## 4 Simulation Results  
In this section, all optimal controllers are compared with one another and with a standard PID controller in the frequency and time domains, as well as on the torque/speed plane. Figure 9 depicts the measured output at each mean wind speed (i.e., generator speed). Figure 9a shows the time responses for the MPC, LQG, and \(H_{\infty}\) controllers in mode 1. It is difficult to confirm if the \(C_{pmax}\) curve is tracked, which is the control objective in the time domain as discussed in Section 3.1, i.e., in Figure 9a. However, tracking of the \(C_{pmax}\) curve is better presented on the torque/speed plane in Figure 10. The figure demonstrates how all the optimal controllers, MPC (in red), \(H_{\infty}\) (in blue), and LQG (in black), perform on the torque/speed plane. They all appear to be satisfactory in closely tracking the \(C_{pmax}\) curve. The corresponding time responses of the power output for the MPC, LQG, and \(H_{\infty}\) controllers at a mean wind speed of 8 m/s (below-rated wind speed, i.e., mode 1) are illustrated in Figure 11a. Figure 9: Time responses of control output (generator speed) in modes 1, 2, and 3: MPC, LQG, and \(H_{\infty}\) based on the Bladed linearization model; (**a**) mode 1, (**b**) mode 2, (**c**) mode 3. Controller performance is also interpreted in the frequency domain, i.e., open-loop frequency response (Figure 12). The open-loop system [22] (\(y/e\)) in Figure 8 is considered the product of the controller (open-loop) and the process (i.e., wind turbine ). The controllers should achieve a gain crossover frequency in the range of 0.6 to 2 rad/s due to wind characteristics [23]. Gain crossover frequencies below 0.6 rad/s could cause an extremely slow control action, whereas those above 2 rad/s could cause an aggressive control action, potentially resulting in saturation of actuator, particularly in high wind speeds, i.e., in the above-rated region. In mode 1, Figure 12 shows that the gain crossover frequencies of the controllers (MPC, LQG, and \(H_{\infty}\)) are approximately between 0.6 and 2 rad/s. Notably, that PID controller is included in this study for comparison purposes only. The implication is that these controllers are neither too aggressive nor too slow. In summary, the time response in Figure 9a, results on torque/speed plane in Figure 10, and open-loop frequency responses in Figure 12 all show that the overall performance of these optimal controllers are satisfactory. Figure 13a shows the corresponding control input, i.e., torque demand in mode 1. Control of generator speed is achieved by adjusting torque in mode 1, as aforementioned. As depicted in Figure 9b the time responses of the MPC, LQG, and \(H_{\infty}\) controllers in mode 2 are adequate in terms of fluctuations, as they persist under 12%. Table 1 summarizes the results of generator speed fluctuations of all optimal controllers in mode 2; it shows the mean fluctuations for each controller in mode 2. The results show that fluctuations appear satisfactory because they remain below 12%. The corresponding time responses of the power output of the MPC, LQG, and \(H_{\infty}\) controllers at a mean wind speed of 10 m/s (just below-rated wind speeds, i.e., mode 2) are illustrated in Figure 11b. The controller performance is also interpreted in the frequency domain. Figure 14 shows that at a mean wind speed of 10 m/s and the gain crossover frequencies of the controllers (MPC, LQG, and \(H_{\infty}\)) are in the range of 0.6 to 2 rad/s, which is satisfactory as aforementioned. Note that, again, the PID controller is also included here for comparison purposes only. The results imply that these controllers are neither extremely aggressive nor slow. However, for the PID controller, the peaks at about 14.13 rad/s persist mostly above 0 dB, which corresponds to the drive-train mode, indicating that the controller would  
Figure 10: Behaviour of MPC, LQG, and \(H_{\infty}\) controllers in mode 1 on torque/speed plane. be sensitive to noise and uncertainty; hence, it is not robust.