||Chaotic Synchronization and Its Application to AWGN Communication Systems
||Department of Engineering Science
Chaotic communication system
Nonlinear output feedback
Lyapunov stability theorem
Largest Lyapunov exponent
Linear matrix inequality (LMI)
本論文以混沌理論為基礎，探討混沌訊號用於通訊系統時的效能分析。首先，藉由一具有混沌特性的水平平台之電子化實現，設計非同調(non-coherent)的通訊系統。在接收端以非線性輸出迴授方式設計控制器，使接收端與發射端完成訊號同步，並以最大李亞普諾夫指數確認系統之渾沌特性未被控制器破壞。第二部份，針對主僕混沌系統中，制動器失效問題，提出以李亞普諾夫穩定定理與線性矩陣不等式為基礎所設計之可靠度控制器，並探討未發生與發生趨動誤差之系統動態，以確保主僕混沌系統之間能保持同步。第三部份，以非同調混沌通訊系統為基礎，設計MIMO架構，分別以一/二/三組的羅倫斯電路經過可加性高斯白雜訊(Additive White Gaussian Noise, AWGN)通道傳輸。系統中於發射端加入資訊符元，在接收端進行羅倫斯電路同步與資訊符元的解碼，並以位元錯誤率(Bit-Error-Rate, BER)作為通訊系統的效能指標並進行分析。
High information correctness and security enhancement are the main objectives of information processing and telecommunication engineering. Information streaming is propagated from one point to another by diversified transmission technologies and mechanisms with electric signals being either wired or wireless. The most important element in communication is synchronization between transmitter and receiver. In other words, excellent synchronization brings high efficiency for decreasing transmission failures. However, considering the demands from a variety of environmental noises, the challenges of design and control for synchronization problems become increasingly difficult. Simultaneously, the reliability of received information can be seriously compromised due to the ceaselessly increasing multi-user and wireless transmissions. As such, this decade has been witness to the rapid growth of various technologies that has generated tremendous changes in synchronization technologies for telecommunications. Currently, the most popular area for synchronization research has been aimed at ensuring and increasing the reliability of information under transmission between two different points under various conditions including time-varying delays of propagation channels and interference from multi-users, among others.
In this dissertation, the stability and performance analysis of a chaos-based communication system has been studied in three separate parts. Firstly, a corresponding electronic horizontal platform system (HPS) with chaotic behavior is designed and implemented; further, a non-coherent coupled chaos-based communication is built based on the proposed electronic design method. Additionally, an output feedback controller is proposed for global synchronization between coupled electronic HPS systems, the stability condition of which is derived by employing the Lyapunov stability theory. Then, the chaotic behavior of the electronic HPS is verified with the largest Lyapunov exponent. In the second part, by combining the Lyapunov stability theory with the linear matrix inequality (LMI) optimization technique, a reliable feedback controller is established to guarantee synchronization between the master and slave chaotic systems despite the occurrence of some control component (actuator) failures. Following this, an illustrated example is provided to demonstrate the effectiveness of the developed system, both with and without the actuator failures. In the third part, numbers of non-coherent coupled chaotic communication systems denoted by the Lorenz system work as a Multi-Input-Multi-Output (MIMO) structure in a wireless channel with the Additive White Gaussian Noise (AWGN) effect. In this structure, information symbols are covered and encrypted with chaotic dynamics of the Lorenz system at the transmitter side. Then, information decryption processing is completed by synchronization control design of the Lorenz system at the receiver side. And finally, an SNR (Signal to Noise Rate)-to-BER (Bit-Error Rate) diagram is illustrated as a correctness indicator of the proposed structure.
