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 global 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 to the equivariant error state yields a filter we term the equivariant filter. The geometry of the state-space manifold appears naturally as a curvature modification to the classical Riccati equation for extended Kalman filtering. The equivariant filter exploits the symmetry and respects the geometry of an equivariant system model, and thus yields high-performance, robust filters for a wide range of mechatronic and navigation systems.