## 3 Design of the Control Scheme
### Design of Adaptive Linear Active Disturbance Rejection Control  
The structure of the ALADRC is shown in Figure 3. LADRC mainly consists of three parts: LTD, LESO, and PD controller. On the basis of IPSO parameter adjustment, APC can adjust \(k_{p}\) and \(k_{d}\) in the PD controller in real-time and compensate the observation error of LESO through the tracking error of the HST at the same time. LTD is to arrange the transition process for the system. Its function is to make the input signal and its differential signal output smooth, to ensure the system can realize fast tracking and reduce overshooting. The design of LTD for carriage \(i\) draws on [35]:  
\[\left\{ \begin{array}{l}\dot{\gamma}_{1i}=\gamma_{2}\\ \dot{\gamma}_{2i}=-1.76\alpha\gamma_{2i}-\alpha^{2}(\gamma_{1i}-x_{d})\end{array} \right. \tag{8}\]  
where \(\alpha\) is an adjustable parameter to ensure that an appropriate transition process is arranged, whilst \(\gamma_{1i}\) and \(\gamma_{2i}\) are the signals that arrange the transition process for the ideal displacement signal \(x_{d}\) of carriage \(i\). Due to the observation ability of LESO, LADRC has a natural decoupling function, which makes it possible to design a separate control law for a carriage with traction unit. According to (4), the dynamic model of the carriage \(i\) can be transformed into:  
\[\left\{ \begin{array}{l}\dot{\chi}_{1i}=\chi_{2i}\\ \dot{\chi}_{2i}=G_{i}(t)+f_{i}(t)+d_{i}u_{i}(t)\\ y_{i}=\chi_{1i}\end{array} \right. \tag{9}\]  
Figure 3: ALADRC control structure diagram. where, \(\chi_{1i}\) and \(\chi_{2i}\) are measurable state variables, representing the position and speed of carriage \(i\), respectively. \(G_{i}(t)\) is an unknown nonlinear function of the system and the interaction force between the carriages, \(f_{i}(t)\) is the basic running resistance, the additional running resistance and the disturbance caused by the time-varying HST coefficients. \(d_{i}\) is partially known and the known part is denoted as \(d_{0i}\). The sum of the internal disturbance and external disturbance is defined as the total disturbance \(F_{i}=G_{i}(t)+f_{i}(t)+(d_{i}-d_{0i})u_{i}(t)\). Rewrite (9) as follows:  
\[\left\{ \begin{array}{l}\dot{\chi}_{1i}=\chi_{2i}\\ \dot{\chi}_{2i}=G_{i}(t)+f_{i}(t)+(d_{i}-d_{0i})u_{i}(t)+d_{0i}u_{i}(t)=d_{0i}u _{i}(t)+F_{i}\\ y_{i}=\chi_{1i}\end{array} \right. \tag{10}\]  
Add an extended state variable \(\chi_{3i}=F_{i}\) to represent the disturbance of the system. Thus, \(\chi_{i}=\left[\chi_{1i}\;\chi_{2i}\;\chi_{3i}\right]^{T}\) is the extended state including the disturbance. Equation (10) can be translated into a state extension space description as:  
\[\left\{ \begin{array}{l}\dot{\chi}_{i}=Q\chi_{i}+M_{i}u_{i}+N\dot{F}_{i}\\ y_{i}=U\chi_{i}\end{array} \right. \tag{11}\]  
where \(Q=\left[ \begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&0\end{array} \right]\), \(M_{i}=\left[ \begin{array}{c}0\\ d_{0i}\\ 0\end{array} \right]\), \(N=\left[ \begin{array}{c}0\\ 0\\ 1\end{array} \right]\) and \(U=[1\;0\;0]\). According to (11), the design of LESO for carriage \(i\) is as follows:  
\[\left\{ \begin{array}{l}\dot{z}_{i}=Qz_{i}+M_{i}u_{i}+P(y_{i}-\hat{y}_{i})\\ \hat{y}_{i}=Uz_{i}\end{array} \right. \tag{12}\]  
where \(z_{i}=[z_{1i}\;z_{2i}\;z_{3i}]^{T}\) is the state vector of the observer, which is used to observe the state of system \(\chi_{i}=\left[\chi_{1i}\;\chi_{2i}\;\chi_{3i}\right]^{T}\). \(P=\left[P_{1}\;P_{2}\;P_{3}\right]^{T}\) is the observation gain matrix of LESO. The observation gain is parameterized by placing all eigenvalues at the observation bandwidth and the observer gain is obtained from its characteristic equation:  
\[\lambda(s)=(s+\omega)^{3}=s^{3}+P_{1}s^{2}+P_{2}s+P_{3} \tag{13}\]  
the observer gain can be obtained as \(P=\left[3\omega\;3\omega^{2}\;\omega^{3}\right]^{T}\), where \(\omega\) is the bandwidth of LESO.