A large scale dynamical system can have a large number of modes. Like a general square matrix can be approximated by its largest eigenvalues, i.e. by projecting it onto the space spanned by the eigenvalues corresponding to the largest eigenvalues, a dynamical system can be approximated by its dominant modes: a reduced order model, called the modal equivalent, can be obtained by projecting the state space on the subspace spanned by the dominant modes. This technique, modal approximation or modal model reduction, has been successfully applied to transfer functions of large-scale power systems, with applications such as stability analysis and controller design, see [16] and references therein.
The dominant modes, and the corresponding dominant poles of the system transfer function, are specific eigenvectors and eigenvalues of the state matrix. Because the systems are very large in practice, it is not feasible to compute all modes and to select the dominant ones. This chapter is concerned with the efficient computation of these dominant poles and modes specifically, and their use in reduced order modeling. In Sect. 2 the concept of dominant poles and modal approximation is explained in more detail. Dominant poles can be computed with specialized eigensolution methods, as is described in Sect. 3. Some generalizations of the presented algorithms are shown in Sect. 4. The theory is illustrated with numerical examples in Sect. 5 and 6 concludes.
Part of the contents of this chapter is based on [15, 16]. The pseudocode algorithms presented in this chapter are written using Matlab-like [21] notation.
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Rommes, J. (2008). Modal Approximation and Computation of Dominant Poles. In: Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78841-6_9
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