Functional T-observers

This article is a contribution to behavioral observer theory which was started by Valcher and Willems in 1999 and which was recently exposed by Fuhrmann in a comprehensive survey article. It is also a further development of the article on T-observers by Oberst and the author. For a given continuous or discrete time linear time-invariant behavior we assume that a linear function of a trajectory (e.g., some components) can be measured. We are interested in estimating another linear function of this trajectory.We generalize the notions of T-observability and T-observers introduced by Oberst and the author. T denotes a multiplicatively closed subset of the ring of operators. For different choices of T,T-observability coincides with observability, reconstructibility, trackability, or detectability, a T-observer is an exact, dead-beat, tracking, or asymptotic observer. We show the equivalence of T-observability and the existence of T-observers and give a constructive parametrization of all T-observers. Corresponding results for proper T-observers are also presented.Partial observation of the state of a Kalman state space system (compare e.g. Fuhrmann’s work) is a special case of our setting, and so are the observers of certain unknown components of a behavior studied by Bisiacco, Valcher, and Willems. The first result on functional observers in context with Rosenbrock equations or polynomial matrix descriptions is due to Wolovich (1974).