10 research outputs found
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Properties and Classification of Generalized Resultants and Polynomial Combinants
Polynomial combinants define the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the frequency assignment problems in Linear Systems. The theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Dynamic combinants are linked to the theory of “Generalised Resultants”, which provide the matrix representation of polynomial combinants. We consider coprime set polynomials for which assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. The complete parameterization of combinants and coresponding Generalised Resultants is prerequisite to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved
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Generalised resultants, dynamic polynomial combinants and the minimal design problem
The theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems (DAP), which provides the unifying description of the dynamic, as well as static pole and zero dynamic assignment problems in linear systems. The assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original DAP. This paper demonstrates the origin of dynamic polynomial combinants from linear systems, examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree, and examines their spectral assignment. Central to this study is the link of dynamic combinants to the theory of generalised resultants, which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterisation of combinants and respective generalised resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, which is referred to as the dynamic combinant minimal design (DCMD) problem. An algorithmic approach based on rank tests of Sylvester matrices is given, which produces the minimal order and degree solution in a finite number of steps. Such solutions provide low bounds for the respective dynamic assignment control problems
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Approximate zero polynomials of polynomial matrices and linear systems
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials"
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Approximate zero polynomials of polynomial matrices and linear systems
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials"
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Structure assignment problems in linear systems: Algebraic and geometric methods
The Determinantal Assignment Problem (DAP) is a family of synthesis methods that has emerged as the abstract formulation of pole, zero assignment of linear systems. This unifies the study of frequency assignment problems of multivariable systems under constant, dynamic centralized, or decentralized control structure. The DAP approach is relying on exterior algebra and introduces new system invariants of rational vector spaces, the Grassmann vectors and Plücker matrices. The approach can handle both generic and non-generic cases, provides solvability conditions, enables the structuring of decentralisation schemes using structural indicators and leads to a novel computational framework based on the technique of Global Linearisation. DAP introduces a new approach for the computation of exact solutions, as well as approximate solutions, when exact solutions do not exist using new results for the solution of exterior equations. The paper provides a review of the tools, concepts and results of the DAP framework and a research agenda based on open problems
On Hermite's invariant for binary quintics
The Hermite invariant H is the defining equation for the hypersurface of
binary quintics in involution. This paper analyses the geometry and invariant
theory of H. We determine the singular locus of this hypersurface and show that
it is a complete intersection of a linear covariant of quintics. The projective
dual of this hypersurface can be identified with itself via an involution. It
is shown that the Jacobian ideal of H is perfect of height two, and we describe
its SL_2-equivariant minimal resolution. The last section develops a general
formalism for evectants of covariants of binary forms, which is then used to
calculate the evectant of H
On the Stability of Random Matrices
Η διδακτορική διατριβή αποσκοπεί σε μια πολυδιάστατη οικογένεια προβλημάτων στα μαθηματικά. Επιπλέον ένα από τα τρία θέματα που παρουσιάζονται στη διατριβή αυτή είναι η διαδικασία της διάχυσης σε δυναμικά χρηματοοικονομικά δίκτυα. Τα γραφήματα αυτά είναι συνεκτικά, κατευθυνόμενα και σταθμισμένα δίκτυα τραπεζών (κόμβοι), όπου τα αρχικά κεφάλαια και η ποσότητα δανείων που μεταφέρονται από την μία τράπεζα (κόμβο) στην άλλη θεωρούνται γνωστά. Με άλλα λόγια προσπαθούμε να θεραπέυσουμε προβλήματα στο χρηματοοικονομικό τομέα όπου αρχικά έχουμε τραπεζικά δυναμικά δίκτυα που χαρακτηρίζονται από διαφορετικά επίπεδα αρχικών μοχλεύσεων και καταλήγουμε να έχουμε ένα σταθερό τραπεζικό δίκτυο χάρη στη διαδικασία της διάχυσης. Επίσης, παρουσιάζουμε αρκετά παραδείγματα όπου επιβεβαιώνετε η θεωρία μας και μέσω πρακτικών παραδειγμάτων. Τέλος, έχουμε την πεποίθηση ότι η δουλειά αυτή μπορεί να επεκταθεί κάνοντας χρήση της μαθηματικής θεωρίας ελέγχου με σκοπό να κάνουμε αυτά τα δίκτυα πιο εύκολα ελέγξιμα.Επιπλέον, ένα άλλο πρόβλημα που παρουσιάζετε στη διατριβή αυτή είναι η έννοια των almost zeros πολυωνυμικών διανυσμάτων ή πινάκων. Στο θέμα αυτό το κίνητρό μας ήταν η κατανόηση αλλά και να παρουσιάσουμε πως η έννοια των almost zeros εξαρτάται από την τυχαιότητα. Πιο συγκεκριμένα μελετάμε και παρουσιάζουμε τα στατιστικά χαρακτηριστικά των almost zero και την τιμή τους για δοθέντες τυχαίους πολυωνυμικούς πίνακες. Τέλος, διάφορα παραδείγματα παρουσιάζονται για να είναι πιο κατονοητή η μελέτη αυτού του θέματος.Τέλος, το τελευταίο θέμα στο οποίο πραγματευόμαστε είναι η μελέτη σε γραμμικά, χρονικά-αμετάβλητα πολυμεταβλητά συστήματα που περιγράφονται από συστήματα εξισώσεων. Πιο συγκεκριμένα εστιάζουμε σε τυχαία συστήματα όπου οι παράμετροι έχουν αντικατασταθεί από τυχαίους πίνακες. Επιπροσθέτως, ορίζουμε την έννοια της ε-ελεγξιμότητας για τυχαία συστήματα για δοθέν θετικό αριθμό ε. Επίσης, θεωρούμε τη στοιχειώδη κατηγορία τυχαίων πινάκων Gaussian ortogonal ensemble (GOE) και υπολογίζουμε ότι το ε-μη-ελεγξιμότητα ενός τέτοιου τυχαίου συστήματος εξαρτάται από την κατανομή των στοιχείων του. Στην περίπτωση αυτή η ε-μη-ελεγξιμότητα υπολογίζεται.The aim of this thesis is multiple. Initially we study the diffusion process on dynamicalfinancial networks. To be more precise, we study the effect of diffusion method to interbanknetworks in relation to connected, directed and weighted networks. We consider networks ofn different banks which exchange funds (loans) and the main feature is how the leverage'sof banks can be chosen to improve the financial stability of the network. This is done byconsidering differential equations of diffusion type.Furthermore, we investigate the problem