On the partial realization problem
Andrea Gombani · 2010
owner or a Hankel matrix. We consider an application to the usual partial realization problem. The results are quite general and no particular assumption on the location of the interpolating nodes are needed. I. INTRODUCTION AND PRELIMINARIES We consider here a two sided interpolation problem where we allow non disjoint interpolation nodes. We first consider the case when the interpolation points are disjoint from the poles of the interpolating function and show then how this restriction can be lifted. The problem has a long history (starting, in some sense, with the Ho-Kalman algorithm, see [8]) and was investigated by Rissanen [10] , Gragg and Lindquist [7] and others. In the interpolation formulation it was studied by Anderson and Antoulas [1] using L¨ owner matrices and later by Anotoulas, Ball, Kang and Willems [2] using linear fractional transformations. An approach which led to these results was developed in a special case by Kimura [9] and Georgiou [5] and generalized by the authors We show how a state space approach to the problem yields simple formulas for constructing the interpolants which do not require a specific structure of the interpolation nodes (e.g. all equal or all disjoint) and allows for a generalization to the case when the interpolant has poles also at the interpolation nodes. If M is a complex matrix, Tr shall denote its trace, M T its transpose and M its transpose conjugate. (M) denotes its spectrum. The inclusion (M1) (M2) expresses the fact the spectrum of M1 forms a subset of that of M2 including multiplicities. Let F be a rational p ◊ m matrix of McMillan degree N with realization F(z) = D+C (sI A) 1 B. We are going to use Rosenbrock’s notation F A B