Numerical modelling of coupled linear dynamical systems

Abstract: Numerical modelling of several coupled passive linear dynamical systems (LDS) is considered. Since such component systems may arise from partial differential equations, transfer function descriptions, lumped systems, measurement data, etc., the first step is to discretise them into finite-dimensional LDSs using, e.g., the finite element method, autoregressive techniques, and interpolation. The finite-dimensional component systems may satisfy various types of energy (in)equalities due to passivity that require translation into a common form such as the scattering passive representation. Only then can the component systems be coupled in a desired feedback configuration by computing pairwise Redheffer star products of LDSs. Unfortunately, a straightforward approach may fail due to ill-posedness of feedback loops between component systems. Adversities are particularly likely if some component systems have no energy dissipation at all, and this may happen even if the fully coupled system could be described by a finite-dimensional LDS. An approach is proposed for obtaining the coupled system that is based on passivity preserving regularisation. Two practical examples are given to illuminate the challenges and the proposed methods to overcome them: the Butterworth low-pass filter and the termination of an acoustic waveguide to an irrational impedance.