# Speed-Gradient Adaptive Control for Parametrically Uncertain UAVs in Formation
## 2 Point-Mass Model of UAV Dynamics
### Consensus Algorithm for Multi-Agent Formation Control
#### 4.2.2 UAV Formation Control based on the Consensus Algorithm
Reachability condition: for any set of values \(m_{i}\), \(i=1,2,...,N\) from the ranges of possible values \(m_{\min i}\leqslant m_{i}\leqslant m_{\max i}\), there exists a set of parameters \(\hat{m}_{*}=[\hat{m}_{*1},\hat{m}_{*2},\ldots,\hat{m}_{*N}]^{\mathrm{T}rn}\), \(\hat{k}_{xx}=[\hat{k}_{x*1},\hat{k}_{x*2},\ldots,\hat{k}_{x*N}]\), \(\hat{k}_{y*}=[\hat{k}_{y*1},\hat{k}_{y*2},\ldots,\hat{k}_{y*N}]^{\mathrm{T}}\), \(\hat{k}_{k*}=[\hat{k}_{h*1},\hat{k}_{h*2},\ldots,\hat{k}_{h*N}]^{\mathrm{T}}\) and function \(f(Q_{tF})\) (\(f(Q_{tF})>0\) as \(Q_{tF}>0\)) for which for all \(x,y\), \(h\), \(t\) hold: \[\hat{Q}_{tF}(e_{xyh},y_{xyh},\hat{m}_{*},\hat{k}_{x*},\hat{k}_{y*},\hat{k}_{h*}) \leqslant-f(Q_{tF}(e_{xyh},y_{xyh})).\] (64) To prove this, we can consider the Lyapunov function: \[V_{F}(t)=Q_{tF}(e_{xyh},y_{xyh}).\] (65) Because \(Q_{tF}(e_{xyh},y_{xyh})\) is the sum of the control objective functions for each UAV and the control signal based on the consensus algorithm enters additively into the overall control signal (61), by taking into account expressions (27) and (41) the proof of Proposition 1 implies \[\hat{V}_{F}(t)=-\sum_{i=1}^{N}\rho_{i}Q_{ti}(e_{xyh},y_{xyh})+f_{c}(e_{xyh},y_{ xyh}),\] (66)where \(\rho_{i}>0,\ i=1,2,\ldots,N\) and \(f_{c}(e_{xyh},y_{xyh})\) is an addition to the derivative of the consensus algorithm (63). This function is as follows:  
\[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\]0  
By assumption, the associated communication digraph \(\mathcal{G}\) contains a spanning tree; therefore (see [76]), the Laplace matrix \(L\) is positive semi-definite, and hence all the matrices inside the parentheses of expression (67 ) are positive definite. Then, \(f_{c}(e_{xyh},y_{xyh})<0\), and for (66) there always exists \(\rho_{*}>0\) such that  
\[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\]1  
is valid. Therefore, condition (64) is satisfied. This completes the proof. **Remark 6**: _In practice, it is preferable to use the robust form of the tuning algorithm (57), cf. [53]:_  
\[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\]2 \[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\]3 \[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\]4 \[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\]5 \[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\]6  
_where \(\sigma_{mi}>0\), \(\sigma_{xi}>0\), \(\sigma_{yi}>0\), \(\sigma_{hi}>0\); \(m_{0i}\), \(\hat{k}_{x0i}\), \(\hat{k}_{y0i}\), \(\hat{k}_{h0i}\) are some \"guessed\" values of the adjustable control law parameters._  
**Remark 7**: _The obtained adaptation and control algorithms for a single UAV in (23) and (47) for group control of UAVs in (61) and (62), as well as the tuning algorithms (69), are applicable for UAV models with the form (1) and expressions (12)-(14) when conditions (8) are met._