Abstract
Control of chaos generally refers to realize a desired behavior of chaotic system output and its states. In this manner, we design a fractional high-order differential feedback controller (FHODFC) to increase tracking performance of a nonlinear system output and its differentials for a desired trajectory signal. The proposed controller is based on fractional calculus and high-order extracted differentials of error signal. The suggested fractional approach is applied to a single-input–single-output affine Duffing-Holmes dynamical system in matlab/simulink environment. Duffing-Holmes system is analyzed for two different problems: estimation and control problems. The simulation results clearly demonstrate superior dynamic behavior of the FHODFC compared to the classical high-order differential feedback controller (HODFC) version for both estimation and control problems.
Issue Section:
Research Papers
References
1.
Ott
,
E.
,
Grebogi
,
C.
, and
Yorke
,
J. A.
, 1990
, “
Controlling Chaos
,” Phys. Rev. Lett.
,
64
(11
), p. 1196
.10.1103/PhysRevLett.64.11962.
Zhang
,
H.
,
Liu
,
D.
, and
Wang
,
Z.
, 2009
, Controlling Chaos: Suppression, Synchronization and Chaotification
,
Springer Science & Business Media
, London.3.
Pyragas
,
K.
, 1992
, “
Continuous Control of Chaos by Self-Controlling Feedback
,” Phys. Lett. A
,
170
(6
), pp. 421
–428
.10.1016/0375-9601(92)90745-84.
Cho
,
S.-J.
,
Jin
,
M.
,
Kuc
,
T.-Y.
, and
Lee
,
J. S.
, 2014
, “
Control and Synchronization of Chaos Systems Using Time-Delay Estimation and Supervising Switching Control
,” Nonlinear Dyn.
,
75
(3
), pp. 549
–560
.10.1007/s11071-013-1084-45.
Hua
,
C.
, and
Guan
,
X.
, 2004
, “
Adaptive Control for Chaotic Systems
,” Chaos, Solitons Fractals
,
22
(1
), pp. 55
–60
.10.1016/j.chaos.2003.12.0716.
Jeong
,
S.
,
Ji
,
D.
,
Park
,
J. H.
, and
Won
,
S.
, 2013
, “
Adaptive Synchronization for Uncertain Complex Dynamical Network Using Fuzzy Disturbance Observer
,” Nonlinear Dyn.
,
71
(1–2
), pp. 223
–234
.10.1007/s11071-012-0654-17.
Li
,
X.-J.
, and
Yang
,
G.-H.
, 2017
, “
Fuzzy Approximation-Based Global Pinning Synchronization Control of Uncertain Complex Dynamical Networks
,” IEEE Trans. Cybern.
,
47
(4
), pp. 873
–883
.10.1109/TCYB.2016.25307928.
Gholami
,
A.
, and
Markazi
,
A. H.
, 2012
, “
A New Adaptive Fuzzy Sliding Mode Observer for a Class of Mimo Nonlinear Systems
,” Nonlinear Dyn.
,
70
(3
), pp. 2095
–2105
.10.1007/s11071-012-0602-09.
Pan
,
I.
,
Korre
,
A.
,
Das
,
S.
, and
Durucan
,
S.
, 2012
, “
Chaos Suppression in a Fractional Order Financial System Using Intelligent Regrouping PSO Based Fractional Fuzzy Control Policy in the Presence of Fractional Gaussian Noise
,” Nonlinear Dyn.
,
70
(4
), pp. 2445
–2461
.10.1007/s11071-012-0632-710.
Lin
,
C.-T.
, and
Jou
,
C.-P.
, 1999
, “
Controlling Chaos by GA-Based Reinforcement Learning Neural Network
,” IEEE Trans. Neural Networks
,
10
(4
), pp. 846
–859
. 10.1109/72.77423611.
Yu
,
J.
,
Yu
,
H.
,
Chen
,
B.
,
Gao
,
J.
, and
Qin
,
Y.
, 2012
, “
Direct Adaptive Neural Control of Chaos in the Permanent Magnet Synchronous Motor
,” Nonlinear Dynamics
,
70
(3
), pp. 1879
–1887
.10.1007/s11071-012-0580-212.
Das
,
S.
,
Acharya
,
A.
, and
Pan
,
I.
, 2014
, “
Simulation Studies on the Design of Optimum PID Controllers to Suppress Chaotic Oscillations in a Family of Lorenz-Like Multi-Wing Attractors
,” Math. Comput. Simul.
