In this paper, two fractional-order linear controllers are proposed to stabilize unstable equilibrium points of a chaotic fractional-order system. The first controller is based on the dynamic output feedback control idea and requires detectability of the linearized model of the fractional-order system on the equilibrium point. The second controller is a dynamic state feedback controller and requires observability of the linearized model. In both considered cases, the stabilizability of the model is assumed. The number of inner states in the second controller is one and therefore its structure is much simpler than the first controller. To illustrate the applicability, these controllers are applied to control chaos in the fractional-order Chen system. Numerical simulations results are presented to evaluate the performance of the proposed controllers.
Issue Section:
Research Papers
1.
2007,
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
, J.
Sabatier
, O.
Agrawal
, and J. T.
Machado
, eds., Springer
, New York
.2.
Agrawal
, O.
, Machado
, J. T.
, and Sabatier
, J.
, 2004, “Special Issue on Fractional Derivatives and Their Applications
,” Nonlinear Dyn.
0924-090X, 38
(1–2
), pp. 1
–489
.3.
Ortigueira
, M.
, and Machado
, J. T.
, 2006, “Special Issue on Fractional Calculus Applications in Signals and Systems
,” Signal Process.
0165-1684, 86
(10
), pp. 2503
–3094
.4.
Bagley
, R. L.
, and Calico
, R. A.
, 1991, “Fractional-Order State Equations for the Control of Visco-Elastically Damped Structures
,” J. Guid. Control Dyn.
0731-5090, 14
, pp. 304
–311
.5.
Rossikhin
, Y. A.
, and Shitikova
, M. V.
, 1997, “Application of Fractional Derivatives to the Analysis of Damped Vibrations of Viscoelastic Single Mass System
,” Acta Mech.
0001-5970, 120
, pp. 109
–125
.6.
Engheta
, N.
, 1996, “On Fractional Calculus and Fractional Multipoles in Electromagnetism
,” IEEE Trans. Antennas Propag.
0018-926X, 44
(4
), pp. 554
–566
.7.
Arkhincheev
, V. E.
, 1993, “Anomalous Diffusion in Inhomogeneous Media: Some Exact Results
,” Modeling, Measurement and Control A: General Physics Electronics and Electrical Engineering
, 26
(2
), pp. 11
–29
.8.
El-Sayed
, A. M. A.
, 1996, “Fractional-Order Diffusion-Wave Equation
,” Int. J. Theor. Phys.
0020-7748, 35
(2
), pp. 311
–322
.9.
Le Méhauté
, A.
and Crepy
, G.
, 1983, “Introduction to Transfer and Motion in Fractal Media: The Geometry of Kinetics
,” Solid State Ionics
0167-2738, 9–10
, pp. 17
–30
.10.
Nakagawa
, M.
, and Sorimachi
, K.
, 1992, “Basic Characteristics of a Fractance Device
,” IEICE Trans. Fundamentals
0916-8508, 75-A
(12
), pp. 1814
–1819
.11.
Oldham
, K. B.
, and Zoski
, C. G.
, 1983, “Analogue Instrumentation for Processing Polarographic Data
,” J. Electroanal. Chem.
0022-0728, 157
, pp. 27
–51
.12.
Chen
, G.
, and Friedman
, G.
, 2005, “An RLC Interconnect Model Based on Fourier Analysis
,” IEEE Trans. Comput.-Aided Des.
0278-0070, 24
(2
), pp. 170
–183
.13.
Jenson
, V. G.
, and Jeffreys
, G. V.
, 1977, Mathematical Methods in Chemical Engineering
, 2nd ed., Academic
, New York
.14.
Cole
, K. S.
, 1993, “Electric Conductance of Biological Systems
,” Proceedings of Cold Spring Harbor Symposium Quant. Biol.
, Cold Spring Harbor, NY, pp. 107
–116
.15.
Anastasio
, T. J.
, 1994, “The Fractional-Order Dynamics of Brainstem Vestibule-Oculumotor Neurons
,” Biol. Cybern.
0340-1200, 72
, pp. 69
–79
.16.
Laskin
, N.
, 2000, “Fractional Market Dynamics
,” Physica A
0378-4371, 287
, pp. 482
–492
.17.
2000,
Applications of Fractional Calculus in Physics
, R.
Hilfer
, ed., World Scientific
, Singapore
.18.
Podlubny
, I.
, 1999, “Fractional-Order Systems and PIλDμ-Controllers
,” IEEE Trans. Autom. Control
0018-9286, 44
(1
), pp. 208
–214
.19.
Lune
, B. J.
, 1994, “Three-Parameter Tunable Tilt-Integral Derivative (TID) Controller
,” U.S. Patent No. US5371,670.20.
Oustaloup
, A.
, Sabatier
, J.
, and Lanusse
, P.
, 1999, “From Fractal Robustness to CRONE Control
,” Fractional Calculus Appl. Anal.
1311-0454, 2
(1
), pp. 1
–30
.21.
Oustaloup
, A.
, Moreau
, X.
