401 research outputs found
On sets defining few ordinary lines
Let P be a set of n points in the plane, not all on a line. We show that if n
is large then there are at least n/2 ordinary lines, that is to say lines
passing through exactly two points of P. This confirms, for large n, a
conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for
this problem, as well as all sets having fewer than n - C ordinary lines for
some absolute constant C. We also solve, for large n, the "orchard-planting
problem", which asks for the maximum number of lines through exactly 3 points
of P. Underlying these results is a structure theorem which states that if P
has at most Kn ordinary lines then all but O(K) points of P lie on a cubic
curve, if n is sufficiently large depending on K.Comment: 72 pages, 16 figures. Third version prepared to take account of
suggestions made in a detailed referee repor
Effizientes Lösen von großskaligen Riccati-Gleichungen und ein ODE-Framework für lineare Matrixgleichungen
This work considers the iterative solution of large-scale matrix equations. Due to the size of the system matrices in large-scale Riccati equations the solution can not be calculated directly but is approximated by a low rank matrix ZYZ^*. Herein Z is a basis of a low-dimensional rational Krylov subspace. The inner matrix Y is a small square matrix. Two ways to choose this inner matrix are examined: By imposing a rank condition on the Riccati residual and by projecting the Riccati residual onto the Krylov subspace generated by Z. The rank condition is motivated by the well-known ADI iteration. The ADI solutions span a rational Krylov subspace and yield a rank-p residual. It is proven that the rank-p condition guarantees existence and uniqueness of such an approximate solution. Known projection methods are generalized to oblique projections and a new formulation of the Riccati residual is derived, which allows for an efficient evaluation of the residual norm. Further a truncated approximate solution is characterized as the solution of a Riccati equation, which is projected to a subspace of the Krylov subspace generated by Z. For the approximate solution of Lyapunov equations a system of ordinary differential equations (ODEs) is solved via Runge-Kutta methods. It is shown that the space spanned by the approximate solution is a rational Krylov subspace with poles determined by the time step sizes and the eigenvalues of the matrices of the Butcher tableau of the used Runge-Kutta method. The method is applied to a model order reduction problem. The analytical solution of the system of ODEs satisfies an algebraic invariant. Those Runge-Kutta methods which preserve this algebraic invariant are characterized by a simple condition on the corresponding Butcher tableau. It is proven that these methods are equivalent to the ADI iteration. The invariance approach is transferred to Sylvester equations.Diese Arbeit befasst sich mit der numerischen Lösung hochdimensionaler Matrixgleichungen mittels iterativer Verfahren. Aufgrund der Größe der Systemmatrizen in großskaligen algebraischen Riccati-Gleichung kann die Lösung nicht direkt bestimmt werden, sondern wird durch eine approximative Lösung ZYZ^* von geringem Rang angenähert. Hierbei wird Z als Basis eines rationalen Krylovraums gewählt und enthält nur wenige Spalten. Die innere Matrix Y ist klein und quadratisch. Es werden zwei Wege untersucht, die Matrix Y zu wählen: Durch eine Rang-Bedingung an das Riccati-Residuum und durch Projektion des Riccati-Residuums auf den von Z erzeugten Krylovraum. Die Rang-Bedingung wird durch die wohlbekannten ADI-Verfahren motiviert. Die approximativen ADI-Lösungen spannen einen Krylovraum auf und führen zu einem Riccati-Residuum vom Rang p. Es wird bewiesen, dass die Rang-p-Bedingung Existenz und Eindeutigkeit einer solchen approximativen Lösung impliziert. Aus diesem Ergebnis werden effiziente iterative Verfahren abgeleitet, die eine solche approximative Lösung erzeugen. Bisher bekannte Projektionsverfahren werden auf schiefe Projektionen erweitert und es wird eine neue Formulierung des Riccati-Residuums hergeleitet, die eine effiziente Berechnung der Norm erlaubt. Weiter wird eine abgeschnittene approximative Lösung als Lösung einer Riccati-Gleichung charakterisiert, die auf einen Unterraum des von Z erzeugten Krylovraums projiziert wird. Um die Lösung der Lyapunov-Gleichung zu approximieren wird ein System gewöhnlicher Differentialgleichungen mittels Runge-Kutta-Verfahren numerisch gelöst. Es wird gezeigt, dass der von der approximativen Lösung aufgespannte Raum ein rationaler Krylovraum ist, dessen Pole von den Zeitschrittweiten der Integration und den Eigenwerten der Koeffizientenmatrix aus dem Butcher-Tableau des verwendeten Runge-Kutta-Verfahrens abhängen. Das Verfahren wird auf ein Problem der Modellreduktion angewendet. Die analytische Lösung des Differentialgleichungssystems erfüllt eine algebraische Invariante. Diejenigen Runge-Kutta-Verfahren, die diese Invariante erhalten, werden durch eine Bedingung an die zugehörigen Butcher-Tableaus charakterisiert. Es wird gezeigt, dass diese speziellen Verfahren äquivalent zur ADI-Iteration sind. Der Invarianten-Ansatz wird auf Sylvester-Gleichungen übertragen
Control of fluid flows and other systems governed by partial differential-algebraic equations
The motion of fluids, such as air or water, is central to many engineering systems of significant
economic and environmental importance. Examples range from air/fuel mixing in combustion engines
to turbulence induced noise and fatigue on aircraft. Recent advances in novel sensor/actuator
technologies have raised the intriguing prospect of actively sensing and manipulating the motion
of the fluid within these systems, making them ripe for feedback control, provided a suitable control
model exists. Unfortunately, the models for many of these systems are described by nonlinear,
partial differential-algebraic equations for which few, if any, controller synthesis techniques exist.
