Technical Brief

Sliding Mode Control of Vibration in Single-Degree-of-Freedom Fractional Oscillators

[+] Author and Article Information
Jian Yuan

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: yuanjianscar@gmail.com

Youan Zhang

Institute of Technology,
Yantai Nanshan University,
Yantai 265713, China
e-mail: zhangya63@sina.com

Jingmao Liu

Shandong Nanshan International Flight Co., Ltd.,
Yantai 265713, China
e-mail: liujingmao@nanshan.com.cn

Bao Shi

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: baoshi781@sohu.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 12, 2016; final manuscript received April 2, 2017; published online July 10, 2017. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 139(11), 114503 (Jul 10, 2017) (6 pages) Paper No: DS-16-1183; doi: 10.1115/1.4036665 History: Received April 12, 2016; Revised April 02, 2017

This paper proposes sliding mode control of vibration in three types of single-degree-of-freedom (SDOF) fractional oscillators: the Kelvin–Voigt type, the modified Kelvin–Voigt type, and the Duffing type. The dynamical behaviors are all described by second-order differential equations involving fractional derivatives. By introducing state variables of physical significance, the differential equations of motion are transformed into noncommensurate fractional-order state equations. Fractional sliding mode surfaces are constructed and the stability of the sliding mode dynamics is proved by means of the diffusive representation and Lyapunov stability theory. Then, sliding mode control laws are designed for fractional oscillators, respectively, in cases where the bound of the external exciting force is known or unknown. Furthermore, sliding mode control laws for nonzero initialization case are designed. Finally, numerical simulations are carried out to validate the above control designs.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Park, S. , 2001, “ Analytical Modeling of Viscoelastic Dampers for Structural and Vibration Control,” Int. J. Solids Struct., 38(44), pp. 8065–8092. [CrossRef]
Lima, A. M. G. D. , Bouhaddi, N. , Rade, D. A. , and Belonsi, M. , 2015, “ A Time-Domain Finite Element Model Reduction Method for Viscoelastic Linear and Nonlinear Systems,” Lat. Am. J. Solids Struct., 12(6), pp. 1182–1201. [CrossRef]
Bagley, R. L. , 1979, “ Applications of Generalized Derivatives to Viscoelasticity,” Ph.D. thesis, Air Force Institute of Technology, Wright-Patterson AFB, OH.
Bagley, R. L. , and Torvik, J. , 1983, “ Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures,” AIAA J., 21(5), pp. 741–748. [CrossRef]
Bagley, R. L. , and Torvik, P. , 1983, “ A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol. (1978-Present), 27(3), pp. 201–210. [CrossRef]
Xu, Z. D. , Xu, C. , and Hu, J. , 2015, “ Equivalent Fractional Kelvin Model and Experimental Study on Viscoelastic Damper,” J. Vib. Control, 21(13), pp. 2536–2552.
Moreau, X. , Ramus-Serment, C. , and Oustaloup, A. , 2002, “ Fractional Differentiation in Passive Vibration Control,” Nonlinear Dyn., 29(1–4), pp. 343–362. [CrossRef]
Pritz, T. , 2003, “ Five-Parameter Fractional Derivative Model for Polymeric Damping Materials,” J. Sound Vib., 265(5), pp. 935–952. [CrossRef]
Rossikhin, Y. A. , and Shitikova, M. V. , 2010, “ Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results,” ASME Appl. Mech. Rev., 63(1), p. 010801. [CrossRef]
Barone, G. , Di Paola, M. , Iacono, F. L. , and Navarra, G. , 2015, “ Viscoelastic Bearings With Fractional Constitutive Law for Fractional Tuned Mass Dampers,” J. Sound Vib., 344, pp. 18–27. [CrossRef]
Rüdinger, F. , 2006, “ Tuned Mass Damper With Fractional Derivative Damping,” Eng. Struct., 28(13), pp. 1774–1779. [CrossRef]
Lewandowski, R. , and Pawlak, Z. , 2011, “ Dynamic Analysis of Frames With Viscoelastic Dampers Modelled by Rheological Models With Fractional Derivatives,” J. Sound Vib., 330(5), pp. 923–936. [CrossRef]
Deng, R. , Davies, P. , and Bajaj, A. , 2006, “ A Nonlinear Fractional Derivative Model for Large Uni-Axial Deformation Behavior of Polyurethane Foam,” Signal Process., 86(10), pp. 2728–2743. [CrossRef]
Padovan, J. , Chung, S. , and Guo, Y. H. , 1987, “ Asymptotic Steady State Behavior of Fractionally Damped Systems,” J. Franklin Inst., 324(3), pp. 491–511. [CrossRef]
Padovan, J. , and Guo, Y. , 1988, “ General Response of Viscoelastic Systems Modelled by Fractional Operators,” J. Franklin Inst., 325(2), pp. 247–275. [CrossRef]
