Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Name: Analysis, Modeling and Stability of Fractional Order Differential Systems 1 | The Infinite State Approach
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The Infinite State Approach

Jean-Claude Trigeassou Nezha Maamri(Autor*in)

Wiley (Verlag)

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Erschienen am 6. August 2019

320 Seiten

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978-1-119-64884-0 (ISBN)

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Inhalt

Cover
Half-Title Page
Dedication
Title Page
Copyright Page
Contents
Foreword
Preface
PART 1: Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs)
1. The Fractional Integrator
1.1. Introduction
1.2. Simulation and modeling of integer order ordinary differential equations
1.2.1. Simulation with analog computers
1.2.2. Simulation with digital computers
1.2.3. Initial conditions
1.2.4. State space representation and simulation diagram
1.2.5. Concluding remarks
1.3. Origin of fractional integration: repeated integration
1.4. Riemann-Liouville integration
1.4.1. Definition
1.4.2. Laplace transform of the Riemann-Liouville integral
1.4.3. Fractional integration operator
1.4.4. Fractional differentiation
1.5. Simulation of FDEs with a fractional integrator
1.5.1. Simulation of a one-derivative FDE
1.5.2. FDE
1.5.3. Simulation of the general linear FDE
A.1. Appendix
A.1.1. Lord Kelvin's principle
A.1.2. A brief history of analog computing
A.1.3. Interpretation of the RK2 algorithm
A.1.4. The gamma function
2. Frequency Approach to the Synthesis of the Fractional Integrator
2.1. Introduction
2.2. Frequency synthesis of the fractional derivator
2.3. Frequency synthesis of the fractional integrator
2.3.1. Objective
2.3.2. Direct method
2.3.3. Indirect method
2.3.4. Frequency synthesis of 1/Sn
2.4. State space representation of I nd (S)
2.5. Modal representation of I nd (S)
2.6. Numerical algorithm
2.7. Frequency validation
2.8. Time validation
2.9. Internal state variables
A.2. Appendix: design of fractional integrator parameters
A.2.1. Definition of Gn
A.2.2. Definition of a and ?
3. Comparison of Two Simulation Techniques
3.1. Introduction
3.2. Simulation with the Grünwald-Letnikov approach
3.2.1. Euler's technique
3.2.2. The Grünwald-Letnikov fractional derivative
3.2.3. Numerical simulation with the Grünwald-Letnikov integrator
3.2.4. Some specificities of the Grünwald-Letnikov integrator
3.2.5. Short memory principle
3.3. Simulation with infinite state approach
3.4. Caputo's initialization
3.5. Numerical simulations
3.5.1. Introduction
3.5.2. Comparison of discrete impulse responses (DIRs)
3.5.3. Simulation accuracy
3.5.4. Static error caused by the short memory principle
3.5.5. Caputo's initialization
3.5.6. Conclusion
A.3. Appendix: Mittag-Leffler function
A.3.1. Definition
A.3.2. Laplace transform
A.3.3. Unit step response of 1/sn + a
A.3.4. Caputo's initialization
4. Fractional Modeling of the Diffusive Interface
4.1. Introduction
4.2. Heat transfer and diffusive model of the plane wall
4.2.1. Heat transfer
4.2.2. Physical model of the diffusive interface
4.2.3. Frequency analysis of the diffusive phenomenon
4.2.4. Time analysis of the diffusive phenomenon
4.2.5. Conclusion
4.3. Fractional commensurate order models
4.3.1. Physical origin
4.3.2. Analysis of physical commensurate order models
4.4. Optimization of the fractional commensurate order model
4.4.1. The proposed frequency approach
4.4.2. Conclusion
4.5. Fractional non-commensurate order models
4.5.1. Justification
4.5.2. Parameter estimation of Hn1,n2(s)
4.5.3. Numerical examples
4.5.4. Conclusion
4.6. Conclusion
A.4. Appendix: estimation of frequency responses - the least-squares approach
A.4.1. Identification of the commensurate order model HN-1,N(j?)
A.4.2. Parameter estimation of the non-commensurate model Hn1,n2 (j?)
5. Modeling of Physical Systems with Fractional Models: an Illustrative Example
5.1. Introduction
5.2. Modeling with mathematical models: some basic principles
5.3. Modeling of the induction motor
5.3.1. Construction of the induction motor
5.3.2. Principle of operation
5.3.3. Induction motor knowledge model
5.3.4. Park's model
5.3.5. Fractional Park's model
5.4. Identification of fractional Park's model
5.4.1. Simplified model
5.4.2. Identification algorithm
5.4.3. Nonlinear optimization
5.4.4. Simulation of yk and sk
5.4.5. Comments
5.4.6. Application to the identification of fractional Park's model
