Quantized Online LQR

A team from Stanford and Caltech has published a breakthrough paper on arXiv titled "Quantized Online LQR," introducing a new algorithm that fundamentally changes how systems can be controlled over communication-constrained networks. The researchers—Barron Han, Victoria Kostina, and Babak Hassibi—address the classic problem of linear-quadratic regulation (LQR) where a plant needs to be controlled by a remote controller with limited bandwidth. Traditional approaches quantize and transmit the plant's raw state at every time step, requiring O(T) total bits over T steps and injecting persistent quantization noise that degrades control performance.

Their novel approach, called the Quantized Certainty Equivalent (QCE-LQR) algorithm, flips this paradigm. Instead of sending raw state data, the plant locally observes its state, estimates the system dynamics using ordinary least squares, and transmits only these learned dynamics estimates over the rate-limited uplink. The remote controller—which knows the control cost—then computes and returns the optimal control policy. This allows the plant to compute control actions locally using its superior state knowledge. The researchers proved this approach is fundamentally optimal: any scheme achieving O(T^α) regret for α ∈ [1/2,1) must transmit at least Ω(log T) bits, and QCE-LQR matches this bound.

The practical implications are significant. In numerical experiments across four benchmark systems—from a simple scalar unstable plant to a complex 24-parameter Boeing 747 lateral model—a variant of QCE-LQR achieved regret comparable to an unquantized certainty equivalent controller over horizons of T=10,000 steps. The regret bound contains inflation factors Q_slow(ϱ) and Q_fast(ϱ) that vanish as codebook resolution increases, smoothly recovering the unquantized baseline performance. This represents orders of magnitude reduction in communication requirements while maintaining control performance, enabling efficient remote control of complex physical systems where bandwidth is limited.