So I have the systems
$$\Sigma_{1}:p_{1}(\frac{d}{dt})y_{1}=q_{1}(\frac{d}{dt})u_{1}$$
$$\Sigma_{2}:p_{2}(\frac{d}{dt})y_{2}=q_{2}(\frac{d}{dt})u_{2}$$
And the equations $u_{2}=y_{1}$, $u_{1}=u+y_{2}$ and $y=y_{1}$.
Clearly $u_{2}=y_{1}=y$, but I still have to eliminate $u_{1}$ and $y_{2}$.
Define $l:=\begin{bmatrix}u_{1}\\y_{2}\end{bmatrix}$ and $w:=\begin{bmatrix}u\\y\end{bmatrix}$
I want to define $R(\xi)$ and $M(\xi)$ such that $$R(\frac{d}{dt})\begin{bmatrix}u\\y\end{bmatrix}=M(\frac{d}{dt})\begin{bmatrix}u_{1}\\y_{2}\end{bmatrix}$$
Now, this wouldn't be so hard if it were not for the fact that $w$ is defined with manifest variables as opposed to latent variables. So I'm stuck, really.
By omitting $d/dt$ and changing the variables we can obtain the following equations:
$$\begin{align} p_1 y &= q_1 (u + y_2) \\ p_2 y_2 &= q_2 y \\ \\ p_1 y - q_1 u &= q_1 y_2 \\ q_2 y &= p_2 y_2 \\ \\ \begin{bmatrix}-q_1 & p_1 \\ 0 & q_2\end{bmatrix} \begin{bmatrix}u \\ y\end{bmatrix} &= \begin{bmatrix}0 & q_1 \\ 0 & p_2\end{bmatrix} \begin{bmatrix}u_1 \\ y_2\end{bmatrix} \end{align}$$