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

This paper is the third and final component of a three-part effort on observers contracting a Riemannian distance between the state of the system and its estimate. In Part I, we showed that such a contraction property holds if the system dynamics and the Riemannian metric satisfy two key conditions: a differential detectability property and a geodesic monotonicity property. With the former condition being the focus of Part II, in this Part III, we study the latter condition in relationship to the nullity of the second fundamental form of the output function. We formulate sufficient and necessary conditions for it to hold. We establish a link between it and the infinite gain margin property, and we provide a systematic way for constructing a metric satisfying this condition. Finally, we illustrate cases where both conditions hold.