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2012 | Book

Introduction Read chapter Read first chapter

Distributed-Order Dynamic Systems

Stability, Simulation, Applications and Perspectives

Authors: Zhuang Jiao, YangQuan Chen, Igor Podlubny

Publisher: Springer London

Book Series : SpringerBriefs in Electrical and Computer Engineering

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

Distributed-order differential equations, a generalization of fractional calculus, are of increasing importance in many fields of science and engineering from the behaviour of complex dielectric media to the modelling of nonlinear systems.

This Brief will broaden the toolbox available to researchers interested in modeling, analysis, control and filtering. It contains contextual material outlining the progression from integer-order, through fractional-order to distributed-order systems. Stability issues are addressed with graphical and numerical results highlighting the fundamental differences between constant-, integer-, and distributed-order treatments. The power of the distributed-order model is demonstrated with work on the stability of noncommensurate-order linear time-invariant systems. Generic applications of the distributed-order operator follow: signal processing and viscoelastic damping of a mass–spring set up.

A new general approach to discretization of distributed-order derivatives and integrals is described. The Brief is rounded out with a consideration of likely future research and applications and with a number of MATLAB® codes to reduce repetitive coding tasks and encourage new workers in distributed-order systems.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
As a branch of mathematics, calculus includes differential calculus and integral calculus. Calculus is the study of change, and has widespread applications in science, economics and engineering, and can solve many real world problems. It is well known that a system’s dynamical properties can be described by an ordinary differential equation (ODE) which contains functions of an independent variable.
Zhuang Jiao, YangQuan Chen, Igor Podlubny
Chapter 2. Distributed-Order Linear Time-Invariant System (DOLTIS) and Its Stability Analysis
Abstract
By using distributed-order concept, we can describe the dynamical properties of real world system more accurately, so distributed-order system identification problem was studied. In the following sections, the stability analysis of distributed-order linear time-invariant systems in four cases are first studied, then the frequency-domain responses are presented, and time-domain responses on the basis of numerical inverse Laplace transform technique are shown in details.
Zhuang Jiao, YangQuan Chen, Igor Podlubny
Chapter 3. Noncommensurate Constant Orders as Special Cases of DOLTIS
Abstract
Stability is a minimum requirement for control systems, certainly including fractional-order systems. The stability results on fractional-order linear time-invariant (FO-LTI) systems with commensurate orders were presented for the first time, it permits to check the asymptotically stability through the location of the system matrix eigenvalues of the pseudo state space representation of fractional-order system in the Complex plane. Henceforth, there were some systematic results on the robust stability of interval uncertain FO-LTI systems. The BIBO-stability of fractional-order delay systems of retarded and neutral types was studied and sufficient conditions were presented for retarded type, and only sufficient conditions were provided for neutral type. Sufficient conditions of stability were provided for an important special case fractional-order delay system of neutral type. However, such theorems don’t permit to conclude the system stability without computing the system’s poles, which constitutes tedious work, so based on Cauchy’s integral theorem and by solving an initial-value problem, an effective numerical algorithm for testing the BIBO stability of fractional delay systems was presented.
Zhuang Jiao, YangQuan Chen, Igor Podlubny
Chapter 4. Distributed-Order Filtering and Distributed-Order Optimal Damping
Abstract
The idea of using the distributed-order differential equation first proposed by Caputo in (1969) is at least mathematically interesting as demonstrated in the previous chapters. However, people may question its usefulness in engineering practice. In this chapter, we included two generic applications. One is on distributed order signal processing and the other is on optimal distributed damping. We hope these two initial applications can serve as motivating examples to further the investigation in distributed order dynamics systems, signal processing, modeling and controls.
Zhuang Jiao, YangQuan Chen, Igor Podlubny
Chapter 5. Numerical Solution of Differential Equations of Distributed Order
Abstract
In this chapter we present a general approach to numerical solution to discretization of distributed-order derivatives and integrals, and to numerical solution of ordinary and partial differential equations of distributed order.
Zhuang Jiao, YangQuan Chen, Igor Podlubny
Chapter 6. Future Topics
Abstract
In previous chapters, methods and tools for the modeling of distributed-order systems were discussed, which include stability analysis of distributed-order systems in four cases of the weighting function of order, and two special cases: double noncommensurate orders and \(N\)-term noncommensurate orders. Distributed-order signal processing technique and optimal distributed-order damping strategies were studied. A general approach to numerical solution to discretization of distributed-order derivatives and integrals, and to numerical solution of ordinary and partial differential equations of distributed order was proposed.
Zhuang Jiao, YangQuan Chen, Igor Podlubny
Backmatter
Metadata
Title
Distributed-Order Dynamic Systems
Authors
Zhuang Jiao
YangQuan Chen
Igor Podlubny
Copyright Year
2012
Publisher
Springer London
Electronic ISBN
978-1-4471-2852-6
Print ISBN
978-1-4471-2851-9
DOI
https://doi.org/10.1007/978-1-4471-2852-6