# Adaptive Deterministic Vibration Control of a Piezo-Actuated Active-Passive Isolation Structure
## 1 Introduction
As long as the frequency of the reference signal obtained through the sensor is accurate enough, the feedforward control can realize the effective compensation of residual vibration. However, in some cases, the reference signal of the vibration source is difficult to obtain accurately, which will degrade the feedforward control performance. Feedback control only needs to detect the residual vibration of the controlled object, without paying attention to the source of vibration. Reference [14] presented an investigation on the performance of active vibration control of the rib-stiffened plate by using only residual velocity feedback signal. An adaptive feedback control system is designed for the unknown vibration disturbance suppression of a lathe system in Reference [15], in which an adaptive algorithm is introduced to adjust the feedback controller in real time according to the unknown residual acceleration feedback signal. Besides, compared with feedback control, feedforward control will use more sensors that will not be conducive to the light weight and reliability of the vibration isolation system. Therefore, feedback control should be considered. The control approaches of lead/lag, H\({}_{\infty}\), and linear quadratic Gaussian (LQG) are common feedback controls for AVC systems. [PERSON] and [PERSON] applied the proportion integral differential(PID) control approach to active vibration control of building structures and analyzed the system stability [16]. [PERSON] et al. designed an optimal robust minimax LQG controller by the proper choice of the weight parameters and applied it to the vibration control of flexible beams [17]. [PERSON] et al. developed an H\({}_{\infty}\) controller to suppress frequency-varying vibration disturbances of the floating raft isolation system [18]. [PERSON] studied a simultaneous mixed LQR/H\({}_{\infty}\) control approach based on the vibration suppression of quarter-car active suspension systems [19]. The feedback controller mentioned above reduces the sensitivity function of the desired frequency range by obtaining a high open-loop gain, and then realizes the vibration suppression in the corresponding frequency range. However, [PERSON]'s sensitivity integral theorem indicates that the sensitivity function \(S(jw)\) should satisfy \(\int_{0}^{\infty}\ln\lvert S(jw)\rvert\,dw=0\) for the stable plant, and the area of sensitivity reduction (\(\ln\lvert S\rvert\) negative) should equal the area of sensitivity increase (\(\ln\lvert S\rvert\) positive) [20]. For the active vibration control system, the concrete manifestation is that the vibration will be inevitably amplified outside the frequency range where the vibration is expected to be suppressed, which is also known as the \"waterbed effect\". In addition to this important limitation, if the open-loop gain is too high, it will also reduce the phase margin and decrease the robustness. Therefore, these controllers cannot completely suppress the deterministic vibration disturbance, especially with multifrequency and time-varying characteristics. The feedback control based on the disturbance observer (DOB) [21] and the internal model principle (IMP) [22] is an effective solution for the suppression of deterministic vibration disturbances. The related feedback control methods based on the DOB were compared and summarized in Reference [23], and the design of the DOB was comprehensively analyzed. The reference indicated that the key to the design of the feedback control system based on the DOB is the selection of its low-pass filter with a proper cutoff frequency, which is related to the external disturbance's property. When the disturbance changes in a wide range due to time-varying, the disturbance suppression effect is limited. [PERSON] et al. designed a zero-phase odd repetitive controller that was used for the suppression of deterministic vibration in a magnetically suspended centrifugal compressor based on IMP [24].