# Nonlinear Aeroelastic System Identification Based on Neural Network
## 3 Nonlinear aeroelastic System Model  
A typical two-dimensional wing section with a control surface is illustrated in Figure 3. This includes the plunge (\(h\)), the pitch (\(\alpha\)) of the main wing, and the flap (\(\beta\)) of the control surface. Figure 2: Flowchart for neural network identification. SNR: Signal-to-noise ratio. Other parameters of the model are shown in Table 1[12]. Governing equations of motion for the structure of the nonlinear aeroelastic system are derived to be:  
\[\left[ \begin{array}{cc}m&mx_{m}b\\ mx_{m}b&l_{a}\end{array} \right]\left\{ \begin{array}{cc}\ddot{h}\\ \ddot{\alpha}\end{array} \right\}+\left[ \begin{array}{cc}c_{h}&0\\ 0&c_{a}\end{array} \right]\left\{ \begin{array}{cc}\dot{h}\\ \dot{\alpha}\end{array} \right\}+\left[ \begin{array}{cc}k_{h}&0\\ 0&k_{a}\end{array} \right]\left\{ \begin{array}{cc}h\\ \alpha\end{array} \right\}+\left[ \begin{array}{cc}0\\ M_{\alpha}(\alpha)-k_{a}\alpha\end{array} \right]=\left\{ \begin{array}{cc}-L \\ M_{a}\end{array} \right\} \tag{3}\]  
Constants are defined as follows: \(m\) is the mass of the wing section; \(b\) is the semi-chord of the airfoil; \(l_{a}\) is the moment of inertia about the elastic axis; \(x_{m}\) is the non-dimensional distance between the mass center and elastic axis; \(c_{h}\) and \(c_{a}\) represent structural damping coefficients of the wing in plunge and pitch, respectively; and \(k_{h}\) and \(k_{a}\) are stiffness coefficients of the wing in plunge and pitch, respectively. \(L\) and \(M_{\alpha}\) are the aerodynamic lift and moment, which can be defined as follows [19]:  
\[L=\rho V^{2}bc_{l_{\alpha}}\left[\alpha+\frac{\dot{h}}{V}+\left(\frac{1}{2}-x_ {b}\right)b\frac{\dot{\alpha}}{V}\right]+\rho V^{2}bc_{l_{\beta}}\beta \tag{4}\]  
\[M_{\alpha}=\rho V^{2}b^{2}c_{m\alpha}\left[\alpha+\frac{\dot{h}}{V}+\left( \frac{1}{2}-x_{b}\right)b\frac{\dot{\alpha}}{V}\right]+\rho V^{2}b^{2}c_{m \beta}\beta \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(V\) & \(6\) m/s \\ \(\rho\) & \(1.225\) kg/m\({}^{3}\) \\ \(b\) & \(0.135\) m \\ \(m\) & \(12.387\) kg \\ \(x_{m}\) & \(0.2466\) \\ \(x_{b}\) & \(-0.6\) \\ \(l_{a}\) & \(0.065\) kg/m\({}^{2}\) \\ \(c_{h}\) & \(27.43\) kg/s \\ \(c_{a}\) & \(0.180\) kg/m\({}^{2}\)/s \\ \(k_{h}\) & \(2844.2\) N/m \\ \(k_{a}\) & \(2.82\) Nm/rad \\ \(c_{la}\) & \(2\pi\) \\ \(c_{l\beta}\) & \(3.358\) \\ \(c_{ma}\) & \(-0.628\) \\ \(c_{m\beta}\) & \(-0.635\) \\ \hline \end{tabular}
\end{table} 
Table 1: Nonlinear aeroelastic system parameters. Figure 3: Typical two-dimensional wing section with control surface. where \(\rho\) is the free-stream density; \(V\) is the inflow velocity; \(c_{l\alpha}\) and \(c_{m\alpha}\) are the lift and moment coefficient per angle of attack, respectively; \(c_{l\beta}\) and \(c_{m\beta}\) are the lift and moment coefficient per control surface deflection (\(\beta\)), respectively; \(x_{b}\) is the non-dimensional distance between the mid-chord and the elastic axis; \(\beta\) is the flap displacement of the control surface; and \(M_{\alpha}(\alpha)\) is the moment-rotation relationship of pitch angle (\(\alpha\)). Substituting Equations (4) and (5) into Equation (3), the following is obtained:  
\[ \begin{split}&\left[\begin{array}{cc}m&mx_{m}b\\ mx_{m}b&l_{a}\end{array}\right]\left\{\begin{array}{c}\ddot{h}\\ \ddot{\alpha}\end{array}\right\}+\left[\begin{array}{cc}c_{h}+\rho Vbc_{l \alpha}&\rho Vb^{2}c_{l\alpha}\left(\frac{1}{2}-x_{b}\right)\\ -\rho Vb^{2}c_{m\alpha}&c_{a}-\rho Vb^{3}c_{m\alpha}\left(\frac{1}{2}-x_{b} \right)\end{array}\right]\left\{\begin{array}{c}\dot{h}\\ \dot{\alpha}\end{array}\right\}\\ &+\left[\begin{array}{cc}k_{h}&\rho V^{2}bc_{l\alpha}\\ 0&k_{\alpha}-\rho V^{2}b^{2}c_{m\alpha}\end{array}\right]\left\{\begin{array} []{c}h\\ \alpha\end{array}\right\}+\left[\begin{array}{cc}0\\ M_{\alpha}(\alpha)-k_{\alpha}\alpha\end{array}\right]=\left\{\begin{array}[ ]{c}-\rho V^{2}bc_{l\beta}\\ \rho V^{2}b^{2}c_{m\beta}\end{array}\right\}\beta\end{split}  \tag{6}\]  
Defining:  
\[