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

Abstract

SUMMARY 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‐Ibañez 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 © 2011 John Wiley & Sons, Ltd.