We study a scenario where an aircraft has multiple heterogeneous sensors collecting measurements to track a target vehicle of unknown location. The measurements are sampled along the flight path and our goals to optimize sensor placement to minimize estimation error. We select as a metric the Fisher Information Matrix (FIM), as "minimizing" the inverse of the FIM is required to achieve small estimation error. We propose to generate the optimal path from the Hamilton-Jacobi (HJ) partial differential equation (PDE) as it is the necessary and sufficient condition for optimality. A traditional method of lines (MOL) approach, based on a spatial grid, lends itself well to the highly non-linear and non-convex structure of the problem induced by the FIM matrix. However, the sensor placement problem results in a state space dimension that renders a naive MOL approach intractable. We present a new hybrid approach, whereby we decompose the state space into two parts: a smaller subspace that still uses a grid and takes advantage of the robustness to non-linearities and non-convexities, and the remaining state space that can by found efficiently from a system of ODEs, avoiding formation of a spatial grid.
翻译:我们研究的情景是,飞机拥有多种不同的传感器,收集测量结果,以跟踪位置不明的目标飞行器。测量结果在飞行路径和我们优化传感器定位以尽量减少估计误差的目标上取样。我们选择了“最小化”FIM反面的渔业信息矩阵(FIM)作为衡量标准,以达到小估计误差。我们提议从汉密尔顿-贾科比(HJ)部分差分方程(PDE)中产生最佳路径,因为它是最佳化的必要和充分条件。基于空间网格的传统线法(MOL)方法本身很适合FIM矩阵引发的问题高度非线性和非电离子结构。然而,传感器布置问题导致一个使天真的MOL方法难以解决的状态空间层面。我们提出了一种新的混合方法,将国家空间分为两个部分:一个小的子空间,仍然使用电网,并利用非线性和非电解的强性,以及从空间轨系中高效地发现从空间轨系形成一个空间轨迹中可以找到的剩余空间状态。