# Adaptive Deterministic Vibration Control of a Piezo-Actuated Active-Passive Isolation Structure
## 3 Design of Adaptive Active Vibration Controller
### Youla Parameterization of the Inner-Loop Central Controller  
The central controller cannot completely suppress the low-frequency deterministic residual vibration, especially with multifrequency and time-varying characteristics. Then, the Youla parameterized controller is formulated by augmenting the central controller with the \(Q\) parameter based on the internal model principle (IMP), and the adaptive algorithm is developed to online tune the free \(Q\) parameter according to the variable residual vibration. The Youla parameterization of the controller consists of the interconnection of two blocks: a fixed block \(J\) and a stable system \(Q\) that can be chosen as desired [31]. The \(J\) block can be written as:  
\[J:\left\{ \begin{array}{l}\hat{x}(k+1)=(\widetilde{A}+\bar{B}F+L\widetilde{C}_{y} )\hat{x}(k)-Ly(k)+\bar{B}y_{Q}(k)\quad\hat{x}(0)=\hat{x}_{0}\\ u(k)=F\hat{x}(k)+y_{Q}(k)\\ y(k)-\hat{y}(k)=y(k)-\widetilde{C}_{y}\hat{x}(k),\end{array} \right. \tag{6}\]  
The \(Q\) block can be written as:  
\[Q:\left\{ \begin{array}{l}x_{Q}(k+1)=A_{Q}x_{Q}(k)+B_{Q}(y(k)-\hat{y}(k)), \quad x_{Q}(0)=x_{Q,0}\\ y_{Q}(k)=C_{Q}x_{Q}(k)\end{array} \right. \tag{7}\]  
Then, the Youla parameterized controller \(K(Q)\) consists of two interconnected blocks \(J\) and \(Q\) as shown in Figure 4.  
To realize the design of the adaptive regulator, the augmented plant \(G\) and the controller \(J\) are integrated into \(T\) as shown in Figure 5.  
Therefore, \(T\) can be represented as follows:  
\[\left[ \begin{array}{c}e(k)\\ y(k)-\hat{y}(k)\end{array} \right]=T\left[ \begin{array}{c}w(k)\\ y_{Q}(k)\end{array} \right] \tag{8}\]  
where  
\[T=\left[ \begin{array}{cc}T_{11}&T_{12}\\ T_{21}&T_{22}\end{array} \right] \tag{9}\]  
\[T_{11}:\left[ \begin{array}{cc}\widetilde{A}+\widetilde{B}_{1}F&\widetilde{B }F&\widetilde{E}_{x}\\ 0&\widetilde{A}+L\widetilde{C}_{e}&-\widetilde{E}_{x}-L\widetilde{E}_{e}\\ \widetilde{C}_{e}&0&\widetilde{E}_{e}\end{array} \right] \tag{10}\]  
\[T_{12}:\left[ \begin{array}{cc}\widetilde{A}+\bar{B}F&\widetilde{B}\\ \widetilde{C}_{e}&0\end{array} \right] \tag{11}\]  
\[T_{21}:\left[ \begin{array}{cc}\widetilde{A}+L\widetilde{C}_{e}&-\widetilde {E}_{x}-L\widetilde{E}_{e}\\ -\widetilde{C}_{e}&\widetilde{E}_{e}\end{array} \right] \tag{12}\]  
\[T_{22}:\left[ \begin{array}{cc}\widetilde{A}+L\widetilde{C}_{e}&0\\ -\widetilde{C}_{e}&0\end{array} \right] \tag{13}\]  
Figure 4: The closed-loop system with \(Q\) parameterized controller. Figure 5: The closed-loop system with the integration of \(G\) and \(J\) blocks. Let \(W(z)\) and \(E(z)\) denote the \(Z\) transform of the disturbance input \(w\) and the system error \(e\). The closed-loop transfer function from \(w\) to \(e\) can be obtained by linear fractional transformation as follows:  
\[E(z)=[T_{11}(z)+T_{12}(z)Q(z)T_{21}(z)]W(z) \tag{14}\]  
To adjust the adaptive regulator for multifrequency deterministic residual vibration and take into account the stability requirement of the closed-loop system, the \(Q\) parameter is considered as:  
\[Q(z)=\sum\limits_{q=1}^{n_{q}}\theta_{q}z^{1-q}F(z) \tag{15}\]  
where \(F(z)=\frac{b_{1}z^{m-1}+\cdots b_{m}}{z^{m}+a_{2}z^{m-1}+\cdots a_{m}}\) is a stable weighting function used to adjust the dynamic properties of \(Q(z)\). By properly choosing \(F(z)\), the robustness of the closed-loop system can be improved. According to the internal model principle, let \(\theta=\left[ \begin{array}{cccc}\theta_{1}&\theta_{2}&\cdots&\theta_{n_{q}- 1}&\theta_{n_{q}}\end{array} \right]^{\mathsf{T}}\) be the free parameter, then the attenuation conditions of the deterministic residual vibration can be written as follows [32]:  
\[A_{\theta}\theta+B_{\theta}=0 \tag{16}\]  
where  
\[A_{\theta}=\left[ \begin{array}{cccc}V_{1,re}(p_{1})&\cdots&V_{n_{q},re}(p_{ 1})\\ V_{1,im}(p_{1})&\cdots&V_{n_{q},im}(p_{1})\\ \vdots&\cdots&\vdots\\ V_{1,re}(p_{k_{0}})&\cdots&V_{n_{q},re}(p_{k_{0}})\\ V_{1,im}(p_{k_{0}})&\cdots&V_{n_{q},im}(p_{k_{0}})\end{array} \right] \tag{17}\]  
\[B_{\theta}=\left[