Non-Integer Order Calculus and its Applications

Name: Non-Integer Order Calculus and its Applications | 9th International Conference on Non-Integer Order Calculus and Its Applications, Lódz, Poland
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9th International Conference on Non-Integer Order Calculus and Its Applications, Lódz, Poland

Piotr Ostalczyk Dominik Sankowski Jacek Nowakowski(Editor)

Springer (Publisher)

1st Edition

Published on 22. March 2018

X, 229 pages

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978-3-319-78458-8 (ISBN)

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This book focuses on fractional calculus, presenting novel advances in both the theory and applications of non-integer order systems. At the end of the twentieth century it was predicted that it would be the calculus of the twenty-first century, and that prophecy is confirmed year after year. Now this mathematical tool is successfully used in a variety of research areas, like engineering (e.g. electrical, mechanical, chemical), dynamical systems modeling, analysis and synthesis (e.g technical, biological, economical) as well as in multidisciplinary areas (e.g. biochemistry, electrochemistry).As well as the mathematical foundations the book concentrates on the technical applications of continuous-time and discrete-time fractional calculus, investigating the identification, analysis and control of electrical circuits and dynamical systems. It also presents the latest results.Although some scientific centers and scientists are skeptical and actively criticize the applicability of fractional calculus, it is worth breaking through the scientific and technological walls. Because the "fractional community" is growing rapidly there is a pressing need for the exchange of scientific results. The book includes papers presented at the 9th International Conference on Non-integer Order Calculus and Its Applications and is divided into three parts: Mathematical foundations Fractional systems analysis and synthesis System modelingSeven papers discuss the mathematical foundations, twelve papers address fractional order analysis and synthesis and three focus on dynamical system modeling by the fractional order differential and difference equations. It is a useful resource for fractional calculus scientific community.

