Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms

Existence, stability and robustness analysis of limit cycles in hybrid anti-lock braking systems

“…an estimate derived only from wheel speed measurements, we characterise the maximum allowable measurements error which guarantees limit cycle stability and we discuss how to design the controller based on the confidence in the quality of the wheel slip measure. In this respect, is worth mentioning that, as the braking pressure is not always available for measurements, while the angular wheel speed always is in ABS, the threshold on the braking torque T b might be replaced by thresholds on the wheel deceleration (Kuo and Yeh 1992;Pasillas-Le´pine 2006).…”

“…Johansen et al 2003;Pasillas-Le´pine 2006) that the longitudinal dynamics of the vehicle (expressed by the state variable v) are much slower than the rotational dynamics of the wheel (expressed by the state variable or !) due to large differences in inertia.…”

“…Other works focused on the study of rule-based ABS are (Ouwerkerk and Guntur 1972;Kuo and Yeh 1992;Fu, Fang and Zhao 2001;Pasillas-Le´pine 2006) each of which considers a different number of discrete states in the ABS controller. In particular, in Kuo and Yeh (1992) an ABS controller with four discrete states is studied, and the switching between the different states is governed by wheel deceleration thresholds and the elapsed time intervals spent in the different states.…”

“…The tuning of the threshold values is made by analysing the quarter car dynamics, even if no formal existence and stability analysis of the closed-loop trajectories are given. Recently, Pasillas-Le´pine (2006) has presented a five states controller, where the threshold values are based on wheel deceleration, and a stability analysis is performed via Poincare´maps and notions from hybrid systems theory. From the industrial viewpoint, rule-based ABS were introduced on commercial cars by Bosch (2008) and the discussion in Kiencke and Nielsen (2000), Chapter 8, provides a qualitative description of such algorithms.…”

“…In fact, in the structural stability analysis all main system and measurement uncertainties are taken into account and tools are provided to formally consider their action also in the design phase. Furthermore, with respect to the works in Kuo and Yeh (1992); Fu et al (2001) and Pasillas-Le´pine (2006); where simplified tire-road models are used, we work on a widely used friction model without resorting to any simplification.…”