Abstract
In this study, a new controller for chaos synchronization is proposed. It consists of a state feedback controller and a robust control term using Legendre polynomials to compensate for uncertainties. The truncation error is also considered. Due to the orthogonal functions theorem, Legendre polynomials can approximate nonlinear functions with arbitrarily small approximation errors. As a result, they can replace fuzzy systems and neural networks to estimate and compensate for uncertainties in control systems. Legendre polynomials have fewer tuning parameters than fuzzy systems and neural networks. Thus, their tuning process is simpler. Similar to the parameters of fuzzy systems, Legendre coefficients are estimated online using the adaptation rule obtained from the stability analysis. It is assumed that the master and slave systems are the Lorenz and Chen chaotic systems, respectively. In secure communication systems, observer-based synchronization is required since only one state variable of the master system is sent through the channel. The use of observer-based synchronization to obtain other state variables is discussed. Simulation results reveal the effectiveness of the proposed approach. A comparison with a fuzzy sliding mode controller shows that the proposed controller provides a superior transient response. The problem of secure communications is explained and the controller performance in secure communications is examined.
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References
Bagheri P, Shahrokhi M, Salarieh H, 2015. Adaptive observerbased synchronization of two non-identical chaotic systems with unknown parameters. J Vibr Contr, 23(3): 389–399. https://doi.org/10.1177/1077546315580052
Chen CS, 2009. Quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems. Exp Syst Appl, 36(9):11827–11835. https://doi.org/10.1016/j.eswa.2009.04.007
Cherrier E, Boutayeb M, Ragot J, 2006. Observers-based synchronization and input recovery for a class of nonlinear chaotic models. IEEE Trans Circ Syst I, 53(9): 1977–1988. https://doi.org/10.1109/TCSI.2006.882817
Chien MC, Huang AC, 2012. Adaptive impedance controller design for flexible-joint electrically-driven robots without computation of the regressor matrix. Robotica, 30(1): 133–144. https://doi.org/10.1017/S0263574711000403
Effa JY, Essimbi BZ, Ngundam JM, 2009. Synchronization of improved chaotic Colpitts oscillators using nonlinear feedback control. Nonl Dynam, 58(1-2):39–48. https://doi.org/10.1007/s11071-008-9459-7
Fard MB, Khorashadizadeh S, 2015. Model free robust impedance control of robot manipulators using Fourier series expansion. AI & Robotics, p.1–7. https://doi.org/10.1109/RIOS.2015.7270740
Fateh MM, Khorashadizadeh S, 2012a. Robust control of electrically driven robots by adaptive fuzzy estimation of uncertainty. Nonl Dynam, 69(3):1465–1477. https://doi.org/10.1007/s11071-012-0362-x
Fateh MM, Khorashadizadeh S, 2012b. Optimal robust voltage control of electrically driven robot manipulators. Nonl Dynam, 70(2):1445–1458. https://doi.org/10.1007/s11071-012-0546-4
Fateh MM, Ahmadi SM, Khorashadizadeh S, 2014a. Adaptive RBF network control for robot manipulators. J AI Data Min, 2(2):159–166. https://doi.org/10.22044/jadm.2014.246
Fateh MM, Azargoshasb S, Khorashadizadeh S, 2014b. Modelfree discrete control for robot manipulators using a fuzzy estimator. Int J Comput Math Electr Electron Eng, 33(3): 1051–1067. https://doi.org/10.1108/COMPEL-05-2013-0185
Grassi G, Mascolo S, 1997. Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal. IEEE Trans Circ Syst I, 44(10):1011–1014. https://doi.org/10.1109/81.633891
Grassi G, Mascolo S, 1998. Design of nonlinear observers for hyperchaos synchronization using a scalar signal. IEEE Int Symp on Circuits and Systems, p.283–286. https://doi.org/10.1109/ISCAS.1998.704006
Grzybowski JMV, Rafikov M, Balthazar JM, 2009. Synchronization of the unified chaotic system and application in secure communication. Commun Nonl Sci Numer Simul, 14(6):2793–2806. https://doi.org/10.1016/j.cnsns.2008.09.028
