## 5 Results and Discussion
### Piezoelectric Vibration Control  
This study examines the active vibration suppression effects of the used three control algorithms: AFSMC, RC, and hybrid AFSMC/RC. The design parameters of the AFSMC controller were set as follows: \(gu=0.2\), \(gs=0.4\), and \(\gamma_{\mathrm{A}}=0.2\). Using the time-domain system identification method, we obtained the mathematical model of the thin flexible plate system for the repetitive controller design. The relevant parameter setting was \(k_{\mathrm{r}}=0.15\), and a low-pass \(q\) filter with a 10 Hz cutoff frequency was adopted to ensure the robustness of the repetitive controller. Figure 12 shows the vibration response results of the thin plate with active vibration-suppression control performed by various controllers. The AFSMC controller proved highly effective in suppressing vibrations in the thin plate; however, periodic responses remained while in steady state. RC proved more effective in reducing the periodic output in steady state; however, the initial rate of convergence was slow (40 s to reach steady state), and the amplitudes were greater. This can be attributed to an overly conservative learning gain \(k_{\mathrm{r}}\) and low-pass filter \(q\), the purpose of which was to ensure that the unmodeled dynamics in the system did not cause instability in the RC system. The hybrid AFSMC/RC control algorithm provided the benefits of AFSMC as well as RC, resulting in swift convergence (at approximately 25 s) as well as improved output responses during steady state. Figure 13 presents a comparison of the performance of the three controllers with regard to the active suppression of vibrations. The frequency band was divided into three ranges to facilitate analysis: (1) low-frequency peaks were dominated by the dynamic response of the mechanical system; (2) peak values at 1 Hz were caused by periodic excitations of the motor; and (3) peak values at 1.4 Hz were associated with the natural resonance frequency of the thin plate. AFSMC is clearly able to reduce the effects of disturbances at low frequencies and largely eliminate disturbances at 1 Hz resulting from the periodic motions of the motor. RC proved effective in suppressing vibration responses at 1 Hz; however, it was shown to amplify responses at lower frequencies as well as at natural resonance frequency of the thin plate. The hybrid AFSMC/RC control was shown to reduce output responses in all three frequency ranges. Figure 11: Flexible plate output response and FFT analysis without vibration suppression: (**a**) time response of uncontrolled system output, (**b**) FFT results at 1 Hz, and (**c**) FFT results at resonance frequency. Figure 14 shows the control input for each type of control. As can be seen, the control input of AFSMC shows continuous fluctuations. This is because the rule bank changes with each sampling point, and the continuous switching back and forth produces this chattering. The RC maintains periodic control input after the output converges to cope with the continuous excitation. As for the  
Figure 12: Piezoelectric vibration control in flexible structure: Controlled output responses obtained using: (**top**) AFSMC; (**middle**) RC; and (**bottom**) hybrid AFSMC/RC method. Figure 13: Piezoelectric vibration control for the flexible structure system: FFT analysis (**left**) low frequency range; (**middle**) 1 Hz; and (**right**) resonance frequency of 1.4 Hz. control input of the hybrid control, the graph shows overlapping effects of the swift rule bank switching in AFSMC and the learning process in RC, which also presents greater control efforts. Figures 15 and 16, respectively, present the variations in the rule banks of AFSMC and hybrid control. The resulting variations in the two rule banks are similar; i.e., they both converge to within a stable range because they use the same AFSMC design parameters. When using the hybrid control, rules C3, C4, and C5 resulted in smaller fluctuations.