## 1 Introduction  
Vibration, a ubiquitous physical phenomenon in the field of engineering and technology, has brought many inconveniences and hazards to daily life, scientific research, and production. For example, vibrations may reduce the processing accuracy and the surface finish of the workpieces, affect the accuracy of precision equipments, cause the chatter of aircrafts and ship hulls, and decrease the life of mechanical systems [1,2]. In recent years, high requirement on vibration control has been paid in processing equipments, measuring instruments, mechanical systems, and other engineering applications. Therefore, an effective vibration isolation system should be developed [3,4]. In recent years, a lot of vibration isolation systems have been developed and exploited effectively to control vibration. Vibration isolation systems can be divided into two kinds in general: passive isolation systems and active isolation systems [5; 6]. Passive isolation systems cannot require external power sources and control systems during the action, which operate with negative work to consume the kinetic energy for vibration isolation systems [7; 8; 9]. They have been extensively used to isolate vibration because of the simple structure and the low cost [10; 11; 12]. The aforementioned passive isolation systems can effectively isolate vibration without any external power and control systems. However, the fixed-structure parameters (stiffness and damping coefficient) of passive isolation systems fail to meet the high requirements on various dynamic performances. In comparison with passive isolation systems, active electromagnetic isolation systems can effectively solve these problems with controllable stiffness and damping coefficient to achieve dynamic performance indexes based on various control methods. Lots of active electromagnetic isolation systems have been designed and utilized to control vibration. [PERSON] et al. designed the vibration isolation system to provide the electro-magnetic levitation force based on the magnetic actuator and the pneumatic elastic chamber for semiconductor processing equipments [13]. [PERSON] et al. developed a novel vibration isolation system with a mechanical spring, two electromagnets, and a permanent magnet which are built for a magnetic spring [14]. In order to overcome the problems of the passive isolation systems, [PERSON] et al. presented a hybrid-type isolation system based on the electromagnetic and pneumatic force for mechanical systems [15]. Although many methods have been developed and used effectively for designing active isolation systems, active control methods should be studied due to the fact that the controller can improve the performance of vibration isolation systems. According to different application conditions and requirements on vibration control accuracy, different control strategies and algorithms are needed to design the controller of an active vibration isolation system. At present, the control methods of active vibration isolation systems mainly include direct output feedback, adaptive control, optimal control, neural network control, and so on. [PERSON] et al. proposed a fuzzy controller for a vibration isolation system disturbed by regular excitation signals [16]. [PERSON] et al. employed linear quadratic gaussian control and \(\mu\)-synthesis in the controller for the active isolation system [17]. [PERSON] et al. employed an adaptive controller for actively controlling an actuator for vibration isolation, and the experimental results indicate the validity of the adaptive controller for active vibration isolation [18]. [PERSON] et al. utilized linear parameter-varying control techniques to design adaptive active vibration isolation systems for a single degree-of-freedom vibration isolation system [19]. [PERSON] et al. adopted neural network control methods to attenuate external disturbance and isolate vibration in an isolation platform [20]. [PERSON] et al.