For a typical state space model, i.e.
$$ \dot{x} = Ax + Bu $$ $$ y = Cx $$
how can you augment the system such that a feedback control system could track a specific frequency reference (say, 400Hz) while completely rejecting constant disturbances?
By Integral Controller I mean something like:
where
$ A_{a} = \begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & w_{r} & 0 & 0 \\ 0 & -w_{r} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & w_{t} \\ 0 & 0 & 0 & -w_{t} & 0 \\ \end{matrix} $
and
$ B_{a} = \begin{matrix} 1\\ 0 \\ 1\\ 0 \\ 1 \\ \end{matrix} $
where $w_{r}$ is the frequency being rejected and $w_{t}$ is the frequency being tracked. If there is no frequency being rejected, What should the $A_{a}$ and $B_{a}$ matrix look like? Should I use a 3x3 matrix with the $w_{r}$ removed or do I just set $w_{r}$ to zero? i.e.
$ A_{a} = \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & w_{t} \\ 0 & -w_{t} & 0 \\ \end{matrix} $
and
$ B_{a} = \begin{matrix} 1\\ 0 \\ 1 \\ \end{matrix} $
OR
$ A_{a} = \begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & w_{t} \\ 0 & 0 & 0 & -w_{t} & 0 \\ \end{matrix} $
and
$ B_{a} = \begin{matrix} 1\\ 0 \\ 1\\ 0 \\ 1 \\ \end{matrix} $
thanks for the help!