Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year


IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

Online
ISSN
2191-0294
See all formats and pricing
More options …
Volume 19, Issue 3-4

Issues

A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay

José António Tenreiro Machado
  • Department of Electrical Engineering, Institute of Engineering, Rua Dr. António Bernardino de Almeida, 431, Porto 4249-015, Portugal
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Behrouz Parsa Moghaddam
Published Online: 2018-03-14 | DOI: https://doi.org/10.1515/ijnsns-2016-0094

Abstract

In this paper, we propose a high-accuracy linear B-spline finite-difference approximation for variable-order (VO) derivative. We consider VO fractional differentiation as a control parameter for improving the stability in systems exhibiting vibrations. The method is applied to nonlinear feedback with VO fractional derivative. The results demonstrate the efficiency and high accuracy of the novel algorithm.

Keywords: variable-order fractional calculus; fractional calculus; finite-difference method; B-spline; feedback; nonlinear control systems

PACS: 02.70.Bf; 05.45.Pq; 05.45.Gg; 02.30.Ks

References

  • [1]

    Chen Y. M., Liu Q. X., Liu J. K., Steady state response analysis for fractional dynamic systems based on memory-free principle and harmonic balancing, Int. J. Non-Linear Mech. 81 (2016), 154–164.CrossrefGoogle Scholar

  • [2]

    Liu X., Hong L., Jiang J., Tang D., Yang L., Global dynamics of fractional-order systems with an extended generalized cell mapping method, Nonlinear Dyn. 83 (3) (2016), 1419–1428.CrossrefGoogle Scholar

  • [3]

    Machado J. T., Kiryakova V., Mainardi F., Recent History of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (3) (2011), 1140–1153.CrossrefGoogle Scholar

  • [4]

    Machado J. T., Galhano A. M., Trujillo J. J., Science metrics on fractional calculus development since 1966, Fractional Calculus Appl. Anal. 16 (2) (2013), 479–500.Google Scholar

  • [5]

    Machado J. T., Mainardi F., Kiryakova V., Fractional Calculus: Quo Vadimus? (Where Are We Going?) Contributions to round table discussion held at ICFDA 2014, Fractional Calculus Appl. Anal. 18 (2) (2015), 495–526.Google Scholar

  • [6]

    Lopes A. M., Machado J. T., Application of fractional techniques in the analysis of forest fires, Nonlinear Sci Int. J.. Numer. Simul. 17 (7–8) (2016), 381–390.Google Scholar

  • [7]

    De la Sen M., About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory Appl. 2011 (1) (2011).Google Scholar

  • [8]

    De la Sen M., Hedayatib V., Atanib Y. G., Rezapourb S., The existence and numerical solution for a k-dimensional system of multi-term fractional integro-differential equations, Nonlinear Anal.–Model. Control 22(2) (2017), 188–209.CrossrefGoogle Scholar

  • [9]

    Momani S., Qaralleh R., An efficient method for solving systems of fractional integro–differential equations, Comput. Math. Appl. 52 (3) (2006), 459–70.Google Scholar

  • [10]

    De la Sen M., On Nonnegative Solutions of fractional-linear time-varying dynamic systems with delayed dynamics, Abstr. Appl. Anal. 2014 (2014).Google Scholar

  • [11]

    Yang X. J., Local fractional functional analysis and its applications, Asian Academic Publisher, Hong Kong, 2011.Google Scholar

  • [12]

    Yang X. J., Advanced local fractional calculus and its applications, World Science, New York, NY, USA, 2012.Google Scholar

  • [13]

    Dabiri A., Butcher E. A., Nazari M., One-dimensional impact problem in fractional viscoelastic models, ASME 2016 International Design Engineering Technical Conferences & Computers and Information in Conference Engineering, IDETC/CIE, 2016.Google Scholar

  • [14]

    Dabiri A., Butcher E. A., Nazari M., Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation, Sound Vibr J.. 388 (2017), 230–244.CrossrefGoogle Scholar

  • [15]

    Dabiri A., Nazari M., Butcher E. A., The spectral parameter estimation method for parameter identification of linear fractional order systems, In Conference American Control (ACC), (2016), 2772–2777.Google Scholar

  • [16]

