# Finite-Time Height Control of Quadrotor UAVs
## 3 Inner-Loop Control Scheme Design
### Finite-Time ESO Design
 \tag{9}\]  
It can be derived that when \(\beta_{1}>0\), \(\beta_{2}>0\), \(\beta_{3}>0\) and \(0<n<1\) hold, the characteristic roots of matrix \(\boldsymbol{P}_{1}\) have negative real parts, i.e., \(\boldsymbol{P}_{1}\) is a Hurwitz matrix. Taking the derivative of \(V_{1}(t)\), we have  
\[\begin{array}{l}\dot{V}_{1}(t)&=\dot{\boldsymbol{h}}(t)^{T}\mathbf{L}_{1} \boldsymbol{h}(t)+\boldsymbol{h}(t)^{T}\mathbf{L}_{1}\dot{\boldsymbol{h}}(t) \\ &=(\boldsymbol{P}_{1}\boldsymbol{h}(t)-\mathbf{Q}_{1}\varpi(t))^{T}\mathbf{L}_ {1}\boldsymbol{h}(t)+\boldsymbol{h}(t)^{T}\mathbf{L}_{1}(\boldsymbol{P}_{1} \boldsymbol{  
where \(\widetilde{\widetilde{Q}_{1}}^{T}=-\mathbf{Q}_{1}^{T}\mathbf{L}_{1}=[\beta_{3},-2]\) with \(\mathring{Q}_{1}=\|\widetilde{\widetilde{Q}}_{1}\|=\sqrt{\beta_{3}^{2}+4}\). Since \(\mathbf{P}_{1}\) is a Hurwitz matrix and \(\mathbf{L}_{1}\) is a positive definite matrix, there exists a positive definite symmetric matrix \(\mathbf{T}_{1}\) such that \(\mathbf{P}_{1}^{T}\mathbf{L}_{1}+\mathbf{L}_{1}\mathbf{P}_{1}=-\mathbf{T}_{1}\) holds. Additionally, there is  
\[\lambda_{\min}\{\mathbf{L}_{1}\}\|\boldsymbol{h}(t)\|^{2}\leq V_{1}(t)\leq \lambda_{\max}\{\mathbf{L}_{1}\}\|\boldsymbol{h}(t)\|^{2} \tag{11}\]  
As a result, we have  
\[ \begin{array}{rl}\dot{V}_{1}(t)&=-\boldsymbol{h}(t)^{T}\mathbf{T}_{1} \boldsymbol{h}(t)+2\varpi(t)\widetilde{\widetilde{Q}_{1}}^{T}\boldsymbol{h}(t )\\ &\leq-\lambda_{\min}\{\mathbf{T}_{1}\}\|\boldsymbol{h}(t)\|^{2}+2M\mathring{Q }_{1}\|\boldsymbol{h}(t)\|\\ &=-\frac{1}{2}(\lambda_{\min}\{\mathbf{T}_{1}\}\|\boldsymbol{h}(t)\|-4M \mathring{Q}_{1})\|\boldsymbol{h}(t)\|-\frac{1}{2}\lambda_{\min}\{\mathbf{T}_ {1}\}\|\boldsymbol{h}(t)\|^{2}\end{array}  \tag{12}\]  
For Formula (12), when \(\|\boldsymbol{h}(t)\|\geq\frac{4M\mathring{Q}_{1}}{\lambda_{\min}\{\mathbf{T}_ {1}\}}\) is satisfied, \(\dot{V}_{1}(t)\leq 0\) holds true. Moreover, introduce the shrinkage factor \(0<\delta<1\). When \(\|\boldsymbol{h}(t)\|\geq\frac{4M\mathring{Q}_{1}}{\delta\lambda_{\min}\{ \mathbf{T}_{1}\}}\) is fulfilled, we have  
\[\dot{V}_{1}(t)\leq-\frac{1}{2}\lambda_{\min}\{\mathbf{T}_{1}\}\|\boldsymbol{h} (t)\|^{2}-\frac{2M\mathring{Q}_{1}(1-\delta)}{\delta}\|\boldsymbol{h}(t)\| \tag{13}\]  
Letting \(d_{1}=\frac{1}{2}\lambda_{\min}\{\mathbf{T}_{1}\}\) and \(d_{2}=\frac{\mathring{2M\mathring{Q}_{1}}(1-\delta)}{\delta}\), the Formula (13) is rewritten as  
\[\dot{V}_{1}(t)\leq-d_{1}\|\boldsymbol{h}(t)\|^{2}-d_{2}\|\boldsymbol{h}(t)\| \leq-\frac{d_{1}}{\lambda_{\max}\{\mathbf{L}_{1}\}}V_{1}(t)-\frac{d_{2}}{ \sqrt{\lambda_{\max}\{\mathbf{L}_{1}\}}}\sqrt{V_{1}(t)} \tag{14}\]  
According to Lemma 1, \(\boldsymbol{h}(t)\) converges to \(\left\{\boldsymbol{h}(t)\|\boldsymbol{h}(t)\|\leq\frac{4M\mathring{Q}_{1}}{ \lambda_{\min}\{\mathbf{T}_{1}\}}\right\}\) in finite time \(T_{m}\). Further adjusting parameters \(\beta_{1}\),\(\beta_{2}\),\(\beta_{3}\) and \(n\) can make \(\lambda_{\min}\{\mathbf{T}_{1}\}\) sufficiently small, and the estimation error can converge to the very small neighborhood of zero. The convergence time of the finite-time ESO can be calculated by the following formula as  
\[T_{m}\leq\frac{2\lambda_{\max}\{\mathbf{L}_{1}\}}{d_{1}}\ln\frac{\frac{d_{1}} {\lambda_{\max}\{\mathbf{L}_{1}\}}\sqrt{V_{1}(0)}+\frac{d_{2}}{\sqrt{\lambda_ {\max}\{\mathbf{L}_{1}\}}}}{\frac{d_{2}}{\sqrt{\lambda_{\max}\{\mathbf{L}_{1} \}}}} \tag{15}\]  
\(\Box\)