## 2 Article Structure Description and Aircraft Dynamics Modeling
### Real-Time Aerodynamic Parameter Identification of Aircraft  
The pitch moment coefficient of the aircraft can be simplified into the following form by linearization. Here, the moment is mainly concerned. It can be regarded as related to some flight state variables and satisfies an instantaneous linear relationship:  
\[\left\{\begin{array}{c}C_{l}=C_{l_{o}}+C_{l_{g}}\beta+C_{l_{g}}\delta_{\rm ail }+C_{l_{g}}\frac{\omega_{l}b}{2V}+C_{l_{g}}\frac{rb}{2V}\\ C_{m}=C_{m_{o}}+C_{m_{o}}\alpha+C_{m_{o}}\delta_{ele}+C_{m_{o}}\frac{\omega_{ e}v^{2}}{2V}\\ C_{n}=C_{n_{o}}+C_{n_{g}}\beta+C_{n_{g}}\delta_{ntr}+C_{n_{p}}\frac{\omega_{ p}v}{2V}+C_{n_{p}}\frac{rb}{2V}\end{array}\right. \tag{16}\]  
In the formula, \(C_{l}\), \(C_{m}\), and \(C_{n}\) are pitching, yawing, and rolling moment coefficients. The lower right corner of the three variables represents the derivative of the moment coefficient of a variable to the lower right variable. The lower right corner is 0, representing the moment coefficient caused by the body configuration, \(b\) represents the wingspan length, and \(\overline{c}\) represents the average aerodynamic chord length.  
Sometimes, the modeling term on the right side of the equation also includes some high-order terms or coupling terms of different variables. Appropriate adjustments need to be made for different configurations of aircraft to accurately model, and a trade-off between the number of modeling variables and the amount of dynamic corresponding information is made, so as to achieve the purpose of accurately identifying and minimizing the amount of calculation.  
Here, the pitch moment is taken as an example for analysis. The variable is defined as \(X\), which is a vector composed of data obtained directly from the aircraft sensor. It contains the following contents:  
\[X=\left[\alpha,\delta,\frac{\omega_{y}c}{2V}\right] \tag{17}\]  
The detrending method introduced in the previous section can realize the recursive detrending of the data. The vector after the detrending is defined as \(X_{d}\), the variable in the lower right corner \"\(d\)\" represents the detrended data of this variable, and \(X_{d}\) is expressed as follows:  
\[X_{d}=\left[\alpha_{d},\delta_{d},\frac{\omega_{y}c}{2V}\right]_{d} \tag{18}\]The frequency domain sensor data \(\widetilde{X}\) can be obtained by recursive Fourier transform, the superscript \"\(\sim\)\" represents the frequency domain data, and the expression of \(\widetilde{X}\) is as follows:  
\[\widetilde{X}_{d}=\begin{bmatrix}\alpha_{d}(1)&\delta_{d}(1)&\frac{\omega_{d} \varsigma}{2\
abla}|a(1)\\ \alpha_{d}(2)&\delta_{d}(2)&\frac{\omega_{d}\varsigma}{2\
abla}|a(2)\\ \cdots&\cdots&\cdots\\ \alpha_{d}(M)&\delta_{d}(M)&\frac{\omega_{d}\varsigma}{2\
abla}|_{d}(M)\\ \end{bmatrix} \tag{19}\]  
In the frequency domain, the problem of a recursive solution can be expressed in the form of recursive least squares. The expression is as follows:  
\[\widetilde{z}_{d}=\widetilde{X}_{d}\theta+\widetilde{e} \tag{20}\]  
\[\widetilde{z}_{d}=\begin{bmatrix}\widetilde{C}_{m}(1)&\widetilde{C}_{m}(2)& \vdots&\widetilde{C}_{m}(M)\\ \end{bmatrix}^{T} \tag{21}\]  
\[\theta=\begin{bmatrix}C_{m_{\alpha}}&C_{m_{\delta_{\epsilon}}}&C_{m_{\alpha \omega_{d}\varsigma}}\\ \end{bmatrix}^{T} \tag{22}\]  
The value of theta can be obtained by the following expression [16]:  
\[\theta=\begin{bmatrix}\text{Re}\Big{(}\widetilde{X}^{\dagger}\widetilde{X} \Big{)}\end{bmatrix}^{-1}\text{Re}\Big{(}\widetilde{X}^{+}\widetilde{z}\Big{)} \tag{23}\]  
In the formula, the variable with upper right corner \"+\" represents the complex conjugate transpose of this variable, and \"Re\" represents the real part of a complex number.  
Eventually, we need to recover the static information \(C_{m0}\) filtered out of the trending process:  
\[C_{m0}=C_{m}-\hat{\theta}X \tag{24}\]