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 $E^{\prime }=PE$,  $P=1$, and  $K^{\prime }$ is a new controller calculated for this system then the controlls noise becomes

$$S_c^{\prime }={\left|\frac{EP}{1-K^{\prime }}\right|}^2{S}_{\mathrm{env}}+{\left|\frac{M{K}^{\prime }}{1-K^{\prime }}\right|}^2{S}_{\mathrm{meas}}$$

In order for these two ststments to be equlivilant

$$K^{\prime }=KP$$

I think this means there needs to be some sort of elimination of the $P$ from the $K^{\prime }$ since what we are looking for is $K$ and not $K^{\prime }$.