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Synchronization of uncertain chaotic systems with minimal parametric information. (English) Zbl 1536.93837

Summary: The chaotic synchronization is mainly hampered by uncertain system dynamics in terms of underlying parameters. To obtain accurate parameter estimates for the proper synchronization of uncertain chaotic systems (UCSs) through adaptive control, it is necessary to satisfy the persistence of excitation (PE) condition. Furthermore, the challenges imposed by the explosion of complexity in sequential stabilization procedures, slow performances, and lack of information in UCSs hinder the synchronization process through existing control techniques. Therefore, the contribution of this research is two-fold: First, a systematic stabilization approach with minimal calculations is proposed to achieve proper synchronization between chaotic drive and response systems. The key concept of the proposed theory is to obtain an invariant manifold by immersing the error dynamics (resulting from the mismatch between the drive and response systems) into lower-order target dynamics. From there, the control law for synchronizing these chaotic systems can be derived by defining passive outputs and associated storage functions. The proposed framework, which combines the concepts of immersion and passivity, is referred to as the passivity and immersion (P&I) approach. Second, unlike adaptive control, the synchronization of UCSs is made possible by selecting the appropriate target dynamics without any parameter estimation (or with minimal parametric information in certain cases). The simplicity and superiority of the proposed approach over existing techniques are preserved and demonstrated by application to a number of well-known chaotic systems.

MSC:

93D99 Stability of control systems
93C15 Control/observation systems governed by ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
Full Text: DOI

References:

