通过测距测量进行集合成员定位
Set-Membership Localization via Range Measurements
AI总结：
本文提出了一种基于集合成员方法的定位算法，通过凸优化计算定位集的紧外界集，以保证位置估计的准确性。
摘要：
本文讨论了一个经典的几何问题，即从多个带有噪声的欧几里得距离测量中估计未知点的位置。我们通过集合成员方法来解决这个问题，在此方法中，我们假设距离测量受到未知但有界的误差影响，并且我们表征了所有与测量和其假设误差模型一致的点的集合。这个集合是非凸的，但本文证明该集合包含在由某些闭球和多面体的交集给出的区域中，我们称之为定位集。然后，我们开发了基于凸优化的高效方法，用于计算定位集的紧外界集，该外界集具有简单的结构（一个框或一个椭球），然后作为保证的集值位置估计。% 绑定集的中心也用作位置估计点。相关问题，即通过球体和椭球体对定位集进行内近似，也作为凸优化问题提出。与基于非凸成本最小化问题半正定松弛的现有方法不同，我们的方法是直接的、几何的，并基于一组满足测量方程成对差值的多面体点。
英文摘要：
In this paper we discuss a classical geometrical problem of estimating an unknown point's location in $\Real{n}$ from several noisy measurements of the Euclidean distances from this point to a set of known reference points (anchors). We approach the problem via a set-mem\-ber\-ship methodology, in which we assume the distance measurements to be affected by unknown-but-bounded errors, and we characterize the set of all points that are consistent with the measurements and their assumed error model. This set is nonconvex, but we show in the paper that it is contained in a region given by the intersection of certain closed balls and a polytope, which we call the {\em localization set}. Then, we develop efficient methods, based on convex programming, for computing a tight outer-bounding set of simple structure (a box, or an ellipsoid) for the localization set, which then acts as a guaranteed set-valued location estimate. % The center of the bounding set also serves as a point location estimate. Related problems of inner approximation of the localization set via balls and ellipsoids are also posed as convex programming problems. Different from existing methods based on semidefinite programming relaxations of a nonconvex cost minimization problem, our approach is direct, geometric and based on a polyhedral set of points that satisfy pairwise differences of the measurement equations.
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