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State-space Realization and Generalized Popov Belevitch Hautus Criterion for High-order Linear systems—The Singular Case

Guang-Ren Duan* and Ya-Jun Gao
International Journal of Control, Automation, and Systems, vol. 18, no. 8, pp.2038-2047, 2020

Abstract : In this paper, based on the inverse of block companion matrix pencils associated with matrix polynomials, a first-order state-space realization in descriptor system form for a general high-order linear system is derived. An important feature of the proposed realization is that it allows the high-order linear system to be singular. While in the case that the high-order system is nonsingular, two realizations in normal state-space system forms are directly deduced. With the help of the proposed realization, a generalized Popov Belevitch Hautus (PBH) criterion for highorder systems is then established in terms of the original system matrix polynomials. Examples are studied which well demonstrate the proposed theories.

Keyword : Companion matrices, continuous-time systems, discrete-time systems, high-order systems, PBH criterion, system realizations.

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