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Estimators for a Class of Commensurate Fractional-Order Systems with Caputo Derivative

  • Rafael Martínez-GuerraEmail author
  • Claudia Alejandra Pérez-Pinacho
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

Using some concepts of observability, based on algebraic properties of fractional-order systems, we can synthesize observers for nonlinear fractional-order systems. The technique used in this chapter is based on Rafael Martínez-Guerra et al., Fractional Algebraic Observability property Synchronization of Integral and Fractional Order Chaotic Systems: A Differential Algebraic and Differential Geometric Approach With Selected Applications in Real-Time, Springer 2015, [1], Rafael Martínez-Guerra and Christopher D. Cruz-Ancona. Algorithms of Estimation for Nonlinear Systems: A Differential and Algebraic Viewpoint, Springer 2017, [2]. The former verifies whether a given state of a system can be estimated from a function that depends on the output, input and their finite number of fractional-order derivatives, i.e., if state is reconstructible from output and input measurements. The methodology proposed consist in finding a canonical form for the original system, and this is obtained through a mapping given by the output of the system and its successive fractional-order derivatives. Then, we can design an observer to estimate the state of the transformed system, so-called Fractional Generalized Observability Canonical Form Rafael Martínez-Guerra et al., Fractional Algebraic Observability property Synchronization of Integral and Fractional-Order Chaotic Systems: A Differential Algebraic and Differential Geometric Approach With Selected Applications in Real-Time, Springer 2015, [1]. Finally, from an inverse mapping, obtain the estimates of the original state.

References

  1. 1.
    Rafael Martínez-Guerra, Claudia A. Pérez-Pinacho & Gian Carlo Gómez-Cortés. Synchronization of Integral and Fractional Order Chaotic Systems: A Differential Algebraic and Differential Geometric Approach With Selected Applications in Real-Time, Springer 2015.Google Scholar
  2. 2.
    Rafael Martínez-Guerra & Christopher D. Cruz-Ancona. Algorithms of Estimation for Nonlinear Systems: A Differential and Algebraic Viewpoint, Springer 2017.Google Scholar
  3. 3.
    D. Matignon & B. d’Andrea-Novel. Observer-based controllers for fractional differential systems, Proc. of the 36th CDC, San Diego, Ca. USA, 5: 4967–4972, (December) 1997.Google Scholar
  4. 4.
    Kilbas, A., Srivastava, H., and Trujillo, J., Theory and Applications of Fractional Differential Equations., Elsevier B. V., (2006).CrossRefGoogle Scholar
  5. 5.
    Kai Diethelm. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Springer 2004.zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Rafael Martínez-Guerra
    • 1
    Email author
  • Claudia Alejandra Pérez-Pinacho
    • 1
  1. 1.Automatic ControlCINVESTAV-IPNMexico CityMexico

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