Discrete Fractional Calculus: Applications In Control And Image Processing

Applications in Control and Image Processing

Piotr Ostalczyk

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814725682
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

The main subject of the monograph is the fractional calculus in the discrete version. The volume is divided into three main parts. Part one contains a theoretical introduction to the classical and fractional-order discrete calculus where the fundamental role is played by the backward difference and sum. In the second part, selected applications of the discrete fractional calculus in the discrete system control theory are presented. In the discrete system identification, analysis and synthesis, one can consider integer or fractional models based on the fractional-order difference equations. The third part of the book is devoted to digital image processing.

The main subject of the monograph is the fractional calculus in the discrete version. The volume is divided into three main parts. Part one contains a theoretical introduction to the classical and fractional-order discrete calculus where the fundamental role is played by the backward difference and sum. In the second part, selected applications of the discrete fractional calculus in the discrete system control theory are presented. In the discrete system identification, analysis and synthesis, one can consider integer or fractional models based on the fractional-order difference equations. The third part of the book is devoted to digital image processing.

Readership: Researchers, academics, professionals and graduate students in pattern recognition/image analysis, robotics and automated systems, systems engineering and mathematical modeling.

Chapter 1

Discrete-variable real functions

In this chapter a notion of one discrete-variable real-valued functions is introduced. Fundamental operations are defined. A great emphasis is put on the discrete convolution of two functions [Ostalczyk (2001); O’Flynn and Moriarty (1987); Proakis (2007); Weeks (2011)]. Two special discrete-variable functions denoted as a^(ν)(k) and c^(ν)(k) are defined and discussed. The mentioned functions play an important role in fractional-order differences and sums. The link between two discrete-variable function and a digital image finishes this chapter.

1.1Notation

Firstly one defines some sets:

– the set of real numbers,

– set of real non-negative numbers,

– set of real negative numbers,

– subset of real nonnegative numbers,

– finite interval,

– set of complex numbers,

– set of integers,

– finite set of integers,

– subset of integers greater or equal to k₀ > 0,

– set of non-negative integers,

– finite set of integers greater or equal to k₀ > 0 and less or equal to k,

– finite set of equally spaced real numbers

– set of rational numbers,

Re{.} – real part of a complex number,

Im{.} – imaginary part of a complex number,

– complex conjugate to z,

h > 0 – constant (sampling period),

t – continuous time,

kh – discrete time (k for h = 1),

A_k – real (k + 1) × (k + 1) matrix,

1_k – real (k + 1) × (k + 1) unit matrix,

0_k – real (k + 1) × (k + 1) zero matrix,

a_k, b_k, c_k – real (k + 1) × 1 column vectors,

T – matrix (vector) transposition,

* – discrete convolution operation,

tr – trase of the matrix,

|.| – absolute value,

[rad] – radian,

[dB] – decibel,

[dec] – decade,

[V] – volt,

[s] – second.

1.2Class of one discrete-variable functions

Different signals encountered in science and engineering are described by real-valued functions of a single independent discrete real variable (discrete time instants, discrete distance),

is denoted as

where

is the f...

Table of contents

• Cover page
• Copyright page
• Title page
• Preface
• Contents
• List of Figures
• List of Tables
• 1. Discrete-variable real functions
• 2. The n-th order backward difference/sum of the discrete-variable function
• 3. Fractional-order backward differ-sum
• 4. The FOBD-S graphical interpretation
• 5. The FOBD/S selected properties
• 6. The FO dynamic system description
• 7. Linear FO system analysis
• 8. The linear FO discrete-time fundamental elements
• 9. FO discrete-time system structures
• 10. Fractional discrete-time PID controller
• 11. FOS approximation problems
• 12. Fractional potential
• 13. FO image filtering and edge detection
• Appendix A Selected linear algebra formulae and discrete-variable special functions
• Bibliography
• Index