Abstract
Signals that evolve over multiple variables or indices occur in all fields of science and engineering. Measurements of the distribution of temperature across the globe during a certain period of time are an example of such a signal. Multivariable systems describe the evolution of signals over a spatialtemporal domain. The mathematical equations involved in such a description are called a model and this model dictates which values the signals can obtain as a function of time and space. In an industrial production setting, such mathematical models may be used to monitor the process or determine the control action required to reach a certain setpoint. Since their evolution is over both space and time, multivariable systems are described by Partial Differential Equations (PDEs).
Generally, it is not the signals or systems themselves one is interested in, but the information they carry. The main numerical tools to extract system trajectories from the PDE description are Finite Element (FE) methods. FE models allow simulation of the model via a discretization scheme. The main problem with FE models is their complexity, which leads to large simulation time, making them not suitable for applications such as online monitoring of the process or modelbased control design. Model reduction techniques aim to derive lowcomplexity replacement models from complex process models, in the setting of this work, from FE models. The approximations are achieved by projection on lowerdimensional subspaces of the signals and their dynamic laws. This work considers the computation of empirical projection spaces for signals and systems evolving over multidimensional domains. Formally, signal approximation may be viewed as a lowrank approximation problem. Whenever the signal under consideration is a function of multiple variables, lowrank approximations can be obtained via multilinear functionals, tensors. It has been explained in this work that approximation of multivariable systems also boils down to lowrank approximation problems.The first problem under consideration was that of finding lowrank approximations to tensors. For order2 tensors, matrices, this problem is well understood. Generalization of these results to higherorder tensors is not straightforward. Finding tensor decompositions that allow suitable approximations after truncation is an active area of research. In this work a concept of rank for tensors, referred to as multilinear or modal rank, has been considered. A new method has been defined to obtain modal rank decompositions to tensors, referred to as Tensor Singular Value Decomposition (TSVD). Properties of the TSVD that reflect its sparsity structure have been derived and lowrank approximation error bounds have been obtained for certain specific cases. An adaptation of the TSVD method has been proposed that may give better approximation results when not all modal directions are approximated. A numerical algorithm has been presented for the computation of the (dedicated) TSVD, which with a small adaptation can also be used to compute successive rankone approximation to tensors. Finally, a simulation example has been included which demonstrates the methods proposed in this work and compares them to a wellknown existing method.
The concepts that were introduced and discussed with regard to signal approximation
have been used in a system approximation context.We have considered the wellknown model reduction method of Proper Orthogonal Decompositions (POD). We have shown how the basis functions inferred from the TSVD can be used to define projection spaces in POD. This adaptation is both a generalization and a restriction. It is a generalization because it allows POD to be used in a scalable fashion for problems with an arbitrary number of dependent and independent variables. However, it is also a restriction, since the projection spaces require a Cartesian product structure of the domain. The model reduction method that is thus obtained has been demonstrated on a benchmark example from chemical engineering. This application shows that the method is indeed feasible, and that the accuracy is comparable to existing methods for this example. In the final part of the thesis the problem of reconstruction and approximation of multidimensional signals was considered. Specifically, the problem of sampling and signal reconstruction for multivariable signals with nonuniformly distributed sensors on a Cartesian domain has been considered. The central question of this chapter was that of finding a reconstruction of the original signal from its samples. A specific reconstruction map has been examined and conditions for exact reconstruction have been presented. In case that exact reconstruction was not possible, we have derived an expression for the reconstruction error.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  6 Jun 2011 
Place of Publication  Eindhoven 
Publisher  
Print ISBNs  9789038624891 
DOIs  
Publication status  Published  2011 
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Belzen, van, F. (2011). Approximation of multivariable signals and systems : a tensor decomposition approach. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR712120