Spatio-temporal transfer function conditions of positive realness for translation invariant lattice networks of interacting linear systems

TL;DR

This paper investigates conditions for positive realness in translation invariant lattice networks of linear systems, linking energy dissipation, stability, and dispersion relations using spatio-temporal transfer functions.

Contribution

It introduces new conditions for positive realness in lattice networks, connecting transfer function properties with stability and dispersion analysis.

Findings

Derived conditions for positive realness using block Toeplitz forms

Established links between energy dissipation and system stability

Analyzed phonon dispersion relations in the network context

Abstract

This paper is concerned with networks of interacting linear systems at sites of a multidimensional lattice. The systems are governed by linear ODEs with constant coefficients driven by external inputs, and their internal dynamics and coupling with the other component systems are translation invariant. Such systems occur, for example, in finite-difference models of large-scale flexible structures manufactured from homogeneous materials. Using the spatio-temporal transfer function of this translation invariant network, we establish conditions for its positive realness in the sense of energy dissipation. The latter is formulated in terms of block Toeplitz bilinear forms of the input and output variables of the composite system. We also discuss quadratic stability of the network in isolation from the environment and phonon theoretic dispersion relations.