[1] Tan, Z.; Sun, J.; Zhang, H.; Xie, X., Chaos synchronization control for stochastic nonlinear systems of interior PMSMs based on fixed-time stability theorem, Appl. Math. Comput., 430, Article 127115 pp., 2022 · Zbl 1510.93253
[2] Sun, J.; Shan, Z.; Liu, P.; Wang, Y., Backstepping synchronization control for three-dimensional chaotic oscillatory system via DNA strand displacement, IEEE Trans. NanoBiosci., 2022
[3] Carroll, T.; Pecora, L., Synchronizing chaotic circuits, IEEE Trans. Circ. Syst., 38, 4, 453-456, 1991
[4] Elgar, S.; Chandran, V., Higher order spectral analysis of Chua’s circuit, IEEE Trans. Circuits Syst. I, 40, 10, 689-692, 1993 · Zbl 0844.58055
[5] Li, G.; Zhang, B., A novel weak signal detection method via chaotic synchronization using chua’s circuit, IEEE Trans. Ind. Electron., 64, 3, 2255-2265, 2016
[6] Vaidyanathan, S.; Azar, A. T., Backstepping Control of Nonlinear Dynamical Systems, 2020, Academic Press
[7] Ivanov, I., Chaotic synchronization in models of impulsive power systems with delay, Int. Appl. Mech., 54, 1, 94-104, 2018 · Zbl 1398.34106
[8] Cuomo, K. M.; Oppenheim, A. V.; Strogatz, S. H., Synchronization of lorenz-based chaotic circuits with applications to communications, IEEE Trans. Circ. Syst. II, 40, 10, 626-633, 1993
[9] Genesio, R.; Tesi, A., Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica, 28, 3, 531-548, 1992 · Zbl 0765.93030
[10] Tan, X.; Zhang, J.; Yang, Y., Synchronizing chaotic systems using backstepping design, Chaos Solitons Fractals, 16, 1, 37-45, 2003 · Zbl 1035.34025
[11] Rössler, O. E., An equation for continuous chaos, Phys. Lett. A, 57, 5, 397-398, 1976 · Zbl 1371.37062
[12] Wang, H. O.; Tanaka, K.; Ikeda, T., Fuzzy modeling and control of chaotic systems, (1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World, Vol. 3. 1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World, Vol. 3, ISCAS 96, 1996, IEEE), 209-212
[13] Xia, Y.; Yang, Z.; Han, M., Lag synchronization of unknown chaotic delayed Yang-Yang-type fuzzy neural networks with noise perturbation based on adaptive control and parameter identification, IEEE Trans. Neural Netw., 20, 7, 1165-1180, 2009
[14] Yau, H.-T.; Wu, S.-Y.; Chen, C.-L.; Li, Y.-C., Fractional-order chaotic self-synchronization-based tracking faults diagnosis of ball bearing systems, IEEE Trans. Ind. Electron., 63, 6, 3824-3833, 2016
[15] Sepulchre, R.; Jankovic, M.; Kokotovic, P. V., Constructive Nonlinear Control, 2012, Springer Science & Business Media · Zbl 1067.93500
[16] Shadab Nayyer, S.; Wagh, S. R.; Singh, N. M., Passivity and immersion (P&I) approach for constructive stabilization and control of nonlinear systems, IEEE Control Syst. Lett., 7, 817-822, 2023
[17] Pozo, F.; Ikhouane, F.; Rodellar, J., Numerical issues in backstepping control: Sensitivity and parameter tuning, J. Franklin Inst. B, 345, 8, 891-905, 2008 · Zbl 1201.93043
[18] Yassen, M., Chaos synchronization between two different chaotic systems using active control, Chaos Solitons Fractals, 23, 1, 131-140, 2005 · Zbl 1091.93520
[19] Joshi, S. K., Synchronization of chaotic dynamical systems, Int. J. Dyn. Control, 9, 3, 1285-1302, 2021
[20] Deng, W.; Li, C., Synchronization of chaotic fractional chen system, J. Phys. Soc. Japan, 74, 6, 1645-1648, 2005 · Zbl 1080.34537
[21] Kharabian, B.; Mirinejad, H., Synchronization of rossler chaotic systems via hybrid adaptive backstepping/sliding mode control, Results Control Optim., 4, Article 100020 pp., 2021
[22] Park, J. H., Synchronization of genesio chaotic system via backstepping approach, Chaos Solitons Fractals, 27, 5, 1369-1375, 2006 · Zbl 1091.93028
[23] Dedieu, H.; Ogorzalek, M., Identifiability and identification of chaotic systems based on adaptive synchronization, IEEE Trans. Circuits Syst. I, 44, 10, 948-962, 1997
[24] Chaitali, C.; Swapnil, J.; Shadab, S.; Wagh, S. R.; Singh, N. M., Adaptive control of feedback linearizable systems with finite-time convergence, (2022 Australian & New Zealand Control Conference. 2022 Australian & New Zealand Control Conference, ANZCC, 2022), 75-80
[25] Vaidyanathan, S.; Volos, C.; Pham, V.-T.; Madhavan, K., Analysis, adaptive control and synchronization of a novel 4-d hyperchaotic hyperjerk system and its SPICE implementation, Arch. Control Sci., 25, 1, 135-158, 2015 · Zbl 1446.93045
[26] Shadab, S.; Revati, G.; Wagh, S.; Singh, N., Finite-time parameter estimation for an online monitoring of transformer: A system identification perspective, Int. J. Electr. Power Energy Syst., 145, Article 108639 pp., 2023
[27] Gevers, M.; Bazanella, A. S.; Coutinho, D. F.; Dasgupta, S., Identifiability and excitation of linearly parametrized rational systems, Automatica, 63, 38-46, 2016 · Zbl 1329.93050
[28] Shadab Nayyer, S.; Wagh, S. R.; Singh, N. M., Towards a constructive framework for stabilization and control of nonlinear systems: Passivity and immersion (PI) approach, 2022, arXiv e-prints, arXiv:2208.10539
[29] Nayyer, S. S.; Hozefa, J.; Gunjal, R.; Bhil, S. K.; Wagh, S. R.; Singh, N. M., Passivity and immersion (P&I) approach with Gaussian process for stabilization and control of nonlinear systems, IEEE Access, 10, 132621-132634, 2022
[30] Nijmeijer, H.; Berghuis, H., On Lyapunov control of the duffing equation, IEEE Trans. Circ. Syst. I, 42, 8, 473-477, 1995 · Zbl 0845.93040
[31] Vaidyanathan, S.; Azar, A. T., Hybrid synchronization of identical chaotic systems using sliding mode control and an application to vaidyanathan chaotic systems, (Advances and Applications in Sliding Mode Control Systems, 2014, Springer), 549-569
[32] Nayyer, S. S.; Revati, G.; Wagh, S.; Singh, N., Passivity and immersion based-modified gradient estimator: A control perspective in parameter estimation, 2022, arXiv preprint arXiv:2211.10674
[33] Sepestanaki, M. A.; Barhaghtalab, M. H.; Mobayen, S.; Jalilvand, A.; Fekih, A.; Skruch, P., Chattering-free terminal sliding mode control based on adaptive barrier function for chaotic systems with unknown uncertainties, IEEE Access, 10, 103469-103484, 2022
[34] Yao, Q., Synchronization of second-order chaotic systems with uncertainties and disturbances using fixed-time adaptive sliding mode control, Chaos Solitons Fractals, 142, Article 110372 pp., 2021 · Zbl 1496.93095
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