# An Improved Model-Free Adaptive Nonlinear Control and Its Automatic Application
## 4 Simulation Experiment and Analysis
### Experiment 2: Circulating Fluidized Bed Boiler Drum Water Level Control
Maintain the amount of water supply within a reasonable working range, thus ensuring that the economizer and water supply pipe work normally without damage. Because of the time-varying physical characteristics of convection tube bundles, the valve position changes according to the boiler drum capacity and requires a certain amount of time to reflect the water level changes, so a certain time delay exists in the system. Moreover, there are large disturbances in the system due to the influence of temperature, environment, and equipment losses [21; 31]. To sum up, the drum water level system has no exact mathematical model, and it is accompanied by disturbance. Thus, the application of the MFAC method is a right choice. Due to the impact on the efficiency, time consumption, and production of the unit, the on-site debugging of the drum water level control system is avoided. Instead, we choose to construct a model of the actual system. MATLAB software, along with its integrated Simulink toolbox, is utilized for simulation purposes. The fuzzy logic toolbox within MATLAB provides powerful capabilities, including fuzzy logic inference systems and graphical user interfaces. This facilitates the rapid design of fuzzy controllers and system generation, ensuring convenience, speed, and visualization. The feedwater in the economizer is preheated by the flue gas before being introduced into the steam drum's water space. At this point, the boiler water, located at the top of the boiler, flows downward through downcomer pipes outside the furnace due to its own weight. After entering the water-wall tubes through the lower header box, the boiler water undergoes heat absorption and vaporization. During this period, a large amount of steam-water mixture forms in the drum, while the simultaneous separation of the steam and water occurs. The aforementioned water-wall tubes are laid around the furnace and are commonly referred to as upcomers. Subsequently, the boiler water circulates and vaporizes continuously within the downcomer pipes, upcomers, and drum. The saturated steam accumulated in the upper part of the drum passes through the superheater, where it is continually heated, ultimately becoming superheated steam. Finally, the superheated steam enters the main steam pipeline and goes into the turbine. The specific operation process is shown in the following Figure 6.  
In this case, the simulation model for the drum water level three-equalizing control system is based on the DHX25-1.25-AII circulating fluidized bed boiler, which has a rated evaporation capacity of 25 t/h and a rated pressure of 1.25 Mpa. Through extensive experimental analysis and verification, the technical parameters of this boiler were used. Considering the previous description of the dynamic characteristics of the water level and referencing relevant literature, the transfer function expression for the drum water level under disturbances from the steam flow and feedwater flow can be calculated [21; 31]:  
\[G(s)=\frac{k}{Ts(T_{d}s+1)} \tag{38}\]where \(T\) is the pure lag time under the disturbance of water supply; \(T_{d}\) is the water level reaction time; \(G(s)\) is the transfer function of the control system; \(s\) is a complex variable in the transfer function; and \(k\) is the proportional coefficient. **Remark 5**. : _The transfer function of the system can be transformed into a difference equation. The system parameters are identified using the least square method with the forgetting factor [32; 33], and the mathematical model of the system is obtained._  
In this experiment, with the parameters \(T=50\)\(s\), \(T_{d}=30\)\(s\), the model is only used to provide input and output data. Given the reference signal \(y_{d}=4,t\leq 500;y_{d}=7,t>500\); the MFAC and MFAC-OER controller parameters are set to \(\phi_{1}(1)=0.8\), \(\phi_{1}(1)=0.8\), \(\phi_{2}(1)=0.2\), \(\lambda=5\), \(\rho=0.85\), \(\mu=1\), \(\beta=0.2\), \(\theta=0.2\). The PID controller parameters are set to \(K_{p}=1.2\), \(K_{i}=0.8\), \(K_{d}=0.4\).