A Comparative Study of Synchronization Methods of Rucklidge Chaotic Systems with Design of Active Control and Backstepping Methods — Oak Academic Publishing
Research ArticleOpen AccessGoogle Scholar indexed
A Comparative Study of Synchronization Methods of Rucklidge Chaotic Systems with Design of Active Control and Backstepping Methods
Department of Mathematics, Chittagong University of Engineering & Technology (CUET), Chattogram, Bangladesh
,
Department of Mathematics, Chittagong University of Engineering & Technology (CUET), Chattogram, Bangladesh
1 Department of Mathematics, Chittagong University of Engineering & Technology (CUET), Chattogram, Bangladesh
2 Department of Mathematics, Chittagong University of Engineering & Technology (CUET), Chattogram, Bangladesh
The performance of two widely used chaos synchronization approaches, active control and backstepping control, is investigated in this study. These two methods are projected to synchronize two chaotic systems (Master/Drive of Rucklidge Systems) that are identical but have different initial conditions. The paper’s significant feature is that based on error dynamics, controllers are designed using the appropriate variable and the time synchronization between master Rucklidge and drive Rucklidge systems using both methods. The control function of the active control method is designed on the proper selection of matrices. The chaotic behavior is controlled using a recursive backstepping design based on the Lyapunov stability theory with a validated Lyapunov function. The effectiveness of the controller in eradicating the chaotic behavior from the state trajectories is also revealed using numerical simulations with Matlab. The backstepping method is superior to the active control method for synchronization of the measured pair of systems, as it takes less time to synchronize while exhausting the first one than the second one with great performance, according to numerical simulation and graphical outcomes.
KeywordsChaotic SystemSynchronizationActive ControlBackstepping ControlLyapunov Function
Pourmahmood, M., Khanmohammadi, S. and Alizadeh, G. (2011) Synchronization of Two Different Uncertain Chaotic Systems with Unknown Parameters Using a Robust Adaptive Sliding Mode Controller. Communications in Nonlinear Science and Numerical Simulation, 16, 2853-2868. https://doi.org/10.1016/j.cnsns.2010.09.038
Kyrtsou, C. and Labys, W.C. (2006) Evidence for Chaotic Dependence between Us Inflation and Commodity Prices. Journal of Macroeconomics, 28, 256-266. https://doi.org/10.1016/j.jmacro.2005.10.019
Kyrtsou, C. and Labys, W.C. (2007) Detecting Positive Feedback in Multivariate Time Series: The Case of Metal Prices and Us Inflation. Physica A: Statistical Mechanics and Its Applications, 377, 227-229. https://doi.org/10.1016/j.physa.2006.11.002
Alligood, K.T., Sauer, T.D. and Yorke, J.A. (1996) Chaos. Springer, Berlin. https://doi.org/10.1007/b97589
Luo, R., Su, H. and Zeng, Y. (2017) Chaos Control and Synchronization via Switched Output Control Strategy. Complexity, 2017, Article ID: 6125102. https://doi.org/10.1155/2017/6125102
Lei, Y., Xu, W. and Xie, W. (2007) Synchronization of Two Chaotic Four-Dimensional Systems Using Active Control. Chaos, Solitons & Fractals, 32, 1823-1829. https://doi.org/10.1016/j.chaos.2005.12.014
Matouk, A. and Elsadany, A. (2016) Dynamical Analysis, Stabilization and Discretization of a Chaotic Fractional-Order GLV Model. Nonlinear Dynamics, 85, 1597-1612. https://doi.org/10.1007/s11071-016-2781-6
Zhang, H., Chen, D., Xu, B. and Wang, F. (2015) Nonlinear Modeling and Dynamic Analysis of Hydro-Turbine Governing System in the Process of Load Rejection Transient. Energy Conversion and Management, 90, 128-137. https://doi.org/10.1016/j.enconman.2014.11.020
Zhang, H., Chen, D., Wu, C., Wang, X., Lee, J.-M. and Jung, K.-H. (2017) Dynamic Modeling and Dynamical Analysis of Pump-Turbines in S-Shaped Regions during Runaway Operation. Energy Conversion and Management, 138, 375-382. https://doi.org/10.1016/j.enconman.2017.01.053
