Should the kalmanfilter have disturbance as input?

Question

This is a kalmanfilter

As you can see, the process noise(disturbance) is not going to the kalmanfilter. But the state space model for the state feedback system is written as this: https://youtu.be/H4_hFazBGxU?t=5m43s

So what is right and what is wrong? Should a kalmanfilter have disturbance as input too?

Like this. With disturbance as input:

$$ \ \begin{bmatrix} \dot{x} \\ \\\dot{\tilde{x}} \end{bmatrix} =\begin{bmatrix} A - BL& BL \\ 0 & A-KC \end{bmatrix} \begin{bmatrix} x\\ \tilde{x} \end{bmatrix}+\begin{bmatrix} I & 0\\ I & K \end{bmatrix}\begin{bmatrix} d\\ n \end{bmatrix}\ $$

Or this. Without disturbance as input:

$$ \ \begin{bmatrix} \dot{x} \\ \\\dot{\tilde{x}} \end{bmatrix} =\begin{bmatrix} A - BL& BL \\ 0 & A-KC \end{bmatrix} \begin{bmatrix} x\\ \tilde{x} \end{bmatrix}+\begin{bmatrix} I & 0\\ 0 & K \end{bmatrix}\begin{bmatrix} d\\ n \end{bmatrix}\ $$

Answer 1

I don't like the format you're using, with $\tilde{x}$, because it masks the fact that it is actually $x-\hat{x}$. In generally, you don't actually care about the state error; what you really should care about are the states themselves. The state error drives your state estimates to be equal to the actual states (eventually), but beyond that they shouldn't really be providing any useful information.

I say this because, as written, it looks like the actual system states $x$ aren't a part of your state error calculations because there's a big fat zero in the bottom-left of the system matrix. In reality though, again, you've masked the fact that you're actually using the system states twice - once directly as $x$ and then again as $x - \hat{x}$. There's a zero in the bottom-left of the system matrix because it's redundant.

All this is a preface to two points:

1. Your input disturbances $d$ will propagate to the Kalman filter naturally because they affect the system states, which in turn are measured and used as part of the input to the Kalman filter, AND
2. You can't include input disturbances in a Kalman filter in real life because you can't measure them, in the same way you can't "know" what the measurement noise is.

Use the first form.