## 3 Design of an Adaptive Compound Controller for Reference Model
### Adaptive-Layer Controller  
The optimization of vehicle steering angles aims to achieve a balance between stability and energy savings. The adaptive-layer controller is employed to reduce the excessive yaw moment addition in the upper-layer controller, which works in conjunction with the lower-layer controller to achieve energy savings. The vehicle stability is evaluated through a phase plane diagram. In the present analysis, we focus on the vehicle stability evaluation index \(\rho\), which ranges from 0 to 1. The external yaw moment \(M_{z}\) obtained from the upper-layer controller is adjusted based on the evaluation index \(\rho\) to produce the modified external yaw moment \(M_{z}^{\prime}\). We use the side slip angle and its rate of change for phase plane analysis, i.e., \((\beta-\hat{\beta})\), which effectively addresses accuracy issues on low-adhesion road surfaces or with large side slip angles [40; 41]. The fitted boundary of the stability region can be represented as follows:  
\[\lambda_{3}\leq\lambda_{1}\hat{\beta}+\beta\leq\lambda_{2} \tag{37}\]  
where \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) represent the stability region boundary coefficients. Figure 6 illustrates the saddle point positions and boundary curves in the phase plane diagram. Two grey dashed lines represent right and left boundaries, and a blue solid line and a green solid line represent the minimum and maximum yaw rates. The region enclosed by these lines is the stability region. The phase trajectories within the stability region will converge to the equilibrium point. The coefficients of \(\lambda_{1}\), \(\  
The above method defines the boundary of the stability region; however, vehicle stability differs near the node and saddle boundaries. Therefore, an evaluation index \(\rho\) is introduced to quantitatively describe the vehicle's current stability state [42]:  
\[I_{\beta}=1-\text{sign}[(\beta_{\text{max}}-\beta)(\beta-\beta_{\text{min}})] \frac{\min(|\beta_{\text{max}}-\beta|,|\beta-\beta_{\text{min}}|)}{(\beta_{\text {max}}-\beta_{\text{min}})/2} \tag{41}\]  
\[I_{\gamma}=1-\text{sign}[(\gamma_{\text{max}}-\gamma)(\gamma-\gamma_{\text{ min}})]\frac{\min(|\gamma_{\text{max}}-\gamma|,|\gamma-\gamma_{\text{min}}|)}{( \gamma_{\text{max}}-\gamma_{\text{min}})/2} \tag{42}\]  
\[\rho=\left\{ \begin{array}{ll}0,&0\leq I_{\text{actual}}<z/2\\ \frac{1}{2}\{1-\cos[\pi(I_{\text{actual}}-z/2)/(1-z/2)]\},&z/2\leq I_{\text{ actual}}<1\end{array} \right. \tag{43}\]  
where \(\beta_{\text{max}}=\lambda_{2}-\lambda_{1}\hat{\beta}\), \(\beta_{\text{min}}=\lambda_{3}-\lambda_{1}\hat{\beta}\), \(\gamma_{\text{max}}=\mu\text{g}/v_{x}\), and \(\gamma_{\text{min}}=-\mu\text{g}/v_{x}\); \(z\) is the adaptive coefficient; \(I_{\beta}\) is used to describe the level of the side slip angle; \(I_{\gamma}\) is used to describe the level of the yaw rate; and \(I_{\beta}\) and \(I_{\gamma}\) are indices ranging from 0 to 1, determined by the shortest distance from the vehicle's current state to the upper and lower boundaries of the stability region. If we define \(I_{\text{actual}}=\max\{I_{\beta},I_{\gamma}\}\), then \(I_{\beta}\) and \(I_{\gamma}\) are unified to obtain the evaluation index \(\rho\) in Equation (43). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(p_{1}\) & \(-0.0005\) \\ \(p_{2}\) & \(0.0372\) \\ \(p_{3}\) & \(-1.1254\) \\ \(p_{4}\) & \(19.1114\) \\ \(p_{5}\) & \(0.1079\) \\ \(p_{6}\) & \(0.0001\) \\ \(p_{7}\) & \(79.9997\) \\ \(p_{8}\) & \(-1.8113\) \\ \(p_{9}\) & \(0.7331\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the self-stable boundary. Figure 6: Phase plane analysis with the simulation parameters \(v_{x}=72\) km/h, \(\mu=0.85\), and \(\delta=0\) rad. The red circle represents the saddle point, the two grey dashed lines represent the left and right stability boundaries, and a blue solid line and a green solid line represent minimum and maximum yaw rates, respectively. By analyzing the phase plane diagram, the vehicle's stability level can be determined.