# Cooperative Guidance Law for the Mother-Cabin of the Anti-UAV Cluster Mother-Son Missile
## 3 Multi-Missile Cooperative Guidance Law Design
### Guidance Law Design in the Normal Direction of LOS
\begin{array}{l}\dot{z}_{0}=v_{0}+u,v_{0}=h_{0}(z_{0}-\sigma)+z_{1}\\ \dot{z}_{1}=v_{1},\qquad v_{1}=h_{1}(z_{1}-v_{0})+z_{2}\\ \vdots\\ \dot{z}_{m-1}=v_{m-1},v_{m-1}=h_{m-1}(z_{m-1}-v_{m-2})+z_{m}\\ \dot{z}_{m}=h(z_{m}-v_{m-1})\end{array} \right. \tag{17}\]  
_where the expression of function \(h\) is as follows:_  
\[h_{i}(\cdot)=-\lambda_{i}L^{1/(m-i+1)}\Big{|}\cdot\Big{|}^{(m-i)/(m-i+1)}\text {sgn}(\cdot)-\mu_{i}(\cdot) \tag{18}\]  
_where \(\lambda_{i}>0\), \(\mu_{i}>0\), \(i=0,1,\ldots,\text{m}\), and \(L\) is the Lipschitz constant. Based on the observer, the estimation error can be eliminated in a finite time, i.e., \(z_{0}=\sigma\), \(z_{1}=d\), \(\ldots\), \(z_{m}=\sigma_{m}-d^{(m-1)}\)._  
Combined with system (10), Lemma 3 is applied to design an inhomogeneous disturbance observer (see Equation (19)) to estimate \(w_{qi}/x_{1i}\):  
\[\left\{ \begin{array}{l}\dot{z}_{0i}=v_{0i}-\frac{2x_{2i}}{x_{1i}}x_{4i}- \frac{u_{qi}}{x_{1i}}\\ v_{0i}=-\lambda_{2i}L_{i}^{\frac{1}{3}}|z_{0i}-x_{4i}|^{\frac{2}{3}}\text{sgn} (z_{0i}-x_{4i})-\mu_{2i}(z_{0i}-x_{4i})+z_{1i}\\ \dot{z}_{1i}=v_{1i}\\ v_{1i}=-\lambda_{1i}L_{i}^{\frac{1}{3}}|z_{1i}-v_{0i}|^{\frac{1}{3}}\text{sgn} (z_{1i}-v_{0i})-\mu_{1i}(z_{1i}-v_{0i})+z_{2i}\\ \dot{z}_{2i}=-\lambda_{0i}L_{i}\text{sgn}(z_{2i}-v_{1i})-\mu_{0i}(z_{2i}-v_{1i} )\\ \dot{d}_{qi}=z_{1i}\end{array} \right. \tag{19}\]  
where \(d_{qi}=w_{qi}/x_{1i}\) represents the system disturbance term, \(L_{i}\) is the constant satisfying \(L_{i}>\left|\dot{d}_{qi}\right|\), and \(\hat{d}_{qi}\) is the estimated value of the system disturbance. **Proof of the Controller (15). ** To prove the stability and finite-time convergence of controller (15), we introduce the following lemma. \(\square\)  
**Lemma 4** ([19]). : _Suppose \(V(t)\) is a positive definite function, and there are constants \(\mu,\eta>0\) and \(0<\lambda<1\) such that \(V(t)\) satisfies the following differential inequalities:_  
\[\dot{V}(t)+\mu V(t)+\eta V^{\lambda}(t)\leq 0 \tag{20}\]Then the system will converge to zero in a finite time, and the convergence time satisfies the following constraint:  
\[t_{f}\leq t_{0}+\frac{1}{\mu(1-\lambda)}\ln\frac{\mu V^{1-\lambda}(t_{0})+\eta}{\eta} \tag{21}\]  
The Lyapunov function \(V\) is selected as:  
\[V=\frac{1}{2}s^{2} \tag{22}\]  
By differentiating Equation (22), we obtain:  
\[ \begin{array}{ll}\dot{V}&=s\dot{s}=-k_{1}s^{2}-k_{2}|s|^{\beta_{1}}\text{ ssgn}(s)\\ &=-k_{1}s^{2}-k_{2}|s|^{\beta_{1}+1}\\ &=-\mu V-\eta V^{\eta}\leq 0\end{array}  \tag{23}\]  
where \(\mu=2k_{1}>0\), \(\eta=2\frac{\beta_{1}+1}{2}k_{2}\), and \(0<\lambda=\frac{\beta_{1}+1}{2}<1\). According to Lemma 4, the sliding mode surface \(s\) is reachable in a limited time, and the LOS and LOS rate of each missile converges to zero in a limited time. **Theorem 1**. : _Considering the multi-missile system (3), when the communication topology graph \(G\) is undirected and connected, the controller (9) and the controller (15) can make the multi-missile system impact the target at the same time. The LOS of each missile converges to zero in a finite time, and the speed converges at the impact time._  
**Remark 4**. : _The controllers designed in this paper contain the symbolic function \(\text{sgn}(\cdot)\), which may cause chattering in the output of the controller. Therefore, the following function \(h(\cdot)\)is designed to replace the original symbolic function:_  
\[h(x)=\frac{2}{1+e^{-10x}}-1 \tag{24}\]