# Terminal Distributed Cooperative Guidance Law for Multiple UAVs Based on Consistency Theory
## 4 The Upper-Level Distributed Coordination Strategy Based on Consistency
Therefore, whether \(a_{M\_2,i}\) is zero or not, the derivative of the time-to-go can be expressed as:  
\[\dot{t}_{go,i}=-1+f_{2,i}(r_{i},V_{M,i},\sigma_{M,i})a_{M\_2,i} \tag{42}\]  
The dynamic change of the UAV's impact time can be expressed as:  
\[\dot{t}_{f,i}=f_{2,i}(r_{i},V_{M,i},\sigma_{M,i})a_{M\_2,i} \tag{43}\]  
For the dynamic system described in Equation (43), according to the cooperative control theory, the following cooperative control algorithm is designed:  
\[a_{M\_2,i}=f_{2,i}^{-1}(r_{i},V_{M,i},\sigma_{M,i})\left(\sum_{j=1}^{n}\frac {s_{ij}t_{f,j}}{\sum_{j=1}^{n}s_{ij}}-t_{f,i}\right)=f_{2,i}^{-1}(r_{i},V_{M,i },\sigma_{M,i})\sum_{j=1}^{n}\frac{s_{ij}}{\sum_{j=1}^{n}s_{ij}}\Big{(}t_{f,j} -t_{f,i}\Big{)} \tag{44}\]  
By using this algorithm, the operational requirement of impact time cooperative guidance can be satisfied. Substitute Equation (44) into Equation (43) to obtain,  
\[\dot{t}_{f,i}=\sum_{j=1}^{n}\frac{s_{ij}}{\sum_{j=1}^{n}s_{ij}}\Big{(}t_{f,j }-t_{f,i}\Big{)}=\sum_{j=1}^{n}d_{ij}\Big{(}t_{f,j}-t_{f,i}\Big{)} \tag{45}\]  
where, \(d_{ij}=s_{ij}/\sum_{j=1}^{n}s_{ij}\)  
For the first-order closed-loop cooperative control system described in Equation (45), the research results of the consistent cooperative control theory show that the necessary and sufficient conditions for the communication network topological system to converge to consistency are as follows: if and only if the communication network topology of the system is strongly connected, that is, there is connectivity between any two nodes in the communication network structure. Therefore, for the cooperative guidance system composed of multiple UAVs, the ultimate goal of the \(ith\) UAV can be expressed as:  
\[t_{f,i}\rightarrow\sum\limits_{j=1}^{n}s_{ij}t_{f,j}/\sum\limits_{j=1}^{n}s_{ij} \tag{46}\]  
A non-singular sliding mode guidance law with impact time constraint is designed in Section 3. When the desired impact time is designated in advance, the guidance law can be used to attack the target at the designated time. Based on this, the collaborative control is designed:  
\[a_{M,2,i}=k\epsilon_{i}\sin\sigma_{M,i}+M(p\text{sign}(K_{2,i}+K_{3,i})+1) \text{sign}(\epsilon_{i}) \tag{47}\]  
where, \(\epsilon_{i}=\sum\limits_{j=1}^{n}d_{ij}\Big{(}t_{f,j}-t_{f,i}\Big{)}=\sum \limits_{j=1}^{n}d_{ij}\big{(}t_{go,j}-t_{go,i}\big{)}\)  
To sum up, the distributed time cooperative guidance law with time constraint designed in this paper can be expressed as:  
\[ \begin{array}{ll}a_{M,i}&=a_{M,1,i}+a_{M,2,i}\\ &=-\frac{V_{M,i}}{K_{2,i}+K_{3,i}}\Big{(}(1+K_{1,i})+(K_{2,i}+K_{3,i})\frac{V _{M,i}\sin\sigma_{M,i}}{r_{i}}\Big{)}+\\ &k\epsilon_{i}\sin\sigma_{M,i}+M(p\text{sign}(K_{2,i}+K_{3,i})+1)\text{sign}( \epsilon_{i})\end{array}  \tag{48}\]  
It can be seen from Equation (48) that the architecture of the distributed time cooperative guidance law is a two-layer cooperative guidance architecture. The bottom layer is the guidance law based on sliding mode control theory, and the upper layer is the distributed cooperative strategy based on consistency.