Stabilizing of a Class of Underactuated Euler Lagrange System Using an Approximate Model

TL;DR

This paper introduces an approximate method for solving matching conditions in energy shaping control of under-actuated Euler Lagrange systems, simplifying the stabilization process and demonstrated on an inverted pendulum.

Contribution

The paper proposes a novel approximate solution approach for the matching PDEs in energy shaping control, reducing complexity in stabilizing under-actuated EL systems.

Findings

Successfully stabilizes an inverted pendulum on a cart

Provides a new method to approximate solutions of matching conditions

Ensures asymptotic stability with the proposed control rule

Abstract

The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the under-actuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear, nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of under-actuated EL systems. To develop the method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. So the assignable potential energy function and the controlled inertia matrix, both are constructed as a common solution of these PDEs. Afterwards, the gyroscopic and dissipative forces are found as the solution of the kinetic energy matching condition. Finally, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed loop system which used the control rule derived with the proposed method is also given. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.