Linear System Matrices of Rational Transfer Functions

Abstract

AbstractIn this paper we derive new sufficient conditions for a linear system matrix \( S(\lambda ):=\begin{bmatrix} T(\lambda ) \;&{}\; -U(\lambda ) \\ V(\lambda ) \;&{}\; W(\lambda ) \end{bmatrix}, \) where \(T(\lambda )\) 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 \(R(\lambda ):= W(\lambda ) + V(\lambda )T(\lambda )^{-1}U(\lambda ),\) which is also known as the transfer function connected to the system matrix \(S(\lambda )\). The pole structure, zero structure and null space structure of \(R(\lambda )\) can be then computed with the staircase algorithm and the QZ algorithm applied to pencils derived from \(S(\lambda )\). 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.KeywordsSystem matrixStrong minimalityStrong irreducibility