Algorithms for Optimal Control with Fixed-Rate Feedback

TL;DR

This paper develops a greedy quantization algorithm for fixed-rate networked control of scalar systems, minimizing control cost efficiently and nearly optimally, with extensions to event-triggered control and rate-cost tradeoffs.

Contribution

It introduces a practical greedy quantizer design for fixed-rate control, connecting it to scalar successive refinement and extending to event-triggered scenarios.

Findings

The greedy quantizer nearly matches the globally optimal scheme in performance.

The proposed method is computationally tractable and effective for rate-limited control.

Extensions enable tradeoffs between transmission rate and control cost in event-triggered control.

Abstract

We consider a discrete-time linear quadratic Gaussian networked control setting where the (full information) observer and controller are separated by a fixed-rate noiseless channel. The minimal rate required to stabilize such a system has been well studied. However, for a given fixed rate, how to quantize the states so as to optimize performance is an open question of great theoretical and practical significance. We concentrate on minimizing the control cost for first-order scalar systems. To that end, we use the Lloyd-Max algorithm and leverage properties of logarithmically-concave functions and sequential Bayesian filtering to construct the optimal quantizer that greedily minimizes the cost at every time instant. By connecting the globally optimal scheme to the problem of scalar successive refinement, we argue that its gain over the proposed greedy algorithm is negligible. This is significant since the globally optimal scheme is often computationally intractable. All the results are proven for the more general case of disturbances with logarithmically-concave distributions and rate-limited time-varying noiseless channels. We further extend the framework to event-triggered control by allowing to convey information via an additional "silent symbol", i.e., by avoiding transmitting bits; by constraining the minimal probability of silence we attain a tradeoff between the transmission rate and the control cost for rates below one bit per sample.