# Finite-Time Height Control of Quadrotor UAVs
## 3 Inner-Loop Control Scheme Design
### Design of Super-Twisting Controller  
In this section, a finite-time super-twisting controller of the height velocity inner loop (2) is designed according to the finite-time ESO (4). If \(\tau(t)\) is the output of the outer-loop control system and \(\mu(t)\) is the tracking error of the height velocity inner loop (2), then \(\mu(t)=\tau(t)-x_{1}(t)\). Combined with the dynamics equation of the height velocity inner loop (2), it is obtained that  
\[ \begin{array}{rl}\dot{\mu}(t)&=\dot{\tau}(t)-\dot{x}_{1}(t)\\ &=\dot{\tau}(t)-b_{0}u(t)-x_{2}(t)+g\end{array}  \tag{16}\]  
The finite-time super-twisting controller is designed as  
\[ \begin{array}{rl}u(t)=\frac{1}{b_{0}}(\alpha_{1}|\mu(t)|^{\frac{1}{2}}\text{ sign}(\mu(t))+\
u(t)+\dot{\tau}(t)-z_{2}(t)+g)\\ \dot{\
u}(t)=\alpha_{2}\text{sign}(\mu(t))\end{array}  \tag{17}\]where \(\alpha_{1}\) and \(\alpha_{2}\) are positive adjustable parameters. By inserting the finite-time super-twisting controller (17) into the Formula (16), the error system of the height velocity inner loop (2) is written as  
\[ \begin{array}{l}\dot{\mu}(t)=\
u(t)-\alpha_{1}|\mu(t)|^{\frac{1}{2}}\text{ sign}(\mu(t))+\varsigma_{2}(t)\\ \dot{\
u}(t)=-\alpha_{2}\text{sign}(\mu(t))\end{array}  \tag{18}\]  
Considering the finite-time super-twisting controller, the stability proof of the inner loop is given as follows. **Theorem 2. **_Considering the dynamics equation of the inner loop and the finite-time super-twisting controller, if there are parameters \(\alpha_{1}>0\) and \(\alpha_{2}>0\) satisfying \(\lambda_{\min}\{\textbf{T}_{2}\}-2\dot{Q}_{2}>0\), then the error of the inner-loop system \(\mu(t)\) will converge to a small neighborhood of the origin in a finite time. And convergence time \(T_{s}\) can be calculated by the following formula as_  
\[T_{s}\leq\frac{2\sqrt{V_{2}(0)}}{q}+\frac{2\lambda_{\max}\{\textbf{L}_{1}\}}{ d_{1}}\ln\frac{\frac{d_{1}}{\lambda_{\max}\{\textbf{L}_{1}\}}\sqrt{V_{1}(0)}+ \frac{d_{2}}{\sqrt{\lambda_{\max}\{\textbf{L}_{1}\}}}}{\frac{d_{2}}{\sqrt{ \lambda_{\max}\{\textbf{L}_{1}\}}}} \tag{19}\]  
**Proof of Theorem 2. ** Construct an error vector \(\boldsymbol{\eta}(t)=\left[|\mu(t)|^{\frac{1}{2}}\text{sign}(\mu(t)),\
u(t) \right]^{\text{T}}\), and Lyapunov function \(V_{2}(t)\) is selected as  
\[V_{2}(t)=\boldsymbol{\eta}(t)^{\text{T}}\textbf{L}_{2}\boldsymbol{\eta}(t) \tag{20}\]  
where the positive definite symmetric matrix \(\textbf{L}_{2}\) is selected as  
\[\textbf{L}_{2}= \begin{bmatrix}4\alpha_{2}+\alpha_{1}^{2}&-\alpha_{1}\\ -\alpha_{1}&2\end{bmatrix}  \tag{21}\]  
Take the derivative of \(\boldsymbol{\eta}(t)\), and we have  
\[ \begin{array}{l}\dot{\boldsymbol{\eta}}(t)=\left[\begin{array}{cc}\frac{1 }{2}|\mu(t)|^{-\frac{1}{2}}(\
u(t)-\alpha_{1}|\mu(t)|^{\frac{1}{2}}\text{sign}( \mu(t))+\varsigma_{2}(t))\\ -\alpha_{2}\text{sign}(\mu(t))\end{array}\right]\\ =|\mu(t)|^{-\frac{1}{2}}\Big{\{}\left[\begin{array}{cc}-\frac{1}{2}\alpha_{1 }&\frac{1}{2}\\ -\alpha_{2}&0\end{array}\right]\boldsymbol{\eta}(t)+\left[\begin{array}{cc}1 \\ 0\end{array}\right]\varsigma_{2}(t)\Big{\}}\\ =|\mu(t)|^{-\frac{1}{2}}\big{\{}\textbf{P}_{2}\boldsymbol{\eta}(t)+\textbf{Q }_{2}\varsigma_{2}(t)\big{\}}\end{array}  \tag{22}\]  
It is concluded from Formula (22) that all eigenvalues of \(\textbf{P}_{2}\) have negative real parts when \(\alpha_{1}>0\) and \(\alpha_{2}>0\) are satisfied, i.e., the matrix \(\textbf{P}_{2}\) is a Hurwitz matrix. According to the Formula (22), the derivative of the Formula (20) is calculated as  
\[\begin{array}{l}\dot{V}_{2}(t)=\dot{\boldsymbol{\eta}}(t)^{\text{T}}\textbf{ L}_{2}\boldsymbol{\eta}(t)+\boldsymbol{\eta}(t)^{\text{T}}\textbf{L}_{2} \dot{\boldsymbol{\eta}}(t  
With the Formulas (23) and (24), it is obtained that  
\[