Observers, invariance and autonomy

Jochen Trumpf, based on joint work with C. Lageman and R. Mahony

In this talk I will discuss observers for affine systems without drift whose state evolves on a finite dimensional, connected Lie group with outputs in a homogeneous space. In many cases these systems have a representation as bilinear systems in a matrix space plus a possibly nonlinear output equation. The discussion will reveal a surprising connection between invariance of the system equations and autonomy of the observation error. The closeness of the resulting theory to the corresponding theory for linear systems is quite astonishing, albeit predicted by Brockett in his 1972 paper "System theory on group manifolds and coset spaces". Inspired by these results, I will take a fresh look at the Kalman filter and present a (to the best of my knowledge) new interpretation as a gradient observer.