If I have a state-space model, so that matrices $A$, $B$, $C$ and $D$ are known, how can I design the right input $u$, so that $y$ is a desired signal, say, a sine wave with constant amplitude?
$$\begin{aligned} \dot{x} &= A x + B u \\ y &= C x + D u \end{aligned}$$
You can design an asymptotic output tracker based on feedback linearization if the residual dynamics are stable. The theory for this can be found in the book 'Nonlinear Control Systems' by Isidori [Springer]. You can find examples worked out using Mathematica here and here.
Another way is to develop a LQR tracking controller. See, for example, Chapter 4 of the book 'Optimal Control' by Anderson and Moore [available online].
