The problem of constructing data-based, predictive, reduced models for the Kuramoto–Sivashinsky equation is considered, under circumstances where one has observation data only for a small subset of the dynamical variables. Accurate prediction is achieved by developing a discrete-time stochastic reduced system, based on a NARMAX (Nonlinear Autoregressive Moving Average with eXogenous input) representation. The practical issue, with the NARMAX representation as with any other, is to identify an efficient structure, i.e., one with a small number of terms and coefficients. This is accomplished here by estimating coefficients for an approximate inertial form. The broader significance of the results is discussed.

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Στην παρούσα διπλωματική υλοποιήθηκε ένα προγραμματιστικό περιβάλλον σε Matlab το οποίο θα μας επιτρέπει να αναλύσουμε σήματα (χρονοσειρές) με δύο τεχνικές: (Ι) με την τεχνική της εμβάπτισης του σήματος σε χώρους φάσεων υψηλών διαστάσεων με σκοπό τον υπολογισμό της φράκταλ (μορφοκλασματικής) διάστασης του ελκυστή που παράγεται στο χώρο φάσεων, χρησιμοποιώντας το θεώρημα εμβάπτισης του Takens και τη μέθοδο Grassberger & Procaccia, και (ΙΙ) με μοντελοποίηση του σήματος με τη μέθοδο NARMAX, ενσωματώνοντας Extended Kalman Filters και τη Θεωρία Διαμελισμού του Λαϊνιώτη για την εύρεση της τάξης (βαθμού πολυπλοκότητας) των NARMAX μοντέλων. Σκοπός της διπλωματικής είναι η σύγκριση των αντίστοιχων αποτελεσμάτων για διάφορες κατηγορίες σημάτων, με σκοπό να διαπιστωθεί κατά πόσο η φράκταλ διάσταση του σήματος σχετίζεται με την τάξη των NARMAX μοντέλων του σήματος. / In this thesis, we implemented in Matlab, a programming environment which allows us to analyze signals (time series) with two techniques: (I) with the technique of immersion of the signal in high-dimensional phase space to calculate the fractal dimension of the attractor generated in phase space, using the theorem of Takens and the method of Grassberger & Procaccia, and (II) signal modeling method NARMAX, incorporating Extended Kalman Filters and the Laynioti partition theorem for finding the degree of complexity of NARMAX models. The aim of the thesis is the comparison of the results for various categories of signals, in order to determine if the fractal dimension of the signal is associated with the degree of complexity of NARMAX models of the signal.

Submitted by Silvana Teresinha Dornelles Studzinski (sstudzinski) on 2016-12-23T10:42:58Z No. of bitstreams: 1 Cassio Binkowski_.pdf: 1965327 bytes, checksum: 87b7380f1bab367237fb868e0de20388 (MD5) / Made available in DSpace on 2016-12-23T10:42:59Z (GMT). No. of bitstreams: 1 Cassio Binkowski_.pdf: 1965327 bytes, checksum: 87b7380f1bab367237fb868e0de20388 (MD5) Previous issue date: 2016-09-14 / Nenhuma / A identificação de sistemas está longe de ser uma tarefa nova. Sendo inicialmente proposta na metade do século XX, foi extensamente desenvolvida para sistemas lineares, devido às exigências da época relacionadas à complexidade dos sistemas e também do poder computacional, atingindo excelente resultados. No entanto, com o aumento da complexidade dos sistemas e das exigências de controle, os modelos lineares não mais conseguiam representar os sistemas em toda a faixa de operação exigida, sendo assim requerendo uma aplicação dos modelos não-lineares. Visto que todos os sistemas presentes na natureza possuem certo grau de não linearidade, é correto afirmar que um modelo não-linear é capaz de representar as dinâmicas dos sistemas de forma mais compreensiva que um modelo linear. A identificação de sistemas não lineares foi então estudada e diversos modelos foram propostos, atingindo ótimos resultados. Nesse trabalho foi realizado um estudo de dois modelos não-lineares, NARMAX e Hammerstein-Wiener, aplicando esses modelos a dois processos simulados. Foram então derivados dois algoritmos para realizar a estimação dos parâmetros dos modelos NARMAX e Hammerstein-Wiener, utilizando um estimador ortogonal, e também um algoritmo para geração de sinais de entrada multinível. Os modelos foram então estimados para os sistemas simulados, e comparados utilizando os critérios AIC, FPE, Lipschitz e de correlação cruzada de alta ordem. Os melhores resultados foram obtidos com os modelos Hammerstein-Wiener-OLS e NARMAX-OLS, ao contrário do modelo NARMAX-RLS. No entanto, devido a resultados bastante divergentes entre os modelos, pode-se concluir que essa área ainda carece de desenvolvimento de técnicas precisas para comparação e avaliação de modelos, bem como quanto à quantificação do nível de não-linearidade do sistema em questão. / The task of system identification is far from being a new one. It was initially proposed in the mid of the 20th century, and had then been extensively developed for linear systems, due to the demands of that time concerning computational power, systems complexity and control requirements. It has achieved excellent results in this approach. However, due to the rise of systems complexity and control requirements, linear models were no longer able to meet the desired accuracy and larger operating range, and therefore the usage nonlinear models were pursued. As all systems in nature are nonlinear to some extent, it is correct to state that nonlinear models can represent a whole lot more of systems’ dynamics than linear models. Nonlinear models were then studied, and several techniques were presented, being able to achieve very good results. In this work, two of the available nonlinear models were studied, namely NARMAX and Hammerstein-Wiener, applying these models in two simulated systems. Two algorithms were then derived to estimate parameters for NARMAX and Hammerstein-Wiener models using an orthogonal estimator, and also an algorithm for generating multi-level input signals. The models were then estimated to the simulated systems, and compared using the AIC, FPE, Lipschitz and high-order cross-correlation criteria. The best results were obtained for the Hammerstein-Wiener-OLS and NARMAX-OLS models, as opposed to the NARMAX-RLS model. However, due to divergent observed results between models, it can be concluded that precise methods for model comparison and validation still needs to be developed, as well as a method for nonlinearity quantification for the system in hand.