Observer Design for Nonlinear Systems with Equivariance

Equivariance is a common and natural property of many nonlinear control systems, especially those associated with models of mechatronic and navigation systems. Such systems admit a symmetry, associated with the equivariance, that provides structure enabling the design of robust and high performance observers. A key insight is to pose the observer state to lie in the symmetry group rather than on the system state space. This allows one to define a globally defined intrinsic equivariant error but poses a challenge in defining internal dynamics for the observer. By choosing an equivariant lift of the system dynamics for the observer internal model we show that the error dynamics have a particularly nice form. Applying the methodology of Extended Kalman Filtering (EKF) to the equivariant error state yields the Equivariant Filter (EqF). The geometry of the state-space manifold appears naturally as a curvature modification to the classical EKF Riccati equation. The equivariant filter exploits the symmetry and respects the geometry of an equivariant system model and yields high performance robust filters for a wide range of mechatronic and navigation systems.