Pseudo-energy shaping for the stabilization of a class of second-order systems

Abstract

A Lyapunov direct method is presented for a class of second-order systems that includes mechanical systems. This method shall be called a pseudo-energy shaping method because it reduces to the energy shaping method when a given second-order system is a mechanical system. The pseudo-energy shaping method comprehends both the Lyapunov direct method for mechanical systems proposed by Aguilar-Ibanez and the controlled Lagrangian method that has been successfully applied to stabilize mechanical systems. A class of second-order systems including mechanical systems is defined first. For this class, matching conditions are derived for the construction of an energy-like Lyapunov function that shall be called a pseudo-energy function. Easily verifiable conditions are then presented for stabilizability by the pseudo-energy shaping method for a class of second-order linear systems and for a class of second-order nonlinear systems with one degree of under-actuation. These results are applied to stabilize a two-dimensional overhead crane system and a three-link robot arm system. Copyright (C) 2011 John Wiley & Sons, Ltd.