# Self-Anti-Disturbance Control of a Hydraulic System Subjected to Variable Static Loads
## 3 Simulation Analysis of Optimal Control Strategy
### Pole Optimized Configuration  
We introduce the state feedback gain matrix \(K=\begin{bmatrix}K_{1}&K_{2}&K_{3}\end{bmatrix}\); then, the characteristic polynomial can be expressed as  
\[\begin{split}&\alpha(\lambda)=\det(\lambda I-A+bK)=\left|\left[ \begin{array}{ccc}\lambda&0&0\\ 0&\lambda&0\\ 0&0&\lambda\end{array}\right]-\left[\begin{array}{ccc}0&1&0\\ 0&0&1\\ -17,036&-59,625&-98\end{array}\right]+\left[\begin{array}{ccc}0&0&0\\ 0&0&0\\ k_{1}&k_{2}&k_{3}\end{array}\right]\right|\\ &=\lambda^{3}+(k_{3}+98)\lambda^{2}+(k_{2}+59,625)\lambda+(k_{1}+17,036) \end{split} \tag{14}\]  
The core content of analyzing the performance of the control system is to determine the dominant pole, where the effect of the far pole is ignored. Therefore, the EHS control system is considered to be equivalent to a second-order control system containing a main pair of poles. Determining the location of the expected dominant pole from the representation of the dynamic indicators \(\sigma_{p}\%\) and \(t_{s}\) is as follows  
\[\sigma_{p}\%=e^{-\pi\xi/\sqrt{1-\xi^{2}}}\times 100\% \tag{15}\]  
where \(\sigma_{p}\%\) represents the maximum overshoot, and \(\xi\) is the damping (\(0<\xi<1\)). Assuming that the allowable error of the EHS control system is set to \(5\%\), there is  
\[t_{s}=\frac{3}{\xi\omega_{n}} \tag{16}\]  
where \(t_{s}\) is the adjustment time (s), and \(\omega_{n}\) denotes the undamped frequency (rad/s). The expected dominant pole of an EHS system can be expressed as  
\[\lambda_{1,2}=-\xi\omega_{n}\pm j\omega_{n}\sqrt{1-\xi^{2}} \tag{17}\]  
where the system expects \(\lambda_{1,2}\) to dominate the poles. Considering that the maximum overshoot of the EHS control system is limited to \(\sigma_{p}\%\leq 5\%\), and the adjustment time is set to \(t_{s}\leq 0.5s\), the expected dominant poles (\(\lambda_{1}\) and \(\lambda_{2}\)) of the EHS system can be given by the following Equation (18). Analyzing this equation, the following settings are specified  
\[\left\{\begin{array}{l}\sigma_{p}\%=e^{-\pi\xi/\sqrt{1-\xi^{2}}}\times 10 0\%\leq 5\%\\ t_{s}=\frac{3}{\xi\omega_{n}}\leq 0.5,\\ where,\xi\geq 0.69,\xi\omega_{n}\geq 6.0\end{array}\right. \tag{18}\]  
Herein, the clear value is \(\xi=0.707\), \(\xi\omega_{n}=6.0=6\); then, there is  
\[\lambda_{1,2}=-\xi\omega_{n}\pm j\omega_{n}\sqrt{1-\xi^{2}}=-6\pm j6 \tag{19}\]  
Figure 5: Step response curve of the EHS system after quadratic optimal control with linearity.  
Let the third pole \(\lambda_{3}=10R_{e}[\lambda_{1}]=-60\) be set; so, the following expression is  
\[\alpha^{*}(s)=(s+60)(s+6-j6)(s+6+j6)=s^{3}+72s^{2}+792s+4,320 \tag{20}\]  
We establish \(\alpha(s)=\alpha^{*}(s)\), and then obtain the following expression  
\[K=\begin{bmatrix}k_{1}&k_{2}&k_{3}\end{bmatrix}=\begin{bmatrix}-27&-58,836&-12,7 32\end{bmatrix}\]  
which shows the step response curve of the pole optimal configured EHS control system obtained by the Simulink module simulation. As can be seen from Figure 6, the EHS system can be stabilized within 1.3 s. After the EHS control system was optimized by the pole configuration, the stabilization time was shortened by 20.3 s, and the control performance was significantly improved; however, the overshoot phenomenon occurred prematurely.