---
title: "扩展 Kalman 滤波"
created: 2026-06-22
updated: 2026-06-22
type: concept
tags: [state-estimation, filtering, nonlinear-systems]
sources: [nano-filter]
---

# 扩展 Kalman 滤波

Extended Kalman Filter (EKF) 是最早的非线性 [[kalman-filter|KF]] 扩展，通过对非线性函数做一阶 Taylor 展开实现局部线性化。

## 核心机制

对非线性系统 $x_t = f(x_{t-1}) + \xi_t$, $y_t = g(x_t) + \zeta_t$：
- 在**当前状态估计点**处计算 Jacobian: $F_t = \frac{\partial f}{\partial x}\big|_{\hat{x}_{t-1}}$, $G_t = \frac{\partial g}{\partial x}\big|_{\hat{x}_{t|t-1}}$
- 用线性化模型 $x_t \approx F_t x_{t-1} + c$, $y_t \approx G_t x_t + d$ 运行标准 KF

## 局限性

- Taylor 展开仅为一阶近似，强非线性下误差大
- 需要计算 Jacobian 矩阵（对高维系统代价高）
- [[nano-filter|NANO]] 证明 EKF 的线性化误差是 Gaussian 滤波次优性的根源，提出直接优化的替代方案

## 参考
- [[kalman-filter|Kalman Filter]]
- [[gaussian-filtering|Gaussian Filtering]]
- [[unscented-kalman-filter|UKF]]
- [[nano-filter|NANO Filter]]