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Στη διπλωματική εργασία αυτή ασχοληθήκαμε με την πρόβλεψη της εξόδου μη-γραμμικών συστημάτων με τη χρήση αναδρομικών αλγορίθμων που χρησιμοποιούν συναρτήσεις πυρήνα. Παρουσιάζεται ο δικός μας αναδρομικός αλγόριθμος πρόβλεψης και βλέπουμε πως αποδίδει σε σχέση με έναν άλλο ήδη υπάρχων και ιδιαίτερα δημοφιλή αλγόριθμο. Στο πρώτο κεφάλαιο δίνουμε μια σύντομη περιγραφή του προβλήματος που καλούμαστε να λύσουμε. Στη συνέχεια δείχνουμε πως οι συναρτήσεις πυρήνα μπορούν να χρησιμοποιηθούν για να μας βοηθήσουν να λύσουμε το πρόβλημα αυτό. Στο δεύτερο κεφάλαιο αναλύουμε περισσότερο τις συναρτήσεις πυρήνα και τις ιδιότητες που τις χαρακτηρίζουν. Παρουσιάζουμε τα βασικά θεωρήματα και βλέπουμε πώς διαμορφώνεται το πρόβλημα της πρόβλεψης με την εφαρμογή αυτών. Επιπλέον παρουσιάζουμε πως το πρόβλημα μας μετατρέπεται στο γνωστό πρόβλημα γραμμικών ελαχίστων τετραγώνων στην περίπτωση που χρησιμοποιήσουμε γραμμικό πυρήνα. Στο τρίτο κεφάλαιο παρουσιάζουμε τον αλγόριθμο μας, αναλύοντας το συλλογισμό που μας οδήγησε σε αυτόν. Δίνουμε επίσης μια περιγραφή ενός άλλου αλγορίθμου που χρησιμοποιείται ήδη για την επίλυση τέτοιων προβλημάτων. Στο τέταρτο κεφάλαιο γίνονται μια σειρά από προσομοιώσεις σε MATLAB οπού βλέπουμε πόσο καλά μπορεί να κάνει την πρόβλεψη των εξόδων μη-γραμικών συστημάτων ο αλγόριθμός μας. Επίσης αντιπαραθέτουμε και την απόδοση του ανταγωνιστικού αλγορίθμου. Στα πειράματα μας εξετάζουμε το σφάλμα πρόβλεψης των προαναφερθέντων αλγορίθμων, την ταχύτητα σύγκλισης τους καθώς και την σθεναρότητα τους. Τέλος παρουσιάζουμε τα συμπεράσματα μας εξηγώντας γιατί πιστεύουμε ότι η δικία μας προσέγγιση υπερτερεί της άλλης. / This dissertation deals with the problem of predicting the output of non-linear systems using recursive kernel methods. We will present our own prediction algorithm and see how it performs in relation to a widely used alternative algorithm. In the first chapter we provide a short description of the problem of non-linear prediction. We then describe how kernel methods could help us solve this problem. In the second chapter we further analyze kernel functions and their properties. We present the basic theorems and see how these affect and transform the problem at hand. Furthermore, we explain how this problem results in the linear least squares problem in case we use the linear kernel. In the third chapter we present our algorithm and reasoning that led to it. We also describe a different algorithm that is already used to predict such signals. In the fourth chapter we perform a series of simulations in the Matlab environment were we evaluate how well the two approaches predict the output. In this evaluation we consider the complexity, the error and robustness of the algorithms. Finally we present our conclusion and explain why our algorithm is superior to the alternative.

Ο έλεγχος μη γραμμικών δειγματοληπτικών συστημάτων αποτελεί μια σημαντική πρόκληση, στην ανάλυση και σχεδιασμό ελεγκτών συστημάτων, με μεγάλη πρακτική εφαρμογή. Σε αυτή την μελέτη παρουσιάζουμε ένα από τα πιο σημαντικά εργαλεία ανάλυσης και σχεδιασμού, για μια γενική κατηγορία μη γραμικών δειγματοληπτικών συστημάτων (nonlinear sampled-data control). Παρουσιάζεται μια γενίκευση και ενοποίηση των γνωστών αποτελεσμάτων σχεδίασης εξομοίωσης (Emulation Design), διαθέσιμων στην βιβλιογραφία, θεωρώντας την διατήρηση των γενικευμένων ανισώσεων διάχυσης (Dissipation inequalities) υπό δειγματοληψίας, στο πλαίσιο του σχεδιασμού εξομοιωτών ελεγκτών δυναμικής κατάστασης ανάδρασης (dynamic state feedback controllers). Στην συνέχεια παρουσιάζονται δύο παραδείγματα μελέτης εφαρμογής των αποτελεσμάτων της γενικής μελέτης σχετικά με την διατήρηση των ιδιοτήτων διαχυτικότητας (dissipativity properties), ευστάθειας (stability), εισόδου-σε-κατάσταση σταθερότητας (input-to-state stability) και παθητικότητας (passivity), όταν ο ελεγκτής διακριτού χρόνου ενός συστήματος σχεδιάζεται με εξομοίωση. / The current diploma thesis is about a general and unified framework of a sampled-data nonlinear system using the emulation design.