Gupta MM, Jin L, Homma N, 2005. Static and Dynamic Neural Networks: from Fundamentals to Advanced Theory. Wiley-IEEE Press, New York, USA. https://doi.org/10.1002/0471427950
Hsu CF, 2011. Adaptive fuzzy wavelet neural controller design for chaos synchronization. Expert Syst Appl, 38(8): 10475–10483. https://doi.org/10.1016/j.eswa.2011.02.092
Huang AC, Wu SC, Ting WF, 2006. A FAT-based adaptive controller for robot manipulators without regressor matrix: theory and experiments. Robotica, 24(2):205–210. https://doi.org/10.1017/S0263574705002031
Izadbakhsh A, Khorashadizadeh S, 2017. Robust task-space control of robot manipulators using differential equations for uncertainty estimation. Robotica, 35(9):1923–1938. https://doi.org/10.1017/S0263574716000588
Kai CY, Huang AC, 2013. A regressor-free adaptive controller for robot manipulators without Slotine and Li’s modification. Robotica, 31(7):105058. https://doi.org/10.1017/S0263574713000301
Khorashadizadeh S, Fateh MM, 2013. Adaptive Fourier seriesbased control of electrically driven robot manipulators. 3rd Int Conf on Control, Instrumentation, and Automation, p.213–218. https://doi.org/10.1109/ICCIAutom.2013.6912837
Khorashadizadeh S, Fateh MM, 2015. Robust task-space control of robot manipulators using Legendre polynomials for uncertainty estimation. Nonl Dynam, 79(2):1151–1161. https://doi.org/10.1007/s11071-014-1730-5
Khorashadizadeh S, Fateh MM, 2017. Uncertainty estimation in robust tracking control of robot manipulators using the Fourier series expansion. Robotica, 35(2):310–336. https://doi.org/10.1017/S026357471500051X
Khorashadizadeh S, Mahdian M, 2016. Voltage tracking control of DC-DC boost converter using brain emotional learning. 4th Int Conf on Control, Instrumentation, and Automation, p.268–272. https://doi.org/10.1109/ICCIAutom.2016.7483172
Kreyszig E, 2007. Advanced Engineering Mathematics. John Wiley & Sons, Hoboken, USA.
Kuo CL, 2011. Design of a fuzzy sliding-mode synchronization controller for two different chaos systems. Comput Math Appl, 61(8):2090–2095. https://doi.org/10.1016/j.camwa.2010.08.080
Laoye JA, Vincent UE, Kareem SO, 2009. Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller. Chaos Sol Fract, 39(1):356–362. https://doi.org/10.1016/j.chaos.2007.04.020
Lee SM, Ji DH, Park JH, et al., 2008. [H∞synchronization of chaotic systems via dynamic feedback approach. Phys Lett A, 372(29):4905–4912. https://doi.org/10.1016/j.physleta.2008.05.047
Li LL, Liu Y, Yao QG, 2014. Robust synchronization of chaotic systems using slidingmode and feedback control. J Zhejiang Univ-Sci C (Comput & Electron), 15(3):211–222. https://doi.org/10.1631/jzus.C1300266
Li XR, Zhao LY, Zhao GZ, 2005. Sliding mode control for synchronization of chaotic systems with structure or parameters mismatching. J Zhejiang Univ-Sci, 6A(6):571–576. https://doi.org/10.1631/jzus.2005.A0571
Liao TL, Tsai SH, 2000. Adaptive synchronization of chaotic systems and its application to secure communications. Chaos Sol Fract, 11(9):1387–1396. https://doi.org/10.1016/S0960-0779(99)00051-X
Lin TC, Huang FY, Du ZB, et al., 2015. Synchronization of fuzzy modeling chaotic time delay memristor-based Chua’s circuits with application to secure communication. Int J Fuzzy Syst, 17(2):206–214. https://doi.org/10.1007/s40815-015-0024-5
Liu MQ, Zhang JH, 2008. Exponential synchronization of general chaotic delayed neural networks via hybrid feedback. J Zhejiang Univ-Sci A, 9(2):262–270. https://doi.org/10.1631/jzus.A071336
Lu JA, Wu XQ, Han XP, et al., 2004. Adaptive feedback synchronization of a unified chaotic system. Phys Lett A, 329(4-5):327–333. https://doi.org/10.1016/j.physleta.2004.07.024
Naseh MR, Haeri M, 2009. Robustness and robust stability of the active sliding mode synchronization. Chaos Sol Fract, 39(1):196–203. https://doi.org/10.1016/j.chaos.2007.01.123
Nijsure YA, Kaddoum G, Gagnon G, et al., 2016. Adaptive air-to-ground secure communication system based on ADS-B and wide-area multilateration. IEEE Trans Veh Technol, 65(5):3150–3165. https://doi.org/10.1109/TVT.2015.2438171
Ogata K, 1995. Discrete-Time Control Systems (2nd Ed.). Prentice Hall, Englewood Cliffs, USA.
