Abstract
This paper mainly focuses on the robust synchronization of switched fractional-order improved Rikitake two-disk dynamo system which is a more general topology of the original Rikitake system. In particular, we first explore and design a simplified model very close to a real geophysical process which is described by an orbiting point, primarily using concepts from dynamical system theory. Dynamical properties, including symmetry, dissipation, stability of equilibria, Lyapunov exponents, and bifurcation, are analyzed on the basis of theoretical analysis and numerical simulation. Secondly, some less stringent conditions for adaptive control are derived when the system is subjected to external disturbances, parametric uncertainties and nonlinear input. The convergence of the state space trajectories with the overall robust asymptotic stability of the closed-loop system is obtained using the Lyapunov theory. To highlight our contribution, an illustrative example is presented to show the effectiveness and applicability of the proposed synchronization scheme. This proposal derives a systematic procedure for the design and control of a wide range of electromechanical systems. The results obtained in this work have not yet been reported in the literature to the best of the authors’ knowledge and thus deserve dissemination.
Similar content being viewed by others
References
Matignon D (1996) Stability results for fractional differential equations with applications to control processing. Comput Eng Syst Appl 2:963–968
Wang Y (2018) Dynamic analysis and synchronization of conformable fractional-order chaotic systems. Eur Phys J Plus. https://doi.org/10.1140/epjp/i2018-12300-y
Li C, Deng W (2007) Remarks on fractional derivatives. Appl Math Comput 187:777–784. https://doi.org/10.1016/j.amc.2006.08.163
Petréas I (2011) Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media, Berlin
Mahmoud GM, Mansour EA, Tarek MAE (2016) Active control technique of fractional-order chaotic complex systems. Eur Phys J Plus. https://doi.org/10.1140/epjp/i2016-16200-x
Heaviside O (2008) Electromagnetic theory, cosimo. Cambridge University, Cambridge
Montesinos-Garcia JJ, Guerra RM (2017) A fractional exponential polinomial state observer in secure communications. In: 2017 14th International conference on electrical engineering, computing science and automatic control (CCE), pp 1–6. IEEE
Hilfer R (2000) Applications of fractional calculus in physics. World Scientific
Xu Y, Wang H, Li Y, Pei B (2014) Image encryption based on synchronization of fractional chaotic systems. Commun Nonlinear Sci Numer Simul 19:3735–3744. https://doi.org/10.1016/j.cnsns.2014.02.029
Li C, Chen G (2004) Chaos in the fractional order chen system and its control. Chaos Soli Fract 22:549–554. https://doi.org/10.1016/j.chaos.2004.02.035
Pourmahmood MA (2012) Robust finite-time stabilization of fractional-order chaotic systems based on fractional Lyapunov stability theory. J Comput Nonlinear Dyn 7:021010. https://doi.org/10.1115/1.4005323
Atanackovic T, Budincevic M, Pilipovic S (2005) On a fractional distributed-order oscillator. J Phys A Math Gen 38:6703. https://doi.org/10.1088/0305-4470/38/30/006
Tavazoei MS, Haeri M (2007) A necessary condition for double scroll attractor existence in fractional-order systems. Phys Lett A 367:102–113. https://doi.org/10.1016/j.physleta.2007.05.081
Li C (2006) Projective synchronization in fractional order chaotic systems and its control. Prog Theor Phys 115(3):661–666
Peng Y, Sun K, He S (2020) Synchronization for the integer-order and fractional-order chaotic maps based on parameter estimation with JAYA-IPSO algorithm. Eur Phys J Plus. https://doi.org/10.1140/epjp/s13360-020-00340-9
