# A Novel Disturbance Rejection Control of Roll Channel for Small Air-to-Surface Missiles
## 2 Dynamic Model
\begin{bmatrix}\delta_{x}&\delta_{y}\end{bmatrix} ^{T}\), which is the rolling rudder and yaw rudder. \[A= \begin{bmatrix}b_{11}&b_{12}&b_{14}&0\\ b_{21}+b_{24}^{\prime}\alpha&b_{22}-b_{24}^{\prime}b_{32}-b_{24}^{\prime} \alpha\tan v&b_{24}-b_{24}^{\prime}(b_{34}-a_{33})&-b_{36}b_{24}^{\prime}\\ \alpha&-(\alpha\tan v+b_{32})&-b_{34}+a_{33}&-b_{36}\\ 1&-\tan v&0&0\end{bmatrix} , \tag{4}\]  
\[B= \begin{bmatrix}b_{17}&b_{15}\\ 0&b_{25}-b_{24}^{\prime}b_{35}\\ 0&-b_{35}\\ 0&0\end{bmatrix} . \tag{5}\]  
For small axisymmetric missiles, the influence of internal and external disturbances leads to uncertainty in the model of the control system and greater uncertainty in the real aerodynamic shape and nominal. These characteristics are particularly prominent in the rolling attitude control loop. The attitude dynamics of the rolling channel is as follows:  
\[J_{x}\frac{d\omega_{x}}{dt}+\big{(}J_{z}-J_{y}\big{)}\omega_{z}\omega_{y}=M_{x}, \tag{6}\]  
\[\frac{d\gamma}{dt}=\omega_{x}-\tan\theta\big{(}\omega_{y}\cos\gamma-\omega_{x} \sin\gamma\big{)}, \tag{7}\]  
where \(\omega_{x}\), \(\omega_{y}\) and \(\omega_{z}\) are the roll rate, yaw rate and pitch rate in the airframe coordinate system, which are measured by the navigation system, but there will be certain deviation and noise sigma due to the influence of sensor characteristics. \(M_{x}\) is the aerodynamic rolling moment caused by the lateral centroid deviation of the projectile body, and its size increases linearly with the increase of lift force, which can be expressed as follows:  
\[M_{x}=m_{x}qS_{ref}L_{ref}=M_{x_{0}}+M_{x}^{\beta}\beta+M_{x}^{\delta_{x}} \delta_{x}+M_{x}^{\delta_{y}}\delta_{y}+M_{x}^{\omega_{x}}\omega_{x}+M_{x}^{ \omega_{y}}\omega_{y}. \tag{8}\]  
The torque coefficient can be expressed as:  
\[m_{x}=m_{x0}+m_{x}^{\delta_{x}}\delta_{x}+m_{x}^{\delta_{y}}\delta_{y}+m_{x}^{ \omega_{x}}\omega_{x}+m_{x}^{\omega_{y}}\omega_{y}+m_{x}^{\beta}\beta. \tag{9}\]  
Equation (6) can be simplified as:  
\[J_{x}\frac{d\omega_{x}}{dt}=m_{x}qS_{ref}L_{ref}+\big{(}J_{y}-J_{z}\big{)} \big{(}\omega_{y}-\omega_{z}\big{)}+M_{xd}, \tag{10}\]  
where \(J_{x}\), \(J_{y}\) and \(J_{z}\) are the rotational inertia of the projectile body around the longitudinal axis, normal axis and side axis, respectively. \(\big{(}J_{y}-J_{z}\big{)}\big{(}\omega_{y}-\omega_{z}\big{)}\) is inertial sympathetic coupling. \(q\) is the dynamic pressure. \(S_{ref}\) is the reference area of the projectile body. \(L_{ref}\) is the reference length. \(m_{x}\) is the rolling moment coefficient. \(m_{x0}\) is the error caused by the machining accuracy and assembly accuracy of the projectile wing, which is manifested as strong internal disturbance. \(m_{x}^{\delta_{x}}\delta_{x}\) is the rolling moment caused by rolling rudder \(\delta_{x}\). \(m_{x}^{\delta_{y}}\delta_{y}\) is the rolling moment caused by yaw control surface, and is the internal disturbance. \(m_{x}^{\omega_{x}}\omega_{x}\) is rolling damping caused by rolling angle rate \(\omega_{x}\). \(m_{x}^{\omega_{y}}\omega_{y}\) is rolling damping caused by yaw rate \(\omega_{y}\), and is internal disturbance. \(m_{x}^{\beta}\beta\) is the oblique blowing moment caused by sideslip angle \(\beta\), and is the strong internal disturbance. \(M_{xd}\) is the disturbance torque due to centroid deviation. To sum up, the rolling moment can be expressed as the following three parts:  
\[m_{x}=m_{x}(\delta_{x})+m_{x}(\omega_{x})+m_{xd}, \tag{11}\]where \(m_{x}(\delta_{x})=m_{x}^{\delta_{x}}\delta_{x}\) is the control torque caused by the control quantity \(u=\delta_{x}\). \(m_{x}(\omega_{x})=m_{x}^{\omega_{x}}\omega_{x}\) is the damping moment caused by the rolling angular rate \(\omega_{x}\). \(m_{xd}=m_{x0}+m_{x}^{\delta_{x}}\delta_{x}+m_{x}^{\delta_{y}}\delta_{y}+m_{x}^ {\omega_{x}}\omega_{x}+m_{x}^{\omega_{y}}\omega_{y}+m_{x}^{\beta}\beta\) and the disturbances in the second and third terms on the right-hand side of Equation (8) can be viewed in a broad sense as internal disturbance moments.