Stabilizing Dynamical Systems with Fixed-Rate Feedback using Constrained Quantizers

The stabilization of unstable dynamical systems using rate-limited feedback links is investigated. In the scenario of a constant-rate link and a noise with unbounded support, the fundamental limit of communication is known, but no simple algorithm to achieve it exists. The main challenge in constructing an optimal scheme is to fully exploit the communication resources while occasionally signaling the controller that a special operation needs to be taken due to a large noise observation. In this work, we present a simple and explicit algorithm that stabilizes the dynamical system and achieves the fundamental limits of communication. The new idea is to use a constrained quantizer in which certain patterns of sequences are avoided throughout the quantization process. These patterns are preserved to signal the controller that a zoom-out operation should be initiated due to large noise observation. We show that the constrained quantizer has a negligible effect on the rate, so it achieves the fundamental limit of communication. Specifically, the rate-optimal algorithm is shown to stabilize any β-moment of the state if the noise has a bounded absolute (β +ϵ)-moment for some ϵ > 0 regardless of the other noise characteristics.