# Drum Water Level Control Based on Improved ADRC
## 2 ILADRC of Drum Water Level
### I.2.1.2.1.2.2.3  
[PERSON] [30] proved that two same low-order state observers are equivalent to a high-order ESO in series, and the low-order state observers adopt uniform parameters. This method not only guarantees the quality of control, but also reduces the number of parameters that need to be adjusted for the extended state observer, which reduces the difficulty of parameter setting. At present, the LADRC has simple structure, relatively mature parameter setting rules, and more convenient application than nonlinear ADRC. Inspired by the paper [30], in order to improve the disturbance rejection performance of ADRC, an IADRC for drum water level is designed by adding the same LESO and introducing the estimated error value of total disturbance on the basis of the structure of LADRC. The added extended state observer (LESO2) and the original extended state observer (LESO1) adopt uniform parameters, which not only ensure the control quality, but also reduce the number of parameters to be tuned, and simplify the difficulty of parameter tuning.  
The structure of the ILADRC is shown in Figure 2.  
If \(u\) in Equation (7) is substituted into Equation (1), the original system can be reduced to:  
\[\dot{y}=f(\dot{y},y,w)+u_{0}-z_{3}, \tag{9}\]  
Figure 1: Structural diagram of linear active disturbance rejection control (LADRC).  
Figure 2: Structural diagram of the improved linear active disturbance rejection controller (ILADRC).  
where \(z_{3}\) is the estimate of \(f(\dot{y},y,w)\). When \(z_{3}\) accurately estimates \(f(\dot{y},y,w)\), then  
\[z_{3}{=}\hat{f}\approx f(\dot{y},y,w){=}x_{3}. \tag{10}\]  
Substituting Equation (10) into Equation (9), we can get:  
\[\ddot{y}=f(\dot{y},y,w)+u_{0}-\hat{f}\approx u_{0}. \tag{11}\]  
Therefore, the original system is transformed into an integral series form. \(ef\) is defined as the estimated error of the linear observer for the total disturbance:  
\[ef=f(\dot{y},y,w)-\hat{f}. \tag{12}\]  
Substitute Equation (12) into Equation (11) to simplify:  
\[\dot{y}=f(\dot{y},y,w)+u_{0}-\hat{f}\approx ef+u_{0}. \tag{13}\]  
\(ef\) can be observed by the actual output quantity \(y\) and the virtual control quantity \(u_{0}\) in the control process. Therefore, its observed value \(\hat{ef}\) can be obtained by adding an observer with the same structure before the existing LESO. \(u_{01}\) is a new virtual control quantity after adding observer LESO2. It is defined as:  
\[u_{01}=k_{p}(r-y)-k_{d}\dot{y}. \tag{14}\]  
The relationship between the control variable \(u_{0}\) and the virtual control variable \(u_{01}\) is shown in Equation (15). The control law \(u\) can be obtained by adjusting the observed \(\hat{ef}\) to \(u_{01}\).  
\[\left\{\begin{array}{l}u_{0}=-\hat{ef}+u_{01}\\ u=\frac{u_{0}-\hat{f}}{b_{0}}\end{array}\right.. \tag{15}\]  
Equation (13) is transformed into the integral series form of Equation (16).  
\[\ddot{y}=ef-\hat{ef}+u_{01}{=}u_{01}. \tag{16}\]  
Thus, the ILADRC structure introduces the estimation error \(ef\) of the total disturbance and compensates for its observation. It can enhance the control system's ability to restrain the unknown disturbance.