Exponential Stability and Tuning for a Class of Mechanical Systems

TL;DR

This paper establishes exponential stability for certain mechanical systems using a novel Lyapunov function and demonstrates how this analysis guides the tuning of passivity-based controllers, including PID and IDA, with an application to robotic arm stabilization.

Contribution

It introduces a new Lyapunov function for stability analysis and applies it to tune passivity-based controllers for nonlinear mechanical systems.

Findings

Proves exponential stability for a class of port-Hamiltonian systems.

Shows how the analysis guides controller tuning for improved convergence.

Validates the approach with a robotic arm example.

Abstract

In this paper, we prove the exponential stability property of a class of mechanical systems represented in the port-Hamiltonian framework. To this end, we propose a Lyapunov candidate function different from the Hamiltonian of the system. Moreover, we study how the proposed analysis can be used to determine the exponential stability and the rate of convergence of some (nonlinear)-mechanical systems stabilized by two passivity-based control techniques, namely, PID passivity-based control and interconnection and damping assignment. We implement the former control approach to stabilize a three degrees-of-freedom robotic arm at the desired equilibrium point to illustrate the mentioned analysis.