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Sliding Mode Control on Finite-Time Synchronization of Nonlinear Hyper-mechanical Fractional Systems

  • Research Article-Systems Engineering
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Abstract

This study delves into finite-time projective synchronization within a thermal mechanical fractional-order system in the presence of external disruptions and uncertainties. The investigation utilizes a developed sliding mode surface and employs the Lyapunov approach to synchronize trajectories. Furthermore, the study explores three alternative fractional complex and real control rules employing Lyapunov functions, resulting in the formulation of a hybrid controller based on switching laws. Recognizing the escalating demand for efficient controllers, the research addresses the imperative to consider interruptions, uncertainties, and control input limits. To address this challenge, a novel disturbance-observer-based fractional-order sliding mode control technique is introduced, specifically designed to stabilize a fractional-order hyper-chaotic system. This method accommodates uncertainty, disturbances, and control input saturation. The research analyzes the system’s characteristics, including Lyapunov exponents, bifurcation diagrams, phase portraits, and time series of state vectors. Finally, numerical simulations underscore the potential and robustness of the proposed technique for uncertain five-dimensional systems, confirming its effectiveness.

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Abbreviations

\(\beta \) :

Define operator of fractional order

v :

Nanofluid velocity

P :

Represents that pressure

\({\tilde{\rho }}\) :

Represents density

\({\tilde{\mu }}\) :

Represents viscosity

\({\tilde{T}}\) :

Temperature

\({\tilde{D}}_{\text {B}}\) :

Defined Brownian diffusion term

\({\tilde{D}}_{{\tilde{\text {T}}}}\) :

Thermophoretic diffusion term

Pr:

Prandtl number

\(R_n\) :

Concentration of Rayleigh number

\({\mathbb {N}}_B\) :

Modified density ratio

\({\mathbb {N}}_A\) :

Modified diffusivity ratio

\({\mathbb {V}}_1\) :

Lyapunov function

\(\digamma \) :

Positive finite gain

\(\textbf{q}\) :

Velocity vector

\({{\text {Rn}}_{f}}\) :

Rescaled thermal Rayleigh number

t :

Dimensionless time components

(u, v, w):

Component of velocity vector

x, y, z :

Cartesian coordinates

References

  1. Kheshti, M.; Taghvaei, S.; Salehi, M.; Baraijany, A.M.: Bifurcation analysis and poincare’ map of a hyperchaotic system. J. Appl. Nonlinear Dyn. 12(01), 125–131 (2023)

    Article  MathSciNet  Google Scholar 

  2. Iskakova, K.; Alam, M.M.; Ahmad, S.; Saifullah, S.; Akgul, A.; Yilmaz, G.: Dynamical study of a novel 4d hyperchaotic system: an integer and fractional order analysis. Math. Comput. Simul. 208, 219–245 (2023)

    Article  MathSciNet  Google Scholar 

  3. Ozpolat, E.; Gulten, A.: A novel 4d hyperchaotic system with its dynamical analysis and synchronization. in 2023 11th International Symposium on Digital Forensics and Security (ISDFS). pp. 1–5. IEEE, (2023)

  4. Dezfuli, M.A.; Zeinoddini, M.; Nazari, S.A.; Pasand, M.S.; Harati, R.M.: A probe into the fatigue crack growth in mechanical systems with hyperchaotic/chaotic dynamics. Mech. Syst. Signal Process. 191, 110184 (2023)

    Article  Google Scholar 

  5. Kundu, P.K.; Chatterjee, S.: Nonlinear feedback synthesis and control of periodic, quasiperiodic, chaotic and hyper-chaotic oscillations in mechanical systems. Nonlinear Dyn. 111(12), 11559–11591 (2023)

    Article  Google Scholar 

  6. Mohammadi, M.; Farajpour, A.; Rastgoo, A.: Coriolis effects on the thermo-mechanical vibration analysis of the rotating multilayer piezoelectric nanobeam. Acta Mech. 234(2), 751–774 (2023)

    Article  MathSciNet  Google Scholar 

  7. Rashid, S.; Jarad, F.; Ahmad, A.G.: A novel fractal-fractional order model for the understanding of an oscillatory and complex behavior of human liver with non-singular kernel. Results Phys. 35, 105292 (2022)

    Article  Google Scholar 

  8. Malik, N.A.; Chang, C.-L.; Chaudhary, N.I.; Raja, M.A.Z.; Cheema, K.M.; Shu, C.-M.; Alshamrani, S.S.: Knacks of fractional order swarming intelligence for parameter estimation of harmonics in electrical systems. Mathematics 10(9), 1570 (2022)

    Article  Google Scholar 

  9. Alabedalhadi, M.: Exact travelling wave solutions for nonlinear system of spatiotemporal fractional quantum mechanics equations. Alex. Eng. J. 61(2), 1033–1044 (2022)

