Research Papers

Numerical Stability Analysis of Linear Incommensurate Fractional Order Systems

[+] Author and Article Information
Sambit Das

Department of Mechanical Engineering,
IIT Kharagpur 721302, India
e-mail: sambitiitkgp@gmail.com

Anindya Chatterjee

Department of Mechanical Engineering,
IIT Kanpur 208016,India
e-mail: anindya100@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received November 9, 2012; final manuscript received March 5, 2013; published online xx xx, xxxx. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(4), 041012 (May 31, 2013) (6 pages) Paper No: CND-12-1196; doi: 10.1115/1.4023966 History: Received November 09, 2012; Revised March 05, 2013

We present a method for detecting right half plane (RHP) roots of fractional order polynomials. It is based on a Nyquist-like criterion with a system-dependent contour which includes all RHP roots. We numerically count the number of origin encirclements of the mapped contour to determine the number of RHP roots. The method is implemented in Matlab, and a simple code is given. For validation, we use a Galerkin based strategy, which numerically computes system eigenvalues (Matlab code is given). We discuss how, unlike integer order polynomials, fractional order polynomials can sometimes have exponentially large roots. For computing such roots we suggest using asymptotics, which provide intuition but require human inputs (several examples are given).

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Monje, C. A., Chen, Y. Q., Vinagre, B. M., Xue, D., and Feliu, V., 2010, Fractional Order Systems and Control – Fundamentals and Applications, Advanced Industrial Control Series, Springer, Berlin.
Caponetto, R., Dongola, G., Fortuna, L., and Petras, I., 2010, Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore.
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, San Diego.
Sabatier, J., Moze, M., and Farges, C., 2010, “LMI Stability Conditions for Fractional Order Systems,” Comput. Math. Appl., 59(5), pp. 1594–1609. [CrossRef]
Trigeassou, J. C., Maamri, N., Sabatier, J., and Oustaloup, A., 2011, “A Lyapunov Approach to the Stability of Fractional Differential Equations,” Signal Process., 91(3), pp. 437–445. [CrossRef]
Trigeassou, J. C., Benchellal, A., Maamri, N., and Poinot, T., 2009, “A Frequency Approach to the Stability of Fractional Differential Equations,” Trans. Syst. Signals Dev., 4(1), pp. 1–26.
Ogata, K., 2001, Modern Control Engineering, 4th ed., Prentice Hall, Englewood Cliffs, NJ.
Bonnet, C., and Partington, J. R., 2002, “Analysis of Fractional Delay Systems of Retarded and Neutral Type,” Automatica, 38, pp. 1133–1138. [CrossRef]
Petras, I., 2009, “Stability of Fractional Order Systems With Rational Orders: A Survey,” Fract. Calc. Appl. Anal., 12(3), pp. 269–298.
Buslowicz, M., 2008, “Stability of Linear Continuous-Time Fractional Order Systems With Delays of the Retarded Type,” Bull. Polish Acad. Sci. Techn. Sci., 56(4), pp. 319–324.
Petras, I., 2011, “Stability Test Procedure for a Certain Class of the Fractional-Order Systems,” 12th International Carpathian Control Conference (ICCC), Velke Karlovice, pp. 303–307.
Hwang, C., and Cheng, Y.-C., 2006, “A Numerical Algorithm for Stability Testing of Fractional Delay Systems,” Automatica, 42(5), pp. 825–831. [CrossRef]
Chatterjee, A., 2005, “Statistical Origins of Fractional Derivatives in Viscoelasticity,” J. Sound Vib., 284(3–5), pp. 1239–1245. [CrossRef]
Singh, S. J., and Chatterjee, A., 2006, “Galerkin Projections and Finite Elements for Fractional Order Derivatives,” Nonlinear Dyn., 45(1–2), pp. 183–206. [CrossRef]
Singh, S. J., and Chatterjee, A., 2011, “Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems,” ASME J. Comput. Nonlinear Dyn., 6(2), p. 021010. [CrossRef]
Das, S., and Chatterjee, A., 2013, “Simple Recipe for Accurate Solution of Fractional Order Equations,” ASME J. Comput. Nonlinear Dyn., 8(3). [CrossRef]
Moornani, K. A., and Haeri, M., 2010, “On Robust Stability of LTI Fractional Order Delay Systems of Retarded and Neutral Type,” Automatica, 46(2), pp. 362–368. [CrossRef]
Hinch, E. J., 1991, Perturbation Methods, Cambridge University Press, Cambridge.
Maamri, N., Trigeassou, J. C., and Mehdi, D., 2009, “A Frequency Approach to Analyze the Stability of Delayed Fractional Differential Equations,” Proceedings of the European Control Conference ECC, Budapest, Hungary.


Grahic Jump Location
Fig. 1

Clockwise oriented contour on complex s plane that includes all RHP zeros of f(s)

Grahic Jump Location
Fig. 2

This closed loop system gives the same output as the open loop system of Eq. (7), provided we take G(s)=1/ansαn and H(s)=a0+∑k=1n-1aksαk



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 Journal Articles
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