# Research on Tracking Control of Urban Rail Trains Based on Improved Disturbance Observer
## 2 Problem Statement
### The In-Train Force and Running Resistance of the URTs  
The two adjacent carriages of URTs are connected by a complex coupler, which plays a vital role in preventing collisions between carriages and absorbing longitudinal impact forces [22]. Besides, the couplers are subjected to excessive compression and tension, which leads to irreversible deformation and seriously threatens the safe operation of the URTs. Therefore, the consideration of the train coupling force holds tremendous significance. The simplified coupler of the connection between adjacent carriages is shown in Figure 1.  
The internal structure and working principle of the couplers device are very complex. When analyzing the in-train force of the URTs, the connection device between the carriages can be simplified to a \"spring\" system, and the coupling forces between the \(i\)th carriage and the \((i+1)\)th carriage can be expressed as  
\[f_{in_{(i+1)}}(t)=\kappa\Delta\dot{\sigma}_{i(i+1)}(t) \tag{1}\]  
where \(\kappa\) denotes the elastic coefficient of the coupler. According to the direction of the URTs operation shown in Figure 1, \(\Delta\dot{\sigma}_{i(i+1)}(t)\) can be represented as  
\[\Delta\dot{\sigma}_{i(i+1)}(t)=x_{i}(t)-x_{i+1}(t)-S \tag{2}\]  
where \(S=2l+\dot{\theta}_{i}+\dot{\theta}_{i+1}\) is a constant. Based on [PERSON]'s law of mechanics, the force analysis of the \(i\)th carriage is carried out. \[m_{i}\dot{v}_{i}(t)=u_{i}(t)-f_{i}(t)-f_{in_{(i+1)}}(t)+f_{in_{(i-1)i}}(t) \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\dot{d}_{L}(t)\) & The estimate of lumped disturbance \\ \(e_{d}(t)\) & The disturbance estimation error \\ \(\rho\) & The actuator health factor \\ \(\sigma_{\text{max}}(\Psi)\),\(\sigma_{\text{min}}(\Psi)\) & The maximum and minimum singular values of the \(\Psi\) \\ \(\|\Psi\|\) & The Euclidean norm of the matrix \(\Psi\) \\ \(Sym(\Psi)\) & The matrix and its transpose sum \(\Psi+\Psi^{T}\) \\ \(I_{n}\) & \(n\times n\) dimensional unit matrix \\ \(\ast\) & The symmetric part of the matrix \\ \hline \hline \end{tabular} where \(x_{i}(t)\), and \(\dot{v}_{i}(t)\) denote the position and speed of the \(i\)th carriage. \(f_{in_{(i+1)}}(t)\) and \(f_{in_{(i-1)i}}(t)\) represent that the carriage is subject to the coupling force of the \((i+1)\)th carriage and \((i-1)\)th carriage. \(\Delta\dot{\sigma}_{i(i+1)}(t)\) indicates the length of the coupler being stretched or compressed. \(\dot{\theta}_{i}+\dot{\theta}_{i+1}\) denotes the original length of the coupler when no deformation has occurred. \(2l\) is the length of one carriage.  
\end{table} 
Table 1: _Cont._  
Figure 1: The in-train force analysis structure diagram of the URTs. where \(m_{i}\), \(\hat{v}_{i}(t)\), and \(u_{i}(t)\) denote the mass, control input, and acceleration of the \(i\)th carriage, respectively. \(f_{i}(t)\) denotes the running resistance of the \(i\)th carriage, including basic resistance, additional resistance, and unknown external disturbance, which can be described as  
\[f_{i}(t)=a_{i}+b_{i}\hat{v}_{i}(t)+c_{i}\hat{v}_{i}^{2}(t)+f_{ai}(t)+O_{i}(\cdot) \tag{4}\]  
where \(a_{i}\), \(b_{i}\), and \(c_{i}\) denote the Davis coefficients of the \(i\)th carriage. The drag coefficients vary from different train types and are generally derived from experimental tests in wind tunnels. \(a_{i}+b_{i}\hat{v}_{i}(t)\) represents the rolling mechanical resistance between the wheels and rails, and \(c_{i}\hat{v}_{i}^{2}(t)\) represents the aerodynamic resistance. \(O_{i}(\cdot)\) is the unknown external disturbance to which the URTs subject and is highly influenced by operation environmental factors. \(f_{ai}(t)\) denotes additional resistance, usually determined by the running track line, which can be described as  
\[f_{ai}(t)=d_{c}+d_{r}+d_{t} \tag{5}\]  
where \(d_{c}=10.5\alpha m_{i}g/(10^{3}l_{r})\), \(d_{r}=m_{i}g\sin(\theta)\), and \(d_{t}=1.3\times 10^{-4}l_{s}m_{i}g/10^{3}\) represent the curve resistance, ramp resistance, and tunnel resistance, respectively [23].