# Data-Driven Model-Free Adaptive Sliding Mode Control Based on FFDL for Electric Multiple Units
## 3 Model-Free Adaptive Sliding Mode Controller for EMUs
### Stability Analysis  
**Assumption 3**. : _The EMUs iare not affected by serious external factors during operation (such as derailment caused by debris flow and rail damage). Since the train system is subject to strict speed regulation, the speed variation of the train tends to be infinitely small when the system is stable and the sampling time is small enough._  
**Remark 5**. : _For the convenience of description, let \(L_{v}=L_{u}=1\), other cases are similar._  
**Theorem 2**. : _If the system (6) satisfying Assumptions 1-3 is controlled by FFDL-MFASMC schemes (12), (13), and (19) with the desired output signal \(\mathbf{v}_{d}(t+1)\)=\(\mathbf{v}_{d}\)= const, then there exists \(\lambda_{\min}\), when \(\lambda>\lambda_{\min}\), the control system guarantees:_  
1. _The estimated value_ \(\mathbf{\Phi}(t)=[\mathbf{\hat{\Phi}}_{1}(t),\mathbf{\Phi}_{2}(t)]\) _of PPJM is bounded;_
2. \(\lim\limits_{t\rightarrow\infty}\lVert\mathbf{e}(t)\rVert=0\)_;_
3. _The output sequence and the input sequence are bounded._  
Proof. : \(\mathbf{\hat{\Phi}}(t)=[\mathbf{\hat{\Phi}}_{1}(t),\mathbf{\hat{\Phi}}_{2}(t)]\) is bounded. Firstly, rewrite (13) into the following form:  
\[\mathbf{\Phi}(t)=\mathbf{\Phi}(t-1)+\frac{\beta(\mathbf{v}(t)-\mathbf{v}(t-1))\Delta\mathbf{H}^{T }(t-1)}{\mu+\ \parallel\Delta\overline{\mathbf{H}}(t-1)\|^{2}}-\frac{\beta\mathbf{\Phi}(t-1) \Delta\mathbf{H}(t-1)\Delta\mathbf{H}^{T}(t-1)}{\mu+\ \parallel\Delta \overline{\mathbf{H}}(t-1)\|^{2}}. \tag{20}\]  
Subtracting \(\mathbf{\Phi}(t)\) from both sides of (20) yields:  
\[ \begin{split}\delta(t)=&\delta(t-1)+\mathbf{\Phi}(t-1 )-\mathbf{\Phi}(t)+\frac{\beta(\mathbf{v}(t)-\mathbf{v}(t-1))\Delta\overline{\mathbf{H}}^{T}( t-1)}{\mu+\ \parallel\Delta\overline{\mathbf{H}}(t-1)\|^{2}}\\ -&\frac{\beta\mathbf{\hat{\Phi}}(t-1)\Delta\mathbf{H}(t-1) \Delta\mathbf{H}^{T}(t-1)}{\mu+\ \parallel\Delta\overline{\mathbf{H}}(t-1)\|^{2}}.\end{split}  \tag{21}\]  
Figure 3: An improved block diagram of MFASMC structure for EMUs. Substituting \(\mathbf{\hat{\Phi}}(t-1)\)=\(\delta(t)\)+\(\mathbf{\Phi}(t-1)\) into (21) yields:  
\[ \begin{split}\delta(t)=&\delta(t-1)+\mathbf{\Phi}(t- 1)-\mathbf{\Phi}(t)+\frac{\beta(\mathbf{v}(t)-\mathbf{v}(t-1))\Delta\mathbf{\hat{H}}^{T }(t-1)}{\mu+\parallel\Delta\mathbf{\hat{H}}(t-1)\parallel^{2}}\\ -&\frac{\beta(\delta(t-1)+\mathbf{\Phi}(t-1))\Delta \mathbf{\hat{H}}(t-1)\Delta\mathbf{\hat{H}}^{T}(t-1)}{\mu+\parallel\Delta \mathbf{\hat{H}}(t-1)\parallel^{2}}.\end{split}  \tag{22}\]  
Substituting data model (7) into (22) yields:  
\[\delta(t)= \delta(t-1)\left(I-\frac{\beta\Delta\mathbf{\hat{H}}(t-1)\Delta \mathbf{\hat{H}}^{T}(t-1)}{\mu+\parallel\Delta\mathbf{\hat{H}}(t-1)\parallel^ {2}}\right)+\mathbf{\Phi}(t-1)-\mathbf{\Phi}(t). \tag{23}\]  
From Theorem 1, \(b\) is a positive constant and \(\|\mathbf{\Phi}(t)\|\leq b\). Taking the norm on both sides of (23), we have:  
\[\|\delta(t)\|= \left\|\delta(t-1)\left(I-\frac{\beta\Delta\mathbf{\hat{H}}(t-1) \Delta\mathbf{\hat{H}}^{T}(t-1)}{\mu+\parallel\Delta\mathbf{\hat{H}}(t-1) \parallel^{2}}\right)\right\|+2b. \tag{24}\]  
Squaring the first term on the right side of (24), one obtains:  
\[ \begin{split}&\left\|\delta(t-1)\left(I-\frac{\beta\Delta \mathbf{\hat{H}}(t-1)\Delta\mathbf{\hat{H}}^{T}(t-1)}{\mu+\parallel\Delta \mathbf{\hat{H}}(t-1)\parallel^{2}}\right)\right\|^{2}=\left\|\delta(t-1) \right\|^{2}\\ &+\left(\frac{\beta\parallel\Delta\mathbf{\hat{H}}(t-1) \parallel^{2}}{\mu+\parallel\Delta\mathbf{\hat{H}}(t-1)\parallel^{2}}-2 \right)\frac{\beta\left\|\delta(t-1)\right\|^{2}\parallel\Delta\mathbf{\hat{ H}}(t-1)\parallel^{2}}{\mu+\parallel\Delta\mathbf{\hat{H}}(t-1)\parallel^{2}}. \end{split}