In the Buzz code we move the plant to shape the seismic noise without shifting the controls noise is:
\[ S_c = \left |\frac{EP}{1-KP}\right |^2 S_\mathrm{env} + \left |\frac{MKP}{1-KP}\right |^2 S_\mathrm{meas} \]
where \(K\) is the controller, \(P\) is the plant, \(E\) is the environmental noise shaping filter, \(M\) is the measurement noise shaping filter, \(S_c\) is the controls noise PSD, and \(S_\mathrm{meas} \) and \(S_\mathrm{env} \) are the unshaped PSDs of the respective noise. If we move the plant to the env noise shaping filter such that \(E'=PE\), \(P=1\), and \(K'\) is a new controller calculated for this system then the controlls noise becomes
\[ S_c' = \left |\frac{EP}{1-K'}\right |^2 S_\mathrm{env} + \left |\frac{MK'}{1-K'}\right |^2 S_\mathrm{meas} \]
In order for these two ststments to be equlivilant
\[ K'=KP \]
I think this means there needs to be some sort of elimination of the \(P\) from the \(K'\) since what we are looking for is \(K\) and not \(K'\).
Assuming that \(P' \neq1 \) . i.e. the plant has some shape left in it, then to find the controller to use in the system use:
\[ K=\frac{K'P'}{P} ,\]
where \(K\) is the controller to use in the system, \(P \) is the origional unmodified plant, \(P' \) is the modified plant, and \(K' \) is the controller found by buzz.