## 3 Results
### Experimental Closed-Loop Validation  
The last step of the control design process was the validation of the derived compensator. The linear closed-loop simulation using linearised ROM was compared with the result of full-scale nonlinear FEM simulation and experiment with the physical setup. As introduced in Section 2.3, one of the benefits of having an accurate FEM model is that it can be used to check the controller (step 3). The controller was designed on a simplified model in which several phenomena were neglected--centrifugal force, for example. The designed controller can be included in the FEM software to check if it still functions as intended. In Abaqus, this has been done by implementing the discrete time version of the controller in an Abaqus user subroutine (UAMP). The transient dynamic model is solved using fixed time stepping at the controller sampling rate. This will take a while for large models, but in this case it is manageable. For larger models, if necessary, the efficiency can be increased by using intermediate-fidelity models with substructures, for example. The first test of the controller was unsuccessful. The controller became unstable. The reason is that a small amount of damping is needed for a stable closed loop system. The full model did not yet include any damping, and the numerical damping of the time integration was too small, with the small time steps that are used. Adding damping solved these issues. There is no direct equivalent in transient simulations of viscous model damping (which was used to make the reduced model match the measurements). A solution is to apply beta material damping (stiffness proportional) and match the damping ratio of the viscous modal damping at the unstable frequency observed in the simulation without damping (171 Hz). This solution is not ideal, because higher frequencies will be damped more and lower frequencies less. Nevertheless, the damping at the problematic frequency is accurate, and this matters most. This approach worked well for our purposes. As discussed in Section 2.3, more accurate modelling of damping is possible but not simple. Figure 22: Closed-loop performance achieved with the derived robust controller (47) and the set of five plant models obtained for varying load position (46): reference command following (**left**) and input disturbance rejection (**right**). Figure 23 shows a comparison of linear and nonlinear numerical simulations with actual experiment with the real motion stage for a particular case of load mass position of \(l=200\) mm that was identified as a worst case for achievable performance in the controller design phase. There is a very good match between the actual and predicted closed-loop behaviour, both for the linear and nonlinear simulation. The detail of the initial part of the transient on the right plot reveals a better fidelity of the nonlinear model. The linear model response shows an initial peaking phenomenon during the first ten milliseconds which is actually not observed in real plant data. The corresponding frequency of approximately 155 Hz indicates its relevance with the second bending mode of the flexible arm. The discrepancy in the observed behaviour might be caused by nonlinear effects not captured in the linear model or slight mismatch in the damping of the eigenmode in the linear model. Furthermore, further evolution of the initial oscillations during the first seventy milliseconds are captured better by the nonlinear simulation. On the other hand, the computational complexity aspect for both models should also be considered. While the linear simulation takes few seconds to numerically integrate the associated ordinary differential equations, the nonlinear simulation requires several hours and specialised numerical solvers. The overall shape of the response shows that both reduced and full-scale nonlinear models provide very good predictions of closed-loop performance. Very similar results were obtained for other mass positions and for varying amplitude of the step reference change.