# Terminal Distributed Cooperative Guidance Law for Multiple UAVs Based on Consistency Theory
## 3 Design of Bottom Guidance Law Based on Sliding Mode Control Theory
### Design of the Impact Time Control Guidance Law
#### 3.2.2 Proof of Stability  
Choose \(V=(1/2)s^{2}\) as the Lyapunov function, then,  
\[\dot{V}=s\dot{s}=-\frac{K_{2}+K_{3}}{V_{M}}ks^{2}\sin\sigma_{M}-\frac{M}{V_{M} }(p|K_{2}+K_{3}|+(K_{2}+K_{3}))|s| \tag{27}\]  
It can be seen from \((K_{2}+K_{3})\sin\sigma_{M}\geq 0\) and \(p|K_{2}+K_{3}|+(K_{2}+K_{3})\geq 0\) that \(\dot{V}\) is negative semidefinite, which means that when the leading angle, the sliding surface \(s=0\) may not be satisfied. In order to satisfy the attack time control constraints and make the sliding surface \(s=0\), it is necessary to explain that the leading angle \(\sigma_{M}=0\) is not an attractor.  
It can be seen from Equation (4) that:  
\[\dot{\sigma}_{M}=\dot{\gamma}_{M}-\dot{\lambda} \tag{28}\]  
When the leading angle \(\sigma_{M}=0\), it can be seen from Equation (2) that the rate of change of line of sight angle is as follows:  
\[\dot{\lambda}=-\frac{V_{M}\sin\sigma_{M}}{r}=0 \tag{29}\]  
Equivalent control term of the guidance law:  
\[a_{M}^{eq}=-\frac{V_{M}}{K_{2}+K_{3}}\bigg{(}(1+K_{1})+(K_{2}+K_{3})\frac{V_{M }\sin\sigma_{M}}{r}\bigg{)}=0 \tag{30}\]  
Switching term of the guidance law:  
\[a_{M}^{sup}=-k\sin\sigma_{M}-M(p\text{sign}(K_{2}+K_{3})+1)\text{sign}(s)=-M \text{sign}(s) \tag{31}\]  
It can be obtained from Equation (3) that:  
\[\dot{\gamma}_{M}=\frac{a_{M}}{V_{M}}=\frac{a_{M}^{eq}+a_{M}^{sup}}{V_{M}}=- \frac{M}{V_{M}}\text{sign}(s) \tag{32}\]  
Then the change rate of the leading angle satisfies,  
\[\dot{\sigma}_{M}=\dot{\gamma}_{M}-\dot{\lambda}=-\frac{M}{V_{M}}\text{sign}(s) \tag{33}\]  
Therefore, when the leading angle \(\sigma_{M}=0\) but \(s\
eq 0\), \(\dot{\sigma}_{M}\
eq 0\), it means that the leading angle is not an attractor. At the same time, it can be seen from Equation (33) that when the sliding surface \(s>0\), the change rate of the leading angle \(\dot{\sigma}_{M}<0\), the leading angle decreases; when the sliding surface \(s<0\), the change rate of the leading angle \(\dot{\sigma}_{M}>0\), the leading angle increases. This means that only when the sliding surface \(s=0\) and the leading angle \(\sigma_{M}=0\), the leading angle \(\sigma_{M}=0\) is an attractor of the system. For the leading angle \(\sigma_{M}\
eq 0\), the stability of Lyapunov function has been proved by Equation (27).  
At the same time, it should be noted that when the leading angle \(\sigma_{M}=0\), this can be known according to the law of [PERSON]:  
\[\lim_{\sigma_{M}\to 0}\frac{\sin\sigma_{M}-\sigma_{M}\cos\sigma_{M}}{\sin^{2} \sigma_{M}}=\lim_{\sigma_{M}\to 0}\frac{\sigma_{M}}{2\cos\sigma_{M}}=0 \tag{34}\]  
\[\lim_{\sigma_{M}\to 0}\frac{\sigma_{M}}{\sin\sigma_{M}}=\lim_{\sigma_{M}\to 0}\frac{1}{\cos \sigma_{M}}=1 \tag{35}\]  
Therefore, the coefficients \(K_{2}\) and \(K_{3}\) are not singular. When the UAV's leading angle is zero, the guidance law can also be activated. According to the above analysis, the guidance law, Equation (24), is a non-singular sliding mode guidance law with impact time control, which is recorded as NSMG.