These lecture notes give a completely self-contained introduction to the control theory of linear time-invariant systems. No prior knowledge is requried apart from linear algebra and some basic familiarity with ordinary differential equations. Thus, the course is suited for students of mathematics in their second or third year, and for theoretically inclined engineering students. Because of its appealing simplicity and elegance, the behavioral approch has been adopted to a large extend. A short list of recommended text books on the subject has been added, as a suggestion for further reading.
Die Theorie der mehrdimensionalen Systeme ist ein relativ junges Forschungsgebiet innerhalb der Systemtheorie, erste Arbeiten stammen aus den 70er Jahren. Hauptmotiv für das Studium multidimensionaler Systeme war die Notwendigkeit einer Erweiterung der Theorie der digitalen Filter, die in der klassischen, eindimensionalen Signalverarbeitung (zeitabhängige Signale) Anwendung finden, auf den Bereich der Bildverarbeitung, also auf zweidimensionale Signale.; Die Vorlesung beschäftigt sich daher in ihrem ersten Teil mit skalaren zweidimensionalen Systemen und beschränkt sich im wesentlichen auf den linearen Fall. Untersucht werden zweidimensionale Filter, ihre wichtigsten Eigenschaften, Kausalität und Stabilität, sowie ihre Zustandsraum- realisierungen, etwa die Modelle von Roesser und Fornasini-Marchesini. Parallelen und Unterschiede zur eindimensionalen Systemtheorie werden betont.; Im zweiten Teil der Vorlesung werden allgemeine höherdimensionale und multivariable Systeme behandelt. Für diese Systeme erweist sich der von Jan Willems begründete Zugang zur Systemtheorie, der sogenannte behavioral approach, als zweckmäßig. Grundlegende Ideen dieses Ansatzes sowie eine der wichtigsten Methoden zum Rechnen mit Polynomen in mehreren Variablen, die Theorie der Gröbnerbasen, werden vorgestellt.
We study a model for learning periodic signals in recurrent neural networks proposed by Doya and Yoshizawa  that can be considered as a model for temporal pattern memory in animal motoric systems. A network receives an external oscillatory input and adjusts its weights so that this signal can be reproduced approximately as the network output after some time. We use tools from adaptive control theory to derive an algorithm for weight matrices with special structure. If the input is generated by a network of the same structure the algorithm converges globally and does not exhibit the deficiencies of the back-propagation based approach of Doya and Yoshizawa under a persistency of excitation condition. This simple algorithm can also be used for open loop identification under quite restructive assumptions. The persistency of excitation condition cannot be proven even for the matrices with special structure but for a 3d system. For higher dimensional systems we give connections to the theory of linear time-varying systems where this condition is generically true (under assumption which are also needed in the time-invariant case). However, we cannot show that the linearized system related to the nonlinear neural network fulfills these generic assumptions.
The edge enhancement property of a nonlinear diffusion equation with a suitable expression for the diffusivity is an important feature for image processing. We present an algorithm to solve this equation in a wavelet basis and discuss its one dimensional version in some detail. Sample calculations demonstrate principle effects and treat in particular the case of highly noise perturbed signals. The results are discussed with respect to performance, efficiency, choice of parameters and are illustrated by a large number of figures. Finally, a comparison with a Fourier method and a finite volume method is performed.
In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in the recent years. The goal of the present paper is to provide a mathematical foundation for nonlinear diffusion filtering as a scale-space transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By stuying the Lyapunow functional, it is shown that nonlinear diffusion reduces Lp norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it fulfills an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m-dimensional vector-valued images. The results are juxtaposed to linear and morphological scale-spaces.
A way to derive consistently kinetic models for vehicular traffic from microscopic follow the leader models is presented. The obtained class of kinetic equations is investigated. Explicit examples for kinetic models are developed with a particular emphasis on obtaining models, that give realistic results. For space homogeneous traffic flow situations numerical examples are given including stationary distributions and fundamental diagrams.
The efficient numerical treatment of the Boltzmann equation is a very important task in many fields of application. Most of the practically relevant numerical schemes are based on the simulation of large particle systems that approximate the evolution of the distribution function described by the Boltzmann equation. In particular, stochastic particle systems play an important role in the construction of various numerical algorithms.