## 3 Active Disturbance Rejection Control Design
### Controller Design with Prescribed Tracking Performance
A Lyapunov function is chosen as \(V_{1}=1/2v_{1}^{2}\), taking time derivative of it then combining with (42) and, (43) we have  
\[ \begin{split}\dot{V}_{1}&=v_{1}\dot{v}_{1}\\ &=\zeta z_{1}\bigg{(}\zeta x_{2}-\zeta\dot{x}_{1d}-\zeta\frac{ \epsilon_{1}\underline{\rho}}{\rho}+k_{1}\xi_{1}-\zeta x_{2c}+\zeta\alpha_{1}- \zeta\xi_{2}\bigg{)}\end{split}  \tag{44}\]Define \(e_{2}=x_{2}-x_{2c}\) and \(v_{2}=e_{2}-\xi_{2}\) as an auxiliary tracking error and compensated tracking error for the second step, respectively. The virtual control law \(\alpha_{1}\) is chosen as  
\[\alpha_{1}=\dot{x}_{1d}+\frac{e_{1}\dot{\rho}}{\rho}-\frac{1}{\zeta}k_{1}e_{1} \tag{45}\]  
Substituting (45) into (44) leads to  
\[\dot{V}_{1}=-k_{1}v_{1}^{2}+\zeta v_{1}v_{2} \tag{46}\]  
The dynamic of \(\xi_{2}\) is designed as  
\[\dot{\xi}_{2}=-k_{2}\xi_{2}+(x_{3c}-\alpha_{2})+\xi_{3} \tag{47}\]  
with \(\xi_{2}(0)=0\) and \(x_{3c}\) is the filtered signal of \(\alpha_{2}\) determined by  
\[T_{f}\dot{x}_{3c}+x_{3c}=\alpha_{2}\]  
where the filter initial conditions are \(x_{3c}(0)=\alpha_{2}(0)\) and \(\dot{x}_{3c}(0)=0\). A candidate Lyapunov function is chosen as  
\[V_{2}=V_{1}+\frac{1}{2}v_{2}^{2} \tag{48}\]  
Taking derivative of \(V_{2}\) then combining with (4), (46) and (47), one obtains  
\[\dot{V}_{2} =\dot{V}_{1}+v_{2}\dot{v}_{2} \tag{49}\] \[=-k_{1}v_{1}^{2}+\zeta v_{1}v_{2}+v_{2}(f_{2}+(x_{3}-x_{3c}-\xi_{ 3})+d_{1}-\dot{x}_{2c}+k_{2}\xi_{2}+\alpha_{2})\]  
Hence, the virtual control \(\alpha_{2}\) is designed as follows:  
\[\alpha_{2}=-f_{2}-d_{1}+\dot{x}_{2c}-\zeta v_{1}-k_{2}e_{2} \tag{50}\]  
Since the terms \(x_{2}\) and \(d_{1}\) are immeasurable, their estimated value will be used, the virtual control \(\alpha_{2}\) is reconstructed as  
\[\alpha_{2}=-\dot{f}_{2}-\dot{x}_{e1}+\dot{x}_{2c}-\zeta v_{1}-k_{2}(\dot{x}_{2 }-x_{2c}) \tag{51}\]  
The derivative of \(V_{2}\) is rewritten as  
\[\dot{V}_{2} =-k_{1}v_{1}^{2}+\zeta v_{1}v_{2}+v_{2}(f_{2}+v_{3}+d_{1}-\dot{x} _{2c}+k_{2}\xi_{2}+\alpha_{2}) \tag{52}\] \[\dot{V}_{1}=-k_{1}v_{1}^{2}+\zeta v_{1}v_{2} \tag{46}\]0  
The tracking error of \(x_{3}\) is given by  
\[\dot{V}_{1}=-k_{1}v_{1}^{2}+\zeta v_{1}v_{2} \tag{46}\]1  
The compensated tracking error is determined as  
\[\dot{V}_{1}=-k_{1}v_{1}^{2}+\zeta v_{1}v_{2} \tag{46}\]2  
where the dynamic of \(\xi_{3}\) is determined as  
\[\dot{V}_{1}=-k_{1}v_{1}^{2}+\zeta v_{1}v_{2} \tag{46}\]3  
Define the Lyapunov function as  
\[\dot{V}_{1}=-k_{1}v_{1}^{2}+\zeta v_{1}v_{2} \tag{46}\]4Taking derivative of \(V_{3}\), we have  
\[ \begin{split}\dot{V}_{3}&=\dot{V}_{2}+v_{3}\dot{v}_{3}\\ &=-k_{1}v_{1}^{2}-k_{2}v_{2}^{2}+v_{2}v_{3}+v_{3}\big{(}f_{3}+g_{3 }u+d_{2}-\dot{x}_{3c}-\dot{\xi}_{3}\big{)}\end{split}  \tag{57}\]  
The actual control voltage applied to the system is designed as  
\[u=\frac{1}{g_{3}}\Big{(}-\hat{f}_{3}-\hat{d}_{2}+\dot{x}_{3c}-(\dot{x}_{2}-x_{2 c}-\xi_{2})-k_{3}e_{3}\Big{)} \tag{58}\]  
Substituting (58) and (55) into (57) yields  
\[\dot{V}_{3}=-k_{1}v_{1}^{2}-k_{2}v_{2}^{2}-k_{3}v_{3}^{2}+v_{2}\hat{f}_{2}+v_{2 }\hat{x}_{e1}+k_{2}v_{2}\hat{x}_{2}+v_{3}\hat{f}_{3}+v_{3}\hat{x}_{e2}-v_{3} \hat{x}_{2} \tag{59}\]  
Theorem 4.2: _By employing control laws (45), (51), and (58) with regard to the CF approach with error compensation mechanisms (43), (47), (55), and ESMO (12), ESO (24), the closed-loop system output tracking capability satisfies the prescribed tracking performance defined in (36)._  
Proof of Theorem 4.2: See Appendix A.