## 3 The Switch Control Strategy of the Hydro-Pneumatic Suspension
### Ride Comfort Control  
Through many years of investigation and practice, the quarter-car suspension nonlinear model is suitable for the research on the ride comfort control [27]. In this paper, the ride comfort controller of the hydro-pneumatic suspension is built based on the quarter-car model, which makes it easy to compute. Through this quarter-car model, the controller output (\(f_{ul}\)) of one side of the suspension can be calculated, and the other side controller output is expressed as \(f_{ur}=\zeta f_{ul}\), where \(\zeta\) is the correlation coefficient that connects the control quantity of left and right sides' suspensions. The value of \(\zeta\) can be obtained through the optimization algorithm which will be introduced in the Section 4. According to [PERSON]'s law, the state space representation can be written as follows. \[\left\{ \begin{array}{ll}\dot{X}=AX+BU+GW\\ Y=CX+DU\end{array} \right. \tag{16}\]  
where  
\[A=\left[ \begin{array}{cccc}2_{ul}&2_{s}&\dot{z}_{ul}&\dot{z}_{s}\end{array}  \right]^{T},U=f_{ul},W=z_{ql}\]  
\[C=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ -\frac{k_{x}+k_{y}}{m_{s}}&\frac{k_{z}}{m_{s}}&-\frac{c_{s}}{m_{s}}&\frac{c_{s} }{m_{s}}\\ \frac{k_{x}}{m_{s}}&-\frac{k_{y}}{m_{s}}&\frac{c_{s}}{m_{s}}&-\frac{c_{s}}{m_{s }}\\ \end{array} \right]D=\left[ \begin{array}{c}0\\ 0\\ 0\\ \frac{1}{m_{s}}\\ \end{array} \right]\]  
Define the sliding surface as  
\[S(X)=n_{1}x_{1}+n_{2}x_{2}+n_{3}x_{3}+x_{4} \tag{17}\]  
where \(X\) is the state vector, and the polynomial \(p^{3}+n_{3}p^{2}+n_{2}p+n_{1}\) should satisfy the Hurwitz law based on the design criteria of the SMC. [28] On the basis of modern control theory, the variable structure systems' controllability should be constant for any nonsingular transformation. Because of the controllability of the matrices \(A\) and \(B\), the nonsingular transformation is carried on \(X\). \[\widetilde{X}=T_{s}X \tag{17}\]  
\[\left[ \begin{array}{c}\widetilde{X}_{1}\\ \widetilde{X}_{2}\end{array} \right]=\left[ \begin{array}{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\end{array} \right]\left[ \begin{array}{c}\widetilde{X}_{1}\\ \widetilde{X}_{2}\end{array} \right]+\left[ \begin{array}{cc}0&K\\ B_{2}&0\end{array} \right]\left[ \begin{array}{c}f_{ul}\\ z_{ql}\end{array} \right] \tag{18}\]  
The values of the matrices in Equations (17) and (18) are shown as follows. \[ \begin{array}{c}T_{s}=\left[\begin{array}{ccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&\frac{m_{s}}{m_{u}}\\ 0&0&0&1\end{array}\right]\widetilde{X}_{1}=\left[\begin{array}{cc}x_{1}&x_{2 }&\dot{x}_{1}+\frac{m_{s}}{m_{u}}\dot{x}_{2}\end{array}\right]\widetilde{X}_{2 }=\left[\dot{x}_{2}\right]\\ A_{11}=\left[\begin{array}{ccc}0&0&1\\ 0&0&0\\ -\frac{k_{s}}{m_{u}}&0\end{array}\right]A_{12}=\left[\begin{array}{cc}- \frac{m_{s}}{m_{u}}\\ 1\end{array}\right]A_{21}=\left[\begin{array}{cc}\frac{k_{s}}{m_{s}}&-\frac{ k_{s}}{m_{s}}&\frac{c_{s}}{m_{s}}\end{array}\right]\\ A_{22}=-c_{s}(\frac{1}{m_{u}}+\frac{1}{m_{s}}),B_{2}=-\frac{1}{m_{u}},K=\left[ \begin{array}{cc}0&0&\frac{k_{s}}{m_{u}}\end{array}\right]^{T}\end{array} \]  
The sliding surface can be rewritten according to Equations (16) to (18). \[S(X)=\left[ \begin{array}{cc}N_{1}&N_{2}\end{array} \right]\left[ \begin{array} []{c}\widetilde{X}_{1}\\ \widetilde{X}_{2}\end{array} \right]=N_{1}\widetilde{X}_{1}+N_{2}\widetilde{X}_ {2} \tag{19}\]  
where  
\[N_{1}=\left[ \begin{array}{ccc}n_{1}&n_{2}&n_{3}\end{array} \right],N_{2}= \left[\frac{m_{u}}{m_{u}-m_{s}n_{3}}\right]m_{u}-m_{s}n_{3}\
eq 0\]  
\[\widetilde{X}_{2}=\frac{m_{u}-m_{s}n_{3}}{m_{u}}N_{1}\widetilde{X}_{1} \widetilde{X}_{1}=\left(A_{11}-A_{12}N_{2}^{-1}N_{1}\right)X_{1}\]  
For the purpose of satisfying the Hurwitz criteria to ensure the stability according to the SMC theory, Equation (21), the characteristic polynomial of Equation (20), should ensure the roots are negative. In other words, the value of \(p\) must be greater than zero. As a consequence, the bounds of \(n_{1},n_{2}\), and \(n_{3}\) are confirmed, as shown in Equation (22).