# 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  
Consider the following extended control algorithm for system (53) and (54):  
\[ \begin{split}\bar{\
u}_{1i}&=-\hat{n}_{i}(k_{xi}y_{ xi}-\hat{r}_{xi}(t))+\hat{k}_{xi}y_{xi}+\bar{\
u}_{1 ci},\\ \bar{\
u}_{2i}&=-\hat{n}_{i}(k_{yi}y_{yi}-\hat{\
u}_ {yi}(t))+\hat{k}_{yi}y_{yi}+\bar{\
u}_{2 ci},\\ \bar{\
u}_{3i}&=-\hat{n}_{i}(k_{hi}y_{hi}-\hat{\
u}_ {hi}(t))+k_{hi}y_{hi}+\bar{\
u}_{3 ci},\end{split}  \tag{61}\]  
Figure 1: Communication graph. where \(\vartheta_{1 ci}\), \(\vartheta_{2 ci}\), \(\vartheta_{3 ci}\) are additional control signals based on the consensus algorithm by [PERSON] [PERSON] and [PERSON] (2016):  
\[\vartheta_{1 ci} =-k_{xc}y_{xi}-\rho_{x}\sum_{j=1}^{N}a_{ij}\big{(}y_{xi}-y_{xj} \big{)}, \tag{62}\] \[\bar{\upsilon}_{2 ci} =-k_{yc}y_{yi}-\rho_{y}\sum_{j=1}^{N}a_{ij}\big{(}y_{yi}-y_{yi} \big{)},\] \[\vartheta_{3 ci} =-k_{hc}y_{hi}-\rho_{h}\sum_{j=1}^{N}a_{ij}\big{(}y_{hi}-y_{hi} \big{)},\]  
where \(k_{xc}>0\), \(k_{yc}>0\), \(k_{hc}>0\), \(\rho_{x}>0\), \(\rho_{y}>0\), \(\rho_{h}>0\) denote the consensus algorithm parameters and \(a_{ij}\) represents the adjacency matrix elements, thereby defining a directed connections graph (digraph) \(\mathcal{G}\) between the UAVs for the consensus algorithms (59) and (60). In matrix form, (62) can be written as  
\[\vartheta_{1c} =-(k_{xc}I_{N}+\rho_{x}L)y_{x}, \tag{63}\] \[\bar{\upsilon}_{2c} =-(k_{yc}I_{N}+\rho_{y}L)y_{y},\] \[\bar{\upsilon}_{3c} =-(k_{hc}I_{N}+\rho_{h}L)y_{h},\]  
where \(I_{N}\) denotes the \(N\times N\) identity matrix, \(L\) is the Laplace matrix corresponding to the adjacency matrix and the directed connection graph \(\mathcal{G}\), and the following vector notations are used: \(y_{x}=[y_{x1},y_{x2},\ldots,y_{xN}]^{\mathrm{T}}\), \(y_{y}=[y_{y1},y_{y2},\ldots,y_{yN}]^{\mathrm{T}}\), \(y_{h}=[y_{h1},y_{h2},\ldots,y_{hN}]^{\mathrm{T}}\), \(\vartheta_{1c}=[\vartheta_{1c1},\vartheta_{1c2},\ldots,\vartheta_{1 cN}]^{ \mathrm{T}}\), \(\vartheta_{2c}=[\vartheta_{2c1},\vartheta_{2c2},\ldots,\vartheta_{2 cN}]^{ \mathrm{T}}\), \(\vartheta_{3c}=[\vartheta_{3c1},\vartheta_{3c2},\ldots,\vartheta_{3 cN}]^{ \mathrm{T}}\). The following Proposition can now be stated. **Proposition 3**. : _System (53) and (54) with control (61) and (62), adaptation algorithms (57), control goal (58), and the settings \(\hat{m}_{i}\), \(\hat{k}_{xi}\), \(\hat{k}_{y}\), \(\hat{k}_{hi}\), and \(i=1,2,\ldots,N\) is stable if the associated communication digraph \(\mathcal{G}\) contains a spanning tree._  
**Sketch of Proof for Proposition 3. ** We can prove this proposition, using the SG scheme by checking the following conditions (cf. the proof of Proposition 2):  
1. The scalar objective function \(Q_{tF}(e_{xyh},y_{xyh})\) is non-negative and satisfies the growth condition \(\inf\limits_{t\geqslant 0}Q_{tF}(e_{xyh},y_{xyh})\to\infty\) as \(\|(e_{xyh},y_{xyh})\|\to\infty\). This is true for function (58). 2. Function \(Q_{tF}\) should be convex on the adjustable parameters \(\hat{m}_{i}\), \(\hat{k}_{xi}\), \(\hat{k}_{yi}\), \(\hat{k}_{hi}\), \(i=1,2,\ldots,N\). This condition is met because \(\hat{Q}_{tF}\) is linear with respect to the parameters. 3.