# Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes
## 3 Control System Configuration
### Control System with Active Front Steering Saturation  
Let \(\mathbf{e}_{v}=\mathbf{x}_{p}-\mathbf{x}_{ref}\) be the error between the vehicle state and reference state. The speed-variable-dependent dynamical error equation \(\mathbf{e}_{v}\) can be obtained from (1) as  
\[\dot{\mathbf{e}}_{v}=\dot{\mathbf{x}}_{v}-\dot{\mathbf{x}}_{ref}=\mathbf{A}_{v }(V_{0})\mathbf{e}_{v}+\mathbf{B}_{m}M_{z}+\mathbf{A}_{v}(V_{0})\mathbf{x}_{ ref}+\mathbf{B}_{v}(V_{0})\delta_{f}+\mathbf{d}_{v}-\dot{\mathbf{x}}_{ref},\ V_{0}\in[V_{\min},V_{\max}]\]  
Substituting (31) into the equation above yields  
\[\dot{\mathbf{e}}_{v}=\mathbf{A}_{v}(V_{0})\mathbf{e}_{v}+\mathbf{B}_{m}M_{z}+ \overline{\mathbf{d}}_{v}(t),\ V_{0}\in[V_{\min},V_{\max}] \tag{32}\]  
where  
\[\overline{\mathbf{d}}_{v}(t)=\mathbf{B}_{v}(V_{0})v+\mathbf{d}_{v}-\dot{ \mathbf{x}}_{ref} \tag{33a}\] \[|v|=\left|\delta_{f}-\delta_{d,ref}\right|\to 0 \tag{33b}\]  
If the yaw moment \(M_{z}\) is used alone, i.e., \(\delta_{sat}=0\) and \(\delta_{f}=\delta_{d}\), then the disturbance \(v\) in (33) becomes \(v=\delta_{d}-\delta_{d,ref}\) with  
\[|v|=\left|\delta_{d}-\delta_{d,ref}\right|=\left\{ \begin{array}{cl}0&,if\ | \delta_{d}|\leq\widetilde{\delta}_{\lim}\\ |\delta_{d}|-\widetilde{\delta}_{\lim}&,if\ |\delta_{d}|>\widetilde{\delta}_{\lim} \end{array} \right. \tag{34}\]  
Under the condition (34), when the driver steering angle \(\delta_{d}\) surpasses the threshold, i.e., \(|\delta_{d}|>\widetilde{\delta}_{\lim}\), the related disturbance magnitude \(|v|\) begins to grow in proportion to the difference \(|\delta_{d}|-\widetilde{\delta}_{\lim}\), and this can lead to tire/road friction saturating if the driver steering angle \(\delta_{d}\) becomes large enough. In this case of excessive front-wheel steering, the stabilizing yaw moment \(M_{z}\) in (32) operates in the tire/road friction saturation zone with its capacity almost used in suppressing the instability effect of the difference \(|\delta_{d}|-\widetilde{\delta}_{\lim}\). In contrast, if the auxiliary saturation angle \(\delta_{sat}\) is used to limit the wheel steering angle \(\delta_{f}\) in the presence of \(|\delta_{d}|>\widetilde{\delta}_{\lim}\), then the resulting disturbance \(v\) in (33) can be effectively reduced and the tire/road friction can remain within the linear region. This strategy also frees up more capacity of the yaw moment \(M_{z}\) to carry out the reference state tracking and mitigate other potential instability effects as shown in (32) and (33). It is possible to represent the total uncertain disturbance \(\overline{\mathbf{d}}_{v}(t)\) in (32) by using the full-column-rank property of the augmented matrix \(\ [\mathbf{B}_{v}(V_{0})\mathbf{B}_{m}]\) where \(\mathbf{B}_{v}(V_{0})\) and \(\mathbf{B}_{m}\) are the column vectors related to the front-wheel steering angle \(\delta_{f}\) and yaw moment \(M_{z}\), respectively. Based on the concept of Lyapunov redesign and the structure of the system input matrix \(\mathbf{B}_{m}\), the uncertain disturbance \(\overline{\mathbf{d}}_{v}(t)\) can be decomposed into two parts [17; 18; 19], namely, a matching condition part \(\mathbf{B}_{m}\xi(t)\) and an unmatching condition part \(\mathbf{B}_{v}(V_{0})\xi(t)\) described as follows:  
\[\mathbf{B}_{v}(V_{0})v+\mathbf{d}_{v}-\dot{\mathbf{x}}_{ref}=\overline{ \mathbf{d}}_{v}(t)=\underbrace{\mathbf{B}_{v}(V_{0})\xi(t)}_{\text{unmatching condition part}}+\underbrace{\mathbf{B}_{m}\xi(t)}_{\text{matching condition part}} \tag{35}\]  
The unmatching condition part \(\mathbf{B}_{v}(V_{0})\varsigma\) on the right side of (35) represents a disturbance in the direction of \(\mathbf{B}_{v}(V_{0})\) with an uncertain magnitude \(\varsigma\). This magnitude is primarily caused by the term \(v\) on the left side of (35), which arises from the wheel steering angle as described in (33) or (34), and is expected to be on the order of \(10^{-2}\) radians (thesame order with the wheel steering angle \(\delta_{f}\)).