, and Nouillant
, M.
, 1996, “The CRONE Suspension
,” Control Eng. Pract.
0967-0661, 4
(8
), pp. 1101
–1108
.22.
Oustaloup
, A.
, Mathieu
, B.
, and Lanusse
, P.
, 1995, “The CRONE Control of Resonant Plants: Application to a Flexible Transmission
,” Eur. J. Control
0947-3580, 1
(2
), pp. 113
–121
.23.
Raynaud
, H. F.
, and Inoh
, A. Z.
, 2000, “State-Space Representation for Fractional-Order Controllers
,” Automatica
0005-1098, 36
, pp. 1017
–1021
.24.
Monje
, C. A.
, and Feliu
, V.
, 2004, “The Fractional-Order Lead Compensator
,” IEEE International Conference on Computational Cybernetics
, Vienna, Austria, Aug. 30–Sep. 1.25.
Hartley
, T. T.
, Lorenzo
, C. F.
, and Qammer
, H. K.
, 1995, “Chaos in a Fractional-Order Chua’s System
,” IEEE Trans. Circuits Syst., I
, 42
, pp. 485
–490
. 0002-782026.
Arena
, P.
, Caponetto
, R.
, Fortuna
, L.
, and Porto
, D.
, 1997, “Chaos in a Fractional-Order Duffing System
,” Proceedings ECCTD
, Budapest, pp. 1259
–1262
.27.
Ahmad
, W. M.
, and Sprott
, J. C.
, 2003, “Chaos in Fractional-Order Autonomous Nonlinear Systems
,” Chaos, Solitons Fractals
0960-0779, 16
, pp. 339
–351
.28.
Lu
, J. G.
, and Chen
, G.
, 2006, “A Note on the Fractional-Order Chen System
,” Chaos, Solitons Fractals
0960-0779, 27
(3
), pp. 685
–688
.29.
Lu
, J. G.
, 2006, “Chaotic Dynamics of the Fractional-Order Lü System and Its Synchronization
,” Phys. Lett. A
0375-9601, 354
(4
), pp. 305
–311
.30.
Li
, C.
and Chen
, G.
, 2004, “Chaos and Hyperchaos in the Fractional-Order Rössler Equations
,” Physica A
0378-4371, 341
, pp. 55
–61
.31.
Lu
, J. G.
, 2005, “Chaotic Dynamics and Synchronization of Fractional-Order Arneodo’s Systems
,” Chaos, Solitons Fractals
0960-0779, 26
(4
), pp. 1125
–1133
.32.
Sheu
, L. J.
, Chen
, H. K.
, Chen
, J. H.
, Tam
, L. M.
, Chen
, W. C.
, Lin
, K. T.
, and Kang
, Y.
, 2008, “Chaos in the Newton–Leipnik System With Fractional-Order
,” Chaos, Solitons Fractals
0960-0779, 36
(1
), pp. 98
–103
.33.
Andrievskii
, B. R.
, and Fradkov
, A. L.
, 2003, “Control of Chaos: Methods and Applications. Part I: Methods
,” Autom. Remote Control (Engl. Transl.)
0005-1179, 64
(5
), pp. 673
–713
.34.
2003,
Chaos Control: Theory and Applications
, G.
Chen
and X.
Yu
, eds., Springer-Verlag
, Berlin, Germany
.35.
Lin
, W.
, 2007, “Global Existence Theory and Chaos Control of Fractional Differential Equations
,” J. Math. Anal. Appl.
0022-247X, 332
(1
), pp. 709
–726
.36.
Tsai
, J. S.
, Chien
, T. H.
, Guo
, S. M.
, Chang
, Y. P.
, and Shieh
, L. S.
, 2007, “State-Space Self-Tuning Control for Stochastic Fractional-Order Chaotic Systems
,” IEEE Trans. Circuits Syst., I
, 54
(3
), pp. 632
–642
.37.
Ahmad
, W. M.
, El-Khazali
, R.
, and Al-Assaf
, Y.
, 2004, “Stabilization of Generalized Fractional-Order Chaotic Systems Using State Feedback Control
,” Chaos, Solitons Fractals
0960-0779, 22
, pp. 141
–150
.38.
Podlubny
, I.
, 1999, Fractional Differential Equations
, Academic
, New York
.39.
Matignon
, D.
, 1996 “Stability Results for Fractional Differential Equations With Applications to Control Processing
,” Computational Engineering in Systems and Application Multi-Conference, IMACS, IEEE-SMC Proceedings
, Lille, France, Vol. 2
, pp. 963
–968
.40.
Ahmed
, E.
, El-Sayed
, A. M. A.
, and El-Saka
, H. A. A.
, 2007, “Equilibrium Points, Stability and Numerical Solutions of Fractional-Order Predator-Prey and Rabies Models
,” J. Math. Anal. Appl.
0022-247X, 325
, pp. 542
–553
.41.
Matignon
, D.
and d’Andréa Novel
, B.