Zhang, H., Chen, D., Xu, B., Patelli, E. and Tolo, S. (2018) Dynamic Analysis of a Pumped-Storage Hydropower Plant with Random Power Load. Mechanical Systems and Signal Processing, 100, 524-533. https://doi.org/10.1016/j.ymssp.2017.07.052
Zhang, H., Chen, D., Wu, C. and Wang, X. (2018) Dynamics Analysis of the Fast-Slow Hydro-Turbine Governing System with Different Time-Scale Coupling. Communications in Nonlinear Science and Numerical Simulation, 54, 136-147. https://doi.org/10.1016/j.cnsns.2017.05.020
Zhang, H., Chen, D., Guo, P., Luo, X. and George, A. (2018) A Novel Surface-Cluster Approach towards Transient Modeling of Hydro-Turbine Governing Systems in the Start-Up Process. Energy Conversion and Management, 165, 861-868. https://doi.org/10.1016/j.enconman.2018.03.097
Hamidzadeh, S. and Esmaelzadeh, R. (2014) Control and Synchronization Chaotic Satellite Using Active Control. International Journal of Computer Applications, 94, 29-33. https://doi.org/10.5120/16380-5887
Shahverdiev, E. and Shore, K. (2009) Impact of Modulated Multiple Optical Feedback Time Delays on Laser Diode Chaos Synchronization. Optics Communications, 282, 3568-3572. https://doi.org/10.1016/j.optcom.2009.05.068
Robert, B., Feki, M. and Iu, H.H. (2006) Control of a PWM Inverter Using Proportional plus Extended Time-Delayed Feedback. International Journal of Bifurcation and Chaos, 16, 113-128. https://doi.org/10.1142/S0218127406014629
Pecora, L.M. and Carroll, T.L. (1991) Driving Systems with Chaotic Signals. Physical Review A, 44, 2374-2383. https://doi.org/10.1103/PhysRevA.44.2374
Pecora, L.M. and Carroll, T.L. (1990) Synchronization in Chaotic Systems. Physical Review Letters, 64, 821-823. https://doi.org/10.1103/PhysRevLett.64.821
Olusola, O., Vincent, U. and Njah, A. (2010) Synchronization, Multistability and Basin Crisis in Coupled Pendula. Journal of Sound and Vibration, 329, 443-456. https://doi.org/10.1016/j.jsv.2009.09.030
Idowu, B.A., Olusola, O.I., Onma, O.S., Vaidyanathan, S., Ogabi, C.O. and Adejo, O.A. (2019) Chaotic Financial System with Uncertain Parameters—Its Control and Synchronisation. International Journal of Nonlinear Dynamics, and Control, 1, 271-286. https://doi.org/10.1504/IJNDC.2019.098682
Njah, A. and Vincent, U. (2008) Chaos Synchronization between Single and Double Wells Duffing-Van der Pol Oscillators Using Active Control. Chaos, Solitons & Fractals, 37, 1356-1361. https://doi.org/10.1016/j.chaos.2006.10.038
Vincent, U. (2008) Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysis. Nonlinear Analysis: Modelling and Control, 13, 253-261. https://doi.org/10.15388/NA.2008.13.2.14583
Li, C., Liao, X. and Wong, K.-W. (2005) Lag Synchronization of Hyperchaos with Application to Secure Communications. Chaos, Solitons & Fractals, 23, 183-193. https://doi.org/10.1016/j.chaos.2004.04.025
Peng, C.-C., Hsue, A.W.-J. and Chen, C.-L. (2011) Variable Structure Based Robust Backstepping Controller Design for Nonlinear Systems. Nonlinear Dynamics, 63, 253-262. https://doi.org/10.1007/s11071-010-9801-8
Yu, Y. and Zhang, S. (2004) Adaptive Backstepping Synchronization of Uncertain Chaotic System. Chaos, Solitons & Fractals, 21, 643-649. https://doi.org/10.1016/j.chaos.2003.12.067
Yan, J.-J., Yang, Y.-S., Chiang, T.-Y. and Chen, C.-Y. (2007) Robust Synchronization of Unified Chaotic Systems via Sliding Mode Control. Chaos, Solitons & Fractals, 34, 947-954. https://doi.org/10.1016/j.chaos.2006.04.003
Yao, Q. (2021) Synchronization of Second-Order Chaotic Systems with Uncertainties and Disturbances Using Fixed-Time Adaptive Sliding Mode Control. Chaos, Solitons & Fractals, 142, Article ID: 110372. https://doi.org/10.1016/j.chaos.2020.110372
Zhang, H., Huang, W., Wang, Z. and Chai, T. (2006) Adaptive Synchronization between Two Different Chaotic Systems with Unknown Parameters. Physics Letters A, 350, 363-366. https://doi.org/10.1016/j.physleta.2005.10.033
Elabbasy, E., Agiza, H. and El-Dessoky, M. (2006) Global Chaos Synchronization for Four-Scroll Attractor by Nonlinear Control. Scientific Research and Essays, 1, 65-71.
