Based on the Lyapunov stability theorem, a decentralized adaptive sliding mode control scheme is proposed in this thesis for a class of mismatched perturbed large-scale systems containing dead-zone input to solve regulation problems. The main idea is that some adaptive mechanisms are embedded both in the sliding surface and in the controllers, so that not only the mismatched perturbations are suppressed during the sliding mode, but also the information of upper bound of perturbations is not required. The sliding surface function is firstly designed through the usage of a pseudo controller which is capable of stabilizing the reduced-order systems. The second step is to design the controllers so that the trajectories of the controlled systems are able to reach sliding surface in a finite time. Once the controlled system enters the sliding mode, the asymptotical stability is guaranteed for each subsystem even the mismatched perturbations exist. A numerical example and a practical example are given to demonstrate the feasibility of the proposed design technique.

Based on the Lyapunov stability theorem, an adaptive sliding mode control scheme is proposed in this thesis for a class of systems with mismatched state-delayed perturbations to solve regulation problems. The main idea is that some adaptive mechanisms are embedded both in the sliding surfaces and in the controllers, so that not only the mismatched perturbations are suppressed during the sliding mode, but also the information of upper bound of perturbations is not required. The sliding surface functions are firstly designed through the usage of designed pseudo controllers, which is capable of stabilizing the reduced-order systems. The number of the sliding surface functions required by the proposed control scheme depends on the relationship between systems's dimension and number of inputs. The second step is to design the controllers so that the trajectories of the controlled system are able to reach sliding surface in a finite time. Once the controlled system enters the sliding mode, the asymptotical stability is guaranteed. Two numerical examples and one practical experiment are given for demonstrating the feasibility of the proposed control scheme.

Based on the Lyapunov stability theorem, an adaptive sliding mode control scheme is proposed in this thesis for a class of mismatched perturbed multi-input multi-output (MIMO) dynamic systems to solve regualtion problems. The sliding surface function is firstly designed by treating some state variables as a pseudo controllers through the usage of sliding function to stabilize the rest of state variables. In this thesis the number of these pseudo controllers is less than that of the state variables to be stabilized. The second step is to design the controllers so that the trajectories of the controlled systems are able to reach sliding surface in a finite time. Some adaptive mechanisms are embedded in the sliding surface function and sliding mode controllers, so that not only the mismatched perturbations can be suppressed during the sliding mode, but also the information of upper bounds of some perturbations are not required when designing the sliding surface function and controllers. Once the controlled system enters the sliding mode, the state trajectories can achieve asymptotical stability under certain conditions. A numerical example and a practical example are given to demonstrate the feasibility of the proposed design technique.

Based on the Lyapunov stability theorem, an adaptive sliding mode control scheme is proposed in this thesis for a class of mismatched perturbed multi-input multi-output (MIMO) dynamic systems to solve stabilization problems. In order to suppress the perturbations in the control systems, adaptive mechanisms are employed both in sliding function and control effort, so that the information of upperbound of some perturbations is not required when designing the proposed control scheme. Due to the novel design of sliding function, the state trajectories of this system can achieve asymptotical stability in the sliding mode even if mismatched perturbations exist. In addition, with an adaptive mechanism embedded in the proposed control scheme, the controller can drive the state's trajectory into the designated sliding surface in a finite time. A numerical example is demonstrated for showing the applicability of the proposed design technique.

Based on the Lyapunov stability theorem, an adaptive variable structure observer and a controller are proposed in this thesis for a class of mismatched perturbed multi-input multi-output (MIMO) dynamic systems with unmeasurable states to solve regulation and tracking problems. In order to estimate the unmeasurable states, a design methodology of variable structure observers is presented first. Then the controller is designed so that the trajectories of the controlled systems are able to reach sliding surface in a finite time. Some adaptive mechanisms are embedded in the sliding surface function and sliding mode controllers, so that not only the mismatched perturbations are suppressed effectively during the sliding mode, but also the information of upper bounds of some perturbations are not required. When the controlled system is the sliding mode, the stability or asymptotical stability is guaranteed. A numerical example and a practical example are given to demonstrate the feasibility of the proposed design technique.

