State Space representation
You can just write $(2)$ in $y_t$, so it becomes

$$y_t = \begin{bmatrix} b_1-a_1 &amp;&amp; -a_2\end{bmatrix} \begin{bmatrix} x_{a,t}\\ x_{b,t}\end{bmatrix} +\begin{bmatrix} 1 \end{bmatrix}u_t $$

Actually, this representation is called controllable canonical form. However, your system is not strictly proper, so you need to split it as

$$G(z)=\frac{z^2 + b_1 z}{z^2 + a_1 z + a_2} = 1 + \frac{(b_1 - a_1) z - a_2}{z^2 + a_1 z + a_2}$$

See how the coefficients appear?