How is a state disturbance matrix constructed?

Question

Consider the system:

$\dot{x}$ = Ax + Bu

y = Cx + Du

Where x contains 4 states, we have 2 inputs $u = \begin{bmatrix}u_1\\u_2\end{bmatrix}$ and A, B, C & D are known.

Now if 2 separate noise sources are added to each input the state equation can be written as:

$\dot{x}$ = Ax + Bu + N$\omega$

y = Cx + Du + E$\omega$

Where $\omega = \begin{bmatrix}\omega_1\\\omega_2\end{bmatrix}$

How can N and E be constructed assuming the noise is white noise?

Answer 1

If the noise is directly affecting the input then N=B because you get "the same" amount of input as noise.

Answer 2

In general this matrix describe natural phenomena, that is, the system as a disturbance that affect some of the states and outputs. This disturbance has an intensity as well. If this disturbance can be modeled as a white noise, then we must adapt our system such that it includes the disturbance.

E.g. consider the disturbance $w_1$ affects directly the states $x_1$ and $x_3$, but none of the outputs. For simplicity, we assume $w_2$ affects only the outputs. The matrix $N$ could be represented as

$$N=\begin{bmatrix}c_{11} & 0\\ 0 & 0\\ c_{13} & 0\\0 & 0 \end{bmatrix}$$ where $c_{11}$ and $c_{13}$ are constants that depend on the intensity of the white noise being modeled. For instance, if the disturbance affects states $x_1$ and $x_3$ with the same intensity, then $c_{11}=c_{13}$. Similar derivation can be made for the matrix $E$. So, for modeling this disturbed system, there is freedom to choose the constants and the strength of the disturbance (those are the parameters of the disturbance), but in the end of the day the chosen parameters should provide a model that represents the disturbed system accurately.