# Adaptive Backstepping Control of Quadrotor UAVs with Output Constraints and Input Saturation
## 3 Adaptive Backstepping Controller Design
### Stability Analysis  
The control law (31) and the adaptive laws (32), (33) are selected based on Assumptions 1-4. The following properties hold if the system satisfies the state constraints \(-k_{bi}<x_{i1}(0)<k_{bi}\) and input saturation.  
Theorem 1: _All closed-loop signals are bounded._  
Theorem 2: _The attitude system state does not exceed the set constrained condition \(-k_{bi}<x_{i1}(t)<k_{bi},\forall t>0\)._  
Theorem 3: _The output tracking error can be made to converge to the domain of zero by selecting suitable parameters and the convergence accuracy is affected by the parameters._To facilitate the subsequent proof, first define  
\[K_{i}=\frac{1}{\tau_{i}}-\frac{a_{i}}{4}-\frac{\varrho_{i}^{*2}}{2\sigma_{i}} \tag{35}\]  
Substituting (35) into (34), then  
\[\dot{V}_{i2}\leq-a_{i}(k_{i1}-1)e_{i1}^{2}-K_{i}T_{i}^{2}-\frac{\gamma_{i}}{2 \beta_{i}}\tilde{\Theta}_{i}^{2}-\frac{v_{i}}{2\rho_{i}}\tilde{w}_{i}^{2}-k_{ 12}e_{i2}^{2}+\frac{3\sigma_{i}}{2}+0.2785\varsigma_{i}w_{i}^{*} \tag{36}\]  
According to Lemma 2, it can be deduced that  
\[-\ln\frac{k_{ai}^{2}}{k_{ai}^{2}-e_{i1}^{2}}\geq-\frac{e_{i1}^{2}}{k_{ai}^{2}- e_{i1}^{2}} \tag{37}\]  
According to Equation (37), for the attitude control subsystem that requires output constrained, we can further obtain  
\[\dot{V}_{i2}\leq-(k_{i1}-1)\ln\frac{k_{\text{bi}}^{2}}{k_{\text{bi}}^{2}-e_{1i }^{2}}-K_{i}T_{i}^{2}-\frac{\gamma_{i}}{2\beta_{i}}\tilde{\Theta}_{i}^{2}- \frac{v_{i}}{2\rho_{i}}\tilde{w}_{i}^{2}-k_{i2}e_{i2}^{2}+\frac{3\sigma_{i}}{ 2}+0.2785\varsigma_{i}w_{i}^{*} \tag{38}\]  
Define \(D_{i}=min\{2(k_{i1}-1),2k_{i2},2K_{i},\gamma_{i},v_{i}\}\), \(H_{i}=\frac{3\sigma_{i}}{2}+0.2785\varsigma_{i}w_{i2}^{*}\). Based on (26), we can obtain  
\[\dot{V}_{i2}\leq-D_{i}V_{i2}+\text{H}_{i} \tag{39}\]  
**Proof of Theorem 1.** Multiplying both sides of (39) by \(e^{tD_{i}}\)at the same time, the following inequality is obtained  
\[e^{D_{i}t}\dot{V}_{i2}\leq(-D_{i}V_{i2}+\text{H}_{i})e^{D_{i}t} \tag{40}\]  
Perform the integration operation on Equation (40):  
\[\begin{split}&\frac{d}{dt}\Big{(}e^{D_{i}t}V_{i2}\Big{)}\leq \text{H}_{i}e^{D_{i}t}\\ & e^{D_{i}t}V_{i2}-V_{i2}(0)\leq\frac{\text{H}_{i2}}{D_{i}} \Big{(}e^{D_{i}t}-1\Big{)}\\ & 0\leq V_{i2}\leq V_{i2}(0)e^{-D_{i}t}+\frac{\text{H}_{i}}{D_{i} }\Big{(}1-e^{-D_{i}t}\Big{)}\leq V_{i2}(0)+\frac{\text{H}_{i}}{D_{i}}\end{split} \tag{41}\]  
If \(V_{i2}(0)\leq\ell\), we can obtain \(V_{i2}(0)\leq\ell+\frac{\text{H}_{i}}{D_{i}}\). Based on (39), it is known that \(V_{i2}\) is bounded. It can be determined that \(e_{i1}e_{i2}\), \(a_{i},s_{i}\) are bounded. Also, \(u_{i},x_{i1}\) are bounded. Then, all variables are in the closed-loop control system. \(\square\)  
**Proof of Theorem 2.**\(-k_{bi}<x_{i1}(0)<k_{bi}\) can be obtained according to Lemma 1. Since \(x_{i1}(t)=e_{i1}(t)+x_{i1d}(t)\), it can be deduced that  
\[-k_{ai}+x_{i1d}(t)<x_{i1}(t)<k_{ai}+x_{i1d}(t),\forall t>0 \tag{42}\]  
Then we infer that \(-k_{bi}<x_{i1}(t)<k_{bi},\forall t>0\). \(\square\)  
**Proof of Theorem 3.** Based on Equation (26), it can be obtained that  
\[\frac{1}{2}ln\frac{k_{ai}^{2}}{k_{ai}^{2}-e_{i1}^{2}}\leq\bigg{(}V_{i2}(0)- \frac{H_{i}}{D_{i}}\bigg{)}e^{-D_{i}t}+\frac{H_{i}}{D_{i}} \tag{43}\]Transforming both sides of Equation (43) into an exponent with \(e\) as the base  
\[\frac{k_{zi}^{2}}{k_{zi}^{2}-e_{i1}^{2}}\leq e^{2\left[(V_{2}(0)-H_{i}/D_{i})e^{- D_{i}t}+H_{i}/D_{i}\right]} \tag{44}\]  
Since \(|e_{i1}|<k_{zi}\), it is known that \(k_{zi}^{2}-e_{i1}^{2}>0\), and it can be obtained that  
\[|e_{i1}(t)|\leq k_{zi}(t)\sqrt{1-e^{-2\left[(V_{2}(0)-H_{i}/D_{i})e^{-D_{i}t}+H_ {i}/D_{i}\right]}} \tag{45}\]  
When \(t\rightarrow\infty\), \(e_{i1}(t)\leq k_{bi}(t)\sqrt{1-e^{-2H_{i}/D_{i}}}\), we obtain  
\[|x_{i1}(t)-x_{i1d}(t)|\leq k_{zi}(t)\sqrt{1-e^{-2H_{i}/D_{i}}} \tag{46}\]  
According to the definitions of \(H_{i}\) and \(D_{i}\), the output tracking error converges to the domain of 0 when \(t\rightarrow\infty\), and the convergence accuracy is affected by the parameters.