## 2 Control Strategies
### Sliding Mode Lateral Stability Control
#### 2.2.3 Effort Application: Low-Level Layer  
Most of the current control solutions use differential braking between the left and right wheels to induce a correcting yaw moment on the vehicle chassis. However, non-conventional vehicle architecture, such as direct wheel-driven cars, allows one to benefit also from traction torques, independently delivered to each wheel. Indeed, in more sophisticated vehicle layouts, the availability of multi-quadrant electric motors (IWM) or active differential consent to take advantage also of traction torque, resulting in a higher value of applicable momentum around the vehicle's z-axis. Moreover, the presence of SBW and BBW systems ensures simple interfaceability with higher-level CU, allowing the implementation of advanced and complex control solutions. Even if optimal control signals (42) secure the controlled states converge to desired ones, an allocation problem still need to be solved. Indeed, the controller should stabilize the vehicle as well as produce control effort that is in accordance with the driver's intention, keeping unaltered the driving path. To avoid the typical chattering problem of the Sliding Mode controller [24], saturation is adopted in the definition of the control dead zone. Note that using these functions to deal with SMC output discontinuities could be detrimental since the controller formulation loses its robustness properties [33]. To solve this, specific smoothing transfer functions are used. Adopted criteria are the values of the state errors and vehicle longitudinal speed. This solution provides a useful tool to avoid the fast transient yaw moment demanded by the upper controller during abrupt cornering maneuvers. This leads to the use of relatively high control gains while averting undesired spikes during the wheel torque allocation phase. This low-level controller is parametrized to be easily applied to different traction architectures. This ensures the portability and flexibility of the whole proposed strategy, allowing the application of advanced TV techniques, regardless of vehicle layout. Optimal steer angle is applied as a correction with respect to the desired trajectory: it should not directly overwrite the driver command. For doing so, \(\delta_{cor}\) is limited to 10% of the steering angle command to avoid deviation, which could lead to unpredictable vehicle dynamics. Indeed, during fast transient phases, a greater wheel steering angle correction can bring the tires outside the linear behavior boundaries, saturating the available friction in the contact patch (high lateral and longitudinal slips). Moreover, the rate of the steering correction \(\delta_{cor}\) is limited to 1500 deg/s to not exceed the maximum driver capability of the steering angle speed application [38]. **Yaw Moment**  
The task of this controller is to map the desired yaw moment into equivalent torque demand for each wheel. In this case, the desired \(M_{z}z\) is applied to the vehicle using an advanced TV technique. However, this approach contemplates a constrained control problem because of the multi-DoF solutions which are available. Indeed, independent wheel torque control implicates 4-DoF. This leads to \(\infty^{4}\) possible solutions [9]. For this reason, an optimal allocation strategy based on Moore-Penrose Pseudo-inverse is proposed [18], where TV efforts are limited to the maximum deliverable torques of each wheel. Even in this case, a good trade-off between vehicle stability and driver intention must be satisfied. The optimal yaw moment \(M_{yaw}\) calculated by (42) is the external yaw moment \(M_{zz}\) defined in (37). However, some modifications are assumed. Firstly, the last term of (37) is neglected. This assumption is coherent with the proposed testing procedure and condition \((a)\). As will be explained later, the maximum road angle during the reference maneuvers is about 6 deg, which limits the contribution of the sine component to a maximum of 10%. For further activities, this term can be accounted.