# Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons
## 2 The Anti-Lock Braking System Model  
The ABS braking system equipment from [15] (Figure 1) consists of two vertically superimposed wheels that simulate the relative motion between the road (lower wheel) and the quarter-car model (upper wheel). Throughout the braking process, the two wheels remain in contact and interaction occurs at the contact surface between them. The upper wheel is equipped with a disk braking system actuated by a braking lever, which is itself actuated by an inextensible string tensioned with the help of a pulley attached to a small flat DC motor [15]. The lower wheel is spin by the bigger flat DC motor, which is power-supplied to accelerate the wheel. After the two wheels in contact reach a threshold angular speed, the braking phase starts with the bigger DC motor left in free running mode (no power) while the smaller DC motor actuates the braking mechanism. Both DC motors are electronically controlled by pulse-width modulation (PWM) signals of 3.5 kHz. The varying pulse-width of the upper wheel is the control input variable denoted as \(u\), in the following equations of motion [15; 33]:  
\[ \begin{split}\dot{x}_{1}&=S(\lambda)(c_{11}x_{1}+c_{ 12})+c_{13}x_{1}+c_{14}+(c_{15}S(\lambda)+c_{16})s_{1}M_{1},\\ \dot{x}_{2}&=S(\lambda)(c_{21}x_{1}+c_{22})+c_{23}x_{ 2}+c_{24}+c_{25}S(\lambda)s_{1}M_{1},\\ \dot{M}_{1}&=c_{31}(b(u)-M_{1}),\\ b(u)&=\left\{\begin{array}{c}b_{1}u+b_{2},u\geq u _{0},\\ 0,u<u_{0},\end{array}\right.\end{split}  \tag{1}\]  
where the angular speed for the upper wheel is denoted as \(x_{1}(t)\) (measured in rad/s), and for the lower wheel is denoted \(x_{2}(t)\). The time argument is omitted next for simplified notation. Additionally, \(s=s_{1}=1\) in braking mode, and \(c_{ij}\) are given constant parameters. \(M_{1}(t)\) is the braking torque evolving according to the third differential equation, where constants \(b_{1}\) and \(b_{2}\) model the actuator dead-zone dynamics whose threshold is \(u_{0}\), while \(S(\lambda)=\frac{s(\lambda)}{L(\sin\phi-s\mu(\lambda)\cos\phi)}\) depends on the constant angle \(\phi\) and \(\mu(\lambda)=w_{4}\lambda^{p}/(a+\lambda^{p})+w_{3}\lambda^{3}+w_{2}\lambda^{2 }+w_{1}\lambda\) is the friction coefficient (\(\mu\))-slip coefficient (\(\lambda\)) static characteristic with \(w_{i},i=\overline{1,4},a,p\) being known constants. Notice at this point that the model (1) is not obtained directly, but its coefficients depend on many physical  
Figure 1: The anti-lock braking system (ABS) model diagram. parameters of the ABS system and can be recovered in the original documentation [15]. Some of them are properly presented throughout the paper when needed. In slip control strategy, the aim is to regulate \(\lambda\) to the value of \(\mu\) corresponding to the peak of the \(\mu-\lambda\) curve [15]. This implies for maximum friction coefficient, and favorable minimum braking distance and time. In real world commercial braking systems, the weight load on the car, the tire inflation state and wear, the road-tire contact surface and the road surface profile combined with the car speed, all interplay with the braking process and may affect the \(\mu-\lambda\) curve profile significantly. The static characteristics \(\mu(\lambda),S(\lambda)\) are shown in Figure 2.  
In the braking mode, the slip is defined as:  
\[\lambda=(r_{2}x_{2}-r_{1}x_{1})/(r_{2}x_{2}), \tag{2}\]  
where \(r_{1}\), \(r_{2}\) are the radii of the two wheels in contact. For the given laboratory model, since the two radii are approximately equal, it will be considered that \(r_{1}=r_{2}=0.99\) [m], leading to a simpler slip in (2), i.e., \(\lambda=(x_{2}-x_{1})/x_{2}\). From a functional perspective, there are three torques acting on the upper wheel: the braking torque \(M_{1}\), the friction torque in the upper bearing and the friction torque among the wheels. There are two torques acting on the lower wheel: the friction torque in the lower bearing and the friction torque among the wheels.