Thesis (MPhil)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Organisations are faced with ever increasing complexity. While there are many responses to complexity just as there are many definitions of complexity this thesis highlights the use of agile software development as a useful method. The case of agile software development is influenced by its people first approach and minimal process implementation to implement constraints in which phenomena of complexity can be understood. Chapter 1 introduces why complexity is relevant in the organisation today and the issues associated with complexity. A basic introduction to agile software development and why it is a response to complexity is tabled. Chapter 2 investigates the concepts of complexity. To highlight the difference between linear and non-linear systems and reductionistic thinking the ideas that not all things are complex is explored. Key concepts of complex systems are described to come to a better understanding of what entails a complex system. Information theory and dynamic systems are discussed including system attributes such as attractors and bifurcations. Finally the theory of complex adaptive systems is presented and a classification of all complexity theories is tabled. Chapter 3 is centred on agile software development, presenting practices and processes as a understanding of how agile software development is applicable to complexity. Three agile methods are identified. Chapter 4 presents the ideas of modelling and limits to understanding. Models of complex systems are useful but are limited, due to the properties of complex systems. The concept of mental models leads to models of organisations and how leaders need to help with transformation of these models towards models that are more aligned to agile type thinking. Chapter 5 looks at the core concepts, and practices of agile applied to complexity and why these are relevant in responding to complexity. Towards the end of chapter the role of narrative is explored in the terms of problem definition and solutioning in complex systems. Chapter 6 concludes the thesis with insights where agile software development is an appropriate response to complexity and the conditions in which it is not. / AFRIKAANSE OPSOMMING: Organisasies staar toenemende kompleksiteit in die gesig en alhoewel daar verskeie maniere bestaan om kompleksiteit teen te werk, asook verskeie definisies van wat kompleksiteit is, focus die tesis op aanpasbare ("agile") sagteware ontwikkeling as n bruikbare metode. Met eienskappe soos “mense/verbruiker/gebruiker – eerste” benadering asook minimale proses implementering, verskaf aanpasbare sagteware ontwikkeling die raamwerk waar binne die konsep kompleksiteit verstaan kan word. Hoofstuk 1 bespreek die kwessies rondom kompleksiteit en die relevansie daarvan in organisasies vandag. Verder word aanpasbare sagteware ontwikkeling kortliks bespreek en hoe dit gebruik kan word om kompleksiteit te verstaan, word getoon. Hoofstuk 2 ondersoek die verskillende konsepte random kompleksiteit. Daar word gekyk na linêre en nie-linêre stelsels asook die oortuiging dat nie alle dinge kompleks is nie word geondersoek. Sleutel kenmerke van komplekse stelsels word beskryf om sodoende ‘n better begrip te kry van wat ‘n komplekse stelsel behels. Informasie teorie en dinamiese stelsels word bespreek, insluitend kenmerke soos “attraktors” en “bifurkasies”. Laastens, word die teorie rondom komplekse aanpasbare stelsels bespreek en ‘n klassifikasie van alle komplekse teorieë word uiteengesit. Hoofstuk 3 fokus op “agile software development” en hoe sulke praktyke en prosesse toepaslik is op kompleksiteit. Daaropvolgend word drie aanpasbare metodes ge-identifiseer en individueel bespreek. Hoofstuk 4 verduidelik die idees rondom modellering en grense van begrip. Modelle van komplekse stelsels kan nuttig wees, maar weens die eienskappe van komplekse stelsels, is die nuttigheid beperk. Die konsep van mentale modelle lei tot modelle van organisasies en hoe leiers hierdie modelle moet transformer na modelle wat meer in-lyn is met agile denkwyses. Hoofstuk 5 kyk na die kern konsepte en praktyke waar “agile” toegepas word tot kompleksiteit en waarom dit relevant is om kompleksiteit teen te werk. In die tweede helfde van die hoofstuk word die rol van narratief ondersoek in terme van probleem-identifisering en die soek na oplossings in komplekse sisteme. Hoofstuk 6 sluit af met insigte tot aanpasbare sagteware ontwikkelingsmetodes as toepaslike oplossing tot kompleksiteit asook die toestande waaronder dit geld.

