# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer
## 3 Control Design
### Adaptive Extended State Observer Design
Then, there exists a positive definite matrix \(P_{\delta}\) satisfying the following Lyapunov equation:  
\[A_{\delta}^{T}P_{\delta}+P_{\delta}A_{\delta}=-Q_{\delta}, \tag{12}\]  
where \(Q_{\delta}\) is a positive definite symmetric matrix. **Remark 2**. : _In order to reduce the number of parameters to be adjusted in the observer, the characteristic polynomial is selected as \(\left(s+1\right)^{3}=s^{3}+3s^{2}+3s+1\), so the parameters of the observer are \( \begin{bmatrix}\beta_{1}&\beta_{2}&\beta_{3}\end{bmatrix} = \begin{bmatrix}3&3&1 \end{bmatrix} \). Then, \(\omega_{0}\) can be used to adjust the bandwidth of the adaptive observer (8). The commonly used bandwidth parameterization method is adopted [26]._  
The following Lyapunov candidate functions are considered:  
\[V_{\delta}=\omega_{0}\delta^{T}P_{\delta}\delta+\frac{1}{\kappa_{1}}\widetilde {m}^{2}+\frac{1}{\kappa_{2}}\widetilde{n}^{2}+\frac{1}{\kappa_{3}}\widetilde{ b}^{2}. \tag{13}\]  
The time derivative of Equation (13) is as follows:  
\[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\] \[=\omega_{0}\bigg{(}\omega_{0}\delta^{T}A_{\delta}^{T}+B^{T}\frac{ 1}{\omega_{0}}\Big{(}m\widetilde{\kappa}_{2}+\widetilde{m}\hat{x}_{2}+ \widetilde{n}\alpha+\widetilde{b}v\Big{)}+\frac{h}{\omega_{0}^{2}}E^{T}\bigg{)} P_{\delta}\delta\] \[+\omega_{0}\delta^{T}P_{\delta}\bigg{(}\omega_{0}A_{\delta}\delta +B\frac{1}{\omega_{0}}\Big{(}m\widetilde{\kappa}_{2}+\widetilde{m}\hat{x}_{2}+ \widetilde{n}\alpha+\widetilde{b}v\Big{)}+E\frac{h}{\omega_{0}^{2}}\bigg{)}+ \frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{2}} \widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}}\widetilde{b}\dot{ \widetilde{b}}\] \[=\omega_{0}^{2}\delta^{T}A_{\delta}^{T}P_{\delta}\delta+B^{T}P_{ \delta}\delta\Big{(}m\widetilde{\kappa}_{2}+\widetilde{m}\hat{x}_{2}+ \widetilde{n}\alpha+\widetilde{b}v\Big{)}+\frac{h}{\omega_{0}^{2}}E^{T}P_{ \delta}\delta\] \[+\omega_{0}^{2}\delta^{T}P_{\delta}A_{\delta}\delta+\delta^{T}P_{ \delta}B\Big{(}m\widetilde{\kappa}_{2}+\widetilde{m}\hat{x}_{2}+\widetilde{n} \alpha+\widetilde{b}v\Big{)}+\delta^{T}P_{\delta}E\frac{h}{\omega_{0}^{2}}+ \frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{2}} \widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}}\widetilde{b}\dot{ \widetilde{b}}\] \[=\omega_{0}^{2}\delta^{T}\big{(}A_{\delta}^{T}P_{\delta}+P_{ \delta}A_{\delta}\big{)}\delta+2B^{T}P_{\delta}\delta\Big{(}m\widetilde{\kappa} _{2}+\widetilde{m}\hat{x}_{2}+\widetilde{m}\hat{x}_{2}+\widetilde{n}\alpha+ \widetilde{b}v\Big{)}\] \[+2\frac{h}{\omega_{0}^{2}}E^{T}P_{\delta}\delta+\frac{2}{\kappa_ {1}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{2}}\widetilde{n}\dot{ \widetilde{m}}+\frac{2}{\kappa_{3}}\widetilde{b}\dot{\widetilde{b}}.\]  
Because \(m,n,b\) is a constant, \(\dot{m},\dot{n},\dot{b}\) is equal to zero at any time; then, \(\dot{\widetilde{m}}=-\dot{\hat{m}},\dot{\widetilde{m}}=-\dot{\hat{n}},\dot{ \widetilde{b}}=-\dot{\hat{b}}\).