Reference request: Controllable and Observable form for transform function I came across some online material a year ago that claimed that a the ABCD matrix of a transfer function $$G(z) = \frac{b_1 z+b_2}{z^2+a_1z + a_2}$$ can be directly computed from the coefficients of the transfer function
Can someone provide a reference to some material that contains the proof to this concept?

Consider the generic transfer function
$$
H(s) = \frac{Y(s)}{X(s)} = \frac{b_0s^n + \cdots + b_ns^0}{s^n + a_1s^{n-1} + \cdots + a_n}
$$
Then the state space model is
\begin{align}
\dot{\mathbf{q}} &= \mathbf{Aq} + \mathbf{B}u\\
y &= \mathbf{Cq} + Du
\end{align}
where 
\begin{align}
\mathbf{A} &= 
\begin{bmatrix}
-a_1 & 1 & 0 & \cdots & 0\\
-a_2 & 0 & 1 & 0 & \vdots\\
\vdots & \vdots & & \ddots & 0\\
-a_{n-1} & 0 &\cdots & & 1\\
-a_n & 0 & 0&\cdots & 0
\end{bmatrix}\\
\mathbf{B} &=
\begin{bmatrix}
b_1 - a_1b_0\\
b_2 - a_2b_0\\
\vdots\\
b_{n-1} - a_{n-1}b_0\\
b_n - a_nb_0
\end{bmatrix}\\
\mathbf{C} &= \begin{bmatrix}1 & 0 & \cdots & 0\end{bmatrix}\\
D &= b_0
\end{align}
You can go through the full derivation here.
