## 3 Controllable and Uncontrollable Modes for AVC
### Theoretical Analysis  
In case of a persistent sinusoidal disturbance, the open-loop and closed-loop vibration energies are obtained through a covariance analysis [28]. The covariance matrix of the sinusoidal disturbance is denoted as:  
\[\mathrm{E}[F(t)F(t+\tau)]=|F|^{2}\mathrm{e}^{i\omega\tau}, \tag{7}\]  
where \(|F|\) is the amplitude of the persistent disturbance and \(\omega\) is the angular frequency of disturbance. Based on the Lyapunov equation [28], the total open-loop vibration energy is:  
\[\mathrm{E}_{o}=\frac{1}{2}\sum_{l=1}^{MN}(x_{2i-1,2i-1}+x_{2i,2i})=\frac{1}{2} \sum_{m=1}^{M}\sum_{n=1}^{N}\frac{\beta_{mn}}{\left(\omega_{mn}^{2}+\mathrm{i }\eta_{p}\omega\omega_{mn}-\omega^{2}\right)^{2}} \tag{8}\]  
where \(l=mn\) and \(\beta_{mn}=|F|^{2}e^{i\omega\tau}\varrho_{mn}^{2}(x_{F},y_{F})/\rho_{p}^{2}h_ {p}^{2}\). Figure 4: Schematic diagram of a typical smart structure. Figure 5: (**a**) Description of the sensing voltage of PZT and the acceleration in a controllable mode; (**b**) description of the sensing voltage of PZT and the acceleration in an uncontrolled mode. For the AVC system, \(e(t)\) tends to zero due to the effect of control signal \(y(t)\) in FxLMS. Based on Equation (5), the ideal control signal (y(t)) is:  
\[y(t)=-\frac{\mathbf{\Phi}_{s}^{{}^{\prime\prime}}\mathbf{\Pi}\mathbf{\Phi}_{F}^{ \mathrm{T}}}{\mathbf{\Phi}_{s}^{{}^{\prime\prime}}\mathbf{\Pi}\mathbf{\Phi}_{a }^{{}^{\prime\prime}}\mathbf{\Gamma}}F(t) \tag{9}\]  
where \(\mathbf{\Pi}=\mathrm{diag}\big{[}1/\left(\omega_{mn}^{2}+\mathsf{j}\eta_{p} \omega\omega_{mn}-\omega^{2}\right)\big{]}\). Substituting Equation (9) into Equation (5) yields:  
\[\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\widetilde{\mathbf{C}}\mathbf{B}_{F}F(t) \tag{10}\]  
\[\mathrm{where}\quad\widetilde{\mathbf{C}}\quad=\quad\left[ \begin{array}{ ccc}\widetilde{\mathbf{C}}_{1,1}&\widetilde{\mathbf{C}}_{1,2}&\cdots& \widetilde{\mathbf{C}}_{1,MN}\\ \widetilde{\mathbf{C}}_{2,1}&\widetilde{\mathbf{C}}_{2,2}&\cdots&\widetilde {\mathbf{C}}_{2,MN}\\ \vdots&\vdots&&\vdots\\ \widetilde{\mathbf{C}}_{MN,1}&\widetilde{\mathbf{C}}_{MN,2}&\cdots&\widetilde {\mathbf{C}}_{MN,MN}\end{array} \right];\quad\widetilde{\mathbf{C}}_{i,j}= \quad\left[ \begin{array}{c}\widetilde{c}_{i,j}&0\\ 0&0\end{array} \right];\]  
\[\widetilde{c}_{i,j}=\left\{ \begin{array}{c}-\frac{\alpha_{j}\mathsf{j} \mathsf{j}\left(x_{x},y_{a},\Delta_{x},\Delta_{y}\right)}{\sum\limits_{k=1}^{ MN}\alpha_{k}\mathsf{j}\left(x_{x}y_{a},\Delta_{x},\Delta_{y}\right)}\quad i \
eq j\\ 1-\frac{\alpha_{j}\mathsf{j}\mathsf{j}\left(x_{x},y_{a},\Delta_{x},\Delta_{y} \right)}{\sum\limits_{k=1}^{MN}\alpha_{k}\mathsf{j}\left(x_{x}y_{a},\Delta_{x },\Delta_{y}\right)}\quad i=j\end{array} \right. ;\quad\text{the term $\alpha_{j}$ is given by:}\\ \alpha_{j}=\phi_{j}\big{(}x_{s},y_{s},\Delta_{x},\Delta_{y}\big{)}\,/\left( \omega_{j}^{2}+\mathsf{j}\eta_{p}\omega\omega_{j}-\omega^{2}\right) \tag{11}\]  
Based on Equation (10), the total closed-loop energy is:  
\[E_{c}=\frac{1}{2}\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}\frac{\beta_{mn} \left(1-\gamma_{mn}\right)^{2}}{\left(\omega_{mn}^{2}+\mathsf{j}\eta_{p}\omega \omega_{mn}-\omega^{2}\right)^{2}} \tag{12}\]  
in which the term \(\gamma_{mn}\) is given by:  
\[\gamma_{mn}=\frac{\phi_{mn}\big{(}x_{a},y_{a},\Delta_{x},\Delta_{y}\big{)} \sum\limits_{j=1}^{MN}\alpha_{j}\mathsf{\varphi}_{j}\left(x_{F},y_{F}\right) /\phi_{mn}\left(x_{F},y_{F}\right)}{\sum\limits_{k=1}^{MN}\alpha_{k}\phi_{k} \big{(}x_{a},y_{a},\Delta_{x},\Delta_{y}\big{)}} \tag{13}\]  
Obviously, the value of \(\gamma_{mn}\) is influenced by the disturbance frequency, the mode shape, and the modal strain. To specify the controllable mode, the ratio of the open-loop energy to the closed-loop energy is defined as:  
\[\gamma=\frac{E_{c}}{E_{o}} \tag{14}\]  
The defined ratio is used as a criterion to predict the AVC effect.