For a SISO, linear, continuos time system, Lagrange formula applied to the output signal is: $$ y(t)= \mathbf c^T(\Phi \mathbf x(0) +\mathbf\Psi u) +du=\mathbf c^T\Phi \mathbf x(0) +\mathbf c^T\mathbf\Psi u + du $$ And is also verified this property: $$ \frac{d}{dt}(\mathbf\Psi u)=A\mathbf\Psi u +\mathbf bu$$ By deriving Lagrange formula and using the previous property we obtain: $$ \dot y(t)= \mathbf c^T A \mathbf\Psi u + \mathbf c^T \mathbf bu +d\dot u $$
$$ \ddot y(t)= \mathbf c^T A^2 \mathbf\Psi u + \mathbf c^T A \mathbf bu + \mathbf {c^T b }\dot u +d\ddot u $$ $$...$$ By doing a linear combination of the derivates of the output signal: $$\alpha y(t)+\beta \dot y(t) +\ddot y(t)+... $$ Is it possible to obtain the ARMA model of the system? Thanks.