How to put an ARMA(2,2) model in state-space form I am interested in an ARMA$(2,2)$ model with an additional input variable, which I want to put in state-space form.   If $w_t$ is white noise, and $x_t$ is a known input, the model is given by:
$$y_t = \beta_0 + \beta_1 \cdot x_{t-1} + \alpha_1 \cdot y_{t-1} + \alpha_2 \cdot y_{t-2} + w_t + \theta_1 \cdot w_{t-1} + \theta_2 \cdot w_{t-2}.$$
Can someone please show how to write this in state-space form?

One way to do it is to define the state vector as
$$
\xi_t = \begin{pmatrix}
y_t \\
y_{t-1} \\
w_{t} \\
w_{t-1} \\
1 \\
\end{pmatrix}
$$
The measurement equation is just
$$
y_t = \begin{pmatrix}
1 & 0 & 0 & 0   & 0
\end{pmatrix} \, \xi_t
$$
i.e. there is no noise term. The state transition equation is then
$$
\underbrace{\begin{pmatrix}
y_t \\
y_{t-1} \\
w_{t} \\
w_{t-1} \\
1 \\
\end{pmatrix}}_{\xi_t}
 = 
\begin{pmatrix}
\alpha_1 & \alpha_2 & \theta_1 & \theta_2 & \beta_0+\beta_1 x_{t-1} \\
1 & 0 & 0 & 0 & 0  \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
\underbrace{\begin{pmatrix}
y_{t-1} \\
y_{t-2} \\
w_{t-1} \\
w_{t-2} \\
1 \\
\end{pmatrix}}_{\xi_{t-1}}
+
\begin{pmatrix}
1 \\
0 \\
1 \\
0 \\
0 \\
\end{pmatrix} w_t
$$
