# Data-Driven Model-Free Adaptive Sliding Mode Control Based on FFDL for Electric Multiple Units
## 2 Dynamic Analysis of EMUs Operation Process  
The traction/braking system of EMUs is composed of several relatively independent power units [21]. Each power unit is associated with an adjacent power unit and couples each other, its longitudinal dynamic is described in Figure 1.  
Figure 1: Dynamic description of EMUs operation process.  
In the Figure 1, the middle carriage is the powered locomotive equipped with a traction unit, and the first and last carriages are trailers. According to [PERSON]'s law of kinematics, the forces on the train power units are analyzed, and the traction force or braking force, basic resistance, and carriage force are applied to each power unit. According to the analysis of the dynamic process of each power unit, the mathematical model of the three power units of the train can be expressed as [22]:  
\[\left\{\begin{array}{c}m_{1}\dot{\varphi}_{1}(t)=u_{1}-F_{N1}(t)-F_{Z1}(t),\\ \vdots\\ m_{i}\dot{v}_{i}(t)=u_{i}-F_{Ni}(t)+F_{Z(i-1)}(t)-F_{Zi}(t),\\ \vdots\\ m_{n}\dot{v}_{n}(t)=u_{n}-\dot{F}_{Nn}(t)+F_{Z(n-1)}(t),\\ \dot{x}_{i}(t)=v_{i}(t).\end{array}\right. \tag{1}\]  
where  
\[F_{Ni}(t)=a_{i}+b_{i}v_{i}(t)+c_{i}v_{i}^{2}(t), \tag{2}\]  
\[F_{Zi}(t)=k(x_{i}(t)-x_{i+1}(t))+d(v_{i}(t)-v_{i+1}(t)), \tag{3}\]  
\(u_{i}(i=1,\cdots,n)\) is the traction/braking force generated by the power unit \(i\) under different working conditions of the train; \(m_{i}(t),v_{i}(t),x_{i}(t)\) is the mass, speed, and displacement of power unit \(i\), respectively; \(F_{Ni}(t)\) is the basic resistance of power unit \(i\); \(a_{i},b_{i},c_{i}\) are the operating resistance coefficients; \(F_{Zi}(t)\) is the force between power unit \(i\) and power unit \(i+1\), \(k\) is the elastic coefficient of the adjacent power unit, and \(d\) is the damping coefficient.  
Taking the CRH380A EMUs (with six motors and two trailers) as the object for research, which has three independent traction power units (T1 + M1 + M2, M3 + M4, M5 + M6 + T2), and the grouping is shown in Figure 2.  
Substituting \(F_{Zi}(t),F_{Ni}(t)\) into (1), the longitudinal dynamics model of the high-speed EMUs are obtained as follows:  
\[\begin{array}{c}m_{1}\dot{\varphi}_{1}(t)=u_{1}-\left(a_{1}+b_{1}v_{1}(t)+c _{1}v_{1}^{2}(t)\right)\\ -\left(k(x_{1}(t)-x_{2}(t))+d(v_{1}(t)-v_{2}(t))\right),\\ m_{2}\dot{\varphi}_{2}(t)=u_{2}-\left(a_{2}+b_{2}v_{2}(t)+c_{2}v_{2}^{2}(t) \right)\\ +\left(k(x_{1}(t)-x_{2}(t))+d(v_{1}(t)-v_{2}(t))\right)\\ -k(x_{2}(t)-x_{3}(t))-d(v_{2}(t)-v_{3}(t)),\\ m_{3}\dot{\varphi}_{3}(t)=u_{3}-\left(a_{3}+b_{3}v_{3}(t)+c_{3 are uncertainties in the mass parameters, drag coefficient, spring coefficient, and damping coefficient in the train model, and there are also nonlinear terms in the model [23]. If the data-driven control method is not used to design the controller, the coupling relationship of each power unit needs to be considered, and the amount of calculation is huge. Therefore, a model-free adaptive sliding mode controller is designed in this paper to perform speed tracking control for EMUs.  
The common traction/braking models of trains are divided into empirical models [1; 2; 3], fluid dynamics [4; 24; 25], and fluid-empirical dynamics models [24]. Empirical models are widely used in train dynamics simulation. A fluid dynamics model is a better model to study the behavior of brake systems, but it is more complex and slow in calculation. The fluid-empirical dynamics model combines the fluid dynamics brake pipeline model and the empirical brake valve model. The model of this paper is the empirical model.