Analysis, Modeling and Stability of Fractional Order Differential Systems 1

Book description

This book introduces an original fractional calculus methodology (‘the infinite state approach’) which is applied to the modeling of fractional order differential equations (FDEs) and systems (FDSs). Its modeling is based on the frequency distributed fractional integrator, while the resulting model corresponds to an integer order and infinite dimension state space representation. This original modeling allows the theoretical concepts of integer order systems to be generalized to fractional systems, with a particular emphasis on a convolution formulation.

Table of contents

  1. Cover
  2. Foreword
  3. Preface
  4. PART 1: Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs)
    1. 1 The Fractional Integrator
      1. 1.1. Introduction
      2. 1.2. Simulation and modeling of integer order ordinary differential equations
      3. 1.3. Origin of fractional integration: repeated integration
      4. 1.4. Riemann–Liouville integration
      5. 1.5. Simulation of FDEs with a fractional integrator
      6. A.1. Appendix
    2. 2 Frequency Approach to the Synthesis of the Fractional Integrator
      1. 2.1. Introduction
      2. 2.2. Frequency synthesis of the fractional derivator
      3. 2.3. Frequency synthesis of the fractional integrator
      4. 2.4. State space representation of
      5. 2.5. Modal representation of
      6. 2.6. Numerical algorithm
      7. 2.7. Frequency validation
      8. 2.8. Time validation
      9. 2.9. Internal state variables
      10. A.2. Appendix: design of fractional integrator parameters
    3. 3 Comparison of Two Simulation Techniques
      1. 3.1. Introduction
      2. 3.2. Simulation with the Grünwald–Letnikov approach
      3. 3.3. Simulation with infinite state approach
      4. 3.4. Caputo’s initialization
      5. 3.5. Numerical simulations
      6. A.3. Appendix: Mittag-Leffler function
    4. 4 Fractional Modeling of the Diffusive Interface
      1. 4.1. Introduction
      2. 4.2. Heat transfer and diffusive model of the plane wall
      3. 4.3. Fractional commensurate order models
      4. 4.4. Optimization of the fractional commensurate order model
      5. 4.5. Fractional non-commensurate order models
      6. 4.6. Conclusion
      7. A.4. Appendix: estimation of frequency responses – the least-squares approach
    5. 5 Modeling of Physical Systems with Fractional Models: an Illustrative Example
      1. 5.1. Introduction
      2. 5.2. Modeling with mathematical models: some basic principles
      3. 5.3. Modeling of the induction motor
      4. 5.4. Identification of fractional Park’s model
  5. PART 2: The Infinite State Approach
    1. 6 The Distributed Model of the Fractional Integrator
      1. 6.1. Introduction
      2. 6.2. Origin of the frequency distributed model
      3. 6.3. Frequency distributed model
      4. 6.4. Finite dimension approximation of the fractional integrator
      5. 6.5. Frequency synthesis and distributed model
      6. 6.6. Numerical validation of the distributed model
      7. 6.7. Riemann–Liouville integration and convolution
      8. 6.8. Physical interpretation of the frequency distributed model
      9. A.6. Appendix: inverse Laplace transform of the fractional integrator
    2. 7 Modeling of FDEs and FDSs
      1. 7.1. Introduction
      2. 7.2. Closed-loop modeling of an elementary FDS
      3. 7.3. Closed-loop modeling of an FDS
      4. 7.4. Transients of the one-derivative FDS
      5. 7.5. Transients of a two-derivative FDS
      6. 7.6. External or open-loop modeling of commensurate fractional order FDSs
      7. 7.7. External and internal representations of an FDS
      8. 7.8. Computation of the Mittag-Leffler function
      9. A.7. Appendix: inverse Laplace transform of
    3. 8 Fractional Differentiation
      1. 8.1. Introduction
      2. 8.2. Implicit fractional differentiation
      3. 8.3. Explicit Riemann–Liouville and Caputo fractional derivatives
      4. 8.4. Initial conditions of fractional derivatives
      5. 8.5. Initial conditions in the general case
      6. 8.6. Unicity of FDS transients
      7. 8.7. Numerical simulation of Caputo and Riemann–Liouville transients
    4. 9 Analytical Expressions of FDS Transients
      1. 9.1. Introduction
      2. 9.2. Mittag-Leffler approach
      3. 9.3. Distributed exponential approach
      4. 9.4. Numerical computation of analytical transients
    5. 10 Infinite State and Fractional Differentiation of Functions
      1. 10.1. Introduction
      2. 10.2. Calculation of the Caputo derivative
      3. 10.3. Initial conditions of the Caputo derivative
      4. 10.4. Transients of fractional derivatives
      5. 10.5. Calculation of fractional derivatives with the implicit derivative
      6. 10.6. Numerical validation of Caputo derivative transients
      7. A.10. Appendix: convolution lemma
  6. References
  7. Index
  8. End User License Agreement

Product information

  • Title: Analysis, Modeling and Stability of Fractional Order Differential Systems 1
  • Author(s): Jean-Claude Trigeassou, Nezha Maamri
  • Release date: September 2019
  • Publisher(s): Wiley-ISTE
  • ISBN: 9781786302694