## 3 Design of the Sliding-Mode Active Disturbance Rejection Controllers  
Consider the second-order differential model of the system, let \(\mathrm{z}_{f}=\mathrm{f}=-\frac{\mathrm{c}}{\mathrm{m_{m}}}\dot{\mathrm{z}}_ {\mathrm{m}}+\mathrm{\omega}\) be the generalized perturbation, which includes internal and external system perturbations. Then, the system equation of the state can be expressed as:  
\[\left\{ \begin{array}{c}\left(\begin{array}{c}\dot{\mathrm{z}}_{\mathrm{m} }\\ \ddot{\mathrm{z}}_{\mathrm{m}}\\ \dot{\mathrm{z}}_{f}\end{array}\right)=\left(\begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&0\end{array}\right)\left(\begin{array}{c}\dot{\mathrm{z}}_{\mathrm{m}}\\ \dot{\mathrm{z}}_{\mathrm{m}}\\ \mathrm{z}_{f}\end{array}\right)+\left(\begin{array}{c}0\\ \frac{\mathrm{k_{m}}\mathrm{\alpha}}{\mathrm{m_{m}}}\\ 0\end{array}\right)\mathrm{V_{r}}+\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\dot{f}\\ \mathrm{z_{m}}=\left(1&0&0\right)\left(\begin{array}{c}\dot{\mathrm{z}}_{ \mathrm{m}}\\ \dot{\mathrm{z}}_{f}\end{array}\right)\end{array} ,\right. \tag{6}\]  
Designing the observer equation:  
\[\left\{ \begin{array}{c}\dot{\mathrm{z}}=\mathrm{a}\mathrm{z}+\mathrm{bu}+ \mathrm{A}(\mathrm{z_{m}}-\ \dot{\mathrm{z}}_{\mathrm{m}})\\ \dot{\mathrm{z}}_{\mathrm{m}}{=}\mathrm{c}\mathrm{z}\end{array} \right., \tag{7}\]  
where z is the observed value of \(\dot{\mathrm{z}}_{\mathrm{m}}\), \(\ddot{\mathrm{z}}_{\mathrm{m}}\), \(\dot{\mathrm{z}}_{f}\), and \(\dot{\mathrm{z}}_{\mathrm{m}}\) is the observed value of \(\mathrm{z_{m}}\). Simplifying Equation (7) as:  
\[\dot{\mathrm{z}}= (\mathrm{a}-\mathrm{Ac})\mathrm{z}+\mathrm{bu}+\mathrm{A}\mathrm{z }_{\mathrm{m}}, \tag{8}\]\[\text{where a}= \begin{pmatrix}0&1&0\\ 0&0&1\\ 0&0&0\end{pmatrix} ,\text{A}= \begin{pmatrix}\alpha_{1}\\ \alpha_{2}\\ \alpha_{3}\end{pmatrix} ,\text{c}= \begin{pmatrix}1&0&0\end{pmatrix} .\]  
Equation (9) can be obtained as:  
\[\text{a}-\text{Ac}= \begin{pmatrix}-\alpha_{1}&1&0\\ -\alpha_{2}&0&1\\ -\alpha_{3}&0&0\end{pmatrix} , \tag{9}\]  
Substituting Equation (9) into Equation (8), we have:  
\[\dot{\text{z}}= \begin{pmatrix}-\alpha_{1}&1&0\\ -\alpha_{2}&0&1\\ -\alpha_{3}&0&0\end{pmatrix} \text{z}+ \begin{pmatrix}\text{b}&\text{A}\end{pmatrix} \begin{pmatrix}\text{u}\\ \text{z}_{\text{m}}\end{pmatrix} = \begin{pmatrix}-\alpha_{1}&1&0\\ -\alpha_{2}&0&1\\ -\alpha_{3}&0&0\end{pmatrix} \text{z}+ \begin{pmatrix}0&\alpha_{1}\\ \text{b}_{\text{o}}&\alpha_{2}\\ 0&\alpha_{3}\end{pmatrix}\begin{pmatrix}\text{u}\\ \text{z}_{\text{m}}\end{pmatrix} , \tag{10}\]  
Designing the following linear active disturbance rejection observer:  
\[\left\{ \begin{array}{l}\dot{\text{z}}_{1}=\text{z}_{2}+\alpha_{1}(\text{z}_ {\text{m}}-\text{z}_{1})\\ \dot{\text{z}}_{2}=\text{z}_{3}+\alpha_{2}(\text{z}_{\text{m}}-\text{z}_{1}) +\text{b}_{\text{o}}\text{u}\\ \dot{\text{z}}_{3}=\text{z}_{3}(\text{z}_{\text{m}}-\text{z}_{1})+\dot{z}_{f} \end{array} \right., \tag{11}\]  
Then, we select the suitable gains \(\alpha_{1},\alpha_{2},\alpha_{3}\), which enable real-time tracking of each variable in the system, which means let \(\text{z}_{1}\to\text{z}_{\text{m}},\text{z}_{2}\to\text{z}_{m},\text{z}_{3} \to\text{z}_{f}\). To obtain \(\alpha_{1},\alpha_{2},\alpha_{3}\) according to the characteristic equations of a linear active disturbance rejection observer:  
\[\text{SI}-\text{a}= \begin{pmatrix}\text{S}&0&0\\ 0&\text{S}&0\\ 0&0&\text{S}\end{pmatrix} - \begin{pmatrix}-\alpha_{1}&1&0\\ -\alpha_{2}&0&1\\ -\alpha_{3}&0&0\end{pmatrix} = \begin{pmatrix}\text{S}+\alpha_{1}&-1&0\\ \alpha_{2}&\text{S}&-1\\ \alpha_{3}&0&\text{S}\end{pmatrix} , \tag{12}\]  
The characteristic polynomial is:  
\[\lambda(\text{S})=|\text{SI}-\text{a}|=\text{S}^{2}(\text{S}+\alpha_{1})+ \alpha_{3}+\alpha_{2}=\text{S}^{3}+\alpha_{1}\text{S}^{2}+\alpha_{2}\text{S} +\alpha_{3}, \tag{13}\]  
For simple adjustment, the observer is parameterized by choosing an observer bandwidth at \(\omega_{\text{o}}\).