距离测量全局渐近有效定位
Global and Asymptotically Efficient Localization from Range Measurements
摘要：
我们认为基于距离的定位问题，其中涉及到估计一个对象的位置，通过使用$m$传感器，希望随着传感器的数量$m$的增加，估计收敛到真实的位置与最小的方差。在一定的传感器配置和测量噪声条件下，证明了LS估计的强相合性和渐近正态性。然而，LS问题是非光滑和非凸的，因此很难解决。然后，我们设计可实现的估计，具有相同的渐近性质的LS之一。这些估计器基于两步估计架构，即任何$\sqrt{m}$一致的估计，然后进行一步高斯-牛顿迭代，可以产生具有与LS相同的渐近性质的解决方案。两步法的关键是在第一步中构造一个$\sqrt{m}$ -相容估计。根据测量噪声方差是否已知，分别提出了Bias-Eli估计（涉及求解广义信赖域子问题）和Noise-Est估计（通过求解凸问题得到）.证明了它们是$\sqrt{m}$ -相容的。此外，我们表明，通过丢弃上述两个优化问题中的约束，得到的封闭形式的估计（称为偏差Eli-Lin和噪声Est-Lin）也是$\sqrt{m}$ -一致的。大量的仿真结果验证了我们的理论主张的正确性，表明所提出的两步估计可以渐近达到Cramer-Rao下界。
英文摘要：
We consider the range-based localization problem, which involves estimating an object's position by using $m$ sensors, hoping that as the number $m$ of sensors increases, the estimate converges to the true position with the minimum variance. We show that under some conditions on the sensor deployment and measurement noises, the LS estimator is strongly consistent and asymptotically normal. However, the LS problem is nonsmooth and nonconvex, and therefore hard to solve. We then devise realizable estimators that possess the same asymptotic properties as the LS one. These estimators are based on a two-step estimation architecture, which says that any $\sqrt{m}$-consistent estimate followed by a one-step Gauss-Newton iteration can yield a solution that possesses the same asymptotic property as the LS one. The keypoint of the two-step scheme is to construct a $\sqrt{m}$-consistent estimate in the first step. In terms of whether the variance of measurement noises is known or not, we propose the Bias-Eli estimator (which involves solving a generalized trust region subproblem) and the Noise-Est estimator (which is obtained by solving a convex problem), respectively. Both of them are proved to be $\sqrt{m}$-consistent. Moreover, we show that by discarding the constraints in the above two optimization problems, the resulting closed-form estimators (called Bias-Eli-Lin and Noise-Est-Lin) are also $\sqrt{m}$-consistent. Plenty of simulations verify the correctness of our theoretical claims, showing that the proposed two-step estimators can asymptotically achieve the Cramer-Rao lower bound.
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