Reformulation of Matching Equation in Potential Energy Shaping

TL;DR

This paper presents a reformulation of the potential energy PDE in energy shaping control, simplifying the solution process under certain conditions and converting it into a linear matrix inequality, demonstrated on benchmark systems.

Contribution

It introduces a new reformulation of the potential energy PDE that simplifies solving it by focusing on the homogeneous part under specific conditions.

Findings

The reformulation allows easier PDE solving.

The condition can be expressed as a linear matrix inequality.

Validated on multiple benchmark systems.

Abstract

Stabilization of an underactuated mechanical system may be accomplished by energy shaping. Interconnection and damping assignment passivity-based control is an approach based on total energy shaping by assigning desired kinetic and potential energy to the system. This method requires solving a partial differential equation (PDE) related to he potential energy shaping of the system. In this short paper, we focus on the reformulation of this PDE to be solved easier. For this purpose, under a certain condition that depends on the physical parameters and the controller gains, it is possible to merely solve the homogeneous part of potential energy PDE. Furthermore, it is shown that the condition may be reduced into a linear matrix inequality form. The results are applied to a number of benchmark systems.