A major motivation for Lie group observers is their application as sensor fusion algorithms for an inertial measurement unit which can be used to estimate the orientation of a rigid-body. In the first part of this thesis we propose several types of nonlinear, deterministic, locally exponentially convergent, state observers for systems with all, or part, of their states evolving on the general linear Lie group of invertible matrices. Our proposed Lie group observer with full-state measurement is applicable to left-invariant systems on linear Lie groups and yields linear estimation error dynamics. We also propose a way to extend our full-state observer, to build observers with partial-state measurement, i.e., only a proper subset of the states are available for measurement. Our proposed Lie group observer with partial-state measurement is applicable when the measured states are evolving on a Lie group and the rest of the states are evolving on the Lie algebra of this Lie group. We illustrate our observer designs on various examples, including rigid-body orientation estimation and dynamic homography estimation. In the second part of this thesis we propose a nonlinear, deterministic state observer, for systems that evolve on real, finite-dimensional vector spaces. This observer uses the property of high-gain observers, that they are approximate differentiators of the output signal of a plant. Our new observer is called a composite high-gain observer because it consists of a chain of two or more sub-observers. The first sub-observer in the chain differentiates the output of the plant. The second sub-observer in the chain differentiates a certain function of the states of the first sub-observer. Effectiveness of the composite observer is demonstrated via simulation.

This thesis provides an analytical and two numerical methods for solving a parabolic equation of two-dimensional mean curvature flow with some applications. In analytical method, this equation is solved by Lie group analysis method, and in numerical method, two algorithms are implemented in MATLAB for solving this equation. A geometric algorithm and a step-wise algorithm; both are based on a deterministic game theoretic representation for parabolic partial differential equations, originally proposed in the genial work of Kohn-Serfaty [1]. / +46-767165881

A major motivation for Lie group observers is their application as sensor fusion algorithms for an inertial measurement unit which can be used to estimate the orientation of a rigid-body. In the first part of this thesis we propose several types of nonlinear, deterministic, locally exponentially convergent, state observers for systems with all, or part, of their states evolving on the general linear Lie group of invertible matrices. Our proposed Lie group observer with full-state measurement is applicable to left-invariant systems on linear Lie groups and yields linear estimation error dynamics. We also propose a way to extend our full-state observer, to build observers with partial-state measurement, i.e., only a proper subset of the states are available for measurement. Our proposed Lie group observer with partial-state measurement is applicable when the measured states are evolving on a Lie group and the rest of the states are evolving on the Lie algebra of this Lie group. We illustrate our observer designs on various examples, including rigid-body orientation estimation and dynamic homography estimation. In the second part of this thesis we propose a nonlinear, deterministic state observer, for systems that evolve on real, finite-dimensional vector spaces. This observer uses the property of high-gain observers, that they are approximate differentiators of the output signal of a plant. Our new observer is called a composite high-gain observer because it consists of a chain of two or more sub-observers. The first sub-observer in the chain differentiates the output of the plant. The second sub-observer in the chain differentiates a certain function of the states of the first sub-observer. Effectiveness of the composite observer is demonstrated via simulation.