    Article  Google Scholar 

  10. Tlelo-Cuautle, E.; Gonzalez-Zapata, A.M.; Diaz-Munoz, J.D.; de la Fraga, L.G.; Cruz-Vega, I.: Optimization of fractional-order chaotic cellular neural networks by metaheuristics. Eur. Phys. J. Spec. Top. 231(10), 2037–2043 (2022)

    Article  Google Scholar 

  11. Shi, J.; He, K.; Fang, H.: Chaos, hopf bifurcation and control of a fractional-order delay financial system. Math. Comput. Simul. 194, 348–364 (2022)

    Article  MathSciNet  Google Scholar 

  12. Zhang, S.; Wang, C.; Zhang, H.; Ma, P.; Li, X.: Dynamic analysis and bursting oscillation control of fractional-order permanent magnet synchronous motor system. Chaos Solitons Fractal 156, 111809 (2022)

    Article  MathSciNet  Google Scholar 

  13. Youyou, W.; Yang, Y.; Uzzi, B.: A discipline-wide investigation of the replicability of psychology papers over the past two decades. Proc. Natl. Acad. Sci. 120(6), e2208863120 (2023)

    Article  Google Scholar 

  14. Turkyilmazoglu, M.: Hyperthermia therapy of cancerous tumor sitting in breast via analytical fractional model. Comput. Biol. Med. 164, 107271 (2023)

    Article  Google Scholar 

  15. Turkyilmazoglu, M.; Altanji, M.: Fractional models of falling object with linear and quadratic frictional forces considering caputo derivative. Chaos Solitons Fractals 166, 112980 (2023)

    Article  MathSciNet  Google Scholar 

  16. Turkyilmazoglu, M.: Transient and passage to steady state in fluid flow and heat transfer within fractional models. Int. J. Numer. Methods Heat Fluid Flow 33(2), 728–750 (2023)

    Article  Google Scholar 

  17. Surendar, R.; Muthtamilselvan, M.: Helical force with a two-phase cattaneo ltne model on hyper-chaotic convection in the presence of magnetic field. Eur. Phys. J. Plus 138(7), 658 (2023)

    Article  Google Scholar 

  18. Escalante-Martinez, J.; Gomez-Aguilar, J.; Calderon-Ramon, C.; Aguilar-Melendez, A.; Padilla-Longoria, P.: Synchronized bioluminescence behavior of a set of fireflies involving fractional operators of liouville-caputo type. Int. J. Biomath. 11(03), 1850041 (2018)

    Article  MathSciNet  Google Scholar 

  19. Escalante-Martinez, J.; Gomez-Aguilar, J.; Calderon-Ramon, C.; Aguilar-Melendez, A.; Padilla-Longoria, P.: A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der pol oscillators. Int. J. Biomath. 11(01), 1850014 (2018)

    Article  MathSciNet  Google Scholar 

  20. Dhobale, S.M.; Chatterjee, S.: A general class of optimal nonlinear resonant controllers of fractional order with time-delay for active vibration control-theory and experiment. Mech. Syst. Signal Process. 182, 109580 (2023)

    Article  Google Scholar 

  21. Akpado, E.; Monwanou, A.: Nonlinear dynamics, adaptive control and synchronization of a system modeled by a chemical reaction with integer-and fractional-order derivatives. Int. J. Dyn. Control 1–18 (2023)

  22. Zheng, W.; Chen, Y.; Wang, X.; Chen, Y.; Lin, M.: Enhanced fractional order sliding mode control for a class of fractional order uncertain systems with multiple mismatched disturbances. ISA Trans. 133, 147–159 (2023)

    Article  Google Scholar 

  23. Lin, Y.-T.; Wang, J.-L.; Liu, C.-G.: Output synchronization analysis and pd control for coupled fractional-order neural networks with multiple weights. Neurocomputing 519, 17–25 (2023)

    Article  Google Scholar 

  24. Peng, Q.; Jian, J.: Asymptotic synchronization of second-fractional-order fuzzy neural networks with impulsive effects. Chaos Solitons Fractals 168, 113150 (2023)

    Article  MathSciNet  Google Scholar 

  25. Coronel-Escamilla, A.; Gomez-Aguilar, J.; Torres, L.; Escobar-Jimenez, R.; Valtierra-Rodriguez, M.: Synchronization of chaotic systems involving fractional operators of liouville-caputo type with variable-order. Phys. A 487, 1–21 (2017)

    Article  MathSciNet  Google Scholar 

  26. Li, J.-F.; Jahanshahi, H.; Kacar, S.; Chu, Y.-M.; Gomez-Aguilar, J.; Alotaibi, N.D.; Alharbi, K.H.: On the variable-order fractional memristor oscillator: data security applications and synchronization using a type-2 fuzzy disturbance observer-based robust control. Chaos Solitons Fractals 145, 110681 (2021)

    Article  MathSciNet  Google Scholar 

  27. Martinez-Fuentes, O.; Montesinos-Garcia, J.J.; Gomez-Aguilar, J.F.: Generalized synchronization of commensurate fractional-order chaotic systems: applications in secure information transmission. Digital Signal Process. 126, 103494 (2022)

