Nonlinear feedback, double bracket dissipation and port control of Lie-Poisson systems

TL;DR

This paper develops nonlinear feedback control methods for Lie-Poisson systems, stabilizing unstable equilibria using techniques like controlled Lagrangians, double bracket dissipation, and IDA-PBC, with applications to satellite and plasma flow models.

Contribution

It introduces novel nonlinear feedback control strategies for Lie-Poisson systems, combining multiple existing methods to achieve asymptotic stabilization of unstable equilibria.

Findings

Stabilized a rotor-driven satellite model.

Achieved stabilization of Hall magnetohydrodynamic flow.

Demonstrated effectiveness of combined control techniques.

Abstract

Methods from controlled Lagrangians, double bracket dissipation and interconnection and damping assignment -- passivity based control (IDA-PBC) are used to construct nonlinear feedback controls which (asymptotically) stabilize previously unstable equilibria of Lie-Poisson Hamiltonian systems. The results are applied to find an asymptotically stabilizing control for the rotor driven satellite, and a stabilizing control for Hall magnetohydrodynamic flow.