## 3 Controller Design
### Model Predictive Controller  
The predicted state vectors in the prediction horizon \(N_{p}\) with the future control commands in the control horizon \(N_{c}\) can be expressed as follows. \[ \begin{array}{c}\mathbf{x}_{k,i+1}=\mathbf{\Phi}_{m}\mathbf{x}_{k,i}+ \mathbf{\Gamma}_{m}\mathbf{u}_{k,i}\\ \mathbf{x}_{k,i+2}=\mathbf{\Phi}_{m}^{2}\mathbf{x}_{k,i}+\mathbf{\Phi}_{m} \mathbf{\Gamma}_{m}\mathbf{u}_{k,i}+\mathbf{\Gamma}_{m}\mathbf{u}_{k,i+1}\\ \vdots\\ \mathbf{x}_{k,i+N_{p}}=\mathbf{\Phi}_{m}^{N_{p}}\mathbf{x}_{k,i}+\mathbf{ \Phi}_{m}^{N_{p}-1}\mathbf{\Gamma}_{m}\mathbf{u}_{k,i}+\cdots+\mathbf{\Phi}_{ m}^{N_{p}-N_{c}}\mathbf{\Gamma}_{m}\mathbf{u}_{k,i+N_{c}-1}\end{array}  \tag{13}\]  
where the initial condition of the state vector \(\mathbf{x}_{k,i}\) can be expressed as follows. \[\mathbf{x}_{k,i}= \begin{bmatrix}0&\vartheta_{y}&0&r&\delta_{f}&\hat{M}_{z} \end{bmatrix} ^{T} \tag{14}\]  
where the initial lateral displacement and heading angle are zero in the body coordinates of the vehicle; \(r\) and \(\delta_{f}\) can be measured using yaw rate and steering angle sensors, respectively; \(\vartheta_{y}\) is the estimated lateral velocity that can be estimated using localization-based [40] \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Value** \\ \hline Vehicle mass (\(m\)) & 1530 \(kg\) \\ Yaw-plane rotational inertia of vehicle (\(l_{z}\)) & 2315 \(kg\)-\(m^{2}\) \\ Distances from the C.G. to the front axes (\(l_{f}\)) & 1.1 \(m\) \\ Wheel base (\(L\)) & 2.78 \(m\) \\ Cornering stiffness of the front axes (\(C_{\alpha\beta}\)) & 150.3 \(k\)N/_nd_ \\ Cornering stiffness of the rear axes (\(C_{\alpha r}\)) & 104.9 \(k\)N/_nd_ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vehicle parameters. Figure 1: Block diagram of the proposed algorithm. or inertial-measurement-unit-based [41] approaches; \(\hat{M}_{z}\) can be estimated using estimated logitudinal tire forces as follows. \[\hat{M}_{z}=\textbf{B}_{CA}\hat{\textbf{F}}_{x} \tag{15}\]  
where \(\textbf{F}_{x}\) is the estimated longitudinal tire force that can be estimated using a closed-loop disturbance estimator [42]; \(\textbf{B}_{CA}\) is the effective matrix as shown below. \[\textbf{B}_{CA}=\left[-\frac{d}{2}cos\delta_{f}+l_{f}sin\delta_{f}\ \ -\frac{d}{2}\ \frac{d}{2} cos\delta_{f}+l_{f}sin\delta_{f}\ \ \ \frac{d}{2}\right] \tag{16}\]  
where \(d\) is the track width of the vehicle. The predicted output vectors in the prediction horizon \(N_{p}\) can be expressed as follows. \[ \begin{array}{c}\textbf{y}_{p,k,i+1}=\textbf{H}_{m}\boldsymbol{\Phi}_{m} \textbf{x}_{k,i}+\textbf{H}_{m}\boldsymbol{\Gamma}_{m}\textbf{u}_{k,j}\\ \textbf{y}_{p,k,i+2}=\textbf{H}_{m}\boldsymbol{\Phi}_{m}^{N}\textbf{x}_{k,i}+ \textbf{H}_{m}\boldsymbol{\Phi}_{m}\boldsymbol{\Gamma}_{m}\textbf{u}_{k,i}+ \textbf{H}_{m}\boldsymbol{\Gamma}_{m}\textbf{u}_{k,i+1}\\ \vdots\\ \textbf{y}_{p,k,i+N_{p}}=\textbf{H}_{m}\boldsymbol{\Phi}_{m}^{N_{p}}\textbf{x} _{k,i}+\textbf{H}_{m}\boldsymbol{\Phi}_{m}^{N_{p}-1}\boldsymbol{\Gamma}_{m} \textbf{u}_{k,i}+\cdots+\textbf{H}_{m}\boldsymbol{\Phi}_{m}^{N_{p}-N_{c}} \boldsymbol{\Gamma}_{m}\textbf{u}_{k,i+N_{c}-1}\end{array}  \tag{17}\]  
From Equations (11) and (15), the augmented prediction model can be formulated as follows. \[\textbf{Y}_{p}=\textbf{F}\textbf{x}_{k,i}+\textbf{GU}_{p} \tag{18}\]  
where \(\textbf{U}_{p}\) is the input vector; \(\textbf{Y}_{p}\) is the output vector; **F** and **G** are the system and input matrices, respectively. They are defined as follows. \[\textbf{U}_{p}=\left[\textbf{u}_{k,i}^{T}\ \ \ \textbf{u}_{k,i+1}^{T}\ \ \ \cdots\ \ \ \textbf{u}_{k,i+N_{c}-1}^{T}\right]^{T} \tag{19}\]  
\[\textbf{Y}_{p}=\left[\textbf{Y}_{p,k,i+1}^{T}\ \ \textbf{y}_{p,k,i+2}^{T}\ \ \ \cdots\ \ \ \textbf{y}_{p,k,i+N_{p}}^{T}\right]^{T} \tag{20}\]  
\[\textbf{F}=\left[\textbf{H}_{m}\boldsymbol{\Phi}_{m}\ \ \ \textbf{H}_{m} \boldsymbol{\Phi}_{m}^{2}\ \ \ \cdots\ \ \ \textbf{H}_{m}\boldsymbol{\Phi}_{m}^{N_{p}}\right]^{T} \tag{21}\]  
\[\textbf{G}=\left[