## 4 Experimental Verification
### Discussion  
In the MHL model, both the rate-dependent term and the dynamic term possess the ability to describe dynamic characteristics. Therefore, defining the frequency range for each term becomes an important question. For the rate-dependent term, a corresponding inverse model can be established and utilized for feedforward compensation to achieve favorable open-loop frequency characteristics. Based on this line of thinking, it would be ideal to have the rate-dependent term describe as wide a frequency range as possible. However, in reality, as the frequency range described by the rate-dependent term widens, its modeling accuracy decreases. Thus, a more reasonable method is to design the rate-dependent term based on the system's application requirements. Taking FSM modeling as an example, its application background is satellite laser communication. Since the vibration spectrum of satellite platforms is mainly concentrated below 100 Hz, if the rate-dependent term can cover the frequency range up to 100 Hz, the response characteristics within this frequency range are not necessarily the same. Figure 17: Tracking results of dual-axis spiral trajectory for FSM: (**a**) dual-axis spiral scan curves; (**b**) \(x\)-axis tracking results; (**c**) \(y\)-axis tracking results. Figure 18: Frequency–response curves of closed-loop control system. band can be significantly improved during inverse compensation. As shown in Figure 18, at 100 Hz, the closed-loop system exhibits an amplitude-frequency characteristic of 0.2 dB and a phase angle delay of 0.69\({}^{\circ}\), indicating that the closed-loop system can effectively track signals below 100 Hz and suppress disturbances within this frequency range. This study investigates a fast steering mirror (FSM) system employing a two-dimensional (\(x\)-/\(y\)-axes) piezoelectric actuation architecture for satellite laser communication beam control, achieving a 2 mrad deflection range with a 0.05 \(\mu\)rad resolution. As shown in Figure 18, the proposed composite control method demonstrates a closed-loop bandwidth of 1.5 KHz and a settling time of 2 ms. The tracking test results in Section 4.4 reveal outstanding control performance with a normalized maximum error (NME) consistently below 1% and a normalized root-mean-square error (NRMSE) maintained within 0.6% under various test signals, confirming the superior capabilities in high-bandwidth, rapid-response, and high-precision control. For satellite laser communication applications, the positioning accuracy of the beam control system may be affected by two primary factors: temperature-induced parameter drift in the piezoelectric ceramics and external mechanical disturbances from satellite platform vibrations. However, the temperature dependence of piezoelectric parameters typically exhibits slow time-varying characteristics, while platform vibration energy is predominantly concentrated below 100 Hz. Consequently, the developed composite control method with a 1.5 KHz closed-loop bandwidth effectively suppresses both the temperature-dependent parameter variations and platform vibration disturbances below 100 Hz. The MHL model identified in the experiments is constructed based on an FSM system equipped with a high-voltage driver. Due to the integration of a notch filter in the driver, the frequency characteristics of the signal within the relevant range are adjusted. This results in the resonance phenomenon, originally anticipated near 2 kHz, not being displayed in Figure 10. This also explains why the modeling errors in Figures 12 and 13 do not exhibit a monotonically increasing trend with the input frequency. However, it should be clarified that this adjustment does not alter the inherent resonant frequency of the FSM system itself. It is noteworthy that compared with the rate-dependent hysteresis model presented in [23], the proposed model, which combines the time delay term and the linear dynamic term, effectively describes the dynamic characteristics of the system in the frequency range of 4 kHz.