Tracking poles and representing Hankel operators directly from data
This paper addresses the estimation of the poles of a rational function near the unit circle \(\partial D\) sample values of the function on a grid of points on \(\partial D\). It is motivated by numerical difficulties in the solution of the Nehari problem in the presence of poles close to \(\partial D\), and this in turn is motivated by sundry standard problems in \(H^{\infty}\) control theory, such as robust stabilization. Suppose given a rational function G restricted to \(\partial D\) and bounded there. We seek the rational function \(\hat G,\) bounded and analytic in D which minimises \(\| G-\hat G\|_{\infty}\). Mathematically the problem is solved once we know the singular vectors of the Hankel operator with symbol G. It transpires that a straightforward numerical approximation to the matrix of \(H_ G\) (based on Fourier coefficients) is effective as long as the poles of G are not close to \(\partial D\). However, unfavourably placed poles do occur in applications, and so a refinement of the obvious representation was proposed and analysed in \textit{J. W. Helton}, and \textit{N. J. Young} [Signal processing, Pt. II: Control theory and applications, Lect., Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 23, 115-137 (1990; Zbl 0711.93030)]. In broad terms, we use orthonormal bases for the domain and range of \(H_ G\) which are better tuned to the action of \(H_ G\) than are the standard bases \((e^{in\theta})\), and to this end we need estimates for the unfavourable poles of G. The present paper puts forward a method of obtaining such estimates based on a local best \(L^ 2\)-fit by rational functions of degree 1. A convergence theorem is proved and numerical examples are given.
scientific article; zbMATH DE number 4198390
scientific article