# 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
With the enhanced scheme (56), the system (37) turns into  
\[\dot{\mathbf{e}}=\mathbf{A}_{v}(V_{0})\mathbf{e}+\mathbf{B}_{m}Sat_{M}( \mathbf{Ke}-\gamma_{H}\mathbf{B}_{m}^{T}\mathbf{Pe})+\mathbf{B}_{w}(V_{0}) \mathbf{w},\ \gamma_{H}>0,\ V_{0}\in[V_{\text{min}},V_{\text{max}}] \tag{57a}\] \[\mathbf{w}^{T}\mathbf{Hw}\leq 1,\ \mathbf{H}=diag(\rho_{\xi}^{-2}\rho_{\xi}^{ -2}) \tag{57b}\]  
Due to the fact of Item (II) in Lemma 1, there exits \(\widetilde{\gamma}_{H}(\mathbf{e})\geq 0\) depending on the state \(\mathbf{e}\) such that  
\[\forall\mathbf{e}\in L_{V}(g_{c})\Rightarrow Sat_{M}(\mathbf{Ke}-\gamma_{H} \mathbf{B}_{m}^{T}\mathbf{Pe})=\mathbf{Ke}-\widetilde{\gamma}_{H}(\mathbf{e} )\mathbf{B}_{m}^{T}\mathbf{Pe},\ \widetilde{\gamma}_{H}(\mathbf{e})\geq 0 \tag{58}\]  
Choose the Lyapunov function \(V(\mathbf{e})=\mathbf{e}^{T}\mathbf{Pe}\), \(\mathbf{P}=\mathbf{Q}^{-1}>\mathbf{0}\) in Lemma 1 for the state feedback system (57). The Lyapunov derivative along the trajectories of (57) is obtained as  
\[\dot{V}(\mathbf{e})=\dot{\mathbf{e}}^{T}\mathbf{Pe}+\mathbf{e}^{T} \mathbf{P}\dot{\mathbf{e}}\\ =\mathbf{e}^{T}(\mathbf{A}_{v}^{T}(V_{0})\mathbf{P}+\mathbf{PA}_{ v}(V_{0}))\mathbf{e}+Sat_{M}(\mathbf{Ke}-\gamma_{H}\mathbf{B}_{m}^{T}\mathbf{Pe})^{T} \mathbf{B}_{m}^{T}\mathbf{Pe}+\mathbf{e}^{T}\mathbf{PB}_{m}Sat_{M}(\mathbf{Ke} -\gamma_{H}\mathbf{B}_{m}^{T}\mathbf{Pe})+\mathbf{e}^{T}\mathbf{PB}_{m}(V_{0} )\mathbf{w}+\mathbf{w}^{T}\mathbf{B}_{w}^{T}(V_{0})\mathbf{Pe},\\ \forall V_{0}\in[V_{\text{min}},V_{\text{max}}]\]  
By using (58), the Lyapunov derivative (59) has the property  
\[\forall\mathbf{e}\in L_{V}(g_{c})\Rightarrow\dot{V}(\mathbf{e})= \mathbf{e}^{T}(\mathbf{A}_{v}^{T}(V_{0})\mathbf{P}+\mathbf{PA}_{v}(V_{0})+ \mathbf{K}^{T}\mathbf{B}_{m}^{T}\mathbf{P}+\mathbf{PB}_{m}\mathbf{K})\mathbf{e }+\mathbf{e}^{T}\mathbf{PB}_{w}(V_{0})\mathbf{w}+\mathbf{w}^{T}\mathbf{B}_{w}^ {T}(V_{0})\mathbf{Pe} \tag{60}\] \[-2\widetilde{\gamma}_{H}(\mathbf{e})\mathbf{e}^{T}\mathbf{PB}_{m} \mathbf{B}_{m}^{T}\mathbf{Pe},\ \forall V_{0}\in[V_{\text{min}},V_{\text{max}}]\]  
**Theorem 1**. : _Consider the state feedback system (57). The Lyapunov derivative (60) can be estimated as_  
\[\textbf{(I)}\ \dot{V}(\mathbf{e})\leq-\mu_{c}V(\mathbf{e})-2 \widetilde{\gamma}_{H}(\mathbf{e})\mathbf{e}^{T}\mathbf{PB}_{m}\mathbf{B}_{m}^ {T}\mathbf{Pe}\leq-\mu_{c}V(\mathbf{e}),\ \forall V_{0}\in[V_{\text{min}},V_{\text{max}}],\ \text{if}\\ 1\leq\mathbf{e}^{T}\mathbf{Pe}\leq g_{c}\] (61a) _Moreover,_ \[\textbf{(II)}\ |M_{z}|=\left|Sat_{M}(\mathbf{Ke}-\gamma_{H}\mathbf{B}_{m}^ {T}\mathbf{Pe})\right|\leq\widetilde{M}_{z,allow},\ \forall\mathbf{e}\in\Re^{2}\] \[\textbf{(III)}\ \underset{\mathbf{e}\in L_{V}(1)}{Max}\mathbf{e}^{T} \mathbf{e}<\gamma_{c}^{2}\] (61b) _Accordingly,_ \[\textbf{(a)}\ \  
\[1\leq\mathbf{e}^{T}\mathbf{\mathrm{Pe}}\Rightarrow\mathbf{e}^{T}(\mathbf{A}_{v}^{T}(V_{0})\mathbf{ \mathrm{P}}+\mathbf{\mathrm{PA}}_{v}(V_{0})+\mathbf{K}^{T}\mathbf{\mathrm{B}}_{m}^{T}\mathbf{ \mathrm{P}}+\mathbf{\mathrm{PB}}_{m}\mathbf{\mathrm{K}})\mathbf{\mathrm{e}}+\mathbf{e}^{T}\mathbf{ \mathrm{PB}}_{m}(V_{0})\mathbf{\mathrm{w}}+\mathbf{w}^{T}\mathbf{\mathrm{B}}_{m}^{T}(V_{0} )\mathbf{\mathrm{Pe}}\leq-\mu_{c}\mathbf{\mathrm{e}}^{T}\mathbf{\mathrm{Pe}},\ \forall V_{0}\in[V_{\min},V_{\max}] \tag{62}\]  
Thus, by combining the properties in (60) and (62), the Lyapunov derivative (59) can be estimated as  
\[\mathbf{w}^{T}\mathbf{Hw}\leq 1,\ \mathbf{H}=diag(\rho_{\xi}^{-2}\rho_{\xi}^{ -2}) \tag{57b}\]0  
Item (II): The result can be immediately obtained by the definition in (56). Item (III): Same as in the proof of Lemma 1. The property provided in Item (I) and Item (II) leads to the results of Item (a) and Item (b). The result of Item (c) comes from the property in Item (III). \(\square\)  
In emergency situations, the enhanced control (56) can make the system response faster. By calculating the enhanced control (56), we obtain the longitudinal slip magnitude \(|s_{L}|\) in (43), and proceed to calculate the control torque of each wheel using (40).