## 2 Control Strategies  
Controlling the lateral behavior of automotive vehicles is fundamental to achieving higher safety and comfort levels for passengers. Typically, Electronic Stability Controls (ESCs) act on brake actuators to prevent undesired operative conditions, such as under-over steering and rollover. The recent availability of BW systems opens the way to the adoption of more sophisticated control policies, able to exploit new possibilities. Indeed, BBW actuators consent to the application of different braking torques on each wheel, increasing the overall yaw moment which can be applied to the vehicle body. In addition, SBW can be easily interfaced with a high-level Control Unit (CU), introducing additional steering angle and assisting the driver during dangerous and risky maneuvers. In the next subsections, the proposed model-based control methodologies are described in detail, which rely on the LQR theory and SMC approach. Both strategies have in common the control objective definition, which is explained here. **Assumption**  
A single-track model with 3-DoF in longitudinal \(x\), lateral \(y\), and yaw \(\gamma\) motion directions is considered. \(F_{xi}\) and \(F_{yi}\) are the longitudinal and lateral forces generated in the tire-road contact patch; \(\delta\) is the front wheel steering angles; \(V_{x}\) and \(V_{y}\) are longitudinal and lateral speed; \(\beta\) is the side-slip angle; \(a\) and \(b\) are the front and rear wheelbase. In all the equations, the same convention is adopted in which \(i\in\{f,r\}\) stands for front or rear axle, respectively. The controllers are based on the following assumptions, fundamental to ensure the validity of the adopted modeling approach, both concerning dynamics and kinematics variables:  
* Steering angle is small: \(\delta<10\) deg;
* Vehicle longitudinal speed is constant: \(V_{x}=cost\);
* Tire slip angles are small: \(\alpha_{i}<<1\) deg;
* Tire cornering stiffness is known and constant: \(C_{yf}=const;C_{yr}=const\). System (1) represents conditions \((a)\) and \((b)\), leading to (2). \[ \begin{cases}\beta=\frac{V_{y}}{V_{x}}=\frac{\dot{y}}{V_{x}}\\ \dot{y}=\quad V_{x}\cdot\dot{\beta}\end{cases} \Rightarrow\beta=\frac{V_{y}}{V_ {x}}=\frac{\dot{y}}{V_{x}} \tag{1}\]  
\[a_{y}=\ddot{y}+V_{x}\cdot\dot{\gamma}=\ddot{y}+V_{x}\cdot r=V_{x}\cdot(\frac{d \beta}{dt}+\dot{\gamma}) \tag{2}\]  
Small tire slip angle \((c)\) and constant cornering stiffness \((d)\), instead, are abstracted through (3) and (4), assuming a linear behavior of the wheel's lateral forces with respect to side-slip angles \(\alpha_{i}\). \[F_{yi}=2C_{\alpha_{i}}\cdot\alpha_{i} \tag{3}\]  
\[ \begin{cases}\alpha_{f}=\delta-\theta_{yf}=\delta-\beta-\frac{\dot{q}\cdot r}{ V_{x}}\\ \alpha_{r}=-\theta_{or}=-\beta+\frac{b\cdot r}{V_{x}}\end{cases}  \tag{4}\]  
**Vehicle Dynamics**  
Lateral and yaw motion dynamics are described by (5), where \(m\) is the mass, \(a_{y}\) lateral acceleration, and \(M_{zz}\) the external yaw moment, typically applied through differential torque allocation between left and right wheels. \[ \begin{cases}m\cdot a_{y}=F_{yf}+F_{yr}\\ l_{z}\cdot\dot{\gamma}=I_{z}\cdot\dot{r}=a\cdot F_{yf}-b\cdot F_{yr}+M_{zz} \end{cases}  \tag{5}\]Combining (2), (3), and (4) in the system (5) yields to (6), making \(\beta\) and \(r\) explicit. \[ \begin{cases}\frac{d\beta}{dt}=-r+\frac{2C_{sf}}{m\cdot V_{x}}(\delta-\beta- \frac{ar}{V_{x}})+\frac{2C_{sf}}{m\cdot V_{x}}(-\beta+\frac{br}{V_{x}})\\ \frac{dr}{dt}=2a\frac{C_{sf}}{t_{x}}(\delta-\beta-\frac{ar}{V_{x}})-2b\frac{2 ar }{t_{x}}(-\beta+\frac{br}{V_{x}})+\frac{M_{ex}}{t_{x}}\end{cases}  \tag{6}\]  
This approach is fundamental to defining the state space of the dynamic vehicle behavior. **Target Vehicle Kinematics**  
Desired side-slip angle (7) and yaw rate (8) appear linearly dependent by \(\delta\) if it is assumed that the front steering vehicle is in steady state conditions. Indeed, in this situation, (9) is the steering angle for negotiating a curve radius \(R\), with (10) being the vehicle under-steer gradient \(K_{V}\).