A scaling theory for linear systems

In this paper we develop a theory of scaling for rational (transfer) functions in terms of transformation groups. In particular, we identify two different four-parameter scaling groups which play natural roles in studying linear systems and investigate the effect of scaling on Fisher information and related statistical measures in system identification. The scalings considered include change of time scale, feedback, exponential scaling, magnitude scaling, etc. The scaling action of the groups studied in this paper is tied to the geometry of transfer functions in a rather strong way as becomes apparent in our examination of the invariants of scaling. As a result, the scaling process also provides new insight into the parameterization question for rational functions.