedure that yields the optimal filters and is guaranteed to converge to R it c (D). Finally, by establishing a connection to feedback quantization we design a causal and a zerodelay coding scheme which, for Gaussian sources, achieves an operational rate lower than R it c (D)+0.254 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.

This paper presents novel results on perfect reconstruction feedback quantizers (PRFQs), i.e., noise-shaping, predictive and sigma-delta A/D converters whose signal transfer function is unity. Our analysis of this class of converters is based upon an additive white noise model of quantization errors. Our key result is a formula that relates the minimum achievable MSE of such converters to the signal-to-noise ratio (SNR) of the scalar quantizer embedded in the feedback loop. This result allows us to obtain analytical expressions that characterize the corresponding optimal filters. We also show that, for a fixed SNR of the scalar quantizer, the end-to-end MSE of an optimal PRFQ which uses the optimal filters (which for this case turn out to be IIR) decreases exponentially with increasing oversampling ratio. Key departures from earlier work include the fact that fed back quantization noise is explicitly taken into account and that the order of the converter filters is not a priori restricted.