would Kalman Filter capture consistent drift in observation model?

Question

I am studying Kalman filter and I wonder how it handles the case of consistent bias in observation model?

Let's take this example from wikipedia: https://en.wikipedia.org/wiki/Kalman_filter#Example_application,_technical

I highlight the question I have below:

I wonder if in a practical case, the measurement starts to develop a consistent bias over time. so the $H$ matrix is no longer $[1, 0]$, but slowly becomes like $[1 + f(t), 0]$ would Kalman Filter capture that drift of $f(t)$? How exactly? I am puzzled since $v_k$ has a mean $0$.

Answer 1

Sensor biases, of the form of an additive offset, are often modelled as $z_k=H\,x_k+b_k+v_k$ with $b_k$ the bias at time $k$. One often either tries to calibrate the value of the bias or extend the state of the system by adding $b_k$ to the state vector and let the Kalman filter find the value of the bias.

For the second case it can be noted that this allows for a drifting bias over time as well, but do make sure that the extended system is observable.

If instead the value of $H$ is drifting with time then your model becomes nonlinear and you could try using an extended Kalman filter. But depending on the exact system and modelling, the extended Kalman filter could diverge.