A manifold structure on the set of functional observers

Jochen Trumpf, Uwe Helmke

Assume, a linear function f of the state x of a linear control system in state space form is given. An observer for this function is another linear control system taking the input and the output of the observed system as its input and generating from it an estimate for f. The observer is called a tracking observer if its state z satisfies the tracking property: whenever the observer is started with an exact estimate z(0)=f(x(0)) it will continue to produce an exact estimate z(t)=f(x(t)) for all future times t.

It is well known that a tracking observer for f exists if and only if the kernel of f is a conditioned invariant subspace with respect to the observed system. The set M of tracking observer parameters of fixed size, i.e. tracking observers of fixed order together with the functions they are tracking, is shown to be a smooth manifold. Furthermore, the set of conditioned invariant subspaces of fixed codimension together with their friends, i.e. the output injections making the subspace invariant, is shown to carry a vector bundle structure over M.

Potential applications are in the area of pole placement by dynamic output feedback as well as the analysis of the convergence behaviour of numerical algorithms for observer design.