## 2 System Modeling  
We developed a mathematical model for the large flexible structure used in this study to facilitate subsequent analysis. The structure comprises a large and flexible truss-like thin plate with a servomotor as the driving force. A transmission mechanism enables the rotation of the thin plate around the roll and yaw axes. Piezoelectric actuators and piezoelectric sensors were placed at the root of the vertical thin plate, as shown in Figure 1.  
In formulating the dynamic equations, we took into account the transverse vibrations in the thin flexible plate. The dynamic equations used for the overall system can be regarded as a combination of the dynamic equations pertaining to a cantilevered thin flexible plate and a servo motor. The thin plate system is treated as linear and the rotation system is assumed to be rigid. The assumptions and derivation procedures used in modeling the overall structure are similar to those in [13]; however, we also included nonlinearities and coupling terms in the current model. When the motor introduces rotary motions, coupling generates disturbances in the thin flexible plate. Vibrations in the thin plate also influence the rotary motions. Based on the effects of coupling, the dynamic equations of the overall system can be written as follows:  
\[\mathbf{M}\ddot{\mathbf{\eta}}+\mathbf{C}^{\mathbf{D}}\ddot{\mathbf{\eta}}+ \mathbf{K}\mathbf{\eta}=\mathbf{B}^{\mathbf{P}}u_{\mathbf{p}}+\mathbf{\Phi}_ {\mathbf{\theta}}+\mathbf{F} \tag{1}\]  
Figure 1: Motor-driven system for the rotation of thin flexible plate held in a vertical position. \[\mathbf{L_{a}}\mathbf{i_{a}}+\mathbf{R_{a}}\mathbf{i_{a}}=\mathbf{e_{a}}-\mathbf{K _{b}}\mathbf{\dot{\theta}_{m}} \tag{2}\]  
\[\mathbf{J_{m}}\mathbf{\ddot{\theta}_{m}}+\mathbf{B_{m}}\mathbf{\dot{\theta}_{m}} +\mathbf{\Phi_{n}}=\mathbf{K_{i}}\mathbf{i_{a}} \tag{3}\]  
\[\mathbf{L_{a}}=\left[ \begin{array}{cc}L_{\mathrm{ar}}&0\\ 0&L_{\mathrm{ay}}\end{array} \right];\quad\mathbf{R_{a}}=\left[ \begin{array}{ cc}R_{\mathrm{ar}}&0\\ 0&R_{\mathrm{ay}}\end{array} \right];\quad\mathbf{e_{a}}=\left[ \begin{array}{ cc}e_{\mathrm{ar}}&0\\ 0&e_{\mathrm{ay}}\end{array} \right];\quad\mathbf{K_{b}}=\left[ \begin{array}{ cc}K_{\mathrm{br}}&0\\ 0&K_{\mathrm{by}}\end{array} \right] \tag{4}\]  
where \(\mathbf{M}\) and \(\mathbf{K}\) denote the mass and stiffness coefficient matrices, respectively; both are ij \(\times\) ij diagonal matrices with no coupling between the modes; \(\mathbf{\eta}\) is a vector containing modal participation factors with dimensions ij \(\times\) 1; \(\mathbf{C^{D}}\) is the damping coefficient matrix; \(\mathbf{B^{P}}\) signifies the piezoelectric coefficient matrix; \(u_{\mathrm{p}}\) indicates the control input from the piezoelectric actuator; and \(\mathbf{F}\) represents the external disturbance. The parameters associated with motor dynamics are as follows. \(\mathbf{L_{a}}\) denotes the inductance constant matrix; \(\mathbf{R_{a}}\) indicates the resistance matrix; \(\mathbf{i_{a}}\) is the current vector; \(\mathbf{e_{a}}\) signifies the motor input voltage vector; \(\mathbf{K_{b}}\) is the back-emf constant matrix; \(\mathbf{j_{m}}\) denotes the inertia matrix; \(\mathbf{B_{m}}\) is the viscous friction coefficient matrix; \(\mathbf{K_{i}}\) is the torque constant matrix; and \(\mathbf{\dot{\theta}_{m}}\) denotes the motor angle vector. The subscripts \(\mathbf{r}\) and \(\mathbf{y}\) in the above matrices, respectively, refer to servomotor parameters of the roll and yaw axes.