Wang, X. and Wang, Y. (2011) Adaptive Control for Synchronization of a Four-Dimensional Chaotic System via a Single Variable. Nonlinear Dynamics, 65, 311-316. https://doi.org/10.1007/s11071-010-9893-1
Olusola, O.I., Vincent, E., Njah, A.N. and Ali, E. (2011) Control and Synchronization of Chaos in Biological Systems via Backsteping Design. International Journal of Nonlinear Science, 11, 121-128.
Stenflo, L. (1996) Generalized Lorenz Equations for Acoustic-Gravity Waves in the Atmosphere. Physica Scripta, 53, 83-84. https://doi.org/10.1088/0031-8949/53/1/015
Mata-Machuca, J., Martinez-Guerra, R. and Aguilar-Lopez, R. (2010) An Exponential Polynomial Observer for Synchronization of Chaotic Systems. Communications in Nonlinear Science and Numerical Simulation, 15, 4114-4130. https://doi.org/10.1016/j.cnsns.2010.01.040
Bai, E.-W. and Lonngren, K.E. (1997) Synchronization of Two Lorenz Systems Using Active Control. Chaos, Solitons & Fractals, 8, 51-58. https://doi.org/10.1016/S0960-0779(96)00060-4
Ho, M.-C., Hung, Y.-C. and Chou, C.-H. (2002) Phase and Anti-Phase Synchronization of Two Chaotic Systems by Using Active Control. Physics Letters A, 296, 43-48. https://doi.org/10.1016/S0375-9601(02)00074-9
Codreanu, S. (2003) Synchronization of Spatiotemporal Nonlinear Dynamical Systems by an Active Control. Chaos, Solitons & Fractals, 15, 507-510. https://doi.org/10.1016/S0960-0779(02)00128-5
Jankovic, M., Sepulchre, R. and Kokotovic, P.V. (1996) Constructive Lyapunov Stabilization of Nonlinear Cascade Systems. IEEE Transactions on Automatic Control, 41, 1723-1735. https://doi.org/10.1109/9.545712
Yassen, M. (2006) Chaos Control of Chaotic Dynamical Systems Using Backstepping Design. Chaos, Solitons & Fractals, 27, 537-548. https://doi.org/10.1016/j.chaos.2005.03.046
Tan, X., Zhang, J. and Yang, Y. (2003) Synchronizing Chaotic Systems Using Backstepping Design. Chaos, Solitons & Fractals, 16, 37-45. https://doi.org/10.1016/S0960-0779(02)00153-4
Tee, L.S. and Salleh, Z. (2013) Dynamical Analysis of a Modified Lorenz System. Journal of Mathematics, 2013, Article ID: 820946. https://doi.org/10.1063/1.4801119
Vincent, U. (2008) Synchronization of Identical and Non-Identical 4-D Chaotic Systems Using Active Control. Chaos, Solitons & Fractals, 37, 1065-1075. https://doi.org/10.1016/j.chaos.2006.10.005
Yang, T., Li, X.F. and Shao, H.H. (2001) Chaotic Synchronization Using Backstepping Method with Application to the Chua’s Circuit and Lorenz System. Proceedings of the 2001 American Control Conference, Arlington, VA, USA, 25-27 June 2001, Vol. 3, 2299-2300. https://doi.org/10.1109/ACC.2001.946094
Krstic, M., Kokotovic, P.V. and Kanellakopoulos, I. (1995) Nonlinear and Adaptive Control Design. John Wiley & Sons, Inc., Hoboken.
Khalil, H.K. (1996) Adaptive Output Feedback Control of Nonlinear Systems Represented by Input-Output Models. IEEE Transactions on Automatic Control, 41, 177-188. https://doi.org/10.1109/9.481517