# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer
## 3 Control Design
### Controller Design Based on an Extended State Observer
#### 3.2.1 Extended State Observer Design and Stability Analysis  
According to the estimated values of the model parameters, system (7) can be rewritten as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\dot{m}x_{2}+\dot{n}\alpha+\hat{b}v+f\\ \dot{f}=l.\end{array} \right. \tag{19}\]  
In order to estimate the system uncertainties and external disturbances, an ESO is designed as follows:  
\[\left\{ \begin{array}{l}\dot{\hat{x}}_{1}=\hat{x}_{2}+\eta_{1}(x_{1}-\hat{x}_ {1})\\ \dot{\hat{x}}_{2}=\dot{m}\hat{x}_{2}+\dot{n}\alpha+\hat{b}v+\dot{f}+\eta_{2}(x_ {1}-\hat{x}_{1})\\ \dot{\hat{f}}=\eta_{3}(x_{1}-\hat{x}_{1}),\end{array} \right. \tag{20}\]  
where \(\eta_{1},\eta_{2},\eta_{3}\) are adjustable parameters. From (19) and (20), the dynamic error can be obtained as follows:  
\[\left\{ \begin{array}{l}\dot{\widetilde{x}}_{1}=\widetilde{x}_{2}-\eta_{1} \widetilde{x}_{1}\\ \dot{\widetilde{x}}_{2}=\dot{m}\widetilde{x}_{2}+\widetilde{f}-\eta_{2} \widetilde{x}_{1}\\ \dot{\widetilde{f}}=l-\eta_{3}\widetilde{x}_{1}.\end{array} \right. \tag{21}\]  
We define that \(A_{\epsilon}= \begin{bmatrix}-\eta_{1}&1&0\\ -\eta_{2}&\dot{m}&1\\ -\eta_{3}&0&0\end{bmatrix} ,\epsilon= \begin{bmatrix}\widetilde{x}_{1}\\ \widetilde{x}_{2}\\ \widetilde{x}_{3}\end{bmatrix} ,E= \begin{bmatrix}0\\ 0\\ 1\end{bmatrix} \). The above formula can be rewritten as a state space form such as follows:  
\[\dot{\epsilon}=A_{\epsilon}\epsilon+El. \tag{22}\]  
**Theorem 2**. : _Considering the ESO (20), there exist appropriate observer parameters, which satisfy_  
\[\left\{ \begin{array}{l}\eta_{1}>\dot{m}\\ (\eta_{1}-\dot{m})(\eta_{2}-\eta_{1}\dot{m})-\eta_{3}>0\\ \eta_{3}>0,\end{array} \right. \tag{23}\]  
_such that the errors of observer converge to the neighborhood of zero in finite-time._  
**Proof. ** Selecting a candidate [PERSON] function as follows:  
\[V_{\varepsilon}=\varepsilon^{T}P_{\varepsilon}\varepsilon. \tag{24}\]  
The characteristic equation of \(A_{\varepsilon}\) can be expressed as follows:  
\[ \begin{array}{l}D=|sI_{3}-A_{\varepsilon}|\\ =\left|\begin{array}{cccc}s+\eta_{1}&-1&0\\ \eta_{2}&s-\hat{m}&-1\\ \eta_{3}&0&s\end{array}\right|\\ =s^{3}+(\eta_{1}-\hat{m})s^{2}+(\eta_{2}-\eta_{1}\hat{m})s+\eta_{3},\end{array}  \tag{25}\]  
where \(s\) is the Laplace variable and \(I_{3}\) is the three-dimensional unit matrix. The adjustable parameters \(\eta_{1},\eta_{2},\eta_{3}\) satisfy  
\[\left\{ \begin{array}{l}\eta_{1}>\hat{m}\\ (\eta_{1}-\hat{m})(\eta_{2}-\eta_{1}\hat{m})-\eta_{3}>0\\ \eta_{3}>0.\end{array} \right. \tag{26}\]  
Therefore, \(A_{\varepsilon}\) is a Hurwitz matrix, and all eigenvalues of \(A_{\varepsilon}\) have negative real parts. Since \(A_{\varepsilon}\) is a Hurwitz matrix, there exists a positive definite matrix \(P_{\varepsilon}\) satisfying the following Lyapunov equation:  
\[A_{\varepsilon}^{T}P_{\varepsilon}+P_{\varepsilon}A_{\varepsilon}=-Q_{ \varepsilon}, \tag{27}\]  
where \(Q_{\varepsilon}\) is a positive definite symmetric matrix. The time derivative of \(V_{\varepsilon}\) is as follows:  
\[ \begin{array}{l}\dot{V}_{\varepsilon}&=\dot{\varepsilon}^{T}P_{ \varepsilon}\varepsilon+\varepsilon^{T}P_{\varepsilon}\dot{\varepsilon}\\ &=(A_{\varepsilon}\varepsilon+El)^{T}P_{\varepsilon}\varepsilon+\varepsilon^ {T}P_{\varepsilon}(A_{\varepsilon}\varepsilon+El)\\ &=\varepsilon^{T}\big{(}A_{\varepsilon}^{T}P_{\varepsilon}+PA_{\varepsilon} \big{)}\varepsilon+2 lE^{T}P_{\varepsilon}\varepsilon.\end{array}  \tag{28}\]  
According to (28) and Remark 4, we have  
\[ \begin{array}{l}\dot{V}_{\varepsilon}&=-\varepsilon^{T}Q_{\varepsilon} \varepsilon+2 lE^{T}P_{\varepsilon}\varepsilon\\ &\leq-\lambda_{\min}\{Q_{\varepsilon}\}\|\varepsilon\|^{2}+2l_{up}\|E^{T}P_{ \varepsilon}\|\|\varepsilon\|\\ &=-q\lambda_{\min}\{Q_{\varepsilon}\}\|\varepsilon\|^{2}-\big{(}(1-q)\lambda_ {\min}\{Q_{\varepsilon}\}\|\varepsilon\|-2l_{up}\big{\|}E^{T}P_{\varepsilon} \big{\|}\big{)}\|\varepsilon\|,\end{array}  \tag{29}\]  
where \(0<q<1\).