## 3 Stability Based Coordinated Control of Active Suspension and DYC
### Genetic Sliding Mode DYC  
The DYC is able to produce an extra yaw moment by adjusting the longitudinal force of each wheel to improve the yaw's stability. The flow chart of the proposed DYC controller is shown in Figure 3.  
First, according to the 2-DOF model, the ideal vehicle yaw rate \(r_{d}\) is:  
\[r_{d}=\min\left\{\left|\frac{V_{x}}{l(1+K_{u}V_{x}^{2})}\delta\right|\left| \frac{l\text{g}}{V_{x}}\right|\right\}\cdot\text{sgn}(\delta). \tag{22}\]  
where \(l\) is the wheelbase and \(K_{u}\) is the stability factor. The ideal value of the sideslip angle \(\beta_{d}\) is 0. Figure 3: Flow chart of genetic sliding mode DYC. Figure 2: Optimization results of fitness. The sliding mode control is used to track the ideal yaw rate and sideslip angle. The switching function is set as:  
\[s=\frac{w}{\left|\Delta r\right|_{max}}|r_{d}-r|+\frac{1-w}{\left|\Delta\beta \right|_{max}}|\beta_{d}-\beta| \tag{23}\]  
where \(\left|\Delta r\right|_{max}\) and \(\left|\Delta\beta\right|_{max}\) are the maximum allowable yaw rate error and the sideslip angle error, and \(w\) is the weight coefficient ranging from 0 to 1. Equation (23) uses variable normalization and absolute values for the two errors, solving the problem that the dimension of the weight coefficient is meaningless and falsely reaches the switching surface. The derivative of the above expression is:  
\[\dot{s}=\frac{w}{\left|\Delta r\right|_{max}}(\dot{r}_{d}-\dot{r})\text{sgn}( r_{d}-r)+\frac{1-w}{\left|\Delta\beta\right|_{max}}\dot{\beta}\text{sgn}( \beta). \tag{24}\]  
To avoid excessive chattering, the exponential approaching law is taken:  
\[\dot{s}=-\varepsilon\text{sgn}(s)-k_{d}s. \tag{25}\]  
where \(\varepsilon\) and \(k_{d}\) are the approaching law parameters which are both greater than 0. The total yaw moment is:  
\[M_{req}=I_{z}\dot{r}. \tag{26}\]  
The total required yaw moment can be obtained from Equations (24)-(26):  
\[M_{req}=I_{z}\{\dot{r}_{d}+\frac{\left|\Delta r\right|_{max}}{w\text{sgn}(r_ {d}-r)}[\frac{1-w}{\left|\Delta\beta\right|_{max}}\dot{\beta}\text{sgn}(\beta )+\varepsilon\text{sgn}(s)+k_{d}s]\}. \tag{27}\]  
To avoid chattering caused by the sign function, the saturation function is used to replace it. \[\text{sat}(s)=\left\{ \begin{array}{ll}\frac{s}{\tau}&\left|s\right|\leq\tau \\ \text{sgn}(s)&\left|s\right|>\tau\end{array} \right. \tag{28}\]  
where \(\tau\) is the thickness of the boundary layer. In order to further suppress the chattering of the sliding mode control, the genetic algorithm is used to optimize the \(\varepsilon\) and \(k_{d}\) of the approaching law. The fitness function is:  
\[F=\sum_{i=1}^{n}(A_{1}|s|+A_{2}|ds|+A_{3}|M_{req}|). \tag{29}\]  
where \(n\) is the amount of simulation sampling points, and \(A1\) to \(A3\) are the weight coefficients. The required yaw moment can be achieved by the longitudinal forces of four wheels. Additionally, the longitudinal force of the front and rear wheels is distributed according to the proportion of the vertical load of the front and rear wheels. For the computing of the driving/braking torque of the four wheels, we take the slip ratio as the intermediate variable. The target slip ratio is computed by the target longitudinal force. Then, track the target value by controlling the slip ratio to obtain the wheel torque. According to the magic formula, the tire longitudinal force \(F_{xi}\) is affected by the tire slip angle \(\alpha_{i}\), vertical load \(F_{zi}\), tire-road friction coefficient \(\mu\), and slip ratio \(\lambda_{i}\). The relationship is:  
\[F_{xi}=h(F_{zi},\alpha_{i},\lambda_{i},\mu). \tag{30}\]  
If \(F_{zi}\), \(\alpha_{i}\) and \(\mu\) are known, the relationship between \(F_{xi}\) and \(\lambda_{i}\) can be expressed as a univariate function.