    Diethelm K., Freed A. D., The FracPECE subroutine for the numerical solution of differential equations of fractional order, Forschung und wissenschaftliches Rechnen 1999 (1998), 57–71.Google Scholar

  • [17]

    Weilbeer M., Efficient numerical methods for fractional differential equations and their analytical background, Papierflieger, 2005.Google Scholar

  • [18]

    Dabiri A., Butcher E. A., Efficient modified Chebyshev differentiation matrices for fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 50 (2017), 284–310.CrossrefGoogle Scholar

  • [19]

    Samko S. G., Fractional integration and differentiation of variable order, Anal. Math. 21 (3) (1995), 213–236.Google Scholar

  • [20]

    Lorenzo C. F., Hartley T. T., Variable order and distributed order fractional operators, Nonlinear Dyn. 29 (1–4) (2002), 57–98.CrossrefGoogle Scholar

  • [21]

    .Sun H. G, Chen W., Chen Y. Q., Variable-order fractional differential operators in anomalous diffusion modeling, Physica A 388 (21) (2009), 4586–4592.CrossrefGoogle Scholar

  • [22]

    Moghaddam B. P., Machado J. A. T., Behforooz H., An integro quadratic spline approach for a class of variable-order fractional initial value problems, Chaos, Solitons Fractals, 2017.Google Scholar

  • [23]

    Coimbra C. F. M., Mechanics with variable order differential operators, Annalen der Physik 12 (11–12) (2003), 692–703.CrossrefGoogle Scholar

  • [24]

    Soon S. C. M., Coimbra C. F. M., Kobayashi M. H., The variable viscoelasticity oscillator, Annalen der Physik 14 (6) (2005), 378–389.CrossrefGoogle Scholar

  • [25]

    Yang X. J., Machado J. A. T., A new fractional operator of variable order: application in the description of anomalous diffusion, Physica A 481 (2017), 276–283.CrossrefGoogle Scholar

  • [26]

    Sun H. W. H. G., Chen W., Chen Y., A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top. Perspect. Fractional. Dynam. Control 193 (185) (2011), 185–192.Google Scholar

  • [27]

    Sheng H., Sun H., Coopmans C., Chen Y., Bohannan G. W., A physical experimental study of variable-order fractional integrator and differentiator, Eur. Phys. JSpec. Top 193 (1) (2011), 93–104.CrossrefGoogle Scholar

  • [28]

    Tang H., Wang D., Huang R., Pei X., Chen W., A new rock creep model based on variable-order fractional derivatives and continuum damage mechanics, Bull. Eng. Geol. Environ. (2017).Google Scholar

  • [29]

    Sheng H., Sun H., Coopmans C., Chen Y., Bohannan G. W., Physical experimental study of variable-order fractional integrator and differentiator, Proceedings of The 4th IFAC Workshop Fractional Differentiation and its Applications FDA’10 (2010).Google Scholar

  • [30]

    Ramirez L., Coimbra C., On the variable order dynamics of the nonlinear wake caused by a sedimenting particle, Physica D. 240 (13) (2011), 1111–1118.CrossrefGoogle Scholar

  • [31]

    Moghaddam B. P., Machado J. A. T., Extended algorithms for approximating variable order fractional derivatives with applications, J. Sci. Comput. 71 (3) (2017), 1351–1374.CrossrefGoogle Scholar

  • [32]

    Moghaddam B. P., Machado J. A. T., SM-algorithms for approximating the variable-order fractional derivative of high order, Fundamenta Informaticae 151 (1–4) (2017), 293–311.CrossrefGoogle Scholar

  • [33]

    Moghaddam B. P., Mostaghim Z. S., Modified finite difference method for solving fractional delay differential equations, Bol. Sociedade Paranaense Matemtica 35 (2) (2017), 49–58.CrossrefGoogle Scholar

  • [34]

    Moghaddam B. P., Yaghoobi S., Machado J. T., An extended predictor-corrector algorithm for variable-order fractional delay differential equations, J. Comput. Nonlinear Dyn. (2016).Google Scholar

  • [35]

    Moghaddam B. P., Mostaghim Z. S., A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations, Ain Shams Eng. J. 5 (2) (2014), 585–594.CrossrefGoogle Scholar

