## 3 Active Disturbance Rejection Control Design
### Controller Design with Prescribed Tracking Performance  
The primary control objective is to design a control law that ensures the output tracking error, i.e., \(e_{1}=x_{1}-x_{1d}\), satisfies the prescribed tracking performances including convergence rate, overshoot, and steady-state error under both smooth and abruptly changing reference trajectories. The desired performance function is defined by  
\[\rho(t)=(\rho_{0}-\rho_{\infty})e^{-\phi t}+\rho_{\infty} \tag{34}\]  
where \(\phi>0\) signifies the convergence rate, \(\rho_{0}>0\) and \(\rho_{\infty}>0\) are determined as  
\[\lim_{t\to 0}\rho(t)=\rho_{0};\lim_{t\to\infty}\rho(t)=\rho_{\infty}; \text{and}\;\rho_{0}>\rho_{\infty} \tag{35}\]  
**Remark 4**. : _The function (34) is strictly positive monotonically decreasing and bounded. This function converges to a small value defined by \(\rho_{\infty}\) which can be regarded to be the steady-state error, whereas \(\rho_{0}\) can be thought as the initial error and \(\phi\) designates the convergence rate of the error._  
The design control algorithm must guarantee that the output tracking error \(e_{1}\) satisfies the inequality, i.e., prescribed tracking performance, presented as follows:  
\[-\underline{v}\rho(t)<e_{1}<\overline{v}\rho(t),\forall t\geq 0 \tag{36}\]  
where \(\underline{v}\) and \(\overline{v}\) are the positive constants selected by designers. To do so, the initial tracking error must satisfy \(e_{1}(0)>-\underline{v}\rho(0)\) and \(e_{1}(0)<\overline{v}\rho(0)\). The figurative illustration of the prescribed tracking performance control is depicted in Figure 4.  
Figure 4: An example of the prescribed time-varying tracking performance with \(\rho_{0}=1\), \(\rho_{\infty}=0.03\), \(\overline{v}=1\), \(\underline{v}=1\), and \(\phi=10\). The above condition can be transformed into  
\[\epsilon_{1}=\rho(t)S(z_{1}) \tag{37}\]  
where \(S(z_{1})\) is a smooth and monotonically increasing function that is mathematically represented as  
\[S(z_{1})=\frac{\overline{v}e^{z_{1}}-\underline{v}e^{-z_{1}}}{e^{z_{1}}+e^{-z_{ 1}}} \tag{38}\]  
From the definition of the function \(S(z_{1})\), some interesting features of it can be listed as follows:  
1. \(-\underline{v}<S(z_{1})<\overline{v}\)
2. \(\lim\limits_{z_{1}\rightarrow+\infty}S(z_{1})=\overline{v}\) and \(\lim\limits_{z_{1}\rightarrow-\infty}S(z_{1})=\underline{v}\)  
From (37) and (38), the conversion error \(z_{1}\) of the actual tracking error \(e_{1}\) can be expressed by  
\[z_{1}=\frac{1}{2}\log\frac{\lambda+\underline{v}}{\overline{v}-\lambda} \tag{39}\]  
where \(\lambda=\epsilon_{1}/\rho(t)\) and \(\log(\bullet)\) is the natural logarithm operator of \(\bullet\). **Remark 5**. : _The prescribed tracking performance (36) is equivalent to the bounded tracking of the conversion error \(z_{1}\). It means that when \(z_{1}\) is bounded, the prescribed performance (36) is always satisfied._  
Taking time derivative of (39) yields  
\[\dot{z}_{1}=\frac{1}{2}\bigg{(}\frac{1}{\lambda+\underline{v}}+\frac{1}{ \overline{v}-\lambda}\bigg{)}\bigg{(}\frac{\dot{e}_{1}}{\rho}-\frac{e_{1} \underline{\rho}}{\rho^{2}}\bigg{)} \tag{40}\]  
Considering the system dynamics (4), one obtains  
\[\dot{z}_{1}=\zeta\bigg{(}x_{2}-\dot{x}_{1d}-\frac{e_{1}\underline{\rho}}{\rho} \bigg{)} \tag{41}\]  
where \(\zeta=\frac{1}{2\rho}\Big{(}\frac{1}{\overline{v}-\lambda}+\frac{1}{\overline {v}+\underline{\rho}}\Big{)}\). To applying CF approach, the tracking error and compensated tracking error signal for the first step are designed as  
\[ \begin{split} e_{1}&=z_{1}\\ v_{1}&=e_{1}-\zeta_{1}\end{split}  \tag{42}\]  
where the dynamic of \(\zeta_{1}\) is given by  
\[\dot{\xi}_{1}=-k_{1}\xi_{1}+\zeta(x_{2c}-\alpha_{1})+\zeta\xi_{2} \tag{43}\]  
with \(\xi_{1}(0)=0\) and \(x_{2c}\) is the filtered signal of \(\alpha_{1}\) determined by  
\[T_{f}\dot{x}_{2c}+x_{2c}=\alpha_{1}\]  
where the filter initial conditions are \(x_{2c}(0)=\alpha_{1}(0)\) and \(\dot{x}_{2c}(0)=0\), and \(T_{f}\) is time constant designed later.