# Data-Driven Model-Free Adaptive Sliding Mode Control Based on FFDL for Electric Multiple Units
## 4 Simulation Experiment and Analysis
### Simulation 2: External Disturbance  
The second group of experiments incorporates disturbance signals to verify the robustness of the proposed algorithm and compares it with MFAC and DITSMC methods. During the experiment, white noise is introduced to simulate the external disturbance encountered in the actual operation of the train. The control strategy and vehicle information are input into the simulation experimental bench, then to record the train operation speed, position, and control force of the three methods.  
(1) MFASMC: The initial conditions of the system are set as \(\mathbf{v}_{1}(1)=\mathbf{v}_{2}(1)=\mathbf{v}_{3}(1)=\mathbf{v}_{d}\), \(\mathbf{\Phi}_{1}(1)=\mathbf{\Phi}_{2}(1)=diag\{0.5,0.5,0.5\}\). The controller parameters are set to \(\lambda=1\), \(\rho=0.85\), \(\mu=1\), \(\beta=0.2\), \(\varepsilon=10\), \(q=0.5\), \(\delta_{1}(1)=\delta_{2}(1)=-0.6\), \(\mathbf{P}=diag\{2,2,2\}\).  
Figure 8: The control force distribution of each power unit of MFASMC.  
Figure 9: The acceleration distribution of each power unit of MFASMC.  
(2) MFAC: The FFDL-MFAC method control law scheme is as follows: \[\mathbf{u}(t)=\mathbf{u}(t-1)+\frac{\rho\mathbf{\hat{\Phi}}_{2}^{T}(t)(\mathbf{v}_{d}(t+1)-\mathbf{v }(t)-\mathbf{\hat{\Phi}}_{1}(t)\Delta\mathbf{v}(t))}{\lambda+\left\|\mathbf{\Phi}_{2}(t) \right\|^{2}}.\] (35) The initial conditions of the system are set as \(\mathbf{v}_{1}(1)=\mathbf{v}_{2}(1)=\mathbf{v}_{3}(1)=\mathbf{v}_{d}\), \(\mathbf{\Phi}_{1}(1)=\mathbf{\Phi}_{2}(1)=diag\{0.5,0.5,0.5\}\). The controller parameters are set to \(\lambda=1\), \(\rho=0.85\), \(\mu=1\), \(\beta=0.2\), \(\varepsilon=10\). (3) DITSMC [28]: The CFDL-DITSMC method control law scheme is as follows: \[\Delta\mathbf{u}(t)=\frac{1}{\mathbf{\hat{\Psi}}(t)}[\mathbf{K}_{1}^{-1}\mathbf{s}(t)-\mathbf{K}_{ 1}^{-1}\mathbf{K}_{2}\mathbf{E}(t)+\mathbf{v}_{d}(t+1)-\mathbf{v}(t)-\text{sgn}(\mathbf{s}(t))]\]  
Figure 11: Speed tracking error curves of each power unit for the three methods.  
Figure 10: Speed tracking curves of each power unit for the three methods.  
Further, in order to analyze the control performance of each controller more intuitively, the following several performance indicators are considered to evaluate the controller.  
(1) Mean square error performance (MSE)  
\[MSE=\frac{1}{nT}\sum_{i=1}^{n}\sum_{t=1}^{T}|v_{d}-v_{i}(t)|^{2}. \tag{39}\]  
(2) Integral of absolute value of error criterion (IAE)  
Figure 12: Control force curves of each power unit for the three methods.  
Figure 13: Acceleration curves of each power unit for the three methods.  
\[IAE=\frac{\sum\limits_{i=1}^{n}\int_{0}^{\infty}|e_{i}(t)|dt}{n}.\] (40) (3) Maximum acceleration/deceleration (MAXA) \[MAXA=\sup\limits_{t\in[2,T]i\in[1,n]}|v_{i}(t)-v_{i}(t-1)|. \tag{41}\]  
The performance index values for the three control methods are recorded in Table 2. The MSE indicator measures the deviation between the observed value and the true value and is more sensitive to outliers in the data. The smaller the value, the better the system tracking effect, and the IAE index is similar to the MSE measurement effect. The MSE and IAE values of MFAC and DITSMC were greater than those of MFASMC. Input stability of the maximum acceleration/deceleration reaction system. It can be seen that the maximum acceleration/deceleration of the train using the MFASMC control scheme is 0.942 m/s\({}^{2}\), and the change is small; the maximum acceleration/deceleration of MFASMC and DITSMC is 1.045 m/s\({}^{2}\) and 1.056 m/s\({}^{2}\), respectively, which means that MFAC and DITSMC are not conducive to the comfort requirements of passengers, while MFASMC can meet the requirements.  
In summary, this section introduces external disturbances, and verifies the robustness and superiority of the proposed algorithm through velocity tracking error, control force changes, and acceleration changes.