## 3 Design of Multi-Objective Fuzzy Continuous Damping Controller
### Multi-Objective Fuzzy Continuous Damping Controller
#### 3.2.2 Fuzzy Rule Reasoning Process  
The triangular membership function is easy to operate on the premise of meeting the accuracy requirements and it is more conducive to simulation; the language variables are almost all triangular membership functions (NB and PB of the yaw angle rate are trapezoidal membership functions).  
The expression of triangular membership function is  
\[f(x,a_{1},a_{2},a_{3})=\left\{\begin{array}{cc}0,&x\leq a_{1}\\ \dfrac{x-a_{1}}{a_{2}-a_{1}},&a_{1}<x\leq a_{2}\\ \dfrac{d_{3}-x}{a_{3}-a_{3}},&a_{2}<x<a_{3}\\ 0,&x\geq a_{3}\end{array}\right. \tag{31}\]  
where \(a_{2}\) is the vertex coordinate value of the triangle; \(a_{1}\) and \(a_{3}\) are the intersection coordinates of the left and right sides of the triangle and the bottom edge, respectively.  
The input and output variables and the rules of fuzzy control are shown in Figure 8.  
The basic idea of multi-objective fuzzy continuous damping control is described as follows. Under good road conditions (RI and yaw rate values are extremely small), the additional damping force generated by fuzzy control should be as small as possible to prevent any impact on sky-hook control. When the RI reaches a relatively large degree, the compressed and damping reaction should be increased to prevent excessive roll moment and thereby produce a damping force with a compensation effect to resist rollover. When the RI value is relatively small and the yaw rate is large, the prevention of instability should be the top consideration, and the ground adhesion should be increased by boosting the damping force of relevant wheels so as to aid the ESP in producing satisfactory effects. When the RI and yaw rate are large but show opposite effects (this state is rare and is generally caused by tripped rollover), a moderate damping force can be considered to build a certain complementary relationship between stability and rollover.  
The fuzzy implication relation that consists of a series of \"if... then... \" is composed of linguistic fuzzy rules, which can be converted into tabular fuzzy rules [46]. The tabular fuzzy rules are intuitive, simple, and easy to check, which are often used in the design of fuzzy controllers. The fuzzy rule  
Figure 8: Multi-objective fuzzy continuous damping control model. (**a–c**) are diagrams of the membership functions of the inputs RI and yaw rate and the output damping reference coefficient, respectively. (**d–g**) are the diagrams of the rule surface of each wheel (CL1/CR1/CL2/CR2: damping reference coefficient of front left/front right/near left/rear right wheel, respectively).  
function of fuzzy set and the center of area surrounded by abscissa is centroid method [42]. The solution \(x_{0}\) and the mathematical expression of centroid method is as follows:  
\[x_{0}=\frac{\int xf(x,a_{1},a_{2},a_{3})dx}{\int f(x,a_{1},a_{2},a_{3})dx} \tag{32}\]  
The quantization factors are defined as \(N_{u}=1\). The actual damping forces of suspension are calculated by the range of CDC and the damping reference coefficients of fuzzy control.