# Robust Speed Control of a Multi-Mass System: Analytical Tuning and Sensitivity Analysis
## 4 Simulation Results
### Time-Domain Analysis  
In Figure 14, the figures at the left show the high robustness of the PI-controlled system with the time delay ignored (i.e., in the ideal case, G\({}_{\Delta}\) = 1). The response also showed an excellent tracking performance. Nevertheless, when a 0.9 ms time delay exists, the PI controller begins to perform poorly after the powertrain moment value is inverted, as shown in the figures at the right. This outcome confirms the accuracy of the frequency analysis. The LQ-controlled system displays high robustness even with a time delay, as shown in Figure 15. However, its time response has a steady-state deviation and no significant influence of a 0.9 ms delay on the system.  
Figure 16 shows that the LQI-controlled system has the best tracking performance with zero steady-state error and an insignificant influence of a 0.9 ms delay on the system. Figure 17 presents each controller's angular speed tracking error in case of a time delay. The PI controller has an analogous tracking response since the dominant poles of PI were  
Figure 14: Time response of the PI-controlled system: (**Left**) without time delay, (**Right**) with time delay.  
Figure 13: Desired reference signal and powertrain moment.  
placed close to the dominant poles of LQI. However, the difference in robustness is apparent. Although the LQ controller is robust, it has an absolute steady-state error of 0.37 rad/s. This error occurs as a consequence of applying the powertrain moment. LQI controller, in contrast, demonstrates both high robustness and tracking performance, and it could achieve a zero steady-state error in a short time.  
In order to confirm the accuracy of the estimated noise rejection performance, demonstrated in Figure 12, the previous simulation test is executed again. However, this time, white measurement noise is added to the angular speed feedback signal. The previous simulation test is executed again, but this time, white measurement noise is added to the angular speed feedback signal. The white noise has an amplitude of \(\pm\)0.5 rad/s and a  
Figure 16: Time response of the optimal LQI-controlled system: (**Left**) without time delay, (**Right**) with time delay.  
Figure 15: Time response of the optimal LQ-controlled system: (**Left**) without time delay, (**Right**) with time delay.  
sampling time of \(1\times 10^{-5}\) s. The noise amplification magnitudes for PI, LQ, and LQI are 4.6778, 0.01878, and 0.01832, respectively. So, the control command is expected to have high noise amplification when the PI controller is implemented. In contrast, LQ and LQI should handle the measurement noise with low influence in the control signal. These expectations are proven in Figures 18-20.  
The simulation results meet the expectations of the noise model, where the PI controller leads to high noise amplification that causes noise in the controller moment that exceeds \(\pm\)1000 N.m, compared to the case of the noise-free measurements. Moreover, with both time delay and noise, the PI controller drove the system to instability directly after attempting to settle the angular speed to 30 rad/s. On the contrary, both LQ and LQI controllers had a low noise of \(\pm\)3 N.m in the controller signal.  
Figure 17: Influence of the time delay on the angular speed error: (**Top**) PI-controlled, (**Middle**) LQ-controlled, (**Bottom**) LQI-controlled.  
Figure 18: Measurement noise amplification of PI controller: (**Left**) without time delay, (**Right**) with time delay.  
In summary, the PI controller has fewer closed-loop poles than both LQ and LQI controllers, making it less efficient in controlling complex systems. LQ and LQI demonstrate high robustness against a time delay and measurement noise. PI, conversely, could not handle highly dynamic reference signals in the presence of time delay. In addition, PI has bad noise rejection quality compared to both LQ and LQI. Based on these results, the PI controller option will not be considered for further testing in the rest of this work.