Submitted by Ricardo Sampaio null (cobraxms@yahoo.com.br) on 2018-04-02T17:22:08Z No. of bitstreams: 1 Ricardo Sampaio - Matrizes no estudo e na resolução de sistemas lineares.pdf: 1339231 bytes, checksum: c27b73e8517bd73345ca823cfeecf3e5 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-04-03T12:04:18Z (GMT) No. of bitstreams: 1 sampaio_r_me_sjrp.pdf: 1130997 bytes, checksum: 82f22c665b2241146e03d978ffb2e926 (MD5) / Made available in DSpace on 2018-04-03T12:04:18Z (GMT). No. of bitstreams: 1 sampaio_r_me_sjrp.pdf: 1130997 bytes, checksum: 82f22c665b2241146e03d978ffb2e926 (MD5) Previous issue date: 2018-03-01 / Os sistemas de equações lineares são muito úteis, pois podem modelar matematicamente diversos problemas em Estatística, Física, Química, Engenharia, Administração, Economia, enfim, em várias áreas do conhecimento. Historicamente, o cálculo de soluções desse tipo de sistema pelos chineses motivou o surgimento das matrizes, que, grosseiramente falando, são tabelas de elementos distribuídos em linhas e colunas. O uso de matrizes facilita o estudo e também a resolução de sistemas lineares, pois simplificam a notação e padronizam os procedimentos. O método do escalonamento de matrizes, por exemplo, é uma técnica que pode ser utilizada em sistemas lineares em geral, além de ser facilmente automatizada devido ao seu algoritmo. O objetivo deste trabalho é apresentar alguns conceitos e resultados sobre matrizes e sistemas lineares e abordar a relação entre eles, além de propor alguns problemas que podem ser resolvidos utilizando esses resultados. Professores do Ensino Médio podem utilizar tais problemas em sala de aula para trabalhar esse assunto com seus alunos. / The systems of linear equations are very useful because they can mathematically model several problems in Statistics, Physics, Chemistry, Engineering, Administration, Economics, and finally, in various areas of knowledge. Historically, the computation of solutions of this type of system by the Chinese motivated the emergence of matrices, which, roughly speaking, are tables of elements distributed in rows and columns. The use of matrices facilitates the study and the resolution of linear systems because they simplify the notation and standardize the procedures. The Echelonment method, for example, is a technique that can always be used in linear systems in general, besides being easily automated due to its algorithm. The objective of this work is to present some concepts and results about matrices and linear systems and to approach the relation between them, besides proposing some problems that can be solved using these results. High school teachers may use such problems in the classroom to work on this subject with their students.

[pt] A tese apresenta uma revisão do problema de estabilidade robusta estruturada de sistemas lineares invariantes no tempo, isto é, da procura de condições para que as raízes de polinômios e autovalores de matrizes estejam contidas em uma região predeterminada do plano complexo caso os seus coeficientes variem dentro de uma região dada no espaço paramétrico. Além de revisitar resultados já estabelecidos a tese apresenta novos resultados que podem ser de interesse teórico e/ou prático. A tese discute o problema da análise, como descrito acima, e da síntese de sistemas com realimentação unitária. / [en] This thesis presents a review of the problem of structured robust stabilization of linear time invariant systems, that is, search of conditions for the roots of polynomials and eigenvalue of matrices to stay in a region of the complex palne when the coefficients change inside a fixed region in the parametric space. Besides revisiting already estabished results, the thesis presents new results which can be of theoretical and/or practical interest. The thesis discuss the problem of analysis as described above, and the synthesis with unity feedback.

