A Simple Intrinsic Reduced-Observer for Geodesic Flow

Aghannan and Rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. The (position and velocity) observer is based on the Riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic. They proved local convergence. When the system is conservative, we propose an intrinsic reduced order (velocity) observer based on the Jacobi metric, which can be initialized such that it converges exponentially for any initial true velocity. For non-conservative systems the observer can be used as a complement to the one of Aghannan and Rouchon. More generally the reduced observer provides velocity estimation for geodesic flow with position measurements. Thus it can be (formally) used as a fluid flow soft sensor in the case of a perfect incompressible fluid. When the curvature is negative in all planes the geodesic flow is sensitive to initial conditions. Surprisingly in this case we have global exponential convergence and the more unstable the flow is, faster is the convergence.