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. In the aforementioned quantizer design, it was established that the quantizer 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 probability distributions of the quantizer outputs, 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 version of these algorithms that operates with fixed precision arithmetic. We evaluate the performance of this algorithm numerically, and demonstrate competitive results with respect to fundamental tradeoffs between bitrate and LQG control performance.

翻译：我们提出一种自适应编码方法，通过接近最小比特率的前缀自由反馈实现线性二次型高斯(LQG)控制。该方法将近期针对最小速率LQG控制的量化器设计与可数字母表上通用无损源编码的研究相结合。在上述量化器设计中，已证明量化器输出是渐近平稳遍历过程。为实现可证明近最小比特率的LQG控制，需将量化器输出高效编码为二进制码字。若已知量化器输出的概率分布或其极限分布，即可实现该目标。然而获取此类分布极具挑战性——这些分布难以获得闭合表达式。这促使我们引入通用源编码。本文的主要理论贡献在于证明（经可逆变换后）量化器输出属于指数或幂律包络类随机变量（取决于被控对象维度）。基于包络类通用编码思想，我们开发了采用定点精度的实用零延迟算法。通过数值实验评估算法性能，验证了其在比特率与LQG控制性能基本权衡关系上的竞争性表现。