Parameter estimation and synchronization of hyper chaotic Lu system with disturbance input and uncertainty using two under-actuated control signals

Abstract

In this paper, synchronization of two hyper chaotic Lu systems with unknown factors, such as uncertainty and disturbance, and less control signals with respect to system’s dimensions by applying an adaptive sliding mode controller is presented. In this research, firstly, an integral sliding mode controller is designed for synchronization of two hyper chaotic Lu systems with known parameters. Secondly, a controller is used to synchronize two hyper chaotic Lu systems with unknown parameters; in this case, unknown parameters are estimated using an adaptive rule. As a measure of performance, stability of the proposed control method is proved using the Lyapunov stability method. Numerical simulations validate the stabilization and synchronization.

Keywords

Chaos and uncertainty hyper chaotic Lu system adaptive control parameter estimation

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References

Ahmad

et al . (2016) The synchronization of chaotic systems with different dimensions by a robust generalized active control. Optik-International Journal for Light and Electron Optics 127(11): 4859–4871.

Ahmad

(2015) A research on the synchronization of two novel chaotic systems based on a nonlinear active control algorithm. Engineering, Technology & Applied Science Research 5(1): 739–747.

Almatroud Othman

Noorani

MSM

Al-Sawalhab

(2016) Dual synchronization of chaotic and hyperchaotic systems. Journal of Nonlinear Sciences and Applications 9: 4666–4677.

et al . (2012) Design of a new multistage chaos synchronized system for secure communications and study on noise perturbation. Mathematical and Computer Modelling 54(1–2): 7–18.

Barrio

Angeles Martínez

Serrano

Wilczak

(2015) When chaos meets hyperchaos: 4D Rossler model. Physics Letters A 379: 2300–2305.

Daltzis

et al . (2018) Analysis, synchronization and circuit design of a 4D Hyperchaotic hyperjerk system. Computation 6(1): 1–19.

Durdu

Turan Özcerit

(2015) A novel chaotic system for secure communication applications. Information Technology and Control 44(3): 271–278.

Elabbasy

Agiza

EL-Dessoky

(2006) Adaptive synchronization of a hyperchaotic system with uncertain parameter. Chaos, Solitons and Fractals 30(5): 1133–1142.

Ferreira

Jr Lino

Porto

(1999) Uncovering oscillations, complexity, and chaos in chemical kinetics using mathematica. Journal of Chemical Education 76(6): 861–866.

10.

Gaspard

(1999) Microscopic chaos and chemical reactions. Physica A 263(1–4): 315–328.

11.

Huang

Wang

(2014) Synchronization and Lag Synchronization of Hyperchaotic Memristor-Based Chua’s Circuits, Mathematical Problems in Engineering 2014: 1–7.

12.

Jiang

G-Y

Huang

(2013) Chaos and its control in semiconductor laser with delayed negative optoelectronic feedback. Turkish Journal of Physics 37: 296–303.

13.

Kandangath

Krishnamoorthy

Ying-Cheng

Gaudet

(2007) Inducing chaos in electronic circuits by resonant perturbations. IEEE Transactions on Circuits and Systems I 54(5): 1109–1119.

14.

Kuo

C-L

(2011) Design of a fuzzy sliding-mode synchronization controller for two different chaos systems. Computers & Mathematics with Applications 61(8): 2090–2095.

15.

S-Y

et al . (2013) Chaotic motions in the real fuzzy electronic circuits. Journal of Abstract and Applied Analysis: 2013: 1–8.

16.

Martínez-Guerra

Pérez-Pinacho

Gómez-Cortés

(2015) Synchronization of Integral and Fractional Order Chaotic Systems. Switzerland: Springer.

17.

et al . (2015) Chaos suppression for a four-dimensional fundamental power system model using adaptive feedback control. Transactions of the Institute of Measurement and Control 39(2): 194–207.

18.

Pecora

Carroll

(1990) Synchronization in chaotic systems. Physical Review Letters 64(8–19): 821–825.

19.

Ren

H-P

Bai

Huang

Z-Z

Grebogi

(2017) Secure communication based on hyperchaotic chen system with time-delay. International Journal of Bifurcation and Chaos 27(5): 15.

20.

Sandr

(1996) Numerical calculation of Lyapunov exponents. The Mathematica Journal, 6(3): 78–84.

21.

Sun

(2012) Hyperchaos and hyperchaos control of the sinusoidally forced simpliﬁed Lorenz system. Nonlinear Dynamics, 69(3): 1383–1391.

22.

Vaidyanathan

(2013) Adaptive design of controller and synchronizer for Lu-xiao chaotic system with unknown parameters. International Journal of Computer Science & Information Technology (IJCSIT) 5(1): 197–210.

23.

Wang

Dong

(2015) Secure Communication based on a hyperchaotic system with disturbances. Journal of Mathematical Problems in Engineering 2015: 7.

24.

Wang

Zhong

Dong

(2016) On the novel chaotic secure communication scheme design. Communications in Nonlinear Science and Numerical Simulation 39: 108–117.

25.

Wang

Zhang

Zheng

(2006) A new modified hyperchaotic Lu system. Physica A: Statistical Mechanics and its Applications 371(2): 260–272.

26.

Wang

(2015) Impulsive Synchronization of hyperchaotic Lü systems with two methods. International Journal of Modern Nonlinear Theory and Application 4(1): 1–9.

27.

Wei

Zhang

(2014) Hidden hyperchaotic attractors in a modiﬁed Lorenz–Stenﬂo system with only one stable equilibrium. International Journal of Bifurcation and Chaos 24(10): 14.

28.

Wei

Zhang

(2015a) Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system. Nonlinear Dynamics 82(1–2): 131–141.

29.

Wei

Zhang

Wang

Yao

(2015b) Hidden attractors and dynamical behaviors in an extended rikitake system. International Journal of Bifurcation and Chaos 25(2): 1–11.

30.

Wei

Wang

Liu

(2014) A new ﬁnding of the existence of hyperchaotic attractors with no equilibria. Mathematics and Computers in Simulation, 100: 13–23.

31.

Wei

et al . (2016) Dynamics at inﬁnity, degenerate hopf and zero-hopf bifurcation for Kingni–Jafari system with hidden attractors. International Journal of Bifurcation and Chaos 26(7): 1–16.

32.

Kurths

(2015) A secure communication scheme based generalized function projective synchronization of a new 5D hyperchaotic syste. Physica Scripta 90(4): 1–12.

33.

Jia

Zhao

(2015) Maximum wind power tracking for PMSG chaos systems – ADHDP method. Applied Soft Computing 36: 204–209.

34.

Xiong

et al . (2016) Design and hardware implementation of a new chaotic secure communication technique. PLoS ONE 11(8): e0158348; 1–19; doi: 10.1371/journal.pone.0158348

35.

Mobayen

Ren

Jafari

(2017) Robust finite-time synchronization of a class of chaotic systems via adaptive global sliding mode control. Journal of Vibration and Control 1–13. doi: 10.1177/1077546317713532.

36.

Zang

et al . (2016) Applications of chaotic dynamics in robotics. International Journal of Advanced Robotic Systems 13(2): 1–17. doi:10.5772/62796

37.

Zhao

J, DZ

(2012) A new impulsive synchronization of chen hyper-chaotic systm and Lu hyper-chaotic system. Journal of Vibration and Control 19(12): 1773–1778.

38.

Zhao

Y-B

Zhang

X-Z

Sun

X-Y

(2014) Adaptive sliding mode output synchronization tracking for hyper-chaotic Lü system based on adaptive PWL filters. Applied Mechanics and Materials: 644: 3610–3615.