# Vehicle Control Strategy Evaluation Based on the Driving Stability Region
## 3 The Driving Stability Region for Control Strategy Evaluation
### The Solution of the Driving Stability Region  
For any nonlinear system:  
\[\dot{x}_{i}=f_{i}(x_{1},x_{2},\ldots,x_{n}),\hskip 14.226378 pti=1,2,\ldots,n \tag{12}\]  
Let the initial condition of its state variable be:  
\[t=0:x_{i}(0)=x_{i0} \tag{13}\]  
In the phase plane (or phase space):  
\[\frac{d\dot{x}_{i}}{dx_{i}}=\frac{d\dot{x}_{i}}{dt}/\frac{dx_{i}}{dt} \tag{14}\]  
when the numerator and denominator on the right in Equation (14) are both zero; such a point is called the singular point. The physical meaning of a singular point represents a state of equilibrium in a system; therefore, it is also known as an equilibrium point. The stability of nonlinear system can be judged according to the dynamic characteristics of the system equilibrium points, especially the bifurcation law of the system equilibrium points with the variation of parameters. For a vehicle dynamics system, the equilibrium point is the point where the rate of change of the state variable is simultaneously zero. If a vehicle is regarded as a nonlinear  
Figure 2: The dynamic characteristic diagrams with or without 4 WS (four\(-\)wheel steering) control: (**a**) Phase space trajectory; (**b**) projection of phase space trajectory on \(v_{y}-\omega\) plane; (**c**) system state variables; and (**d**) vehicle operation attitudes. system composed of a body and tires, the driving torque and front wheel steering angle are regarded as the control parameters of the nonlinear system. The dynamic process of losing the stability of the vehicle is a dynamic bifurcation phenomenon of the nonlinear system that changes with the control parameters. The dynamic process of vehicle handling stability meets the definition of nonlinear dynamics as follows. Generally, the nonlinear dynamic control system composed of vehicle body and tires can be expressed in the form of the following differential algebraic equation [28]:  
\[F(x)=\left\{ \begin{array}{l}\dot{x}=f(x,u)\\ y=g(x,u)\\ z=h(x,u)\end{array} \right. \tag{15}\]  
In the equation, \(x\in R^{n}\) is the state variable of the system, such as the longitudinal velocity \(v_{x}\) of the vehicle body, lateral velocity \(v_{y}\), and yaw rate \(\omega\) etc. \(u\in R^{m}\) is the control variable of the system, such as the driving torque \(T_{d}\) and front wheel angle \(\delta_{f}\), etc. \(y\in R^{q}\) is the observed variable of the system, such as the acceleration of each state variable and corresponding changing rate. \(z\in R^{p}\) is the output variable of the system after control, such as various state variables. We assume that \(f,g\), and \(h\) are smoothly differentiable within the defined domain and that \(n\), \(m\), \(q\), and \(p\) are positive integers. When point \((x^{*},u^{*})\) satisfies:  
\[F(x^{*},u^{*})= \begin{bmatrix}f(x^{*},u^{*})\\ g(x^{*},u^{*})\end{bmatrix} =0 \tag{16}\]  
then this point is the equilibrium point of the vehicle dynamics system. In the vehicle nonlinear dynamic system, driving and steering are considered as bifurcation parameters. The set of bifurcation parameters is defined as the set of two-dimensional bifurcation vectors corresponding to the sudden change of phase trajectory, change in the number of equilibrium points, or change in the properties of equilibrium points in the dynamic characteristics of the vehicle system under the coupled effect of driving and steering. The region enclosed by the bifurcation boundary and coordinate axis is called the driving stability region of the vehicle [29]. The bifurcation boundary is determined by the set of bifurcation parameters in the driving torque and steering angle amplitude space. Based on the above definition, the solution method for the driving stability region proposed in this paper is shown in Figure 3[29]. First, the initial longitudinal velocity and front wheel steering angle are selected and input into the 5 DOF vehicle system dynamics model, and the system equilibrium point is solved by using the hybrid algorithm based on the Genetic Algorithm (GA) and Sequential Quadratic Programming (SQP) methods.