A Lower-bound for Variable-length Source Coding in Linear-Quadratic-Gaussian Control with Shared Randomness

TL;DR

This paper establishes fundamental lower bounds on the communication rate needed for variable-length source coding in LQG control systems with shared randomness, considering prefix-free constraints and rate-distortion trade-offs.

Contribution

It derives new lower bounds on communication costs in LQG control with shared randomness and variable-length coding, extending previous bounds to relaxed prefix constraints.

Findings

Lower bounds expressed via directed information.

Bound holds with shared randomness and prefix-free constraints.

Generalization to relaxed prefix-free coding and rate-distortion analysis.

Abstract

In this letter, we consider a Linear Quadratic Gaussian (LQG) control system where feedback occurs over a noiseless binary channel and derive lower bounds on the minimum communication cost (quantified via the channel bitrate) required to attain a given control performance. We assume that at every time step an encoder can convey a packet containing a variable number of bits over the channel to a decoder at the controller. Our system model provides for the possibility that the encoder and decoder have shared randomness, as is the case in systems using dithered quantizers. We define two extremal prefix-free requirements that may be imposed on the message packets; such constraints are useful in that they allow the decoder, and potentially other agents to uniquely identify the end of a transmission in an online fashion. We then derive a lower bound on the rate of prefix-free coding in terms of directed information; in particular we show that a previously known bound still holds in the case with shared randomness. We generalize the bound for when prefix constraints are relaxed, and conclude with a rate-distortion formulation.