Table of Contents V
List of Figures VIII
Chapter 1 Introduction 1
1.1 Motivation 1
1.2 Overview of Previous Works and Brief Sketch of the Contents 4
Chapter 2 Design and Implementation of the Master-Slave Electronic Horizontal Platform System and its Synchronization 9
2.1. The Mechanical and Corresponding Electronic HPS Model 10
2.1.1 Mechanical Horizontal Platform System 10
22.214.171.124 The HPS Model Formulation 10
126.96.36.199 The Simulation Results of HPS 12
2.1.2 The Corresponding Electronic HPS Model and its Chaotic Behavior 14
188.8.131.52 The Corresponding Electronic HPS Model 14
184.108.40.206 The Simulation Results of Corresponding Electronic HPS Model 17
220.127.116.11 The Largest Lyapunov exponent 18
2.2. Design and Synchronization of Master-Slave Electronic Horizontal Platform System.. 20
2.2.1 Output Feedback Design for Synchronizing the Master-Slave Electronic Horizontal Platform Systems 21
2.3. Numerical Simulation Results of Output Feedback Design for Synchronizing the Master-Slave Electronic Horizontal Platform System 25
Chapter 3 Reliable Synchronization Approach of Nonlinear Coupled Chaotic Systems 30
3.1. Problem Formulation 30
3.2. The Reliable Feedback Controller Design 32
3.3. Numerical Simulation 36
Chapter 4 Chaotic Security Communication in Wireless AWGN Channel 41
4.1. Problem Formulation of Master-Slave Chaotic Secure Communication 41
4.2. Chaotic Secure Communication in Wireless AWGN Channel 46
4.3. The MIMO-like Structure of Chaotic Wireless Communication 47
4.4. Numerical Simulation 48
Chapter 5 Conclusions and Future Works 54
5.1. Conclusions 54
5.2. Future Works 56
 E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, vol. 20, pp. 130-141, 1963.
 S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering, Westview Press, 1994.
 L. Fortuna, M. Frasca and M.G. Xibilia, “Chua’s circuit implementations: yesterday, today and tomorrow,” World Scientific Series on Nonlinear Science, Series A, 2009.
 P.J. Holmes and D.A. Rand , “The bifurcations of Duffing's equation: An application of catastrophe theory,” Journal of Sound and Vibration, vol. 44, pp. 237-253, 1976.
 O. E. Rössler, “An equation for continuous chaos,” Physics Letter A, vol. 57, pp. 397-398, 1976.
 G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos, vol. 9, pp. 1465-1466, 1999.
 M. Hénon, “A two-dimensional mapping with a strange attractor,” Communications in Mathematical Physics, vol. 50, pp. 69–77, 1976.
 C. Knudsen, Lyapunov exponents, Technical University of Denmark Press, 1999.
 J. C. Sprott, Chaos and time-series analysis, Oxford University Press, pp. 116-117, 2003.
 P. Glendinning, Stability, instability and chaos, Cambridge University Press, 1994.
 S. Strogatz, Non-linear dynamics and chaos: with applications to physics, biology, chemistry and engineering, Perseus Books, 2000.
 L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, pp. 821-824, 1990.
 H. Kebriaei and M. J. Yazdanpanah, “Robust adaptive synchronization of different uncertain chaotic systems subject to input nonlinearity Original Research Article,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, pp. 430-441, 2010.
 L. Zhang, L. Huang, Z. Zhang and Z. Wang, “Fuzzy adaptive synchronization of uncertain chaotic systems via delayed feedback control,” Physics Letter A, vol. 372, pp. 6082-6086, 2008.
 Z. Li and S. Shi, “Robust adaptive synchronization of Rossler and Chen chaotic systems via slide technique,” Physics Letter A, vol. 311, pp. 389-395, 2003.
 Z. P. Yu, “Asymptotic and time-limited synchronization of uncertain unified chaotic system based on Lyapunov function,” Proceedings of IEEE ICDMA’2011, pp. 236-240, 2011.
 F. Dachselt and W. Schwarz, “Chaos and cryptography,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, pp. 1498-1509, 2001.
 S. Chen and H. Leung, “Chaotic watermarking for video authentication in surveillance applications,” IEEE Transactions on Circuits and Systems for Video Technology, pp. 704-709,2008.
 W. Xiao, Z. Ji, X. Zhang and W. Wu, “A watermarking algorithm based on chaotic encryption,” Proceedings of IEEE TENCON’02, pp. 545-548, 2002.
 D. Que, L. Lu, H. Wang and Y. Ding, “A digital voice secure communication system based on logistic-mapping in wavelet domain chaotic modulation,” Proceedings of IEEE ICSP '04, pp. 2397-2400, 2004.
 C.H. Shih, Implementation of a chaos-based multimedia encryption software, National Taiwan Ocean University Press, 2011.
