## 3 Results
### Experimental Closed-Loop Validation
They are not presented here for the sake of brevity. Additionally, a disturbance rejection case was modelled using a similar simulation setup in Abaqus and compared with the linear simulation and experimental data. Step torque was injected using the actuator by adding it to the controller output to excite the system and evaluate its disturbance rejection capability. The results are shown in Figure 24. The overall shape of the response reveals well-behaved closed-loop performance. The feedback loop is capable of attenuating all the unwanted flexible arm vibrations and settle the whole system. There are some transient oscillations during the initial phase as shown in the right detailed plot of Figure 24. This part is mainly affected by the high-frequency bending modes of the motor-arm coupling and base plate dynamics. The comparison of linear and nonlinear simulation confirms the previous results obtained from the reference-following test. The nonlinear simulation captures the high-frequency dynamics more closely at the cost of much higher computational demands. From the control perspective, the reduced-order linearised model is still capable providing high-fidelity predictions of both open- and closed-loop plant responses. Figure 23: Validation of closed-loop performance: comparison of real experimental data of the plant with reduced-order linear simulation and full-scale nonlinear FEM transient simulation. Left: whole step setpoint reference change response, right: detail of the initial transient. Figure 25 shows a comparison of linear and nonlinear simulation in terms of prediction error between the modelled closed-loop response and experimental results. In both cases, the full-scale FEM transient modelling provided more accurate predictions. Table 3 quantifies average and worst-case values of the prediction error in terms of the root-mean-square and peak-to-peak indices  
\[e_{rms}=\sqrt{\sum_{k=1}^{N}e^{2}(k)/N},\;e_{pk-pk}=\underset{\forall k}{max}(e (k))-\underset{\forall k}{min}(e(k)). \tag{50}\]  
The difference is more pronounced in the disturbance rejection test because of more significant excitation of high-frequency dynamics of the machine frame and actuator-arm coupling. We conclude that from the control-relevance perspective, the reduced-order  
Figure 24: Validation of closed-loop performance: comparison of real plant experimental data with reduced order linear simulation and full-scale nonlinear FEM transient simulation. Left: whole step input disturbance rejection, right: detail of the initial transient. Figure 25: Comparison of reduced order linear simulation with full scale FEM model transient simulations: evolution of prediction error in the reference-following (**left**) and disturbance-rejection (**right**) experiments. linear model also proved to be very capable for both numerical simulations and robust controller design. The nonlinear dynamic effects would be pronounced more for higher rotational velocities and driving torque amplitudes. Nevertheless, the difference is minor for the assumed operational regimes of the physical plant due to practical actuator and construction constraints. In order to obtain the results for Figures 24 and 25, we needed to redistribute the stiffness between the disk coupling and the shaft. Otherwise, the disk coupling would buckle. The linear model in these figures has not been updated, but a linearised version of the updated full model would be very similar. The buckling problem was not present at the excitation levels for Figure 23. If the full model should be able to predict a loss of linearity, then the disk-coupling part of the model needs further validation. However, this was not necessary for our purposes of control-relevant modelling. These issues show that the full model does not need to be perfect in order to be valuable. Incremental model updates can be made when the situation demands them.