A Separation Principle on Lie Groups
Authors: S. Bonnabel, P. Martin, P. Rouchon, E. Salaün, IFAC world congress 2011, pp. 8004-8009, August 28 -September 2, Milano DOI: 10.3182/20110828-6-IT-1002.03353
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.
Download PDF
BibTeX:
@Proceedings{,
author = {S. Bonnabel, P. Martin, P. Rouchon, E. Salaün},
editor = {},
title = {A Separation Principle on Lie Groups},
booktitle = {18th IFAC world congress 2011},
volume = {},
publisher = {},
address = {Milano, Italy},
pages = {8004-8009},
year = {2011},
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.},
keywords = {Lie groups, Separation principle, Non-holonomic systems, Mechanical systems}}
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.
Download PDF
BibTeX:
@Proceedings{,
author = {S. Bonnabel, P. Martin, P. Rouchon, E. Salaün},
editor = {},
title = {A Separation Principle on Lie Groups},
booktitle = {18th IFAC world congress 2011},
volume = {},
publisher = {},
address = {Milano, Italy},
pages = {8004-8009},
year = {2011},
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.},
keywords = {Lie groups, Separation principle, Non-holonomic systems, Mechanical systems}}