## 4 Model Simulation Results  
This section aims to demonstrate the effectiveness of the PID-CTOPC approach numerically through designing autopilots for a missile under external disturbances. Comparison with the PCTOPC and the [PERSON]'s approach is carried out to highlight the superiorities of the PID-CTOPC. [PERSON]'s approach can enable the closed-loop system better robustness than the one based upon the PCTOPC and has made many achievements [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45]. Introduction of [PERSON]'s approach targeting system (1) is given in Appendix A.  
Longitudinal dynamics of a missile flying at Mach 3 and an altitude of 6096 m (20,000 ft) perturbed by disturbance is given by [19; 40]:  
\[\left\{ \begin{array}{l}\dot{\alpha}=\frac{0.7B_{1}S_{2}}{m_{3}}M_{a}C_{n} \cos\alpha+q\\ \dot{q}=\frac{0.7B_{2}S_{3}d_{3}}{l_{y}}M_{a}^{2}C_{m}\\ \dot{\delta_{1}}=\delta_{2}\\ \dot{\delta_{2}}=-\omega_{a}^{2}\delta_{1}-2\xi_{a}\omega_{a}\delta_{2}+\omega_ {a}^{2}\delta_{c}+d(t)\end{array} \right. \tag{27}\]  
Figure 1: Connections and comparisons between the PCTOPC and the PID-CTOPC. where, \(d(t)\) represents the unknown external disturbance. The force coefficient \(C_{n}\) and the torque coefficient \(C_{m}\) are given by:  
\[\left\{ \begin{array}{l}C_{n}=a_{n}\alpha^{3}k^{3}+b_{n}\alpha|\alpha|k^{2}+c_{n }\Big{(}2-\frac{M_{n}}{3}\Big{)}\alpha k+d_{n}k\delta_{1}\\ C_{m}=a_{m}\alpha^{3}k^{3}+b_{m}\alpha|\alpha|k^{2}+c_{m}\Big{(}-7+\frac{8M_{n }}{3}\Big{)}\alpha k+d_{m}k\delta_{1}\end{array} \right. \tag{28}\]  
where, \(\alpha\), \(q\), \(\delta_{1}\), and \(\delta_{2}\) are system states. \(\alpha\) and \(q\) represent the attack angle (rad) and the pitch rate (rad/s), respectively. Differential equations of \(\delta_{1}\) and \(\delta_{2}\) describe the actuator dynamics. \(\delta_{c}\) represents the tail deflection (rad) as well as the control input, which ranges from \(-30\) degree to \(30\) degree. The attack angle \(\alpha\) is taken as the system output, which ranges from \(-20\) degree to \(20\) degree. Values of the model parameters in Formulas (27) and (28) can be seen in [19; 46]. Two simplifications in [19] are adopted. Firstly, \(\cos\alpha\approx 1\). Secondly, since the control surface mainly generates rotational torque, thus, its contribution in the force equation \(C_{n}\), namely, the term \(d_{n}k\delta_{1}\), is ignored. Hence, in the controller design, \(\overline{C}_{n}=a_{n}\alpha^{3}k^{3}+b_{n}\alpha|\alpha|k^{2}+c_{n}\Big{(}2 -\frac{M_{n}}{3}\Big{)}\alpha k\) is used to replace \(C_{n}\). In addition, to write the missile dynamics into the form in system, let \(x=[x_{1},x_{2},x_{3},x_{4}]^{T}=[\alpha,q,\delta_{1},\delta_{2}]^{T}\) and denote:  
\[\left\{ \begin{array}{l}[f_{1}(x),f_{2}(x),f_{3}(x),f_{4}(x)]^{T}=\Big{[} \frac{0.7B_{0}\zeta}{m_{0}}M_{a}\overline{C}_{n}+q,\frac{0.7B_{0}\zeta d_{1}}{ l_{y}}M_{a}^{2}C_{m},\delta_{2},-\omega_{a}^{2}\delta_{1}-2\zeta_{a}^{2} \omega_{a}\delta_{2}\Big{]}^{T}\\ [g_{1}(x),g_{2}(x),g_{3}(x),g_{4}(x)]^{T}=\big{[}0,0,0,\omega_{a}^{2}\big{]}^{ T}\end{array} \right. \tag{29}\]  
Next, to design the controller, the unknown external disturbance \(d(t)\) is assumed to be zero. Then, in the simulation, \(d(t)\) is considered. Differentiating the attack angle \(\alpha\) four times yields:  
\[\left\{ \begin{array}{l}\dot{\alpha}=f_{1}(x)\\ \ddot{\alpha}=\frac{\partial f_{1}(x)}{\partial x_{1}}f_{1}(x)+f_{2}(x)\\ \alpha^{[3]}=\Big{[}\frac{\partial^{2}f_{1}(x)}{\partial x_{1}^{2}}f_{1}(x)+ \Big{(}\frac{\partial f_{1}(x)}{\partial x_{1}}\Big{)}^{2}+\frac{\partial f_{ 2}(x)}{\partial x_{1}}\Big{]}f_{1}(x)+\frac{\partial f_{1}(x)}{\partial x_{1} }f_{2}(x)+\frac{\partial f_{2}(x)}{\partial x_{3}}f_{3}(x)\\ \alpha^{[4]}=\rho_{1}(x)\cdot f_{1}(x)+\rho_{2}(x)\cdot f_{2}(x)+\frac{\partial f _{1}(x)}{\partial x_{1}}\cdot\frac{\partial f_{2}(x)}{\partial x_{3}}f_{3}(x )+\frac{\partial f_{2}(x)}{\partial x_{3}}f_{4}(x)+\frac{\partial f_{2}(x)}{ \partial x_{3}}g_{4}(x)\cdot u\end{array} \right.