# Adaptive Deterministic Vibration Control of a Piezo-Actuated Active-Passive Isolation Structure
## 1 Introduction  
With the improvement of the performance of optical equipment carried by on-orbit spacecraft, the requirement of vibration isolation on the space instrument platform is increasing in the space microgravity environment. For example, the vibration caused by the mass imbalance of the momentum wheels should be suppressed to a micrometers level to ensure the image quality of satellites [1]. The main vibration disturbance of on-orbit spacecraft is often caused by reaction/momentum wheel assemblies [2]. Additionally, with the introduction of the large flexible appendage systems of spacecraft with a cantilever structure, such as solar panel, communication antenna, and space robotic arm, the resonant mode may be excited during the operation of the flexible system, and the deterministic vibration disturbance will be formed [3]. Particularly, the static imbalance, dynamic imbalance, and bearing imperfection of assemblies will excite a large amplitude deterministic vibration disturbance [4]. These deterministic vibration disturbances will have multifrequency and time-varying characteristics, mainly distributed in the low-frequency range below 150 Hz [5], and even ultralight and large size flexible structures may produce vibration around 10 Hz [6]. To suppress the vibration disturbance from the base of the optical equipment placement platform, the passive isolation platform is widely used to carry the precision opticalequipment on spacecraft [2]. With the higher performance requirement of vibration isolation, the limitations of the passive isolation platform have made the active vibration control system more attractive [7]. One of the limitations that cannot be ignored is that the low-frequency deterministic vibration disturbance is not sufficiently isolated, and especially near the system natural frequency of the isolation system, the disturbance may be amplified [8]. Moreover, there is an unavoidable design contradiction between the structural bearing capability and low-frequency vibration isolation. Therefore, some active vibration control strategies are proposed to improve the low-frequency vibration isolation performance. Among many vibration isolation platforms on spacecraft, the Stewart-like parallel kinematic platform not only has advantages of stability, high precision, and high payload-bearing capability, but also can provide support for different active control methods, and the related vibration isolation systems have been discussed in reference [9]. The key to the isolation performance of the Stewart-like platform is the design of the single support axis vibration isolation system [10]. Therefore, this paper will focus on the design of an adaptive active control system for the single-axis isolation structure, which can be used to suppress the low-frequency unknown deterministic residual vibration with multifrequency and time-varying characteristics. Generally, the main methods of active vibration control (AVC) are feedforward and feedback control. Feedforward control directly uses the reference signal of the vibration source to eliminate the residual vibration. [PERSON] et al. discussed the characteristics of feedforward and feedback systems for the active vibration control, and pointed out that the feedforward system relies on the existence of some prior knowledge of the vibration disturbance to be controlled, and the prior knowledge is contained in a reference signal that drives the secondary source through the controller [11]. [PERSON] et al. present a novel feedforward control algorithm with the orthogonal pair-wise reference frequency regulator, and the multiple-frequency vibration was dramatically suppressed with an improved convergence rate [12]. An adaptive feedforward vibration control system with variable step size is developed in Reference [13], which was successfully implemented to suppress the vibration of a kind of thin-walled structure.