# Data-Driven Model-Free Adaptive Sliding Mode Control Based on FFDL for Electric Multiple Units
## 3 Model-Free Adaptive Sliding Mode Controller for EMUs
### Model-Free Adaptive Controller  
Consider the following criterion function:  
\[J\big{(}\mathbf{u}(t)\big{)}=\|\mathbf{v}_{d}(t+1)-\mathbf{v}(t+1)\|^{2}+\lambda\|\mathbf{u}(t) -\mathbf{u}(t-1)\|^{2}, \tag{8}\]  
where \(\mathbf{v}_{d}\) is the desired output, and \(\lambda>0\) is a weighting factor that limits the change in control input. Substituting (7) into (8) yields:  
\[J\big{(}\mathbf{u}(t)\big{)}=\|\mathbf{v}_{d}(t+1)-\mathbf{v}(t)-\mathbf{\Phi}(t)\Delta\bar{ \mathbf{H}}(t)\|^{2}+\lambda\|\mathbf{u}(t)-\mathbf{u}(t-1)\|^{2}. \tag{9}\]  
Minimizing (9) with respect to the input signal, yields the following control law:  
\[\begin{split}\mathbf{u}_{MFA}(t)=&\mathbf{u}_{MFA}(t-1)+ \frac{\mathbf{\Phi}_{L_{\
u}+1}^{T}(t)\rho_{L_{\
u}+1}(\mathbf{v}_{d}(t+1)-\mathbf{v}(t))} {\lambda+\|\mathbf{\[PERSON]}_{L_{\
u}+1}(t)\|^{2}}\\ -&\frac{\mathbf{\[PERSON]}_{L_{\
u}+1}^{T}(t)\left(\sum\limits_ {i=1}^{L_{\
u}}\rho_{i}\mathbf{\[PERSON]}_{i}(t)\Delta\mathbf{v}(t-i+1)+\sum\limits_{i=L_{ \
u}+2}^{L_{\
u}+L_{u}}\rho_{i}\mathbf{\[PERSON]}_{i}(t)\Delta\mathbf{u}_{MFA}(t-i+1)\right) }{\lambda+\|\mathbf{\[PERSON]}_{L_{\
u}+1}(t)\|^{2}}.\end{split} \tag{10}\]  
where \(\rho_{i}\in(0,1]\) is a step-size constant, which is used to make the FFDL-MFAC algorithm general.  
Then, we need to estimate the unknown PPJM \(\mathbf{\Phi}(t)\). Introduce the following PPJM criterion function:  
\[J\big{(}\mathbf{\Phi}(t)\big{)}=\big{\|}\Delta\mathbf{v}(t)-\mathbf{\Phi}(t)\Delta\bar{ \mathbf{H}}(t-1)\big{\|}^{2}+\mu\big{\|}\mathbf{\Phi}(t)-\mathbf{\Phi}(t)\big{\|}^{2}. \tag{11}\]  
where \(\mu>0\) is a weighting factor that limits the rate of change of adjacent parameters.  
Minimizing (11) with respect to \(\mathbf{\Phi}(t)\) yields the following parameter estimation algorithms:  
\[\mathbf{\hat{\Phi}}(t)=\mathbf{\hat{\Phi}}(t-1)+\big{(}\Delta\mathbf{v}(t)-\mathbf{\hat{\Phi} }(t-1)\Delta\bar{\mathbf{H}}(t-1)\big{)}\Delta\bar{\mathbf{H}}(t-1)^{T}\Big{(}\mu+ \Delta\bar{\mathbf{H}}(t-1)\Delta\bar{\mathbf{H}}(t-1)^{T}\Big{)}^{-1}, \tag{12}\]  
The PPJM estimation algorithm (12) without inversion can be further given:  
\[\mathbf{\Phi}(t)=\mathbf{\Phi}(t-1)+\frac{\beta\big{(}\Delta\mathbf{v}(t)-\mathbf{\hat{\Phi}} (t-1)\Delta\bar{\mathbf{H}}(t-1)\big{)}\Delta\bar{\mathbf{H}}(t-1)^{T}}{\mu+\big{\|} \Delta\bar{\mathbf{H}}(t-1)\big{\|}^{2}}. \tag{13}\]  
where \(\mathbf{\hat{\Phi}}(t)\) is the estimation value of \(\mathbf{\Phi}(t)\); \(\beta\in(0,2]\) is a step-size constant.  
In summary, control law (10) can be rewritten as:  
\[\begin{split}\mathbf{u}_{MFA}(t)=&\mathbf{u}_{MFA}(t-1)+\frac{( \mathbf{\[PERSON]}_{L_{\varphi}+1}(t)-\delta_{L_{\varphi}+1}(t))^{\mathrm{T}}\rho_{L_{ \varphi}+1}(\mathbf{v}_{d}(t+1)-\mathbf{v}(t))}{\lambda+\left\|\mathbf{\[PERSON]}_{L_{\varphi}+1 }(t)-\delta_{L_{\varphi}+1}(t)\right\|^{2}}\\ -&\frac{\left(\mathbf{\[PERSON]}_{L_{\varphi}+1}(t)-\delta_{L_{ \varphi}+1}(t)\right)^{\mathrm{T}}\left(\sum\limits_{i=1}^{L_{\varphi}}\rho_{i }(\mathbf{\[PERSON]}_{i}(t)-\delta_{i}(t))^{\mathrm{T}}\Delta\mathbf{v}(t-i+1)\right)}{ \lambda+\left\|\mathbf{\[PERSON]}_{L_{\varphi}+1}(t)-\delta_{L_{\varphi}+1}(t)\right\|^ {2}}\\ -&\frac{\left(\mathbf{\[PERSON]}_{L_{\varphi}+1}(t)-\delta_{L_{ \varphi}+1}(t)\right)^{\mathrm{T}}\left(\sum\limits_{i=L_{\varphi}+2}^{L_{ \varphi}+L_{\varphi}}\rho_{i}(\mathbf{\[PERSON]}_{i}(t)-\delta_{i}(t))^{\mathrm{T}} \Delta\mathbf{u}_{MFA}(t-i+1)\right)}{\lambda+\left\|\mathbf{\Phi}_{L_{\varphi}+1}(t)- \delta_{L_{\varphi}+1}(t)\right\|^{2}}.\end{split} \tag{14}\]  
where \(\delta_{i}\) is parameter estimation errors.  
**Remark 3**: _The PPJM \(\mathbf{\Phi}(t)\) in (7) is considered as differential signals in a certain sense and is bounded at any time \(t\). The interaction between \(\mathbf{\Phi}(t)\) and \(\Delta\mathbf{u}(t)\) can be neglected when the system sampling period and \(\Delta\mathbf{u}(t)\) both are small [28]._  
**Remark 4**: _Equations (12) and (13) are independent of the mathematical model of the controlled system, and the PPJM estimate is related only to the online I/O data of the system._