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I used the measured position to estimate the velocity using a Kalman filter.

But in simulations, the Kalman gain changes quickly and then remains constant when position and velocity continue to change. For example, position and velocity change at times 0 to 0.5(s) and 3 to 4(s), respectively. But the Kalman gain only changes 0 to 0.1(s) and then remain constant.

My question is: Why is the Kalman gain constant when position and velocity change?

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The Kalman Gain $K_k$ for a discrete, linear system is computed using the state covariance matrix $P_k$, measurement matrix $H_k$ and measurement noise covariance matrix $R_k$:

$$ K_k = P_kH_k^T(H_kP_kH_k^T+R_k)^{-1} $$

If the measurement matrix and the measurement noise covariance matrix are constant and the state covariance matrix converges to a steady-state value $P_\infty$, the Kalman Gain also converges to a steady-state value $K_\infty$, regardless of actual measurements or state.

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