Esta dissertação aborda a síntese de leis de controle baseadas em compensação dinâmica de saída para sistemas lineares contínuos invariantes no tempo, sujeitos a restrições no atuador. Baseados em condições suficientes expressas em LMIs, obtidas a partir de uma nova condição de setor para a função zona-morta, dois métodos são propostos. O primeiro método propõe a síntese de compensadores dinâmicos de anti-windup de ordem plena e reduzida para um sistema com um controlador dado a priori desprezando limites do atuador. A síntese dos compensadores considera um atuador restrito em amplitude e o sistema sujeito a perturbações limitadas em norma L2. A verificação das condições enunciadas garante a estabilidade da origem em malha fechada, ganho L2 limitado da perturbação à saída controlada do sistema, e um conjunto de inicialização dos estados do sistema tolerante à perturbação de norma L2 máxima, conhecida. Na ausência de perturbação a origem é garantida assintoticamente estável. Para o compensador de ordem plena, condições em LMIs garantem a estabilidade local e global da origem. Para o caso de ordem reduzida, inicialmente obtém-se estas condições expressas em termos de BMIs, sobre as quais aplica-se o Lema de Finsler, e pela escolha apropriada de seus multiplicadores, condições expressas em LMIs são obtidas. Dois métodos baseados em BMIs são propostos para estender os resultados obtidos das otimizações, em esquemas de relaxação. O segundométodo proposto aborda a síntese de controladores dinâmicos, para um atuador restrito em amplitude e taxa de variação, sujeito a perturbações limitadas em norma L2. Propõe-se um controlador dinâmico não-linear, composto por um compensador dinâmico linear, um integrador saturante e laços anti-windup. Esta metodologia possibilita a síntese simultânea de um controlador, e da malha de compensação anti-windup. A verificação das condições enunciadas garante a estabilidade da origem em malha fechada, ganho L2 limitado da perturbação à saída controlada do sistema, e um conjunto de inicialização do sistema tolerante à perturbação de norma L2 máxima, conhecida. Na ausência de perturbação a origem é garantida assintoticamente estável. As condições para estabilidade local e global são formuladas em LMIs a partir da aplicação do lema de Finsler e da escolha apropriada de multiplicadores. Dois métodos também baseados em BMIs podem estender os resultados obtidos das otimizações em esquemas de relaxação. Casos especiais são propostos a partir do método geral obtido; nestes casos condições diretamente em LMI podem ser obtidas. Problemas de otimização são propostos para ambos os métodos para maximizar a tolerância à perturbação e a minimização do ganho L2 da perturbação à saída controlada. Exemplos numéricos são apresentados para ilustrar o efeito de cada método na solução do problema de máxima tolerância à perturbação. / This thesis addresses control law synthesis based upon dynamic output compensation of continuous time invariant linear systems. By sufficient conditions expressed in LMIs, obtained from a new sector condition to the dead-zone function, two methods are proposed. The first one, comprises the synthesis of full and reduced order dynamic antiwindup compensators for a system with a linear controller a priori given, regardless the actuators limits. The compensator synthesis considers an amplitude limited actuator and a system subjected to L2-norm limited perturbation. The verification of the announced conditions assures the closed-loop origin stability, limited L2 gain for the perturbation at the system controlled output, and a system states initialization set tolerating the known L2-norm disturbance. In the absence of disturbance the origin is asymptotically stable guaranteed. For the full-order compensator, LMI conditions assure the local and global origin closed-loop stability. For the reduced order case, at first, BMI conditions are obtained, which, by the Finsler Lemma and proper choosing of itsmultipliers, become LMIs. Two methods are presented based on BMIs in order to improve the obtained results, by relaxation schemes. The second method addresses the synthesis of dynamic controllers considering a linear plant with the actuator restricted both in amplitude and rate, the system is meant to be subjected to L2 norm disturbances. The proposed methodology issues a non-linear dynamic controller, composed by a dynamic linear compensator, a saturating integrator and anti-windup loops. This approach allows the simultaneous synthesis of both the controller and the anti-windup loops. The validity of its announced conditions guarantees the closedloop origin stability, an upper-bound to the disturbances L2 gain on the plant’s controlled output, and an initialization set for the system that stands the specified disturbance. In the absence of disturbances the system’s origin is asymptotically stable guaranteed. Both local and global stability conditions are given in terms of LMIs. Two BMI-based methods arise in order to improve the results obtained on the referred optimizations problems by relaxation schemes. Special cases are derived from the general method; in these cases directly LMI conditions may be obtained. Optimization problems arise in both methods in order to maximize disturbance tolerance and rejection, by means of respectively its L2-norm, and L2-gain at the controlled output. Numerical Examples are presented in order to illustrate the effect of each method in the solution of the disturbance toleration problem.

