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.

翻译：本文研究了在广泛非线性观测类别下，无噪声线性动力系统的控制与感知的基本信息论极限。通过刻画稳定性所需的最小信息流，我们分析了控制与感知组件之间的相互作用。具体而言，我们推导了均方可观性与可镇定性所需的必要条件，证明从状态到观测的平均定向信息率必须超过不稳定动力学的内在扩张率。此外，为应对非线性观测通道固有的非高斯分布带来的挑战，我们通过对系统概率分量施加正则性假设（具体为对数凹性）建立了充分条件。我们证明，在这些条件下，微分熵的发散意味着估计误差的收敛，从而弥合了信息论界限与估计性能之间的差距。通过建立这些结果，我们揭示了感知层所施加的基本性能极限，将经典数据率约束推广至更具挑战性的非线性观测模型范畴。