## 3 Modeling and Analysis of Attitude Control of Missile
### Design of ADRC Attitude Controller  
Based on Equations (22)-(27), in order to construct a model that is suitable for ADRC, it is necessary to transform the non-linear dynamic model. Deriving (23), substituting (24) into (23), and considering \(\omega_{z}\approx\dot{\vartheta}\), the model can be given by  
Figure 16: Coordinates and angle definitions of the missile under wind disturbance.  
\[\ddot{\vartheta}=(\frac{M_{z}^{\oplus z}\cos\gamma}{I_{z}}-\dot{\gamma}\sin\gamma) \dot{\vartheta}+\frac{M_{z}^{\oplus z}\cos\gamma}{I_{z}}\dot{\delta}_{z}+\frac{ M_{z}^{\oplus z}\cos\gamma}{I_{z}}\delta_{z}+\dot{\omega}_{y}\sin\gamma+\dot{\gamma}\cos \gamma\omega_{y}+ \tag{28}\] \[\frac{(M_{z}^{\oplus z}\alpha_{y}+M_{z}^{\oplus z}\dot{\alpha}_{y} )\cos\gamma+(J_{x}-J_{y})\omega_{x}\omega_{y}\cos\gamma}{J_{z}}\]  
where \(M_{z}^{\oplus z}\), \(M_{z}^{\oplus z}\), \(M_{z}^{\oplus z}\), \(M_{z}^{\oplus z}\), \(M_{z}^{\oplus z}\) are the corresponding torque coefficients generated by corresponding parameter unit changes respectively.  
Equation (28) can be rewritten as  
\[\ddot{\vartheta}=c_{1}\dot{\vartheta}+c_{2}\dot{\delta}_{z}+p\delta_{z}+q_{0} \tag{29}\]  
where  
\[c_{1}=\frac{M_{z}^{\oplus z}\cos\gamma}{I_{z}}-\dot{\gamma}\sin\gamma,\ c_{2}= \frac{M_{z}^{\oplus z}\cos\gamma}{J_{z}},\]  
\(p=\frac{M_{z}^{\oplus z}\cos\gamma}{I_{z}}\), \(q_{0}=\frac{(M_{z}^{\oplus z}\alpha+M_{z}^{\oplus z}\dot{\alpha})\cos\gamma+(I_ {z}-J_{y})\omega_{x}\omega_{y}\cos\gamma}{J_{z}}+\dot{\omega}_{y}\sin\gamma+ \dot{\gamma}\omega_{y}\cos\gamma\) Take the input regulation parameter \(p_{0}\), and take the term that does not obviously contain \(\delta_{z}\) as the total disturbance \(q\), (29) can be reduced to  
\[\ddot{\vartheta}=p_{0}\delta_{z}+q \tag{30}\]  
where  
\[q=c_{1}\dot{\vartheta}+(p-p_{0})\delta_{z}+q_{0}+c_{2}\dot{\delta}_{z} \tag{31}\]  
For the model (30), take \(\mathbf{s}=[s_{1}\ s_{2}\ s_{3}]^{\mathsf{T}}\) as the extended state variable, and design the ESO according to Equations (11)-(13). Take the observer bandwidth \(\omega_{O1}\) and the observation parameters of the third-order state observer as \(\mathbf{L}_{1}=\left[\begin{array}{cc}3\omega_{O1}&3\omega_{O1}^{2}&\omega_{ O1}^{3}\end{array}\right]^{\mathsf{T}}\), then the expansion state \(s_{3}\) is the estimated value \(\hat{q}\) of the total disturbance \(q\).  
Following the method of (14)-(18), take the PD controller and make a reasonable selection of the controller frequency and damping ratio, where \(k_{p1}\), \(k_{p1}\) are the parameters of the controller, the values are related to the natural oscillation frequency \(\omega_{n1}\) and the damping coefficient \(\xi_{1}\) of the rocket. Under the action of wind, consider the non-linearity of the dead-zone of the actuator. The simulation block diagram is shown in Figure 17.  
In the figure, \(\vartheta_{r}\) is the pitch angle given value and \(\mathbf{W}\) is the wind, which is considered as the external disturbance of the non-linear model.  
In order to compare with the traditional controller, a PID gain scheduling controller is designed. The control parameters of the method are predesigned under little disturbance theory at the selected different characteristic points thought the flight trajectory, and the control parameters cannot be changed according to large external disturbances in real-time. So, when encountering relatively large wind disturbances, the system will have greater parameter uncertainty, and the system may be out of control.  
Figure 17: Pitch attitude ADRC model including actuator non-linearity and wind disturbance.