251 research outputs found
Robust control of quasi-linear parameter-varying L2 point formation flying with uncertain parameters
Robust high precision control of spacecraft formation flying is one of the most important techniques required for high-resolution interferometry missions in the complex deep-space environment. The thesis is focussed on the design of an invariant stringent performance controller for the Sun-Earth L2 point formation flying system over a wide range of conditions while maintaining system robust stability in the presence of parametric uncertainties. A Quasi-Linear Parameter-Varying (QLPV) model, generated without approximation from the exact nonlinear model, is developed in this study. With this QLPV form, the model preserves the transparency of linear controller design while reflecting the nonlinearity of the system dynamics. The Polynomial Eigenstructure Assignment (PEA) approach used for Linear Time-Invariant (LTI) and Linear Parameter-Varying (LPV ) models is extended to use the QLPV model to perform a form of dynamic inversion for a broader class of nonlinear systems which guarantees specific system performance. The resulting approach is applied to the formation flying QLPV model to design a PEA controller which ensures that the closed-loop performance is independent of the operating point. Due to variation in system parameters, the performance of most closed-loop systems are subject to model uncertainties. This leads naturally to the need to assess the robust stability of nonlinear and uncertain systems. This thesis presents two approaches to this problem, in the first approach, a polynomial matrix method to analyse the robustness of Multiple-Input and Multiple-Output (MIMO) systems for an intersectingD-region,which can copewith time-invariant uncertain systems is developed. In the second approach, an affine parameterdependent Lyapunov function based Linear Matrix Inequality (LMI) condition is developed to check the robust D-stability of QLPV uncertain systems.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Quantum Monodromy in the Isotropic 3-Dimensional Harmonic Oscillator
The isotropic harmonic oscillator in dimension 3 separates in several
different coordinate systems. Separating in a particular coordinate system
defines a system of three commuting operators, one of which is the Hamiltonian.
We show that the joint spectrum of the Hamilton operator, the component of
the angular momentum, and a quartic integral obtained from separation in
prolate spheroidal coordinates has quantum monodromy for sufficiently large
energies. This means that one cannot globally assign quantum numbers to the
joint spectrum. The effect can be classically explained by showing that the
corresponding Liouville integrable system has a non-degenerate focus-focus
point, and hence Hamiltonian monodromy.Comment: 15 pages, 8 figure
Convex and Algebraic Geometry
The subjects of convex and algebraic geometry meet primarily in the theory of toric varieties. Toric geometry is the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Recent developments in toric geometry that were discussed during the workshop include applications to mirror symmetry, motivic integration and hypergeometric systems of PDE’s, as well as deformations of (unions of) toric varieties and relations to tropical geometry
Dimer Models from Mirror Symmetry and Quivering Amoebae
Dimer models are 2-dimensional combinatorial systems that have been shown to
encode the gauge groups, matter content and tree-level superpotential of the
world-volume quiver gauge theories obtained by placing D3-branes at the tip of
a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the
quiver graph. However, the string theoretic explanation of this was unclear. In
this paper we use mirror symmetry to shed light on this: the dimer models live
on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is
wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the
singular point, and geometrically encode the same quiver theory on their
world-volume.Comment: 55 pages, 27 figures, LaTeX2
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Dimer models from mirror symmetry and quivering amoebae
Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume
Hamiltonian dynamics and spectral theory for spin-oscillators
We study the Hamiltonian dynamics and spectral theory of spin-oscillators.
Because of their rich structure, spin-oscillators display fairly general
properties of integrable systems with two degrees of freedom. Spin-oscillators
have infinitely many transversally elliptic singularities, exactly one
elliptic-elliptic singularity and one focus-focus singularity. The most
interesting dynamical features of integrable systems, and in particular of
spin-oscillators, are encoded in their singularities. In the first part of the
paper we study the symplectic dynamics around the focus-focus singularity. In
the second part of the paper we quantize the coupled spin-oscillators systems
and study their spectral theory. The paper combines techniques from
semiclassical analysis with differential geometric methods.Comment: 32 page
Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
We further exploit the relation between tropical Grassmannians and
cluster algebras in order to make and refine
predictions for the singularities of scattering amplitudes in
planar super Yang-Mills theory at higher multiplicity . As a
mathematical foundation that provides access to square-root symbol letters in
principle for any , we analyse infinite mutation sequences in cluster
algebras with general coefficients. First specialising our analysis to the
eight-particle amplitude, and comparing it with a recent, closely related
approach based on scattering diagrams, we find that the only additional letters
the latter provides are the two square roots associated to the four-mass box.
In combination with a tropical rule for selecting a finite subset of variables
of the infinite cluster algebra, we then apply our
results to obtain a collection of rational and square-root
letters expected to appear in the nine-particle amplitude. In particular these
contain the alphabet found in an explicit 2-loop NMHV symbol calculation at
this multiplicity.Comment: v2: corrected minor typos, added references and acknowledgements,
improved conclusion, version to appear in JHE
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