Abstract
To achieve control and synchronization of chaotic electronic circuits, a nonlinear optimal (H-infinity) control method is developed and is tested on Chua’s circuit. Although this electronic circuit is deterministic, for specific values of its parameters its phase diagrams may change in a random-like manner, thus exhibiting a chaotic behavior. In the article’s control approach, an approximate linearization procedure is applied first to the dynamic model of the circuit. The linearization takes place around a temporary operating point which is recomputed at each iteration of the control method. It actually uses Taylor series expansion and the computation of the system’s Jacobian matrices. At a next stage, an H-infinity feedback controller is developed for the approximately linearized model of the circuit. This controller is obtained after solving an algebraic Riccati equation at each time step of the control method. To prove the stability properties of the control scheme and the elimination of the synchronization error, Lyapunov analysis is used. The proposed control scheme is demonstrated to satisfy the H-infinity tracking performance condition, and this indicates elevated robustness against model uncertainty and external perturbations. Moreover, the global asymptotic properties of the control method are proven. Finally, under the proposed nonlinear optimal control approach, it is shown that different Chua’s circuits can get synchronized and that chaotic behavior can be replicated by them.
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References
Kolumban, G., Kennedy, M.P., Chua, L.: The role of synchronization in digital communications using chaos—part II: chaotic modulation and chaotic synchronization. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 45(11), 1129–1140 (1998)
Suykens, J., Curran, P., Yang, T., Vandewalle, J., Chua, L.: Nonlinear \(H_\infty\) synchronization of Lur’e systems: dynamic output feedback case. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 44(11), 1089–1092 (1997)
Zhou, Y., Hua, Z., Run, C., Chen, C.P.: Cascade chaotic systems with applications. IEEE Trans. Cybern. 45(9), 2001–2012 (2015)
Hu, G., Pivka, L., Zheleznyak, A.L.: Synchronization of a one-dimensional array of Chua’s circuits by feedback control and noise. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 42(10), 736–740 (1995)
Rigatos, G., Abbaszadeh, M.: Synchronization of chaotic electronic circuits using nonlinear optimal control. In: IEEE 28th International Symposium on Industrial Electronics. IEEE ISIE 2019, Vancouver, Canada (2019)
Chen, Y., Wu, X., and Gui, Z.: Global chaos synchronization for modified Chua’s circuit systems via linear state error feedback control. In: Proceedings of the 27th Chinese Control Conference, July 2006, Kuming Yunnan, China
Hua, C., Ge, C., Guan, X.: Synchronization of chaotic Lur’e systems with time-delays using sampled-data control. IEEE Trans. Neural Netw. Learn. Syst. 26(6), 1214–1221 (2015)
Martinez-Guerra, R., and Mata Machuka, J.L.: An observer for the synchronization of chaotic Liouvillian systems: a real-time application to Chua’s oscillator. In: 51st IEEE Conference on Decision and Control, Maui, Hawai, USA (2012)
Rigatos, G.: A chaotic communication system of improved performance based on the derivative-free nonlinear Kalman filter. Int. J. Syst. Sci. 47(3), 2152–2168 (2016)
Xiao, M., Cao, J.: Synchronization of a chaotic electronic circuit system, with cubic term via adaptive feedback control. Commun. Nonlinear Sci. Numer. Simul. 14, 3379–3388 (2009)
Yassen, M.T.: Adaptive control and synchronization of a modified Chua’s circuit system. Appl. Math. Comput. 136, 113–126 (2005)
Zuppa, L.A.: Direct adaptive control design and synchronization of Chua’s circuits. In: 2003 European Control Conference, Cambridge, UK (2003)
Li, X.J., Yang, G.H.: FLS-based adaptive synchronization control of complex dynamical networks with nonlinear couplings and state-dependent uncertainties. IEEE Trans. Cybern. 46(1), 171180 (2016)
Siderskiy, V., Kapila, V.: Parameter matching using adaptive synchronization of two Chua’s oscillators. In: American Control Conference, June 2014. Portland, Oregon, USA (2014)
Salarieh, H., Alasty, A.: Adaptive chaos synchronization in Chua’s systems with noisy parameters. Math. Comput. Simul. 79, 233–241 (2008)
Yan, J.J., Lin, J.S., Liao, T.L.: Synchronization of a modified Chua’s circuit system via adaptive sliding-mode control. Chaos Solitons Fractals 36, 45–52 (2008)
Agiza, H.N., Matouk, A.E.: Adaptive synchronization of Chua’s circuits with fully unknown parameters. Chaos Solitons Fractals 28, 219–227 (2006)
Batmart, T., Niamsup, P.: Adaptive control and synchronization of the perturbed Chua’s system. Math. Comput. Simul. 75, 37–55 (2007)
Zhang, T., Feng, G.: Output tracking and synchronization of chaotic Chua’s circuit with disturbances via model predictive regulator. Chaos Solitons Fractals 39, 810–820 (2009)