 L. Kocarev and A. V. Oppenheim, “General approach for chaotic synchronization with applications to communication,” Physical Review Letters, vol. 74, pp. 5028-5031, 1995.
 P. Stavroulakis, Chaos applications in telecommunications, CRC Press, New York Press, 2006.
 J. B. Andersen, T. S. Rappaport and S. Yoshida, “Propagation measurements and models for wireless communications channels,” IEEE Communication Magazine, vol. 33, pp. 42-49, 1995.
 G. Heidari-Bateni and C. D. McGillem, “Chaotic sequences for spread spectrum: an alternative to pn-sequences,” proceedings of IEEE ICSTWC’92, pp. 437-440, 1992.
 W. Stallingsm, Wireless communications and networks, Prentice-Hall Press, 2002.
 K. Wesolowshi, Mobile communication systems, Wiley Press,2002.
 A.N. Miliou, I.P. Antoniades, S.G. Stavrinides, and A.N. Anagnostopoulos, “Secure communication by chaotic synchronization: robustness under noisy conditions,” Nonlinear Analysis: Real World Applications, vol. 8, pp. 1003-1012, 2007.
 M. Chen and W. Min, “Unknown input observer based chaotic secure communication,” Physics Letter A, vol. 372, pp. 1595–1600, 2007.
 C. Zhang and H.Zheng, “Dynamic properties of Coupled Maps,” Discrete Dynamics in Nature and society, vol. 2010, Article ID 905102, 10 pages, 2010.
 X.Y. Wang and J.M. Song, “Synchronization of the unified chaotic system,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, pp. 3409–3416, 2008.
 Z.M. Ge, L.H. Li, S.Y. Li, and C.H. Chang, “Chaos synchronization of double Duffing systems with parameters excited by a chaotic signal,” Journal of Sound and Vibration, vol. 317, pp. 449–455, 2008.
 H. Sun and H. Cao, “Chaos control and synchronization of a modified chaotic system,” Chaos, Solitons and Fractals, vol. 37, pp. 1442–1455, 2008.
 C.K Huang, S.C. Tsay, and T.R. Wu, “Implementation of chaotic secure communication systems based on OPA circuits,” Choas, Solitons and Fractals, vol. 23, pp. 345–377, 2005.
 V. Astakhov, A. Shabunin, A. Klimshin, and V. Anishchenko, “In-phase and antiphase complete chaotic synchronization in symmetrically coupled discrete maps,” Discrete Dynamics in Nature and Society, vol. 7, no. 4, pp. 215–229, 2002.
 S.C. Tsay, C.K. Huang, D.L. Qiu, and W.T. Chen, “Implementation of bidirectional chaotic communication systems based on Lorenz circuits,” Chaos, Solitons and Fractals, vol. 20, pp. 567–579, 2004.
 M. Itoh, “Synthesis of electronic circuits for simulation nonlinear dynamics,” International Journal of Bifurcation and Chaos, vol. 11, pp. 605–653, 2001.
 C.L. Huang, Nonlinear dynamics of the horizontal platform, National Chiao Tung University Press, 1996.
 G. Gandhi, Electronic realizations of chaotic circuits: from breadboard to nanotechnology, Péter Pázmány Catholic University Press, 2008.
 E. Ott, C. Grebogi and J.A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, pp. 1196-1199, 1990.
 T. Ueta and G. Chen, “Bifurcation analysis of Chen’s attractor,” International Journal of Bifurcation and Chaos, vol. 10, pp. 1917-1931, 2000.
 T. Wada, M. Ikeda, Y. Ohta and D.D. Siljak, “Parametric absolute stability of Lur’e systems,” IEEE Transaction on Automatic Control, vol. 43, pp. 1649-1653, 1998.
 Y. Zhang and J. Sun, “Controlling chaotic Lu systems using impulsive control,” Physics Letter A, vol. 342, pp. 256-262, 2005.
 X.S. Yang and Q. Yuan, “Chaos and transient chaos in simple Hopfield neural networks,” Neurocomputing, vol. 69, pp. 232-241, 2005.
