Implicit Network Descriptions of RLC Networks and the Problem of Re-Engineering

City Research Online City, University of London Institutional Repository Citation: Livada, M. (2017). Implicit network descriptions of RLC networks and the problem of re-engineering. (Unpublished Doctoral thesis, City, University of London) This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: https://openaccess.city.ac.uk/id/eprint/17916/ Link to published version: Copyright: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. Reuse: Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. City Research Online: http://openaccess.city.ac.uk/ [email protected] Implicit Network Descriptions of RLC Networks and the Problem of Re-engineering by Maria Livada A thesis submitted for the degree of Doctor of Philosophy (Ph.D.) in Control Theory in the Systems and Control Research Centre School of Mathematics, Computer Science and Engineering July 2017 Abstract This thesis introduces the general problem of Systems Re-engineering and focuses to the special case of passive electrical networks. Re-engineering differs from classical control problems and involves the adjustment of systems to new requirements by intervening in an early stage of system design, affecting various aspects of the underlined system structure that affect the final control design problem. Addressing problems of re-engineering requires the development of a system representation able to embody these structural changes. In the case of Re-engineering in passive electrical networks, certain types of re- engineering transformations involve alterations of values or nature of existing elements, modification of network's topology and possible evolution of the network. We resort to the Implicit Network Description W (s) as a unifying representation, which stems from the Impedance/Admittance integral-differential models, since it enables the representation of such parametric and structural changes of the system as perturbations on it. By using tools and results from classical network theory and algebraic systems theory, the thesis deals with the development and study of fundamental system aspects of this new description in terms of McMillan degree, regularity and other system properties of the implicit network description. The thesis also examines the effect of transformations that preserve network cardinality on the Implicit Network Description and particularly in the natural frequencies of the network. This leads to the formulation of Determinantal Frequency Assignment Problems for natural frequency improvements. Using the exterior algebra, algebraic geometry framework we prove sufficient conditions for complex frequency assignability for a special case of network transformations and we examine whether real solutions to the problem exist. Additionally, transformations linked to the variation of network cardinality, are represented as augmentation or reduction in terms of dimension of the Implicit Network Description and by identifying those that remain intact we are in position to define fixed dynamics, enabling the formulation of partial i structure assignment problems. The results derived in this thesis provide the means for addressing the general systems re-engineering problem in a rather structured setup. Maria Livada City, University of London, July 18, 2017 ii Acknowledgements Undertaking this PhD has been a truly life-changing experience for me and it would not have been possible to do without the support and guidance that I received from many people. First and foremost, I would like to express my deepest gratitude to my supervisor Profes- sor N. Karcanias for giving me inspiration, motivation and guidance during my research studies. Without his supervision, immense knowledge and support the development of my Thesis would not have been possible and without his directions and suggestions on several aspects I would not have managed to overcome the difficulties faced during my research studies. In a more personal level, I would also like to acknowledge the moral support he provided me and the fact that he stood always by me throughout my years in London. I also wish to pay special tribute to Dr. J. Leventides for sharing his knowledge with me, for his support and for his constructive suggestions during my research. The important work he has conducted along with his in-depth knowledge in the fields of Mathemat- ics and Control made the discussions we had invaluable and offered me the chance to consider from a different perspective the subject matter of this Thesis. I also would like to thank all the members of the Research Control Centre in City, University of London for their excellent collaboration and feedback during this work. A special thanks to Dr. E. Milonidis and Professor G. Halikias for their collaboration along with moral support and understanding during the best and the worst of times. Moreover, I would like to acknowledge the external examiner, Professor G. Papavasilopoulos, for his comments and suggestions during Viva examination towards the improvement of my Thesis. Finally, I would like to thank my family for standing by me everywhere, anywhere and for everything, regardless the distance and the circumstances and for believing in me. I am and will be grateful for life for everything you are doing for me. Last but not least, a great thanks goes to my friends, especially those who have or had a similar experience with mine, for being there for me when I needed them most, for all the good and bad times we had and for all their understanding and support throughout these years. iii Declaration of Authorship I, Maria Livada, hereby declare that this thesis titled, Ìmplicit Network Descriptions of RLC Networks and the Problem of Re-engineering' and the work presented in it are my own. I also confirm that: The work presented in this Thesis is my own unless stated and referenced in the text accordingly. Where I have consulted the published work of others, this is always clearly at- tributed. I have acknowledged all main sources of help. I grant powers of discretion to the Librarian of City University London to allow single copies of this Thesis for study purposes, subject to normal conditions of acknowledgement. Signed: Date: v Contents List of Figures ix List of Tables xi Abbreviations xiii Notation xv 1 Introduction1 2 Systems and Mathematics Background7 2.1 Introduction...................................7 2.2 Background of Graph Theory and Properties................8 2.3 Polynomial Matrices and Matrix Pencils................... 14 2.4 Determinantal Assignment Problem..................... 20 2.5 Tools from Exterior Algebra.......................... 23 2.6 Laplace Expansion Technique......................... 26 2.7 Complex, Real Varieties and Morphisms................... 27 2.8 Intersection Theory of Complex Algebraic Varieties............. 30 2.9 Conclusions................................... 37 3 Systems Re-engineering and Networks: Problem Statement, Litera- ture Review and Research Agenda 39 3.1 Introduction................................... 39 3.2 The Re-engineering Problem......................... 40 3.3 Review of Network Research.......................... 45 3.4 Research Agenda: Systems Theory and Redesign of Internal Implicit Models 56 3.5 Conclusions................................... 57 4 Implicit Network Descriptions and Their Properties 59 4.1 Introduction................................... 59 4.2 Impedance Modeling, Loop Topology and Selection of Independent Loops 60 4.3 Admittance Modeling and Vertex Topology................. 68 4.4 The Internal Network Operator W(s) and the Implicit Network Description 78 4.5 Relationship Between Impedance and Admittance Operators....... 79 4.6 The Network Pencil and its Relationship to the Internal Network Descrip- tion....................................... 82 vii Contents viii 4.7 Network Regularity and Invertibility of W(s)................ 84 4.8 Natural Frequencies and the Network Pencil................. 96 4.9 Conclusions................................... 102 5 Properties of Implicit Network Descriptions and The McMillan De- gree 103 5.1 Introduction................................... 103 5.2 Implicit McMillan Degree and Its Calculation................ 104 5.3 Necessary and Sufficient Conditions For Determining The Implicit McMil- lan Degree.................................... 108 5.4 Graph Systematic Approach of Necessary and Sufficient Conditions.............................. 115 5.5 The Network Pencil P(s) and Links to the McMillan Degree of the Network121 5.6 Examples.................................... 123 5.7 Conclusions................................... 133 6 System Transformations Preserving or Altering Network Cardinality and Possibly the McMillan Degree 135 6.1 Introduction................................... 135 6.2 RLC Network Transformations Preserving McMillan Degree and Network Cardinality................................... 136 6.3 RLC Network Transformations Preserving Cardinality but Altering McMil- lan Degree...................................

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