Nonlinear gliding stability and control for vehicles with hydrodynamic forcing

This paper presents Lyapunov functions for proving the stability of steady gliding motions for vehicles with hydrodynamic or aerodynamic forces and moments. Because of lifting forces and moments, system energy cannot be used as a Lyapunov function candidate. A Lyapunov function is constructed using a conservation law discovered by Lanchester in his classical work on phugoid-mode dynamics of an airplane. The phugoid-mode dynamics, which are cast here as Hamiltonian dynamics, correspond to the slow dynamics in a multi-time-scale model of a hydro/aerodynamically-forced vehicle in the longitudinal plane. Singular perturbation theory is used in the proof of stability of gliding motions. As an intermediate step, the simplifying assumptions of Lanchester are made rigorous. It is further shown how to design stabilizing control laws for gliding motions using the derived function as a control Lyapunov function and how to compute the corresponding regions of attraction.