Abstract-This study presents a traction control system for electric vehicles with in-wheel motors, based on explicit non-linear model predictive control. The feedback law, available beforehand, is described in detail, together with its variation for different plant conditions. The explicit controller is implemented on a rapid control prototyping unit, which proves the real-time capability of the strategy, with computing times in the order of microseconds. These are significantly lower than the required sampling time for a traction control application. Hence, the explicit model predictive controller can run at the same frequency as a simple traction control system based on Proportional Integral (PI) technology. High-fidelity model simulations provide: i) a performance comparison of the proposed explicit non-linear model predictive controller with a benchmark PI-based traction controller with gain scheduling and anti-windup features; and ii) a performance comparison among two explicit and one implicit non-linear model predictive controllers based on different internal models, with and without consideration of transient tire behavior and load transfers. Experimental test results on an electric vehicle demonstrator are shown for one of the explicit non-linear model predictive controller formulations.Index Terms-traction control, wheel slip, model predictive control, PI control, electric vehicle, in-wheel motors.Sorniotti (email: a.sorniotti@surrey.ac.uk) are with the

This study addresses the development and Hardware-in-the-Loop (HiL) testing of an explicit nonlinear model predictive controller (eNMPC) for an anti-lock braking system (ABS) for passenger cars, actuated through an electro-hydraulic braking (EHB) unit. The control structure includes a compensation strategy to guard against performance degradation due to actuation dead times, identified through experimental tests. The eNMPC is run on an automotive rapid control prototyping unit, which shows its real-time capability with comfortable margin. A validated high-fidelity vehicle simulation model is used for the assessment of the ABS on a HiL rig equipped with the braking system hardware. The eNMPC is tested in 7 emergency braking scenarios, and its performance is benchmarked against a proportional integral derivative (PID) controller. The eNMPC results show: i) the control system robustness with respect to variations of tire-road friction condition and initial vehicle speed; and ii) a consistent and significant improvement of the stopping distance and wheel slip reference tracking, with respect to the vehicle with the PID ABS.

Anti-jerk controllers actively suppress the torsional oscillations of automotive drivetrains, caused by abrupt variations of the traction torque. The main benefits are: i) enhanced passengers' comfort; and ii) increased component life. Extensive literature deals with the design of anti-jerk controllers for electric powertrains with on-board motors, i.e., in which the electric motor is part of the sprung mass of the vehicle, and transmits torque to the wheels through a transmission, half-shafts and constant velocity joints. Nevertheless, a complete and structured comparison of the performance of the different control options is still missing. This study addresses the gap through the assessment of six anti-jerk controllersfive exemplary formulations from the literature, and one novel formulation based on explicit nonlinear model predictive control (eNMPC). All proposed control structures have the potential to be implemented on production vehicles. A set of objective performance indicators is defined to assess the controllers, which are tuned through an optimizationbased routine. Results show that the wheel speed input is critical to enhance controller performance, but may lead to reduced robustness.

Nonlinear model predictive control is proposed in multiple academic studies as an advanced control system technology for vehicle operation at the limits of handling, allowing high tracking performance and formal consideration of system constraints. However, the implementation of implicit nonlinear model predictive control (NMPC), in which the control problem is solved on-line, poses significant challenges in terms of computational load. This issue can be overcome through explicit NMPC, in which the optimization problem is solved off-line, and the resulting explicit solution, with guaranteed level of sub-optimality, is evaluated on-line. Due to the simplicity of the explicit solution, the real-time execution of the controller is possible even on automotive control hardware platforms with low specifications. The explicit nature of the control law facilitates feasibility checks and functional safety validation. This study presents a yaw and lateral stability controller based on explicit NMPC, actuated through the electrohydraulically controlled friction brakes of the vehicle. The controller performance is demonstrated during sine-with-dwell tests simulated with a high-fidelity model. The analysis includes a comparison of implicit and explicit implementations of the control system.

This study discusses vehicle stability control based on explicit nonlinear model predictive control (NMPC) and investigates the influence of prediction model fidelity on controller performance. The explicit solutions are generated through an algorithm using multiparametric quadratic programming (mp-QP) approximations of the multiparametric nonlinear programming (mp-NLP) problems. Controllers with different prediction models are assessed through objective indicators in sine-with-dwell tests. The analysis considers the following prediction model features: 1) nonlinear lateral tire forces as functions of slip angles, which are essential for the operation of the stability controller at the limit of handling. Moreover, a simple nonlinear tire force model with saturation is shown to be an effective alternative to a more complex model based on a simplified version of the Magic Formula; 2) longitudinal and lateral load transfers, playing a crucial role in the accurate prediction of the lateral tire forces and their yaw moment contributions; 3) coupling between longitudinal and lateral tire forces, which has a significant influence on the front-to-rear distribution of the braking forces generated by the controller; and 4) nonlinear peak and stiffness factors of the tire model, with visible yet negligible effects on the results.

Anti-jerk controllers compensate for the torsional oscillations of automotive drivetrains, caused by swift variations of the traction torque. In the literature model predictive control (MPC) technology has been applied to anti-jerk control problems, by using a variety of prediction models. However, an analysis of the influence of the prediction model complexity on anti-jerk control performance is still missing. To cover the gap, this study proposes six anti-jerk MPC formulations, which are based on different prediction models and are fine-tuned through a unified optimization routine. Their performance is assessed over multiple tip-in and tip-out maneuvers by means of an objective indicator. Results show that: i) low number of prediction steps and short discretization time provide the best performance in the considered nominal tip-in test; ii) the consideration of the drivetrain backlash in the prediction model is beneficial in all test cases; iii) the inclusion of tire slip formulations makes the system more robust with respect to vehicle speed variations and enhances the vehicle behavior in tip-out tests; however, it deteriorates performance in the other scenarios; and iv) the inclusion of a simplified tire relaxation formulation does not bring any particular benefit.

This paper presents a traction controller for combined driving and cornering conditions, based on explicit nonlinear model predictive control. The prediction model includes a nonlinear tire force model using a simplified version of the Pacejka Magic Formula, incorporating the effect of combined longitudinal and lateral slips. Simulations of a front-wheel-drive electric vehicle with multiple motors highlight the benefits of the proposed formulation with respect to a controller with a tire model for pure longitudinal slip. Objective performance indicators provide a performance assessment in traction control scenarios.

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