Optimal Periodic Impulsive Control for Orbital Stabilization of Underactuated Systems

Abstract

The orbital stabilization problem for underactuated systems with a single passive degree-of-freedom is revisited. The impulse controlled Poincaré map (ICPM) approach, in which stabilizing impulsive inputs are applied on a Poincaré section, has distinct advantages over existing methods but feedback compensation once every oscillation limits the rate of convergence to the desired orbit. To overcome these limitations, we propose stabilization through application of multiple impulsive inputs during each oscillation. An optimal control problem is formulated to minimize a quadratic cost functional and the optimal inputs are obtained by solving a discrete periodic Riccati equation. Simulation results for a Pendubot are presented, highlighting the advantages of the control design over the ICPM method in terms of convergence rate and robustness to parameter uncertainty.