## 3 Design of Trajectory Tracking Control System
### Vehicle Instability Judgment and Proportional Controller Allocation Based on the Phase Plane Method  
The longitudinal speed, the adhesion coefficient of the ground, and the front wheel angle affect the vehicle's stability during the driving process. The phase plane method is crucial to judge whether the system is stable when these three parameters change. The \(\beta-\hat{\beta}\) phase plane of the vehicle can identify its unsteady condition. The second-order autonomous system shown in Equation (29) is the basis for realizing the phase plane. \[\left\{ \begin{array}{l}\dot{z_{1}}=f_{1}(z_{1},z_{2})\\ \dot{z_{2}}=f_{2}(z_{1},z_{2})\end{array} \right. \tag{29}\]  
The slope of each point on the phase diagram trajectory can be obtained from Equation (30). \[\frac{dz_{2}}{dz_{1}}=\frac{dz_{2}/dt}{dz_{1}/dt}=\frac{f_{2}(z_{1},z_{2})}{f_ {1}(z_{1},z_{2})} \tag{30}\]  
where \(z_{1}\) and \(z_{2}\) are the horizontal and vertical coordinates of the phase diagram, respectively. In Equation (30), given the initial state values \(z_{0}(z_{1}(0),z_{2}(0))\), for any time t, the solution \(z(t)=(z_{1}(t),z_{2}(t))\) on the system of state equations is a phase trajectory starting at the initial point \(z_{0}\). In a time-continuous system, the initial value \(z_{0}\) is in the local range, such that the phase trajectory satisfies the following:  
\[\lim_{t\rightarrow\infty}z(t)=z_{e} \tag{31}\]  
At this point, the total energy of the system decays, and the system is in an asymptotically stable state. If there is no external intervention, the system's momentum gradually reduces to zero, the system is stationary, and the state point of convergence is called the equilibrium point. In the phase plane diagram, the phase trajectories in the stable region will converge to the equilibrium point, and the unstable phase trajectories will diverge. Equations (1) and (2) are expressed as second-order autonomous systems, as follows:  
\[\left\{ \begin{array}{l}\dot{\beta}=f_{1}(\beta,\omega)\\ \dot{\omega}=f_{2}(\beta,\omega)\end{array} \right. \tag{32}\]  
Under the given longitudinal velocity, the adhesion coefficient of the ground, and the front wheel angle, different initial values are assigned to Equation (32), the phase trajectory of the system is drawn, and the \(\beta-\hat{\beta}\) phase plan is obtained. Figure 5 shows the five-parameter diamond phase plan under the condition of \(\
u_{\text{x}}=60\) km/h, \(\mu=0.8\), and \(\delta_{\text{f}}=0\). The five parameter values of the diamond region are the \(\hat{\beta}\) values of the upper and lower boundaries, the \(\beta\) values of the left and right boundaries, and the \(\beta\) value of the equilibrium point. We denote such by \(\dot{\beta}_{\text{lim}}^{+}\), \(\dot{\beta}_{\text{lim}}^{-}\), \(\beta_{\text{lim}}^{+}\), \(\beta_{\text{lim}}\), and \(\beta_{\text{e}}\), respectively. The step size was selected under the given range of working conditions shown in Table 2, and the simulation, as shown in Figure 5, was carried out in order to establish a relatively complete table look-up database of five parameter values. In addition, the boundary equations of the stable region under each working condition were obtained. Table 3 shows some of the database parameter values obtained with the simulation. In the \(\beta-\dot{\beta}\) phase plan, the shortest distance between the vehicle state point in the stability domain and the boundary of the stability domain is defined as the stability degree. The degree of stability can characterize the degree of stability of the vehicle; moreover, the state points outside of the boundary of the stability domain are already in an unstable state, and their stability is 0. The calculation model of stability degree is as follows:  
\[S_{\beta}=\left\{ \begin{array}{l}0,B_{i}\leq\dot{\beta}+A_{i}\beta(i=1,2),B_ {i}\geq\dot{\beta}+A_{i}\beta(i=3,4)\\ \min\biggl{[}\Big{|}\frac{\dot{\beta}+A_{i}\beta-B_{i}}{\sqrt{A_{i}^{2}+1}} \Big{]}(i=1,2,3,4),\\ B_{i}\geq\dot{\beta}+A_{i}\beta(i=1,2),B_{i}\leq\dot{\beta}+A_{i}\beta(i=3,4) \end{array} \right.