A rotating flexible beam undergoing large deformation is known to exhibit chaotic motion for certain parameter values. This work deals with an approach for control of chaos known as chaos synchronization. A nonlinear controller based on the Lyapunov stability theory is developed, and it is shown that such a controller can avoid the sensitive dependence of initial conditions seen in all chaotic systems. The proposed controller ensures that the error between the controlled and the original system, for different initial conditions, asymptotically goes to zero. A numerical example using the parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.
Issue Section:
Technical Brief
References
1.
Guckenheimer
, J.
, and Holmes
, P.
, 1983
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
(Applied Mathematical Sciences), Vol. 42
, Springer-Verlag
, New York
.2.
Kovacic
, I.
, and Brennan
, M. J.
, 2011
, The Duffing Equation: Nonlinear Oscillators and Their Behaviour
, 1st ed., Wiley
, Hoboken, NJ.3.
Burov
, A. A.
, 1986
, “On the Non-Existence of a Supplementary Integral in the Problem of a Heavy Two-Link Plane Pendulum
,” Prikl. Mat. Mekh., USSR
, 50
(1
), pp. 123
–125
.4.
Lankalapalli
, S.
, and Ghosal
, A.
, 1996
, “Possible Chaotic Motion in a Feedback Controlled 2R Robot
,” IEEE International Conference on Robotics and Automation
, Minneapolis, MN, pp. 1241
–1246
.5.
Reddy
, B. S.
, and Ghosal
, A.
, 2015
, “Nonlinear Dynamics of a Rotating Flexible Link
,” ASME J. Comput. Nonlinear Dyn.
, 10
(6
), p. 061014
.6.
Ott
, E.
, Grebogi
, C.
, and Yorke
, J. A.
, 1990
, “Controlling Chaos
,” Phys. Rev. Lett.
, 64
(11
), pp. 1196
–1199
.7.
Pinto
, F. H. I. P.
, Ferreira
, A. M.
, and Savi
, M.
, 2004
, “Chaos Control in a Nonlinear Pendulum Using a Semi-Continuous Method
,” Chaos, Solitons Fractals
, 22
(3
), pp. 653
–668
.8.
De Paula
, A. S.
, and Savi
, M. A.
, 2009
, “Controlling Chaos in a Nonlinear Pendulum Using an Extended Time-Delayed Feedback Control Method
,” Chaos, Solitons Fractals
, 42
(5
), pp. 2981
–2988
.9.
Jahromi
, S. A. Z.
, Haji
, A. H.
, and Mahzoon
, M.
, 2005
, “Non-Linear Dynamics and Chaos Control of a Physical Pendulum With Rotating Mass
,” 13th Annual (International) Mechanical Engineering Conference
, Isfahan University of Technology, Isfahan, Iran.10.
Starett
, J.
, and Tagg
, R.
, 1995
, “Control of a Chaotic Parametrically Driven Pendulum
,” Phys. Rev. Lett.
, 74
(11
), pp. 1974
–1977
.11.
Pyragas
, K.
, 2001
, “Control of Chaos Via an Unstable Delayed Feedback Controller
,” Phys. Rev. Lett.
, 86
(11
), pp. 2265
–2268
.12.
Fradkov
, L. A.
, and Evans
, J. R.
, 2005
, “Control of Chaos: Methods and Applications in Engineering
,” Annu. Rev. Control
, 29
(1
), pp. 33
–56
.13.
Ruiqi
, W.
, and Zhujun
, J.
, 2004
, “Chaos Control of Chaotic Pendulum System
,” Chaos, Solitons Fractals
, 21
(1), pp. 201
–207
.14.
Pecora
, L. M.
, and Carroll
, T. L.
, 1990
, “Synchronization in Chaotic Systems
,” Phys. Rev. Lett.
, 64
(8
), pp. 821
–824
.15.
Pecora
, L. M.
, and Carroll
, T. L.
, 1991
, “Synchronizing Chaotic Circuits
,” IEEE Trans. Circuits Syst.
, 38
(4), pp. 453
–456
.16.
Wu
, X.
, and Lu
, J.
, 2003
, “Parameter Identification and Backstepping Control of Uncertain Lu System
,” Chaos, Solitons Fractals
, 18
(4), pp. 721
–729
.17.
Yu
, Y. G.
, and Zhang
, S. C.
, 2004
, “Adaptive Backstepping Synchronization of Uncertain Chaotic Systems
,” Chaos, Solitons Fractals
, 21
(3), pp. 643
–649
.18.
Yau
, H. T.
, 2004
, “Design of Adaptive Sliding Mode Controller for Chaos Synchronization With Uncertainties
,” Chaos, Solitons Fractals
, 22
(2
), pp. 341
–347
.19.
Wu
, X.
, Wang
, L.
, and Zhang
, J.
, 2012
, “Synchronisation of Unified Chaotic Systems With Uncertain Parameters in Finite Time
,” Int. J. Modell., Identif. Control
, 17
(4
), pp. 295
–301
.20.
Vaidyanathan
, S.
, 2014
, “Global Chaos Synchronisation of Identical Li-Wu Chaotic Systems Via Sliding Mode Control
,” Int. J. Modell., Identif. Control
, 22
(2
), pp. 170
–177
.21.
Sundarapandian
, V.
, Sivaperumal
, S.
, and Ahmad
, T. A.
, 2015
, “Global Chaos Synchronisation of Identical Chaotic Systems Via Novel Sliding Mode Control Method and Its Application to Zhu System
,” Int. J. Modell., Identif. Control
, 23
(1
), pp. 92
–100
.22.
Handa
, H.
, and Sharma
, B. B.
, 2014
, “Simple Synchronisation Scheme of Chaotic Chua's Systems With Cubic Nonlinearity in Complex Coupled Networks
,” Int. J. Appl. Nonlinear Sci.
, 1
(4
), pp. 300
–311
.23.
Saha
, P.
, Ghosh
, D.
, and Chowdhury
, A. R.
, 2014
, “Modified Projective Synchronisation of Different Order Chaotic Systems With Adaptive Scaling Factor
,” Int. J. Appl. Nonlinear Sci.
, 1
(3
), pp. 230
–246
.24.
Shaker
, M. C.
, and Ghosal
, A.
, 2006
, “Nonlinear Modeling of Flexible Link Manipulators Using Non-Dimensional Variables
,” ASME J. Comput. Nonlinear Dyn.
, 1
(2
), pp. 123
–134
.25.
Nayfeh
, A. H.
, 1993
, Introduction to Perturbation Techniques
, Wiley
, Hoboken, NJ.26.
Endurance Wind Power Inc., 2016, Endurance Wind Power Inc., Surrey, BC, Canada, accessed Oct. 6, 2016, http://www.endurancewindpower.com/e3120.html
27.
MATLAB
, 2012
, “Version 8.0 (R2012b)
,” The MathWorks, Natick, MA.Copyright © 2017 by ASME
You do not currently have access to this content.