  In [1], it is 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 [1]. 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.
