Control of Electric Vehicles Using a Model Predictive Controller With Closed Form Solution

In this paper, the problem of vehicle stability control using model predictive technique is addressed. The vehicle under consideration is an electric vehicle with an electric motor driving each wheel independently. For the purpose of stability control, it is required that the vehicle tracks a desired yaw rate at all times therefore, extending the linear range of the vehicle dynamics. The desired yaw rate is defined based on vehicle speed, steering wheel angle and road surface friction. The vehicle stability control system considered in this paper consists of a high-level controller that compares the current states of the vehicle with its desired states to determine the required forces and moments at the center of mass, and a low-level controller to track those C.G. forces and moments by adjusting the motor torques on each wheel. It will be shown that a non-predictive low-level controller can have a closed form solution. In order to avoid saturation of the tires, the low-level controller has a penalty function that increases exponentially when the tire forces are close to the limits of saturation to reduce tire forces to keep them within the tires force capacity. In this paper, a model predictive controller is designed as the low-level controller to predict the tire forces and the yaw moment at the C.G. to minimize the tracking error of desired C.G. forces and moments. To keep the tire forces within the tires capacity limit, a penalty function is used at each sample time to penalize control actions that result in excessive tire forces. This adds a level of anticipation to the low-level controller to detect in advance when tires are about to saturate and to choose control actions to prevent that from happening. Since tire capacity limit is treated with an analytical penalty function, it is still possible to find a closed form solution for the model predictive low-level controller. The proposed controller is tested with simulations and the results are compared with a similar non-predictive controller.