Šiame darbe sprendžiamas multiplikatyviųjų funkcijų reikšmių sumavimo uždavinys. Nagrinėjama klasė , kuriai priklauso funkcijos, tenkinančios keletą sąlygų. Svarbiausia iš jų: . Čia C – konstanta, o M – pakankamai didelis, bet fiksuotas teigiamas realusis skaičius. Šios sąlygos prasmė: klasės funkcijos pirminių skaičių aibėje yra artimos vienetui. Darbe įrodyta teorema: jei , tai kai , teisinga asimptotinė formulė . Čia tam tikra konstanta, priklausanti nuo funkcijos . / The problem of an asymptotical behaviour of values of multiplicative functions is solved in this work. The class is defined with some conditions. The most important condition is: , C is a constant, M is a sufficiently large real positive number here. The following theorem is proved: if function belongs to the class then when . A constant depends on function .

Die vorliegende Diplomarbeit beschäftigt sich mit zwei Aspekten der statistischen Eigenschaften von Clusterverfahren. Zum einen geht die Arbeit auf die Frage der Existenz von unterschiedlichen Clusteranalysemethoden zur Strukturfindung und deren unterschiedlichen Vorgehensweisen ein. Die Methode des Abstandes zwischen Mannigfaltigkeiten und die K-means Methode liefern ausgehend von gleichen Daten unterschiedliche Endclusterungen. Der zweite Teil dieser Arbeit beschäftigt sich näher mit den asymptotischen Eigenschaften des K-means Verfahrens. Hierbei ist die Menge der optimalen Clusterzentren konsistent. Bei Vergrößerung des Stichprobenumfangs gegen Unendlich konvergiert diese in Wahrscheinlichkeit gegen die Menge der Clusterzentren, die das Varianzkriterium minimiert. Ebenfalls konvergiert die Menge der optimalen Clusterzentren für n gegen Unendlich gegen eine Normalverteilung. Es hat sich dabei ergeben, dass die einzelnen Clusterzentren voneinander abhängen. / The following thesis describes two different views onto the statistical characterics of clustering procedures. At first it adresses the questions whether different clustering methods exist to ascertain the structure of clusters and in what ays the strategies of these methods differ from each other. The method of distance between the manifolds as well as the k-means method provide different final clusters based on equal initial data. The second part of the thesis concentrates on asymptotic properties of the k-means procedure. Here the amount of optimal clustering centres is consistent. If the size of the sample range is enlarged towards infinity, it also converges in probability towards the amount of clustering centres which minimized the whithin cluster sum of squares. Likewise the amount of optimal clustering centres converges for infinity towards the normal distribution. The main result shows that the individual clustering centres are dependent on each other.

Wir betrachten ein System nichtlinearer parabolischer partieller Differentialgleichungen zur Modellierung des biologischen Phänomens Chemotaxis, das unter anderem in Aggregationsprozessen in Lebenszyklen bestimmter Einzeller eine wichtige Rolle spielt. Unser Chemotaxismodell benutzt Sensitivitäts funktionen, die die vorkommenden biologischen Prozesse genauer spezifizieren. Trotz der durch die Sensitivitätsfunktionen eingebrachten, zusätzlichen Nichtlinearitäten in den Gleichungen erhalten wir zeitlich globale Existenz von Lösungen für verschiedene biologisch realistische Klassen von Sensitivitätsfunktionen und können unter unterschiedlichen Bedingungen an die Systemdaten Konvergenz der Lösungen zu trivialen und nicht-trivialen stationären Punkten beweisen. / We consider a system of non-linear parabolic partial differential equations modeling chemotaxis, a biological phenomenon which plays a crucial role in aggregation processes in the life cycle of certain unicellular organisms. Our chemotaxis model introduces sensitivity functions which help describe the biological processes more accurately. In spite of the additional non-linearities introduced by the sensitivity functions into the equations, we obtain global existence of solutions for different classes of biologically realistic sensitivity functions and can prove convergence of the solutions to trivial and non-trivial steady states.

