## 3 Design of Optimal Distribution Strategy of Damping Force Controller
### Optimal Distribution Strategy of Damping Force
\begin{array}{l}\frac{\partial L_{multiplier}}{\partial F_{D1}}= \frac{-2\alpha_{1}}{(F_{D1}+\beta_{1})^{3}}-\gamma-F_{D1}\text{Sign}(\dot{Z}_ {D1})=0\\ \frac{\partial L_{multiplier}}{\partial F_{D2}}=\frac{-2\alpha_{2}}{(F_{D2}+ \beta_{2})^{3}}-\gamma-F_{D2}\text{Sign}(\dot{Z}_{D2})=0\\ \frac{\partial L_{multiplier}}{\partial F_{D1}}=\frac{-2\alpha_{1}}{(F_{D1}+ \beta_{1})^{3}}+\gamma-F_{Dr1}\text{Sign}(\dot{Z}_{Dr1})=0\\ \frac{\partial L_{multiplier}}{\partial F_{D2}}=\frac{-2\alpha_{2}}{(F_{D2}+ \beta_{2})^{3}}+\gamma-F_{Dr2}\text{Sign}(\dot{Z}_{Dr2})=0\\ \qquad\qquad\qquad\frac{\partial L_{multiplier}}{\partial F_{ij}}=-F_{Dij} \text{Sign}(\dot{Z}_{Dij})=0\\ \qquad\qquad\qquad\frac{\partial L_{multiplier}}{\partial F_{cij}}=2\lambda_{ij }\varepsilon_{ij}=0\end{array} \right. \tag{27}\]  
In one solution, all \(\varepsilon_{ij}\) values that are equal to 0 are excluded because the variable leads to \(F_{dij}=0\), thus conflicting with semi-active control theory. Another solution in which all \(\lambda_{ij}\) are 0 is as follows:  
\[\gamma=\frac{-2\alpha_{lj}}{(F_{dlj}+\beta_{lj})^{3}} \tag{28}\]  
\[\gamma=\frac{2\alpha_{rj}}{(F_{dry}+\beta_{rj})^{3}} \tag{29}\]  
Equations (29) and (30) may not be able to solve for \(\gamma\). Instead, the solution should be  
\[\left\{ \begin{array}{l}\lambda_{lj}=0,\quad\varepsilon_{rj}=0\\ \lambda_{rj}=0,\quad\varepsilon_{lj}=0\end{array} \right. \tag{30}\]  
For the un-tripped rollover investigated in this study, the movement of the left and right sides of the vehicle suspension is always on the contrary, and the damping force on the left and right sides of the suspension is always reversed. The front and rear wheel distance is usually the same or similar, supposing \(B=B_{1}\) because the wheel distance should be determined by the outermost wheel in the anti-rollover control [42]. The range of feasible solutions is shown in Figure 5. The theoretical extreme value of the performance index \(J_{vd}\) exists only when the damping force of the shock absorber is in the range of the feasible solution shown in Figure 5. In addition, the damping force of the shock absorber may be distributed in quadrants I and III. However, for the un-tripped rollover investigated in this research, the damping force is distributed in quadrants II and IV. The constraint equation of Equation (19) can be written as  
\[\left\{ \begin{array}{l}F_{Dlj}\text{Sign}(\dot{\varphi})\leq 0\\ F_{Dr}\text{Sign}(\dot{\varphi})\geq 0\end{array} \right. \tag{31}\]When \(\lambda_{\eta j}=0\), Equations (29) and (31) are substituted into Equation (27):  
\[\frac{\partial L_{multiplier}}{\partial F_{Dlj}}=\frac{-2\alpha_{lj}}{\left(F_{ Dlj}+\beta_{lj}\right)^{3}}+\frac{-2\alpha_{rj}}{\left(F_{D\eta j}+\beta_{rj} \right)^{3}}+F_{Dlj}\text{Sign}(\dot{\varphi})=0 \tag{32}\]  
When \(\lambda_{\eta j}\) and \(\alpha_{\eta j}\) are not negative, then \(\dot{\varphi}>0\). By the same token, when \(\lambda_{lj}=0\), then \(\dot{\varphi}<0\). Substituting Equations (28) and (31) into Equation (27):  
\[\frac{\partial L_{multiplier}}{\partial F_{D\eta j}}=\frac{-2\alpha_{rj}}{ \left(F_{D\eta j}+\beta_{rj}\right)^{3}}+\frac{-2\alpha_{lj}}{\left(F_{D\eta j }+\beta_{jj}\right)^{3}}+F_{D\eta j}\text{Sign}(\dot{\varphi})=0 \tag{33}\]  
The theoretical solution of the optimization problem in Equation (22) can be expressed as  
\[\left\{ \begin{array}{l}F_{Dlj-t}=\frac{-\frac{2\sqrt[3]{\alpha_{l}}}{\dot{ \varphi}}M_{xDn-roll}+\sqrt[3]{\alpha_{lj}}\sum\beta_{lj}-\beta_{lj}\sum\sqrt[ 3]{\alpha_{lj}}}{\sum\sqrt[3]{\alpha_{lj}}}\left[\frac{1-Sign(M_{xDn-roll})}{2} \right]\\ F_{D\eta j-t}=\frac{-\frac{2\sqrt[3]{\alpha_{rj}}}{\dot{\beta}}M_{xDn-roll}+ \sqrt[3]{\alpha_{rj}}\sum\beta_{rj}-\beta_{rj}\sum\sqrt[3]{\alpha_{rj}}}{\sum \sqrt[3]{\alpha_{rj}}}\left[\frac{1+Sign(M_{xDn-roll})}{2}\right]\end{array} \right. \tag{34}\]  
Considering the limitation of the CDC shock absorber, the damper on the other side needs to output a certain compensation damping force when the theoretical solution exceeds the shock absorber limitation.