# An Improved Data-Driven Integral Sliding-Mode Control and Its Automation Application
## 3 Improved Model-Free Adaptive Sliding-Mode Controller  
### Integral Sliding-Mode Controller  
Define the system output error as  
\[e(i)=I(i)-I_{d}(i) \tag{13}\]  
Inspired by the algorithm in [27], consider designing the following sliding-mode function:  
\[\
u(i)=e(i)+c_{1}E(i-1)+c_{2}F(i-1) \tag{14}\]  
where \(0<c_{1}<1\), \(0<c_{2}<1\). The two integral error terms can be defined as follows:  
\[E(i)=\sum\limits_{k=0}^{i}e(k)=E(i-1)+e(i) \tag{15}\]  
\[F(i)=\sum\limits_{k=0}^{i}\text{sig}^{\text{g}}e(k)=F(i-1)+\text{sig}^{\text{g }}e(i) \tag{16}\]  
Subsequently, the following reaching law is given:  
\[\Delta\
u(i+1)=\
u(i+1)-\
u(i)=0 \tag{17}\]Combining (13), (14) and (17), it is not difficult to obtain  
\[\begin{array}{l}\
u(i)=\
u(i+1)\\ =e(i+1)+c_{1}E(i)+c_{2}F(i)\\ =I(i+1)-I_{d}(i+1)+c_{1}E(i)+c_{2}F(i)\end{array} \tag{18}\]  
Substituting the data model (6) into (18), the following can be obtained:  
\[\begin{array}{l}\
u(i)=I(i)+\hat{h}_{1}(i)\Delta I(i)+\hat{h}_{2}(i)\Delta O (i)\ -I_{d}(i+1)+c_{1}E(i)+c_{2}F(i)\end{array} \tag{19}\]  
From (19), the equivalent control-law expression can be derived:  
\[\begin{array}{l}\Delta O_{eq}(i)=\hat{h}_{2}^{-1}(i)\Big{(}\
u(i)-c_{1}E(i) -c_{2}F(i)+I_{d}(i+1)-I(i)-\hat{h}_{1}(i)\Delta I(i)\Big{)}\end{array} \tag{20}\]  
To improve the robustness of the system and alleviate the chattering phenomenon, the following switching law is introduced [26; 27]:  
\[\begin{array}{l}\Delta O_{sw}(i)=\hat{h}_{2}^{-1}(i)\beta\text{sgn}(\
u(i)) \end{array} \tag{21}\]  
where \(0<\beta\) denotes the switching gain.  
Based on the above analysis, the data-driven ISMC scheme can be obtained as follows:  
\[\begin{array}{l}\left\{\begin{array}{l}\Delta O(i)=\Delta O_{eq}(i)+\Delta O _{sw}(i)\\ \Delta O_{eq}(i)=\hat{h}_{2}^{-1}(i)\Big{(}\
u(i)-c_{1}E(i)-c_{2}F(i)+I_{d}(i+ 1)-I(i)-\hat{h}_{1}(i)\Delta I(i)\Big{)}\\ \Delta O_{sw}(i)=\hat{h}_{2}^{-1}(i)\beta\text{sgn}(\
u(i))\\ \
u(i)=e(i)+c_{1}E(i-1)+c_{2}F(i-1)\\ E(i)=\sum\limits_{k=0}^{i}e(k)=E(i-1)+e(i)\\ F(i)=\sum\limits_{k=0}^{i}\text{sig}^{k}e(k)=F(i-1)+\text{sig}^{k}e(i)\end{array}\right.\end{array} \tag{22}\]  
**Remark 2**.: _The following saturation function can be considered instead of the sign function:_  
\[\begin{array}{l}\text{sat}(\sigma)=\left\{\begin{array}{l}\text{sgn}( \sigma)),|\sigma|>\text{X}\\ \sigma/\text{X},|\sigma|<\text{X}\end{array}\right.\end{array} \tag{23}\]  
_where_ X \(>\) 0_._  
The control block diagram is shown in Figure 1.  
Figure 1: The FPDL-ISMC algorithm.  
From (32), it follows that \(\hat{\mathbf{H}}(i)\) is bounded. Since \(\mathbf{H}(i)\) is bounded, it follows that \(\hat{\mathbf{H}}(i)\) is bounded.  
Proof.: The system will reach quasisliding modes.  
Substituting (6) into (13), the following can be obtained:  
\[e(t+1)=y_{d}-y(t+1)=y_{d}-y(t)-\phi_{1}\Delta y(t)-\phi_{2}\Delta u(t) \tag{33}\]  
Substituting (22) into (33), the following can be obtained:  
\[e(i+1)=v(i)-c_{1}E(i)-c_{2}F(i)-\beta\text{sgn}(v(i)) \tag{34}\]  
Substituting (34) into (14), the following can be obtained:  
\[v(i+1)=v(i)-\beta\text{sgn}(v(i)) \tag{35}\]  
From (35), it can be inferred that  
\[|v(i+1)|<|v(i)| \tag{36}\]  
In summary, it can be seen that the system meets the conditions for the existence and arrival of sliding modes. Therefore, \(v(i)\) monotonically decreases and reaches a quasisliding mode.  
Proof.: Substituting (14) into (34), the following can be obtained:  
\[\begin{array}{l}e(i+1)=v(i)-c_{1}E(i)-c_{2}F(i)-\beta\text{sgn}(v(i))\\ =e(i)-c_{1}e(i)-c_{2}\text{sig}^{a}e(i)-\beta\text{sgn}(v(i))\end{array} \tag{37}\]  
According to the Lemma 1 and (37), the convergence expression can be obtained:  
\[\begin{array}{l}|e(i)|_{\infty}\leq\left(1+r^{\frac{r}{1-r}}-r^{\frac{1}{1- r}}\right)\\ \cdot\max\left\{\left(\frac{\beta}{c_{2}}\right)^{\frac{1}{r}},\left(\frac{c_ {2}}{1-c_{1}}\right)^{\frac{1}{1-r}}\right\}\end{array} \tag{38}\]  
From (38), the tracking error is bounded.  
In summary, using the control scheme (22), the output of the system can successfully track the desired output.