# Adaptive Backstepping Control of Quadrotor UAVs with Output Constraints and Input Saturation
## 2 Problem Formulation and Preliminaries
The control gain is rewritten as \(\bar{b}_{i}=b_{i}+\left(\bar{b}_{i}-b_{i}\right)\). The deterministic part \(b_{i}= \begin{bmatrix}U_{1}/m_{s},U_{1}/m_{s},\cos\phi\text{cos}\theta/m_{s},1/J _{x},1/J_{y},1/J_{z}\end{bmatrix} ^{T}\) is involved in the UAV controller design, and the uncertain part is divided into a unknown function. \(w_{i}\) denotes the bounded total perturbation of each channel, and the perturbation will be compensated by designing a robust term. \(u_{i}\) indicates the control input for each channel. The attitude control subsystem requires output constraints. When \(i=\phi,\theta,\psi\), the attitude is constrained to the set \(|x_{i1}|<k_{bi}\), where \(k_{bi}\) is a constant. In addition, the control input obeys the following saturation properties:  
\[u(v)=\text{sat}(v)= \begin{cases}u_{M},&v\geq u_{M}\\ v,&u_{m}<v<u_{M}\\ u_{m},&v\leq u_{m}\end{cases}  \tag{3}\]  
where \(v\) denotes the unconstrained actual input. \(u_{M}>0\) and \(u_{m}<0\) denote the maximum and minimum of the control input, respectively. **Assumption 1**. : _For all \(t>0\), the desired tracking trajectory \(x_{i1d}\) and its first and second order derivative are bounded, and they satisfies \(\Omega_{d}=\{x_{i1d}\mid|x_{i1d}|\leq E_{0},|\dot{x}_{i1d}|\leq E_{1},|\dot{x}_ {i1d}|\leq E_{2}\}\), where \(E_{0},E_{1},E_{2}\) are all positive constants._  
**Assumption 2**. : _The approximation error \(\varepsilon\) is bounded, i.e., \(|\varepsilon|<b\), and \(0<b<1\) is of arbitrary accuracy._  
**Assumption 3**. : _The position and speed of the UAV are measurable._  
**Assumption 4**. : _For all \(t>0\), both the uncertain internal part of the model and the external disturbances are bounded and there exists an unknown parameter \(d\) such that \(\|w_{i}\|\leq d\)._  
**Lemma 1** ([39]). : _For any constant \(k_{ai}\), let the open set \(N=R\times Z\subset R^{l+1}\) and the interval \(Z=\{e_{i1}\in R:-k_{ai}<e_{i1}<k_{ai}\}\subset R,k_{ai}\in R^{+}\), and consider the following system_  
\[\dot{\zeta}=\phi(t,\zeta) \tag{4}\]_where \(\zeta=[\omega,e_{1i}]^{\mathsf{T}}\in N\) and \(\vartheta=R^{+}\times N\to R^{l+1}\) are segmentally continuous at the time \(t\) and locally Lipschitz in the system \(\zeta\), and consistent with \(t\) and \(R^{+}\times N\). Assuming that there exist functions \(U:R^{l}\to R^{+}\) and \(V_{l1}:Z\to R^{+}\), continuously differentiable and positive definite in their respective domains of definition. We obtain \(V_{l1}(e_{i1})\to\infty\) when \(e_{i1}\to-k_{ai}\), or \(e_{i1}\to k_{ai}\) and \(\beta_{1}(\parallel\omega\parallel)\leq U(\omega)\leq\beta_{2}(\parallel \omega\parallel)\), where \(\beta_{1}\) and \(\beta_{2}\) are \(K_{\infty}\) class functions. Assume that \(V_{l1}(\zeta)=V_{l1}(x)+U(\omega,t)\), and \(e_{i1}(0)\in Z\). If the inequality holds,_  
\[\dot{V}=\frac{\partial V}{\partial\xi}\vartheta\leq-DV+H \tag{5}\]  
_where \(D\) and \(H\) are positive constants. Then, \(e_{i1}\) remains in the set \(e_{i1}(t)\in Z\), where \(t\in[0,+\infty)\)._  
**Lemma 2** ([34]). : _For any positive constant \(k_{ai}\) and any variable \(e_{i1}\), the equation \(ln\frac{k_{ai}^{2}}{k_{ai}^{2}-e_{i1}^{2}}\leq\frac{e_{i1}^{2}}{k_{ai}^{2}-e_{ i2}}^{2}\) holds when the condition \(|e_{i1}|<k_{ai}\) is satisfied._  
**Lemma 3** ([40]). : _For an initially bounded system, if there exists a \(C^{1}\) continuous and positive definite Lyapunov function \(V(x)\) satisfying \(\pi_{1}(x)\leq V(x)\leq\pi_{2}(x)\) such that \(\dot{V}(x)\leq-r_{1}V(x)+r_{2}\), where \(\pi_{1}(x)\), \(\pi_{2}(x):R^{n}\to R\) are \(K_{\infty}\) like functions and \(r_{1},r_{2}\) are positive constants, then the solution \(x(t)\) of the system is consistently bounded._  
**Lemma 4** ([41]). : _The following inequality holds for any \(\varsigma\) and \(h\in R\) that \(0\leq|\hbar|-\hbar\tanh\left(\frac{h}{\varsigma}\right)\leq k_{p}\varsigma\), where is a constant that satisfies the condition \(k_{p}=e^{-\left(k_{p}+1\right)}\), i.e., \(k_{p}=0.2785\)._  
**Lemma 5** ([42]).