of almost zeros of polynomial matrices as usedin the system theory. It is related to the controllability and observability notion of systems aswell as the determination of Macmillan degree and complexity of systems. We also presentsome new results in this important invariant in the light of randomness and we prove anuncertainty type relation appearing in such ensembles of operators.In addition, we introduce the concept of ε-uncontrollability for random linear systems,i.e. a linear system in which the usual matrices have been replaced by random matrices. Wealso estimate the ε-uncontrollability in the case where the matrices come from the Gaussianorthogonal ensemble. Our proof utilizes tools from systems theory, probability theory andconvex geometry.This thesis is divided into three parts: Introduction and literature review material in PartI, methodological tools which were used in this thesis in Part II, and the rest consists of threeresearch papers among which the first one has been published in a collected volume underSpinger publications and the other two have been published in Institute of Mathematics andits Applications journal (IMA), in Part III.In Part I and especially in Chapter 1 we present an introduction and we provide information related to the motivation and the structure of this thesis. In addition in Chapter 2 we havea literature review about what other scientists have done so far on topics like mathematicalcontrol theory, diffusion process, graphs/networks and random matrix theory.The methodological tools needed for the derivation of the main research results presentedin Part III are given in Part II, Chapter 3. Furthermore, in this part, topics and notions frommathematical control theory, like controllability/observability, are presented. Moreover, wedescribe not only the meaning of RMT but also the the given examples of what a randommatrix is and what makes it so special. Another topic that is covered in this Chapter is graphtheory and examples about connected, directed and weighted graphs/networks and algebraicproperties of graphs are given. Finally, the topic of diffusion process is presented as well.Part III constitutes the most important part of this research study which is presented inthis thesis. Thus, in Chapter 4, we describe the diffusion process on connected, directed andweighted interbank loan networks. We prove that diffusion can drive the bank network to itssteady state where the leverages of all the banks of the network are equal. Diffusion in thebank network is modelled with a system of coupled ordinary linear differential equations. Thelast argument, we ended up with two different methods to the same results and a great success.In Chapter 5, we present the notion of almost zero of polynomial vectors and matrices in termsof a minimization problem. The purpose of this chapter is twofold: (a) Firstly, to explorefurther the algebraic properties and computations of almost zeroes of polynomial vectors ormatrices and present new results of higher order almost zeros; (b) Secondly, to relate thenotion of almost zero to that of randomness i.e. what are the statistical characteristics ofthe almost zero and its norm polynomial value in a given random ensemble of polynomials.In Chapter 6, we consider random systems where the parameters which are matrices havebeen replaced with random matrices. Moreover, we define the ε-uncontrollability for randomsystems and we also describe the GOE. In addition, we give the detailed calculation of theε-uncontrollability of a random system where the matrix A comes from the GOE
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Implicit network descriptions of RLC networks and the problem of re-engineering
The thesis deals with aspects of Systems Re-engineering specialised to the case of passive electrical networks. Re-engineering is a problem different from traditional control problems and this emerges when it is realised that the systems designed in the past cannot perform according to the new performance requirements and such performance cannot be improved by traditional control activities. Re-engineering implies that we intervene in early stages of system design involving sub-processes, values of physical elements, interconnection topology, selection of systems of inputs and outputs and of course retuning of control structures. This is a very challenging problem which has not been addressed before in a systematic way and needs fundamental new thinking, based on understanding of structure evolution during the stages of integrated design. A major challenge in the study of this problem is to have a system representation that allows study of evolution of system properties as well as structural invariants. For linear systems the traditional system representations, such as transfer functions, state space models and polynomial type models do not provide a suitable framework for study structure and property evolutions, since for every change we need to compute again these models and the transformations we have used do not appear in an explicit form in such models. It is for this reason, for a general system, such system representations are not suitable for study of system representations on re-engineering.
It has been recognized that for the special family of systems defined by the passive electrical networks (RLC), there exists a representation introduced by the loop/ nodal analysis, expressed by the impedance/admittance integral-differential models, which have the property of re-engineering transformations of the following type:
1. Changing the values or possible nature of existing elements without changing the network topology,
2. Modifying the network topology without changing network cardinality, that is number of independent loops or nodes,
3. Augmenting or reducing the network by addition or deletion of sub-networks,
4. Combination of all the above transformations.
These kinds of transformations may be represented as perturbations on the original impedance/admittance models. The above indicates that impedance/admittance integral-differential models, which from now on will be referred to as Implicit Network Descriptions is the natural vehicle for studying re-engineering on electrical networks. Although issues related to realisation of impedance/admittance transfer functions within RLC topologies, has been the topic of classical network synthesis, the system aspects of such descriptions have not been properly considered. Addressing problems of network re-engineering requires the development of the fundamental system aspects of such new descriptions in terms of McMillan degree, regularity and a number of other properties. Certain problems of evolution (of system properties) are linked to Frequency Assignment, as far as natural frequencies under re-engineering and this requires use of techniques developed within control theory for Frequency Assignment Problems