Global Stabilisation of Underactuated Mechanical Systems via PID Passivity-Based Control

TL;DR

This paper demonstrates that a broad class of underactuated mechanical systems can be globally stabilized at a desired equilibrium using a PID controller, by leveraging passivity-based control and Lyapunov functions.

Contribution

It introduces a new passivity-based PID control method for global stabilization of underactuated systems, with verifiable conditions and improved robustness over previous approaches.

Findings

Global stabilization achieved for many benchmark systems

New passive outputs enable PID control design

Lyapunov functions ensure stability under weaker assumptions

Abstract

In this note we identify a class of underactuated mechanical systems whose desired constant equilibrium position can be globally stabilised with the ubiquitous PID controller. The class is characterised via some easily verifiable conditions on the systems inertia matrix and potential energy function, which are satisfied by many benchmark examples. The design proceeds in two main steps, first, the definition of two new passive outputs whose weighted sum defines the signal around which the PID is added. Second, the observation that it is possible to construct a Lyapunov function for the desired equilibrium via a suitable choice of the aforementioned weights and the PID gains and initial conditions. The results reported here follow the same research line as [7] and [20]---bridging the gap between the Hamiltonian and the Lagrangian formulations used, correspondingly, in these papers. Two additional improvements to our previous works are the removal of a non-robust cancellation of a potential energy term and the establishment of equilibrium attractivity under weaker assumptions.