, 1996, “Some Results on Controllability and Observability of Finite-Dimensional Fractional Differential Systems
,” Computational Engineering in Systems Applications, IMACS, IEEE-SMC
, Lille, France, Vol. 2
, pp. 952
–956
.42.
Matignon
, D.
, and d’Andréa Novel
, B.
, 1997, “Observer-Based Controllers for Fractional Differential Systems
,” Conference on Decision and Control, IEEE-CSS, SIAM
, San Diego, CA, pp. 4967
–4972
.43.
Chen
, Y. Q.
, Ahn
, H. S.
, and Xue
, D.
, 2006, “Robust Controllability of Interval Fractional-Order Linear Time Invariant Systems
,” Signal Process.
0165-1684, 86
(10
), pp. 2794
–2802
.44.
Wang
, Z.
, Cao
, G.
, and Zhu
, X.
, 2006, “Research on the Stability, Controllability and Observability for Fractional-Order LTI Systems
,” J. Harbin Inst. Technol.
1005-9113, 13
(5
), pp. 580
–583
.45.
Zeng
, Q.
, Cao
, G.
, and Zhu
, X.
, 2005, “Research on Controllability for a Class of Fractional-Order Linear Control Systems
,” J. Syst. Eng. Electron.
, 16
(2
), pp. 376
–381
.46.
Bettayeb
, M.
, and Djennoune
, S.
, 2008, “New Results on Controllability and Observability of Fractional Dynamical Systems
,” J. Vib. Control
1077-5463, 14
(9–10
), pp. 1531
–1541
.47.
Micu
, S.
, and Zuazua
, E.
, 2006, “On the Controllability of a Fractional-Order Parabolic Equation
,” SIAM J. Control Optim.
0363-0129, 44
(6
), pp. 1950
–1972
.48.
El-Ghaffar
, A. A.
, 2001, “Observability for Singular Control Systems of Fractional-Order
,” J. Fractional Calculus
0918-5402, 19
, pp. 79
–88
.49.
Abd El-Ghaffar
, A.
, Moubarak
, M. R. A.
, and Shamardan
, A. B.
, 2000, “Controllability of Fractional Nonlinear Control System
,” J. Fractional Calculus
0918-5402, 17
, pp. 59
–69
.50.
Moubarak
, M. R. A.
, 1999, “Singular Control Systems of Fractional-Order
,” J. Fractional Calculus
0918-5402, 16
, pp. 19
–26
.51.
Hotzel
, R.
, and Fliess
, M.
, 1998, “On Linear Systems With a Fractional Derivation: Introductory Theory and Examples
,” Math. Comput. Simul.
0378-4754, 45
, pp. 385
–395
.52.
Kailath
, T.
, 1980, Linear Systems
, Prentice-Hall
, Englewood Cliffs, NJ
.53.
Huijberts
, H.
, 2006, “Linear Controllers for the Stabilization of Unknown Steady States of Chaotic Systems
,” IEEE Trans. Circuits Syst., I
, 53
(10
), pp. 2246
–2254
.54.
Diethelm
, K.
, Ford
, N. J.
, and Freed
, A. D.
, 2002, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,” Nonlinear Dyn.
0924-090X, 29
, pp. 3
–22
.55.
Tavazoei
, M. S.
, and Haeri
, M.
, 2007, “Unreliability of Frequency Domain Approximation in Recognising Chaos in Fractional Order Systems
,” IET Signal Processing
, 1
(4
), pp. 171
–181
.56.
Tavazoei
, M. S.
, and Haeri
, M.
, 2007, “A Necessary Condition for Double Scroll Attractor Existence in Fractional Order Systems
,” Phys. Lett. A
0375-9601, 367
(1–2
), pp. 102
–113
.57.
Chen
, G.
, and Ueta
, T.
, 1999, “Yet Another Attractor
,” Int. J. Bifurcat. Chaos Appl. Sci. Eng.
, 9
(7
), pp. 1465
–1466
.58.
Deng
, W.
, and Li
, C.
, 2005, “Synchronization of Chaotic Fractional Chen System
,” J. Phys. Soc. Jpn.
0031-9015, 74
(6
), pp. 1645
–1648
.59.
Muir
, T. A.
, 1960, Treatise on the Theory of Determinants
, Dover
, New York
.60.
Ott
, E.
, Grebogi
, C.
, and Yorke
, J. A.
, 1990, “Controlling Chaos
,” Phys. Rev. Lett.
0031-9007, 64
, pp. 1196
–1199
.61.
Chanfreau
, P.
, and Lyyjynen
, H.
, 1999, “Viewing the Efficiency of Chaos Control
,” J. Nonlinear Math. Phys.
1402-9251, 6
(3
), pp. 314
–331
.62.
Tavazoei
, M. S.
, Haeri
, M.
, Bolouki
, S.
, and Siami
, M.
, 2008, “Stability Preservation Analysis for Frequency Based Methods in Numerical Simulation of Fractional Order Systems
,” SIAM J. Numer. Anal.
, 47
(1
), pp. 321
–338
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