Research Papers

On the Stability of Fractional-Order Systems of Neutral Type

[+] Author and Article Information
Mohammad Ali Pakzad

Department of Electrical Engineering,
Science and Research Branch,
Islamic Azad University,
Tehran 14778-93855, Iran
e-mail: m.pakzad@srbiau.ac.ir

Sara Pakzad

Department of Electrical Engineering,
South Tehran Branch,
Islamic Azad University,
Tehran 14778-93855, Iran
e-mail: spakzad@pedc.ir

Mohammad Ali Nekoui

Faculty of Electrical and Computer Engineering,
K. N. Toosi University of Technology,
Seyed-Khandan, P. O. Box 16315-1355,
Tehran, Iran
e-mail: manekoui@eetd.kntu.ac.ir

For interpretation of the references to color in figures 3 and 4, the reader is referred to the web version of this article

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 21, 2014; final manuscript received April 28, 2014; published online April 6, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(5), 051013 (Sep 01, 2015) (7 pages) Paper No: CND-14-1079; doi: 10.1115/1.4027593 History: Received March 21, 2014; Revised April 28, 2014; Online April 06, 2015

The aim of this study is to offer a new analytical method for the stability testing of neutral type linear time-invariant (LTI) time-delayed fractional-order systems with commensurate orders and multiple commensurate delays. It is evident from the literature that the stability assessment of this class of dynamics remains unsolved yet and this is the first attempt to take up this challenging problem. The method starts with the determination of all possible purely imaginary characteristic roots for any positive time delay. To achieve this, the Rekasius transformation is used for the transcendental terms in the characteristic equation. An explicit analytical expression in terms of the system parameters which reveals the stability regions (pockets) in the domain of time delay has been presented. The number of unstable roots in each delay interval is calculated with the definition of root tendency (RT) on the boundary of each interval. Two example case studies are also provided, which are not possible to analyze using any other methodology known to the authors.

Copyright © 2015 by ASME
Topics: Stability , Delays
Your Session has timed out. Please sign back in to continue.


