Entropy 2019, 21(2), 156; https://doi.org/10.3390/e21020156
Entropy Analysis and Neural Network-Based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors
Hadi Jahanshahi 1
, Maryam Shahriari-Kahkeshi 2, Raúl Alcaraz 3, Xiong Wang 4, Vijay P. Singh 5 and Viet-Thanh Pham 6,*
, Maryam Shahriari-Kahkeshi 2, Raúl Alcaraz 3, Xiong Wang 4, Vijay P. Singh 5 and Viet-Thanh Pham 6,*
1
Department of Aerospace Engineering, Faculty of New Sciences and Technologies, University of Tehran, Tehran 14395-1561, Iran
2
Faculty of Engineering, Shahrekord University, Shahrekord 64165478, Iran
3
Research Group in Electronic, Biomedical and Telecommunication Engineering, University of Castilla-La Mancha (UCLM), 16071 Cuenca, Spain
4
Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
5
Department of Biological & Agricultural Engineering and Zachry Department of Civil Engineering, Texas A&M University, 2117 TAMU, College Station, TX 77843, USA
6
Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
*
Author to whom correspondence should be addressed.
Received: 21 January 2019 / Revised: 3 February 2019 / Accepted: 4 February 2019 / Published: 7 February 2019
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)
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Abstract
Today, four-dimensional chaotic systems are attracting considerable attention because of their special characteristics. This paper presents a non-equilibrium four-dimensional chaotic system with hidden attractors and investigates its dynamical behavior using a bifurcation diagram, as well as three well-known entropy measures, such as approximate entropy, sample entropy, and Fuzzy entropy. In order to stabilize the proposed chaotic system, an adaptive radial-basis function neural network (RBF-NN)–based control method is proposed to represent the model of the uncertain nonlinear dynamics of the system. The Lyapunov direct method-based stability analysis of the proposed approach guarantees that all of the closed-loop signals are semi-globally uniformly ultimately bounded. Also, adaptive learning laws are proposed to tune the weight coefficients of the RBF-NN. The proposed adaptive control approach requires neither the prior information about the uncertain dynamics nor the parameters value of the considered system. Results of simulation validate the performance of the proposed control method.Keywords:
Non-equilibrium four-dimensional chaotic system; entropy measure; adaptive approximator-based control; neural network; uncertain dynamics
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Jahanshahi, H.; Shahriari-Kahkeshi, M.; Alcaraz, R.; Wang, X.; Singh, V.P.; Pham, V.-T. Entropy Analysis and Neural Network-Based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors. Entropy 2019, 21, 156.
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