# Relationship between transfer functions in a transfer matrix.

I'm taking a MIMO control theory course and this is my first exposure to control theory as an academic topic. Well for this course, I think I understand the concept of finding state space realizations for interconnections of LTIs, but there is a setup to a problem that I'm just not getting.

So we're given a transfer function $$\hat G_1$$ with state space realization $$(A_1, B_1, C_1, D_1)$$ with the state vector $$x_1 \in R^{n_1}$$ and $$\hat G_2$$ with state space realization $$(A_2,B_2,C_2, D_2)$$ with the state vector $$x_2 \in R^{n_2}$$. Additionally, we are given that $$\hat G(s) = [\hat G_1; \hat G_2]$$ by which I mean that $$\hat G(s)$$ is a column vector of the above two transfer functions.

So my question is what relationship exactly does that column vector imply? My goal here is to find the state space realization of the whole system, but I don't know how to start the problem because I don't know what the system's transfer function looks like mathematically. Any help is appreciated.

The expression $$\hat G(s) = [\hat G_1; \hat G_2]$$ is a transfer function matrix, which is generalization of SISO transfer function to MIMO. For example when both $$\hat G_1$$ and $$\hat G_2$$ are SISO, such as

$$\hat G_1(s) = \frac{1}{s}, \quad \hat G_2(s) = \frac{1}{s+1}$$

you get

$$\hat G(s) = \begin{bmatrix} \frac{1}{s} \\ \frac{1}{s+1} \end{bmatrix}$$

which represents a system with one input and two outputs.

You can also stack state space models which is equivalent to $$\hat G(s) = [\hat G_1; \hat G_2]$$ by defining the state vector as $$x=\begin{bmatrix}x_1^\top & x_2^\top\end{bmatrix}^\top\!\!\in\mathbb{R}^{n_1+n_2}$$ with the following state space model

$$A = \begin{bmatrix}A_1 & 0 \\ 0 & A_2\end{bmatrix}, \quad B = \begin{bmatrix}B_1 \\ B_2\end{bmatrix}, \quad C = \begin{bmatrix}C_1 & 0 \\ 0 & C_2\end{bmatrix}, \quad D = \begin{bmatrix}D_1 \\ D_2\end{bmatrix}.$$

If instead you want $$\hat G(s) = [\hat G_1, \hat G_2]$$ (so for SISO $$\hat G_1$$ and $$\hat G_2$$ you would have that $$\hat G$$ has two inputs and one output) you can use

$$A = \begin{bmatrix}A_1 & 0 \\ 0 & A_2\end{bmatrix}, \quad B = \begin{bmatrix}B_1 & 0 \\ 0 & B_2\end{bmatrix}, \quad C = \begin{bmatrix}C_1 & C_2\end{bmatrix}, \quad D = \begin{bmatrix}D_1 & D_2\end{bmatrix}.$$

However this method of combining state space models might not yield minimal models, so they might either be uncontrollable or unobservable. This will for example be the case when $$\hat G_1 = \hat G_2$$.