## 2 Methods
\[G(z)=\frac{0.0003705z^{3}}{z^{4}-3.521z^{3}+4.666z^{2}-2.749z+0.6036} \tag{32}\]  
\[G(s)=\frac{0.4096s^{3}+6506s^{2}+4.437\times 10^{7}s+1.217\times 10^{11}}{s^{4}+20 19s^{3}+1.465\times 10^{6}s^{2}+1.712\times 10^{9}s+1.402\times 10^{10}} \tag{33}\]  
\[\left\{ \begin{array}{l}\left[\begin{array}{l}x_{1}(k+1)\\ x_{2}(k+1)\\ x_{3}(k+1)\\ x_{4}(k+1)\\ \end{array}\right]=\left[\begin{array}{cccc}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ -0.6036&2.749&-4.666&3.521\\ \end{array}\right]\cdot\left[\begin{array}{l}x_{1}(k)\\ x_{2}(k)\\ x_{3}(k)\\ x_{4}(k)\\ \end{array}\right]+\left[\begin{array}{l}0.0004\\ 0.0013\\ 0.0029\\ 0.005\\ \end{array}\right]u\\ y=x_{1}\end{array} \right. \tag{34}\]  
Figure 5: AIC of all sample systems. Figure 6: Bode diagram of open-loop identified system. Based on the above system identification results of the airborne photoelectric pod, the high-order ADRC(HO-ADRC) is designed. The HO-ADRC consists of three parts: tracking differentiator (TD), expansion state observer (ESO), and nonlinear state error feedback (NLSEF) [30]. According to the identified system transfer function, we can determine the fourth-order TD and the fifth-order ESO to establish the HO-ADRC. The fourth-order TD is shown in Equation (35):  
\[\left\{ \begin{array}{l}fs=-r(r(r(r(v_{1}-u(t))+4v_{2})+6v_{3})+4v_{4})\\ v_{1}=v_{1}+h\cdot v_{2}\\ v_{2}=v_{2}+h\cdot v_{3}\\ v_{3}=v_{3}+h\cdot v_{4}\\ v_{4}=v_{4}+h\cdot fs\end{array} \right. \tag{35}\]  
where \(h\) = simulation time interval; \(v_{1}\) = tracking signal input by the system, \(v_{2}\) = input first-order differential signal; \(v_{3}\) = input second-order differential signal; \(v_{4}\) = input third-order differential signal; \(f_{s}\) = input fourth-order differential signal; \(u(t)\) = input signal; and \(r\) = fast factor of the tracking differential. The fifth-order ESO is shown in Equation (36):  
\[\left\{ \begin{array}{l}e=z_{1}-y\\ z_{1}=z_{1}+h(z_{2}-\beta_{01}e+0.0004u(t))\\ z_{2}=z_{2}+h(z_{3}-\beta_{02}e+0.0013u(t))\\ z_{3}=z_{3}+h(z_{4}-\beta_{03}e+0.0029u(t))\\ z_{4}=z_{4}+h(z_{5}-0.6036z_{1}+2.749z_{2}-4.666z_{3}+3.521z_{4}-\beta_{04}e+0. 005u(t))\\ z_{5}=z_{5}-h\beta_{05}e\end{array} \right. \tag{36}\]  
where \(z_{1}\) = estimate of the system output; \(z_{2}\) = estimate of the first-order differentiation of the system output; \(z_{3}\) = estimate of the second-order differentiation of the system output; \(z_{4}\) = estimate of the third-order differentiation of the system output; \(z_{5}\) = estimate of the system disturbance; and \(\beta_{01}\), \(\beta_{02}\), \(\beta_{03}\), \(\beta_{04}\), and \(\beta_{05}\) are the weight factors. The NLSEF is shown in Equation (37):  
\[\left\{ \begin{array}{l}e_{1}=v_{1}-z_{1}\\ e_{2}=v_{2}-z_{2}\\ U_{0}=b_{01}\cdot fal(e_{1},\alpha_{1},\delta)+\\ b_{02}\cdot fal(e_{2},\alpha_{2},\delta)+b_{03}\cdot fal(\int e_{1},\alpha_{ 3},\delta)-200z_{5}\end{array} \right. \tag{37}\]  
where \(b_{01}\), \(b_{02}\), \(b_{03}\) = weight factors of NLSEF; \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\) = nonlinear saturation factors; and \(\delta\) = switching threshold. _fal(e,a,\(\delta\))_ is shown in Equation (38):  
\[fal(e,\alpha,\delta)=\left\{ \begin{array}{l}|e|^{a}\text{sgn}(e)\quad|e|\geq \delta\\ \frac{e}{\delta^{2}-a}\quad|e|<\delta\end{array} \right. \tag{38}\]