Large-scale estimation of dominant poles of a transfer function by an interpolatory framework

Abstract

We focus on the dominant poles of the transfer function of a descriptor system. The transfer function typically exhibits a large norm at and near the imaginary parts of the dominant poles. Consequently, the dominant poles provide information about the points on the imaginary axis where the \scrL \infty norm of the system is attained, and they are also sometimes useful to obtain crude reduced order models. For a large-scale descriptor system, we introduce a subspace framework to estimate a prescribed number of dominant poles. At every iteration, the large-scale system is projected into a small system, whose dominant poles can be computed at ease. Then the projection spaces are expanded so that the projected system after subspace expansion Hermite interpolates the large-scale system at the computed dominant poles. We prove an at-least-quadratic-convergence result for the framework, and provide numerical results confirming this. On real benchmark examples, the proposed frameworks appears to be more reliable than SAMDP in Rommes and Martins [IEEE Trans. Power Syst. 21 (2006), pp. 1471-1483], one of the widely used algorithms for the estimation of the dominant poles, for which we provide a theoretical explanation.