###### Abstract  
The antilock braking system (ABS) is a mechatronic system that helps a driver maintain the maneuverability of a vehicle while braking by preventing wheel lock-ups. However, the design of high-performance controllers for this type of system is complicated because of its highly nonlinear dynamics. The problem becomes even more difficult to resolve when uncertainties in the parameters appear in its dynamics. In this paper, an ABS laboratory setup mimicking a quarter car model is considered. A modified high-order sliding mode (HOSM) controller using a proportional-integral-differential (PID) control as a sliding surface was designed. This controller provides a reference value of a tire slip. The proposed controller uses a tracking error to define the slip surface through the PID controller structure, and the modified HOSM controller holds the system on the previously designed slip surface. The closed-loop system stability has been proven in the sense of Lyapunov. Finally, the ABS laboratory setup allows for experimentally checking the performance of the modified HOSM controller using a PID-sliding surface, showing a considerable increase in the efficiency of the control system compared with a PID-like controller. HOSM; PID-sliding surface; Lyapunov stability; wheeled vehicles; antilock braking system +
Footnote †: journal: Applied  
Article  
A Lyapunov Stability Analysis of Modified HOSM Controllers Using a PID-Sliding Surface Applied to an ABS Laboratory Setup  
[PERSON], [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON], [PERSON]; [PERSON] [PERSON]; [PERSON]. One remarkable feature of the HOSM is that sliding-mode surfaces with relative degrees greater than one can be considered [3]. Since the creation of this technique, a number of articles have been published about the HOSM [4; 5; 6; 7]. Initially, HOSM techniques were used for smooth control systems, but recently, they have been applied to finite-time differentiators. In [8], an interesting comparison of HOSM differentiators and high-gain observers concerning measurement noise was made. Some HOSM application techniques for solving vehicle-control problems can be found in [9; 10; 11], and references [12; 13; 14; 15] presented several fields and practical cases for which these techniques were applied successfully. However, despite the current knowledge about HOSM controllers, in recent years, scientists have proposed a combination of the HOSM controller and PI/PID-sliding surfaces. Some examples can be found in the literature; reference [16] investigated an adaptive fault-tolerant pseudo-PID-SMC scheme for a high-speed train subjected to actuator-fault asymmetric nonlinear actuator saturation. In [17], a combination of SMC with a traditional PID control scheme for a two-degrees-of-freedom planar manipulator was considered. In [18], an SMC scheme for the robust tracking control of the nanopositioning stage composed of piezoceramic stack-actuator-compliant flexure mechanisms using a PID-type sliding surface was presented. In [19], a second-order SMC with a PI-sliding surface for a DC motor-driven control was designed. Reference [20] applied an SMC with a PID-sliding surface to the active vibration damping of pneumatically actuated soft robots. Reference [21] considered the design of a modified HOSM controller using a PI-sliding surface for the attitude control of a ground vehicle. Reference [22] developed an adaptive dynamic PID-sliding surface-based second-order fault-tolerant SMC for the speed control of an electromechanical system. Reference [23] considered a second-order SMC with a PID-sliding surface applied to a modified dynamic model of a quad-rotor UAV with the disturbance of an unbalanced load. In the present work, a modified HOSM using a PID control as a sliding surface is designed to impose a reference value of a tire slip. An ABS setup is used to experimentally test new control laws [24]. It consists of a lower aluminum wheel, simulating a vehicle's longitudinal motion, and an upper plastic wheel, simulating the vehicle's wheel.