# Drum Water Level Control Based on Improved ADRC
## 3 Simulation and Performance Analysis of Drum Water Level ILADRC  
### Performance Analysis under External Disturbance  
In the drum water level control system, a cascade three-impulse controller is composed of three signals of drum water level \(Y(s)\), steam flow \(D(s)\), and feed water flow \(W(s)\). The transfer function model of the drum water level system is shown in Figure 3.  
In Figure 3, the control output of the main circuit is set to \(U(s)\). The measurement error of water level in the main circuit and the loss of boiler connection are set to \(N_{1}(s)\). In the secondary circuit, the measurement error of flow meter and the disturbance of variable working condition are set to \(N_{2}(s)\). \(G_{V}(s)\) is a mass transfer function from W(s) to feed water control valve. \(G_{D}(s)\) is a transfer function from \(D(s)\) to \(Y(s)\). \(G_{W}(s)\) is a transfer function from \(W(s)\) to \(Y(s)\). \(H_{D}(s)\) is the transfer function of steam flow feedforward device. \(H_{W}(S)\) is the transfer function of feed water flow feedback device. \(H_{Y}(s)\) is the transfer function of the water level feedback device. \(G_{c2}(s)\) is a sub-controller. \(G_{c1}(s)\) is the main controller. The initial value of drum water level is 0 and the set water level is \(y_{r}=1\). The transfer function and related parameters of the control object of the drum water level control system in a power plant are as follows:  
\[\left\{ \begin{array}{l}G_{W}(s)=\frac{\lambda}{s(Ts+1)}=\frac{0.037}{s(30s+ 1)}\\ G_{D}s=\frac{3.045s-0.037}{15s^{2}+s}\end{array} \right., \tag{17}\]  
\[\left\{ \begin{array}{l}H_{D}(s)=H_{W}(s)=0.0174\\ G_{V}(s)=20\\ H_{Y}(s)=1\end{array} \right.. \tag{18}\]  
By tuning the controller, when the main controller is LADRC and ILADRC respectively, system parameters are shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **LADRC** & **ILADRC** \\ \hline \multirow{3}{*}{**main-controller**} & \(b_{0}=0.02\) & \(b_{01}=0.92\) \\  & \(\omega_{0}=1.12\) & \(\omega_{01}=\omega_{02}=18\) \\  & \(\omega_{c}=0.28\) & \(\omega_{c1}=\omega_{c2}=0.28\) \\ \hline \hline
**sub-controller** & \(k_{p}=1.2\) & \(k_{i}=0.35\) & \(k_{d}=5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of LADRC and ILADRC. Figure 3: Transfer function model of drum water level system. In order to investigate the performance of the system under external disturbance, when \(t_{d1}=100\) s, the steam flow \(D(s)\) changes from 0.5 to 2. When \(t_{d2}=250\) s, the secondary circuit interference \(N_{2}(s)\) changes from 0 to 2. The comparative simulation of LADRC and ILADRC is shown in Figure 4.  
The comparison of the variations of control input between LADRC and ILADRC is shown in Figure 5. As can be seen from Figure 5, the control input peak of ILADRC is smaller than that of LADRC and the fluctuation is smaller. According to the output energy of the controller input system through the control channel, ILADRC consumes less energy than that of LADRC. The following performance indicators are defined to measure the dynamic performance of the system:  
(1) Settling time  
\[|y(t)-y_{r}|\leq\varepsilon, \tag{19}\]  
where \(y_{r}\) is the reference input and \(y(t)\) is the actual output. The selection of parameter \(\varepsilon\) usually depends on the requirement of control accuracy and needs to be larger than the amplitude of observation noise. At this time, \(t\) is the system Settling time. Figure 4: Comparison of LADRC and ILADRC anti-interference performance. Figure 5: Comparison of control input between LADRC and ILADRC. (2) Overshoot  
Assuming that the time when the system output reaches the first peak is \(t_{p}\), the overshoot \(\theta\) is defined as:  
\[\theta\%=\left|\frac{y(t_{p})-y(\infty)}{y(\infty)}\right|\times 100\%. \tag{20}\]  
The following performance indexes are defined to measure the anti-interference performance of the system.