Analysis, Modeling and Stability of Fractional Order Differential Systems 2

Analysis, Modeling and Stability of Fractional Order Differential Systems 2

by Jean-Claude Trigeassou, Nezha Maamri
Released February 2020
Publisher(s): Wiley-ISTE
ISBN: 9781786304551

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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. With this approach, fundamental issues such as system state interpretation and system initialization – long considered to be major theoretical pitfalls – have been solved easily. Although originally introduced for numerical simulation and identification of FDEs, this approach also provides original solutions to many problems such as the initial conditions of fractional derivatives, the uniqueness of FDS transients, formulation of analytical transients, fractional differentiation of functions, state observation and control, definition of fractional energy, and Lyapunov stability analysis of linear and nonlinear fractional order systems. This second volume focuses on the initialization, observation and control of the distributed state, followed by stability analysis of fractional differential systems.

Table of contents

  1. Cover
  2. Foreword
  3. Preface
  4. PART 1: Initialization, State Observation and Control
    1. 1 Initialization of Fractional Order Systems
      1. 1.1. Introduction
      2. 1.2. Initialization of an integer order differential system
      3. 1.3. Initialization of a fractional differential equation
      4. 1.4. Initialization of a fractional differential system
      5. 1.5. Some initialization examples
    2. 2 Observability and Controllability of FDEs/FDSs
      1. 2.1. Introduction
      2. 2.2. A survey of classical approaches to the observability and controllability of fractional differential systems
      3. 2.3. Pseudo-observability and pseudo-controllability of an FDS
      4. 2.4. Observability and controllability of the distributed state
      5. 2.5. Conclusion
    3. 3 Improved Initialization of Fractional Order Systems
      1. 3.1. Introduction
      2. 3.2. Initialization: problem statement
      3. 3.3. Initialization with a fractional observer
      4. 3.4. Improved initialization
      5. A.3. Appendix
    4. 4 State Control of Fractional Differential Systems
      1. 4.1. Introduction
      2. 4.2. Pseudo-state control of an FDS
      3. 4.3. State control of the elementary FDE
      4. 4.4. State control of an FDS
      5. 4.5. Conclusion
    5. 5 Fractional Model-based Control of the Diffusive RC Line
      1. 5.1. Introduction
      2. 5.2. Identification of the RC line using a fractional model
      3. 5.3. Reset of the RC line
  5. PART 2: Stability of Fractional Differential Equations and Systems
    1. 6 Stability of Linear FDEs Using the Nyquist Criterion
      1. 6.1. Introduction
      2. 6.2. Simulation and stability of fractional differential equations
      3. 6.3. Stability of ordinary differential equations
      4. 6.4. Stability analysis of FDEs
      5. 6.5. Stability analysis of ODEs with time delays
      6. 6.6. Stability analysis of FDEs with time delays
    2. 7 Fractional Energy
      1. 7.1. Introduction
      2. 7.2. Pseudo-energy stored in a fractional integrator
      3. 7.3. Energy stored and dissipated in a fractional integrator
      4. 7.4. Closed-loop and open-loop fractional energies
    3. 8 Lyapunov Stability of Commensurate Order Fractional Systems
      1. 8.1. Introduction
      2. 8.2. Lyapunov stability of a one-derivative FDE
      3. 8.3. Lyapunov stability of an N-derivative FDE
      4. 8.4. Lyapunov stability of a two-derivative commensurate order FDE
      5. 8.5. Lyapunov stability of an N-derivative FDE (N > 2) > 2)
      6. A.8. Appendix
    4. 9 Lyapunov Stability of Non-commensurate Order Fractional Systems
      1. 9.1. Introduction
      2. 9.2. Stored energy, dissipation and energy balance in fractional electrical devices
      3. 9.3. The usual series RLC circuit
      4. 9.4. The series RLC* fractional circuit
      5. 9.5. The series RLL*C* circuit
      6. 9.6. The series RL*C* fractional circuit
      7. 9.7. Stability of a commensurate order FDE: energy balance approach
      8. 9.8. Stability of a commensurate order FDE: physical interpretation of the usual approach
      9. A.9. Appendix
    5. 10 An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems
      1. 10.1. Introduction
      2. 10.2. Indirect Lyapunov method
      3. 10.3. Lyapunov direct method
      4. 10.4. The Van der Pol oscillator
      5. 10.5. Analysis of local stability
      6. 10.6. Large signal analysis
  6. References
  7. Index
  8. End User License Agreement

Product information

  • Title: Analysis, Modeling and Stability of Fractional Order Differential Systems 2
  • Author(s): Jean-Claude Trigeassou, Nezha Maamri
  • Release date: February 2020
  • Publisher(s): Wiley-ISTE
  • ISBN: 9781786304551