Li, S., Yu, D., Chen, H., Cheng, H., Zou, X.: Spontaneous synchronization of two Chua’s circuits based on coupled memristors. In: 14th International Conference on Control Automation Robotics and Vision. ICARCV 2016, Phuket, Thailand (2016)
Niu, H., Zhang, G., and Wang, J.: Chaos synchronization of Chua’s circuit and Lorenz system based on strictly positive realness. In: Proceedings of the 33rd Chinese Control Conference, July 2014, Nanjing, China
Lu, J.G.: Multiple access chaotic digital communications based on generalized synchronization. Chaos Solitons Fractals 25, 221–227 (2005)
Mkouar, H., Boubaker, O.: Chaos synchronization for master-slave piecewise linear systems: applications to Chua’s circuit. Commun. Nonlinear Sci. Numer. Simul. 17, 1292–1302 (2012)
Yang, J., Zhao, L.: Bifurcation analysis and chaos control of the modified Chua’s circuit system. Chaos Solitons Fractals 77, 332–344 (2015)
Rigatos, G.: Modelling and Control for Intelligent Industrial Systems: Adaptive Algorithms in Robotcs and Industrial Engineering. Springer, Berlin (2011)
Rigatos, G.: Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons. Springer, Berlin (2013)
Rigatos, G.: Nonlinear Control and Filtering Using Differential Flatness Approaches: Applications to Electromechanicsl Systems. Springer, Berlin (2015)
Rigatos, G.: Intelligent Renewable Energy Systems: Modelling and Control. Springer, Berlin (2017)
Rigatos, G.: State-Space Approaches for Modelling and Control in Financial Engineering: Systems Theory and Machine Learning Methods. Springer, Berlin (2017)
Rigatos, G.G., Tzafestas, S.G.: Extended Kalman filtering for fuzzy modelling and multi-sensor fusion. Math. Comput. Model. Dyn. Syst. 13, 251–266 (2007)
Basseville, M., Nikiforov, I.: Detection of Abrupt Changes: Theory and Applications. Prentice-Hall, Upper Saddle River (1993)
Rigatos, G., Zhang, Q.: Fuzzy model validation using the local statistical approach. Fuzzy Sets Syst. 60(7), 882–904 (2009)
Granat, R., Kagstrom, B., Kressner, D.: A parallel Schur method for solving continuous-time algebraic Riccati equations. In: 2008 IEEE International Conference on Computer-Aided Control Systems. San Antonio, Texas (2008)
Benner, P., Bujanovi, Z., Krschner, P., Saak, J.: A numerical comparison of different solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems. SIAM J. Sci. Comput. 42(2), A957–A996 (2020)
Toussaint, G.J., Basar, T., and Bullo, F.: \(H_{\infty }\) optimal tracking control techniques for nonlinear underactuated systems. In: Proceedings of the IEEE CDC 2000, 39th IEEE Conference on Decision and Control, Sydney Australia (2000)
Lublin, L., Athans, M.: An experimental comparison of and designs for interferometer testbed. In: Francis, B., Tannenbaum, A. (eds.) Lectures Notes in Control and Information Sciences: Feedback Control, Nonlinear Systems and Complexity, pp. 150–172. Springer, Berlin (1995)
Chithra, A., Mohamed, R.: Synchronization and chaotic communication in nonlinear circuits with nonlinear coupling. J. Comput. Electron. 16, 833–844 (2017)
Chithra, A., Raja Mohamed, I., Rizwana, R.: Observation of chaotic and strange nonchaotic attractors in a simple multi-scroll system. J. Comput. Electron. 17, 51–80 (2017)
Hossam, M.E., Hammuda, M.: A new approach for constrained chaos synchronization with application to secure data communication. J. Frankl. Inst. 356, 6697–6723 (2019)
Tian, K., Ren, H.P., Bai, C.: Synchronization of hyperchaos with time-delay using impulsive control. IEEE Acces 8, 72570–72576 (2020)
Feketa, P., Schauer, A., Meurer, T., Michaelis, D., Ochs, K.: Synchronization of nonlinearly coupled networks of Chua oscillators. In: 11th IFAC Symposium on Nonlinear Control Systems. IFAC NOLCOS 2019, Austria, Vienna (2019)
Yao, Z., Zhou, P., Alsaedi, A., Ma, J.: Energy flow-guided synchronization between chaotic circuits. Appl. Math. Comput. 374, 124998 (2020)
Duan, W., Li, Y., Sun, Y., Shen, J., Yan, X.: Enhanced master–slave synchronization criteria for chaotic Lur’e systems based on time-delayed feedback control. Math. Comput. Simul. 177, 276–294 (2020)
Wu, T., Park, J.H., Xiong, L., Xie, X., Zhang, H.: A novel approach to synchronization conditions for delayed chaoic Lur’e systems with state sampled-data quantized controller. J. Frankl. Inst. 957, 9811–9833 (2020)
Shi, K., Wang, J., Zhang, S., Tang, Y., Cheng, C.: Hybrid-driven finite-time \(H_{\infty }\) sampling synchronization control for coupling memory complex networks with stochastic cyber-attacks. Neurocomputing 387, 241–254 (2020)
Kim, J., Kim, H.: Synchronization of Lur’e type nonlinear systems in linear dynamical networks having fast convergence rate and large DC gains. Syst. Control Lett. 138, 104641 (2020)
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Rigatos, G., Abbaszadeh, M. Nonlinear optimal control and synchronization for chaotic electronic circuits. J Comput Electron 20, 1050–1063 (2021). https://doi.org/10.1007/s10825-021-01655-1
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DOI: https://doi.org/10.1007/s10825-021-01655-1