,
100
, pp. 72
–87
.10.1016/j.matcom.2014.03.00213.
Bucci
,
M. A.
,
Semeraro
,
O.
,
Allauzen
,
A.
,
Wisniewski
,
G.
,
Cordier
,
L.
, and
Mathelin
,
L.
, 2019
, “
Control of Chaotic Systems by Deep Reinforcement Learning
,” preprint arXiv:1906.07672
.https://arxiv.org/abs/1906.0767214.
Song
,
C.
,
Fei
,
S.
,
Cao
,
J.
, and
Huang
,
C.
, 2019
, “
Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control
,” Mathematics
,
7
(7
), p. 599
.10.3390/math707059915.
Liu
,
B.
,
Sun
,
Z.
,
Luo
,
Y.
, and
Zhong
,
Y.
, 2019
, “
Uniform Synchronization for Chaotic Dynamical Systems Via Event-Triggered Impulsive Control
,” Phys. A: Stat. Mech. Appl.
,
531
, p. 121725
.10.1016/j.physa.2019.12172516.
Chen
,
G.
, and
Yu
,
X.
, 2003
, Chaos Control: Theory and Applications
, Vol.
292
,
Springer Science & Business Media
, Berlin.17.
Haghighatnia
,
S.
,
Toossian Shandiz
,
H.
, and
Alfi
,
A.
, 2019
, “
Conformable Fractional Order Sliding Mode Control for a Class of Fractional Order Chaotic Systems
,” Iran. J. Electr. Electron. Eng.
,
2
(3
), pp. 177
–188
.10.22111/IECO.2018.25403.104918.
Boubellouta
,
A.
, and
Boulkroune
,
A.
, 2019
, “
Intelligent Fractional-Order Control-Based Projective Synchronization for Chaotic Optical Systems
,” Soft Comput.
,
23
(14
), pp. 5367
–5384
.10.1007/s00500-018-3490-519.
Rabah
,
K.
,
Ladaci
,
S.
, and
Lashab
,
M.
, 2018
, “
Bifurcation-Based Fractional-Order PIλ Dμ Controller Design Approach for Nonlinear Chaotic Systems
,” Front. Inf. Technol. Electron. Eng.
,
19
(2
), pp. 180
–191
.10.1631/FITEE.160154320.
Ladaci
,
S.
,
Rabah
,
K.
, and
Lashab
,
M.
, 2018
, “
Robust Synchronization of Fractional-Order Arneodo Chaotic Systems Using a Fractional Sliding Mode Control Strategy
,” Advanced Synchronization Control and Bifurcation of Chaotic Fractional-Order Systems, IGI Global, Hershey, PA, pp. 1
–22
.21.
Rabah
,
K.
, and
Ladaci
,
S.
, 2019
, “
A Fractional Adaptive Sliding Mode Control Configuration for Synchronizing Disturbed Fractional-Order Chaotic Systems
,” Circuits, Syst. Signal Process
.
, pp. 1
–21
.10.1007/s00034-019-01205-y22.
Chen
,
D.-Y.
,
Liu
,
Y.-X.
,
Ma
,
X.-Y.
, and
Zhang
,
R.-F.
, 2012
, “
Control of a Class of Fractional-Order Chaotic Systems Via Sliding Mode
,” Nonlinear Dyn.
,
67
(1
), pp. 893
–901
.10.1007/s11071-011-0002-x23.
Wang
,
Z.
,
Liu
,
J.
,
Zhang
,
F.
, and
Leng
,
S.
, 2019
, “
Hidden Chaotic Attractors and Synchronization for a New Fractional-Order Chaotic System
,” ASME J. Comput. Nonlinear Dyn.
,
14
(8
), p. 081010
.10.1115/1.404367024.
Azar
,
A. T.
,
Vaidyanathan
,
S.
, and
Ouannas
,
A.
, 2017
, Fractional Order Control and Synchronization of Chaotic Systems
, Vol.
688
,
Springer
, Cham, Switzerland.25.
Qi
,
G.
,
Chen
,
Z.
, and
Yuan
,
Z.
, 2005
, “
Model-Free Control of Affine Chaotic Systems
,” Phys. Lett. A
,
344
(2–4
), pp. 189
–202
.10.1016/j.physleta.2005.06.07326.
Agee
,
J. T.
,
Bingul
,
Z.
, and
Kizir
,
S.
, 2015
, “
Higher-Order Differential Feedback Control of a Flexible-Joint Manipulator
,” J. Vib. Control
,
21
(10
), pp. 1976
–1986
.10.1177/107754631350497927.
