Linear System Matrices of Rational Transfer Functions

Froilán Dopico (), María del Carmen Quintana () and Paul Van Dooren ()
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Froilán Dopico: Universidad Carlos III de Madrid
María del Carmen Quintana: Universidad Carlos III de Madrid
Paul Van Dooren: Université catholique de Louvain

A chapter in Realization and Model Reduction of Dynamical Systems, 2022, pp 95-113 from Springer

Abstract: Abstract In this paper we derive new sufficient conditions for a linear system matrix $$ S(\lambda ):=\begin{bmatrix} T(\lambda ) \;&{}\; -U(\lambda ) \\ V(\lambda ) \;&{}\; W(\lambda ) \end{bmatrix}, $$ S ( λ ) : = T ( λ ) - U ( λ ) V ( λ ) W ( λ ) , where $$T(\lambda )$$ T ( λ ) is assumed regular, to be strongly irreducible. In particular, we introduce the notion of strong minimality, and the corresponding conditions are shown to be sufficient for a polynomial system matrix to be strongly minimal. A strongly irreducible or minimal system matrix has the same structural elements as the rational matrix $$R(\lambda ):= W(\lambda ) + V(\lambda )T(\lambda )^{-1}U(\lambda ),$$ R ( λ ) : = W ( λ ) + V ( λ ) T ( λ ) - 1 U ( λ ) , which is also known as the transfer function connected to the system matrix $$S(\lambda )$$ S ( λ ) . The pole structure, zero structure and null space structure of $$R(\lambda )$$ R ( λ ) can be then computed with the staircase algorithm and the QZ algorithm applied to pencils derived from $$S(\lambda )$$ S ( λ ) . We also show how to derive a strongly minimal system matrix from an arbitrary linear system matrix by applying to it a reduction procedure, that only uses unitary equivalence transformations. This implies that numerical errors performed during the reduction procedure remain bounded. Since we use unitary transformations in both the reduction procedure and the computation of the eigenstructure, this guarantees that we computed the exact eigenstructure of a perturbed linear system matrix, but where the perturbation is of the order of the machine precision.

Keywords: System matrix; Strong minimality; Strong irreducibility (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-95157-3_6

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DOI: 10.1007/978-3-030-95157-3_6

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