We study localization and control for unstable systems under coarse, single-bit sensing. Motivated by understanding the fundamental limitations imposed by such minimal feedback, we identify sufficient conditions under which the initial state can be recovered despite instability and extremely sparse measurements. Building on these conditions, we develop an active localization algorithm that integrates a set-based estimator with a control strategy derived from Voronoi partitions, which provably estimates the initial state while ensuring the agent remains in informative regions. Under the derived conditions, the proposed approach guarantees exponential contraction of the initial-state uncertainty, and the result is further supported by numerical experiments. These findings can offer theoretical insight into localization in robotics, where sensing is often limited to coarse abstractions such as keyframes, segmentations, or line-based features.

翻译：本研究探讨在粗糙单比特传感条件下的非稳定系统定位与控制问题。受理解此类最小反馈所施加根本限制的动机驱动，我们识别出在非稳定性和极端稀疏测量条件下仍能恢复初始状态的充分条件。基于这些条件，我们开发了一种主动定位算法，该算法将基于集合的估计器与源自Voronoi剖分的控制策略相结合，能够在保证智能体始终处于信息丰富区域的同时，可证明地估计初始状态。在推导所得条件下，所提方法可保证初始状态不确定性的指数收缩，数值实验进一步验证了该结果。这些发现可为机器人学中的定位问题提供理论洞见，因为在机器人领域，传感通常局限于关键帧、分割或基于线特征等粗糙抽象信息。