I want to know how I can get from a set of differential equations to a set of transfer functions for a multi-input multi-output system. I can do this easily with Matlab or by computing $G(s) = C[sI - A]^{-1}B + D$. I have the following two equations: $ \ddot{y}_1 + 2\dot{y}_1 + \dot{y}_2 + u_1 = 0 \\ \dot{y}_2 - y_2 + u_2 - \dot{u}_1 = 0 $There are 2 inputs, $y_i$, and 2 outputs, $u_i$. At first I thought when I want to retrieve the transfer function from $y_1$ to $u_1$ that I had to set $y_2$ and $u_2$ equal to zero. Thus I would have been left with, $\ddot{y}_1 + 2\dot{y}_1 + u_1 = 0$ and $\dot{u}_1 = 0$. However this does not lead to the correct answer, $ y_1 \rightarrow u_1: \frac{-s^2 - s + 1}{s^3 + s^2 - 2 s} $ I also thought about substituting the two formulas in each other. So expressing $y_2$ and $u_2$ in terms of $y_1$ and $u_1$ however this also lead to nothing.
Can someone explain to me how to obtain the 4 transfer functions, $y_1 \rightarrow u_1$, $y_1 \rightarrow u_2$, $y_2 \rightarrow u_1$ and $y_2 \rightarrow u_2$?