Fractional Differential Equations

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Robust and Optimal Control

Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.

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Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

Elements of Probability Theory Uncertain Linear Systems and Robustness Linear Robust Control Design Some Limits of the Robustness Paradigm Probabilistic Methods for Robustness Monte Carlo Methods Randomized Algorithms in Systems and Control Probability Inequalities Statistical Learning Theory and Control Design Sequential Algorithms for Probabilistic Robust Design Sequential Algorithms for LPV Systems Scenario Approach for Probabilistic Robust Design Random Number and Variate Generation Statistical Theory of Radial Random Vectors Vector Randomization Methods Statistical Theory of Radial Random Matrices Matrix Randomization Methods Applications of Randomized Algorithms

Positive 1D and 2D Systems

In the last two decades considerable effort and progress have been made in the two-dimensional (2D) and multidimensional systems theory and its industrial applications. This part of the book summarizes some recent developments in 2D linear systems theory and its applications.

Two-Dimensional Linear Systems

Fractional discrete-time linear systems.- Fractional continuous-time linear systems.- Fractional positive 2D linear systems.- Pointwise completeness and pointwise degeneracy of linear systems.- Pointwise completeness and pointwise degeneracy of linear systems with state-feedbacks.- Realization Problem for positive fractional and continuous-discrete 2D linear systems.- Cone discrete-time and continuous-time linear systems.- Stability of positive fractional 1D and 2D linear systems.- Stability analysis of fractional linear systems in frequency domain.- Stabilization of positive and fractional linear systems.- Singular fractional linear systems.- Positive continuous-discrete linear systems.- Laplace transforms of continuous-time functions and z-transforms of discrete-time functions. "/b>

Selected Problems of Fractional Systems Theory

This monograph covers some selected problems of positive and fractional electrical circuits composed of resistors, coils, capacitors and voltage (current) sources. The book consists of 8 chapters, 4 appendices and a list of references. Chapter 1 is devoted to fractional standard and positive continuous-time and discrete-time linear systems without and with delays. In chapter 2 the standard and positive fractional electrical circuits are considered and the fractional electrical circuits in transient states are analyzed. Descriptor linear electrical circuits and their properties are investigated in chapter 3, while chapter 4 is devoted to the stability of fractional standard and positive linear electrical circuits. The reachability, observability and reconstructability of fractional positive electrical circuits and their decoupling zeros are analyzed in chapter 5. The fractional linear electrical circuits with feedbacks are considered in chapter 6. In chapter 7 solutions of minimum energy control for standard and fractional systems with and without bounded inputs is presented. In chapter 8 the fractional continuous-time 2D linear systems described by the Roesser type models are investigated.

Tadeusz Kaczorek

Papers

Fractional Differential Equations

Positive 1D and 2D Systems

Two-Dimensional Linear Systems

Selected Problems of Fractional Systems Theory

Fractional Linear Systems and Electrical Circuits