# 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.

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**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}}