# Robust Speed Control of a Multi-Mass System: Analytical Tuning and Sensitivity Analysis
## 4 Simulation Results
### Sensitivity Analysis for a Single Parameter Uncertainty  
The selected controllers are designed based on prior knowledge of system model equations and the values of their parameters. The concept of uncertainty will be employed by repeating the same tests several times. Each test will be conducted with a new variation  
Figure 19: Measurement noise amplification of LQ controller: (**Left**) without time delay, (**Right**) with time delay.  
Figure 20: Measurement noise amplification of LQI controller: (**Left**) without time delay, (**Right**) with time delay.  
in a specific parameter. Only LQ and LQI will be investigated. No further analysis will be made for PI since it failed the basic tests. In this section, the sensitivity of each controlled system will be examined concerning time delay sensitivity, disturbance moment effect, single parameters uncertainty, and multiparameter uncertainty. Previously, the time delay sources were introduced: the test bench electrical components delay, which resembled the G\({}_{\text{t}}\) function model (4), and the measurements time delay expressed in Equation (5). The test is performed with a fixed T\({}_{\text{d}}\) at 0.2 s, and T\({}_{\text{m}}\) varies randomly in the range of [0.5, 5] ms. LQ proved highly robust against T\({}_{\text{m}}\), apart from speed tracking efficiency, as shown in Figure 21. However, as expected from the frequency analysis section, LQI is more sensitive to time delay than LQ, as demonstrated in Figure 22. Still, LQI could handle measurement time delay variations until 5 ms.  
Figure 21: Time response of LQ-controlled system at different T\({}_{\text{m}}\) values.  
Figure 22: Time response of LQI-controlled system at different T\({}_{\text{m}}\) values.  
The electrical loading machine perceives the powertrain driving moment (\(M_{drive}\)) as a disturbance moment. Therefore, the loading machine should generate a resistance moment according to the desired rotational speed value. Figures 23 and 24 illustrate the influence of changing the driving moment on LQ- and LQI-controlled systems, respectively. At no load testing (i.e., M\({}_{drive}\) = 0 N.m), LQ followed the desired speed profile perfectly. However, there will be a consequent shift in LQ speed response with any moment applied. Positive moment values (i.e., driving moment) lead to an upward offset in LQ speed response. In comparison, a negative or braking moment causes a downward offset. In contrast, LQI maintained the same speed response at all disturbance moment values. Also, it was able to compensate for the extreme moment change. For instance, when changes were made from 500 to \(-\)500 N.m, the result was an error of 5 rad/s. This error was corrected in about 0.5 s. Therefore, the same LQ- and LQI-tuned parameters will be used for all tests to study the influence of parameter uncertainties on controllers' performance. Furthermore, a large disturbance moment (500 to \(-\)500 N.m) is applied to put the controllers under extreme conditions. Figure 25 illustrates the influence of uncertainty in J\({}_{\rm P2W}\), where the Monte Carlo simulation is utilized to generate random variations in J\({}_{\rm P2W}\). The LQ controller proved to be highly robust against uncertain J\({}_{\rm P2W}\) values. At the same time, no significant change in both angular speed tracking and controller command response is observed.  
Figure 23: Time response of LQ-controlled system at different M\({}_{drive}\) values.  
Figure 24: Time response of LQI-controlled system at different M\({}_{drive}\) values.