There is much interest in the design of feedback controllers for linear systems that minimize the H-infty norm of a specific closed-loop transfer function.  The H-infty optimization problem initiated by Zames (1981), \\cite{zames1981feedback}, has received a lot of interest since its formulation.  In H-infty control theory one uses the H-infty norm of a stable transfer function as a performance measure.  One typically uses approaches in either the frequency domain or a state space formulation to tackle this problem.  Frequency domain approaches use operator theory, J-spectral factorization or polynomial methods while in the state space approach one uses ideas similar to LQ theory and differential games.  One of the key computational issues in the design of H-infty optimal controllers is the determination of the optimal H-infty norm.  That is, determining the infimum of r for which the H-infty norm of the associated transfer function matrix is less than r.  Doyle  et al (1989), presented a state space characterization  for the sub-optimal H-infty control problem.  This characterization requires that the unique stabilizing solutions to  two Algebraic Riccati Equations are positive semi definite as well as satisfying a spectral radius coupling condition.  In this work, we describe an algorithm by Lin et al(1999),  used to calculate the H-infty norm for the state feedback and output feedback control problems.  This algorithm only relies on standard assumptions and divides the problem into three sub-problems. The first two sub-problems rely on algorithms for the state feedback problem formulated in the frequency domain as well as a characterization of the optimal value in terms of the singularity of the upper-half of  a matrix created by the stacked basis vectors of the invariant sub-space of the associated Hamiltonian matrix.  This characterization is verified through a bisection or secant method.  The third sub-problem relies on the geometric nature of the spectral radius of the product of the two solutions to the Algebraic Riccati Equations associated with the first two sub-problems.  Doyle makes an intuitive argument that the spectral radius condition will fail before the conditions involving the Algebraic Riccati Equations fail.  We present numerical results where we demonstrate that the Algebraic Riccati Equation conditions fail before the spectral radius condition fails. / Master of Science

Thesis advisor: Jean-Luc Solère / Thesis advisor: Marius Stan / My dissertation focuses on the problem of matter's divisibility in the 17th-18th centuries. The problem of material divisibility is a focal point at which the metaphysical principle of simplicity and the mathematical principle of space's infinite divisibility conflict. The principle of simplicity is the metaphysical requirement that there must be a simple or indivisible being that is the constitutive foundation of all composite things in nature. Without simple beings, there cannot be composite beings. The mathematical principle of space's infinite divisibility is a staple of Euclidean geometry: space must be divisible into infinitely smaller parts because indivisibles or points cannot compose extension. Without reconciling these metaphysical and mathematical principles, one can call into question the integrity of mathematics and metaphysics. Metaphysical contradiction results from the application of metaphysical simplicity to the composition of material bodies that occupy infinitely divisible space. How can a simple being constitute a material object while occupying a space that lacks a smallest part? Should we assume that a composite material object (such as the paper in front of the reader) exists in an infinitely divisible space, then the simple beings must occupy a space that consists of ever smaller spaces. The simple being thereby appears to consist of parts simpler than itself--a metaphysical contradiction. Philosophers resolve this contradiction by either modifying the metaphysical principle of simplicity to allow for the occupation of infinitely divisible space, or have simply dismissed one principle for the sake of preserving the other principle. The rejection of one principle for preserving the other principle is an undesirable path. Philosophers would either forfeit any attempt to account for the composition of material reality by rejecting simplicity or deny understanding of geometry heretofore via the rejection of space's infinite divisibility. My objective in this dissertation is two-fold: 1.) to provide an historical analysis of various philosophers' attempts to reconcile simplicity and infinite divisibility or to argue for the exclusive nature of the said principles; 2.) to articulate a reconciliation between simplicity and infinite divisibility. Underlying both objectives is my attempt to draw a connection between the metaphysical principle of simplicity and the metaphysical principle of sufficient reason. Having shown in the historical section that each philosopher implicitly references a modified version of the principle of sufficient reason when articulating their theories of metaphysical simplicity, I will use this common principle to develop a Neo-Aristotelian solution to the problem of material divisibility. This Neo-Aristotelian solution differs from other accounts in the historical section by including a potential parts theory of material divisibility while modifying the principle of simplicity: simple beings are no longer conceived as constitutive parts of a material thing, but as the sources of unity for a natural composite being. / Thesis (PhD) — Boston College, 2014. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Philosophy.

