Chaos Synchronization via Linear Matrix Inequalities:A Comparative Analysis


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International Journal on Smart Sensing and Intelligent Systems

Professor Subhas Chandra Mukhopadhyay

Exeley Inc. (New York)

Subject: Computational Science & Engineering, Engineering, Electrical & Electronic


eISSN: 1178-5608



VOLUME 7 , ISSUE 2 (June 2014) > List of articles

Chaos Synchronization via Linear Matrix Inequalities:A Comparative Analysis

Hanéne Mkaouar * / Olfa Boubaker *

Keywords : Chaos synchronization, Linear Matrix Inequalities, Piecewise Affine systems, Lyapunov stability, comparative analysis.

Citation Information : International Journal on Smart Sensing and Intelligent Systems. Volume 7, Issue 2, Pages 553-583, DOI:

License : (CC BY-NC-ND 4.0)

Received Date : 27-March-2014 / Accepted: 03-May-2014 / Published Online: 27-December-2017



In this paper, three chaos synchronization approaches using Linear Matrix Inequality (LMI) tools are evaluated and compared. The comparative analysis is supported by four examples of Piecewise affine (PWA) chaotic systems: The Chua’s original circuit, the Chua’s modified system, the Lur’e like circuit and the five-scroll attractor system. To evaluate the performances of each synchronization approach, we examine first, the practical implementation of the LMIs. We analyze then, by simulation results, the feasibility of each approach for each PWA chaotic system. The elapsed time for solving the predefined LMIs and the influence of their tuning parameters’ domain belonging on the feasibility and the performances of each approach are finally the considered comparative criteria.

