Nonlinear Observer Design with an Appropriate Riemannian Metric

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An observer whose state lives in the same space as the one of the given system and which guarantees a vanishing estimation error exhibits necessarily a symmetric matrix field 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 matrix 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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