# An Improved Data-Driven Integral Sliding-Mode Control and Its Automation Application
## 4 Simulation and Experiment
### Experiment 1: Numerical Simulation  
To increase the applicability of this paper's method, it is extended to an MIMO system. The effectiveness of the proposed algorithm is verified through the simulation of a typical controlled object, and the FFDL-SMC and FFDL-ISMC algorithms are compared under time-varying signals. The first set of simulation objects is a three-input three-output system as follows:  
\[\left\{ \begin{array}{l}I_{1}(i+1)=0.5\times d(i)+1.5\times I_{1}(i)+\\ 0.12\times{I_{1}}^{2}(i-1)+0.8\times O_{1}(i)+\\ 0.1\times O_{2}(i)+0.2\times O_{3}(i)\\ I_{2}(i+1)=0.5\times d(i)+1.6\times I_{2}(i)+\\ 0.14\times{I_{2}}^{2}(i-1)+0.1\times O_{1}(i)+\\ 0.8\times O_{2}(i)+0.2\times O_{3}(i)\\ I_{3}(i+1)=0.5\times d(i)+1.6\times I_{3}(i)+\\ 0.15\times{I_{3}}^{2}(i-1)+0.2\times O_{1}(i)+\\ 0.1\times O_{2}(i)+0.8\times O_{3}(i)\\ \end{array} \right. \tag{39}\]  
The desired output is set as follows:  
\[I_{d}=\left\{ \begin{array}{l}\mathrm{Ins}(0,50,100)\times 5,[0,100]\\ \mathrm{Ins}(250,250,50),[100,150]\\ \mathrm{Ins}(250,290,50),[150,200]\\ \mathrm{Ins}(290,290,525),[200,725]\\ \mathrm{Ins}(290,0,275),[725,1000]\\ \end{array} \right. \tag{40}\]  
where the \(\mathrm{Ins}(k_{1},k_{2},x)\) function is used to generate a vector of \(x\)-point rows between \(k_{1}\) and \(k_{2}\), with the same span of neighboring data, and \(k_{1}\), \(k_{2}\) and \(x\) are the start value, the end value and the number of elements, respectively. The FFDL-ISMC and FFDL-SMC controller parameters are set to \(\mathbf{\hat{H}}_{1}(1)=\mathbf{\hat{H}}_{2}(1)=\mathrm{diag}\{0.6,0.6,0.6\}\), \(\eta=0.75\), \(\mu=1\), \(\beta=0.02\), \(\theta=0.2\), \(c_{1}=0.45\), \(c_{2}=25\), \(L_{I}=1\), \(L_{O}=1\), \(\alpha=0.5\). Figure 2 shows the output curves of FFDL-ISMC and FFDL-SMC, and Figure 3 shows the output error curves. As can be seen from the enlarged output graphs of the output curves, in the first 100 simulation time steps, the traditional FFDL-SMC has a phase delay, and the output cannot be fed back in time, which leads to a certain degree of a tracking-delay phenomenon and large chattering and greatly reduces the good characteristics of the fast response of the SMC. The improved FFDL-ISMC does not have the phase-delay phenomenon and can quickly feedback the output signal, which maintains the fast response of the SMC while removing the noise. In the last 200 simulation time steps, the given signal changes, and the original FFDL-SMC can no longer maintain the tracking state, which further illustrates the adaptability of the proposed algorithm. From the error-related indicators in Table 1, it can be directly observed that the MAE and MSE using the FFDL-SMC method are about three and nine times higher than those using the FFDL-ISMC method, respectively. The tracking effect is more pronounced when the given signal becomes complex. Figure 4 shows the control-input changes. All the control inputs of the FFDL-ISMC scheme change smoothly in each stage, and the control-input ranges are within \([-52,42]\). During the predetermined signal-change stage, the control input can also slow down and change at a certain rate, and the chattering phenomenon is alleviated. The FFDL-SMC has large control-input variations during the scheduled signal-change phase, and the control-input range is within \([-55,47]\), with more frequent and larger variations, and the system-chattering phenomenon is more obvious:  
(1) MAE \[\mathrm{MAE}=\frac{1}{nT}\sum_{s=1}^{n}\sum_{i=1}^{T}\left|I_{s}(i)-I_{d}(i)\right|\] (41) MSE \[\mathrm{MSE}=\frac{1}{nT}\sum_{s=1}^{n}\sum_{i=1}^{T}\left|I_{s}(i)-I_{d}(i) \right|^{2}\] (42)In conclusion, the phase-delay phenomena that exists in the original SMC technique is abolished by the enhanced switching-control term and the new integral sliding-mode function in the improved ISMC. The ISMC approach achieves optimum control effects for systems with noise interference by maintaining both the strong anti-interference capacity of SMC and the quick response qualities of the MFAC method.