# Flattening the Curve of Flexible Space Robotics [PERSON]
## 3 Results
### Gain Stabilization Plus Structural Filters (Bandpass and Notch) for the First Flexible Mode  
Gain stabilization was augmented with structural filters using Equation (33), where structural resonant frequencies were used as filter natural frequencies and filter damping was adjusted to visually \"flatten the curve\" of the frequency response plot. First, a bandpass variant was designed for the first anti-resonant spike. Next, a notch variant was designed for the first resonant spike. Filter details are available in the Appendix B to aid repeatability. Figure 15 displays the results. Notice in the step response plot in Figure 15, the nominal performances are nearly indistinguishable, while in the \"inset\" plot distinct differences are apparent, where quantitative results are displayed in Table 1. Figure 16 displays the four frequency response plots overlaid. The nominal frequency response (prior to gain stabilization) is lowest, indicated by the dashed blue line. The thin-solid red line indicates a gain-stabilized case. Meanwhile, the overlaid dashed gold line indicates bandpass compensation of the first anti-resonance. Finally, the \"flattened\" frequency response plot's thin, dotted black line indicates full mode compensation with the addition of a notch filter at the first structural resonant frequency. Effects on the frequency response phase is displayed in Table 1. Addition of the bandpass filter at the first anti-resonance leads to the highest gain margin, with an excellent phase margin, while addition of the notch filter at the first resonance increases the phase margin substantially further with a slight decrease in gain stability.  
Figure 16: Four computer simulations of the control of rigid body and first flexible mode with frequency in hertz on the abscissa: uncontrolled (narrow, dashed line), PID compensated (thin, solid line), PID compensated with bandpass filter of first antiresonance (thick, dashed line), and PID compensated with bandpass and notch filter of first mode—anti-resonance and resonance (dotted line). (**a**) Response magnitude (first mode zoomed in the inset) on the ordinate; (**b**) phase in degrees on the ordinate.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**System** & **Gain Margin (dB)** & **Phase Margin (Degrees)** & **Stable/Unstable** \\ \hline Uncontrolled & 64.5 & 0.000673 & Stable \\ \hline PID controlled & 4.95 & 17.9 & Stable \\ \hline PID + bandpass & \(-26.2\)1  & 80.2 & Stable \\ \hline PID + bandpass + notch & \(-25.9\)1  & 133 & Stable \\ \hline \hline \end{tabular}
\end{table}
Table 1: Compensation of rigid body plus first flexible mode (resonance and anti-resonance) with step input.  
Figure 15: Five computer simulations of the control of rigid body; the compensated first flexible anti-resonance, compensated first resonance, compensated second anti-resonance, and fully compensated first two modes, with frequency in hertz on the abscissa and response magnitude in degrees on the ordinate. PID (thick, solid red line); PID + bandpass 1 (dotted blue line); and PID + bandpass 1 + notch 1 (thin, solid black line) illustrating substantial improvement by compensating for first anti-resonance, but negligible difference also compensating for first resonance.