Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer functions considered are pointed out based on the existing results of semigroup theory. The influence of the location of the boundary inputs on the transfer function representation is demonstrated. The pole-zero as well as frequency response analyses are also performed. The discussion is illustrated with a practical example of a shell and tube heat exchanger operating in parallel-and countercurrent-flow modes.

This paper discusses irrational transfer function representation for a typical thick-walled double pipe heat exchanger. Closed-form analytical expressions for the individual elements of 2 9 2 transfer function matrix are derived both for the parallel-and the counter-flow configurations using the Laplace transform method. Based on the transfer function representation, its frequency responses are demonstrated both in the form of three-dimensional graphs as well as classical, two-dimensional Bode and Nyquist plots. Finally, steady-state temperature profiles are presented and compared for both flow arrangements considered.

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Abstract. In this article, an approximation of the spatiotemporal response of a distributed parameter system (DPS) with the use of the principal component analysis (PCA) is considered. Based on a data obtained by the numerical solution of a set of partial differential equations, a PCA-based approximation procedure is performed. It consists in the projection of the original data into the subspace spanned by the eigenvectors of the data covariance matrix, corresponding to its highest eigenvalues. The presented approach is carried out using both the classical PCA method as well as two different neural network structures: two-layer feed-forward network with supervised learning (FF-PCA) and single-layer network with unsupervised, generalized Hebbian learning rule (GHA-PCA). In each case considered, the effect of the approximation model structure represented by the number of eigenvectors (or, in the neural case, units in the network projection layer) on the mean square approximation error of the spatiotemporal response and on the data compression ratio is analysed. As shown in the paper, the best approximation quality is obtained for the classical PCA method as well as for the FF-PCA neural approach. On the other hand, an adaptive learning method for the GHA-PCA network allows to use it in e.g. an on-line identification scheme.

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