 W. Yu, J. Cao, “Cryptography based on delayed chaotic neural networks,” Physics Letter A, vol. 356, pp. 333-338, 2006.
 P.K. Das, W.C. Schieve and Z. Zeng, “Chaos in an effective four-neuron neural network,” Physics Letter A, vol. 161,pp. 60-66, 1991.
 W. Xiong, W. Xie and J. Cao, “Adaptive exponential synchronization of delayed chaotic networks,” Physica A: Statistical Mechanics and its Applications, vol. 346 , pp. 832-842, 2006.
 J. Sun and Y. Zhang, “Impulsive control and synchronization of Chua’s oscillators,” Mathematics and Computers in Simulation, vol.66, pp. 499-508, 2004.
 H.T. Yau and C.L. Chen, “Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems,” Chaos, Solitons and Fractals, vol. 30, pp. 709-718, 2006.
 T.L. Liao and N.S. Huang, “An observer-based approach for chaotic synchronization with applications to secure communications,” IEEE Transaction on Circuit and Systems, vol. 46, pp. 1144-1150, 1999.
 H.T. Yau, “Design of adaptive sliding mode controller for chaos synchronization with uncertainties.” Chaos, Solitons and Fractals, vol. 22 , pp. 341-347, 2004.
 B. Yao and F. Wang, “LMI approach to reliable control of linear systems,” Journal of Systems Engineering and Electronics, vol. 17, pp. 381-386, 2006.
 L. Yu, “An LMI approach to reliable guaranteed cost control of discrete-time systems with actuator failure,” Applied Mathematics and Computation, vol. 162, pp. 1325-1331, 2005.
 H.N. Wu and H.Y. Zhang, “Reliable fuzzy control for continuous-time nonlinear system with actuator failures,” IEEE Transactions on Fuzzy System, vol. 14, pp. 609-618, 2006.
 Z. Wang, B. Huang and H. Unbehauen, “Robust reliable control for a class of uncertain nonlinear state-delayed systems,” Automatica, vol. 35, pp. 955-963, 1999.
 J. Wang and H. Shao, “Delay-dependent robust and reliable control for uncertain time-delay systems with actuator failures,” Journal of Franklin Institute, vol. 337, pp. 781-791, 2000.
 H. N. Agiza, “Chaos synchronization of Lü dynamical system,” Nonlinear analysis, vol. 58, pp. 11-20, 2004.
 M. Chen and W. Min, “Unknown input observer based chaotic secure communication,” Physics Letter A, vol. 372, pp. 1595-1600, 2007.
 X. Wu, G. Chen, and J. Cai, “Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control,” Physics D: Nonlinear Phenomena, vol. 229, pp. 52-80, 2007.
 J. M. V. Grzybowski, M. Rafikov, and J. M. Balthazar, “Synchronization of the unified chaotic system and application in secure communication,” Communication of Nonlinear Science Numerical Simulation, vol. 14, pp. 2793-2806, 2004.
 J.S. Lin, N.S. Pai, and H.T. Yau, “Robust controller design for modified projective synchronization of Chen-Lee chaotic systems with nonlinear inputs,” Mathematical Problems in Engineering, vol. 2009, Article ID 649401, 11 pages, 2009.
 A.E. Matouk, “Chaos synchronization between two different fractional systems of Lorenz family,” Mathematical Problems in Engineering, vol. 2009, Article ID 572724, 11 pages, 2009.
 X.F. Li, Y.D. Chu, J.G. Zhang, and Y.X. Chang, “Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor,” Chaos, Solitons and Fractals, vol. 41, pp. 2360-2370, 2009.
 W.D. Chang, “Digital secure communication via chaotic systems,” Digital Signal Process, vol. 19, pp. 693-699, 2009.
 J.S. Lin, C.F. Huang, T.L. Liao, and J.J. Yan, “Design and implementation of digital secure communication based on synchronized chaotic systems,” Digital Signal Process, vol. 20, pp. 229-237, 2010.
 E. N. Sanchez and L. J. Ricalde, “Chaos control and synchronization, with input saturation, via recurrent neural networks,” Neural Networks, vol. 16, pp. 711-717, 2003.
 G. He, Z. Cao, P. Zhu and Hisakazu Ogura, “Controlling chaos in a chaotic neural network,” Neural Networks, vol. 16, pp. 1195-1200, 2003.