PART 2: The Infinite State Approach
6. The Distributed Model of the Fractional Integrator
6.1. Introduction
6.2. Origin of the frequency distributed model
6.3. Frequency distributed model
6.4. Finite dimension approximation of the fractional integrator
6.5. Frequency synthesis and distributed model
6.6. Numerical validation of the distributed model
6.6.1. Reconstruction of the weighting function
6.6.2. Reconstruction of the impulse response
6.7. Riemann-Liouville integration and convolution
6.7.1. Conclusion
6.8. Physical interpretation of the frequency distributed model
6.8.1. The infinite RC transmission line
6.8.2. RC line and spatial Fourier transform
6.8.3. Impulse response of the RC line
6.8.4. General solution
6.8.5. Initialization in the time and spatial domains
A.6. Appendix: inverse Laplace transform of the fractional integrator
7. Modeling of FDEs and FDSs
7.1. Introduction
7.2. Closed-loop modeling of an elementary FDS
7.3. Closed-loop modeling of an FDS
7.3.1. Modeling of an N-derivative FDS
7.3.2. Distributed state
7.4. Transients of the one-derivative FDS
7.4.1. Numerical simulation
7.4.2. Initialization at t = t1
7.4.3. Initialization at different instants
7.5. Transients of a two-derivative FDS
7.6. External or open-loop modeling of commensurate fractional order FDSs
7.6.1. Introduction
7.6.2. External model of an elementary FDE
7.6.3. External representation of a two-derivative FDE
7.6.4. External representation of an N-derivative FDE
7.7. External and internal representations of an FDS
7.8. Computation of the Mittag-Leffler function
7.8.1. Introduction
7.8.2. Divergence of direct computation
7.8.3. Step response approach
7.8.4. Improved step response approach
A.7. Appendix: inverse Laplace transform of 1/Sn + a
8. Fractional Differentiation
8.1. Introduction
8.2. Implicit fractional differentiation
8.3. Explicit Riemann-Liouville and Caputo fractional derivatives
8.3.1. Definitions
8.3.2. Theoretical prerequisites
8.3.3. Comments
8.4. Initial conditions of fractional derivatives
8.4.1. Introduction
8.4.2. Implicit derivative
8.4.3. Caputo derivative
8.4.4. Riemann-Liouville derivative
8.4.5. Relations between initial conditions
8.5. Initial conditions in the general case
8.5.1. Introduction
8.5.2. Implicit derivatives
8.5.3. Caputo derivatives
8.5.4. Riemann-Liouville derivatives
8.5.5. Relations between initial conditions
8.6. Unicity of FDS transients
8.6.1. Transients of the elementary FDE
8.6.2. Unicity of transients
8.6.3. Conclusion
8.7. Numerical simulation of Caputo and Riemann-Liouville transients
8.7.1. Introduction
8.7.2. Simulation of Caputo derivative initialization
8.7.3. Simulation of Riemann-Liouville initialization
9. Analytical Expressions of FDS Transients
9.1. Introduction
9.2. Mittag-Leffler approach
9.2.1. Free response of the elementary FDS
9.2.2. Free response of the N-derivative FDS
9.2.3. Complete solution of the N-derivative FDS
9.3. Distributed exponential approach
9.3.1. Introduction
9.3.2. Solution of Dn (x(t)) = ax(t) using frequency discretization
9.3.3. Solution of Dn (x(t)) = ax(t) using a continuous approach
9.3.4. Solution of Dn (x(t)) = ax(t) using Picard's method
9.3.5. Solution of Dn ( X (t)) = AX (t)
9.3.6. Solution of Dn (X(t)) = AX(t) + Bu(t)
9.4. Numerical computation of analytical transients
9.4.1. Introduction
9.4.2. Computation of the forced response
9.4.3. Step response of a three-derivative FDS
10. Infinite State and Fractional Differentiation of Functions
10.1. Introduction
10.2. Calculation of the Caputo derivative
10.2.1. Fractional derivative of the Heaviside function
10.2.2. Fractional derivative of the power function
10.2.3. Fractional derivative of the exponential function
10.2.4. Fractional derivative of the sine function
10.3. Initial conditions of the Caputo derivative
10.4. Transients of fractional derivatives
10.4.1. Introduction
10.4.2. Heaviside function
10.4.3. Power function
10.4.4. Exponential function
10.4.5. Sine function
10.5. Calculation of fractional derivatives with the implicit derivative
10.5.1. Introduction
10.5.2. Fractional derivative of the Heaviside function
10.5.3. Fractional derivative of the power function
10.5.4. Fractional derivative of the exponential function
10.5.5. Fractional derivative of the sine function
10.5.6. Conclusion
10.6. Numerical validation of Caputo derivative transients
10.6.1. Introduction
10.6.2. Simulation results
A.10. Appendix: convolution lemma
References
Index
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