## 3 Flutter Analysis  
As indicated in Equation (17), the applied voltage that is acted to the piezoelectric actuator can generate the base plate's transverse displacement and acceleration. Meanwhile, the measured transverse displacement and acceleration, in turn, provided the feedback information for the piezoelectric actuator. Then, a control algorithm was utilized to control the vibration of the plate based on the feedback information. The activated piezoelectric layer, therefore, can provide the active stiffness and active mass to the base plate by the appropriate control voltage. Based on reference [21], the control voltage exerted to the piezoelectric actuator at the position \((x_{0},y_{0})\) are derived in terms of the transverse displacement and acceleration  
\[V_{0}(t)=K_{d}w(x_{0},\ y_{0},\ t)-K_{a}\tilde{w}(x_{0},\ y_{0},\ t) \tag{18}\]  
where \(K_{a}\) and \(K_{d}\) are the acceleration and displacement feedback gain of the piezoelectric actuator, respectively. Substituting the expression of \(\mathbf{W}(x,y)\) into Equation (18), and then substituting the resultant expression of voltage \(V_{0}(t)\) into the actuator Equation (17), the motion equation of the plate with the active stiffness and mass is resented as  
\[(\mathbf{M}+\mathbf{M_{p}})\mathbf{\ddot{r}}+(\mathbf{\cdot}\mathbf{F_{Ap2}}) \mathbf{\dot{r}}+(\mathbf{K_{1}+K_{p}\cdot F_{Ap1}})\mathbf{r}=q\mathbf{F_{\dot{ q}}} \tag{19}\]  
where \(\mathbf{M_{p}}\) and \(\mathbf{K_{p}}\) are the active mass and active stiffness matrices related to the piezoelectric layer, which can be obtained as  
\[\mathbf{M_{p}}=-K_{a}\mathbf{K_{2}W^{T}}(x_{0},y_{0}),\ \mathbf{K_{p}}=K_{d} \mathbf{K_{2}W^{T}}(x_{0},y_{0}) \tag{20}\]  
Equation (19) demonstrates the displacement and acceleration feedback control strategy can provide the active stiffness and active mass to the base plate, respectively. By varying the feedback gain of \(K_{a}\) and \(K_{d}\), the active mass \(\mathbf{M_{p}}\) and the active stiffness \(\mathbf{K_{p}}\) are accordingly added to the structure. As a result, it will change the structural nature frequency and damping ratio, which correspondingly alters the structural aeroelastic characteristics. This phenomenon can be observed in the following Section 4 that the active mass and stiffness can effectively enhance the plate's aeroelastic flutter characteristics in supersonic flow. The general solution of Equation (20) can be presented as  
\[\overset{\rightarrow}{\mathbf{r}}(t)=\overset{\rightarrow}{\mathbf{r}}_{0}e ^{\lambda t} \tag{21}\]  
where \(\mathbf{r}_{0}\) and \(\lambda\) represent the eigenvector and eigenvalue of the system. Substituting Equation (21) into the homogeneous differential Equation (19) results in the following eigenvalue problem  
\[[(\mathbf{M}+\mathbf{M_{p}})\lambda^{2}-\mathbf{F_{Ap2}}\lambda+(\mathbf{K_{1 }+K_{p}\cdot F_{Ap1}})]\overset{\rightarrow}{\mathbf{r}}=0 \tag{22}\]  
The nontrivial solution of Equation (22) can be obtained by solving its eigenvalues problem. The solution of the eigenvalue is a complex number that is obtained as  
\[\lambda_{i}=\eta_{i}+i\omega_{i} \tag{23}\]  
where \(\eta_{i}\) is the parameter with respect to the damping ratio \(\zeta_{i}\) and eigenfrequency \(\omega_{i}\) by \(\eta_{i}=\zeta_{i}\omega_{i}\), and \(i=\sqrt{-1}\). Calculating the complex eigenvalues problem of the system, the active aeroelastic flutter properties of the plate with the piezoelectric layer can be numerically analyzed. The non-dimensional flutter velocity is usually obtained in the form of the non-dimensional aerodynamic pressure [2; 43]. However, the variation of the weight fraction of GPLs will change the Young's modulus of a material, which will accordingly alter the non-dimensional aerodynamic pressure. It indicates that the non-dimensional aerodynamic pressure cannot linearly reflect the change of aerodynamic pressure, so the non-dimensional flutter velocity in this paper is obtained in terms of the Mach number.