The objective of this dissertation is to develop a practical methodology for designing full and reduced order H[infinity] filter for plants with polytopic model uncertainty. Because the polytopic model description is convex, it is amenable to a Linear Matrix Inequality (LMI) formulation. Reduced order filters are desirable in applications where fast data processing is necessary. To improve robustness to model uncertainties, this dissertation reformulates an H2 filter design technique as a reduced order H[infinity] filter design methodology. Lyapunov functions are replaced with parameter-dependent Lyapunov functions to provide less conservative results. As the problem is formulated as an LMI, an admissible filter with suitable dynamic behavior can be obtained from the solution of a convex optimization problem. The advantages of this approach over earlier approaches are highlighted in a simple computational example. This filtering technique is used to design a fault detection filter. Robust fault detection filter (RFDF) design is formulated as a multi-objective H[infinity] optimization for a polytopic uncertain system. The order of the RFDF is reduced using LMI techniques and the detection performance is compared with the full order filter. An adaptive threshold is used to reduce the number of false alarms. Examples are presented to illustrate effectiveness of the order reduction. / Thesis (Ph.D.)--Wichita State University, Dept. of Electrical and Computer Engineering. / Includes bibliographic references (leaves 70-76). / "May 2006."

The main topic of this thesis is to exhibit how robust control techniques can be applied to real time systems. Presently, the control techniques used in the industry are very simple even when applied to complex systems; these techniques are intuitive and not necessarily systematic. Moreover, the notion of optimality of robustness is absent. Control design procedures are mostly based on SISO techniques, thus, overlooking the intrinsic multivariable aspect of the design that a MIMO system requires. In this thesis a modern control technique is presented to manipulate a 3DOF birotor helicopter in real time. The objective of this research is to demonstrate the performance of more efficient control algorithms to control these kinds of systems. The robust method proposed in this thesis is an H infinity controller which exhibits robustness to plant model uncertainties, and good disturbance and noise rejection.

Kant's antinomies are exercises designed to illustrate the limits of human reasoning. He skillfully juxtaposes pairs of arguments and exposes the dangerous propensity for human reasoning to stretch beyond the conditioned and into the transcendental ideas of the unconditional. Kant believes this is a natural process and affirms the limits of pure reason in so much as they should prevent us from believing that we can truly know anything about the unconditional. His first antimony addresses the possibility of a beginning in time or no beginning in time. This thesis will focus on this first antinomy and critically assesses it in set theoretic terms. It is this author's belief that the mathematical nuances of infinite sets and the understanding of mathematical objects bear relevance to the proper interpretation of this antinomy. Ultimately, this composition will illustrate that Kant's argument in the first antinomy is flawed because it fails to account for infinite bounded sets and a conceptualization of the infinite as a mathematical object of reason.

Orientador: Alberto Luiz Serpa / Dissertação (mestrado - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-20T02:35:52Z (GMT). No. of bitstreams: 1 Tapias_RenanMoro_M.pdf: 4789122 bytes, checksum: 0c921b95857d6987a5596ae935a85719 (MD5) Previous issue date: 2012 / Resumo: Esta dissertação aborda técnicas de controle robusto H-infinito para sistemas dinâmicos lineares com incertezas paramétricas. Para obtenção do modelo da estrutura em estudo, utiliza-se o método de elementos finitos. A partir do modelo da estrutura, consideram-se incertezas paramétricas, sendo elas, na frequência natural e no fator de amortecimento. As incertezas paramétricas quando consideradas para projeto do controlador H-infinito são tratadas pela abordagem poli tópica. Essa metodologia utiliza o conceito de Desigualdades Matriciais Lineares (LMI). Ainda na fase de projeto do controlador, filtros de ponderação são utilizados para impor uma certa forma em frequência. As incertezas dos sistemas em estudo são consideradas como sendo tanto variantes como invariantes no tempo. O controlador encontrado por essa metodologia se mostrou robusto a incertezas paramétricas, garantindo estabilidade e boa atenuação de vibração dos modos considerados em projeto / Abstract: The aim of this dissertation is to study the H-infinity robust control techniques for linear dynamic systems with parametric uncertainties. The finite element method was employed to find the model of the flexible structure. When dealing with the model, parametric uncertainties were considered for natural frequencies and for damping of the structure. The parametric uncertainties for the H-infinity controller design are handled in the polytopic approach. This methodology uses the concept of Linear Matrix Inequalities (LMI) for the controller project. Weighting filters were used to impose desired frequency response in the controller design. Systems with uncertainties were considered variant and invariant in time. The controller found using this methodology was robust to parametric uncertainties, ensuring stability and good attenuation of vibration in design the considered modes / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

