# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer
## 3 Control Design
### Adaptive Extended State Observer Design
Using [22, 23] theorem \(B^{T}P_{\delta}=C\), (14) can be written as follows:\[\dot{V}_{\delta} =-\omega_{0}^{2}\delta^{T}Q_{\delta}\delta+2C\delta\Big{(}m\widetilde {x}_{2}+\widetilde{m}\hat{x}_{2}+\widetilde{m}\alpha+\widetilde{b}v\Big{)}+2 \frac{h}{\omega_{0}^{2}}E^{T}P_{\delta}\delta+\frac{2}{\kappa_{1}}\,\widetilde{ m}\hat{\widetilde{m}}+\frac{2}{\kappa_{2}}\,\widetilde{m}\hat{\widetilde{m}}+ \frac{2}{\kappa_{3}}\,\widetilde{b}\hat{\widetilde{b}} \tag{15}\] \[=-\omega_{0}^{2}\delta^{T}Q_{\delta}\delta+2C\delta\Big{(}m \widetilde{x}_{2}+\widetilde{m}\hat{x}_{2}+\widetilde{m}\alpha+\widetilde{b}v \Big{)}+2\frac{h}{\omega_{0}^{2}}E^{T}P_{\delta}\delta-\frac{2}{\kappa_{1}} \,\widetilde{m}\hat{m}-\frac{2}{\kappa_{2}}\,\widetilde{m}\hat{\widetilde{m}}- \frac{2}{\kappa_{3}}\,\widetilde{b}\hat{b}\] \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]0 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]1 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]2 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]3 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]4 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]5 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]6 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]7 \[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]8  
The appropriate parameters are selected such that \(\omega_{0}^{2}\lambda_{\min}(Q_{\delta})+\frac{1}{\omega_{0}^{2}}\big{\|}E^{T}P _{\delta}\big{\|}-\omega_{0}|m|\geq 0\), according to Lemma 3.3 in [27], and (15) can be expressed as follows:  
\[\dot{V}_{\delta} =\omega_{0}\dot{\delta}^{T}P_{\delta}\delta+\omega_{0}\delta^{T} P_{\delta}\dot{\delta}+\frac{2}{\kappa_{1}}\widetilde{m}\dot{\widetilde{m}}+ \frac{2}{\kappa_{2}}\widetilde{m}\dot{\widetilde{m}}+\frac{2}{\kappa_{3}} \widetilde{b}\dot{\widetilde{b}} \tag{14}\]9  
We define that \(\chi=\min\left\{\frac{\omega_{0}\lambda_{\min}(Q_{\delta})+\frac{1}{\omega_{0}^{ 2}}\big{\|}E^{T}P_{\delta}\big{\|}-|m|}{\lambda_{\max}(P_{\delta})},\sigma_{1},\sigma_{2},\sigma_{3}\right\}\), \(\vartheta=\frac{\sigma_{1}}{\kappa_{1}}m^{2}+\frac{\sigma_{2}}{\kappa_{2}}n^{2}+ \frac{\sigma_{3}}{\kappa_{3}}p^{2}+\frac{1}{\omega_{0}^{2}}\big{\|}E^{T}P_{ \delta}\big{\|}h_{up}^{2}\).