Fractional Order Processes

Simulation, Identification, and Control

Seshu Kumar Damarla, Madhusree Kundu

• Publisher: CRC Press
• Year: 2018
• eBook ISBN: 9780429996887
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The book presents efficient numerical methods for simulation and analysis of physical processes exhibiting fractional order (FO) dynamics. The book introduces FO system identification method to estimate parameters of a mathematical model under consideration from experimental or simulated data. A simple tuning technique, which aims to produce a robust FO PID controller exhibiting iso-damping property during re-parameterization of a plant, is devised in the book. A new numerical method to find an equivalent finite dimensional integer order system for an infinite dimensional FO system is developed in the book. The book also introduces a numerical method to solve FO optimal control problems.

Key features

• Proposes generalized triangular function operational matrices.
• Shows significant applications of triangular orthogonal functions as well as triangular strip operational matrices in simulation, identification and control of fractional order processes.
• Provides numerical methods for simulation of physical problems involving different types of weakly singular integral equations, Abel's integral equation, fractional order integro-differential equations, fractional order differential and differential-algebraic equations, and fractional order partial differential equations.
• Suggests alternative way to do numerical computation of fractional order signals and systems and control.
• Provides source codes developed in MATLAB for each chapter, allowing the interested reader to take advantage of these codes for broadening and enhancing the scope of the book itself and developing new results.

1

Mathematical Postulations

1.1 Special Functions

In this section, the important functions that are essential for the formulation of fractional-order integral and fractional-order derivative, and that will be used in the following chapters, are provided.

1.1.1 Gamma Function [1]

The gamma function is the generalization of the factorial function to noninteger numbers. It plays a vital role in defining fractional-order integrals and fractional-order derivatives.

The complete gamma function is defined by the Euler limit as

$$\Gamma \left(x\right)=\begin{array}{c}\text{lim}\\ N\to \infty \end{array}\left[\frac{N!N^x}{x\left(x+1\right)\left(x+2\right)\cdots \cdots \left(x+N\right)}\right]$$(1.1)

and by the integral transform as

$$\Gamma \left(x\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-y}{y}^{x-1}dy,x>0.$$ (1.2)

The definition in Equation (1.2) is more practicable than that in Equation (1.1), even tho...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Acknowledgements
• About the Authors
• 1. Mathematical Postulations
• 2. Numerical Method for Simulation of Physical Processes Represented by Weakly Singular Fredholm, Volterra, and Volterra-Fredholm Integral Equations
• 3. Numerical Method for Simulation of Physical Processes Modeled by Abel’s Integral Equations
• 4. Numerical Method for Simulation of Physical Processes Described by Fractional-Order Integro-Differential Equations
• 5. Numerical Method for Simulation of Physical Processes Represented by Stiff and Nonstiff Fractional-Order Differential Equations, and Differential-Algebraic Equations
• 6. Numerical Method for Simulation of Fractional Diffusion-Wave Equation
• 7. Identification of Fractional Order Linear and Nonlinear Systems from Experimental or Simulated Data
• 8. Design of Fractional Order Controllers using Triangular Strip Operational Matrices
• 9. Rational Integer Order System Approximation for Irrational Fractional Order Systems
• 10. Numerical Method for Solving Fractional-Order Optimal Control Problems
• Index