Spectral Entropy Analysis and Synchronization of a Multi-Stable Fractional-Order Chaotic System using a Novel Neural Network-Based Chattering-Free Sliding Mode Technique

Abstract An immense body of research has focused on chaotic systems, mainly because of their interesting applications in a wide variety of fields. A comprehensive understanding and synchronization of chaotic systems play pivotal roles in practical applications. To this end, the present study investigates a multi-stable fractional-order chaotic system. Firstly, some dynamical features of the system are described, and the chaotic behaviour of the system is verified. Then, both spectral entropy and spectral Min-Entropy are computed, and the phenomenon of multi-stability is shown. Besides, the combination of a new chattering-free robust sliding mode controller with a neural network observer is proposed for the synchronization of the fractional-order system. With the neural network estimator, unknown functions of the system are obtained, and the effects of disturbances are completely taken into account. Also, based on the Lyapunov stability theorem, the asymptotical stability of the closed-loop system is confirmed. Lastly, the proposed control technique is applied to the fractional-order system. Numerical results demonstrate the chattering-free and effective performance of the proposed control method for uncertain systems in the presence of unknown time-varying external disturbances.

[1]  Yuming Chu,et al.  Recurrent Neural Network-Based Robust Nonsingular Sliding Mode Control With Input Saturation for a Non-Holonomic Spherical Robot , 2020, IEEE Access.

[2]  Viet-Thanh Pham,et al.  Entropy Analysis and Neural Network-Based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors , 2019, Entropy.

[3]  Moosa Ayati,et al.  Stabilization of nonlinear vibrations of carbon nanotubes using observer-based terminal sliding mode control , 2020, Trans. Inst. Meas. Control.

[4]  Hadi Jahanshahi,et al.  Smooth control of HIV/AIDS infection using a robust adaptive scheme with decoupled sliding mode supervision , 2018, The European Physical Journal Special Topics.

[5]  Oscar Castillo,et al.  A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method , 2020, Appl. Soft Comput..

[6]  Nikolay V. Kuznetsov,et al.  Hidden attractors in Dynamical Systems. From Hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits , 2013, Int. J. Bifurc. Chaos.

[7]  Hadi Jahanshahi,et al.  A financial hyperchaotic system with coexisting attractors: Dynamic investigation, entropy analysis, control and synchronization , 2019, Chaos, Solitons & Fractals.

[8]  Abdon Atangana,et al.  Strange chaotic attractors under fractal-fractional operators using newly proposed numerical methods , 2019, The European Physical Journal Plus.

[9]  Abdon Atangana,et al.  Fractional stochastic modelling illustration with modified Chua attractor , 2019 .

[10]  Dumitru Baleanu,et al.  Fractional modeling of blood ethanol concentration system with real data application. , 2019, Chaos.

[11]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[12]  Anita Alaria,et al.  Applications of Fractional Calculus , 2018 .

[13]  Pengjian Shang,et al.  Permutation complexity and dependence measures of time series , 2013 .

[14]  Jesus M. Munoz-Pacheco,et al.  A fractional-order hyper-chaotic economic system with transient chaos , 2020 .

[15]  Vadim I. Utkin,et al.  On multi-input chattering-free second-order sliding mode control , 2000, IEEE Trans. Autom. Control..

[16]  Yu-Ming Chu,et al.  The effect of market confidence on a financial system from the perspective of fractional calculus: Numerical investigation and circuit realization , 2020 .

[17]  Jiakun Zhao,et al.  A new impulsive synchronization of Chen hyper-chaotic system and Lü hyper-chaotic system , 2013 .

[18]  G. Bartolini,et al.  Chattering avoidance by second-order sliding mode control , 1998, IEEE Trans. Autom. Control..

[19]  Phillip P. A. Staniczenko,et al.  Rapidly detecting disorder in rhythmic biological signals: a spectral entropy measure to identify cardiac arrhythmias. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Behzad Ghanbari,et al.  A new application of fractional Atangana–Baleanu derivatives: Designing ABC-fractional masks in image processing , 2020 .

[21]  Irene M. Moroz,et al.  Hopf bifurcation and synchronization of a five-dimensional self-exciting homopolar disc dynamo using a new fuzzy disturbance-observer-based terminal sliding mode control , 2020, J. Frankl. Inst..

[22]  Marta Borowska,et al.  Entropy-Based Algorithms in the Analysis of Biomedical Signals , 2015 .

[23]  D A Steyn-Ross,et al.  Cortical entropy changes with general anaesthesia: theory and experiment. , 2004, Physiological measurement.

[24]  Amin Yousefpour,et al.  Disturbance observer–based terminal sliding mode control for effective performance of a nonlinear vibration energy harvester , 2020 .

[25]  Sundarapandian Vaidyanathan,et al.  Advances and Applications in Chaotic Systems , 2016, Studies in Computational Intelligence.

[26]  Shaobo He,et al.  Synchronization of fractional time-delayed financial system using a novel type-2 fuzzy active control method , 2020 .

