# Integrated Autopilot Guidance Based on Zero-Effort-Miss Formulation for Tail-Controlled Missiles
## 2 Problem Formulation  
Consider the engagement kinematics of a missile \(M\) that tries to intercept a target \(T\), as shown in Figure 1. In this figure, \(\left(x_{(\cdot)},z_{(\cdot)}\right)\) denotes the position of each one in the \(XOZ\) plane. \(V_{(\cdot)}\), \(a_{(\cdot)}\), and \(\gamma_{(\cdot)}\) represent the speed, normal acceleration, and flight path angle of each one. Then, the equations of motion of the missile and target are given by  
\[\dot{x}_{M}= V_{M}\cos\gamma_{M},\quad\dot{z}_{M}=-V_{M}\sin\gamma_{M},\quad \dot{\gamma}_{M}=\frac{a_{M}}{V_{M}} \tag{1}\] \[\dot{x}_{T}= V_{T}\cos\gamma_{T},\quad\dot{z}_{T}=-V_{T}\sin\gamma_{T},\quad \dot{\gamma}_{T}=\frac{a_{T}}{V_{T}}. \tag{2}\]  
In addition, the relative range \(r\) and line-of-sight (LOS) angle \(\lambda\) between the missile and target are governed by the equations of  
\[\dot{r}= V_{T}\cos\sigma_{T}-V_{M}\cos\sigma_{M} \tag{3}\] \[\dot{r}\dot{\lambda}= V_{T}\sin\sigma_{T}-V_{M}\sin\sigma_{M} \tag{4}\]  
where the look angles \(\sigma_{M}\) and \(\sigma_{T}\) are defined as \(\sigma_{M}=\gamma_{M}-\lambda\) and \(\sigma_{T}=\gamma_{T}-\lambda\), respectively. Combining (1)\(\sim\)(4), we have the linearized kinematics on the initial LOS frame as follows [9]:  
\[\dot{x}_{g}=A_{g}x_{g}+B_{g}a_{M}+C_{g}a_{T} \tag{5}\]where \(x_{g}\) is the guidance state vector given by \(x_{g}= \begin{bmatrix}z&\dot{z}\end{bmatrix} ^{T}\) for the relative range \(z\) defined as \(z=-r\sin(\lambda-\lambda_{0})\). The matrices \(A_{g}\), \(B_{g}\), and \(C_{g}\) are given by  
\[A_{g}= \begin{bmatrix}0&1\\ 0&0\end{bmatrix} ,\quad B_{g}= \begin{bmatrix}0\\ \cos(\sigma_{M0})\end{bmatrix} ,\quad C_{g}= \begin{bmatrix}0\\ -\cos(\sigma_{T0})\end{bmatrix} . \tag{6}\]  
Herein, the missile acceleration \(a_{M}\) in (5) is produced by the aerodynamic force as \(a_{M}=L(\alpha,\delta)/m\), where \(L(\cdot)\), \(m\), \(\alpha\), and \(\delta\) are the lift by the angle-of-attack, mass, angle-of-attack, and tail fin deflection of the missile, respectively. In addition, the short period dynamics of the missile in the \(XZ\) plane are given by  
\[\dot{\alpha}= q-\frac{a_{M}}{V_{M}}=q-\frac{QS}{mV_{M}}\big{(}C_{L_{\alpha}} \alpha+C_{L_{\beta}}\delta\big{)} \tag{7}\] \[\dot{q}= \frac{M_{y}}{I_{yy}}=\frac{QSl}{I_{yy}}\bigg{(}C_{m_{\alpha}} \alpha+C_{m_{\beta}}\frac{l}{2V_{M}}q+C_{m_{\beta}}\delta\bigg{)} \tag{8}\]  
where the variable \(q\) is the pitch rate of the missile. \(Q\), \(S\), \(l\), and \(I_{yy}\) are the dynamic pressure, reference area, reference length, and moment of inertia of the pitch axis, respectively, and \(C_{L_{\alpha}}\), \(C_{L_{\beta}}\), \(C_{m_{\alpha}}\), \(C_{m_{\alpha}}\), and \(C_{m_{\beta}}\) are the aerodynamic coefficients. Herein, if we design a state feedback controller based on (7) and (8), the controller will be challenging to apply to practical situations due to the angle-of-attack being difficult to measure at high-speed conditions. Thus, in terms of the practical application of the controller, it is more advantageous to use the normal acceleration \(a_{M}\), obtainable from the INS sensor, as a state variable rather than the angle-of-attack. Using \(a_{M}\) as a state variable, the equations in (7) and (8) can be rewritten as  
\[\dot{x_{c}}=A_{c}x_{c}+B_{c}\dot{\delta} \tag{9}\]  
where \(x_{c}\) is the redefined missile state vector given by \(x_{c}= \begin{bmatrix}a_{M}&q&\dot{q}\end{bmatrix} ^{T}\) and the system matrices \(A_{c}\) and \(B_{c}\) are given by  
\[A_{c}= \begin{bmatrix}-\frac{QS}{mV_{M}}C_{L_{\alpha}}&\frac{QS}{m}C_{L_{\alpha }}&0\\ 0&0&1\\ -\frac{QSl}{I_{yy}V_{M}}C_{m_{\alpha}}&\frac{QSl}{I_{yy}}C_{m_{\alpha}}&\frac {QSl^{2}}{I_{yy}V_{M}}C_{m_{\beta}}\end{bmatrix} ,\quad B_{c}= \begin{bmatrix} \frac{QS}{m}C_{L_{\beta}}\\ 0\\ \frac{QSl}{I_{yy}}C_{m_{\beta}}\end{bmatrix} . \tag{10}\]  
Combining (5) and (9), we have the integrated equations of the integrated dynamics as  
\[\dot{x}=Ax+B\dot{\delta}+C\alpha_{T} \tag{11}\]  
where \(x\) is the integrated state vector defined as \(x=