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

Pasillas-Lépine, William · 2005 · OpenAlex-citations

DOI: 10.1080/00423110500385873

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Summary

This paper addresses the lack of rigorous mathematical understanding in anti-lock brake system (ABS) algorithms based on wheel deceleration thresholds. While slip-regulation approaches offer clear mathematical frameworks, they require precise slip estimation, which is often difficult. Conversely, deceleration-based methods are robust but typically rely on heuristic tuning. The author proposes a new class of five-phase ABS algorithms and provides a hybrid modeling framework to analytically characterize their behavior, specifically focusing on the existence and stability of limit cycles. The study employs a simplified single-wheel vehicle dynamics model, describing the system through differential equations for wheel slip offset and wheel acceleration offset. The control logic is modeled as a hybrid automaton with five discrete phases, where transitions are triggered by thresholds on the measured wheel acceleration offset. The analysis utilizes two first integrals: one for phases with constant brake torque and an approximate integral for phases with rapidly varying torque. By leveraging these integrals, the author derives an explicit analytical approximation of the Poincaré map (first return map) without requiring numerical integration of the nonlinear differential equations. This approach allows for the determination of conditions under which the algorithm operates correctly and the calculation of bounds for wheel slip. The results demonstrate that the hybrid automaton is deterministic and non-blocking, provided specific threshold conditions are met. The analytical Poincaré map reveals that the system converges to a stable limit cycle, keeping the wheel slip within a small neighborhood of the optimal value. The paper analyzes specific cases, including a "bang-bang" limit and a symmetric case, showing how parameter tuning affects the map's properties. For instance, in the symmetric case, the Poincaré map can be approximated by a simple formula dependent on threshold ratios, allowing for the identification of parameters that minimize the L∞-norm of the map and ensure global attractiveness of the limit cycle. Computer simulations validate these theoretical findings, confirming that the algorithm maintains wheel slip near the optimal point. The significance of this work lies in providing a rigorous mathematical background for deceleration-based ABS algorithms, replacing heuristic tuning with analytical design principles. By enabling the computation of the Poincaré map analytically, the method facilitates efficient calibration of algorithm parameters to achieve optimal braking performance. This framework offers a clearer understanding of the phase-plane behavior of ABS systems, bridging the gap between robust practical implementations and theoretically sound control strategies.

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• braking response