## 3 Aggressive Control of Quadrotors with Extended Finite-Time Observer
### Extended Finite-Time Observer  
An observer of the high-order derivatives guidance vector for each quadrotor is proposed to get the feedrwards. We take \(X_{i}^{d}\) for example, according to \(v_{i1}\) in (17), we have to estimate \(\dddot{X}_{i}^{d}\), \(\dddot{X}_{i}^{d}\), \(\dddot{X}_{i}^{d}\) and \(\dddot{X}_{i}^{d}\). Therefore, we construct an observer to estimate the high-order derivatives of the positions in guidance vector. Before designing the finite-time observer, we introduce the following lemma about finite-time stability. **Lemma 1**. : _[_24_]_ _Consider the system \(\dot{x}=f(x,u)\). Suppose that there exist continuous differentiable functions \(V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}\), real numbers \(c>0\), and \(0<\alpha<1\), such that \(V(0)=0\), \(V(x)>0\) for any \(x\
eq 0\), and \(\dot{V}(x)\leq-cV^{\alpha}(x)\). Then, for all \(x(t_{0})=x_{0}\) in the neighborhood \(\mu\) of the origin, there exists settling time function \(T(x_{0}):\mu\rightarrow\mathbb{R}^{+}\), such that \(T(x_{0})\leq\frac{V^{1-\alpha}(x_{0})}{c(1-\alpha)}\)._Denoting that \(x_{i}^{d}=[X_{i}^{d},\bar{X}_{i}^{d},\bar{X}_{i}^{d},\bar{X}_{i}^{d}]^{T}\), the dynamics of \(x_{i}^{d}\) yields  
\[ \begin{split}\dot{x}_{i}^{d}=& A_{t}x_{i}^{d}+B_{t} \dddot{X}_{i}^{d}\\ y_{i}=& C_{t}x_{i}^{d}\end{split}  \tag{19}\]  
where  
\[A_{t}= \begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix} ,\,B_{t}= \begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix} \text{and}\ C_{t}= \begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix} ^{T}\]  
The extended state is \((x_{i}^{d})_{E}=[(x_{i}^{d})^{T},\bar{X}_{i}^{d}]^{T}\) and its estimation \(\hat{x}_{i}^{d}=[\bar{X}_{i}^{d},\hat{X}_{i}^{d},\hat{X}_{i}^{d},\widehat{X}_ {i}^{d},\widehat{X}_{i}^{d}]^{T}\). Then, we propose the extended observing model as follows  
\[\hat{x}_{i}^{d}= \begin{bmatrix}A_{t}&B_{t}\\ \mathbf{0}_{4}^{T}&0\end{bmatrix} \hat{x}_{i}^{d}-L_{t}(\hat{X}_{i}^{d}-X_{i}^{d}) \tag{20}\]  
where \(L_{t}\) represents the gain matrix which satisfies \(L_{t}= \begin{bmatrix}L_{t1}&L_{t2}\end{bmatrix} ^{T}\), where \(L_{t1}=[k_{o1}&k_{o2}&k_{o3}&k_{o4}]\) and \(L_{t2}=k_{o5}\). Let us denote \(\bar{x}_{i}^{d}=\hat{x}_{i}^{d}-(x_{i}^{d})_{E}\), then, according to (19) and (20), \(\hat{x}_{i}^{d}\) satisfies  
\[\hat{x}_{i}^{d}= \begin{bmatrix}A_{t}&B_{t}\\ \mathbf{0}_{4}^{T}&0\end{bmatrix} \bar{x}_{i}^{d}- \begin{bmatrix}\mathbf{0}_{4}\\ 1\end{bmatrix} (X_{i}^{d})^{(5)}-L_{t}[C_{t},0]\bar{x}_{i}^{d} \tag{21}\]  
Denoting  
\[A_{o}= \begin{bmatrix}A_{t}-L_{t1}^{T}C_{t}&B_{t}\\ -L_{t2}^{T}C_{t}&0\end{bmatrix} ,\quad B_{o}= \begin{bmatrix}\mathbf{0}_{4}\\ 1\end{bmatrix} \]  
Suppose that \(J\) is the Jordan standard form of matrix \(A_{o}\) by implementing a Jordan transformation using matrix \(M\). Theorem 1: _For each agent, the extended guidance vector observer is given in (20). If \(X_{t}^{d}\in C^{5}(\mathbb{R}^{+},\mathbb{R}^{2})\), then,_ **(i)** _the matrix \(A_{o}\) is stabilizable by proper selected gain matrix \(L_{t}\);_ **(ii)** _the observing error \(\hat{x}_{i}^{d}\) converge to the compact sphere \(\|\hat{x}_{i}^{d}\|\leq\beta\), where \(\beta>\frac{\lambda\rho}{\lambda(\gamma-1)}\), within finite-time_  
\[T_{converge}\leq\frac{\|\hat{x}_{i}^{d}(0)\|}{(\gamma-1)\beta\lambda/\bar{ \lambda}-\rho} \tag{22}\]  
_where we denote by \(\underline{\lambda}=\text{min}\{|\lambda_{i}(M)|\}\) and \(\bar{\lambda}=\text{max}\{|\lambda_{i}(M)|\}\), scalar \(\gamma>1\). The scalar \(\rho\) represents the bound of \(|(X_{i}^{d})^{(5)}|\)._  
Proof: (i). The matrix \(A_{o}\) can be rewritten in the following form. \[A_{o}= \begin{pmatrix}-L_{t1}^{T}&I_{4}\\ -L_{t2}&\mathbf{0}_{4}^{T}\end{pmatrix} \]  
which is stable, for some gains \(k_{oi}>0,i=\{1,\ldots,5\}\) in \(L_{t}\). (ii). The control gains in \(L_{t}\) are selected such that the nearest pole of \(A_{o}\) to imaginary axis is placed to \(-\gamma\), where \(\gamma>1\).