Covariance assignment theory, introduced in the late 1980s, provided the only means to directly control the steady-state error properties of a linear system subject to Gaussian white noise and parameter uncertainty. This theory, however, does not extend to control of the transient uncertainties and to date there exist no practical engineering solutions to the problem of directly and optimally controlling the uncertainty in a linear system from one Gaussian distribution to another. In this thesis I design a dual-mode Receding Horizon Controller (RHC) that takes a controllable, deterministic linear system from an arbitrary initial covariance to near a desired stationary covariance in finite time. The RHC solves a sequence of free-time Optimal Control Problems (OCP) that directly controls the fundamental solution matrices of the linear system; each problem is a right-invariant OCP on the matrix Lie group GLn of invertible matrices. A terminal constraint ensures that each OCP takes the system to the desired covariance. I show that, by reducing the Hamiltonian system of each OCP from T?GLn to gln? x GLn, the transversality condition corresponding to the terminal constraint simplifies the two-point Boundary Value Problem (BVP) to a single unknown in the initial or final value of the costate in gln?. These results are applied in the design of a dual-mode RHC. The first mode repeatedly solves the OCPs until the optimal time for the system to reach the de- sired covariance is less than the RHC update time. This triggers the second mode, which applies covariance assignment theory to stabilize the system near the desired covariance. The dual-mode controller is illustrated on a planar system. The BVPs are solved using an indirect shooting method that numerically integrates the fundamental solutions on R4 using an adaptive Runge-Kutta method. I contend that extension of the results of this thesis to higher-dimensional systems using either in- direct or direct methods will require numerical integrators that account for the Lie group structure. I conclude with some remarks on the possible extension of a classic result called Lie?s method of reduction to receding horizon control.

Soit G un groupe algébrique réel simple de rang réel au moins 2 et P un sous-groupe parabolique de G. On montre que tout sous-groupe discret de G intersectant le radical unipotent de P en un réseau est un réseau aritmétique de G, sauf éventuellement lorsque G = SO(2,4n+2) et P est le stabilisateur d'un 2-plan isotrope. Ceci répond partiellement à une conjecture de Margulis, déjà étudiée par Hee Oh. On étudie aussi le cas où G est le produit de plusieurs groupes de rang 1, généralisant des résultats de Selberg, Benoist et Oh. / Let G be a real algebraic group of real rank at least 2 and P a parabolic subgroup of G. We prove that any discrete subgroup of G that intersects the unipotent radical of P in a lattice is an arithmetic lattice of G, except maybe when G=SO(2,4n+2) and P is the stabilizer of an isotropic 2-plane. This provide a partial answer to a conjecture of Margulis that was already studied by Hee Oh. We also study the case where G is a product of several rank 1 groups, generalising results of Selberg, Benoist and Oh.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18044号 / 理博第3922号 / 新制||理||1566(附属図書館) / 30902 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 浅岡 正幸, 教授 加藤 毅, 教授 藤原 耕二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This dissertation develops a variety of structure-preserving algorithms for mechanical systems with external forcing and also extends those methods to systems that evolve on non-Euclidean manifolds. The dissertation is focused on numerical schemes derived from variational principles – schemes that are general enough to apply to a large class of engineering problems. A theoretical framework that encapsulates variational integration for mechanical systems with external forcing and time-dependence and which supports the extension of these methods to systems that evolve on non-Euclidean manifolds is developed. An adaptive time step, energy-preserving variational integrator is developed for mechanical systems with external forcing. It is shown that these methods track the change in energy more accurately than their fixed time step counterparts. This approach is also extended to rigid body systems evolving on Lie groups where the resulting algorithms preserve the geometry of the configuration space in addition to being symplectic as well as energy and momentum-preserving. The advantages of structure-preservation in the numerical simulation are illustrated by various representative examples from engineering applications, which include limit cycle oscillations of an aeroelastic system, dynamics of a neutrally buoyant underwater vehicle, and optimization for spherical shape correlation and matching. / Doctor of Philosophy / Accurate numerical simulation of dynamical systems over long time horizons is essential in applications ranging from particle physics to geophysical fluid flow to space hazard analysis. In many of these applications, the governing physical equations derive from a variational principle and their solutions exhibit physically meaningful invariants such as momentum, energy, or vorticity. Unfortunately, most traditional numerical methods do not account for the underlying geometric structure of the physical system, leading to simulation results that may suggest nonphysical behavior. In this dissertation, tools from geometric mechanics and computational methods are used to develop numerical integrators that respect the qualitative features of the physical system. The research presented here focuses on numerical schemes derived from variational principles– schemes that are general enough to apply to a large class of engineering problems. Energy-preserving algorithms are developed for mechanical systems by exploiting the underlying geometric properties. Numerical performance comparisons demonstrate that these algorithms provide almost exact energy preservation and lead to more accurate prediction. The advantages of these methods in the numerical simulation are illustrated by various representative examples from engineering applications, which include limit cycle oscillations of an aeroelastic system, dynamics of a neutrally buoyant underwater vehicle, and optimization for spherical shape correlation and matching.

