Sensing-Limited Control of Noiseless Linear Systems Under Nonlinear Observations
Ming Li, Fan Liu, Yifeng Xiong, Jie Xu, Tao Liu
TL;DR
This work addresses the fundamental limits of sensing-driven control for noiseless linear systems observed through nonlinear channels. It introduces directed information from the unstable state to the observation sequence as a central metric and derives a necessary condition requiring the average directed information rate to exceed the instability expansion rate $R_{exp}= obreak obracket obreak obreak obreak obreak obreak$ R_{exp} = obreak obreak imes obreak $= log_2 | oldsymbol{ ext{A}_u}| |$ (formatted), ensuring stabilization requires information flow to counteract entropy growth. Under log-concavity and regularity assumptions, the paper proves a sufficiency condition: if the directed information rate strictly exceeds $R_{exp}$, asymptotic mean-square observability and stabilizability are achieved, with posterior error covariances vanishing under certainty-equivalent control. The results extend classical data-rate limits from linear-Gaussian settings to nonlinear sensing models, clarifying how the sensing layer constrains achievable performance in control loops and providing principled guidelines for sensing-aware controller design.
Abstract
This paper investigates the fundamental information-theoretic limits for the control and sensing of noiseless linear dynamical systems subject to a broad class of nonlinear observations. We analyze the interactions between the control and sensing components by characterizing the minimum information flow required for stability. Specifically, we derive necessary conditions for mean-square observability and stabilizability, demonstrating that the average directed information rate from the state to the observations must exceed the intrinsic expansion rate of the unstable dynamics. Furthermore, to address the challenges posed by non-Gaussian distributions inherent to nonlinear observation channels, we establish sufficient conditions by imposing regularity assumptions, specifically log-concavity, on the system's probabilistic components. We show that under these conditions, the divergence of differential entropy implies the convergence of the estimation error, thereby closing the gap between information-theoretic bounds and estimation performance. By establishing these results, we unveil the fundamental performance limits imposed by the sensing layer, extending classical data-rate constraints to the more challenging regime of nonlinear observation models.