This research investigates the harmonic and narrowband disturbance rejection problem for linear time-periodic (LTP) systems. The consequence of disturbances on LTP systems is similar to their linear time-invariant (LTI) counterparts, but is complicated by the interaction of the disturbance and plant acting at different frequencies, which manifests itself in the modulation of the disturbance signal. The result, for an m-periodic plant and disturbance containing a single tone, is that the output contains m tones. Using various disturbance rejection architectures, harmonic and narrowband disturbance rejection is investigated for linear time-periodic plants. Included are classical and multivariable feedback controllers, fixed-gain feedforward designs using finite impulse response (FIR) filters and H-infinity synthesis tools, and adaptive feedforward controllers. The objective of time-periodic, narrowband, disturbance rejection seeks to place a zero in the controlled system's disturbance path and align the zero direction, defined by the null space of the controlled system at the disturbance frequency, with the disturbance. In this research, constraints on controlled system infinity-norms specify nominal performance and robust stability objectives. Periodic controllers are found using existing LTI H-infinity control theory, and causality is satisfied using two techniques which can be added easily to H-infinity solvers: loop-shifting and Q-parameterization. The resulting controllers are high-gain, narrowband-pass, periodic filters; the closed-loop sensitivity has a zero at the disturbance frequency, and the disturbance is in the sensitivity's null space. It is also shown that classical designs do not achieve the same performance levels as periodic controllers. Similar developments are made using the feedforward disturbance rejection architecture. Objectives are given which minimize the weighted infinity-norm of the controlled system. Such feedforward controllers achieve perfect disturbance rejection. A multivariable equivalent of the tapped-delay line is used in the description of periodic FIR filters. In addition, periodic FIR filters are made adaptive using an algorithm similar to filtered-X least mean square (LMS) but modified for periodic systems. / Ph. D.

Nesta dissertação, abordaremos os dois tipos de infinitos existentes: o infinito potencial e o infinito actual. Apresentaremos algumas situações, exemplos que caracterizam cada um desses dois tipos. Focaremo-nos no infinito actual, com o qual discutiremos alguns dos desafios encontrados na teoria criada por Cantor sobre este assunto. Mostraremos também sua importância e a diferença entre este e o infinito potencial. Com isso, buscamos fazer com que o professor compreenda adequadamente os fundamentos matemáticos necessários para que trabalhe, ensine e motive apropriadamente seus alunos no momento em que o infinito e conjuntos infinitos são discutidos em aula. Desta forma, buscamos esclarecer os termos usados e equívocos comuns cometidos por alunos e também professores, muitas vezes enganados ou confundidos pelo senso comum. / In this dissertation, we will discuss the two types of infinities: the potential infinity and the actual infinity. We will present some situations, examples that characterize each of these two types. We will focus on the actual infinity, with which we will discuss some of the challenges found in the theory created by Cantor on this subject. We will also show its importance and the difference between this and the potential infinity. Thus, we seek to make teachers properly understand the mathematical foundations necessary for them to work, teach and properly motivate their students at the time the infinity and infinite sets are discussed in class. In this way, we seek to clarify the terms used and common mistakes made by students and also teachers, so often misguided or confused by common sense.

Nesta dissertação, abordaremos os dois tipos de infinitos existentes: o infinito potencial e o infinito actual. Apresentaremos algumas situações, exemplos que caracterizam cada um desses dois tipos. Focaremo-nos no infinito actual, com o qual discutiremos alguns dos desafios encontrados na teoria criada por Cantor sobre este assunto. Mostraremos também sua importância e a diferença entre este e o infinito potencial. Com isso, buscamos fazer com que o professor compreenda adequadamente os fundamentos matemáticos necessários para que trabalhe, ensine e motive apropriadamente seus alunos no momento em que o infinito e conjuntos infinitos são discutidos em aula. Desta forma, buscamos esclarecer os termos usados e equívocos comuns cometidos por alunos e também professores, muitas vezes enganados ou confundidos pelo senso comum. / In this dissertation, we will discuss the two types of infinities: the potential infinity and the actual infinity. We will present some situations, examples that characterize each of these two types. We will focus on the actual infinity, with which we will discuss some of the challenges found in the theory created by Cantor on this subject. We will also show its importance and the difference between this and the potential infinity. Thus, we seek to make teachers properly understand the mathematical foundations necessary for them to work, teach and properly motivate their students at the time the infinity and infinite sets are discussed in class. In this way, we seek to clarify the terms used and common mistakes made by students and also teachers, so often misguided or confused by common sense.

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