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Content

Intro
Preface
Organization
International Program Committee
Conference Chair
Conference Co-chair
Local Organizing Committee
Local Chair
Members
Referees
Supporting Institutions
Contents
Mathematical Foundations
Decentralized Stabilization of Fractional Positive Descriptor Discrete-Time Linear Systems
1 Introduction - New
2 Fractional Descriptor Linear Systems
3 Decentralized Stabilization of Fractional Descriptor Linear Systems
4 Procedure and Illustrating Example
5 Concluding Remarks
References
Non-invasive Control of the Fractional Hegselmann-Krause Type Model
1 Introduction
2 Preliminaries
3 Optimal Control of the Fractional Hegselmann-Krause Type Model with Leadership
3.1 Existence of Solutions
3.2 Necessary Optimality Conditions
4 Illustrative Examples
References
Differintegarator Based on Fractional Calculus of Convex Functions
1 Introduction
2 Differintegator
2.1 System
2.2 Simulation Results
3 Conclusions
References
The AQM Dropping Packet Probability Function Based on Non-integer Order PID Controller
1 Introduction
2 Fractional Calculus
3 The Packet Dropping Probability Function Based on PID Controller
4 Summary
References
New Numerical Techniques for Solving Fractional Partial Differential Equations in Conformable Sense
1 Introduction
2 Some Basic Motivations
3 Adomian Decomposition Method in Conformable Sense
4 Modified Homotopy Perturbation Method in Conformable Sense
5 Applications and Results
5.1 Example
5.2 Example
5.3 Example
6 Concluding Remarks
References
Implementation of Low-Pass Fractional Filtering for the Purpose of Analysis of Electroencephalographic Signals
1 Introduction
2 Electroencephalography - Fundamentals
3 Literature Review
4 Non-integer Filtering
5 Research Methodology
5.1 Conducted Experiments
5.2 Applied Filters
5.3 Oustaloup Approximation Parameters
6 Results
7 Conclusions and Further Research
References
Grünwald-Letnikov-Laguerre Modeling of Discrete-Time Noncommensurate Fractional-Order State Space LTI MIMO Systems
1 Introduction
2 Fractional Difference and Finite Fractional Difference
3 Laguerre-Based Fractional Differences and Their Approximations
4 Unified Framework for FFLD-Based Models of Multivariable Noncommensurate FD
5 Application to Noncommensurate Fractional Order State Space MIMO Systems
5.1 Unified Framework for Fractional Noncommensurate State Space Models
5.2 Implementation of State Space Models
6 Simulation Example
7 Conclusion
References
Stability Analysis of Fractional Discrete-Time Linear Scalar Systems with Pure Delay
Abstract
1 Introduction
2 Problem Formulation
3 Solution of the Problem
4 Concluding Remarks
Acknowledgement
References
Duality Properties of Variable-Type and -Order Differences
1 Introduction
2 Fractional Constant-Type Variable-Order Differences
2.1 Duality Between Fractional Variable-Order Differences
3 Fractional Variable-Type and -Order Differences
3.1 Switching Strategies for Variable-Type and -Order Differences
4 Duality of Variable-Type and -Order Differences-Main Result
5 Numerical Results
6 Conclusion
References
A PLC Implementation of PSE Approximant for Fractional Order Operator
1 An Introduction
2 Preliminaries
2.1 Elementary Ideas
3 The Experimental System
3.1 General Remarks About Implementation of Special Control Algorithms at PLC Platform
3.2 The Hardware
3.3 The Software
4 Experimental Results
References
Memory-Effective Modifications of PSE Approximation
1 Introduction
2 Preliminaries
3 Main Results
4 Numerical Tests
5 Conclusions
References
Comparison Fixed-Point and Floating-Point Implementation of Noninteger Filter of STM Microcontroller
1 Introduction
2 Implementation Non-integer System in Digital Environment
2.1 Oustaloup Approximation
2.2 Time Domain Approximation
2.3 Hardware Realization
3 Experiment
4 Conclusion
References
Fractional Derivative Approach in Modeling of a Nonlinear Coil for Ferroresonance Analyses
1 Introduction
2 Ferroresonance Analysis - State of the Art
2.1 Ferroresonant Modes
2.2 Analysis Tools
3 Considered Ferroresonance Circuit
3.1 Classical Model
3.2 Fractional Model
4 The Test Circuit and Performed Measurements
5 Computation Results
6 Concluding Remarks
References
Modeling
An Extended Kalman Filter for Time Delays Inspired by a Fractional Order Model
1 Introduction
2 Motivation
3 Modelling
4 EKF Using a First Order Time Delay Model
4.1 EKF Using a Fractional Order Time Delay Models
5 Results
6 Conclusions
References
Fractional Linear Systems with Memory Deficiency and Their State-Space Integer-Order Approximation
1 Introduction
2 Fractional Dynamic Models
2.1 Differential (Difference) Calculus of Non-integer Order
2.2 Continuous-Time Dynamic Models of Fractional Order
2.3 Discrete-Time Dynamic Models of Fractional Order
3 Dynamic Models of Fractional Order with Memory Deficiency
3.1 Switched Derivative of Fractional Order with Memory Impairment
3.2 Switched Difference of Fractional Order with Memory Impairment
3.3 Continuous-Time Models of Fractional Order with Memory Deficiency
3.4 Discrete-Time Models of Fractional Order with Memory Deficiency
4 Approximation of Discrete-Time Fractional Linear Systems with Memory Deficiency
4.1 State-Space Integer-Order Approximation
4.2 Reduced Integer-Order Approximation
5 Numerical Examples
6 Summary
References
Fractional Order System Analysis
Analysis of Fractional Electrical Circuit Using Caputo and Conformable Derivative Definitions
1 Introduction
2 Fractional State-Space Equations
3 Fractional Electrical Circuit and Its State-Space Equation
4 Analysis of the Solutions for Caputo Definition
4.1 Initial Response for Caputo Definition
4.2 Forced Response for Caputo Definition
4.3 General Solution
5 Analysis of the Solutions for CFD Definition
5.1 Initial Response for CFD Definition
5.2 Forced Response for CFD Definition
5.3 General Response
6 Numerical Analysis
7 Final Conclusions
References
Application of Fractional-Order Controller
1 Introduction
1.1 Classic Controllers
1.2 Fractional Order Controllers
2 Methods of Approximation of the Transfer Function
2.1 Taylor Polynomial
2.2 Oustaloup Recursive Approximation (ORA)
3 Model of Inverted Pendulum
4 Simulations
5 Verification of the Controllers
6 Conclusion
References
State Delays Extraction in the Fractional-Order State-Space Model
1 Introduction
2 Mathematical Preliminaries
3 Linear Fractional-Order System Description I
4 Numerical Example
5 Linear Fractional-Order System Description II
5.1 Stability Conditions of Linear Fractional-Order System II
6 Numerical Example
7 Concluding Remarks
References
Solutions of Circuits with Fractional, Nonlinear Elements by Means of a SubIval Solver
1 Introduction
2 Application of SubIval
3 Considered Class of Problems
4 Implicit Time Stepping Solver
5 Computational Example
6 Verification of the Results
7 Summary and Conclusions
A Formulation of State Equations
References
Author Index

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Non-Integer Order Calculus and its Applications

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