几何固有非线性递归滤波器I：算法
Geometrically Intrinsic Nonlinear Recursive Filters I: Algorithms
摘要：
几何固有非线性递归滤波器（GI Filter）被设计用于估计服从非线性离散时间观测的任意连续时间马尔可夫扩散过程X。GI滤波器从根本上不同于常用的扩展卡尔曼滤波器（EKF）及其二阶变体，即使在最简单的非线性情况下也是如此：（i）它使用向量观测的二次函数来更新状态，而不是EKF使用的线性函数。(ii)它是基于更深层次的几何原理，使GI滤波器坐标不变。这意味着，例如，如果线性系经受状态空间的非线性变换f并使用GI滤波器进行分析，则所得到的状态估计和条件方差将是未变换系统的卡尔曼滤波器估计的f下的前推-EKF或其二阶变体不共享的属性。X的噪声协方差和观测协方差本身分别在状态空间和观测空间上诱导几何，以及相关的正则连接。本文的续集发展随机微分几何的结果-基于“内在的位置参数“，一个概念来自调和映射的热流-从中我们推导出坐标自由过滤器更新公式。本文结合一个具体的例子--利用基于运动导弹的传感器跟踪和拦截目标的问题，给出了该算法。计算实验表明，当观测函数是高度非线性的，存在选择的噪声参数GI滤波器显着优于EKF。
英文摘要：
The Geometrically Intrinsic Nonlinear Recursive Filter, or GI Filter, is designed to estimate an arbitrary continuous-time Markov diffusion process X subject to nonlinear discrete-time observations. The GI Filter is fundamentally different from the much-used Extended Kalman Filter (EKF), and its second-order variants, even in the simplest nonlinear case, in that: (i) It uses a quadratic function of a vector observation to update the state, instead of the linear function used by the EKF. (ii) It is based on deeper geometric principles, which make the GI Filter coordinate-invariant. This implies, for example, that if a linear system were subjected to a nonlinear transformation f of the state-space and analyzed using the GI Filter, the resulting state estimates and conditional variances would be the push-forward under f of the Kalman Filter estimates for the untransformed system - a property which is not shared by the EKF or its second-order variants. The noise covariance of X and the observation covariance themselves induce geometries on state space and observation space, respectively, and associated canonical connections. A sequel to this paper develops stochastic differential geometry results - based on "intrinsic location parameters", a notion derived from the heat flow of harmonic mappings - from which we derive the coordinate-free filter update formula. The present article presents the algorithm with reference to a specific example - the problem of tracking and intercepting a target, using sensors based on a moving missile. Computational experiments show that, when the observation function is highly nonlinear, there exist choices of the noise parameters at which the GI Filter significantly outperforms the EKF.
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