## 4 Simulations and Discussion  
In this section, the simulation results of the proposed approach are given. Figure 7 shows the evaluation of the proposed EDS in Figure 3 with a different \(K\) gain. Note that US stands for under-steering, OS represents over-steering, and NS is neutral-steering. Evidently, under \(K\) = 200, the system can achieve neutral-steering, which ensures safe operation. Figure 8 shows all of the performance indexes which concern the evaluation of Figure 7. Figure 8a shows that OS is nimbler and US is more sluggish. OS is suitable for a racecar, and NS is fit for everyday drivers. Figure 8b-d are the longitudinal acceleration, lateral acceleration, and yaw rate of the test, respectively. To sum up, OS is nimble, NS is normal, and US is sluggish. Figure 9 shows the adjusted torque from EDS in these three scenarios. Note that, in the case of \(K\) = 500, the vehicle falls into a state of continuous OS. At 10 s,  
Figure 6: Simulation setup. (**a**) Command of front steering-wheel angle; (**b**) Desired neutral steering circle.  
the vehicle becomes unstable as the torque of the rear left wheel increases and the torque of rear right wheel decreases.  
Figure 8: _Cont._  
Figure 7: EDS test under different \(K\) gain.  
Figure 8: System performance. (**a**) Position of \(X\) coordinate; (**b**) Longitudinal acceleration; (**c**) Lateral acceleration; (**d**) Yaw rate.  
Figure 9: _Cont._  
Note that from the simulation results, if the \(K\) gain is larger than \(K^{*}_{\max}\), the vehicle will respond by obvious OS. This is because over-steering is based on a large speed difference between the left and right wheels. Thus, the torque difference will increase, which leads to an incensement of yaw moment and the side slip angle. It is known that the larger the side slip angle, the bigger the lateral force. However, if the lateral force exceeds its maximum value due to the saturation of the tire's physical conditions, the traction force will decay quickly according to the magic formula of tires. Clearly, because of the loss of traction force, the vehicle slides over the motion surface. Then, the vehicle motion becomes uncontrollable and thus spins. Figure 10 illustrates the analysis diagram of the gain adjusting policy. Under this discovery, adaptively adjusting the \(K\) gain can easily fine-tune motion behaviors of a vehicle based on different steering modes. For example, many racecars have sport and  
Figure 9: Adjusting torque from EDS. (**a**) Front left wheel; (**b**) Front right wheel; (**c**) Rear left wheel; (**d**) Rear right wheel.  
urban driving modes for users to choose. Of course, different steering modes will also result in various fuel economy responses.  
In addition, the performance deteriorations from the larger \(K\) gain can also be found in the analysis of control theory. From the simulations, the proposed approach shows its performance on EDS. However, it also sketches some limitations which should be noted. Figure 11 shows the rearranged control block diagram of the proposed EDS. From this figure, one can obtain the transfer function as  
\[\frac{\omega_{ie}}{V^{*}}=\frac{G(s)K(s)}{1+G(s)K(s)+KG(s)-KM_{T}G(s)} \tag{8}\]  
Figure 11: Control block diagram of proposed scheme.  
Figure 10: Analysis diagram of the gain adjusting policy.  
From Equation (8), one can observe an interesting fact that adequate \(K\) gain can improve the EDS performance. However, too large of a \(K\) gain will lead to an unstable scenario. The analysis concludes the same results discovered from the simulations. Consequently, by using the proposed EDS, the corresponding gain \(K\) should be carefully selected.