Ziel dieser Dissertation ist die Untersuchung der asymptotischen Stabilität numerischer Verfahren für Index-2-Algebro-Differentialgleichungen. Es werden Anfangswertaufgaben für quasilineare Algebro-Differentialgleichungen (ADGln). Die meisten anwendungsrelevanten Aufgaben können damit behandelt werden. Zuerst werden einige Stabilitätsbegrife und Aussagen vorgestellt, die das Fundament für den Rest der Arbeit darstellen. Dies erstreckt sich sowohl auf den kontinuierlichen als auch auf den diskreten Fall. Insbesondere werden Kontraktivitätskonzepte eingeführt und Beziehungen zwischen der Kontraktivität der ADGl und derer der Anwendung eines numerischen Verfahrens. Die eingeführte Kontraktivitätsbegriffe erweitern oder verallgemeinern die bereits bekannten Konzepte. Als wichtigste Aussage in dem Kontraktivitätskontext geht ein Theorem hervor, das allgemeine Bedingungen aufstellt, damit die Anwendung eines IRK(DAE)-Verfahrens auf eine ADGl stabil ist. Bekannte Aussagen für gewöhnliche und Algebro-Differntialgleichungen können als Sonderfälle dieses Ergebnisses gesehen werden. Im weiteren Verlauf der Arbeit wird anhand von neuartigen Index-2-Entkopplungs- und Indexreduktionstechniken die Stabilität von Diskretisierungsverfahren untersucht. Die durchgeführte Analyse erbringt neue Ergebnisse, die eine Verbesserung des Kenntnissstandes in diesem Gebiet darstellen. Die erzielte Aussagen stellen hinreichende Bedingungen, damit ein BDF- oder IRK-Verfahren für eine ADGl das gleiche Stabilitätsverhalten wie für eine gewöhnliche Differentialgleichung besitzt. Diese Ergebnisse werden durch numerishce Beispiele veranschaulicht. Weiterhin stellt man fest, dass eine der gefundenen Voraussetzungen für die Kontraktivität der Anwendung eines algebraisch stabilen IRK(DAE)-Verfahrens, auf eine ebenfalls kontraktive ADGl, genügt. Dieses Ergebnis wurde durch die Anwendung der im ersten Teil dieser Arbeit erzielten Kontraktivitätsaussagen ermöglicht. Die Konsequenzen der soeben genannten Aussage für bestimmte Modelle der Schaltkreissimulation werden ebenfalls erläutert. Aus der oben genannten Analyse, ebenso wie aus der Fachliteratur, geht hervor, dass bei manchen ADGl-Aufgaben die Diskretisierungsverfahren Stabilitätsprobleme aufweisen. Um solche Probleme zu behandeln sind bereits einige Ansätze bekannt. Im letzten Teil der Arbeit werden zwei repräsentativen Ansätze betrachtet und ihre Aussichtschancen für Index-2-Aufgaben anhand eines kritischen Beispieles evaluiert. Des Weiteren wird eine Verallgemeinerung für vollimplizite lineare ADGln des Gear-Gupta-Leimkuhler-Ansatzes (GGL) vorgeschlagen. Der Rest der Arbeit beschäftigt sich mit der Stabilitätsuntersuchung der GGL-Formulierung und der auf sie angewandten numerischen Verfahren. Dafür werden Aussagen dieser Arbeit eingesetzt und man kommt zu der Schlussfolgerung, dass sowohl für die IRK(DAE)- als auch für die BDF-Verfahren die Integration der GGL-Formulierung, natürlich unter bestimmten Voraussetzungen, stabil ist. Dieses Ergebniss wird durch ein numerisches Beispiel belegt. Dabei handelt es um eine Gleichung, die mit einer direkten Anwendung eines Verfahrens Instabilitäten aufweist. Jedoch ist die Integration der entsprechenden GGL stabil. / The purpose of the present PhD work is the asymptotic stability investigation of numerical methods for index 2 differential algebraic equations. Initial value problems are considered for quasi linear differential algebraic equations (DAEs) that cover the most important applications. First some stability concepts and related results are presented, which represent the basis for further investigations. This background concerns both, the continuous and the discreet case. Especially contractivity concepts are introduced and the relationship between the asymptotic stability of the DAE and the numerical method applied to it is established. The new contractivity concepts extend or generalize the already known concepts. The most important result in this context is a theorem that establishes general conditions under which the application of an algebraic stable IRK(DAE) method to a DAE is contractive. Well-known assertions for ordinary and differential algebraic equations can be considered as special cases of this general result. Later on the stability of numerical discretizations applied to index-2 DAEs is investigated. This is made possible by the introduction of new decopling and index reduction techniques. The analysis makes new insights in the asymptotic of numerical methods for DAEs possible. The obtained results state sufficient conditions in order that a BDF or an IRK(DAE) method applying to DAEs shows the same asymptotic stability properties as for ODEs. These results are illustrated by some numerical examples. Moreover, it can be realized that one of the found conditions is sufficient in order to show contractivity of the application of an algebraic stable IRK(DAE) method, supposed the DAE is contractive. This assertion is possible based on the general theorem mentioned in the paragraph above. Further some consequences of the mentioned results for electric network models are shown. According to both, the above mentioned analysis and the specialized literature of this field, the application of numerical methods to some special DAEs shows asymptotic stability problems. A few approaches are known to manage such difficult equations. Two exponents of these techniques are considered and their chances of success for index-2 DAEs are evaluated with the application to a critical example. A generalization of the Gear-Gupta-Leimkuhler (GGL) approach is proposed for full implicit linear DAEs. This generalization is investigated in detail in the rest of the paper, concerning both the analytical and the numerical asymptotic stability of the GGL equation and the numerical methods applied to it correspondingly. The result is, that, if some conditions are fulfilled, IRK(DAE) and BDF methods for the GGL equation will produce stable solutions. This result is illustrated by a numerical example. The application of the methods directly to the considered DAE produces unstable solutions. However, the integration of the corresponding GGL formulation is stable. The obtained result opens new possibility for the numerical treatment of instabilities by differential algebraic equations.

