Nonlinear Dynamics

, Volume 94, Issue 4, pp 3041–3052 | Cite as

A novel PID control with fractional nonlinear integral

  • Aldo-Jonathan Muñoz-VázquezEmail author
  • Anand Sánchez-Orta
  • Vicente Parra-Vega
Original Paper


A nonlinear PID controller for robust tracking of second-order nonlinear systems is proposed, which consists in a classical PD structure plus a fractional-order nonlinear integral action of control (FONLI). This nonlinear integral action of fractional-order induces robustness properties to the closed-loop system in order to withstand for a general class of continuous but not necessarily differentiable disturbances, while inducing a stable sliding motion in finite time, such that, the tracking error converges with exponential rate. The resulting controller is uniformly continuous, preserving the regularity of the control signal. The proposed scheme is extended for higher-order dynamical systems by means of a state-feedback control plus a FONLI, inducing a stable motion in finite time. Numerical results based on simulation are discussed to show the reliability of the proposed scheme.


PID control Fractional-order sliding mode Nonlinear systems Disturbance rejection 



We authors dearly do acknowledge to Editors for handling our paper, and to anonymous reviewers for a thorough review that allowed us to produce an improved paper. Aldo Jonathan also acknowledges CONACYT—Mexico for the project Catedras 1086 “Ambientes Inteligentes”.

Compliance with ethical standards

Conflict of interest

The authors declare they have no conflict of interest regarding the publication of this paper.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Aldo-Jonathan Muñoz-Vázquez
    • 1
    Email author
  • Anand Sánchez-Orta
    • 2
  • Vicente Parra-Vega
    • 2
  1. 1.CONACYT–School of Engineering, Autonomous University of Chihuahua (UACH), Campus IIChihuahuaMexico
  2. 2.Research Center for Advanced StudiesRamos ArizpeMexico

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