# An Improved Data-Driven Integral Sliding-Mode Control and Its Automation Application
## 4 Simulation and Experiment
### Experiment 2: Water-Level Control of CFB
In this example, the simulation model of the three-equilibrium control system of the turbine water level is based on the CFB boiler of type DHX25-1.25-AII, whose rated  
Figure 5: CFB system. Figure 6: The simplified CFB system. evaporation capacity is 25 t/h and rated pressure is 1.25 MPa; the physical platform is shown in Figure 7. It can be seen from the above that the two main factors of the water supply and the amount of steam have the greatest influence on the change in the turbine water level. This paper considers the impact of water supply on the water level of the steam drum, and there exists a dynamic mathematical model for this object as follows [32; 33; 34]:  
\[G(s)=\frac{k}{Ts(T_{d}s+1)} \tag{43}\]  
where \(G(s)\) is the control system's transfer function, \(s\) is a complex variable in the transfer function, \(k\) is the proportional coefficient and \(T\) is the pure lag time under the water-supply disruption. \(T_{d}\) is the water-level reaction time. **Remark 3**. : _It is worth noting that the mathematical model of the boiler water level used in this paper is merely an empirical model, and building a dynamic model that fully represents the system characteristics is actually quite challenging. On the other hand, there are complex actuators present in the boiler system, which have been collectively considered as part of an empirical model. The transfer function of the system can be transformed into a difference equation. Identify the system parameters using the least square method with the forgetting factor, and the mathematical model of the system is obtained._  
The parameters are taken as \(T=40\)s and \(T_{d}=36\)s. The parameters of the FFDL-ISMC and FFDL-SMC controllers are set to \(\mathbf{\hat{H}}_{1}(1)=\mathbf{\hat{H}}_{2}(1)=\text{diag}\{0.46,0.6,0.46\}\), \(\eta=0.75\), \(\mu=0.9\), \(\beta=0.05\), \(\theta=0.2\), \(c_{1}=0.55\), \(c_{2}=0.35\), \(L_{I}=1\), \(L_{O}=1\), \(\alpha=0.5\). The FFDL-MFAC parameters are set to \(\mathbf{\hat{H}}_{1}(1)=\mathbf{\hat{H}}_{2}(1)=\text{diag}\{0.46,0.6,0.46\}\), \(\eta=0.75\), \(\mu=1\), \(\theta=0.2\), \(L_{I}=1\), \(L_{O}=1\). The MFNHTSMC method parameters are set to \(\lambda_{1}=12,\lambda_{2}=2\), \(\frac{\xi_{1}}{t_{1}}=\frac{7}{3},\frac{\xi_{2}}{t_{2}}=\frac{5}{3},\varepsilon _{1}=0.02,\varepsilon_{2}=0.005\)[35]. The principle of FFDL-ISMC drum water-level control is shown in Figure 8.  
Figure 7: CFB boiler experiment platform. Figure 9 shows the boiler bubble water-level output of the FFDL-MFAC, FFDL-SMC, MFNHTSMC and FFDL-ISMC methods, while Figure 10 shows the corresponding water-level output error. The performance indicators are summarized in Table 2. From the two sets of figures and Table 2, it is evident that the proposed FFDL-ISMC method not only has fast tracking characteristics but also effectively suppresses the influence of external disturbances, thereby maintaining a stable water-level height and exhibiting a superior tracking performance compared to the FFDL-MFAC, FFDL-SMC and MFNHTSMC methods. Figure 8: FFDL-ISMC of CFB system. Figure 9: Control-methods system output curve. The specific analyses are as follows: The water-level setting error range of the FFDL-ISMC method is within [\(-0.221\) m, \(0.518\) m], and the tracking accuracy is high, while the water-level errors of the FFDL-MFAC and FFDL-SMC methods are between [\(-1.983\) m, \(1.498\) m] and [\(-1.817\) m, \(1.102\) m], respectively. For the rise time, as shown in Table 2, the FFDL-ISMC method is nearly a quarter of FFDL-SMC and nearly half of FFDL-MFAC. The FFDL-ISMC method's adjustment time is less than that of the FFDL-MFAC method and is almost exactly half that of the FFDL-SMC method. The MAE and MSE obtained by the FFDL-ISMC technique are substantially smaller than those obtained by the FFDL-MFAC and approximately half as large as those obtained by the FFDL-SMC method, indicating that it has more precision and stability. The chattering of the FFDL-ISMC method is reduced compared with that of the FFDL-SMC and MFNHTSMC methods.