Convergence of Nonlinear Observers on R^n with a Riemannian Metric (Part I)

Understand

We study how convergence of an observer whose state lives in a copy of the given system's space can be established using a Riemannian metric.

• We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies that the Lie derivative of the Riemannian metric along the system vector field is conditionally negative.
• Moreover, we establish that the existence of this metric is related to the observability of the system's linearization along its solutions.
• Moreover, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex.