# Rate of Prefix-free Codes in LQG Control Systems

arXiv: https://arxiv.org/abs/1604.01227 · HTML: https://mlsanity.com/paper/1604.01227/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2016.

## Built on by (most cited first, 2 of 2)

- [Time-invariant Prefix Coding for LQG Control](https://mlsanity.com/paper/2204.00588/) (2022)
- [Online variable-length source coding for minimum bitrate LQG control](https://mlsanity.com/paper/2304.00593/) (2023)

## Abstract (arXiv)

In this paper, we consider a discrete time linear quadratic Gaussian (LQG) control problem in which state information of the plant is encoded in a variable-length binary codeword at every time step, and a control input is determined based on the codewords generated in the past. We derive a lower bound of the rate achievable by the class of prefix-free codes attaining the required LQG control performance. This lower bound coincides with the infimum of a certain directed information expression, and is computable by semidefinite programming (SDP). Based on a technique by Silva et al., we also provide an upper bound of the best achievable rate by constructing a controller equipped with a uniform quantizer with subtractive dither and Shannon-Fano coding. The gap between the obtained lower and upper bounds is less than 0.754r+1 bits per time step regardless of the required LQG control performance, where r is the rank of a signal-to-noise ratio matrix obtained by SDP, which is no greater than the dimension of the state.