Pakzad, M. A., and Pakzad, S., 2012, “Stability Map of Fractional Order Time-Delay Systems,” WSEAS Trans. Syst., 10(11), pp. 541–550.
Khader, M. M., 2013, “The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations,” ASME J. Comput. Nonlinear Dyn., 8(4), p. 041018. [CrossRef]
Suh, Y. S., Ro, Y. S., Kang, H. J., and Lee, H. H., 2004, “Necessary and Sufficient Stability Condition of Discrete State Delay Systems,” Int. J. Control Autom. Syst., 2(4), pp. 501–508.
Pakzad, M. A., 2012, “Kalman Filter Design for Time Delay Systems,” WSEAS Trans. Syst., 11(10), pp. 551–560.
Leung, A. Y. T., Yang, H. X., and Zhu, P., 2014, “Periodic Bifurcation of Duffing-van der Pol Oscillators Having Fractional Derivatives and Time Delay,” Commun. Nonlinear Sci. Numer. Simul., 19(4), pp. 1142–1155. [CrossRef]
Gu, K., and Zheng, X., 2014, “Stability Crossing Set for Systems With Three Scalar Delay Channels,” Int. J. Dyn. Control (submitted) [CrossRef]
Pakzad, M. A., and Moaveni, B., 2013, “Delay-Dependent State Estimation for Time Delay Systems,” WSEAS Trans. Syst., 8(1), pp. 1–10.
Gu, K., 2013, “Complete Quadratic Lyapunov-Krasovskii Functional: Limitations, Computational Efficiency, and Convergence,” Advances in Analysis and Control of Time-Delayed Dynamical Systems, New Jersey World Scientific, pp. 1–19. [CrossRef]
Fioravanti, A. R., Bonnet, C., Özbay, H., and Niculescu, S. I., 2012, “A Numerical Method for Stability Windows and Unstable Root-Locus Calculation for Linear Fractional Time-Delay Systems,” Automatica, 48(11), pp. 2824–2830. [CrossRef]
Olgac, N., and Sipahi, R., “An Exact Method for the Stability Analysis of Time Delayed Lineartime-Invariant (LTI) Systems,” IEEE Trans. Autom. Control, 47(5), pp. 793–797. [CrossRef]
Pakzad, M. A., Pakzad, S., and Nekoui, M. A., 2013, “Stability Analysis of Multiple Time Delayed Fractional Order Systems,” American Control Conference, Washington, DC, pp. 170–175.
Sipahi, R., and Delice, I. I., “Extraction of 3D Stability Switching Hypersurfaces of a Time Delay System With Multiple Fixed Delays,” Automatica, 45(6), pp.1449–1454. [CrossRef]
Suh, Y. S., and Lee, M. H., 1999, “Stability of State Delay Systems Based on a Lyapunov Functional,” Proceedings of IEEE International Symposium on Industrial Electronics, Bled, Slovenia, Vol. 3, pp. 1093–1098.
Pakzad, S., and Pakzad, M. A., 2011, “Stability Condition for Discrete Systems With Multiple State Delays,” WSEAS Trans. Syst. Control, 6(11), pp. 417–426.
Thowsen, A., 1981, “The Routh-Hurwitz Method for Stability Determination of Linear Differential-Difference Systems,” Int. J. Control, 33(5), pp. 991–995. [CrossRef]
Walton, K. E., , Marshal, J. E., 1987 “Direct Method for TDS Stability Analysis,” IEE Proc., Part D, 134, pp. 101–107. [CrossRef]
Öztürk, N., and Uraz, A., 1985, “An Analysis Stability Test for a Certain Class of Distributed Parameter Systems With Delays,” IEEE Trans. Circuits Syst., 34(4), pp. 393–396. [CrossRef]
Jury, E. I., and Zeheb, E., 1986, “On a Stability Test for a Class of Distributed System With Delays,” IEEE Trans. Circuits Syst., 37(10), pp. 1027–1028. [CrossRef]
Buslowicz, M., 2008, “Stability of Linear Continuous Time Fractional Order Systems With Delays of the Retarded Type,” Bull. Pol. Acad. Sci.: Tech. Sci., 56(4), pp. 319–324.
Hwang, C., and Cheng, Y. C., 2006, “A Numerical Algorithm for Stability Testing of Fractional Delay Systems,” Automatica, 42(5), pp. 825–831. [CrossRef]
Shi, M., and Wang, Z. H., 2011, “An Effective Analytical Criterion for Stability Testing of Fractional-Delay Systems,” Automatica, 47(9), pp. 2001–2005. [CrossRef]
Tenreiro Machado, J. A., 2011, “Root Locus of Fractional Linear Systems,” Commun. Nonlinear Sci. Numer. Simul., 16(10), pp. 3855–3862. [CrossRef]
Pakzad, M. A., and Nekoui, M. A., 2013, “Direct Method for Stability Analysis of Fractional Delay Systems,” Int. J. Comput. Commun. Control, 7(6), pp. 863–868. [CrossRef]
Pakzad, M. A., and Nekoui, M. A., 2014, “Stability Map of Multiple Time Delayed Fractional Order Systems,” Int. J. Control Autom. Syst., 12(1), pp. 37–43. [CrossRef]
Pakzad, S., Pakzad, M. A., and Nekoui, M. A., “Stability Map of Fractional Delay Systems in the Parametric Space of Delays and Coefficient,” American Control Conference, Washington, DC, pp. 176–181.
Pakzad, M. A., Nekoui, M. A., and Pakzad, S., “Stability Analysis of Time-Delayed Linear Fractional-Order Systems,” Int. J. Control Autom. Syst., 11(3), pp. 519–525. [CrossRef]
Hua, Ch., Liu, D., and Guan, X. P., 2014, “Necessary and Sufficient Stability Criteria for a Class of Fractional-Order Delayed Systems,” IEEE Trans. Circuits Syst., 61(1), pp. 59–63. [CrossRef]
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, New York.
Tavazoei, M. S., and Haeri, M., 2008, “Chaotic Attractors in Incommensurate Fractional Order Systems,” Phys. D, 237(20), pp. 2628–2637. [CrossRef]
Bonnet, C., and Partington, J. R., 2002, “Analysis of Fractional Delay Systems of Retarded and Neutral Type,” Automatica, 38(7), pp. 1133–1138. [CrossRef]
Rekasius, Z. V., 1980, “A Stability Test for Systems With Delays,” Proceedings of the Joint Automation Control Conference, San Francisco, CA.
Collins, G. E., 1971, “The Calculation of Multivariate Polynomial Resultants,” J. Assoc. Comput. Mach., 18(4), pp. 515–532. [CrossRef]


Grahic Jump Location
Fig. 1

Root-locus for C1(s, τ) from τ = 0.4 until τ = 2.4

Grahic Jump Location
Fig. 2

The natural responses of system (32) for different values of the delay

Grahic Jump Location
Fig. 3

Root-locus for C2(s, τ) from τ = 0.1 until τ = 2.4

Grahic Jump Location
Fig. 4

Root-locus for C2(s, τ) from τ = 0.1 until τ = 3.5




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