An internal model principle for observers

Jochen Trumpf and Jan C. Willems

An observer is a device which takes a set of measured signals as input and produces an estimate of another set of signals as output. Here, it is assumed that the two sets of signals are interacting dynamically, and that the dynamical system describing their interaction, the observed system, is known.

The interconnection of the observed system with the observer will give rise to a set of error signals, namely the differences between the to be estimated signals and the actual estimates.

We consider the case of finite dimensional linear, time-invariant, differential (LTID) observed systems and observers. Then, by the elimination theorem, the set of error signals forms a finite dimensional LTID system as well, the error system. Usually, we want the error system to be stable, so that we are guaranteed to get an at least asymptotically accurate estimate. The corresponding observers are usually called asymptotic observers or stable observers.

We prove that in the above setting any observer that leads to an autonomous error system (hence, in particular any asymptotic observer) will contain an internal model of the controllable part of the observed system. This result generalizes a theorem proved in 2002 by Uwe Helmke and Paul Fuhrmann for state space systems.

This internal model principle for observers leads naturally to a parametrization of the set of all asymptotic observers for a given controllable system.