Anytime Reliable Codes for Stabilizing Plants over Erasure Channels

The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in problems of distributed control and networked control systems. Although the work of Schulman and Sahai over the past two decades, and their development of the notions of “tree codes” and “anytime capacity”, provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one needs real-time encoding and real-time decoding and a reliability which increases exponentially with delay, which is what tree codes guarantee. We prove the existence of linear tree codes with high probability and, for erasure channels, give an explicit construction with an expected encoding and decoding complexity that is constant per time instant. We give sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight. This work takes a major step towards controlling plants over noisy channels, and we demonstrate the efficacy of the method through several examples.

💡 Research Summary

The paper addresses the fundamental problem of stabilizing an unstable linear plant when the sensor‑to‑controller communication link is a noisy, rate‑limited channel. Classical information theory guarantees reliable transmission only with arbitrarily large delays, which is unacceptable for feedback control because delay directly translates into instability. The authors therefore focus on “anytime” codes—coding schemes whose decoding error probability decays exponentially with the decoding delay—together with real‑time (causal) encoding and decoding.

The main contributions are threefold. First, they prove that linear tree‑like codes exist with high probability. By imposing a Toeplitz (time‑invariant) structure on the parity‑check matrix and drawing its entries i.i.d. from a Bernoulli(p) distribution, they define an ensemble TZₚ. For any rate R and exponent β satisfying
 R < 1 – log₂(1+ζ) and
 β < H⁻¹(1–R)·