<p>In this thesis we study a scalar uniform and non-uniform quantization of random processes (or signals) in average case setting. Quantization (or discretization) of a signal is a standard task in all nalog/digital devices (e.g., digital recorders, remote sensors etc.). We evaluate the necessary memory capacity (or quantization rate) needed for quantized process realizations by exploiting the correlation structure of the model random process. The thesis consists of an introductory survey of the subject and related theory followed by four included papers (A-D).</p><p>In Paper A we develop a quantization coding method when quantization levels crossings by a process realization are used for its coding. Asymptotical behavior of mean quantization rate is investigated in terms of the correlation structure of the original process. For uniform and non-uniform quantization, we assume that the quantization cellwidth tends to zero and the number of quantization levels tends to infinity, respectively.</p><p>In Papers B and C we focus on an additive noise model for a quantized random process. Stochastic structures of asymptotic quantization errors are derived for some bounded and unbounded non-uniform quantizers when the number of quantization levels tends to infinity. The obtained results can be applied, for instance, to some optimization design problems for quantization levels.</p><p>Random signals are quantized at sampling points with further compression. In Paper D the concern is statistical inference for run-length encoding (RLE) method, one of the compression techniques, applied to quantized stationary Gaussian sequences. This compression method is widely used, for instance, in digital signal and image processing. First, we deal with mean RLE quantization rates for various probabilistic models. For a time series with unknown stochastic structure, we investigate asymptotic properties (e.g., asymptotic normality) of two estimates for the mean RLE quantization rate based on an observed sample when the sample size tends to infinity.</p><p>These results can be used in communication theory, signal processing, coding, and compression applications. Some examples and numerical experiments demonstrating applications of the obtained results for synthetic and real data are presented.</p>