# Adaptive Deterministic Vibration Control of a Piezo-Actuated Active-Passive Isolation Structure
## 1 Introduction
The repetitive control system is to implant the dynamic model of external interference signal into the closed-loop system to realize the suppression of known deterministic vibration disturbance. It can be seen that when the disturbance is time-varying, the dynamic model of disturbance will also change, and it will be difficult for the original repetitive control system to suppress the new disturbance. Recently, a novel compensator capable of minimum-time performance of an in-plane maneuver with zero residual vibration and zero residual vibration-derivative at the end of the maneuver is proposed in Reference [6]. The novel compensator has a whiplash nature of first commanding maneuver states in the opposite direction of the desired end state to achieve the minimum-time performance of the robot manipulation. However, the control performance subjected to the external unknown vibration disturbances needs to be further evaluated. Reference [25] summarized the latest development and application of the Youla parameterization method, in which the parameterization process of the control system is analyzed in detail. The direct or indirect adaptive disturbance suppression algorithm based on Youla parameterization was compared and analyzed in Reference [26], which indicated that the parameterization of the controller is beneficial to the combination of adaptive algorithm. It can be seen that the model of vibration disturbances can be inserted in the controller by the Youla parameterization method; by augmenting the free parameter \(Q\in\mathrm{RH}_{\infty}\), the parameterized controller is designed as a global stable controller that can fit almost any dimension internal model of the multifrequency vibration disturbance. The adaptive algorithm can be introduced to adjust the free parameter \(Q\) of the resulting parameterized controller online according to the vibration disturbance, so the unknown or time-varying deterministic vibration disturbance will be gradually eliminated. [PERSON] et al. designed a Youla parameterized adaptive controller to realize the elimination of the sinusoidal noise disturbance in an acoustic duct [27]. [PERSON] et al. modified the Youla parameterized controller with an adaptive notch filter to compensate for the airflow disturbance and vibration of the disk in the data storage system [28]. However, in this paper, the suppression of the low-frequency deterministic vibration and the band-limited local loop shaping of the closed-loop system near the system's natural frequency will be considered simultaneously. Therefore, an observer-based state feedback central controller with an augmented \(Q\) parameter will be introduced. The central controller will be designed to shape the band-limited local loop of the closed-loop system, and the augmented Youla (\(Q\)) parameterized controller will be adjusted online to eliminate the low-frequency deterministic vibration. The Youla (\(Q\)) parameterized adaptive active vibration controller is designed in two steps. The inner-loop central controller is a linear quadratic Gaussian (LQG) controller that is used to shape the band-limited local loop of the closed-loop system near the system's natural frequency. Then, by augmenting the LQG controller, the Youla parameterized adaptive controller with the recursive least square (RLS) adaptive algorithm is formulated to completely suppress the deterministic residual vibration. The proposed adaptive active vibration control method is applied to the piezo-actuated active-passive isolation structure. The experimental results illustrate that the proposed adaptive active vibration control method can effectively optimize the vibration transmissibility curve near the system's natural frequency and suppress the unknown and time-varying low-frequency deterministic residual vibration. This paper is organized as follows. The piezo-actuated active-passive isolation structure is analyzed in Section 2.