Orientador: José Francisco Gomes / Banca: Galen Mihaylov Sotkov / Banca: Abraham Hirsz Zimerman / Banca: Paulo Teotônio Sobrinho / Banca: Márcio José Martins / Resumo: Nesta fase construísmo e estudamos uma nova classe de modelos integráveis (relativístico e não relativístico) em duas dimensões, associados à álgebra afim 'A IND.3 POT.(1)'. Estes modelos apresentam sólitons tipológicos os quais portam duas cargas elétricas U(1) X U(1). O modelo de Toda afim (relativístico) é construído a partir do modelo WZNW mediante a calibração da ação Swznw e corresponde ao primeiro membro de grau negativo q = -1 de uma hierarquia de modelos cKP do tipo dyon. O modelo mais simples não relativístico dentro desta hierarquia corresponde ao grau q = 2 positivo. As soluções de 1-sóliton para ambos modelos foram construídas e relações explícitas entre ambas soluções (assim como entre as cargas conservadas) foram encontradas. Outro modelo integrável com simetrias não abelianas locais SL(2) X U(1) é introduzido. Numa aproximação à integrabilidade em espaços não-comutativos estudamos generalizações não comutativas no plano dos modelos integráveis bidimensionais sine-, sinh-Gordon e U(N) Quiral Principal. Calculando a amplitude de espalhamento à nível de árvore de um processo de produção de partículas provamos que a versão não-comutativa do modelo de sinh-Gordon que se obtém mediante a deformação Moyal da respectiva ação não é integrável. Por outro lado, a incorporação de vínculos adicionais que são obtidos a partir da generalização da condição de curvatura nula, tornam o modelo integrável. O modelo Quiral Principal generalizado a partir da deformação Moyal da ação, preserva a sua integrabilidade, ao contrário dos modelos sinh-Gordon e sine-Gordon. / Abstract: In this thesis we have constructed and studied a new class of two-dimensional integrable models (relativistic and nonrelativistic), related to the affine algebra 'A IND.3 POT.(1)'. These models admit U(1) X U(1) charged topological solitons. The affine Toda relativistic model is constructed from the gauged WZNW action and corresponds to the first negative grade q = -1 member of a dyonic hierarchy of cKP models. The simplest nonrelativistic model corresponds to the positive grade q = 2 of this hierarchy. The 1-soliton solutions for both models were constructed and explicit relations between them (and the conserved charges as well) were found. Another integrable model with local nonabelian SL(2) X U(1) simetries is introduced. In the context of integrability on noncommutative spaces, we have studied noncommutative generalizations on the plane of the two-dimensional integrable models sine-, sinh-Gordon and U(N) Principal Quiral. By computing for the sinh-Gordon model, the tree-level amplitude of a process of production of particles, we proved that the noncommutative generalization of this model that it is obtained by the Moyal deformation of the corresponding action is not integrable. On the other hand, the addition of extra constraints, obtained by the generalization of the zero-curvature method, renders the integrability of the model. The generalization of the Principal Quiral model by the Moyal deformation of the action preserves the integrability, contrary to the previous case / Doutor

Made available in DSpace on 2014-06-11T19:26:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-22Bitstream added on 2014-06-13T19:54:22Z : No. of bitstreams: 1 moraes_jr_me_sjrp.pdf: 1163228 bytes, checksum: 853fa9bee4a6a3c25b24de14990f3221 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle. / We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems.

Made available in DSpace on 2014-06-11T19:26:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-02-28Bitstream added on 2014-06-13T19:06:23Z : No. of bitstreams: 1 morettijunior_a_me_sjrp.pdf: 762570 bytes, checksum: 59d4b94fad96e41726548c623175fe4e (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho temos como objetivo estudar o número e a distribuição de ciclos limites em sistemas diferenciais lineares por partes. Em particular estudamos o número de ciclos limites do sistema diferencial linear por partes planar ˙x = −y − ε φ ( x) , ˙y = x, onde ε 6= 0 é um parâmetro pequeno e φ é uma função periódica linear por partes ímpar de período 4 . Provamos que dado um inteiro arbitário positivo n, o sistema acima possui exatamente n ciclos limites na faixa |x| ≤ 2 (n + 1 ). Consequentemente, existem sistemas diferenciais lineares por partes contendo uma infinidade de ciclos limites no plano real. Inicialmente obtemos uma quota inferior par a o número destes ciclos limites na faixa | x| ≤ 2 (n + 1 ) via Teoria do Averaging . Em seguida , utilizando a Teoria de Campos de Vetores Rodados, verificamos que o sistema acima tem exatamente n ciclos limites na faixa | x| ≤ 2 (n + 1 ) / The main goal of this work aim to study the number and distribution of limit cycles in piecewise linear differential systems. In particular we consider the planar piecewise linear differential system ˙x = −y − ε φ ( x) , ˙y = x, where ε 6= 0 is a small parameter and φ is an odd piecewise linear periodic function of period 4 . We prove that given an arbitrary positive integer n, the system above has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 ) . Consequently, there are piecewise differential systems containing an infinite number of limit cycles in the real plane. First we get a lower bound on the number of limit cycles in the strip |x| ≤ 2 (n + 1 ) via Averaging Theory. In the following , using the Theory of Rotated Vector Fields, we see that above system has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 )