A Separation Principle on Lie Groups

TL;DR

This paper extends the separation principle to invariant control systems on Lie groups, showing it holds around certain trajectories when symmetry-preserving observers are used, with applications to mobile robots and Lagrangian systems.

Contribution

It proves a local separation principle for nonlinear systems on Lie groups, generalizing classical results to a broader class of geometric control systems.

Findings

Separation principle holds around specific trajectories on Lie groups.

Existence of time-invariant linearized systems with symmetry-preserving observers.

Application to mobile robots and Euler-Poincare systems.

Abstract

For linear time-invariant systems, a separation principle holds: stable observer and stable state feedback can be designed for the time-invariant system, and the combined observer and feedback will be stable. For non-linear systems, a local separation principle holds around steady-states, as the linearized system is time-invariant. This paper addresses the issue of a non-linear separation principle on Lie groups. For invariant systems on Lie groups, we prove there exists a large set of (time-varying) trajectories around which the linearized observer-controler system is time-invariant, as soon as a symmetry-preserving observer is used. Thus a separation principle holds around those trajectories. The theory is illustrated by a mobile robot example, and the developed ideas are then extended to a class of Lagrangian mechanical systems on Lie groups described by Euler-Poincare equations.