## 1 Introduction  
Due to the increasing requirement for accuracy and stability in research and manufacturing, the issue of micro-vibration, which widely exists in engineering applications, has become a critical factor that reduces the performance of high-precision machines. For example, as the state-of-art transmission electron microscope (TEM) has met the VC-G [1, 2] or higher vibration standard (vibration velocity lower than 0.78 \(\upmu\)m/s), the performance requirements of the AVRS are more stringent than before. Active vibration control technology (AVCT) originated from the semi-active control methods represented by skyhook damping [3], for which it is difficult to obtain the absolute inertial reference frame (AIRF) that is required in practice. In a previous paper [4], a virtual skyhook damping isolator was proposed without the AIRF, which used a double cascade damping structure to suppress the resonance peak. In the engineering field, various high-precision instruments are equipped with passive vibration isolation systems, such as pneumatic or coil springs, which make them more sensitive to low-frequency vibrations. Because of the poor performance of passive and semi-active damping technology at low frequencies, the AVCT technique is urgently needed. The combination of AVCT and passive vibration isolators is one of the mainstream methods used today, in which passive isolators attenuate high-frequency vibrations while AVCT reduces the low-frequency vibrations. The AVRS overcomes the disadvantages of passive vibration reduction systems by integrating the active execution components. Control algorithms [5, 6, 7, 8] based on closed-loop PID have been proposed. Most of these methods are aimed at achieving the ideal dynamic capability of skyhook damping while balancing the robustness and stability of the system. Although appropriately increased feedback control (FBC) gain can enhance the vibration reduction effect, the adjustment range of parameters is greatly restricted by factors such as noise, decoupling accuracy,and ground and table disturbance [9]. Furthermore, the FBC response speed is greatly constrained by inherent algorithm delay and the hysteresis of the actuators. The feed-forward control (FFC) strategy was first proposed to suppress the direct disturbances of AVRS, while the compensation control signal can be calculated from its regularity [10]. Under the premise of obtaining an accurate FF model, FFC can theoretically eliminate the influence of ground vibration on the platform. A previous paper [11] presented several prediction methods for FF signals and analyzed their tracking performance using filters. Irrespective of the FF signal prediction error changes with the increase in the frequency, proper parameters in FFC do not affect the closed-loop stability, which is merely determined by the loop transfer function [12]. To further improve the performance of AVRS, the FBC and FFC strategies were incorporated into a six-degree-of-freedom (DOF) isolation system that is based on absolute accelerator measurement [13], where FBC uses a genetic algorithm to suppress payload vibration, and FFC prevents ground vibration from transmitting to the platform. Note that because of the hysteresis and the low response speed of the voice-coil motor, together with the high-order bending mode of the system, not only can the AVRS operating at fixed parameters be unstable, but it is also limited to suppressing the vibration below 100 Hz rather than the entire frequency range [14]. For the frequency above 100 Hz, vibration attenuation of AVRS mainly relies on passive isolation, while the AVCT focuses on the improvement of the low-frequency performance. Furthermore, although the ideal model of AVRS is difficult to obtain, together with the rapid development of AVCT, various advanced control methods have been proposed for tracking the optimal parameters according to its specific applications [15; 16; 17].