Convergence of Nonlinear Observers on ${\mathbb{R}}^{n}$ With a Riemannian Metric (Part II)

In “Convergence of Nonlinear Observers on Rn with a Riemannian Metric (Part I),” we established that a convergent observer with an infinite gain margin can be designed for a given nonlinear system when a Riemannian metric showing that the system is differentially detectable (i.e., the Lie derivative of the Riemannian metric along the system vector field is negative in the space tangent to the output function level sets) and the level sets of the output function are geodesically convex is available. In this paper, we propose techniques for designing a Riemannian metric satisfying the first property in the case where the system is strongly infinitesimally observable (i.e., each time-varying linear system resulting from the linearization along a solution to the system satisfies a uniform observability property) or where it is strongly differentially observable (i.e., the mapping state to output derivatives is an injective immersion) or where it is Lagrangian. Also, we give results that are complementary to those in our previous paper. In particular, we provide a locally convergent observer and make a link to the existence of a reduced order observer. Examples illustrating the results are presented.