# Non-linear Symmetry-preserving Observer on Lie Groups

**Authors**: S. Bonnabel, P. Martin, P. Rouchon, IEEE Trans. Automatic Control, vol 54, no 7, 2009, DOI: 10.1109/TAC.2009.2020646

In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which reminds of the linear stationary case.

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**BibTeX**:

@Article{,

author = {P. Martin S. Bonnabel, P. Rouchon},

title = {Non-linear Symmetry-preserving Observer on Lie Groups},

journal = {IEEE Trans. Automatic Control},

volume = {54},

number = {7},

pages = {1709 - 1713},

year = {2009},

abstract = {In this technical note, we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that the error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which is reminiscent of the linear stationary case.},

location = {},

keywords = {Lie groups, geometry, nonlinear control systems, observers, state-space methods}}