## 3 Design of the Sliding-Mode Active Disturbance Rejection Controllers
\[\text{S}^{3}+\alpha_{1}\text{S}^{2}+\alpha_{2}\text{S}+\alpha_{3}=(\text{S}+ \omega_{\text{o}})^{3}=\text{S}^{3}+3\omega_{\text{o}}\text{S}^{2}+3\omega_{ \text{o}}\text{}^{2}\text{S}+\omega_{\text{o}}\text{}^{3}, \tag{14}\]  
Then, we have \(\alpha_{1}=3\omega_{\text{o}},\alpha_{2}=3\omega_{\text{o}}\text{}^{2},\alpha _{3}=\omega_{\text{o}}\text{}^{3}\), and thus the problem of configuring the control parameters of a linear active disturbance rejection observer can be converted into the selection of the observer bandwidth. The design of the sliding-mode active disturbance rejection control law is:  
\[\text{u}=\text{u}_{\text{smc}}+\text{u}_{\text{eso}}, \tag{15}\]  
where \(\text{u}_{\text{smc}}\) is the sliding-mode control equivalent linear term, and \(\text{u}_{\text{eso}}\) is the control term after the introduction of the state observer, which is expressed as \(\text{u}_{\text{eso}}=-\frac{\text{z}_{3}}{\text{b}}\). Using the state observer shown in Equation (11), \(\text{z}_{3}\to\text{f}\), so the system state equation can be converted to:  
\[\left\{ \begin{array}{l}\begin{pmatrix}\dot{\text{z}}_{\text{m}}\\ \dot{\text{z}}_{\text{m}}\\ \dot{\text{z}}_{f}\end{pmatrix}=\begin{pmatrix}0&1&0\\ 0&0&0\end{pmatrix}\begin{pmatrix}\text{z}_{\text{m}}\\ \dot{\text{z}}_{\text{m}}\\ \text{z}_{f}\end{pmatrix}+\begin{pmatrix}0\\ \text{k}_{\text{m}}\\ 0\end{pmatrix}\text{u}_{\text{SMC}}+\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\dot{f}\\ \text{z}_{\text{m}}=\begin{pmatrix}1&0&0\end{pmatrix}\begin{pmatrix}\text{z}_{ \text{m}}\\ \dot{\text{z}}_{\text{m}}\\ \text{z}_{f}\end{pmatrix}\end{array} \right., \tag{16}\]Now, design the sliding-mode equivalent linear control law \(\mathrm{u}_{\mathrm{smc}}\), using an integral sliding-mode surface:  
\[\mathrm{s=}\gamma_{1}\mathrm{e+}\gamma_{2}\mathrm{\int edt+}\gamma_{3}\dot{ \mathrm{e}}, \tag{17}\]  
where \(\mathrm{e=}z_{m}-z_{\mathrm{d}}\), \(z_{\mathrm{d}}\) is the ideal trajectory. Derive for the sliding-mode surface:  
\[ \begin{array}{l}\dot{\mathrm{s}}\ =\gamma_{1}\dot{\mathrm{e}}+\gamma_{2} \mathrm{e+}\gamma_{3}\dot{\mathrm{e}}\\ \ =\gamma_{1}\big{(}\dot{z}_{m}-\dot{z}_{\mathrm{d}}\big{)}+\gamma_{2}(z_{m}-z_{ \mathrm{d}})+\gamma_{3}\big{(}\ddot{z}_{m}-\ddot{z}_{\mathrm{d}}\big{)}\\ \ =\gamma_{1}\big{(}\dot{z}_{m}-\dot{z}_{\mathrm{d}}\big{)}+\gamma_{2}(z_{m}-z_{ \mathrm{d}})+\gamma_{3}\big{(}\mathrm{b}\mathrm{u}_{\mathrm{SMC}}-\ddot{z}_{ \mathrm{d}}\big{)}\end{array}  \tag{18}\]  
Then, the sliding-mode equivalent linear control law can be obtained as follows:  
\[\mathrm{u}_{\mathrm{SMC}}=\frac{\ddot{z}_{\mathrm{d}}+\frac{\gamma_{2}}{\gamma _{3}}(z_{\mathrm{d}}-z_{m})+\frac{\gamma_{1}}{\gamma_{3}}\big{(}\dot{z}_{ \mathrm{d}}-\dot{z}_{m}\big{)}}{\mathrm{b}}, \tag{19}\]  
Substituting the sliding-mode active disturbance rejection control law:  
\[\mathrm{u=}\mathrm{u}_{\mathrm{smc}}+\mathrm{u}_{\mathrm{eso}}=\frac{\ddot{z}_ {\mathrm{d}}+\frac{\gamma_{2}}{\gamma_{3}}(z_{\mathrm{d}}-z_{m})+\frac{\gamma _{1}}{\gamma_{3}}\big{(}\dot{z}_{\mathrm{d}}-\dot{z}_{m}\big{)}-z_{f}}{ \mathrm{b}}=\frac{\ddot{z}_{\mathrm{d}}+\frac{\gamma_{2}}{\gamma_{3}}(z_{ \mathrm{d}}-z_{1})+\frac{\gamma_{1}}{\gamma_{3}}\big{(}\dot{z}_{\mathrm{d}}- z_{2}\big{)}-z_{3}}{\mathrm{b}} \tag{20}\]  
From above, the structure of the sliding-mode linear active disturbance rejection control is shown in Figure 2.