Fractional Dynamical Systems: Methods, Algorithms and Applications (Studies in Systems, Decision and Control, 402)

✍ Scribed by Piotr Kulczycki (editor), Józef Korbicz (editor), Janusz Kacprzyk (editor)

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 398
• Category: Library

✦ Synopsis

This book presents a wide and comprehensive spectrum of issues and problems related to fractional-order dynamical systems. It is meant to be a full-fledge, comprehensive presentation of many aspects related to the broadly perceived fractional-order dynamical systems which constitute an extension of the traditional integer-order-type descriptions. This implies far-reaching consequences, both analytic and algorithmic, because―in general―properties of the traditional integer-order systems cannot be directly extended by a straightforward generalization to fractional-order systems, modeled by fractional-order differential equations involving derivatives of an non-integer order. This can be useful for describing and analyzing, for instance, anomalies in the behavior of various systems, chaotic behavior, etc. The book contains both analytic contributions with state-of-the-art and theoretical foundations, algorithmic implementation of tools and techniques, and―finally―some examples of relevantand successful practical applications.

✦ Table of Contents

Introduction
Contents
Foundations
Fractional Systems: State-of-the-Art
1 Introduction
2 Foundations
3 Implementations
4 Applications
5 Control Systems
6 Conclusion
References
Fractional Systems: Theoretical Foundations
1 Introduction
2 Fractional Derivatives and Integrals Definitions
2.1 Notation
2.2 Riemann–Liouville Fractional-Order Left-Sided Derivative/Integral
2.3 Grünwald–Letnikov Fractional-Order Left-Sided Derivative/Integral
2.4 Caputo Fractional-Order Left-Sided Derivative/Integral
3 Unilateral (One-Sided) Laplace Transform of the Fractional-Order Integral and Derivative
3.1 The Unilateral Laplace Transform of the Caputo Fractional-Order Left-Side Derivative
3.2 The Unilateral Laplace Transform of the Grünwald–Letnikov Fractional-Order Left-Side Derivative
4 Linear Fractional-Order SISO Systems
4.1 Numerical Solution of the Linear, Time-Invariant Differential Equation with Grünwald–Letnikov Fractional-Order Left-Hand Derivatives
5 Fractional-Order Transfer Function
5.1 Fourier Transform of the Fractional-Order Differential Equation
5.2 Fractional-Order Integrator
5.3 Fractional-Order Inertial Element
6 Elements of the State-Space Equivalence of the Fractional-Order Differential Equation
6.1 Solution of the Fractional-Order State-Space Equation
6.2 Relation Between the State-Space Description of Fractional-Order System and Its Transfer Function
6.3 Realizations of the Fractional-Order Transfer Functions
6.4 Similarity Transformations of Matrices Describing the Fractional-Order Systems
7 Stability of the Fractional-Order Linear Systems
8 Final Remarks
References
Modeling, Behavior and Properties
Mixed Logical Dynamical Modeling of Discrete-Time Hybrid Fractional Systems
1 Introduction
2 Preliminaries
2.1 Fractional-Order Difference
2.2 Discrete-Time Fractional-Order State Space Models
3 Fundamentals of the Concept of Discrete-Time, Integer-Order Hybrid MLD State Space Models
3.1 Boolean Algebra and Equivalent Linear Integer Inequalities
3.2 Functional Components of Discrete-Time Hybrid MLD State Space Models
3.3 Discrete-Time Integer-Order Hybrid MLD State Space Models
4 MLD Model of Discrete-Time Fractional-Order Hybrid Systems
5 Some Examples of Discrete-Time FO MLD Models
6 Conclusions
References
Fractional Variable-Order Derivative and Difference Operators and Their Applications to Dynamical Systems Modelling
1 Introduction
2 Variable Order Operators
3 Variable Order Control Systems
3.1 Approximate Solutions of Variable Order Control Systems
3.2 Including Initial Conditions
3.3 Analytical Solutions of Variable Order Control Systems
4 Application of Variable Order Calculus
4.1 Anti-Windup Algorithm
4.2 Modelling of Heat Transfer Process
5 Conclusions
References
Asymptotic Behavior of Discrete Fractional Systems
1 Introduction
2 Fractional Differences
3 Linear Fractional Forward and Backward Difference Equations
4 Solution Representation with Volterra Convolution Sums
5 Linear Time-Invariant Fractional Systems
6 Asymptotic Properties of Scalar Linear Fractional Equations
6.1 Scalar Time-Invariant Equations
6.2 Scalar Time-Varying Backward Equations
7 Separation of Solutions
8 Conclusions
References
Variable-, Fractional-Order Linear System State-Space Description Transformation