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[1] C. J. Luo, A theory for synchronization of dynamical systems, Communications in Nonlinear Science and Numerical Simulation. 14 (2009) 1901-1951.
[2] A. Pikovsky, M. Rosenblum, J. Kurths, B. Chirikov, P. Cvitanovic, Synchronization: A Universal Concept in Nonlinear Sciences, Second edition, United Kingdom, Cambridge University Press, 2003.
[3] L.M. Pecora, T.L. Carroll, Synchronization in chaotic system. Physical Review Letters. 64 (1990) 821-825.
[4] M. T .Yassen, Chaos synchronization between two different chaotic systems using active control, Chaos, Solitons and Fractals. 23 (2005) 131-140.
[5] W. Yu, Synchronization of three dimensional chaotic systems via a single state feedback,Communications in Nonlinear Science and Numerical Simulation, 16 (2011) 2880- 2886.
[6] U.E. Vincent, R. Guo, Finite-time synchronization for a class of chaotic and hyperchaotic systems via adaptive feedback controller, Physics Letters A, 375 (2009) 3925-3932.
[7] J. Zhang, C. Li, H. Zhang, J. Yu, Chaos synchronization using single variable feedback based on back-stepping method, Chaos, Solitons & Fractals. 21 (2004) 1183-1193.
[8] M. Hu, Y. Yang, Z. Xu, Impulsive control of projective synchronization in chaotic systems,Physics Letters A. 372 (2008) 3228-3233.
[9] T. Yang, L.O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Trans Circuits Systems I: Fundamentals Theory and Applications, 44 (1997) 976-988.
[10] M. Pourmahmood, S. Khanmohammadi, G. Alizadeh, 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 (2011) 2853-2868.
[11] O. Boubaker, R. Dhifaoui, “Robust chaos synchronization for chua’s circuits via active sliding mode control,” In: S. Banerjee and Ş. Ş. Erçetin, Chaos, Complexity and Leadership 2012, (2014) 141-151.
[12] H. Yu, J. Wang, B. Deng, X. Wei, Y. Che, Y.K. Wong, W.L. Chan, K.M. Tsang,Adaptive back-stepping sliding mode control for chaos synchronization of two coupled neurons in the external electrical stimulation, Communications in Nonlinear Science and
Numerical Simulation, 17 (2012)1344-1354,
[13] T. Zhang, G. Feng, Output tracking and synchronization of chaotic Chua’s circuit with disturbances via model predictive regulator. Chaos Solitons & Fractals, 9 (2009) 810-820.
[14] W. Xiaofeng, C. Guanrong, C. Jianping, Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control, Physica D: Nonlinear Phenomena, 229 (2007) 52-80.
[15] Q. Lin, X. Wu, The sufficient criteria for global synchronization of chaotic power systems under linear state-error feedback control, Nonlinear Analysis: Real World Applications, 12(2011) 1500-1509.
[16] G.P. Jiang, W. X. Zheng, An LMI criterion for linear-state-feedback based chaos synchronization of a class of chaotic systems, Chaos, Solitons & Fractals. 26 (2005) 437-443.
[17] F. Chen, W. Zhang, LMI criteria for robust chaos synchronization of a class of chaotic systems, Nonlinear Analysis: Theory, Methods & Applications, 67 (2007) 3384-3393. 
[18] S. Kuntanapreeda, Chaos synchronization of unified chaotic systems via LMI, Physics
Letters A, 373 (2009) 2837-2840.
[19] M. M. Asheghan, M.T.H. Beheshti, An LMI approach to robust synchronization of a class of chaotic systems with gain variations, Chaos, Solitons & Fractals. 42 (2009) 1106-1111.
[20] Y. Chen, X. Wu, Z. Gui, Global synchronization criteria for a class of third-order nonautonomous chaotic systems via linear state error feedback control, Applied Mathematical Modelling. 34 (2010) 4161-4170.
[21] H. Mkaouar, O. Boubaker, Chaos synchronization for master slave piecewise linear systems: Application to Chua’s circuit, Communications in Nonlinear Science and Numerical Simulation, 17 (2012) 1292-1302.
[22] J. Cao, H.X. Li, D.W.C. Ho, Synchronization criteria of Lur’e systems with time-delay feedback control. Chaos Soliton & Fractals 23 (2005) 1285-1298.
[23] Q.L. Han, On designing time-varying delay feedback controllers for master–slave synchronization of Lur’e systems. IEEE Transactions on Circuits and Systems I: Regular Papers 54 (2007) 1573-1583.
[24] J.L. Lu, D.J. Hill, Global asymptotical synchronization of chaotic Lur’e systems using sampled data: a linear matrix inequality approach, IEEE Transactions on Circuits and Systems II: Express Briefs. 55 (2008) 586-590.
[25] C.K. Zhang, Y. He, M. Wu, Improved global asymptotical synchronization of chaotic Lur’e systems with sampled-data control, IEEE Transactions on Circuits and Systems II:Express Briefs. 56 (2009) 320-324.
[26] M. Johansson, A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control, 43 (1998) 555-559.
[27] D. Kumar Ghara, D. Saha, K. Sengupta, Implementation of linear trace moisture sensor by nano porous thin film, International Journal of Smart Sensing and Intelligent Systems, 1(2008) 955-969.
[28] B. Mulliez, E. Moutaye, H. Tap, L.Gatet, F. Gizard, Predistorsion system implementation based on analog neural networks for linearizing high power amplifiers transfer characteristics, International Journal of Smart Sensing and Intelligent Systems, 7 (2014) 400-422.
[29] A. Pavlov, A. Pogromsky, N. Van de Wouw, H. Nijmeijer, On convergence properties of piecewise affine systems. International Journal of Control, 80 (2007) 1233-1247.
[30] O. Boubaker, Gain scheduling control: an LMI approach, International Review of Electrical Engineering, 3 (2008) 378-385.
[31] T. Zhang, G. Feng, Output tracking of piecewise-linear systems via error feedback regulator with application to synchronization of nonlinear Chua’s circuit, IEEE Transactions on Circuits and Systems I, 54 (2007) 1852-1863.
[32] Van de Wouw, N., Pavlov, A.: Tracking and synchronization for a class of PWA systems.Automatica, 44 (2008) 2909-2915.
[33] O. Boubaker, Master-slave synchronization for PWA systems, In Proceedings 3rd IEEE International Conference on Signals, Circuits and Systems, Medenine, Tunisia. 2009, 1-6.
[34] K. Kashima, Y. Kawamura, J.I. Imura, Oscillation analysis of linearly coupled piecewise affine systems: Application to spatio-temporal neuron dynamics, Automatica, 47 (2011)1249-1254.
[35] H. Mkaouar, and O. Boubaker, Sufficient conditions for global synchronization of continuous piecewise affine systems, Lecture Notes in Computer Science, 6752 (2011) 199-211.
[36] A.C.J. Luo, Singularity and dynamics on discontinuous vector fields, Amsterdam, Elsevier, 2006
[37] A.S. Morse, Control using logic-based switching, Lecture notes in control and information sciences, London, Springer, 1997.
[38] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems. IEEE Control Systems, 19 (1999) 59-70.
[39] Chua LO. Wu CW. Huang A. Zhong G. A universal circuit for studying and generating chaos-part I: routes to chaos, IEEE Transactions on circuits and systems. 40 (1993) 732-744. [40] L. O. Chua, M. Komuro, T. Matsumto, The double scroll family, IEEE Transactions on
Circuits and Systems. 33 (1986) 1072-1118.
[41] H. Mkaouar, O. Boubaker, On electronic design of the piecewise linear characteristic of the chua's diode: Application to chaos synchronization, In Proceedings 16th IEEE Mediterranean Electro-technical Conference, Yasmine Hammamet, Tunisia, 25-28 March
2012, 197-200.
[42] M.E. Yalcin, J.A.K. Suykens, J. Vandewalle, Experimental confirmation of 3-scroll and 5-scroll attractors for generalized Chua’s circuit, IEEE Transactions on circuits and systems I:Fundamentals Theory and Applications. 47 (2000), 425-429.
[43] É. Gyurkovics, T. Takács, “Application of a multiplier method to uncertain Lur’e-like systems,” Systems & Control Letters, 60 (2011) 854-862.
[44] G.P. Jiang, K.S. Tang, G.A. Chen, Simple global synchronization criterion for coupled chaotic systems, Chaos, Solitons & Fractals. 15 (2003) 925-935.