Speed Aware Hybrid Adaptive Optimal Control of Bicycle Dynamics With Sliding-Mode Robustness

Abstract The stability of a bicycle changes qualitatively with forward speed: an unstable low-speed weave mode transitions into a lightly damped self-stable region and eventually gives rise to high-speed capsize instability. Because no single controller can deliver uniform performance across these distinct regimes, this article develops a speed aware hybrid control architecture for the benchmark Carvallo–Whipple bicycle model. The velocity domain is partitioned into three physically meaningful regions—low-speed weave, mid-speed self-stability, and high-speed capsize—each governed by a regime-appropriate controller: (1) a high-gain Guard stabilizer at low-speed, (2) a Lyapunov-based model reference adaptive controller (MRAC) in the self-stable band, and (3) a velocity parameterized linear quadratic regulator (LQR), refreshed via recursive least-squares identification, at high-speed. All regimes share a thin-boundary sliding-mode augmentation that provides matched-disturbance robustness with reduced chattering. A multiple Lyapunov functions (MLF) framework formally establishes practical stability under hysteresis and dwell-time-based switching. Controller performance is evaluated on the linearized Whipple model, while open-loop comparisons between linear and nonlinear dynamics confirm that the linear model captures the dominant modal behavior across representative speeds. Simulation results demonstrate improved convergence, disturbance rejection, and stabilization relative to single-strategy controllers, providing a structured, stability-certified methodology for controlling underactuated vehicles with speed-dependent dynamics.