Spatio-temporal transfer function conditions of positive realness for translation invariant lattice networks of interacting linear systems
Provides theoretical conditions for energy dissipation in large-scale flexible structures, but the contribution is incremental as it extends known positive realness concepts to a specific class of systems.
This paper establishes conditions for positive realness (energy dissipation) of translation invariant lattice networks of interacting linear systems using their spatio-temporal transfer function, and discusses quadratic stability and phonon dispersion relations.
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.