# Nonlinear observer design with an appropriate Riemannian metric

**Authors**: R. Sanfelice, L. Praly, 48th IEEE Conference on Decision and Control, pp. 6514-6519, 15-18 Dec. 2009, Shanghai, China DOI: 10.1109/CDC.2009.5400714

An observer whose state lives in a copy of the space of the given system and that guarantees a vanishing estimation error exhibits necessarily a symmetric covariant tensor field of order 2 which is related to the local observability information. A direct construction of this matrix field is possible by solving off-line ordinary differential equations. Using this symmetric covariant tensor field as a Riemannian metric, we prove that geodesic convexity of the level sets of the output function is sufficient to allow the construction of an observer that contracts the geodesic distance between the estimated state and the system's state, globally in the estimated state and semi-globally in the estimation error.

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**BibTeX**:

@Proceedings{,

author = {R. Sanfelice, L. Praly},

editor = {},

title = {Nonlinear observer design with an appropriate Riemannian metric},

booktitle = {48th IEEE Conference on Decision and Control},

volume = {},

publisher = {},

address = {Shanghai},

pages = {6514-6519},

year = {2009},

abstract = {An observer whose state lives in a copy of the space of the given system and that guarantees a vanishing estimation error exhibits necessarily a symmetric covariant tensor field of order 2 which is related to the local observability information. A direct construction of this matrix field is possible by solving off-line ordinary differential equations. Using this symmetric covariant tensor field as a Riemannian metric, we prove that geodesic convexity of the level sets of the output function is sufficient to allow the construction of an observer that contracts the geodesic distance between the estimated state and the system’s state, globally in the estimated state and semi-globally in the estimation error.},

keywords = {}}