Global Minimum Energy State Estimation for Embedded Nonlinear Systems with Symmetry

Choosing a nonlinear state estimator for an application often involves a trade-off between local optimality (such as provided by an extended Kalman filter) and (almost-/semi-) global asymptotic stability (such as provided by a constructive observer design based on Lyapunov principles). This paper proposes a filter design methodology that is both global and optimal for a class of nonlinear systems. In particular, systems for which there is an embedding of the state-manifold into Euclidean space for which the measurement function is linear in the embedding space and for which there is a synchronous error construction. A novel observer is derived using the minimum energy filter design paradigm and exploiting the embedding coordinates to solve for the globally optimal solution exactly. The observer is demonstrated through an application to the problem of unit quaternion attitude estimation, by embedding the 3-dimensional nonlinear system into a 4-dimensional Euclidean space. Simulation results demonstrate that the state estimate remains optimal for all time and converges even with a very large initial error.