# Integrated Autopilot Guidance Based on Zero-Effort-Miss Formulation for Tail-Controlled Missiles
## 3 Integrated Control for Guidance and Autopilot Synthesis
### Design of an Integrated Guidance Autopilot via Proportional Control  
Taking the time derivative to ZEM, we have  
\[\frac{dz_{f}}{dt}= I_{5}(1,\cdot)\frac{d}{dt}\big{\{}\exp\big{(}At_{go}\big{)}x\big{\}}\] \[= I_{5}(1,\cdot)\big{\{}-\exp\big{(}At_{go}\big{)}Ax+\exp\big{(}At_ {go}\big{)}\big{(}Ax+B\dot{\delta}+Ca_{T}\big{)}\big{\}}\] \[= I_{5}(1,\cdot)\exp\big{(}At_{go}\big{)}\big{(}B\dot{\delta}+Ca_{T }\big{)}\] \[= b\big{(}t_{go}\big{)}\dot{\delta}+c\big{(}t_{go}\big{)}a_{T} \tag{19}\]  
where the scalar functions \(b\big{(}t_{go}\big{)}\) and \(c\big{(}t_{go}\big{)}\) are defined as  
\[b\big{(}t_{go}\big{)}=I_{5}(1,\cdot)\exp\big{(}At_{go}\big{)}B \quad\text{and}\quad c\big{(}t_{go}\big{)}=I_{5}(1,\cdot)\exp\big{(}At_{go} \big{)}C, \tag{20}\]  
respectively. From (19), we can present the command for the integrated autopilot guidance as  
\[\dot{\delta}_{c}=-\frac{K_{z}}{b\big{(}t_{go}\big{)}}z_{f} \tag{21}\]  
where \(K_{z}\) is the controller gain, selected as a positive constant. Substituting (21) into (19) leads to  
\[\frac{dz_{f}}{dt}=-K_{z}z_{f}+c\big{(}t_{go}\big{)}a_{T}. \tag{22}\]  
If we consider the Lyapunov candidate function of \(V_{z}=z_{f}^{2}/2\), then its derivative satisfies that  
\[\dot{V}_{z} =-\Big{\{}K_{z}z_{f}+c\big{(}t_{go}\big{)}a_{T}\Big{\}}z_{f}\] \[\leq-K_{z}z_{f}^{2}+\frac{1}{2}\big{|}c\big{(}t_{go}\big{)}\big{|} \Big{(}a_{T}^{2}+z_{f}^{2}\Big{)}\] \[=-\big{(}2K_{z}-|c\big{(}t_{go}\big{)}\big{|}\big{)}V_{z}+\frac{ 1}{2}\big{|}c\big{(}t_{go}\big{)}\big{|}a_{T}^{2} \tag{23}\]  
where [PERSON]'s inequality is used. The inequality in (23) implies that ZEM \(z_{f}\) converges into the ball whose radius is given by  
\[= I_{5}(1,\cdot)\big{\{}-\exp\big{(}At_{go}\big{)}Ax+\exp\big{(}At_ {go}\big{)}\big{(}Ax+B\dot{\delta}+Ca_{T}\big{)}\big{\}}\]0  
if the gain \(K_{z}\) is chosen to satisfy \(K_{z}\geq\Big{|}c\big{(}t_{f}\big{)}\Big{|}/2\). It can also be seen that the convergence radius \(r_{z_{f}}\) in (24) goes to zero as the missile approaches the target. Namely, the command of the tail fin rate in (21) ensures the convergence of ZEM in (18), which means the achievement of the interception against a maneuvering target. In addition, since the proposed control input in (21) stabilizes the ZEM of the integrated dynamics at zero, the closed-loop stability of the missile dynamics is determined by the zero-input stability governed by (7) and (8) with \(\dot{\delta}=0\). To identify the internal stability, let us investigate the dynamics of the angle-of-attack \(\alpha\) as  
\[= I_{5}(1,\cdot)\big{\{}-\exp\big{(}At_{go}\big{)}Ax+\exp\big{(}At_ {go}\big{)}\big{(}Ax+B\dot{\delta}+Ca_{T}\big{)}\big{\}}\]1  
where (7) and (8) are used. In the common flight dynamics of fixed-wing aircraft, the coefficient \(C_{L_{\alpha}}\) has a positive value to produce lift in the upward direction. In addition, for stability in the longitudinal direction, the coefficients \(C_{m_{\alpha}}\) and \(C_{m_{q}}\) are designed to have negative values [15]. As a result, the equation in (25) gives proof that \(\dot{\alpha}\) is asymptotically stable at zero, which implies that \(\alpha\) is marginally stable.