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

The state-of-the-art on generalized (or any order) derivatives in physics and engineering sciences, is outlined for justifying the interest of the noninteger differentiation. The problems subsequent to its use in real-time operations are then set out so as to motivate the idea of synthesizing it by a recursive distribution of zeros and poles. An analysis of the existing work is also proposed to support this idea. A comprehensive study is given of the synthesis of differentiators with integer, noninteger, real or complex orders, and whose action is limited to any given frequency bandwidth. First, a definition, in the operational and frequency domains, of a frequency-band complex noninteger order differentiator, is given in a mathematical space with four dimensions which is a Banach algebra. Then, the determination of its synthesized form, by a recursive distribution of complex zeros and poles characterized by complex recursive factors, is presented. The complex noninteger differentiation order is expressed as a function of these recursive factors. The number of zeros and poles is calculated to be as low as possible while still ensuring the stability of the synthesized differentiator to be synthesized. A time validation is presented. Finally, guidelines are proposed for the conception of the synthesized differentiator.

Frequency-band complex noninteger differentiator: characterization and synthesis

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

Fractional Calculus: Integral and Differential Equations of Fractional Order

http://repositorio.uchile.cl/bitstream/handle/2250/127037/Lyapunov-functions-for-fractional-order-systems.pdf?sequence=1

Lyapunov functions for fractional order systems

Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.

/pdf/fractional-order-control-a-tutorial-4zjusf99pg.pdf

Fractional order control - A tutorial

In this paper, stability results of main concern for control theory are given for finite-dimensional linear fractional differential systems. For fractional differential systems in state-space form, both internal and external stabilities are investigated. For fractional differential systems in polynomial representation, external stability is thoroughly examined. Our main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector |arg(σ)| ≤ απ/2, thus generalizing in a stupendous way the well-known results for the integer case α = 1.

Stability results for fractional differential equations with applications to control processing

In the last decades, fractional differential equations have become popular among scientists in order to model various stable physical phenomena with anomalous decay, say that are not of exponential type. Moreover in discrete-time series analysis, so-called fractional ARMA models have been proposed in the literature in order to model stochastic processes, the autocorrelation of which also exhibits an anomalous decay. Both types of models stem from a common property of complex variable functions: namely, multivalued functions and their behaviour in the neighborhood of the branching point, and asymptotic expansions performed along the cut between branching points. This more abstract point of view proves very much useful in order to extend these models by changing the location of the classical branching points (the origin of the complex plane, for continuous-time systems). Hence, stability properties of and modelling issues by generalized fractional differential systems will be adressed in the present paper: systems will be considered both in the time-domain and in the frequency-domain; when necessary a distinction will be made between fractional differential systems of commensurate and incommensurate.

/pdf/stability-properties-for-generalized-fractional-differential-3rpnlf4c79.pdf

Stability properties for generalized fractional differential systems

Le but de ce travail est l'etablissement de procedures systematiques d'ecriture explicite de modeles physiques de resonateurs d'instruments a vent en vue d'une synthese sonore de qualite. La representation en variables d'etat (ve) est choisie car, s'implantant facilement et permettant une estimation parametrique a posteriori, elle allie les avantages des modeles de signaux et des modeles physiques. Nous developpons une ecriture modulaire des systemes de propagation d'ondes interconnectes, et une procedure de construction recurrente de la representation en ve de reseaux de systemes elementaires. Cette ecriture prend en compte les pertes viscothermiques au cours de la propagation des ondes dans l'air. Cette hypothese acoustique est modelisee par une equation aux derivees partielles fractionnaire (edpf), ce qui nous conduit a envisager deux aspects theoriques interessants: les treillis a plusieurs variables, et la derivation fractionnaire. En premier lieu, nous montrons l'equivalence entre les lignes de transmission avec pertes et des filtres en treillis generalises, constitues d'unites de retard et de filtrage: leur description par des polynomes a plusieurs variables independantes permet de donner une condition suffisante de stabilite. En second lieu, nous approfondissons la theorie de la derivation fractionnaire. Nous donnons une definition au sens des distributions causales, et proposons une autre definition adaptee a des fonctions continues developpables en serie fractionnaire (dsf), ce qui introduit naturellement des coefficients dans le developpement. La caracterisation des fonctions propres de ces deux operateurs de derivation permet de resoudre toute equation differentielle fractionnaire (edf) par des moyens algebriques ; d'un point de vue analytique, nous interpretons les conditions initiales non-entieres par les premiers coefficients du dsf, et donnons le comportement asymptotique des solutions par la position des poles fractionnaires dans un domaine spectral fractionnaire. Enfin, pour une edpf, nous proposons une extension a la dimension infinie: l'ensemble des poles fractionnaires est alors denombrable ; ceci constitue une analyse modale d'ordre fractionnaire

Representations en variables d'etat de modeles de guides d'ondes avec derivation fractionnaire

The goal of this paper is to propose observer-based controllers, either in state-space form or in polynomial representation, for fractional differential systems. As for linear differential systems of integer order, polynomial representation allows us to take advantage of the Youla parametrization in order to asymptotically reject some perturbations. This is illustrated on a worked-out example.

Observer-based controllers for fractional differential systems

A time-domain model for the flexural vibrations of damped plates was presented in a companion paper [Part I, J. Acoust. Soc. Am. 109, 1422-1432 (2001)]. In this paper (Part II), the damped-plate model is extended to impact excitation, using Hertz’s law of contact, and is solved numerically in order to synthesize sounds. The numerical method is based on the use of a finite-difference scheme of second order in time and fourth order in space. As a consequence of the damping terms, the stability and dispersion properties of this scheme are modified, compared to the undamped case. The numerical model is used for the time-domain simulation of vibrations and sounds produced by impact on isotropic and orthotropic plates made of various materials (aluminum, glass, carbon fiber and wood). The efficiency of the method is validated by comparisons with analytical and experimental data. The sounds produced show a high degree of similarity with real sounds and allow a clear recognition of each constitutive material of the plate without ambiguity.

/pdf/time-domain-simulation-of-damped-impacted-plates-ii-1md8xmzx13.pdf

Denis Matignon

Papers

Stability results for fractional differential equations with applications to control processing

Stability properties for generalized fractional differential systems

Representations en variables d'etat de modeles de guides d'ondes avec derivation fractionnaire

Observer-based controllers for fractional differential systems

Time-domain simulation of damped impacted plates. II. Numerical model and results.