    Article  Google Scholar 

  28. Li, H.; Shi, P.; Yao, D.: Adaptive sliding-mode control of markov jump nonlinear systems with actuator faults. IEEE Trans. Autom. Control 62(4), 1933–1939 (2016)

    Article  MathSciNet  Google Scholar 

  29. Su, X.; Liu, X.; Shi, P.; Song, Y.-D.: Sliding mode control of hybrid switched systems via an event-triggered mechanism. Automatica 90, 294–303 (2018)

    Article  MathSciNet  Google Scholar 

  30. Ullah, S.; Mehmood, A.; Khan, Q.; Rehman, S.; Iqbal, J.: Robust integral sliding mode control design for stability enhancement of under-actuated quadcopter. Int. J. Control Autom. Syst. 18, 1671–1678 (2020)

    Article  Google Scholar 

  31. Hu, J.; Zhang, H.; Liu, H.; Yu, X.: A survey on sliding mode control for networked control systems. Int. J. Syst. Sci. 52(6), 1129–1147 (2021)

    Article  MathSciNet  Google Scholar 

  32. Giap, V.N.: Text message secure communication based on fractional-order chaotic systems with takagi–sugeno fuzzy disturbance observer and sliding mode control. Int. J. Dyn. Control 1–15 (2023)

  33. Roy, S.; Baldi, S.; Fridman, L.M.: On adaptive sliding mode control without a priori bounded uncertainty. Automatica 111, 108650 (2020)

    Article  MathSciNet  Google Scholar 

  34. Guha, D.; Roy, P.K.; Banerjee, S.: Adaptive fractional-order sliding-mode disturbance observer-based robust theoretical frequency controller applied to hybrid wind-diesel power system. ISA Trans. 133, 160–183 (2023)

    Article  Google Scholar 

  35. Petravs, I.: The fractional-order lorenz-type systems: a review. Fract. Calculus Appl. Anal. 25(2), 362–377 (2022)

    Article  MathSciNet  Google Scholar 

  36. Daftardar-Gejji, V.: Fractional Calculus and Fractional Differential Equations. Springer, Berlin (2019)

    Book  Google Scholar 

  37. Ibraheem, G.H.; Turkyilmazoglu, M.; Al-Jawary, M.: Novel approximate solution for fractional differential equations by the optimal variational iteration method. J. Comput. Sci. 64, 101841 (2022)

    Article  Google Scholar 

  38. Yin, C.; Dadras, S.; Zhong, S.-M.; Chen, Y.: Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach. Appl. Math. Model. 37(4), 2469–2483 (2013)

    Article  MathSciNet  Google Scholar 

  39. Khalil, H.K.: Control of nonlinear systems

  40. Leine, R.I.; Nijmeijer, H.: Dynamics and Bifurcations of Non-smooth Mechanical Systems. Springer Science & Business Media, Heideiberg (2013)

    Google Scholar 

  41. Buongiorno, J.: Convective transport in nanofluids. (2006)

  42. Kuznetsov, A.; Nield, D.: Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Transp. Porous Media 83, 425–436 (2010)

    Article  Google Scholar 

  43. Agarwal, S.; Sacheti, N.C.; Chandran, P.; Bhadauria, B.; Singh, A.K.: Non-linear convective transport in a binary nanofluid saturated porous layer. Transp. Porous Media 93, 29–49 (2012)

    Article  MathSciNet  Google Scholar 

  44. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MathSciNet  Google Scholar 

  45. Akdemir, A.O.; Dutta, H.; Atangana, A.: Fractional Order Analysis: Theory, Methods and Applications. Wiley, Hoboken (2020)

    Google Scholar 

  46. Luo, R.; Wang, Y.: Finite-time modified projective synchronization between two different chaotic systems with parameter and model uncertainties and external disturbances via sliding control. Indian J. Phys. 88, 301–309 (2014)

    Article  Google Scholar 

  47. Aguila-Camacho, N.; Duarte-Mermoud, M.A.; Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951–2957 (2014)

    Article  MathSciNet  Google Scholar 

  48. Li, C.; Chen, G.: Chaos in the fractional order chen system and its control. Chaos Solitons Fractals 22(3), 549–554 (2004)

    Article  Google Scholar 

  49. Wang, J.; Xiong, X.; Zhang, Y.: Extending synchronization scheme to chaotic fractional-order chen systems. Phys. A 370(2), 279–285 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Ramapuram, Chennai, Tamil Nadu, 600 089, India

    R. Surendar

  2. Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu, 641 046, India

    M. Muthtamilselvan

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Surendar, R., Muthtamilselvan, M. Sliding Mode Control on Finite-Time Synchronization of Nonlinear Hyper-mechanical Fractional Systems. Arab J Sci Eng (2024). https://doi.org/10.1007/s13369-024-08858-1

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