On State Observers—Take 2

Abstract

This is the author’s second attempt to provide a characterization for asymptotic functional state observers in the category of linear time-invariant finite-dimensional systems in input/state/output form in terms of a Sylvester-type matrix equation with a proof that only uses state-space and transfer function methods. The characterizing equation was already proposed in Luenberger’s original work on state observers, but to prove that it is not only sufficient but also necessary when the observed system has no stable uncontrollable modes turns out to be surprisingly hard. The crux of the problem is that in a classical observer interconnection both the output and the input of the observed system enter the observer and hence also the observer error system as separate inputs. They are not independent signals, though, since they are jointly constrained by the equations of the observed system. The first attempt by the author (see the list of references) contained a subtle error in the proof of the main result. To fix this error, some new intermediate results are needed and the final proof is sufficiently different to warrant this paper. As a bonus, details on how to observe stable uncontrollable modes are also provided. The presentation is mostly self contained with only occasional references to standard results in linear system theory. It is an absolute pleasure to dedicate this paper to my friend and colleague Harry Trentelman on the occasion of his 60th birthday. Harry and I have worked together on linear system theory for the last 5 years and our behavioral internal model principle for observers (joined work with Jan Willems) provides an alternative proof for the result reported here (Trumpf et al. IEEE Trans. Autom. Control 59, 1737–1749 (2014)).