## 3 Design of Suspension Controller
### Design of LADRC Controller  
LADRC is simplified from ADRC [25; 26], and mainly consists of a linear extended state observer (LESO) and linear state error feedback (LSEF). Using LADRC can not only ensure that the control accuracy and speed requirements are met, but also reduce the difficulty of parameter adjustment, estimate the sum of system uncertain disturbances in real time and suppress the impact of disturbances on system output. The structure is shown in Figure 4.  
In Figure 4, \(x\) is the given signal of the system and \(y\) is the actual output signal of the control system; \(z_{1},z_{2}\) is the differential of the observed signal and the observed signal of \(y\); \(z_{3}\) is the estimated state observation of uncertain interference; \(b_{0}\) is the compensation factor; \(u_{1}\) is LSEF output and \(u\) is the input of the controlled object after disturbance compensation.  
Figure 4: The structure of LADRC.  
LESO estimates the total unknown factors of the whole system, observes these as the total interference accurately and introduces compensation in the control loop, so as to achieve the purpose of anti-interference. According to the observer theory proposed by [PERSON], the state equation in Equation (10) is reconstructed. The state observer of the expanded system can be designed as follows:  
\[\left\{\begin{array}{l}\varepsilon=z_{1}-y\\ \dot{z}_{1}=z_{2}-\beta_{1}\varepsilon\\ \dot{z}_{2}=z_{3}-\beta_{2}\varepsilon+bu\\ \dot{z}_{3}=-\beta_{3}\varepsilon\end{array}\right. \tag{11}\]  
where \(\varepsilon\) is the tracking output error of the observer, \(z_{1},z_{2},z_{3}\) are the observed values of the system state \(x_{1},x_{2},x_{3}\) and \(\beta_{1},\beta_{2},\beta_{3}\) is the gain parameter of the observer.  
The characteristic equation of the observer is obtained as follows:  
\[\det(\lambda I-A)=\lambda^{3}+\beta_{1}\lambda^{2}+\beta_{2}\lambda+\beta_{3} \tag{12}\]  
where \(\lambda I-A\) is the feature matrix of \(A\), \(A\) is the third-order matrix which represents the observer, \(I\) is a third-order unit matrix and \(\lambda\) is the eigenvalue.  
According to the system stability criterion, the appropriate observer gain parameter is selected, so that the characteristic roots of Equation (12) are all distributed in the left-half plane of the complex plane, and then a stable state observer can be designed. With the bandwidth concept, the three pole points of the observer are all allocated to the real axis \(-\omega_{0}\), so as to determine the observer gain, given as:  
\[\det(\lambda I-A)=\left(\lambda+\omega_{0}\right)^{3} \tag{13}\]  
where \(\omega_{0}\) is the bandwidth of the observer. Then:  
\[\beta_{1}=3\omega_{0},\beta_{2}=3\omega_{0}^{2},\beta_{3}=\omega_{0}^{3} \tag{14}\]  
LSEF uses the target command and the estimated position signal and speed signal to calculate the initial control parameter and transforms the system into an integral series type through dynamic compensation of the estimated total disturbance. LSEF can be designed as:  
\[u_{1}=k_{\rm p}(r-z_{1})+k_{\rm d}(\dot{r}-z_{2}) \tag{15}\]  
where: \(u_{1}\) is the initial control quantity; \(k_{\rm p}\), \(k_{\rm d}\) are the gain coefficients of LSEF and \(r\) and \(\dot{r}\) are the given values of air gap target and speed, respectively.  
Then, combined with LESO to compensate the estimated total disturbance, the final control parameter is:  
\[u=\frac{u_{1}-z_{3}}{b_{0}} \tag{16}\]