Guaranteeing Lanekeeping Performance With Tire Saturation Using Computed Polynomial Lyapunov Functions

Lanekeeping assistance systems hold the promise to save thousands of lives every year by preventing unintended lane departure. The potential field lanekeeping assistance system assists the driver in the lanekeeping task by effectively placing the vehicle in an artificial potential well with minimum at lane center. Previous work mathematically guarantees the performance of the system in the linear region of tire forces, but no guarantees of performance or even stability exist for saturating tires. These guarantees are crucial to ensure safety when the vehicle speed is too high for a given turn or the friction coefficient of the road is low due to surface conditions. Here we explore ways to numerically find Lyapunov functions for a vehicle with lanekeeping assistance and realistic tires. First the nonlinearity is modeled as a sector bounded disturbance, and a Lyapunov function is found for all vehicle trajectories that fit this sector bounded disturbance. Next, a polynomial fit is performed on the HSRI tire model, and a Lyapunov function is found for these polynomial dynamics. Each of these approaches provide Lyapunov functions valid well into the nonlinear region.