In stark contrast, the majority of established control theory assumes plant models of finite (and
typically small) state dimension, expressed as a linear system of ordinary differential equations.
Therefore, this thesis explores the problem of how to apply the mainstream tools of control theory
to the class of systems described by partial differential-algebraic equations, that are either linear,
or for which a linear approximation is valid.
The problems of control system design for infinite-dimensional and algebraically constrained
systems are treated separately in this thesis. With respect to the former, a new method is presented
that enables the computation of a bound on the n-gap between a discretisation of a spatially distributed
plant, and the plant itself, by exploiting the convergence rate of the v-gap metric between
low-order models of successively finer spatial resolution. This bound informs the design, on loworder
models, of H[infinity] loop-shaping controllers that are guaranteed to robustly stabilise the actual
plant. An example is presented on a one-dimensional heat equation.
Controller/estimator synthesis is then discussed for finite-dimensional systems containing algebraic,
as well as differential equations. In the case of fluid flows, algebraic constraints typically
arise from incompressibility and the application of boundary conditions. A numerical algorithm is
presented, suitable for the semi-discrete linearised Navier-Stokes equations, that decouples the differential
and algebraic parts of the system, enabling application of standard control theory without
the need for velocity-vorticity type methods. This algorithm is demonstrated firstly on a simple
electrical circuit, and secondly on the highly non-trivial problem of flow-field estimation in the
transient growth region of a flat-plate boundary layer, using only wall shear measurements.
These separate strands are woven together in the penultimate chapter, where a transient energy
controller is designed for a channel-flow system, using wall mounted sensors and actuators
Distance-based formulations for the position analysis of kinematic chains
This thesis addresses the kinematic analysis of mechanisms, in particular, the position
analysis of kinematic chains, or linkages, that is, mechanisms with rigid bodies (links)
interconnected by kinematic pairs (joints). This problem, of completely geometrical
nature, consists in finding the feasible assembly modes that a kinematic chain can adopt.
An assembly mode is a possible relative transformation between the links of a kinematic
chain. When an assignment of positions and orientations is made for all links with
respect to a given reference frame, an assembly mode is called a configuration. The
methods reported in the literature for solving the position analysis of kinematic chains
can be classified as graphical, analytical, or numerical.
The graphical approaches are mostly geometrical and designed to solve particular
problems. The analytical and numerical methods deal, in general, with kinematic chains
of any topology and translate the original geometric problem into a system of kinematic analysis of all the Assur kinematic chains resulting from replacing some of its revolute
joints by slider joints. Thus, it is concluded that the polynomials of all fully-parallel
planar robots can be derived directly from that of the widely known 3-RPR robot. In
addition to these results, this thesis also presents an efficient procedure, based on distance
and oriented area constraints, and geometrical arguments, to trace coupler curves
of pin-jointed Gr¨ubler kinematic chains. All these techniques and results together are
contributions to theoretical kinematics of mechanisms, robot kinematics, and distance
plane geometry.
equations that defines the location of each link based, mainly, on independent loop
equations. In the analytical approaches, the system of kinematic equations is reduced
to a polynomial, known as the characteristic polynomial of the linkage, using different
elimination methods —e.g., Gr¨obner bases or resultant techniques. In the numerical
approaches, the system of kinematic equations is solved using, for instance, polynomial
continuation or interval-based procedures.
In any case, the use of independent loop equations to solve the position analysis
of kinematic chains, almost a standard in kinematics of mechanisms, has seldom been
questioned despite the resulting system of kinematic equations becomes quite involved
even for simple linkages. Moreover, stating the position analysis of kinematic chains
directly in terms of poses, with or without using independent loop equations, introduces
two major disadvantages: arbitrary reference frames has to be included, and all formulas
involve translations and rotations simultaneously. This thesis departs from this standard
approach by, instead of directly computing Cartesian locations, expressing the original
position problem as a system of distance-based constraints that are then solved using
analytical and numerical procedures adapted to their particularities.
