Approximate state-space and transfer function models for 2x2 linear hyperbolic systems with collocated boundary inputs

Bartecki, Krzysztof

International Journal of Applied Mathematics and Computer Science, Tome 30 (2020) no. 3, pp. 475-491

Résumé

Two approximate representations are proposed for distributed parameter systems described by two linear hyperbolic PDEs with two time- and space-dependent state variables and two collocated boundary inputs. Using the method of lines with the backward difference scheme, the original PDEs are transformed into a set of ODEs and expressed in the form of a finite number of dynamical subsystems (sections). Each section of the approximation model is described by state-space equations with matrix-valued state, input and output operators, or, equivalently, by a rational transfer function matrix. The cascade interconnection of a number of sections results in the overall approximation model expressed in finite-dimensional state-space or rational transfer function domains, respectively. The discussion is illustrated with a practical example of a parallel-flow double-pipe heat exchanger. Its steady-state, frequency and impulse responses obtained from the original infinite-dimensional representation are compared with those resulting from its approximate models of different orders. The results show better approximation quality for the “crossover” input–output channels where the in-domain effects prevail as compared with the “straightforward” channels, where the time-delay phenomena are dominating.

Détail

Keywords: distributed parameter system, hyperbolic equations, approximation model, state space, transfer function
Mots-clés : układ o parametrach rozłożonych, równanie hiperboliczne, model aproksymacyjny, przestrzeń stanu, funkcja przesyłowa

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