# Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes
## 4 Yaw Moment Control Design
### LPV-LMI Yav Moment Control Design with Input Constraint  
The state feedback control for the system (37) is proposed as  
\[M_{z}=\mathbf{Ke},\ \mathbf{K}=\mathbf{YQ}^{-1} \tag{51}\]  
where the matrices \(\mathbf{Y}\) and \(\mathbf{Q}\) satisfy the following LMI conditions  
\[\mathbf{Q}>\mathbf{0} \tag{52a}\] \[ \begin{bmatrix}\mathbf{Q}\mathbf{A}_{vl}^{T}+\mathbf{A}_{vi} \mathbf{Q}+(\alpha_{c}+\mu_{c})\mathbf{Q}+\mathbf{Y}^{T}\mathbf{B}_{m}^{T}+ \mathbf{B}_{m}\mathbf{Y}&\mathbf{B}_{vi}&\mathbf{B}_{m}\\ \mathbf{B}_{vi}^{T}&-\alpha_{c}\rho_{\varsigma}^{-2}&0\\ \mathbf{B}_{m}^{T}&0&-\alpha_{c}\rho_{\varsigma}^{-2}\end{bmatrix} <\mathbf{0},\ i=1,2,3\] (52b) \[ \begin{bmatrix}-\mathbf{Q}&\mathbf{Y}^{T}\\ \mathbf{Y}&-(\widetilde{M}_{z,allow}/g_{c})^{2}\mathbf{I}\end{bmatrix} < \mathbf{0}\] (52c) \[ \begin{bmatrix}-\mathbf{Q}&\mathbf{Q}\\ \mathbf{Q}&-\gamma_{c}^{2}\mathbf{I}\end{bmatrix} <\mathbf{0} \tag{52d}\]  
In (52), the matrices \(\mathbf{A}_{vi},\ \mathbf{B}_{vi},\ \mathbf{B}_{wi},\ i=1,2,3\), and \(\widetilde{M}_{z,allow}\) are defined in (50) and (44), respectively; the parameters \(\alpha_{c}>0,\ \mu_{c}>0,\ \gamma_{c}>0,\ g_{c}>1\) are adjustable via trial and error. To reduce the computation load in emergency situations, the feedback gain \(\mathbf{K}\) in (51) is chosen as constant. For the application of on-line gain scheduling methods, interested readers are referred to the literature (see [6] and references therein). Substituting the feedback control (51) into the system (37) yields the closed-loop error system  
\[\dot{\mathbf{e}}=(\mathbf{A}_{v}(V_{0})+\mathbf{B}_{m}\mathbf{K})\mathbf{e }+\mathbf{B}_{w}(V_{0})\mathbf{w},\ V_{0}\in[V_{\min},V_{\max}] \tag{53a}\] \[\mathbf{w}^{T}\mathbf{Hw}\leq 1,\ \mathbf{H}=diag(\rho_{\varsigma}^{-2}, \rho_{\varsigma}^{-2}) \tag{53b}\]  
Let \(V(\mathbf{e})=\mathbf{e}^{T}\mathbf{Pe},\ \mathbf{P}=\mathbf{Q}^{-1}>\mathbf{0}\) be the Lyapunov function for the system (53). The Lyapunov derivative along the trajectories of (53) can be derived as  
\[\dot{V}(\mathbf{e})=\dot{\mathbf{e}}^{T}\mathbf{Pe}+\mathbf{e}^{ T}\mathbf{P}\dot{\mathbf{e}}\\ =\mathbf{e}^{T}(\mathbf{A}_{v}^{T}(V_{0})\mathbf{P}+\mathbf{P} \mathbf{A}_{v}(V_{0})+\mathbf{K}^{T}\mathbf{B}_{m}^{T}\mathbf{P}+\mathbf{PB} _{m}\mathbf{K})\mathbf{e}+\mathbf{e}^{T}\mathbf{PB}_{w}(V_{0})\mathbf{w}+ \mathbf{w}^{T}\mathbf{B}_{w}^{T}(V_{0})\mathbf{Pe}, \tag{54}\] \[\forall V_{0}\in[V_{\min},V_{\max}]\]  
As shown in Figure 5, we define the stability regions: \(L_{V}(1)=\big{\{}\,\mathbf{e}\in\Re^{2}\big{|}\mathbf{e}^{T}\mathbf{Pe}\leq 1 \big{\}}\), \(\Omega_{ball}(\gamma_{c})=\big{\{}\,\mathbf{e}\in\Re^{2}\big{|}\mathbf{e}^{T} \mathbf{e}\leq\gamma_{c}^{2}\big{\}}\) and \(L_{V}(g_{c})=\big{\{}\,\mathbf{e}\in\Re^{2}\big{|}\mathbf{e}^{T}\mathbf{Pe}\leq \beta_{c}^{2},\ g_{c}>1\big{\}}\) for analysis in Lemma 1 and Theorem 1. By using the Lyapunov derivative (54) and the LMIs (52), we have Lemma 1 below:  
enhances the utilization of the control (51) while working within the input constraint (44). The enhanced scheme is provided as  
\[M_{z}=Sat_{M}(\mathbf{Ke}-\gamma_{H}\mathbf{B}_{m}^{T}\mathbf{Pe}),\ \gamma_{H}>0 \tag{56a}\] \[Sat_{M}(u)= \begin{cases}u,&|u|<\widetilde{M}_{z,allow}\\ \widetilde{M}_{z,allow}&sign(u),&|u|\geq\widetilde{M}_{z,allow},\end{cases}  \tag{56b}\]  
where \(\gamma_{H}>0\) is chosen as sufficiently large. The enhanced scheme (56) can improve the stabilization property of the original control (51) in the region \(L_{V}(g_{c})\) as outlined below.