Optimal control of non-smooth fractional-order systems based on extended Caputo derivative

  • Majid Hallaji
  • Mojtaba Ahmadieh Khanesar
  • Abbas DidebanEmail author
  • Ali Vahidyan Kamyad
Original Paper


In this paper, a novel sub-optimal controller is proposed for a class of non-smooth fractional-order systems. In the proposed approach, a new generalized Bernstein expansion is obtained for the original non-smooth function. This new generalized expansion is used to approximate and extend Caputo fractional-order derivative for non-smooth functions. Finally, by using mentioned concepts, a nonlinear optimal control numerical method is generalized to solve the sub-optimal control problem of a class of non-smooth fractional-order nonlinear systems. Two nonlinear fractional-order dynamic systems (smooth and non-smooth) and Chua fractional-order system (non-smooth chaotic system) are studied to show the feasibility and performance of the proposed approach. The advantage of the proposed approach being able to deal with non-smooth functions can be mentioned. Some examples with exact solutions are selected for the proposed approach. It is observed that the approximate solutions obtained from the proposed approach are very close to the exact solution compared with existing methods.


Fractional optimal control problem (FOCP) Non-smooth fractional-order differential equations Caputo derivative 



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© Springer Nature B.V. 2019

Authors and Affiliations

  • Majid Hallaji
    • 1
  • Mojtaba Ahmadieh Khanesar
    • 1
  • Abbas Dideban
    • 1
    Email author
  • Ali Vahidyan Kamyad
    • 2
  1. 1.Department of Electrical and Control EngineeringSemnan UniversitySemnanIran
  2. 2.Department of MathematicsFerdowsi University of MashhadMashhadIran

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