## 3 Design of the Control Scheme
### Design of Adaptive Linear Active Disturbance Rejection Control
The control law of LADRC for carriage \(i\) is defined as \(\eta_{i}\):  
\[\eta_{i}=u_{pdi}-z_{3i}/b+\zeta_{i} \tag{14}\]  
where \(u_{pdi}=k_{pi}(s_{1i}-z_{1i})+k_{di}(s_{2i}-z_{2i})\) is the control signal output of the PD controller in LADRC for carriage \(i\), and \(k_{pi}\) and \(k_{di}\) are the gain parameters. \(\zeta_{i}\) is the compensation signal of LESO observation error, whilst \(z_{3i}/b\) is the compensation signal of total disturbance. According to the HST dynamic model (4), the dynamic model of a single carriage \(i\) is as follows:  
\[m_{i}\ddot{x}_{i}=u_{i}-\left(c_{oi}+c_{vi}\dot{x}_{i}+c_{ai}\dot{x}_{i}^{2} \right)-f_{di(i+1)}+f_{d(i-1)i}-f_{bi} \tag{15}\]  
Define the displacement and speed tracking error of carriage \(i\) as \(e_{i}=x_{i}-x_{d}\) and \(\dot{e}_{i}=v_{i}-v_{d}\), where \(x_{d}\) and \(v_{d}\) are the given desired displacement and speed signal, respectively. The adaptive value of the LESO observation error is defined as \(\frac{\hat{k}}{\xi_{i}}\), and the adaptive values of parameters \(k_{pi}\) and \(k_{di}\) in the PD controller are defined as \(k_{pi}\) and \(\hat{k}_{di}\), respectively. The adaptive estimation errors are \(\widetilde{\xi}_{i}=\hat{\xi}_{i}-\zeta_{i}\), \(\widetilde{k}_{pi}=\hat{k}_{pi}-k_{pi}\) and \(\widetilde{k}_{di}=\hat{k}_{di}-k_{di}\), respectively. Define a tracking error of carriage \(i\) as follows:  
\[\kappa_{i}(t)=\dot{e}_{i}+\rho e_{i},\;\;\;i=1,2,3,4 \tag{16}\]to make sure that when \(e_{i}\to 0\) and \(\dot{e}_{i}\to 0\), then the \(\zeta_{i}\to 0\), and \(\rho\) is the appropriate constant selected. **Assumption 1. **_The given desired signal \(x_{d}(t)\) and \(v_{d}(t)\) is smooth and bounded, hence \(\|x_{d}(t)\|\leq X_{d}\) and \(\|V_{d}(t)\|\leq V_{d}\), where \(X_{d}\) and \(V_{d}\) are known constants._  
Taking the differential of \(\kappa_{i}\):  
\[\dot{\kappa}_{i}=\ddot{e}_{i}+\dot{\rho}\dot{e}_{i}=\big{(}\ddot{x}_{i}-\ddot{ x}_{d}\big{)}+\rho\big{(}\dot{x}_{i}-\dot{x}_{d}\big{)} \tag{17}\]  
by introducing (15) into the above equation, it can be obtained:  
\[m_{i}\big{(}\dot{\kappa}_{i}+\ddot{x}_{d}-\rho\big{(}\dot{x}_{i}-\dot{x}_{d} \big{)}\big{)}=u_{i}-\Big{(}c_{oi}-c_{vi}\dot{x}_{i}-c_{ai}\dot{x}_{i}^{2}\Big{)} -f_{di(i+1)}+f_{d(i-1)i}-f_{bi} \tag{18}\]  
then  
\[m_{i}\dot{\kappa}_{i}=u_{i}-\Big{(}c_{oi}-c_{oi}\dot{x}_{i}-c_{ai}\dot{x}_{i} ^{2}\Big{)}-m_{i}\big{(}\ddot{x}_{d}-\dot{\rho}\dot{e}_{i}\big{)}-f_{di(i+1)} +f_{d(i-1)i}-f_{bi} \tag{19}\]  
The design of carriage \(i\) control law \(\psi_{i}\) is as follows:  
\[\psi_{i}=\eta_{i}-l\kappa_{i}+m_{i}\big{(}\ddot{x}_{d}-\dot{\rho}\dot{e}_{i} \big{)}+\dot{\xi}_{i} \tag{20}\]  
where \(l\) is an arbitrary positive parameter and the coefficient \(\delta\) in Figure 3 is \(m_{i}(\ddot{x}_{d}-\dot{\rho}\dot{e}_{i})\) in the above equation. Bring the control law of LADRC (14) into above equation:  
\[\psi_{i}=u_{pdi}-z_{3i}/b-l\kappa_{i}+m_{i}\big{(}\ddot{x}_{d}-\dot{\rho}\dot{ e}_{i}\big{)}+\dot{\xi}_{i} \tag{21}\]  
Bring the control law (21) into (19):  
\[m_{i}\dot{\kappa}_{i}=u_{pdi}-l\kappa_{i}+\Big{(}c_{oi}+c_{vi}\dot{x}_{i}+c_{ ai}\dot{x}_{i}^{2}\Big{)}-z_{3i}/b-f_{di(i+1)}+f_{d(i-1)i}-f_{bi}+\dot{\xi}_{i} \tag{22}\]  
due to the observation disturbance ability of LESO, the above equation can be further simplified as:  
\[\eta_{i}=u_{pdi}-z_{3i}/b+\zeta_{i} \tag{14}\]0