  • [36]

    Daftardar-Gejji V., Sukale Y., Bhalekar S., Solving fractional delay differential equations: a new approach, Fractional Calculus Appl. Anal. 18 (2) (2015), 400–418.Google Scholar

  • [37]

    Morgado M. L., Ford N. J., Lima P. M., Analysis and numerical methods for fractional differential equations with delay, J. Comput. Appl. Math. 252 (2013), 159–168.CrossrefGoogle Scholar

  • [38]

    Srinivasan V., Sukavanam N., Sensitivity analysis of nonlinear fractional order control systems with state delay, Int. J. Comput. Math. 93 (1) (2016), 160–178.CrossrefGoogle Scholar

  • [39]

    Muresan C. I., Dutta A., Dulf E. H., Pinar Z., Maxim A., Ionescu C. M., Tuning algorithms for fractional order internal model controllers for time delay processes, Int. J. Control 89 (3) (2016), 579–593.CrossrefGoogle Scholar

  • [40]

    KWON W. H., Lee G. I. W., Kim S. W., Performance improvement using time delays in multivariable controller design, Control Int. J. 52 (6) (1990), 1455–1473.CrossrefGoogle Scholar

  • [41]

    Shanmugathasan N., Johnston R. D., Exploitation of time delays for improved process control, Int. J. Control 48 (3) (1988), 1137–1152.CrossrefGoogle Scholar

  • [42]

    Suh I., Bien Z., Proportional minus delay controller, IEEE Trans. Aut. Control AC-24 (1979), 370–372.Google Scholar

  • [43]

    Suh H., Bien Z., Use of time-delay actions in the controller design, IEEE Trans. Aut. Control AC-25 (1980), 600–603.Google Scholar

  • [44]

    Butcher E. A., Dabiri A., Nazari M., Stability and Control of Fractional Periodic Time-Delayed Systems, pp. 107–125, Springer International Publishing, 2017.Google Scholar

  • [45]

    Machado J. A. T., Fractional-order derivative approximations in discrete-time control systems, Syst. Anal. Modell. Simul. 34 (4) (1999), 419–434.Google Scholar

  • [46]

    Podlubny I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press 198 1988.Google Scholar

  • [47]

    Podlubny I., Fractional-order systems and PI D-controllers, IEEE Trans. Autom. Control 44 (1) (1999), 208–214.CrossrefGoogle Scholar

  • [48]

    Ingman D., J. Suzdalnitsky, M. Zeifman, Constitutive dynamic-order model for nonlinear contact phenomena, J. Appl. Mech. 67 (2) (2000), 383–390.CrossrefGoogle Scholar

  • [49]

    Matignon D., Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl. 2 (1996).Google Scholar

  • [50]

    Machado J. A. T., Lopes A. M., A fractional perspective on the trajectory control of redundant and hyper-redundant robot manipulators, Appl. Math. Modell. (2016).Google Scholar

  • [51]

    Dabiri A., Butcher E. A., Poursina M., Fractional Delayed Control Design for Linear Periodic Systems, In: ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Conference Engineering, (2016), V006T09A063–V006T09A063.Google Scholar

  • [52]

    Wang Z. H., Zheng Y. G., The optimal form of the fractional-order difference feedbacks in enhancing the stability of a sdof vibration system, Sound Vib J.. 326 (3) (2009), 476–488.CrossrefGoogle Scholar

  • [53]

    Smith H., An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2010.Google Scholar

  • [54]

    Henry R. J., Masoud Z. N., Nayfeh A. H., Mook D. T., Cargo pendulation reduction on ship-mounted cranes via boom-luff angle actuation, J. Vib. Control 7 (8) (2001), 1253–1264.CrossrefGoogle Scholar

  • [55]

    Masoud Z. N., Nayfeh A. H., Al-Mousa A., Delayed position feedback controller for the reduction of payload pendulations of rotary cranes, J. Vib. Control 9 (2003), 257–277.CrossrefGoogle Scholar

  • [56]

    Pyragas K., Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170 (6) (1992), 421–428.Google Scholar

About the article

Received: 2016-07-08

Accepted: 2018-02-04

Published Online: 2018-03-14

Published in Print: 2018-06-26


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 231–238, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0094.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in