# Self-Anti-Disturbance Control of a Hydraulic System Subjected to Variable Static Loads
## 3 Simulation Analysis of Optimal Control Strategy
### Simulation Analysis of SAD Control
\begin{array}{l}f{s}=-r(r(r(r(r_{1}-v(t))+3v_{2})+3v_{3})\\ v_{1}=v_{1}+hv_{2}\\ v_{2}=v_{2}+hv_{3}\\ v_{3}=v_{3}+hfs\\ U(t)=v_{1}(t)\end{array} \right. \tag{26}\]  
where \(r\) is a fast factor, and \(h\) denotes the sampling step size. The error feedback Extended State Controller (ESC) follows the output of the control system, monitors the state of each stage of the research object and the total perturbation of the EHS system in real time, and then compensates the perturbation accordingly. If the system suffers from unknown disturbances, the control object is uncertain. This indeterminate object is expressed as  
\[\left\{ \begin{array}{l}x^{(n)}=f(x,\dot{x},\cdots,x^{(n-1)},t)+w(t)+bu\\ y=x(t)\end{array} \right. \tag{27}\]  
where \(f(x,\dot{x},\cdots,x^{(n-1)},t)\) is the system's unknown function, \(w(t)\) is the unknown perturbation, \(b\) is a control gain, \(u\) is the system input, and \(y\) is the system output. Figure 9: Zero-pole diagram of the controlled object exported by Matlab. The error feedback ESC in this paper is fourth-order. Under linear conditions, the fourth-order ESC algorithm of the control system is written as  
\[\left\{ \begin{array}{l}z_{1}=z_{1}+h(z_{2}-\beta_{01}e),z_{2}=z_{2}+h(z_{3}- \beta_{02}e),z_{3}=z_{3}+h(z_{4}-\beta_{03}e+bu),z_{4}=z_{4}+h(-\beta_{04}e)\\ \text{where},e=z_{1}-y\end{array} \right. \tag{28}\]  
where \(\beta_{01},\beta_{02},\beta_{03},\beta_{04}\) are the ESC parameters, and \(e\) is the feedback error term. The transfer relationship from the input to output of the error feedback ESC is expressed as  
\[\left\{ \begin{array}{l}z_{1}=w_{1}(s)y=\frac{\beta_{01}s^{3}+\beta_{02}s^{2} +\beta_{03}s+\beta_{04}}{s^{4}+\beta_{01}s^{5}+\beta_{02}s^{2}+\beta_{03}s+\beta _{04}}y,z_{2}=w_{2}(s)y=\frac{\beta_{03}s^{3}+\beta_{03}s^{2}+\beta_{04}s}{s^{ 4}+\beta_{01}s^{5}+\beta_{02}s^{2}+\beta_{03}s+\beta_{04}}y\\ z_{3}=w_{3}(s)y=\frac{\beta_{03}s^{3}+\beta_{04}s^{2}}{s^{4}+\beta_{01}s^{5}+ \beta_{02}s^{2}+\beta_{03}s+\beta_{04}}y,z_{4}=w_{4}(s)y=\frac{\beta_{04}s^{3} }{s^{4}+\beta_{01}s^{5}+\beta_{02}s^{2}+\beta_{03}s+\beta_{04}}y\end{array} \right. \tag{29}\]  
In the control system, the ESC contains four output variables: \(z_{1}\) follows the system output \(y\), \(z_{2}\) follows \(y^{\prime}\), \(z_{3}\) follows \(y^{\prime\prime}\), \(z_{4}\) follows the system integral disturbance, and the disturbance is compensated by the feedforward method. In order to achieve a predetermined estimation accuracy, a larger gain coefficient was selected, that is, a larger value of the gain coefficients \(\beta_{01},\beta_{02},\beta_{03},\beta_{04}\) was required to satisfy the high-gain ESC mode. Based on the four optimization methods and control strategy system models of the classical PID control, quadratic optimization control, pole configuration, and SAD, the EHS system was simulated by Simulink, and the step response curves of the unit step signal under the four control strategies were compared and analyzed, as shown in Figure 10.  
In Figure 10, the SAD performance was the most stable after the system was optimized. The response speed of PID was 0.6 s slower than that of SAD control, and the response was accompanied by a large amount of overshoot. The response time of the pole configuration was 0.3 s longer than that of the SAD control. Compared with the PID control, although the overshoot was significantly reduced, the overshoot still existed. Both the quadratic optimal control and SAD control did not have overshoot, but the response time of the SAD control was 4.8 s shorter than that of the quadratic optimal control.