Parametric interpolation, H ∞-control and model reduction

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Abstract
Adamjan et al. (1971) established an important equivalence between the optimal Hankel norm model reduction problem (the Schur-Takagi problem) and a classical interpolation problem due to Ahiezer. In two later papers, Ball (1983) and Ball and Helton (1985) investigated a vector version of this interpolation problem, using the sophisticated Ball-Helton operator theory. A parametric approach to the work of Adamjan et al. (1971) was studied by Genin and Kung (1981) in the rational case. We demonstrate that a class of H ∞-optimal control problems and the Hankel norm model reduction problem can be recast as parametric vector interpolation problems (PVIPs), which we solve using the classical Schur construction and results on the inertia of matrices. The connection between parametric vector interpolation and model reduction provides an alternative approach to the PVIP using Glover's state-space formulae (Glover 1984). In both cases, special care needs to be taken at certain ‘non-regular’ values of a gain parameter, a difficulty not recognized in the non-parametric work (e.g. Delsarte et al. 1981).