# Model Reference Adaptive Control of Vehicle Slip Ratio Based on Speed Tracking
## 1 Introduction  
Vehicle braking control is important for driving safety because it can shorten braking distance and avoid losing steering ability and vehicle instability. The braking force is directly dependent on road conditions, which, in turn, influence the wheel slip ratio [1, 2]. Wheel slip ratio, the difference between vehicle longitudinal speed and wheel linear speed, should be controlled at an optimum level, so that the road-surface friction can be utilized to the maximum extent and the wheels can be prevented from locking even on a slippery road. Slip ratio control is an effective method for vehicle braking control, and it is also the core technology associated with anti-lock braking system (ABS) [3, 4, 5]. Slip ratio control usually requires a faster response. Modern electric vehicles or some vehicle brake systems can utilize motors or electromechanical actuators to obtain faster and more precise torque control. In addition, the uncertainty and time variability of the slip control system, especially for real-time road conditions, bring difficulties to control methodologies. Look-up tables based on experimental data are employed in most current vehicles, but their adaptability and robustness are not desirable. Many efforts have been attempted from two aspects to improve the performance. One is the model-free method, such as sliding mode control (SMC) [6, 7, 8], fuzzy logic control (FLC) [9, 10], iterative learning control [11], etc, all of which have strong robustness but with a compromise between performance and complexity. The other is the model-based control method which is relatively simple and has requires a modest time to design and tune [12, 13, 14]. In [13], feedback linearization in combination with the gain scheduling-based Lyapunov approach was used and eliminated the static uncertainty using integral action and dynamic uncertainty with a sufficient stability margin. However, modelingerrors still exist, the stability analysis is valid only locally and the vehicle speed is assumed as a slow time-varying parameter. In [14], a backstepping dynamic surface controller based on the Lyapunov function is designed to regulate wheel slip. It is a simple method but the adopted tire model is difficult to satisfy the varying road conditions. Some road condition estimation methods are proposed, such as a combination of stiffness-based estimation and least squares, a combination of nonlinear Lipschitz observer and modified super-twisting algorithm [15; 16]. However, those methods are more complex or they require more complicated models. In [17], a simple nonlinear disturbance observer combined with a controller is used to track control, but it is based on the assumption that the observer dynamics are much faster than the disturbance. The motivation of this study is to propose a model-based control method for slip control with enhanced braking performance and robustness. Motivated by the literature [17], a simple road adhesion coefficient observer is designed. In contrast, the observer in this paper is simpler in that its design does not require a combination controller and it does not rely on assumptions. In addition, combining the model reference adaptive controller's strong robustness for system nonlinearity and uncertainty [18; 19; 20], a model reference adaptive controller based on a road condition observer is proposed in this paper. The control problem is first formulated from the speed tracking control. The asymptotic and robustness properties are obtained through the adaptive controller based on the Popov hyperstability theory, and a sliding mode observer is constructed to estimate the unknown vehicle velocity. Simulations under different conditions, including different road adhesion coefficient, system uncertainties and modeling errors, are carried out to verify the effectiveness. The paper is organized as follows: in Section 2 the system model and control problem are formulated. Then, a model reference adaptive control strategy-based road adhesion coefficient observer is proposed in Section 3.