# A Novel Disturbance Rejection Control of Roll Channel for Small Air-to-Surface Missiles
## 2 Dynamic Model  
In the design of missile guidance and control system [29], the roll control of the missile body around its longitudinal axis is very important. However, the dynamic environment of the rolling channel is relatively poor under the influence of such interference factors as centroid deviation, processing and installation errors, and high altitude wind. They lead to coupling between pitch, yaw and roll channels and a decrease in guidance accuracy. Therefore, it is required that the roll control should have strong anti-disturbance ability, observe and compensate the above uncertain disturbance, and restrain its influence on attitude control. For small missiles facing the scale, under the assumption that gravity is ignored, the body disturbance motion can be divided into longitudinal disturbance motion and lateral disturbance motion. The lateral movement refers to the lateral centroid movement of the missile body, the angular movement of the missile body around the longitudinal axis and the angular movement of the missile body around the normal axis. The lateral movement parameters are mainly the control parameters [30]. Based on the small disturbance assumption, the lateral disturbance deviation of the body can be obtained according to the dynamic equation and kinematic equation of the body. As shown in Equation (1) below:  
\[\left\{ \begin{array}{l}\frac{d\Delta\omega_{x}}{dt}=\frac{M_{x}^{\alpha x}}{ J_{x}}\Delta\omega_{x}+\frac{M_{y}^{\alpha y}}{J_{x}}\Delta\omega_{y}+\frac{M_{z}^{ \beta}}{J_{x}}\Delta\phi+\frac{M_{y}^{\alpha y}}{J_{x}}\Delta\phi_{y}+\frac{M _{z}^{\alpha x}}{J_{x}}\Delta\varepsilon_{x}+\frac{M_{xx}}{J_{x}},\\ \frac{d\Delta\omega_{y}}{dt}=\frac{M_{y}^{\alpha x}}{J_{y}}\Delta\omega_{x}+ \frac{M_{y}^{\alpha y}}{J_{y}}\Delta\omega_{y}+\frac{M_{y}^{\alpha x}}{J_{y}} \Delta\beta+\frac{M_{y}^{\beta}}{J_{y}}\Delta\beta+\frac{M_{y}^{\alpha y}}{J_ {y}}\Delta\phi_{y}+\frac{M_{y}}{J_{y}},\\ \cos\theta\frac{d\Delta\psi}{dt}=\left(\frac{p-p^{\beta}}{mV}-a_{33}\right) \Delta\beta+\frac{-g\cos\theta}{V}\Delta\gamma+\frac{-Z^{\beta}y}{mV}\Delta \delta_{y}+\frac{F_{xz}}{mV},\\ \frac{d\Delta\psi}{dt}=\frac{1}{\cos\theta}\Delta\omega_{y},\\ \frac{d\Delta\varphi}{dt}=\Delta\omega_{x}-\tan\theta\Delta\omega_{y},\\ \frac{d\Delta z}{dt^{2}}=-V\cos\theta\Delta\psi_{V},\\ \cos\theta\Delta\psi_{V}=\cos\theta\Delta\psi-\Delta\beta+\alpha\Delta\gamma, \end{array} \right. \tag{1}\]  
where \(\alpha\) is the angle of attack; \(\gamma\) is the inclination angle; \(\theta\) is the pitch angle; \(\theta\) is the trajectory inclination angle; \(P\) is the thrust of the engine; \(Z\) is the lateral force; \(V\) is the velocity vector; \(a_{33}=g\sin\theta/V\) is dynamic coefficient of gravity; \(M_{xx}/J_{x}\) is the rolling angular acceleration caused by the rolling disturbance torque \(M_{gx}\); \(M_{gy}/J_{y}\) is the yaw angular acceleration caused by yaw disturbance torque \(M_{gy}\); \(F_{gz}/mV\) is the rate of change of ballistic deflection angle caused by lateral disturbance force \(F_{zg}\). For the convenience of writing, the dynamic coefficients in the equations are written as simplified symbols. \[\left\{ \begin{array}{l}\frac{d\Delta\omega_{x}}{dt}=b_{11}\Delta\omega_{x}+b _{12}\Delta\omega_{y}+b_{14}\Delta\beta+b_{15}\Delta\delta_{y}+b_{17}\Delta \delta_{x}+b_{18}M_{gx},\\ \frac{d\Delta\omega_{y}}{dt}=b_{21}\Delta\omega_{x}+b_{22}\Delta\omega_{y}+b_{ 24}\Delta\beta+b^{\prime}{}_{24}\Delta\beta+b_{25}\Delta\delta_{y}+b_{28}M_{gy },\\ \frac{d\Phi}{dt}=\alpha\omega_{x}-(\alpha\tan\theta-b_{32})\Delta\omega_{y}-(b _{34}-a_{33})\Delta\beta-b_{36}\gamma+b_{35}\Delta\delta_{y}+b_{38}F_{gz},\\ \frac{d\Delta\gamma}{dt}=\Delta\omega_{x}-\tan\theta\Delta\omega_{y}.\end{array} \right. \tag{2}\]  
It is written as the state space equation, namely  
\[\dot{x}=Ax+Bu, \tag{3}\]where \(x= \begin{bmatrix}\omega_{x}&\omega_{y}&\beta&\gamma\end{bmatrix} ^{T}\), the control outputs are, respectively, the longitudinal axis angular rate, normal axis angular rate, sideslip angle and roll angle of the projectile, \(u=