## 7 Time Domain Optimization
### Design of an Adaptive Filter  
As discussed in Sections 5 and 6, in reality, the rotor effective wind speed \(u_{\mathit{eff}}\) cannot be measured perfectly. While the lidar system measures in a three-dimensional wind field, it is only able to yield an estimate of the rotor effective wind speed \(\hat{u}_{\mathit{eff}}\) due to limitations of the measurement principle. The wind field also evolves from the place of measurement to the rotor.  
However, with an Adaptive Filter (AF), the lidar estimate \(\hat{u}_{\mathit{eff}}\) can be matched in the best possible way to the rotor effective wind speed \(u_{\mathit{eff}}\). The transfer function between the rotor effective wind speed measured by the lidar and that sensed by the turbine's rotor can be calculated by  
\[G_{u_{\mathit{eff}}\hat{u}_{\mathit{eff}}}\left(f\right)=\frac{S_{u_{\mathit{ eff}}\hat{u}_{\mathit{eff}}}\left(f\right)}{S_{\hat{u}_{\mathit{eff}}}\left(f \right)}, \tag{15}\]  
where \(S_{u_{\mathit{eff}}\hat{u}_{\mathit{eff}}}\left(f\right)\) is the CPSD between both signals and \(S_{\hat{u}_{\mathit{eff}}\hat{u}_{\mathit{eff}}}\left(f\right)\) is the PSD of the signal from the lidar. As introduced in Section 6 (see Equation (9)), a filter fitting to this transfer function will match the correlated part of \(\hat{u}_{\mathit{eff}}\) to \(u_{\mathit{eff}}\) in an optimal minimum-MSE sense (see [42]). However, due to its simplicity in implementation, a linear low-pass filter can be fitted to the magnitude of the transfer function \(G_{u_{\mathit{eff}}\hat{u}_{\mathit{eff}}}\left(f\right)\) for a given mean wind speed. Here, a first-order Butterworth filter with cutoff frequency \(f_{\mathit{cutoff}}\) is chosen. Because the correlation and thus the transfer function depends, among other factors, on the mean wind speed (e.g., see [8]), the cutoff frequency needs to be adjusted continuously. In this paper, constant correlation and mean wind speed is assumed.  
In addition to the filtering, the timing of the feedforward control action is important. Due to the lidar measuring in front of the turbine, a certain preview time is obtained (depending mainly on the distance and the mean wind speed). A certain fraction of the preview will be consumed by the filtering. However, the feedforward control action \(\beta_{FF}\) needs to be synchronized with the rotor effective wind speed disturbing the system. Thus, an appropriate buffer time \(T_{\mathit{buffer}}\) needs to be found to buffer the filtered signal from the lidar.  
The PSD of the rotor speed is a useful measure to evaluate the benefit of the combined feedback-feedforward controller over the conventional feedback controller. As described by [PERSON] and [PERSON] [42], the PSD for the feedback-only case can be calculated by multiplying the spectrum of the  
Figure 7: Feedback control loops and collective pitch feedforward control loop assuming perfect wind preview (**left**). Static pitch curve (**right**).  
rotor effective wind speed \(S_{u_{eff}u_{eff}}\) with the magnitude of the closed-loop transfer function \(T_{u_{eff}u_{eff}}\) (introduced in Section 6.1):  
\[S_{\omega,FB}\left(f\right)=\left|T_{u_{gm}u_{eff}}\left(f\right)\right|^{2}S_{u _{eff}u_{eff}}\left(f\right). \tag{16}\]  
As introduced in Equation (11), the PSD for the combined feedback-feedforward case with optimal filtering is determined by:  
\[S_{\omega,FBFF}\left(f\right)=S_{\omega,FB}\left(f\right)(1-\gamma_{u_{eff}u_{ eff}}^{2}\left(f\right)). \tag{17}\]  
Thus, the measurement coherence \(\gamma_{u_{eff}u_{eff}}^{2}\left(f\right)\) has a direct impact on the optimal spectrum of the controlled variable. This spectrum can be used to evaluate the implementation of a feedforward controller.