Berwald L (1947) Ueber Systeme von gew´lohnlichen Differentialgleichungen zweiter Ordnung deren Integralkurven mit dem System der geraden Linien topologisch aequivalent sind Ann. Math 48:193–215
Bringuier E (2003) Electrostatic charges in v*B fields and the phenomenon of induction. Eur J Phys 24:21
Rikitake T (1958) Oscillations of a system of disk dynamos. Proc Camb Phil Soc 54:89–105. https://doi.org/10.1017/S0305004100033223
Tudoran RM, Girban A (2010) A hamiltonian look at the Rikitake two-disk dynamo system. Nonlinear Anal Real World Appl 11:2888–2895. https://doi.org/10.1016/j.nonrwa.2009.10.012
Wei Z, Zhang W, Wang Z, Yao M (2015) Hidden attractors and dynamical behaviors in an extended Rikitake system. Int J Bifur Chaos 25:1550028. https://doi.org/10.1142/S0218127415500285
Cortini M, Barton CC (1994) Chaos in geomagnetic reversal records: a comparison between Earth’s magnetic field data and model disk dynamo data. J Geophys Res Solid Earth 99(B9):18021–18033. https://doi.org/10.1029/94jb01237
Gholipour Y, Mola M (2014) Investigation stability of Rikitake system. J Contr Eng Tech 4:82–85
Harb A, Ayoub N (2013) Nonlinear control of chaotic Rikitake two-disk dynamo. Int J Nonlinear Sci 15:45–50
Javid M, Nyamoradi N (2013) Numerical chaotic behavior of the fractional Rikitake system. World J Model Simul 9:120–129
Al-khedhairi A (2020) Dynamical analysis and chaos synchronization of a fractional-order novel financial model based on Caputo-Fabrizio derivative. Eur Phys J Plus 134:532–552. https://doi.org/10.1140/epjp/i2019-12878-4
Xiang-Jun W, Jing-Sen L, Guan-Rong C (2008) Chaos synchronization of Rikitake chaotic attractor using the passive control technique. Nonlinear Dyn 53:45–53. https://doi.org/10.1007/s11071-007-9294-2
Xiao-jun L, Xian-feng L, Ying-xiang C, Jian-gang Z (2008) Chaos and chaos synchronism of the Rikitake two-disk dynamo. In: 2008 Fourth international conference on natural computation. 4, pp 613–617
Wu X, Wang H (2010) A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn 61(3):407–417. https://doi.org/10.1007/s11071-010-9658-x
Vembarasan V, Balasubramaniam P (2013) Chaotic synchronization of Rikitake system based on TS fuzzy control techniques. Nonlinear Dyn 74:31–44
Pang W, Wu Z, Xiao Y, Jiang C (2020) Chaos control and synchronization of a complex rikitake dynamo model. Entropy 22(6):671. https://doi.org/10.3390/e22060671
Wang Y, Lei T, Zhang X, Li C, Jafari S (2020) Hyperchaotic oscillation in the deformed Rikitake two-disc dynamo system induced by memory effect. Complexity. https://doi.org/10.1155/2020/8418041
Weidman CD, Krider EP (1985) The amplitude spectra of lightning radiation fields in the interval from 1 to 20MHz. Radio Sci 21(6):57–60. https://doi.org/10.1029/RS021i006p00964
Podgorski AS, Dunn J, Yeo R (1991) Study of picosecond rise time in human-generated ESD. Proc. IEEE Int. Symp. on EMC, Cherry Hill, NJ, August 13–15
Podgorski AS (2003) Sources of electromagnetic disturbances and emi/emc/emp testing methods in the frequency range up to 100 Ghz. In: XIII International conference on electromagnetic disturbances. pp 1–6
Kammogne STA, Kengne R, Fotsin HB (2017) Dynamics and improved robust adaptive control strategy for the finite time synchronization of uncertain nonlinear systems. Int J Syst Dyn Appl 6(4):34–62
Alain KST, Azar AT, Kengne R, Fotsin HB (2020) Stability analysis and robust synchronization of fractional order modified Colpitts oscillators. Int J Auto Cont 14(1):52–79
Lin H, Antsaklis PJ (2009) Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans Automat Control 54(2):308–322
