Let $f=A\sin{\omega t}=x_1$ and $\dot{f}=A\omega\cos{\omega t}=\dot{x}_1=x_2$. Let the output be $y=cA\omega$, where $c=1$ is a constant. I want to represent this in a state space formulation:
$\dot{x}=\underline{A}x+\underline{B}u$
$y=\underline{C}x+\underline{D}u$
What will $C$ be? I tried: $A\omega=\frac{A}{t}\arcsin{\frac{x_1}{A}}$, but does this make sense? Another possibility is $A\omega=\frac{-x_2}{\cos{\omega t}}$. Which one should I choose?
I make you an example about how to write state-space equations using a dynamic second order equation to let you understand the procedure.
The starting equation is: $$M a + C v + K s = f $$ $C$ is damping, $K$ is stiffness and $M$ is the mass.
Then build your state vector $x =\pmatrix{s\\ v}$
Now write the equation like $ a = 1/M (f - C v - K s)$
$\dot{x} = \pmatrix{v\\ a}$ is the derivative of the state vector.
In the state space:
$\dot{x} = \pmatrix{0& 1, \\-K/M& -C/M} x + \pmatrix{0 \\ 1/M} f$
If you want the displacement as output
$y = \pmatrix{1\\ 0} x $
