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.