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

# Kalman 滤波

Kalman filter (KF) 是线性高斯系统下 [[bayesian-filtering|贝叶斯滤波]]的精确解析解。利用高斯分布在**线性变换下的封闭性**和**条件化下的共轭性**，KF 递归地更新均值和协方差矩阵。

## 基本形式

对于系统 $x_t = A x_{t-1} + \xi_{t-1}$，$y_t = C x_t + \zeta_t$（噪声均为零均值高斯）：

- 预测步：
  - $\hat{x}_{t|t-1} = A \hat{x}_{t-1|t-1}$
  - $P_{t|t-1} = A P_{t-1|t-1} A^\top + Q$

- 更新步（Kalman gain $K_t = P_{t|t-1} C^\top (C P_{t|t-1} C^\top + R)^{-1}$）：
  - $\hat{x}_{t|t} = \hat{x}_{t|t-1} + K_t (y_t - C \hat{x}_{t|t-1})$
  - $P_{t|t} = (I - K_t C) P_{t|t-1}$

## 非线性扩展

对非线性系统，KF 的封闭性不再成立，衍生出：
- [[extended-kalman-filter|EKF]] — Taylor 线性化
- [[unscented-kalman-filter|UKF]] — 无迹变换
- [[nano-filter|NANO]] — 自然梯度优化

## 参考
- [[bayesian-filtering|Bayesian Filtering]]
- [[gaussian-filtering|Gaussian Filtering]]
- [[nano-filter|NANO Filter]]