Milan S. Derpich

A Framework for Control System Design Subject to Average Data-Rate Constraints

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.

On Optimal Perfect Reconstruction Feedback Quantizers

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.

UTILIZING PRIOR KNOWLEDGE IN ROBUST OPTIMAL EXPERIMENT DESIGN

Abstract In this paper we propose a new approach to robust optimal experiment design. The key departure from earlier work is that we specifically account for the fact that, prior to the experiment, we possess only partial knowledge of the system. We also give a detailed analysis of the solution for a simple case and propose a concave optimization algorithm that can be applied more generally.

Simple coding for achieving mean square stability over bit-rate limited channels

The problem of characterizing lower bounds on data-rates needed for closed loop stability has been solved in a variety of settings. However, the available results lead to coding schemes which are very complex and, thus, of limited practical interest. In this paper, we show how simple coding systems comprising only LTI filters and memoryless entropy coded dithered scalar quantizers can be used to stabilize strongly stabilizable SISO LTI plant models over error-free bit-rate limited feedback channels. Despite the simplicity of the building blocks employed, we prove that the data-rates incurred do not exceed absolute lower bounds by more than 1.25 bits per sample.

An Achievable Data-Rate Region Subject to a Stationary Performance Constraint for LTI Plants

This note studies the performance of control systems subject to average data-rate limits. We focus on a situation where a noisy LTI 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 specific source coding scheme, we give a closed-form upper bound on the minimal average data-rate that allows one to attain a given performance level. Instrumental to our main result is the explicit solution of a related signal-to-noise ratio minimization problem, subject to a closed loop performance constraint.

Wireless Access Channels with Near-Ground Level Antennas

In this work we present an empirical study of the added propagation losses that may be associated with providing fixed wireless service from near-ground base-stations to homes in a suburban environment. We present results for various types of environments, classified according to the existence of obstructions in the propagation path and the choice of outdoor-outdoor or outdoor-indoor service. Our results indicate that while on average the additional path-losses associated with lowering the base antenna are relatively small, the variance of these losses will increase at near-ground level, particularly in obstructed links. This has as a result that the power margin required for high availability of a near-ground base antenna may be quite significant.

The Quadratic Gaussian Rate-Distortion Function for Source Uncorrelated Distortions

We characterize the rate-distortion function for zero-mean stationary Gaussian sources under the MSE fidelity criterion and subject to the additional constraint that the distortion is uncorrelated to the input. The solution is given by two equations coupled through a single scalar parameter. This has a structure similar to the well known water-filling solution obtained without the uncorrelated distortion restriction. Our results fully characterize the unique statistics of the optimal distortion. We also show that, for all positive distortions, the minimum achievable rate subject to the uncorrelation constraint is strictly larger than that given by the un-constrained rate-distortion function. This gap increases with the distortion and tends to infinity and zero, respectively, as the distortion tends to zero and infinity.

A Characterization of the Minimal Average Data Rate That Guarantees a Given Closed-Loop Performance Level

This paper studies networked control systems closed over noiseless digital channels. We focus on noisy linear time-invariant (LTI) plants with stationary Gaussian disturbances, Gaussian initial state, scalar-valued control inputs and sensor outputs. For this set-up, we show that the absolute minimal directed information rate that allows one to achieve a prescribed level of performance (not necessarily stationary), over all combinations of encoder-controller-decoder, is achieved when the decoder output is jointly Gaussian with the other signals in the system. This directed information rate lower bounds the achievable operational data rates. When restricting our attention to encoder-controller-decoders which make the random processes in the loop (strongly) asymptotically wide-sense stationary, this bound can be expressed in terms of their asymptotic power spectral densities. Then we show that the directed information rate and stationary performance of any such scheme can be achieved when the concatenated encoder, channel, controller and decoder behave as an AWGN channel with LTI filters. We also present a simple coding scheme that allows one to achieve (operational) average data rates that are at most (approximately) 1.254 bits away from the derived lower bound, while satisfying the performance constraint. A numerical example is presented to illustrate our findings.

On the Minimal Average Data-Rate that Guarantees a Given Closed Loop Performance Level

Abstract This paper deals with control system design subject to average data-rate constraints. By focusing on SISO LTI plants, and a class of source coding schemes, we establish lower and upper bounds on the minimal average data-rate needed to achieve a prescribed performance level. We also guarantee the existence of a specific source coding scheme, within the proposed class, that achieves the desired performance level at average data-rates below our upper bound. Our results are based upon a recently proposed framework to address control problems subject to average data-rate constraints.

Improved upper bounds to the causal quadratic rate-distortion function for Gaussian stationary sources

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 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_c^it(D), for first-order Gauss-Markov processes. R_c^it(D) is a lower bound to the optimal performance theoretically attainable (OPTA) by any causal source code, namely R_c^op(D). We show that, for Gaussian sources, the latter can also be upper bounded as R_c^op(D) ≤ R_c^it(D) + 0.5 log ₂(2πe) bits/sample. In order to analyze R_c^it(D) for arbitrary zero-mean Gaussian stationary sources, we introduce R_c^it̅(D), the information-theoretic causal RDF when the reconstruction error is jointly stationary with the source. Based upon R_c^it̅(D), we derive three closed-form upper bounds to the additive rate loss defined as R_c^it̅(D) - R(D), where R(D) denotes Shannons 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_x², RU(D) can be realized by an additive white Gaussian noise channel surrounded by a unique set of causal pre-, post-, and feed- back niters. 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 niters and is guaranteed to converge to R_c^it̅(D). Finally, by establishing a connection to feedback quantization, we design a causal and a zero-delay coding scheme which, for Gaussian sources, achieves an operational rate lower than R_c^it̅(D) +0.254 and R_c^it̅(D) + 0.754 bits/sample, respectively. This implies that the OPTA among all zero-delay source codes, denoted by R_zd^op(D), is upper bounded as R_zd^op(D) <; R_c^it̅(D) + 1-254 <; R(D) + 1.754 bits/sample.