Stability Theory pp 1-10 | Cite as

# The Hurwitz Matrix and the Computation of second-order Information Indices

## Abstract

An all-pole transfer function *Q*(*s*) = 1/*P*(*s*), where *P*(*s*) is a monic Hurwitz polynomial of degree *n*, is uniquely characterized by the energies (second-order information indices) of *q*(*t*) = *LT* ^{−1}{*Q*(*s*)} and of its first *n* — 1 derivatives. These can be obtained by solving a set of linear equations whose coefficients matrix is the standard Hurwitz matrix for *P*(*s*) or by using the entries of its Routh table. Any strictly proper transfer function *W*(*s*) = *N*(*s*)/*P*(*s*) is characterized by *n* first-order information indices, e.g., Markov parameters, and by *n* second-order information indices, e.g., the energies of the related impulse response and its *n* — 1 successive derivatives; these are simply obtainable from the energies of *q*(*t*). This fact can be exploited to construct reduced-order models that retain both first- and second-order information indices of a given original system. The extension of this approach to multi-input multi-output systems described by a matrix fraction is analysed.

## Keywords

Impulse Response Model Reduction Information Index Lyapunov Equation Hurwitz Matrix## Preview

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