# Research on Tracking Control of Urban Rail Trains Based on Improved Disturbance Observer
## 2 Problem Statement
### The Multiple Point-Mass Model of the URTs  
The URTs is a decentralized power type consisting of motor carriages and trailers. The motor carriage provides power and braking force, and the trailer only provides the braking force. Typically, the URTs is grouped as shown in Figure 2, where the hollow wheels represent trailers and the solid wheels represent motor carriages. The braking and traction system of the URTs is composed of several relatively independent carriages. The modeling process treats the adjacent carriages as one power unit. The advantage of this approach is that it not only reduces the computational complexity of the control process but also takes into account the in-train forces between adjacent carriages [24]. In this way, the URTs containing \(2n\) carriages are uniformly divided into \(n\) power units, each providing control forces. Considering the URTs with \(n\) power units, combined with the coupling forces between adjacent carriages and running resistance, the multiple point-mass dynamics equation of the URTs can be written as  
\[\left\{ \begin{array}{rl}\hat{x}_{i}(t)&=v_{i}(t),i=1,2,\cdot\cdot\cdot,n\\ m_{1}\hat{v}_{1}(t)&=u_{1}(t)-f_{in_{12}}(t)-f_{1}(t)\\ &=u_{1}(t)-\kappa(x_{1}(t)-x_{2}(t)-S)\\ &-m_{1}a_{1}-m_{1}b_{1}\hat{v}_{1}(t)-\sum\limits_{i=1}^{n}m_{i})c_{1}\hat{v}_ {1}(t)^{2}+d_{11}(t)\\ m_{i}\hat{v}_{i}(t)&=u_{i}(t)-f_{in_{(i+1)}}(t)+f_{in_{(i-1)i}}(t)-f_{i}(t)\\ &=u_{i}(t)-\kappa(x_{i}(t)-x_{i+1}(t)-S)+\kappa(x_{i-1}(t)-x_{i}(t)-S)\\ &-m_{i}a_{i}-m_{i}b_{i}\hat{v}_{i}(t)+d_{Li}(t)\\ m_{n}\hat{v}_{n}(t)&=u_{n}(t)+f_{in_{(n-1)n}}(t)-f_{n}(t)\\ &=u_{n}(t)+\kappa(x_{n-1}(t)-x_{n}(t)-S)\\ &-m_{n}a_{n}-m_{n}b_{n}\hat{v}_{n}(t)+d_{Ln}(t)\end{array} \right. \tag{6}\]  
Figure 2: Distribution of the URTs power unit. where \(d_{Li}(t)\) denotes the unknown lumped disturbance consisting of the additional resistance \(f_{ai}(t)\) and the unknown external disturbance \(O_{i}(\cdot)\). According to the kinematic characteristics of the URTs, this paper considers that the aerodynamic resistance is mainly concentrated on the first power unit in the running direction, and the mechanical rolling mechanical resistance acts on all the power units. Assume the first power unit has desired speed and desired position  
\[\dot{x}_{r}(t)=v_{r} \tag{7}\]  
Without loss of generality, it is assumed that each power unit of the URTs in the equilibrium satisfies the desired velocity \(v_{1}^{e}(t)=v_{2}^{e}(t)=\cdots=v_{n}^{e}(t)=v_{r}\) and acceleration \(\dot{v}_{1}^{e}(t)=\dot{v}_{2}^{e}(t)=\cdots=\dot{v}_{n}^{e}(t)=0\). Then, substituting the above assumptions into (6), the position (8) and control inputs (9) of the power unit at equilibrium are obtained. \[x_{i}^{e}(t)-x_{i+1}^{e}(t)=S \tag{8}\]  
\[\left\{ \begin{array}{l}u_{1}^{e}=m_{1}a_{1}+m_{1}b_{1}v_{r}+(\sum\limits_{i =1}^{n}m_{i})c_{1}v_{r}^{2}\\ u_{i}^{e}=m_{i}a_{i}+m_{i}b_{i}v_{r},i=2,\cdots n\end{array} \right. \tag{9}\]  
This means the power units maintain a stable operating distance when the URTs is in equilibrium. At this time, the couplers work in a relaxed state to ensure safe operation between the power units of the URTs. Define the position error variables, speed error variables, and control input error variables as \(\tilde{x}_{i}(t)=x_{i}(t)-x_{i}^{e}(t)+S(i-1)\), \(\tilde{v}_{i}(t)=v_{i}(t)-v_{i}^{e}(t)\), and \(\tilde{u}_{i}(t)=u_{i}(t)-u_{i}^{e}(t)\). Using the same linearization principle as in reference [25], the linear error kinetic equations of the URTs in equilibrium can be written as  
\[\left\{