We study the convergence property of nonlinear observers for nonlinear systems using Riemannian metrics as a measure of the distance between state and estimate. We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing (decreasing) is such that that the Lie derivative of the Riemannian metric along the system vector field is nonpositive (negative) in the tangent space to the output function level sets. Also, if the observer has an infinite gain margin then the level sets of the output function have to be geodesically convex. Conversely, if these two properties are satisfied, then there exists an observer that has an infinite gain margin with Riemannian metric that is decreasing along solutions. We show that the existence of such a contracting metric is related to the observability of the system’s linearization along its solutions. Moreover, the problem of designing a Riemannian metric satisfying the said properties is also addressed for general systems as well as systems that are strongly differentially observable (which leads to reduced-order observers). Examples illustrating the ideas and results are presented.
Ricardo G. Sanfelice is an Associate Professor in the Department of Computer Engineering at the University of California, Santa Cruz. He received his M.S. and Ph.D. degrees from UCSB in 2004 and 2007, respectively. He held a Postdoctoral Associate position at MIT during 2007 and 2008, and afterwards visited the Centre Automatique et Systemes at the Ecole de Mines de Paris for four months. He is the recipient of the 2013 SIAM Control and Systems Theory Prize, the NSF CAREER award, the Air Force Young Investigator Research Award, the 2010 IEEE Control Systems Magazine Outstanding Paper Award, and the 2012 STAR Higher Education Award for his contributions to STEM education. His research interests are in modeling, stability, robust control, observer design, and simulation of nonlinear and hybrid systems with applications to power systems, biology, and aerospace.