Existence, Stability and Robustness Analysis of Limit Cycles in Hybrid Anti-lock Braking Systems

In this work we investigate the stability and robustness properties of Anti-lock Braking Systems based on actuators with discrete dynamics. To this end, we propose a hybrid ABS controller, which can be proved to give rise to an asymptotically stable limit cycle on the wheel slip. Based on this system we will first give necessary conditions for the limit cycle existence, which come from a detailed analysis of the phase plane trajectories of the braking dynamics. Further, we prove the limit cycle existence and its asymptotic stability properties via a Poincar\'e map analysis. The proposed approach allows to evaluate the basins of attraction of the limit cycle in the different discrete states of the controller and to derive exact information on the maximum allowable uncertainty in the measured variables which guarantee the cycle stability. Moreover, a structural stability analysis is performed with respect to different road conditions and to the actuator rate limits.