This paper addresses the Lyapunov functions and sliding mode control design for two degrees-of-freedom (2DOF) and multidegrees-of-freedom (MDOF) fractional oscillators. First, differential equations of motion for 2DOF fractional oscillators are established by adopting the fractional Kelvin–Voigt constitute relation for viscoelastic materials. Second, a Lyapunov function candidate for 2DOF fractional oscillators is suggested, which includes the potential energy stored in fractional derivatives. Third, the differential equations of motion for 2DOF fractional oscillators are transformed into noncommensurate fractional state equations with six dimensions by introducing state variables with physical significance. Sliding mode control design and adaptive sliding mode control design are proposed based on the noncommensurate fractional state equations. Furthermore, the above results are generalized to MDOF fractional oscillators. Finally, numerical simulations are carried out to validate the above control designs.
Issue Section:
Research Papers
References
1.
Park
, S.
, 2001
, “Analytical Modeling of Viscoelastic Dampers for Structural and Vibration Control
,” Int. J. Solids Struct.
, 38
(44
), pp. 8065
–8092
.2.
Diethelm
, K.
, 2010
, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type
, Springer Verlag
, Berlin, Heidelberg.3.
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
.4.
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
.5.
Padovan
, J.
, and Guo
, Y.
, 1988
, “General Response of Viscoelastic Systems Modelled by Fractional Operators
,” J. Franklin Inst.
, 325
(2
), pp. 247
–275
.6.
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
.7.
Kempfle
, S.
, Schäfer
, I.
, and Beyer
, H.
, 2002
, “Fractional Calculus Via Functional Calculus: Theory and Applications
,” Nonlinear Dyn.
, 29
(1–4
), pp. 99
–127
.8.
Schäfer
, I.
, and Kempfle
, S.
, 2004
, “Impulse Responses of Fractional Damped Systems
,” Nonlinear Dyn.
, 38
(1–4
), pp. 61
–68
.9.
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
.10.
Fukunaga
, M.
, 2002
, “On Initial Value Problems of Fractional Differential Equations
,” Int. J. Appl. Math.
, 9
(2
), pp. 219
–236
.11.
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
.12.
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
.13.
Hartley
, T. T.
, and Lorenzo
, C. F.
, 2002
, “Control of Initialized Fractional-Order Systems
,” NASA Technical Report No. 2002-211377
.14.
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
.15.
Hartley
, T. T.
, Veillette
, R. J.
, Lorenzo
, C. F.
, and Adams
, J. L.
, 2013
, “On the Energy Stored in Fractional-Order Electrical Elements
,” ASME
Paper No. DETC2013-13498.16.
Hartley
, T. T.
, Trigeassou
, J.-C.
, Lorenzo
, C. F.
, and Maamri
, N.
, 2015
, “Energy Storage and Loss in Fractional-Order Systems
,” ASME J. Comput. Nonlinear Dyn.
, 10
(6
), p. 061006
.17.
Hartley
, T. T.
, Veillette
, R. J.
, Adams
, J. L.
, and Lorenzo
, C. F.
, 2015
, “Energy Storage and Loss in Fractional-Order Circuit Elements
,” IET Circuits, Devices Syst.
, 9
(3
), pp. 227
–235
.18.
Lorenzo
, C. F.
, and Hartley
, T. T.
, 2015
, “Energy Considerations for Mechanical Fractional-Order Elements
,” ASME J. Comput. Nonlinear Dyn.
, 10
(1
), p. 011014
.19.
Lorenzo
, C. F.
, and Hartley
, T. T.
, 2013
, “Energy Considerations for Fractional Elements
,” ASME
Paper No. DETC2013-13247.20.
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.21.
Oustaloup
, A.
, Moreau
, X.
, and Nouillant
, M.
, 1996
, “The CRONE Suspension
,” Control Eng. Pract.
, 4
(8
), pp. 1101
–1108
.22.
Podlubny
, I.
, 1999
, “Fractional-Order Systems and PIλDμ Controllers
,” IEEE Trans. Autom. Control
, 44
(1
), pp. 208
–214
.23.
Yuan
, J.
, Shi
, B.
, and Ji
, W.
, 2013
, “Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems
,” Adv. Math. Phys.
, 2013
, p. 576709
.24.
Bandyopadhyay
, B.
, and Kamal
, S.
, 2015
, Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach
, Springer
, Switzerland.25.
Ladaci
, S.
, and Charef
, A.
, 2006
, “On Fractional Adaptive Control
,” Nonlinear Dyn.
, 43
(4
), pp. 365
–378
.26.
Shi
, B.
, Yuan
, J.
, and Dong
, C.
, 2014
, “On Fractional Model Reference Adaptive Control
,” TheScientificWorldJournal
, 2014
, p. 521625
.27.
Agrawal
, O. P.
, 2004
, “A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,” Nonlinear Dyn.
, 38
(1–4
), pp. 323
–337
.28.
Baleanu
, D.
, Machado
, J. A. T.
, and Luo
, A. C. J.
, 2012
, “A Formulation and Numerical Scheme for Fractional Optimal Control of Cylindrical Structures Subjected to General Initial Conditions
,” Fractional Dynamics and Control
, Springer Science+Business Media, New York.29.
Li
, Y.
, Chen
, Y.
, and Podlubny
, I.
, 2009
, “Mittag–Leffler Stability of Fractional Order Nonlinear Dynamic Systems
,” Automatica
, 45
(8
), pp. 1965
–1969
.30.
Li
, Y.
, Chen
, Y.
, and Podlubny
, I.
, 2010
, “Stability of Fractional-Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–Leffler Stability
,” Comput. Math. Appl.
, 59
(5
), pp. 1810
–1821
.31.
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
.32.
Montseny
, G.
, 1998
, “Diffusive Representation of Pseudo-Differential Time-Operators
,” Proc. ESSAIM
, 5
, pp. 159
–175
.33.
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
.34.
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
.35.
Trigeassou
, J. C.
, Maamri
, N.
, and Oustaloup
, A.
, 2013
, “Lyapunov Stability of Linear Fractional Systems—Part 1: Definition of Fractional Energy
,” ASME
Paper No. DETC2013-12824.Copyright © 2017 by ASME
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