Online Variable-Length Source Coding for Minimum Bitrate LQG Control

We propose an adaptive coding approach to achieve linear-Quadratic-Gaussian (LQG) control with near-minimum bitrate prefix-free feedback. Our approach combines a recent analysis of a quantizer design for minimum rate LQG control with work on universal lossless source coding for sources on countable alphabets. It was recently demonstrated that the aforementioned quantizer's outputs are an asymptotically stationary, ergodic process. To enable LQG control with provably near-minimum bitrate, the quantizer outputs must be encoded into binary codewords efficiently. This is possible given knowledge of the quantizations' probability distributions, or of their limiting distribution. Obtaining such knowledge is challenging; the distributions do not readily admit closed form descriptions. This motivates the application of universal source coding. Our main theoretical contribution in this work is a proof that (after an invertible transformation), the quantizer outputs are random variables that fall within an exponential or power-law envelope class (depending on the plant dimension). Using ideas from universal coding on envelope classes, we develop a practical, zero-delay, fixed precision source code for the quantizer outputs. We evaluate the performance of this approach numerically, and demonstrate competitive results with respect to fundamental tradeoffs between bitrate and LQG control performance.