# Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons
## 1 Introduction  
The anti-lock braking system (ABS) is a mandatory safety feature in modern vehicles [1] and constitutes an attractive benchmark for control applications. One of the common approaches that has been studied for ABS control is the so-called slip-control. It is well-known that when braking or accelerating, a traction force owed to the friction appears at the contact surface between the tire and the road [2, 3]. This force amplitude proportionally depends on the normal force at the contact point. A friction coefficient called \(\mu\) is representative for the traction behavior, it is surface dependent and well-acknowledged research showed that this friction coefficient is a function of the wheel slip coefficient denoted \(\lambda\)[4]. In principle, the associated control problem is to impose a desired value for the wheel slip \(\lambda\), that is, to set at a desired value for the relative difference between the wheel velocity and the road (or car) velocity [2, 5]. This problem is known as slip control. Among all implemented braking approaches, the slip coefficient control is arguably the most intensively studied in many theoretical and practical implementations [6]. Historically, the wheel lock prevention concept under braking was conceived almost a century ago in the mid-1930s by [PERSON]'s patents [7, 8, 9] and was technically implemented at the end of the 60s when inductive contactless sensors replaced the mechanical ones [9]. The strong electronics development led to powerful integration and robust operation; therefore, hydraulic pressure regulation became possible based on data gathered from the sensors mounted in the proximity of the wheel ensemble. ABSs for serial production became available starting in 1978 [9, 10]. The focus has shifted since then towards assuringmore intelligent control algorithms for the ABS, reviews of such approaches can be checked with earlier works of [11; 12] and more recently with [13; 14; 15; 16]. Slip control is a difficult task for a number of reasons:  
* The ABS dynamics in braking mode are extremely nonlinear and many parameters of the underlying model change with time. * Several factors affect the braking behavior, such as: road slope variation and modifications of the friction-slip characteristic, due to road surface type, tires' material, wheels' sideslip angle, etc. [4; 17; 18]. Among the existing control techniques, sliding mode control (SMC) is an already vastly studied and mature methodology that, among many other control approaches, has proved its success in many industrial applications due to its simplicity and robustness [19; 20; 21; 22; 23]. It has also been applied before to ABS slip control systems in combination with different controllers or just as standalone as presented next, however, without the detailed performance assessment through comparisons with different control methods. An SMC design can be traced back to the work of [PERSON] et al. [11], where a detailed hydraulic actuator model is provided and a friction force observer adds computational complexity to the design. However, an extremum searching algorithm was devised for the optimal slip reference input that maximizes the braking friction force (by maximizing the friction coefficient \(\mu\)). An underlying PID-based slip control was employed in [12] where a sliding-mode optimizer is used for computing the optimal slip reference input but does not benefit from the SMC in the closed-loop. [PERSON] and [PERSON] [3] proposed a nonlinear observer SMC-based slip controller combined with computationally demanding sliding observers for vehicle velocity estimation. A self-learning fuzzy sliding-mode control design was proposed by [PERSON] and [PERSON] [18], where the tuning algorithm of the controller is derived in the Lyapunov sense, but the braking model is a very simplified one. A linearization-based SMC law was proposed in [23] on a simplified quarter-car ABS model without accounting for actuator dynamics.