1 Introduction
2 Mathematical Preliminaries
2.1 Selected Properties of a Matrix k0A[ν(k)]k
3 VFO Linear Time-Variant SISO Systems
3.1 Matrix Fraction Description Like of VFODS
3.2 Transfer-Function-Like Description of the VFODS
3.3 Solution of the VFODE
3.4 Non-commensurate and Commensurate VFODEs
4 VFO Linear Time-Variant State-Space-Like Description
5 VFO Linear Time-Invariant Commensurate State-Space-Like State Vectors Transformation
6 VFO Linear Time-Invariant Commensurate State-Space-Like Iterative Form
7 Final Conclusions
References
Balanced Truncation Model Reduction in Approximation of Nabla Difference-Based Discrete-Time Fractional-Order Systems
1 Introduction
2 System Representation
3 Stability of the Nabla Difference-Based Fractional-Order System
3.1 Commensurate-Order Case
3.2 General Case
3.3 Stability Analysis Examples
4 System Implementation
4.1 FIR-Based Model
4.2 Balanced Truncation Model Order Reduction Method
4.3 Implementation Algorithm
5 Simulation Examples
6 Conclusions
References
State Feedback Law for Discrete-Time Fractional Order Nonlinear Systems
1 Introduction
2 Fractional h-Difference Operators
3 Linear Fractional Order Systems
4 Smart Plate as a Control Plant of a Non-Collocated Control System
5 The Influence of Fractional Order on Vibration Behavior of the Smart Plate
6 Oustaloup Approximation
7 Designing of Optimal LQR Controller
8 Experimental Setup
9 Summary and conclusions
References
Some Specific Properties of Positive Standard and Fractional Interval Systems
1 Introduction
2 Standard and Fractional Linear Systems
3 New Stability Conditions for Different Orders Fractional Positive Linear Systems
4 Transfer Matrices with Nonnegative Coefficients
5 Stability of Fractional Interval Positive Linear Systems
6 Interval Stability of Positive Linear Systems
7 Adjoint Matrix of the Singular Metzler Matrix
8 Concluding Remarks
References
Stability and Controllability
Global Stability of Nonlinear Fractional Dynamical Systems
1 Introduction
2 Positive Continuous-Time Linear Systems
3 Fractional Positive Continuous-Time Linear Systems
4 Fractional Positive Discrete-Time Linear Systems
5 Descriptor Positive Discrete-Time Linear Systems
6 Stability of Fractional Interval Positive Continuous-Time Linear Systems
7 Global Stability of Nonlinear Feedback Systems with Positive Linear Parts
8 Global Stability of Fractional Nonlinear Feedback Continuous-Time Systems
9 Global Stability of Nonlinear Feedback Discrete-Time Systems
10 Global Stability of Descriptor Nonlinear Feedback Discrete-Time Systems
11 Global Stability of Positive Nonlinear Electrical Circuits
12 Summary
References
Controllability of Fractional Linear Systems with Delays in Control
1 Introduction
2 System Description
3 Controllability Conditions
4 Fractional Systems with Multiple Constant Delays in Control
5 Fractional System with a Single Constant Point Delay in Control
6 Example
7 Fractional Systems with Distributed Delays in Control
8 Conclusions
References
Applications
Selected Engineering Applications of Fractional-Order Calculus
1 Introduction
2 RC-Ladder Networks with Supercapacitors
2.1 Basic RC-Ladder Network
2.2 RCR-Uniform Ladder Network
2.3 RCR-Ring Uniform Ladder Network
2.4 RRCr-Uniform Ladder Network
2.5 Exponential RC-Ladder Network
2.6 RC-Plane Network
2.7 Conclusion and Remarks
3 Modeling of Supercapacitors
4 A Chain of Vehicles Driving in Adaptive Cruise Control Mode
5 Modeling of Thermal Processes Inside Buildings
6 Other Models
6.1 Battery Modeling
6.2 Other Possible Applications
7 Final Remarks
References
Fractional Order State Space Models of the One-Dimensional Heat Transfer Process
1 Introduction
2 Preliminaries
2.1 Final Value Theorem
2.2 CFE Approximation
3 The CFE Based Method of Solution for a FO State Equation
3.1 Stability of the CFE Based Discrete Model
3.2 The Cost Function
4 Experimental Heat Transfer System
5 The State Space Model Using the Caputo Operator
5.1 The Spectrum Decomposition of the C Model
5.2 The Step Responses of the C Model
6 The Discrete State Space Model Using the Discrete GL Operator
6.1 Decomposition of the Discrete GL Model
6.2 Stability of the Discrete GL Model
6.3 Accuracy of the Discrete GL Model
6.4 Convergence of the Discrete GL Model
6.5 Experimental Verification of the Discrete GL Model
7 The Discrete State Space Model Using CFE Approximation
7.1 Decomposition of the Discrete CFE Model
7.2 Stability of the Discrete CFE Model
7.3 Accuracy of the Discrete CFE Model
7.4 Convergence of the Discrete CFE Model
7.5 Experimental Verification of the Discrete CFE Model
8 Conclusions
References