The mixed fuzzy controller for anti-braking systems with quasi-nonholonomic constraints

Quasi-constrained dynamics consists of motion not only in constraint space but also in unconstrained or “freedom” space, which may occur when the system escapes or slips away from its constraint manifold during high-speed motion. Traditionally, slippage phenomena, an issue in the automobile industry, is usually ignored because of its high frequency and strong nonlinear features. The conventional Frobenius theorem is focused on holonomic dynamics, which are integrable in freedom space. A complementary Frobenius theorem (CFT) is proposed to release the conventional constraints from “hard” to “soft” models. We derive a geometric formulation, instead of its algebraic counterpart, for acatastatically nonholonomic systems from the topological viewpoint. We also propose a mixed fuzzy controller (MFC) for the nonholonomic system with escaping motion, which includes a traditional controller for the hard subsystem and a non-traditional controller with fuzzy rules for the soft subsystem on the constraint manifold. The closed-loop stability of the nonholonomic system with an MFC scheme is proved under admissible conditions. Finally, the proposed algorithm is applied to a wheeled vehicle with an anti-lock braking system (ABS) under the assumption of Coulomb viscous friction. Computer simulation is used to justify the results