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In this thesis, time-optimal control of the bi-steerable robot is addressed. The bi-steerable robot, a vehicle with two independently steerable axles, is a complex nonholonomic system with applications in many areas of land-based robotics. Motion planning and optimal control are challenging tasks for this system, since standard control schemes do not apply. The model of the bi-steerable robot considered here is a reduced kinematic model with the driving velocity and the steering angles of the front and rear axle as inputs. The steering angles of the two axles can be set independently from each other. The reduced kinematic model is a control system with affine and non-affine inputs, as the driving velocity enters the system linearly, whereas the steering angles enter nonlinearly. In this work, a new approach to solve the time-optimal control problem for the bi-steerable robot is presented. In contrast to most standard methods for time-optimal control, our approach does not exclusively rely on discretization and purely numerical methods. Instead, the Pontryagin Maximum Principle is used to characterize candidates for time-optimal solutions. The resultant boundary value problem is solved by optimization to obtain solutions to the path planning problem over a given time horizon. The time horizon is decreased and the path planning is iterated to approximate a time-optimal solution. An optimality condition is introduced which depends on the number of cusps, i.e., reversals of the driving direction of the robot. This optimality condition allows to single out non-optimal solutions with too many cusps. In general, our approach only gives approximations of time-optimal solutions, since only normal regular extremals are considered as solutions to the path planning problem, and the path planning is terminated when an extremal with minimal number of cusps is found. However, for most desired configurations, normal regular extremals with the minimal number of cusps provide time-optimal solutions for the bi-steerable robot. The convergence of the approach is analyzed and its probabilistic completeness is shown. Moreover, simulation results on time-optimal solutions for the bi-steerable robot are presented.
We introduce some mathematical framework for extreme value theory in the space of continuous functions on compact intervals and provide basic definitions and tools. Continuous max-stable processes on [0,1] are characterized by their “distribution functions” G which can be represented via a norm on function space, called D-norm. The high conformity of this setup with the multivariate case leads to the introduction of a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. We also introduce the concept of “sojourn time transformation” and compare several types of convergence on function space. Again in complete accordance with the uni- or multivariate case it is now possible to get functional generalized Pareto distributions (GPD) W via W = 1 + log(G) in the upper tail. In particular, this enables us to derive characterizations of the functional domain of attraction condition for copula processes. Moreover, we investigate the sojourn time above a high threshold of a continuous stochastic process. It turns out that the limit, as the threshold increases, of the expected sojourn time given that it is positive, exists if the copula process corresponding to Y is in the functional domain of attraction of a max-stable process. If the process is in a certain neighborhood of a generalized Pareto process, then we can replace the constant threshold by a general threshold function and we can compute the asymptotic sojourn time distribution.
On the Fragility Index
(2011)
The Fragility Index captures the amount of risk in a stochastic system of arbitrary dimension. Its main mathematical tool is the asymptotic distribution of exceedance counts within the system which can be derived by use of multivariate extreme value theory. Thereby the basic assumption is that data comes from a distribution which lies in the domain of attraction of a multivariate extreme value distribution. The Fragility Index itself and its extension can serve as a quantitative measure for tail dependence in arbitrary dimensions. It is linked to the well known extremal index for stochastic processes as well the extremal coefficient of an extreme value distribution.
We study reachability matrices R(A, b) = [b,Ab, . . . ,An−1b], where A is an n × n matrix over a field K and b is in Kn. We characterize those matrices that are reachability matrices for some pair (A, b). In the case of a cyclic matrix A and an n-vector of indeterminates x, we derive a factorization of the polynomial det(R(A, x)).
We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of 3-tuples of pairwise incompatible strains. Moreover we construct a five-dimensional continuum of T3s and show that its intersection with the boundary of the symmetrised rank-one convex hull is four-dimensional. We also show that there is another kind of monoclinic-I martensite with qualitatively different semi-convex hulls which, so far as we know, has not been experimentally observed. Our strategy is to combine understanding of the algebraic structure of symmetrised rank-one convex cones with knowledge of the faceting structure of the convex polytope formed by the strains.
