Computation of balanced realisation approximations of delay systems (Q1817715)

Delay systems with transfer functions of the form \(G(s)= e^{-sT}g(s)\) are considered, where \(T>0\) and \(g(s)\) is a scalar strictly proper rational function. One gives numerically reliable algorithms for the computation of a finite-dimensional balanced realization of the stable part of \(G(s)\). Some properties of the Hankel singular values are emphasized, including an upper bound and their asymptotic behaviour; knowledge of this behaviour leads to a reduction in computational effort. An algorithm is proposed for the determination of the Hankel singular values as the zero crossings of a determinant function. The accuracy of these calculations is improved by employing matrix transformations and scaling. Another algorithm gives the initial values of the Schmidt vectors. These results are used in an algorithm which provides approximation of delay based on (Ober type) balanced realization. Some numerical examples illustrate the efficiency of these algorithms in the computation of the balanced realization approximation.