Tanwani A, Liberzon D (2008) Invertibility of nonlinear switched systems. In: Proc. 47th IEEE Conf. on decision and control, pp 286–291
Millerioux G, Daafouz J (2004) Input independent chaos synchronization of switched systems. IEEE Trans Autom Cont 49:1182–1187. https://doi.org/10.1109/TAC.2004.831118
Yu W, Cao J, Yuan K (2008) Synchronization of switched system and application in communication. Phys Lett A 372:4438–4445. https://doi.org/10.1016/j.physleta.2008.04.030
Rui P, Lei Z (2017) Synchronization of nonlinear switched systems based on sampled-data. 36th Chinese Control Conference (CCC)
Park JH, Lee TH (2014) Finite-time adaptive synchronization of one side switching chaotic systems. Comput Sci Comput Intel 2:303–304. https://doi.org/10.1109/CSCI.2014.145
Kammogne STA, Kountchou MN, Kengne R et al (2020) Polynomial robust observer implementation based passive synchronization of nonlinear fractional-order systems with structural disturbances. Frontiers Inf Technol Electron Eng 21(9):1369–1386. https://doi.org/10.1361/FITEE.1900430
Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1):3–22. https://doi.org/10.1023/A:1016592219641
Li C, Hu W, Sprott JC, Wang X (2015) Multistability in symmetric chaotic systems. Eur Phys J Spec Top 224(8):1493–1506. https://doi.org/10.1140/epjst/e2015-02475-x
Sachin B, Varsha DG (2011) A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. J Fract Calc Appl 1(5):1–9
Buscarino A, Fortuna L, Frasca M, Sciuto G (2014) A concise guide to chaotic electronic circuits. Springer, Berlin. https://doi.org/10.1007/978-3-319-05900-6
Hammouch Z, Mekkaoui T (2018) Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system. Complex Intell Syst 4:251–260. https://doi.org/10.1007/s40747-018-0070-3
Liu CX (2011) Fractional-order chaotic circuit theory and applications. Xian Jiaotong University Press, Xian
Daafouz J, Riedinger P, Iung C (2002) Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans Autom Control 47(11):1883–1887. https://doi.org/10.1109/tac.2002.804474
Li J, Yang X, Wu J (2018) Adaptive tracking control approach with prespecified accuracy for uncertain nonlinearly parameterized switching systems. IEEE Access 6:3786–3793. https://doi.org/10.1109/ACCESS.2017.2788446
Marino R, Tomei P (1996) Nonlinear control design: geometric adaptive and robust. Prentice Hall International (UK) Ltd, UK
Wang F, Yang Y (2017) Correction: fractional order barbalat’s lemma and its applications in the stability of fractional order nonlinear systems. Math Model Anal 22(4):503–513. https://doi.org/10.3846/13926292.2017.1329755
Ding SX (2008) Model-based faults diagnosis techniques, design schemes, algorithms, and tools. Springer, Berlin. https://doi.org/10.1007/978-3-540-76304-8
Xiong Z, Qu S, Luo J (2019) Adaptive multi-switching synchronization of high-order memristor-based hyperchaotic system with unknown parameters and its application in secure communication. Complexity 2019:1–18. https://doi.org/10.1155/2019/3827201
Shafiq M, Ahmad I (2019) Multi-switching combination anti-synchronization of unknown hyperchaotic systems. Arab J Sci Eng. https://doi.org/10.1007/s13369-019-03824-8
Khan A, Budhraja M, Ibraheem A (2018) Multi-switching synchronization of four non-identical hyperchaotic systems. Int J Appl Comput Math 4:71. https://doi.org/10.1007/s40819-018-0503-0
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kammogne, A.S.T., Nyiembui, T.P. & Kengne, R. Adaptive observer based-robust synchronization of switched fractional Rikitake systems with input nonlinearity. Int. J. Dynam. Control 10, 162–179 (2022). https://doi.org/10.1007/s40435-021-00796-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-021-00796-2