 S.C. Tsay, C.K. Huang, D.L. Qiu, and W.T. Chen, “Implementation of bidirectional chaotic communication systems based on Lorenz circuits,” Chaos, Solitons and Fractals, vol. 20, pp. 567-579, 2004.
 S.C. Tsay, C.K. Huang, and C.T. Chiang, “Design the hyperchaotic cryptosystems via the Gerschgorin theorem,” Chaos, Solitons and Fractals, vol. 19, pp. 935-948, 2004.
 C. E. Shannon and W. Weaver, The mathematical theory of communication, Illinois University Press, 1963.
 S. J. MacMullan and O. M. Collins, “The capacity of orthogonal and bi-orthogonal codes on the Gaussian channel,” Proceedings of IEEE Information Theory’97, page 338, 1997.
 Z. Lei and T. J. Lim, “Estimation of directions of arrival of multipath signals in CDMA systems,” IEEE Transactions on Communications, vol. 48, pp. 1022-1028, 2000.
 C. Zhao and L. Gan, “Dynamic channel assignment for cellular networks using noisy chaotic neural network,” Proceedings of IEEE ICCASM’2010, pp. 143-147, 2010.
 D. A. Huffman, “A method for the construction of minimum-redundancy codes,” Proceedings of IRE, vol. 40, pp. 1098-1101, 1952.
 A. Moffat, R. Neal, and I. H. Witten. “Arithmetic coding revisited,” ACM Transactions on Information Systems, vol. 16, pp. 256–294, 1998.
 K. R. Rao and P. YIP, Discrete cosine transform: algorithms, advantages, applications, Academic Press, 1990.
 Y. Hwang and H. C. Papadopoulos, “Physical-layer secrecy in AWGN via a class of chaotic DS/SS systems: analysis and design,” IEEE Transactions on Signal Processing, vol. 52, pp. 2637-2649, 2004.
 X. Wu and Z. Wang, “Estimating parameters of chaotic systems under noise-induced synchronization,” Chaos, Solitons and Fractals, vol. 39, pp. 689-696, 2009.
 Y. Zhang, S. Chen and H. Zhou, “Synchronizing the noise-perturbed Lü chaotic system,” Chaos, Solitons and Fractals, vol. 40. pp. 2475-2482, 2009.
 J.Y. Lee, “The corresponding phenomena of mechanical and electronic impact oscillator,” Journal of Sound and Vibration, vol. 311, pp. 579–587, 2008.
 K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Physical Review Letters, vol. 71, pp. 65-68, 1993.
 Specification of AD534 rev. C: internally trimmed precision IC multiplier, Analog Devices, 2011.
 A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano, “Determining Lyapunov exponents from time series,” Physica D: Nonlinear Phenomena, vol. 16, pp. 285–317, 1985.
 X. Wu, J. Cai and M. Wang, “Master–slave chaos synchronization criteria for the horizontal platform systems via linear state error feedback control,” Journal of Sound and Vibration, vol. 295, pp. 378–387, 2006.
 D. I. R. Almedia, J. Alvarez, and J. G. Barajas, “Robust synchronization of Sprott circuits using sliding mode control,” Chaos, Solitons and Fractals, vol. 30, pp. 11–18, 2006.
 G. H. Li, S. P. Zhou, and K. Yang, “Controlling chaos in Colpitts oscillator,” Chaos, Solitons and Fractals, vol. 33, pp. 582–587, 2007.
 J. C. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis, “State-space solutions to standard and control problems,” IEEE Transactions on Automatic Control, vol. 34, pp. 831-846, 1989.
 P.P. Khargonekar, I.R. Petersen and K. Zhou, “Robust stabilization of uncertain linear systems: quadratic stabilizability and control theory,” IEEE Transactions on Automatic Control, vol. 46, pp. 356-361, 1990.
 A.M Lyapunov, The general problem of the stability of motion, University Kharkov Press, 1892.
 S. Boyd, El Ghaoui L, E. Feron and V. Balajrishnsn, “Linear matrix inequalities in system and control theory,” Proceedings Allerton Conference on Communication, Control and Computing, pp. 237-246, 1993.
 D. R. Stinson, Cryptography: theory and practice, CRC Press, 1995.