## 3 Design of Optimal Distribution Strategy of Damping Force Controller
### Optimal Distribution Strategy of Damping Force  
According to the relevant theories of LTR, the ideal anti-rollover damping moment can be expressed as  
\[M_{xDn-roll}=\frac{B_{1}}{2}(F_{D1}-F_{Dr1})+\frac{B_{2}}{2}(F_{Dl2}-F_{Dr2})=-F _{y}h=-M_{x} \tag{16}\]  
where \(M_{xDn-roll}\) is the anti-rollover moment provided by the damping force. \(F_{Dij}\) is the damping force of each wheel. \(M_{x}\) is the roll moment around the X-axis, which can be solved by the vehicle dynamic model constructed by Equations (1)-(8). Considering the theory of semi-active suspension, the power of the shock absorber must be positive. CDC can only convert mechanical energy into thermal energy without the injection of external energy, that is,  
\[P_{Dij-roll}= \begin{cases}F_{Dij}(w-Z_{uij}^{\shortdot})=C_{uij-roll}w(w-Z_{uij} ^{\shortdot}),w(w-Z_{uij}^{\shortdot})>0\\ 0,w(w-Z_{uij}^{\shortdot})\leq 0\end{cases}  \tag{17}\]  
\[F_{Dij}\text{Sign}(Z_{Dij}^{\shortdot})\geq 0 \tag{18}\]  
where the function \(\text{Sign}(x)\) is described as:  
\[\text{Sign}(x)= \begin{cases}1,\ x>0\\ 0,\ x=0\\ -1,\ x<0\end{cases}  \tag{19}\]  
\[\dot{Z_{Dij}^{\shortdot}}=w-Z_{uij}^{\shortdot} \tag{20}\]  
where \(Z_{Dij}\) is the displacement of the shock absorber of each wheel; \(x\) is a variable; \(P_{D\text{-}roll}\) is the power of anti-rollover shock absorber. \(C_{n\text{-}roll}\) is the damping coefficient of the anti-rollover shock absorber. In this paper, the damping force is a free variable to be distributed. Only when the vertical forces are optimized, the distribution result of vertical forces is better, but the damping force are greatly different. Large suspension dynamic displacements caused by damping force control will affect vehicle stability. Thus, the vertical load difference of each wheel between the dynamic and static condition should not be too large to avoid the large wheel dynamic load caused by the semi-active suspension control, which worsens the vehicle handling stability [40; 41]. The optimized performance index of the vertical dynamic load is defined as:  
\[J_{xd}=t_{l1}(\frac{F_{vl10}}{F_{vl1}})^{2}+t_{l2}(\frac{F_{vl20}}{F_{vl2}})^ {2}+t_{r1}(\frac{F_{v10}}{F_{vv1}})^{2}+t_{r2}(\frac{F_{v20}}{F_{vv2}})^{2} \tag{21}\]  
where \(t\) is the weighting factor; \(F_{vij0}\) is the static vertical force of each wheel. The constrained optimization problem can be defined as:  
\[\underset{F_{Dij}}{\text{min}}J_{xd}(M_{xDn-roll},\dot{Z_{Dij}},F_{Dij}) \tag{22}\]  
Equation (19) can be simplified by using the defined variables, \(\alpha_{ij}\) and \(\beta_{ij}\), as  
\[J_{xd}=\frac{\alpha_{l1}}{\left(F_{D1}+\beta_{l1}\right)^{2}}+\frac{\alpha_{ l2}}{\left(F_{D2}+\beta_{l2}\right)^{2}}+\frac{\alpha_{r1}}{\left(F_{Dr1}+ \beta_{r1}\right)^{2}}+\frac{\alpha_{r2}}{\left(F_{Dr2}+\beta_{r2}\right)^{2}} \tag{23}\]  
\[\alpha_{ij}=tF_{vij0}^{2} \tag{24}\]  
\[\beta_{ij}=F_{vij}-F_{Dij} \tag{25}\]The Lagrange multiplier method is used to solve the optimization problem in Equation (22). The optimization objective function considering the constraints of Equations (16) and (18) is constructed as  
\[ \begin{array}{l}L_{multiplier}=J_{vd}+\gamma(M_{xDn-roll}-\frac{B_{1}}{2}(F_{D 1}-F_{Dr1})-\frac{B_{2}}{2}(F_{Dl2}-F_{Dr2}))\\ \qquad\qquad\qquad\qquad\qquad-\sum\lambda_{ij}(F_{Dij}\text{Sign}(Z_{Dij})- \varepsilon_{ij}{}^{2})\end{array}  \tag{26}\]  
where \(\gamma\) and \(\varepsilon_{ij}\) are arbitrary real numbers; \(\lambda_{ij}\) represent nonnegative real numbers. When the objective variable, \(J_{vd}\), takes the extreme value, the partial derivative of each parameter is equal to 0. The partial differential equation can be written as  
\[\left\{