1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 9 achieved displacement t inZ-direction (td,Z and t Z). The RMS error was calculated by rms = f 1000 · 1000 f i=1 td,Z −t Z 2 , (1.5) because the sampling frequency was 1000 Hz and thus the number of measured samples over the time period 1 f was 1000 f . The error in the first iteration corresponds to the error of the P-PI-PI cascade, as learning begins in the second iteration based on the error signal from the first iteration. We see that the algorithm converges and the error decreases over time. Our maximum number of iterations was 26 and convergence was in the order of ≈1 j2 , where j counts the iterations. This means that doubling the number of iterations reduces the error to 1 4 in the considered range. However, there will be a convergence limit due to mechanical limits and maximum motor voltages. Nevertheless, the potential of MFIIC and its convergence looks promising. Using MFIIC, the learned feedforward signal can be saved and used in future experiments. Hence, the results shown in the following sections take the feedforward signal of MFIIC in its converged state, i.e., from the 26th iteration. 1.4.2 Comparison of the Feedforward Control Schemes We first show the representative course over time for A = 1mm and f = 0.5 Hz for the existing P-PI-PI cascade and the three different feedforward control schemes in Fig. 1.6. It can be seen that the achieved displacement t Z comes closer to the desired displacement td,Z for all feedforward control schemes. In Fig. 1.6b, the error between desired and achieved displacement, i.e., tdZ −t Z, is visualized. Here, it is also obvious that all feedforward control schemes reduce the error. The MFIIC and VFF especially reduce the magnitude of the error to approximately 10–20%. The largest magnitude of the error occurs at the turning points of the trajectory, where the velocity is zero and static friction has large influence. In order to make general statements about the tracking performance of all methods for different trajectories, we investigate the results for all amplitudes A∈ {1, 3, 5}mm and frequencies f ∈ {0.25,0.5,1, 2}Hz in a synchronization subspace plot (SSP). The SSP plot can be used to analyze the tracking performance for sine trajectories. The desired displacement td,Z is plotted over the achieved displacement t Z. In the case of ideal tracking, a slope with an incline of 45◦ results. If the incline is lower it means that amplitude overshoot has occurred, if the incline is higher, it means that the Stewart Platform undershot. An ellipse forms in the case of phase errors: In the case of a phase lag, the ellipse is passed through clockwise, and in the case of phase lead the ellipse is passed through counterclockwise. Figure 1.7 visualizes the SSP plots for the measured amplitudes and frequencies. All amplitudes A∈ {1, 3, 5}mmwere only investigated for frequencies f ∈ {0.25,0.5}Hz. For frequency f =1 Hz we investigated amplitudes A ∈ {1, 3}mm 0 0.5 1 1.5 2 –1 –0.5 0 0.5 1 Time in s Displacement inmm Desired P-PI-PI MBDC MFIIC VFF (a) 0 0.5 1 1.5 2 –0.2 –0.1 0 0.1 0.2 Time in s Error inmm P-PI-PI MBDC MFIIC VFF (b) Fig. 1.6 The plots show the results for the P-PI-PI cascade (in black, dashed line), the MBDC (in blue, solid line), the MFIIC (in green, solid line) and the VFF (in orange, solid line) for the desired trajectory withf =0.5Hz andA=1mm. (a) The desired (td,Z) and achieved displacement t Z in Z-direction. (b) The error signal between desired and achieved displacement over time
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