Agee
,
J. T.
,
Bingul
,
Z.
, and
Kizir
,
S.
, 2017
, “
Trajectory and Vibration Control of a Single-Link Flexible-Joint Manipulator Using a Distributed Higher-Order Differential Feedback Controller
,” ASME J. Dyn. Syst., Meas., Control
,
139
(8
), p. 081006
.10.1115/1.403587328.
Du
,
S.
,
van Wyk
,
B. J.
,
Qi
,
G.
, and
Tu
,
C.
, 2009
, “
Chaotic System Synchronization With an Unknown Master Model Using a Hybrid HOD Active Control Approach
,” Chaos, Solitons Fractals
,
42
(3
), pp. 1900
–1913
.10.1016/j.chaos.2009.03.10129.
Nyabundi
,
S.
,
Qi
,
G.
,
Hamam
,
Y.
, and
Munda
,
J.
, 2009
, “
DC Motor Control Via High Order Differential Feedback Control
,” IEEE AFRICON (AFRICON'09
), Nairobi, Kenya, Sept. 23–25, pp. 1
–5
.10.1109/AFRCON.2009.530816530.
Shi
,
X-M.
,
Dai
,
Y.
,
Liu
,
Z-h.
, and
G.-y
,
Q.
, 2010
, “
High Order Differential Feedback Controller and Its Application in Servo Control System of NC Machine Tools
,” International Conference on System Science, Engineering Design and Manufacturing Informatization (ICSEM
), Vol.
1
,
Yichang, China
, Nov. 12–14, pp. 241
–244
.10.1109/ICSEM.2010.7231.
Faradja
,
P.
, and
Qi
,
G.
, 2015
, “
Robustness Based Comparison Between a Sliding Mode Controller and a Model Free Controller With the Approach of Synchronization of Nonlinear Systems
,” 15th International Conference on Control, Automation and Systems
(ICCAS
),
Busan, South Korea, Oct. 13–16, pp.
36
–40
.10.1109/ICCAS.2015.736487532.
Xue
,
W.
,
Xue
,
Y.
,
Qi
,
G.
,
Gao
,
J.
, and
Zheng
,
S.
, 2007
, “
Study of High Order Differential Feedback Control of dc-Link Voltage in Active Power Filter
,” IEEE
International Conference on Automation and Logistics,
Jinan, China, Aug. 18–21, pp.
1063
–1066
.10.1109/ICAL.2007.433872533.
Wei
,
X.
,
Jingjing
,
M.
,
Hongyan
,
J.
, and
Fei
,
Y.
, 2010
, “
The Main Steam Temperature Cascade Control of High Order Differential of Feedback Controller
,” IEEE International Conference on Intelligent System Design and Engineering Application
(ISDEA
), Vol.
2
,
Changsha, China, Oct. 13–14, pp.
683
–687
.10.1109/ISDEA.2010.6834.
Tavazoei
,
M. S.
, 2014
, “
Time Response Analysis of Fractional-Order Control Systems: A Survey on Recent Results
,” Fractional Calculus Appl. Anal.
,
17
(2
), pp. 440
–461
.10.2478/s13540-014-0179-z35.
Oustaloup
,
A.
,
Levron
,
F.
,
Mathieu
,
B.
, and
Nanot
,
F. M.
, 2000
, “
Frequency-Band Complex Noninteger Differentiator: Characterization and Synthesis
,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.
,
47
(1
), pp. 25
–39
.10.1109/81.81738536.
Oustaloup
,
A.
,
Moreau
,
X.
, and
Nouillant
,
M.
, 1996
, “
The Crone Suspension
,” Control Eng. Pract.
,
4
(8
), pp. 1101
–1108
.10.1016/0967-0661(96)00109-837.
Solak
,
E.
,
Morgül
,
Ö.
, and
Ersoy
,
U.
, 2001
, “
Observer-Based Control of a Class of Chaotic Systems
,” Phys. Lett. A
,
279
(1–2
), pp. 47
–55
.10.1016/S0375-9601(00)00808-238.
Qi
,
G.
,
Chen
,
Z.
, and
Yuan
,
Z.
, 2008
, “
Adaptive High Order Differential Feedback Control for Affine Nonlinear System
,” Chaos, Solitons Fractals
,
37
(1
), pp. 308
–315
.10.1016/j.chaos.2006.09.027Copyright © 2020 by ASME
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