# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer
## 3 Control Design  
### Adaptive Extended State Observer Design  
In this section, an adaptive ESO is proposed to estimate the unknown parameters of the system. When the system state is represented as \(x_{1}=\theta,x_{2}=\dot{\theta}\), (5) can be rewritten as the following state space form:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=mx_{2}+n\alpha+bv+d.\end{array} \right. \tag{6}\]  
In order to accomplish the above design tasks, the unmodeled dynamics and the external disturbance \(d\) of the system are extended to an additional state variable, which is defined as \(x_{3}=d\). Using \(h\) as the time derivative of \(x_{3}\), system (6) can be rewritten as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=mx_{2}+n\alpha+bv+x_{3}\\ \dot{x}_{3}=h.\end{array} \right. \tag{7}\]  
In order to estimate the unknown parameters, an adaptive ESO is designed as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=\hat{x}_{2}+\beta_{1}\omega_{0}(x_{1}- \hat{x}_{1})\\ \hat{x}_{2}=\hat{m}\hat{x}_{2}+\hat{n}\alpha+\hat{b}v+\hat{x}_{3}+\beta_{2} \omega_{0}^{2}(x_{1}-\hat{x}_{1})\\ \dot{x}_{3}=\beta_{3}\omega_{0}^{3}(x_{1}-\hat{x}_{1}),\end{array} \right. \tag{8}\]  
where \(\hat{x}_{1},\hat{x}_{2},\hat{x}_{3}\) is the estimated value of \(x_{1},x_{2},x_{3}\) and \(\hat{m},\hat{n},\hat{b}\) is the estimated value of \(m,n,b\). \(\omega_{0}>0\) is the design parameter, \(\beta_{1},\beta_{2},\beta_{3}\) is a positive design parameter, and \(s^{3}+\beta_{1}s^{2}+\beta_{2}s+\beta_{3}\) satisfies the Hurwitz condition. \(\hat{m},\hat{n},\hat{b}\) has the following adaptive law:  
\[\hat{m}=\kappa_{1}\widetilde{x}_{1}\hat{x}_{2}-\sigma_{1}\hat{m},\hat{n}= \kappa_{2}\widetilde{x}_{1}\alpha-\sigma_{2}\hat{n},\hat{b}=\kappa_{3} \widetilde{x}_{1}v-\sigma_{3}\hat{b}, \tag{9}\]  
where \(\widetilde{x}_{1}=x_{1}-\hat{x}_{1}\) is the state estimation error of the observer. \(\kappa_{i}>0,\sigma_{i}>0,i=1,2,3\) is an adjustable parameter. The state estimation error can be obtained by subtracting (8) from (7):  
\[\left\{ \begin{array}{l}\dot{\widetilde{x}}_{1}=\widetilde{x}_{2}-\beta_{1} \omega_{0}\widetilde{x}_{1}\\ \dot{\widetilde{x}}_{2}=m\widetilde{x}_{2}+\widetilde{m}\hat{x}_{2}+\widetilde {m}\alpha+\widetilde{b}v+\widetilde{x}_{3}-\beta_{2}\omega_{0}^{2}\widetilde{x }_{1}\\ \dot{\widetilde{x}}_{3}=h-\beta_{3}\omega_{0}^{3}\widetilde{x}_{1},\end{array} \right. \tag{10}\]  
where \(\widetilde{x}_{2}=x_{2}-\hat{x}_{2},\widetilde{x}_{3}=x_{3}-\hat{x}_{3}, \widetilde{m}=m-\hat{m},\widetilde{n}=n-\hat{n},\widetilde{b}=b-\hat{b}\) is the estimation error of the corresponding parameters. **Theorem 1**. : _Consider system (7) under Assumptions 1 and 2. The observers given by (9) and (10) can ensure that the estimation error can approach any small value by selecting the appropriate design constant \(\kappa_{i}>0,\sigma_{i}>0,i=1,2,3\)._  
Proof. : A new estimation error vector is defined as \(\delta= \begin{bmatrix}\delta_{1}&\delta_{2}&\delta_{3}\end{bmatrix} ^{T}= \begin{bmatrix}\widetilde{\kappa}_{1}&\frac{\widetilde{\kappa}_{2}}{\omega_{0 }}&\frac{\widetilde{\kappa}_{3}}{\omega_{0}^{2}}\end{bmatrix} ^{T}\), \(A_{\delta}= \begin{bmatrix}-\beta_{1}&1&0\\ -\beta_{2}&0&1\\ -\beta_{3}&0&0\end{bmatrix} ,B= \begin{bmatrix}0\\ 1\\ 0\end{bmatrix} ,E= \begin{bmatrix}0\\ 0\\ 1\end{bmatrix} ,C= \begin{bmatrix}1\\ 0\\ 0\end{bmatrix} ^{T}\); then, the estimation error dynamics (11) is transformed into the following:  
\[\left\{ \begin{array}{l}\dot{\delta}=\omega_{0}A_{\delta}\delta+B\frac{1}{ \omega_{0}}\left(m\widetilde{\kappa}_{2}+\widetilde{m}\hat{x}_{2}+\widetilde {n}\alpha+\widetilde{b}\widetilde{v}\right)+E\frac{h}{\omega_{0}^{2}}\\ \widetilde{y}_{\delta}=C\delta.\end{array} \right. \tag{11}\]  
Parameters \(\beta_{1},\beta_{2}\) and \(\beta_{3}\) are selected such that \(A_{\delta}\) is Hurwitz.