Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

A Method for the Hankel-Norm Approximation of Fractional-Order Systems

Journal of Applied Nonlinear Dynamics 6(2) (2017) 153--171 | DOI:10.5890/JAND.2017.06.003

Jay L. Adams; Robert J. Veillette; Tom T. Hartley

Department of Electrical and Computer Engineering, University of Akron, Akron, OH44325-3904, USA

Download Full Text PDF



A model-reduction methodology for fractional-order systems based on the Hankel-norm is presented. The methodology involves the truncation of a Laurent series associated with the fractional-order system in a transformed domain. The truncated Laurent series coefficients are used to construct a finite-order transfer function to approximate the original system. Standard model-reduction techniques are then applied to obtain a final low-order approximation. The Hankel norm of the approximation error can be specified a priori. The approximation method is applied to several fractional-order and other infinite-order systems. It is shown to be more generally applicable than standard finite-order modeling techniques.


  1. [1]  Oldham, K. and Spanier J. (1974), The Fractional Calculus, Integrations and Differentiations of Arbitrary Order (Academic Press, New York, USA).
  2. [2]  Miller, K. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equa- tions (Wiley, New York, USA).
  3. [3]  Samko, S., Kilbas, A., andMarichev, O. (1993), Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Philadelphia, USA).
  4. [4]  Podlubny, I. (1999), Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications (Academic Press, New York, USA).
  5. [5]  Glover, K. (1984), All optimal Hankel-norm approximations of linear multivariable systems and their Lꝏ-error bounds, Intl. J. of Control, 39, 1115-1193.
  6. [6]  Oustaloup, A., Levron, F., Mathieu, B., and Nanot, F. M. (2000), Frequency-Band Complex Noninteger Differentiator: Characterization and Synthesis, IEEE Transactions on Circuits and Systems I, 47, 25-39.
  7. [7]  Mansouri, R., Bettayeb, M., and Djennoune, S. (2008), Non Integer Order System Approximation by an Integer Reduced Model, Journal Européen des Systèmes Automatisés 3.
  8. [8]  Peller, V. (2003), Hankel Operators and Their Applications (Springer, New York, USA).
  9. [9]  Partington, J. R. (1988), An Introduction to Hankel Operators (Cambridge University Press, Cambridge, UK).
  10. [10]  Conway, J. A Course in Functional Analysis, Springer-Verlag, New York, USA.
  11. [11]  Adams, J.L., Veillette, R.J., and Hartley, T.T. (2009), Estimation of Hankel Singular Values for Fractional- Order Systems, Proc of ASME DETC, (San Diego, USA).
  12. [12]  Adams, J.L., Hartley, T.T., and Veillette, R.J. (2010), Hankel Norm Estimation for Fractional-Order Systems Using the Rayleigh-Ritz Method, Computers and Mathematics with Applications.
  13. [13]  Adams, J.L. and Hartley, T.T. (2008), Hankel Operators for Fractional-Order Systems, J. Européen des Systèmes Automatisés, 3.
  14. [14]  Adams, J.L., Veillette R.J., and Hartley, T.T. (2012), Conjugate-Order Systems for Signal Processing: Sta- bility, Causality, Boundedness, Compactness, Signal, Image, and Video Processing, 6, 373-380.