The analysis of real data by means of statistical methods with the aid of a software package common in industry and administration usually is not an integral part of mathematics studies, but it will certainly be part of a future professional work. The present book links up elements from time series analysis with a selection of statistical procedures used in general practice including the statistical software package SAS. Consequently this book addresses students of statistics as well as students of other branches such as economics, demography and engineering, where lectures on statistics belong to their academic training. But it is also intended for the practician who, beyond the use of statistical tools, is interested in their mathematical background. Numerous problems illustrate the applicability of the presented statistical procedures, where SAS gives the solutions. The programs used are explicitly listed and explained. No previous experience is expected neither in SAS nor in a special computer system so that a short training period is guaranteed. This book is meant for a two semester course (lecture, seminar or practical training) where the first three chapters can be dealt within the first semester. They provide the principal components of the analysis of a time series in the time domain. Chapters 4, 5 and 6 deal with its analysis in the frequency domain and can be worked through in the second term. In order to understand the mathematical background some terms are useful such as convergence in distribution, stochastic convergence, maximum likelihood estimator as well as a basic knowledge of the test theory, so that work on the book can start after an introductory lecture on stochastics. Each chapter includes exercises. An exhaustive treatment is recommended. Chapter 7 (case study) deals with a practical case and demonstrates the presented methods. It is possible to use this chapter independent in a seminar or practical training course, if the concepts of time series analysis are already well understood. This book is consecutively subdivided in a statistical part and an SAS-specific part. For better clearness the SAS-specific parts are highlighted. This book is an open source project under the GNU Free Documentation License.
In the verification of positive Harris recurrence of multiclass queueing networks the stability analysis for the class of fluid networks is of vital interest. This thesis addresses stability of fluid networks from a Lyapunov point of view. In particular, the focus is on converse Lyapunov theorems. To gain an unified approach the considerations are based on generic properties that fluid networks under widely used disciplines have in common. It is shown that the class of closed generic fluid network models (closed GFNs) is too wide to provide a reasonable Lyapunov theory. To overcome this fact the class of strict generic fluid network models (strict GFNs) is introduced. In this class it is required that closed GFNs satisfy additionally a concatenation and a lower semicontinuity condition. We show that for strict GFNs a converse Lyapunov theorem is true which provides a continuous Lyapunov function. Moreover, it is shown that for strict GFNs satisfying a trajectory estimate a smooth converse Lyapunov theorem holds. To see that widely used queueing disciplines fulfill the additional conditions, fluid networks are considered from a differential inclusions perspective. Within this approach it turns out that fluid networks under general work-conserving, priority and proportional processor-sharing disciplines define strict GFNs. Furthermore, we provide an alternative proof for the fact that the Markov process underlying a multiclass queueing network is positive Harris recurrent if the associate fluid network defining a strict GFN is stable. The proof explicitely uses the Lyapunov function admitted by the stable strict GFN. Also, the differential inclusions approach shows that first-in-first-out disciplines play a special role.
Bei vielen Fragestellungen, in denen sich eine Grundgesamtheit in verschiedene Klassen unterteilt, ist weniger die relative Klassengröße als vielmehr die Anzahl der Klassen von Bedeutung. So interessiert sich beispielsweise der Biologe dafür, wie viele Spezien einer Gattung es gibt, der Numismatiker dafür, wie viele Münzen oder Münzprägestätten es in einer Epoche gab, der Informatiker dafür, wie viele unterschiedlichen Einträge es in einer sehr großen Datenbank gibt, der Programmierer dafür, wie viele Fehler eine Software enthält oder der Germanist dafür, wie groß der Wortschatz eines Autors war oder ist. Dieser Artenreichtum ist die einfachste und intuitivste Art und Weise eine Population oder Grundgesamtheit zu charakterisieren. Jedoch kann nur in Kollektiven, in denen die Gesamtanzahl der Bestandteile bekannt und relativ klein ist, die Anzahl der verschiedenen Spezien durch Erfassung aller bestimmt werden. In allen anderen Fällen ist es notwendig die Spezienanzahl durch Schätzungen zu bestimmen.
Consider the situation where two or more images are taken from the same object. After taking the first image, the object is moved or rotated so that the second recording depicts it in a different manner. Additionally, take heed of the possibility that the imaging techniques may have also been changed. One of the main problems in image processing is to determine the spatial relation between such images. The corresponding process of finding the spatial alignment is called “registration”. In this work, we study the optimization problem which corresponds to the registration task. Especially, we exploit the Lie group structure of the set of transformations to construct efficient, intrinsic algorithms. We also apply the algorithms to medical registration tasks. However, the methods developed are not restricted to the field of medical image processing. We also have a closer look at more general forms of optimization problems and show connections to related tasks.