Beyer, H. , and Kempfle, S. , 1995, “ Definition of Physically Consistent Damping Laws With Fractional Derivatives,” ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech., 75(8), pp. 623–635. [CrossRef]
Kempfle, S. , Schäfer, I. , and Beyer, H. , 2002, “ Fractional Calculus Via Functional Calculus: Theory and Applications,” Nonlinear Dyn., 29(1–4), pp. 99–127. [CrossRef]
Schäfer, I. , and Kempfle, S. , 2004, “ Impulse Responses of Fractional Damped Systems,” Nonlinear Dyn., 38(1–4), pp. 61–68. [CrossRef]
Rossikhin, Y. A. , and Shitikova, M. V. , 2001, “ Analysis of Rheological Equations Involving More Than One Fractional Parameters by the Use of the Simplest Mechanical Systems Based on These Equations,” Mech. Time-Depend. Mater., 5(2), pp. 131–175. [CrossRef]
Fukunaga, M. , 2002, “ On Initial Value Problems of Fractional Differential Equations,” Int. J. Appl. Math., 9(2), pp. 219–236.
Fukunaga, M. , 2002, “ On Uniqueness of the Solutions of Initial Value Problems of Ordinary Fractional Differential Equations,” Int. J. Appl. Math., 10(2), pp. 177–190.
Fukunaga, M. , 2003, “ A Difference Method for Initial Value Problems for Ordinary Fractional Differential Equations, II,” Int. J. Appl. Math., 11(3), pp. 215–244.
Hartley, T. T. , and Lorenzo, C. F. , 2002, “ Control of Initialized Fractional-Order Systems,” NASA Glenn Research Center, Brook Park, OH, NASA Technical Report No. NASA/TM-2002-211377.
Lorenzo, C. F. , and Hartley, T. T. , 2008, “ Initialization of Fractional-Order Operators and Fractional Differential Equations,” ASME J. Comput. Nonlinear Dyn., 3(2), p. 021101. [CrossRef]
Podlubny, I. , 1999, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, CA.
Podlubny, I. , 1999, “ Fractional-Order Systems and PIλDμ-Controllers,” IEEE Trans. Autom. Control, 44(1), pp. 208–214. [CrossRef]
Wei, Y. H. , Tse, P. W. , Yao, Z. , and Wang, Y. , 2016, “ Adaptive Backstepping Output Feedback Control for a Class of Nonlinear Fractional Order Systems,” Nonlinear Dyn., 86(2), pp. 1047–1056. [CrossRef]
Wei, Y. H. , Chen, Y. Q. , Liang, S. , and Wang, Y. , 2015, “ A Novel Algorithm on Adaptive Backstepping Control of Fractional Order Systems,” Neurocomputing, 165, pp. 395–402. [CrossRef]
Yuan, J. , Shi, B. , and Ji, W. Q. , 2013, “ Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems,” Adv. Math. Phys., 2013, p.576709.
Shi, B. , Yuan, J. , and Dong, C. , 2014, “ On Fractional Model Reference Adaptive Control,” Sci. World J., 2014, p. 521625.
Wei, Y. H. , Sun, Z. Y. , Hu, Y. S. , and Wang, Y. , 2016, “ On Fractional Order Composite Model Reference Adaptive Control,” Int. J. Syst. Sci., 47(11), pp. 2521–2531. [CrossRef]
Agrawal, O. P. , 2004, “ A General Formulation and Solution Scheme for Fractional Optimal Control Problems,” Nonlinear Dyn., 38(1–4), pp. 323–337. [CrossRef]
Agrawal, O. P. , 2008, “ A Formulation and Numerical Scheme for Fractional Optimal Control,” J. Vib. Control, 14(9–10), pp. 1291–1299. [CrossRef]
Bagley, R. L. , and Calico, R. A. , 1991, “ Fractional Order State Equations for the Control of Viscoelastically Damped Structures,” J. Guid. Control Dyn., 14(2), pp. 304–311. [CrossRef]
Fenander, A. , 1996, “ Modal Synthesis When Modeling Damping by Use of Fractional Derivatives,” AIAA J., 34(5), pp. 1051–1058. [CrossRef]
Montseny, G. , 1998, “ Diffusive Representation of Pseudo-Differential Time-Operators,” ESAIM: Proc., 5, pp. 159–175.
Trigeassou, J. C. , Maamri, N. , Sabatier, J. , and Oustaloup, A. , 2012, “ Transients of Fractional-Order Integrator and Derivatives,” Signal Image Video Process., 6(3), pp. 359–372. [CrossRef]
Trigeassou, J. C. , Maamri, N. , Sabatier, J. , and Oustaloup, A. , 2012, “ State Variables and Transients of Fractional Order Differential Systems,” Comput. Math. Appl., 64(10), pp. 3117–3140. [CrossRef]
Wu, C. X. , Yuan, J. , and Shi, B. , 2016, “ Stability of Initialization Response of Fractional Oscillators,” J. Vibroengineering, 18(6), pp. 4148–4154. [CrossRef]
Yuan, J. , Zhang, Y. A. , Liu, J. M. , Shi, B., Gai, M., and Yang, S. , 2017, “ Mechanical Energy and Equivalent Differential Equations of Motion for Single Degree-of-Freedom Fractional Oscillators,” J. Sound Vib., 397, pp. 192–203. [CrossRef]


Grahic Jump Location
Fig. 2

Sliding mode control of vibration in SDOF fractional Kelvin–Voigt oscillators

Grahic Jump Location
Fig. 3

Adaptive sliding mode control of vibration in SDOF fractional Kelvin–Voigt oscillators

Grahic Jump Location
Fig. 1

Vibration response of the SDOF fractional Kelvin–Voigt oscillators




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