# Drum Water Level Control Based on Improved ADRC
## 2 ILADRC of Drum Water Level  
### Traditional LADRC  
In the drum water level control system, a serial three-impulse controller is composed of three signals of drum water level \(Y(s)\), steam flow \(D(s)\), and feed water flow \(W(s)\).  
In order to compensate for the internal disturbance and external disturbance to the control system by feedforward method, the mathematical model of the water level system can be simplified to a second-order system, i.e., \(m=2\), using LESO correlation description:  
\[\ddot{y}=f(\dot{y},y,w)+b_{0}u, \tag{1}\]  
where \(y\) is the output signal of the system, \(u\) is the control input signal of the system, \(b_{0}\) is the gain of the controller, \(w\) is the external disturbance of the system, and \(f(\cdot)\) is the total disturbance of internal disturbance plus external disturbance.  
Define the state as \(x_{1}=y\), \(x_{2}=\dot{y}\) and \(x_{3}=f\). The total perturbation is extended to \(m+1=3\) order linear system, then the state space of Equation (1) is expressed as:  
\[\left\{\begin{array}{l}\dot{x}=\mathrm{A}x+\mathrm{b}_{0}\mathrm{B}u+ \mathrm{E}\dot{f}\\ y=\mathrm{C}x\end{array}\right., \tag{2}\]  
where, \(\mathrm{A}=\left[\begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&0\end{array}\right]\), \(\mathrm{B}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\), \(\mathrm{E}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\), \(\mathrm{C}=[1\quad 0\quad 0]\).  
In LADRC, the LESO is the key link for real-time estimation and on-line compensation of the total disturbances of the system. Its task is to observe the states and total disturbances of the system according to the input and output data.  
As the controller gain \(\hat{b}_{0}=b_{0}\) is greater than zero, the following third-order LESO is established:  
\[\dot{z}=\mathrm{A}z+\hat{b}_{0}\mathrm{B}u+\mathrm{K}e, \tag{3}\]  
where \(e=x-z_{1}=y-z_{1}\) is observer error.  
\[[k_{1}\quad k_{2}\quad k_{3}]=\left[\alpha_{1}\omega_{0}\quad\alpha_{2}\omega_ {0}{}^{2}\quad\alpha_{3}\omega_{0}{}^{3}\right], \tag{4}\]  
\[\alpha_{i}=\frac{\mathrm{m}+1!}{i!\left(\mathrm{m}+1-i\right)!}(i{=}1,2,3), \tag{5}\]  
where \(K=[\begin{array}{ccc}k_{1}&k_{2}&k_{3}\end{array}]^{\mathrm{T}}\) is the observer gain vector. \(K\) is chosen to put all of the observer eigenvalues at \(-\omega_{0}\) for the ease of tuning. Here, \(\omega_{0}\) is referred to as the observer bandwidth. The LESO characteristic equation designed is as follows:  
\[\lambda{=}s^{3}+\omega_{0}\alpha_{1}s^{2}+\omega_{0}{}^{2}\alpha_{2}s^{1}+ \omega_{0}{}^{3}\alpha_{3}{=}\ (s+\omega_{0})^{3}. \tag{6}\]  
Such configuration can not only maintain the stability of the system, but also give a better transition process, and simplify the parameters as the observer bandwidth.  
The linear state error feedback (LSEF) is designed as a PD. The controller is designed as follows:  
\[\left\{\begin{array}{l}u_{0}=k_{p}(r-y)-k_{d}\dot{y}\\ u=\frac{u_{0}-z_{3}}{b_{0}}\end{array}\right., \tag{7}\]since \(Z\approx Y\), the closed-loop dynamic equation of the system can be approximately written as:  
\[\ddot{y}+k_{d}\dot{y}+k_{p}y=k_{p}y_{r}, \tag{8}\]  
where \(k_{p}\) is a proportional coefficient, \(k_{d}\) is a differential coefficient, and \(y_{r}\) is the water level setting value. According to the parameter tuning method of linear ADRC, the proportion and differential coefficients are determined as: \(k_{p}={\omega_{c}}^{2}\), \(k_{d}=2\omega_{c}\). Here, \(\omega_{c}\) is referred to as the controller bandwidth. The structure diagram of the LADRC is shown in Figure 1.