In favor of developing the basics and theory of the proposed approach, this thesis
focuses on the study of the most fundamental planar kinematic chains, namely, Baranov
trusses, Assur kinematic chains, and pin-jointed Gr¨ubler kinematic chains. The results
obtained have shown that the novel developed techniques are promising tools for the
position analysis of kinematic chains and related problems. For example, using these
techniques, the characteristic polynomials of most of the cataloged Baranov trusses can
be obtained without relying on variable eliminations or trigonometric substitutions and
using no other tools than elementary algebra. An outcome in clear contrast with the
complex variable eliminations require when independent loop equations are used to tackle
the problem.
The impact of the above result is actually greater because it is shown that the
characteristic polynomial of a Baranov truss, derived using the proposed distance-based
techniques, contains all the necessary and sufficient information for solving the positionEsta tesis aborda el problema de análisis de posición de cadenas cinemáticas, mecanismos con cuerpos rígidos (enlaces)
interconectados por pares cinemáticos (articulaciones). Este problema, de naturaleza geométrica, consiste en encontrar los
modos de ensamblaje factibles que una cadena cinemática puede adoptar. Un modo de ensamblaje es una transformación
relativa posible entre los enlaces de una cadena cinemática. Los métodos reportados en la literatura para la solución del análisis
de posición de cadenas cinemáticas se pueden clasificar como gráficos, analíticos o numéricos.
Los enfoques gráficos son geométricos y se diseñan para resolver problemas particulares. Los métodos analíticos y numéricos
tratan con cadenas cinemáticas de cualquier topología y traducen el problema geométrico original en un sistema de ecuaciones
cinemáticas que define la ubicación de cada enlace, basado generalmente en ecuaciones de bucle independientes. En los
enfoques analíticos, el sistema de ecuaciones cinemáticas se reduce a un polinomio, conocido como el polinomio característico
de la cadena cinemática, utilizando diferentes métodos de eliminación. En los métodos numéricos, el sistema se resuelve
utilizando, por ejemplo, la continuación polinomial o procedimientos basados en intervalos.
En cualquier caso, el uso de ecuaciones de bucle independientes, un estándar en cinemática de mecanismos, rara vez ha sido
cuestionado a pesar de que el sistema resultante de ecuaciones es bastante complicado, incluso para cadenas simples. Por otra
parte, establecer el análisis de la posición de cadenas cinemáticas directamente en términos de poses, con o sin el uso de
ecuaciones de bucle independientes, presenta dos inconvenientes: sistemas de referencia arbitrarios deben ser introducidos, y
todas las fórmulas implican traslaciones y rotaciones de forma simultánea. Esta tesis se aparta de este enfoque estándar
expresando el problema de posición original como un sistema de restricciones basadas en distancias, en lugar de directamente
calcular posiciones cartesianas. Estas restricciones son posteriormente resueltas con procedimientos analíticos y numéricos
adaptados a sus particularidades.
Con el propósito de desarrollar los conceptos básicos y la teoría del enfoque propuesto, esta tesis se centra en el estudio de las
cadenas cinemáticas planas más fundamentales, a saber, estructuras de Baranov, cadenas cinemáticas de Assur, y cadenas
cinemáticas de Grübler. Los resultados obtenidos han demostrado que las técnicas desarrolladas son herramientas
prometedoras para el análisis de posición de cadenas cinemáticas y problemas relacionados. Por ejemplo, usando dichas
técnicas, los polinomios característicos de la mayoría de las estructuras de Baranov catalogadas se puede obtener sin realizar
eliminaciones de variables o sustituciones trigonométricas, y utilizando solo álgebra elemental. Un resultado en claro contraste
con las complejas eliminaciones de variables que se requieren cuando se utilizan ecuaciones de bucle independientes.
El impacto del resultado anterior es mayor porque se demuestra que el polinomio característico de una estructura de Baranov,
derivado con las técnicas propuestas, contiene toda la información necesaria y suficiente para resolver el análisis de posición de
las cadenas cinemáticas de Assur que resultan de la sustitución de algunas de sus articulaciones de revolución por
articulaciones prismáticas. De esta forma, se concluye que los polinomios de todos los robots planares totalmente paralelos se
pueden derivar directamente del polinomio característico del conocido robot 3-RPR. Adicionalmente, se presenta un
procedimiento eficaz, basado en restricciones de distancias y áreas orientadas, y argumentos geométricos, para trazar curvas
de acoplador de cadenas cinemáticas de Grübler. En conjunto, todas estas técnicas y resultados constituyen contribuciones a la
cinemática teórica de mecanismos, la cinemática de robots, y la geometría plana de distancias.
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