PCA-based approximation of a class of distributed parameter systems: classical vs. neural network approach

Abstract

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.

Steady-state analysis for a class of hyperbolic systems with boundary inputs

“…This class of systems, among which one can mention e.g. heat exchangers, transport pipelines, irrigation channels or electrical transmission lines, is usually described by PDEs of hyperbolic type and known under the common name of hyperbolic systems of conservation laws [3,4,5,6,9,8,15,20,24]. The present paper can be considered as a complement to our recent work [7], where a general transfer function representation for this class of systems has been analyzed.…”

“…where k 11 , k 12 , k 21 , k 22 and λ 1 , λ 2 are constant elements of the matrices K and Λ in (1), respectively. Therefore, it is assumed here that the only external influence on the state variables x 1 and x 2 is given by the boundary conditions (6). Two cases often occurring in practice are considered here: in the first one, both boundary conditions are given for the same edge (l = 0) of Ω and in the second -the input function u(t) acts on the two different edges, l = 0 and l = L, respectively.…”

“…Further analysis is based on the additional assumption that no boundary feedback nor reflection are present in the system, i.e. P 00 , P 01 , P 10 , P 11 in (6) are all zero matrices. The next assumption is that the system is given directly by the two weakly coupled PDEs (8) and (9).…”

Application properties of methods for fault detection and isolation in the diagnosis of complex large-scale processes

“…Another solution are unidirectional, multilayer networks with dynamic neurons [6,18]. There are also used RBF radial networks and GMDH networks [8,10,16,[39][40][41].…”

Transfer function-based analysis of the frequency-domain properties of a double pipe heat exchanger

“…Due to the mathematical complexity, their analysis is more difficult, and possible applications are more limited than in the case of the finitedimensional models. Therefore, in order to enable the implementation of the developed over the years techniques for the synthesis of control systems, the infinite-dimensional DPS models are usually replaced by their finitedimensional approximations [3,4,9,12,16]. Nevertheless, regardless of the approximation method used, the starting point for the synthesis of a control system should be based on the possibly most accurate description of the DPS, taking into account its infinite-dimensional nature, e.g.…”