Exact minimum number of bits to stabilize a linear system
This work provides the exact necessary and sufficient condition for stabilizing an unstable linear system under a rate-limited feedback channel, solving a fundamental problem in control theory.
The paper proves that for an unstable scalar linear stochastic system, the minimum number of control messages per time step needed for moment stability is exactly the integer part of the system gain plus one. This result is shown to generalize to vector systems, dependent noise, and message loss.
We consider an unstable scalar linear stochastic system, $X_{n+1}=a X_n + Z_n - U_n$, where $a \geq 1$ is the system gain, $Z_n$'s are independent random variables with bounded $α$-th moments, and $U_n$'s are the control actions that are chosen by a controller who receives a single element of a finite set $\{1, \ldots, M\}$ as its only information about system state $X_i$. We show new proofs that $M > a$ is necessary and sufficient for $β$-moment stability, for any $β< α$. Our achievable scheme is a uniform quantizer of the zoom-in / zoom-out type that codes over multiple time instants for data rate efficiency; the controller uses its memory of the past to correctly interpret the received bits. We analyze its performance using probabilistic arguments. We show a simple proof of a matching converse using information-theoretic techniques. Our results generalize to vector systems, to systems with dependent Gaussian noise, and to the scenario in which a small fraction of transmitted messages is lost.