A short summary of the linear Kalman filter: the model, the two steps, and the tuning tips I always forget.

System model

\[x_k = F x_{k-1} + B u_k + w_k, \qquad w_k \sim \mathcal{N}(0, Q)\] \[z_k = H x_k + v_k, \qquad v_k \sim \mathcal{N}(0, R)\]

1. Predict

\[\hat{x}_{k|k-1} = F \hat{x}_{k-1} + B u_k, \qquad P_{k|k-1} = F P_{k-1} F^\top + Q\]

2. Update

\[K_k = P_{k|k-1} H^\top \left(H P_{k|k-1} H^\top + R\right)^{-1}\] \[\hat{x}_k = \hat{x}_{k|k-1} + K_k \left(z_k - H \hat{x}_{k|k-1}\right), \qquad P_k = (I - K_k H) P_{k|k-1}\]

Minimal Python implementation

import numpy as np

class KalmanFilter:
    def __init__(self, F, H, Q, R, x0, P0):
        self.F, self.H, self.Q, self.R = F, H, Q, R
        self.x, self.P = x0, P0

    def predict(self):
        self.x = self.F @ self.x
        self.P = self.F @ self.P @ self.F.T + self.Q

    def update(self, z):
        S = self.H @ self.P @ self.H.T + self.R
        K = self.P @ self.H.T @ np.linalg.inv(S)
        self.x = self.x + K @ (z - self.H @ self.x)
        self.P = (np.eye(len(self.x)) - K @ self.H) @ self.P

Tuning tips

Symptom	Likely cause	Fix
Estimate lags behind measurements	$Q$ too small	Increase $Q$
Estimate is noisy	$R$ too small	Increase $R$
Filter diverges	Model mismatch / bad $P_0$	Check $F$, inflate $P_0$

Takeaway: the ratio between $Q$ and $R$ matters more than their absolute values.