# Interplay Between Transmission Delay, Average Data Rate, and Performance in Output Feedback Control over Digital Communication Channels

## TL;DR

This paper analyzes how transmission delay affects the minimal data rate needed for control performance over digital channels, using information theory and demonstrating near-optimal practical coding schemes.

## Contribution

It introduces an information-theoretic framework to quantify the minimal data rate considering delay, and shows how delay increases the data rate requirement for control.

## Key findings

- Delay increases the data rate needed for desired performance.
- Lower bounds are derived using directed information and LTI filters.
- Practical schemes achieve rates close to theoretical bounds.

## Abstract

The performance of a noisy linear time-invariant (LTI) plant, controlled over a noiseless digital channel with transmission delay, is investigated in this paper. The rate-limited channel connects the single measurement output of the plant to its single control input through a causal, but otherwise arbitrary, coder-controller pair. An infomation-theoretic approach is utilized to analyze the minimal average data rate required to attain the quadratic performance when the channel imposes a known constant delay on the transmitted data. This infimum average data rate is shown to be lower bounded by minimizing the directed information rate across a set of LTI filters and an additive white Gaussian noise (AWGN) channel. It is demonstrated that the presence of time delay in the channel increases the data rate needed to achieve a certain level of performance. The applicability of the results is verified through a numerical example. In particular, we show by simulations that when the optimal filters are used but the AWGN channel (used in the lower bound) is replaced by a simple scalar uniform quantizer, the resulting operational data rates are at most around 0.3 bits above the lower bounds.