## 3 Aircraft Elastic Deformation Estimation and Inertial Network Algorithm Design
### Fusion of Inertial Network and FADS  
The inertial navigation system (INS) can respond to the change of angle of attack in time during maneuvering flight, but there is a large error in steady-state flight. The error of FADS is small in steady-state flight, while there is a large error due to the delay of pressure propagation in maneuvering flight. The two are complementary. The angle of attack of FADS and the angle of attack of INS can be fused by complementary filtering. The FADS angle of attack passes through a low-pass filter to take advantage of its precise steady-state component, while the angle of attack of INS is filtered through a high-pass filter, which  
Figure 4: LM algorithm to solve the least square coefficient.  
takes advantage of its timely response to the dynamic change of angle of attack and is then superimposed. The output angle of attack can be written as  
\[\alpha=\frac{1}{\tau S+1}(\alpha_{FADS}-\alpha_{INS})+\alpha_{INS}=\frac{\tau s}{ \tau S+1}\alpha_{NNS}+\frac{1}{\tau s+1}\alpha_{FADS} \tag{42}\]  
However, the filter constant \(\tau\) of the complementary filter has a significant effect on the output. The larger the \(\tau\) is, the less the proportion of the FADS angle of attack component in the output result after fusion is, and the inertial angle of attack component has a great influence on the fusion result. Combined with the above analysis, the angle of attack error of INS is smaller during maneuvering flight, and the angle of attack error of FADS is smaller during steady flight. Therefore, when the aircraft is maneuvering, in order to improve the fusion accuracy during maneuvering, the \(\tau\) can be selected larger, so that the fusion result is more affected by INS. In order to improve the fusion accuracy in level flight, the \(\tau\) can be selected smaller, so that the fusion results are more affected by FADS.  
In order to achieve this goal, a variable gain complementary filtering algorithm is designed. The filter constant \(\tau\) in the complementary filter can change with the change of flight state, and the better angle of attack fusion results can be maintained throughout the flight phase. The change of the gain is most directly based on the change rate of the angle of attack. Considering that as long as the absolute value of the change rate of the angle of attack is large, whether the change rate is positive or negative, the filter constant \(\tau\) is larger. Therefore, the change rule of the filter constant is set to  
\[\tau=k|\dot{\alpha}|=k\bigg{|}\frac{\alpha_{k+1}-\alpha_{k}}{\Delta t}\bigg{|} \tag{43}\]  
Bring the above formula into the complementary filtering Formula (42), and the fusion formula of the variable gain complementary filtering algorithm can be obtained. However, because the inertial angle of attack can quickly respond to the change rate of the angle of attack, while the FADS angle of attack has a large delay, the change rate of the angle of attack is very close to the change rate of the inertial angle of attack, and the change rate of the angle of attack can be replaced by the change rate of the inertial angle of attack.  
The angle of attack based on variable gain complementary filtering can be written as  
\[\begin{split}\alpha_{k+1}&=\frac{2k\big{(}\alpha _{\text{INS,$k+1$}}-\alpha_{\text{INS,$k$}}\big{)}-\Delta t^{2}}{2k\big{(} \alpha_{\text{INS,$k+1$}}-\alpha_{\text{INS,$k$}}\big{)}+\Delta t^{2}}\alpha_ {k}+\frac{\Delta t^{2}}{2k\big{(}\alpha_{\text{INS,$k+1$}}-\alpha_{\text{INS,$ k$}}\big{)}+\Delta t^{2}}\\ &(\alpha_{\text{FADS,$k+1$}}+\alpha_{\text{FADS,$k$}})+\frac{2k \big{(}\alpha_{\text{INS,$k+1$}}-\alpha_{\text{INS,$k$}}\big{)}^{2}}{2k\big{(} \alpha_{\text{INS,$k+1$}}-\alpha_{\text{INS,$k$}}\big{)}+\Delta t^{2}}\end{split} \tag{44}\]