# Flattening the Curve of Flexible Space Robotics [PERSON]
## 3 Results
### Gain Stabilization Plus Structural Filters (Bandpass and Notch) for Two Flexible Modes  
Gain stabilization with the fully compensated first mode was augmented with structural filters using Equation (33), where the structural resonant frequencies were used as filter natural frequencies and filter damping was adjusted to visually \"flatten the curve\" of the frequency response plot at the location of the second mode. First, a bandpass variant was designed for the second anti-resonant spike. Next, a notch variant was designed for the second resonant spike. Figure 17 displays the results. Notice once again in the step response plot in Figure 17 the nominal performances are nearly indistinguishable, while in the \"inset\" plot distinct differences are apparent, where quantitative results are displayed in Table 2. Figure 18 displays the four frequency response plots overlaid. The nominal frequency response (after gain stabilization) is indicated by the dashed blue line. The thin, solid red line indicates the addition of full compensation of the first resonant mode. Meanwhile, the overlaid dashed gold line indicates bandpass compensation of the second 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 two structural resonant frequencies. Effects on frequency response phase is displayed in Table 2. Addition of both bandpass filter at the second anti-resonance and notch filter at the second resonance increases reduces both the gain and phase margin substantially. Compensation of the second resonant structure mode should therefore be avoided, despite an instinct to \"flatten the curve\" of the frequency response plot magnitude. \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 + mode 1** & \(-\)**25.91** & **133** & **Stable** \\ \hline PID + mode 1 + mode 2 & 5.52 & 18.3 & Stable \\ \hline PID + mode 1 + mode 2 & & & \\ + bandpass & 0.625 & 4.01 & Stable \\ \hline \hline \end{tabular}  
* Negative gain margins indicate that stability is lost by decreasing the gain, while positive gain margins indicate that stability is lost by increasing the gain.  
\end{table} 
Table 2: Compensation of rigid body plus first two flexible modes (resonance and anti-resonance) with step input. Figure 17: Five computer simulations of the control of rigid body; the compensated first flexible mode, 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 + mode 1 + bandpass 2 (dotted blue line); PID + mode 1 + bandpass 2 + notch 2 (thin, solid black line), again illustrating substantial improvement by compensating for additional (second) anti-resonance, but negligible difference also compensating for the first additional (second). Table 2 quantitative data corresponding to qualitative presentation in Figure 17.  
Figure 19: Five computer simulations of control of the rigid body; compensated first flexible mode; 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. Table 3 quantitative data corresponding to qualitative presentation in Figure 19. PID (thick, solid red line); PID + modes 1–2 + bandpass 3 (dotted blue line); PID + modes 1–3 (thin, solid black line) again illustrating substantial improvement by compensating for additional (second) anti-resonance, but negligible difference also compensating for first additional (second).