Pecora LM, Carroll TL, 1991. Driving systems with chaotic signals. Phys Rev A, 44(4):2374–2383. https://doi.org/10.1103/PhysRevA.44.2374
Pogromsky A, Nijmeijer H, 1998. Observer-based robust synchronization of dynamical systems. Int J Bifurc Chaos, 8(11):2243–2254.
Shen C, Shi ZG, Ran LX, 2006. Adaptive synchronization of chaotic Colpitts circuits against parameter mismatches and channel distortions. J Zhejiang Univ-Sci A, 7(S2): 228–236. https://doi.org/10.1631/jzus.2006.AS0228
Shi ZG, Hong SH, Chen JM, et al., 2008. Particle filter-based synchronization of chaotic Colpitts circuits combating AWGN channel distortion. Circ Syst Signal Process, 27(6):833–845. https://doi.org/10.1007/s00034-008-9062-7
Shi ZG, Bi SJ, Zhang HT, et al., 2013. Improved auxiliary particle filter-based synchronization of chaotic Colpitts circuit and its application to secure communication. Wirel Commun Mob Comput, 15(10):1456–1470. https://doi.org/10.1002/wcm.2446
Slotine JJE, Li WP, 1991. Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, USA.
Wang C, Ge SS, 2001. Adaptive backstepping control of uncertain Lorenz system. Int J Bifurc Chaos, 11(4):1115–1119. https://doi.org/10.1142/S0218127401002560
Wang H, Ye JM, Miao ZH, et al., 2016. Robust finite-time chaos synchronization of time-delay chaotic systems and its application in secure communication. Trans Inst Meas Contr, 40(4):1177–1187. https://doi.org/10.1177/0142331216678311
Wang LX, 1997. A Course in Fuzzy Systems and Control. Prentice-Hall, New York, USA.
Wang Q, Chen Y, 2006. Generalized Q-S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh-Rose systems. Appl Math Comput, 181(1):48–56. https://doi.org/10.1016/j.amc.2006.01.017
Yang JQ, Chen YT, Zhu FL, 2015. Associated observer-based synchronization for uncertain chaotic systems subject to channel noise and chaos-based secure communication. Neurocomputing, 167:587–595. https://doi.org/10.1016/j.neucom.2015.04.030
Yassen MT, 2005. Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos Sol Fract, 26(3):913–920. https://doi.org/10.1016/j.chaos.2005.01.047
Zadeh SMH, Khorashadizadeh S, Fateh MM, et al., 2016. Optimal sliding mode control of a robot manipulator under uncertainty using PSO. Nonl Dynam, 84(4):2227–2239. https://doi.org/10.1007/s11071-016-2641-4
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Khorashadizadeh, S., Majidi, MH. Synchronization of two different chaotic systems using Legendre polynomials with applications in secure communications. Frontiers Inf Technol Electronic Eng 19, 1180–1190 (2018). https://doi.org/10.1631/FITEE.1601814
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DOI: https://doi.org/10.1631/FITEE.1601814