[27]  Hadi Jahanshahi,et al.  Fast disturbance-observer-based robust integral terminal sliding mode control of a hyperchaotic memristor oscillator , 2019, The European Physical Journal Special Topics.

[28]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[29]  Xiao Yan,et al.  Fuzzy neural network control algorithm for asymmetric building structure with active tuned mass damper , 2020 .

[30]  Zhaojun Li,et al.  A review of entropy measures for uncertainty quantification of stochastic processes , 2019, Advances in Mechanical Engineering.

[31]  Jun Yang,et al.  Disturbance Observer-Based Control: Methods and Applications , 2014 .

[32]  Hadi Jahanshahi,et al.  King algorithm: A novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems , 2020 .

[33]  Abdon Atangana,et al.  Modeling attractors of chaotic dynamical systems with fractal–fractional operators , 2019, Chaos, Solitons & Fractals.

[34]  Jinkun Liu,et al.  Radial Basis Function (RBF) Neural Network Control for Mechanical Systems , 2013 .

[35]  S. Ghasemi,et al.  Application of Fractional Calculus Theory to Robust Controller Design for Wind Turbine Generators , 2014, IEEE Transactions on Energy Conversion.

[36]  Fawaz E. Alsaadi,et al.  A new fractional-order hyperchaotic memristor oscillator: Dynamic analysis, robust adaptive synchronization, and its application to voice encryption , 2020, Appl. Math. Comput..

[37]  Brian Davies.  Exploring Chaos: Theory And Experiment , 1999 .

[38]  Akif Akgul,et al.  Complete analysis and engineering applications of a megastable nonlinear oscillator , 2018, International Journal of Non-Linear Mechanics.

[39]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[40]  Karthikeyan Rajagopal,et al.  Coexisting attractors in a fractional order hydro turbine governing system and fuzzy PID based chaos control , 2020, Asian Journal of Control.

[41]  Devendra Kumar,et al.  Editorial: Fractional Calculus and Its Applications in Physics , 2019, Front. Phys..

[42]  Edris Pouresmaeil,et al.  Finite-Time Disturbance-Observer-Based Integral Terminal Sliding Mode Controller for Three-Phase Synchronous Rectifier , 2020, IEEE Access.

[43]  Seda İğret Araz,et al.  New numerical approximation for Chua attractor with fractional and fractal-fractional operators , 2020 .

[44]  Lei Guo,et al.  Disturbance-Observer-Based Control and Related Methods—An Overview , 2016, IEEE Transactions on Industrial Electronics.

[45]  Yu-Ming Chu,et al.  Antiretroviral therapy of HIV infection using a novel optimal type-2 fuzzy control strategy , 2020 .

[46]  Stefania Santini,et al.  A combined neural network and model predictive control approach for ball transfer unit–magnetorheological elastomer–based vibration isolation of lightweight structures , 2020 .

[47]  Dumitru Baleanu,et al.  Stability of q-fractional non-autonomous systems , 2013 .

[48]  Richard L Magin,et al.  Fractional calculus in bioengineering, part 2. , 2004, Critical reviews in biomedical engineering.

[49]  Neville J. Ford,et al.  Comparison of numerical methods for fractional differential equations , 2006 .

[50]  T. Inouye,et al.  Quantification of EEG irregularity by use of the entropy of the power spectrum. , 1991, Electroencephalography and clinical neurophysiology.

[51]  Yu-Ming Chu,et al.  Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method , 2020, Entropy.

[52]  Kehui Sun,et al.  Fractional symbolic network entropy analysis for the fractional-order chaotic systems , 2020, Physica Scripta.

[53]  Yongli Liu,et al.  A new Barbalat's lemma and Lyapunov stability theorem for fractional order systems , 2017, 2017 29th Chinese Control And Decision Conference (CCDC).

[54]  Shubin Si,et al.  The Entropy Algorithm and Its Variants in the Fault Diagnosis of Rotating Machinery: A Review , 2018, IEEE Access.

[55]  O. Rosso,et al.  Permutation min-entropy: An improved quantifier for unveiling subtle temporal correlations , 2015 .

[56]  Qamar Din.  Stability, bifurcation analysis and chaos control for a predator-prey system , 2018, Journal of Vibration and Control.

[57]  C. Mira,et al.  Chaotic Dynamics: From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism , 1987 .

[58]  Dong Yan-qin.  Applications of Spectral Entropy and Wavelet Entropy Algorithm for Structure Complexity Analysis of Chaotic Sequence , 2014 .

[59]  Hee-Jun Kang,et al.  Adaptive Neural Integral Full-Order Terminal Sliding Mode Control for an Uncertain Nonlinear System , 2019, IEEE Access.

[60]  Mehdi Dalir,et al.  Applications of Fractional Calculus , 2010 .

[61]  Sundarapandian Vaidyanathan,et al.  Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system , 2015, Int. J. Model. Identif. Control..

[62]  Reza Ghaderi,et al.  Sliding mode synchronization of an uncertain fractional order chaotic system , 2010, Comput. Math. Appl..

[63]  Jianfei Huang,et al.  The Grünwald-Letnikov method for fractional differential equations , 2011, Comput. Math. Appl..