# Data-Driven Model-Free Adaptive Sliding Mode Control Based on FFDL for Electric Multiple Units
## 3 Model-Free Adaptive Sliding Mode Controller for EMUs
### Dynamic Linearization  
The CFDL method only considers the relationship between the output change of the system at the next moment and the input change at the current moment [9]. However, the output signal of the EMUs operation system depends not only on the control input at a certain moment. Based on the above considerations, the influence of the input signal and output signal in a fixed length sliding time on the output signal at the next moment can be taken into account when the data is linearized, which is the FFDL data processing method. Theoretically, the complex dynamics existing in the system can be well-obtained by using this method, and the dynamic linearization method can effectively reduce the complexity of the system [26].  
The input and output data set of the EMUs can be equivalent to the following MIMO discrete-time nonlinear system:  
\[\mathbf{v}(t+1)=g(\mathbf{v}(t),\ \cdots,\ \mathbf{v}(t-n_{v}),\mathbf{u}(t),\ \cdots,\ \mathbf{u}(t-n_{u})). \tag{5}\]  
where \(\mathbf{v},\mathbf{u}\) are the input and output of the system at time \(t\), respectively; \(n_{u}\), \(n_{v}\) are the order of input and output, respectively; and \(g(\cdot)\) is an unknown nonlinear time-varying function.  
Define \(\mathbf{H}(t)\in\mathbf{R}^{mL_{v}+mL_{u}}\) as a matrix consisting of control inputs within the sliding time window \([t-L_{u}+1,t]\) and outputs within the sliding time window \([t-L_{v}+1,t]\), as follows:  
\[\mathbf{\bar{H}}(t)=[\mathbf{v}^{T}(t),\cdots,\mathbf{v}^{T}(t-L_{y}+1),\mathbf{u}^{T}(t), \cdots,\mathbf{u}^{T}(t-L_{u}+1)]^{T}, \tag{6}\]  
where \(0\leq L_{v}\leq n_{v},1\leq L_{u}\leq n_{u}\) are the length constants of the linearization of the system output and control input, respectively, which are also called pseudo-orders.  
For the system of the form (5), the following two assumptions are given.  
The partial derivatives of the nonlinear function \(g(\cdot)\) with respect to all variables in the system are continuous.  
System (5) satisfies the following generalized Lipschitz condition, for any \(0\leq t_{1}\
eq t_{2}\) and \(\bar{\mathbf{H}}_{L_{v},L_{u}}(t_{1})\
eq\bar{\mathbf{H}}_{L_{v},L_{u}}(t_{2})\):  
\[\|\mathbf{v}(t_{1}+1)-\mathbf{v}(t_{2}+1)\|\leq b\|\bar{\mathbf{H}}(t_{1})-\bar{\mathbf{H}}(t_ {2})\|,\]  
where \(b\) is a positive constant. Denoting \(\Delta\bar{\mathbf{H}}(t)=\bar{\mathbf{H}}(t)-\bar{\mathbf{H}}(t-1)\.\)  
**Theorem 1** ([27]).: _Consider the discrete-time nonlinear system (5) satisfies Assumptions 1 and 2, for any \(0\leq L_{\
u}\leq n_{\
u},1\leq L_{u}\leq n_{u}\), and \(\|\Delta\bar{\mathbf{H}}(t)\|\
eq 0\). Then, there exists a time-varying matrix \(\mathbf{\Phi}(t)\in\mathbf{R}^{mL_{\
u}+mL_{u}}\), which is called a pseudo-partitioned jacobian matrix (PPJM), such that system (5) can be transformed into the following FFDL data model:_  
\[\mathbf{v}(t+1)=\mathbf{v}(t)+\mathbf{\Phi}(t)\Delta\bar{\mathbf{H}}(t). \tag{7}\]  
_The parameter matrix \(\mathbf{\Phi}(t)=[\mathbf{\Phi}_{1}(t),\cdots,\mathbf{\Phi}_{L_{\
u}+L_{u}}(t)]\) is bounded at any time \(t\), where \(\mathbf{\Phi}_{n}(t)\in\mathbf{R}^{m\times m}\), \(n=1,\cdots,L_{\
u}+L_{u}\)._  
**Remark 2**.: _Theorem 1 has been proved rigorously in [26; 27], and different FFDL data models can be obtained by choosing different linearization length constants \(L_{\
u}\), \(L_{u}\). Reasonable choice of \(\mathbf{\Phi}_{n}(t)\) and \(L_{\
u}\), \(L_{u}\) can improve the flexibility of the data model to describe the system._