0

Full Content is available to subscribers

Subscribe/Learn More  >

Chaos Analysis and Control in Fractional Order Systems Using Fractional Chebyshev Collocation Method

[+] Author Affiliations
Arman Dabiri, Morad Nazari, Eric A. Butcher

University of Arizona, Tucson, AZ

Paper No. IMECE2016-67909, pp. V04BT05A032; 10 pages
doi:10.1115/IMECE2016-67909
From:
  • ASME 2016 International Mechanical Engineering Congress and Exposition
  • Volume 4B: Dynamics, Vibration, and Control
  • Phoenix, Arizona, USA, November 11–17, 2016
  • Conference Sponsors: ASME
  • ISBN: 978-0-7918-5055-8
  • Copyright © 2016 by ASME

abstract

In this paper, fractional Chebyshev collocation method is proposed to study Lyapunov exponents (LEs) and chaos in a fractional order system with nonlinearities. For this purpose, the solution of the fractional order system is discretized by N-degree Gauss-Lobatto-Chebyshev (GLC) polynomials where N is an integer number. Then, the discrete orthogonality relationship for the Chebyshev polynomials is used to obtain the fractional Chebyshev differentiation matrix. The differentiation matrix is then used to convert the nonlinear fractional differential equations to a system of nonlinear algebraic equations with the collocation points as the unknowns. The dominant LE (other than the zero LE) that corresponds to the time dimension is then computed by measuring the exponential rate of the trajectory deviations initiated slightly off the attractor point. The proposed technique is implemented to a damped driven pendulum with fractional order damping and the convergence of the dominant LE is studied versus the number of Chebyshev collocation points. The LE analysis is also verified by studying the system time and frequency responses for different values of the bifurcation parameter. Furthermore, the LE obtained by the proposed method for the analogous integer order system is compared with those obtained by the Jacobian technique and Grüwald-Letnikov approximation. Finally a fractional state feedback controller is designed to control the chaotic system to a desired equilibrium or periodic trajectory such that the error dynamics are time invariant or time periodic, respectively. The numerical example studied is the damped driven pendulum with fractional dampers.

Copyright © 2016 by ASME
Topics: Chaos

Figures

Tables

Interactive Graphics

Video

Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature

Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal

NOTE:
Citing articles are presented as examples only. In non-demo SCM6 implementation, integration with CrossRef’s "Cited By" API will populate this tab (http://www.crossref.org/citedby.html).

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In