To accurately solve the general nuclear spin state function in Nuclear Magnetic Resonance (NMR), a rotation wave approach was employed, allowing the reference frame to rotate in sync with the oscillating magnetic field. The spin state system was analogously treated as a Rubik's Cube, ensuring the diagonalization of only the time-dependent part of the state function. Although Gottfried's equation (1966) aligns with transitions between specific spin states m and m′, his second rotation contradicts the conservation of angular momentum, resulting in inaccuracies for spin states with initial phase shifts or entangled states. Contrarily, Schwinger (1937) efficiently computed the coefficients for each spin state in a frequency range opposite to the Larmor frequency, using an unorthodox approach in quantum mechanics, which unfortunately led to the oversight of his work in subsequent citations. This methodology was also applied to derive the general electron spin state function in Nuclear Magnetic Resonance (NMR) and Electron Paramagnetic Resonance (EPR), enabling the construction of a doubly rotated ground state for time-dependent perturbation theory. This was particularly relevant as the Hamiltonians for magnetic dipole, electric quadrupole, and magnetic octupole moments incorporate powers of I · J terms, necessitating the calculation of sub-state energy levels for perturbation, including those of molecules 14N7 and 7Li3. Furthermore, the study expanded to the general Lie group for 3D rotations along three linearly independent axes, resulting in 12 distinct methods to achieve rotations in any arbitrary direction using these axes, yielding wave function with only one spin operator in each exponent. The ongoing research is now concentrated on generating NMR spectra for 14N7 in amino acids, furthering the understanding of nuclear spin dynamics in complex molecular systems.

Unter der Voraussetzung, dass q keine Einheitswurzel ist und dass die Differentiale duij der Fundamentalmatrix den Linksmodul der 1-Formen erzeugen, werden die bikovarianten Differentialkalküle auf den Quantengruppen SLq(N), Oq(N) und Spq(N) klassifiziert. Es wird gezeigt, dass es auf den Quantengruppen SLq(N), N ≥ 3, abgesehen von eindimensionalen Kalkülen und endlich vielen Werten von q genau 2N bikovariante Differentialkalküle gibt. Diese Kalküle haben die Dimension N². Für die Quantengruppen Oq(N) und Spq(N), N ≥ 3, gibt es unter den genannten Voraussetzungen bis auf endlich viele Werte von q genau zwei bikovariante Differentialkalküle der Dimension N². Die Bimodulstruktur der Kalküle sowie die zugeordneten ad-invarianten Rechtsideale werden explizit angegeben. Für die Quantengruppen SLq(N), N ≥ 3, wird gezeigt, dass es, sofern q keine Einheitwurzel ist, genau 2N² + 2N bikovariante Bimoduln vom Typ (u^c u; f) gibt. / If q is not a root of unity and under the assumption that the differentials duij of the fundamental matrix (uij) generate the left module of 1-forms, all bicovariant differential calculi on quantum groups SLq(N), Oq(N) and Spq(N) are classified. It is shown that on quantum groups SLq(N), N ≥ 3, except of 1-dimensional calculi and finitely many values of q, thre are exactly 2N bicovariant differential calculi. The space of invariant forms has dimension N². For quantum groups Oq(N) and Spq(N), N ≥ 3, under the same assumptions and up to finitely many values of q, there are exactly two bicovariant differential calculi of dimension N². The bimodule structure of the calculi as well as the corresponding ad-invariant right ideals are explicitely described. For quantum groups SLq(N), N ≥ 3, there are exactly 2N² + 2N bicovariant bimodules of type (u^c u; f) provided q is not a root of unity.

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