## 4 Formation Controller Design
### Convergence Analysis  
Now, the guidance vector \([X_{i}^{d},Y_{i}^{d},Z_{i}^{d},\psi_{i}^{d}]^{\mathsf{T}}\) for each quadrotor is obtained (we note that \(Z_{i}^{d}\) and \(\psi_{i}^{d}\) are set constant, so omitted here). Their higher-order derivatives are estimated by using finite-time observer (20) in an aggressive formation. Then, these estimated derivatives are used as feedforwards in Equation (17), composing controller (12). In practice, the estimation errors are usually high at \(t=0\) s, when the initial states of the observer may greatly deviate from the real values. In order to avoid the instability caused by the big initial deviation of the observer, we implement saturation functions to the estimated high-order derivatives to prevent great observation errors at the beginning. To avoid the instability of quadrotors caused by the great initial observation error, we propose a modified controller in Equation (33) using the saturation function as follows  
\[v_{i1}=\sigma_{b4}(\widehat{X}_{i}^{d})-k_{4X}(\theta_{i}g- \sigma_{b3}((\widehat{X}_{i}^{d}))) -k_{3X}(\theta_{i}g-\sigma_{b2}(\hat{X}_{i}^{d})) \tag{33}\] \[-k_{2X}(\hat{X}_{i}-\sigma_{b1}(\hat{X}_{i}^{d}))-k_{1X}(X_{i}- X_{i}^{d})\] \[v_{i2}=\sigma_{b4}(\widehat{Y}_{i}^{d})-k_{4Y}(-\phi_{i}g- \sigma_{b3}(\widehat{Y}_{i}^{d})) -k_{3Y}(-\phi_{i}g-\sigma_{b2}(\hat{Y}_{i}^{d}))\] \[-k_{2Y}(\hat{Y}_{i}-\sigma_{b1}(\hat{Y}_{i}^{d}))-k_{1Y}(Y_{i}- Y_{i}^{d})\]  
The analysis of the convergence is given by the following theorem. **Theorem 2**. : _Consider the quadrotors with dynamics (8) are controlled by the formation controller (12) composed by (33). The bounds are chosen as follows,_  
\[b_{j}= \begin{cases}\mathsf{h}_{j},&\text{when}\quad t\leq T_{convergence}\\ \infty&\text{when}\quad t>T_{convergence}\end{cases}  \tag{34}\]  
_where \(j=\{1,2,3,4\}\) and \(\mathsf{h}_{j}\in\mathbb{R}^{+}\). Then, the formation error for each quadrotor is ultimately bounded and converge to the invariant set as follows._  
\[S_{b}=\{e_{X_{i}}|\|e_{X_{i}}\|\leq\frac{2\kappa\beta\lambda_{\text{max}}(P)} {\lambda_{\text{min}}(Q)}\} \tag{35}\]  
_where \(\tilde{\kappa}>0\). Notations \(P\) and \(Q\) represent some positive-definite matrices._  
Proof. : Using (13), (16), (18) and (33), the closed-loop planar dynamics of a quadrotor yields the following decoupled form. \[ \begin{split}\widetilde{X}_{i}=\sigma_{b4}(\widehat{X}_{i}^{d}) -k_{4X}(\widetilde{X}_{i}-\sigma_{b3}(\widehat{X}_{i}^{d}))-k_{3X}(\bar{X}_{i }-\sigma_{b2}(\hat{X}_{i}^{d}))-k_{2X}(\hat{X}_{i}-\sigma_{b1}(\hat{X}_{i}^{d} ))-k_{1X}(X_{i}-X_{i}^{d})\\ \widetilde{Y}_{i}=\sigma_{b4}(\widehat{Y}_{i}^{d})-k_{4Y}( \widetilde{Y}_{i}-\sigma_{b3}(\widehat{Y}_{i}^{d}))-k_{3Y}(\tilde{Y}_{i}- \sigma_{b2}(\hat{Y}_{i}^{d}))-k_{2Y}(\hat{Y}_{i}-\sigma_{b1}(\hat{Y}_{i}^{d} ))-k_{1Y}(Y_{i}-\tilde{Y}_{i}^{d})\end{split} \]  
Take \(X_{i}\) for example and denote by \(e_{X_{i}}=[X_{i}-X_{i}^{d},\,\bar{X}_{i}-\hat{X}_{i}^{d},\,\bar{X}_{i}-\hat{X} _{i}^{d},\,\overline{X}_{i}-\overline{\hat{X}}_{i}^{d}]^{\mathsf{T}}\). Then, we can write the dynamics of \(e_{X_{i}}\) in state-space form as follows  
\[e_{X_{i}}=A_{c}e_{X_{i}}+\delta_{i} \tag{36}\]  
where  
\[A_{c}= \begin{pmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ -k_{1X}&-k_{2X}&-k_{3X}&-k_{4X}\end{pmatrix} \]  
and \(\delta=[0,0,0,\Delta_{i}]^{\mathsf{T}}\). \[\Delta_{i}=\sigma_{b4}(\widehat{X}_{i}^{d})-\widetilde{X}_{i}^{d}+k_{4X}( \sigma_{b3}(\widehat{X}_{i}^{d})-\widetilde{X}_{i}^{d})+k_{3X}(\sigma_{b2}( \hat{X}_{i}^{d})-\bar{X}_{i})+k_{2X}(\sigma_{b1}(\hat{X}_{i}^{d})-\bar{X}_{i}) \tag{37}\]* When \(t\leq T_{convergence}\), it is trivial to prove that \(\|\delta_{i}\|=|\Delta|\leq\hbar_{4}+|\tilde{X}_{i}^{d}|+k_{4X}(\hbar_{3}+| \tilde{X}_{i}^{d}|)+k_{3X}(\hbar_{2}+|\hat{X}_{i}|)+k_{2X}(\hbar_{1}+|\hat{X}_{ i}|)\leq\mathbf{d}<\infty\) is bounded, since \(r(t)\in\mathbb{C}^{5}(\mathbb{R}^{+},\mathbb{R}^{2})\).