# Speed-Gradient Adaptive Control for Parametrically Uncertain UAVs in Formation
## 2 Point-Mass Model of UAV Dynamics
### Synthesis of UAV Adaptation Law  
To obtain the adaptive tuning law for control law (23), the adjustable parameters \(\hat{m},\hat{k}_{x},\hat{k}_{y},\hat{k}_{h}\) are employed with the Speed-gradient (SG) method, cf. [52; 53; 54; 55]. Let the control aim be as given by (19). The time derivative of \(Q_{t}\) along system (21) and (22) trajectories with control (23) is as follows:  
\[\dot{Q}_{t}=e_{x1}e_{x1}+e_{y1}e_{y1}+e_{h1}e_{h1}+my_{x}\dot{y}_{x}+my_{y}\dot {y}_{y}+my_{h}\dot{y}_{h}. \tag{42}\]  
Taking into account (24), we have  
\[\dot{Q}_{t} =e_{x1}\dot{e}_{x1}+e_{y1}\dot{e}_{y1}+e_{h1}\dot{e}_{h1}+ \tag{43}\] \[+my_{x}\Bigg{(}-a_{x}^{2}e_{x1}+\Bigg{(}\alpha_{x}-\frac{\hat{m} }{m}k_{x}+\frac{\hat{k}_{x}}{m}\Bigg{)}y_{x}+\Bigg{(}\frac{\hat{m}}{m}-1 \Bigg{)}\dot{r}_{x}(t)\Bigg{)}+\] \[+my_{y}\Bigg{(}-a_{y}^{2}e_{y1}+\Bigg{(}\alpha_{y}-\frac{\hat{m} }{m}k_{y}+\frac{\hat{k}_{y}}{m}\Bigg{)}y_{y}+\Bigg{(}\frac{\hat{m}}{m}-1 \Bigg{)}\dot{r}_{y}(t)\Bigg{)}+\] \[+my_{h}\Bigg{(}-a_{h}^{2}e_{h1}+\Bigg{(}\alpha_{h}-\frac{\hat{m} }{m}k_{h}+\frac{\hat{k}_{h}}{m}\Bigg{)}y_{h}+\Bigg{(}\frac{\hat{m}}{m}-1 \Bigg{)}\dot{(}\dot{r}_{h}(t)+g\Bigg{)}\Bigg{)}\] \[=e_{x1}\dot{e}_{x1}+e_{y1}\dot{e}_{y1}+e_{h1}\dot{e}_{h1}\] \[-ma_{x}^{2}e_{x1}y_{x}+\Bigg{(}ma_{x}-\hat{m}k_{x}+\hat{k}_{x} \Bigg{)}y_{x}^{2}+(\hat{m}-m)\dot{r}_{x}(t)y_{x}\] \[-ma_{y}^{2}e_{y1}y_{y}+\Bigg{(}ma_{y}-\hat{m}k_{y}+\hat{k}_{y} \Bigg{)}y_{y}^{2}+(\hat{m}-m)\dot{r}_{y}(t)y_{y}\] \[-ma_{h}^{2}\dot{e}_{h1}y_{h}+\Bigg{(}ma_{h}-\hat{m}k_{h}+\hat{k}_{ h}\Bigg{)}y_{h}^{2}+(\hat{m}-m)(\dot{r}_{h}(t)+g)y_{h}.\]  
The vector of function (43) has a gradient with respect to the adjustable parameters \(\hat{m}\), \(\hat{k}_{x}\), \(\hat{k}_{y}\), \(\hat{k}_{h}\) as follows:  
\[\
abla\dot{Q}_{t}=\left[ \begin{array}{c}-k_{x}y_{x}^{2}+y_{x}\ddot{r}_{x}(t )-k_{y}y_{y}^{2}+y_{y}\ddot{r}_{y}(t)-k_{h}y_{h}^{2}+y_{h}(\ddot{r}_{h}(t)+g) \\ y_{x}^{2}\\ y_{y}^{2}\\ y_{y}^{2}\\ y_{h}^{2}\end{array} \right]. \tag{44}\]  
The SG design procedure leads to the following tuning algorithms for variables \(\hat{m},\hat{k}_{x}\), \(\hat{k}_{y}\), \(\hat{k}_{h}\) in (23):  
\[\hat{m} =-\gamma_{m}\Big{(}-k_{x}y_{x}^{2}+y_{x}\ddot{r}_{x}(t)-k_{y}y_{ y}^{2}+y_{y}\ddot{r}_{y}(t)-k_{h}y_{h}^{2}+y_{h}(\ddot{r}_{h}(t)+g)\Big{)}, \tag{45}\] \[\hat{k}_{x} =-\gamma_{x}y_{x}^{2},\] \[\hat{k}_{y} =-\gamma_{y}y_{y}^{2},\] \[\hat{k}_{h} =-\gamma_{h}y_{h}^{2},\]  
where \(\gamma_{m}>0\), \(\gamma_{x}>0\), \(\gamma_{y}>0\), \(\gamma_{h}>0\) are the adaptation gains (i.e., the design parameters). The following result can be derived. **Proposition 2**. : _In system (21) and (22) with control (23) and tuning algorithms (45), the control goal (19) is achieved and the tuning parameters \(\hat{m}\), \(\hat{k}_{x}\), \(\hat{k}_{y}\), \(\hat{k}_{h}\) are bounded._  
**Proof of Proposition 2. ** To prove this, we can check the conditions under which the SG scheme is operable [52; 53; 54; 55]:  
1. The scalar objective function \(Q_{t}(e_{\text{xyh}},y_{\text{xyh}})\) is non-negative and satisfies the growth condition \(\inf\limits_{t\geqslant 0}Q_{t}(e_{\text{xyh}},y_{\text{xyh}})\to\infty\) as \(\|[e_{\text{xyh}},y_{\text{xyh}}]\|\to\infty\). Note that this condition is met for function (19). 2. Function \(Q_{t}\) should be convex on adjustable parameters \(\hat{m}\), \(\hat{k}_{x}\), \(\hat{k}_{y}\), \(\hat{k}_{h}\). This condition is met because \(Q_{t}\) is linear with respect to the adjustable parameters. 3. The following reachability condition is satisfied: for any \(m\) from the range of its possible values \(m_{\text{min}}\leqslant m\leqslant m_{\text{max}}\), there exists a set of parameters \(\hat{m}_{*}\), \(\hat{k}_{xx}\), \(\hat{k}_{y*}\), \(\hat{k}_{h*}\) and a function \(f(Q_{t})\) (\(f(Q_{t})>0\) as \(Q_{t}>0\)) such that for all \(x\), \(y\), \(h\), \(t\), the inequality \[Q_{t}((e_{\text{xyh}},y_{\text{xyh}},\hat{m}_{*},\hat{k}_{xx},\hat{k}_{y*}, \hat{k}_{h*})\leqslant-f\big{(}Q_{t}(e_{\text{xyh}},y_{\text{xyh}})\big{)},\] (46) is fulfilled, as follows from Proposition 1. This completes the proof.