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

arXiv: https://arxiv.org/abs/2203.12467 · HTML: https://mlsanity.com/paper/2203.12467/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2022.

## Built on by (most cited first, 1 of 1)

- [Online variable-length source coding for minimum bitrate LQG control](https://mlsanity.com/paper/2304.00593/) (2023)

## Abstract (arXiv)

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.