Multidimensional transfer function models

Abstract

Transfer functions are a standard description of onedimensional linear and time-invariant systems. They provide an alternative to the conventional representation by ordinary differential equations and are suitable for computer implementation. This article extends that concept to multidimensional (MD) systems, normally described by partial differential equations (PDEs). Transfer function modeling is presented for scalar and for vector PDEs. Vector PDEs contain multiple dependent output variables, e.g., a potential and a flux quantity. This facilitates the direct formulation of boundary and interface conditions in their physical context. It is shown how carefully constructed transformations for the space variable lead to transfer function models for scalar and vector PDEs. They are the starting point for the derivation of discrete models by standard methods for one-dimensional systems. The presented functional transformation approach is suitable for a number of technical applications, like electromagnetics, optics, acoustics and heat and mass transfer.

Distributed Control of Heat Conduction in Thermal Inductive Materials with 2D Geometrical Isomorphism

“…Rabenstein and Trautmann [38][39] Hong [42] Hong et al [2] ‧On bounded space regions has it been well justified to work.…”

“…Therein, the Double Laplace transform, for example in [36,37], and Double Fourier transform, for example in [15,17,19], were always applied for signal processing and feedback control in semi-infinite and infinite space regions, respectively. The Laplace-Galerkin transform [1,[38][39][40] or Fourier-Galerkin transform [41,42] is justified to model the non-Fourier heat conduction and its controllers for bounded space regions that are of real concern in heat conduction practice. Such an integral transform is obtained through the composite of Laplace transform in time and modal decomposition in space.…”

Modelling and Controller Design of Resonant Thermoacoustic Solar AC Power Generators

“…For control modelling, the wave equation above in conjunction with inhomogeneous boundary conditions of Rijke tube is then transformed into a single spatiotemporal transfer function, which is an extended application of the functional tool known as nD transfer function models (Hong, 2010;Hong, Su, Chou, & Hung, 2011;Rabenstein, 1999;Rabenstein & Trautmann, 2002. Therein the inhomogeneous boundary conditions arise from the von Neumann source of heat-flux at head end and the Robin source of piston-speed at tail end, both of which are transformed into Dirac distribution by way of 2D transferfunction.…”

Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics

“…In this paper, a general class of Sturm-Liouville dynamics is defined with Spatiotemporal Transfer Function (STF) obtained though the composite of modal decomposition and Laplace transform [5][6][7][13][14][15]. The values of STF along imaginary axis thus define mode-frequency responses, whereon a 2D-∞ H norm is created in mode-frequency domain that is employed to prove the small-gain theorem in Class C4.…”