nach oben

Journal of Applied Mathematics and Computing

Erschienen in:

06.10.2021 | Original Research

Fractional integro-differential sliding mode control of a class of distributed-order nonlinear systems

verfasst von: Aldo Jonathan Muñoz-Vázquez, Guillermo Fernández-Anaya, Juan Diego Sánchez-Torres

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 4/2022

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Distributed-order systems arise as a natural generalization of fractional- and integer-order systems, and these are commonly associated with slow and ultra-slow dynamics, which motivates designing robust schemes to enforce fast stabilization. This paper proposes a robust sliding mode controller that induces a stable motion in a finite time, relying on a dynamic extension to produce an integer-order reaching phase. Thus, a continuous fractional sliding mode controller is designed to compensate not necessarily integer-order differentiable disturbances. The theoretical results demonstrate that both the disturbance is exactly compensated and the pseudo-state converges asymptotically to the origin. A numerical simulation is carried out to demonstrate the reliability and efficacy of the proposed method, where a comparison to a conventional sliding mode scheme reveals a superior performance for the proposed fractional sliding mode approach.

Vorheriger Artikel Combined gradient methods for multiobjective optimization

Nächster Artikel A stochastic turbidostat model coupled with distributed delay and degenerate diffusion: dynamics analysis

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Alkasassbeh, M., Omar, Z., Mebarek-Oudina, F., Raza, J., Chamkha, A.: Heat transfer study of convective fin with temperature-dependent internal heat generation by hybrid block method. Heat Transf. Asian Res. 48(4), 1225–1244 (2019)CrossRef

Atanackovic, T., Budincevic, M., Pilipovic, S.: On a fractional distributed-order oscillator. J. Phys. A Math. Gen. 38(30), 6703 (2005)MathSciNetCrossRef

Atanackovic, T.M.: On a distributed derivative model of a viscoelastic body. C.R. Mec. 331(10), 687–692 (2003)CrossRef

Atanacković, T.M., Oparnica, L., Pilipović, S.: On a nonlinear distributed order fractional differential equation. J. Math. Anal. Appl. 328(1), 590–608 (2007)MathSciNetCrossRef

Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, vol. 178. Springer Science & Business Media, Berlin (2008)MATH

Dadkhah, E., Shiri, B., Ghaffarzadeh, H., Baleanu, D.: Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods. J. Appl. Math. Comput. 63(1), 29–57 (2020)MathSciNetCrossRef

Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225(1), 96–104 (2009)MathSciNetCrossRef

Efe, M.Ö.: Integral sliding mode control of a quadrotor with fractional order reaching dynamics. Trans. Inst. Meas. Control. 33(8), 985–1003 (2011)CrossRef

Farhan, M., Omar, Z., Mebarek-Oudina, F., Raza, J., Shah, Z., Choudhari, R., Makinde, O.: Implementation of the one-step one-hybrid block method on the nonlinear equation of a circular sector oscillator. Comput. Math. Model. 31(1), 116–132 (2020)MathSciNetCrossRef

10.

Fernández-Anaya, G., Nava-Antonio, G., Jamous-Galante, J., Muñoz-Vega, R., Hernández-Martínez, E.: Asymptotic stability of distributed order nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 48, 541–549 (2017)MathSciNetCrossRef

11.

Jakovljević, B., Pisano, A., Rapaić, M.R., Usai, E.: On the sliding-mode control of fractional-order nonlinear uncertain dynamics. Int. J. Robust Nonlinear Control 26(4), 782–798 (2016)MathSciNetCrossRef

12.

Jensen, J.L.W.V.: Om konvekse funktioner og uligheder imellem middelvaerdier. Nyt Tidsskrift for Matematik 30, 49–69 (1905) https://www.jstor.org/stable/24528332

13.

Jensen, J.L.W.V.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30, 175–193 (1906). https://doi.org/10.1007/BF02418571MathSciNetCrossRefMATH

14.

Jiao, Z., Chen, Y.Q.: Stability of fractional-order linear time-invariant systems with multiple noncommensurate orders. Comput. Math. Appl. 64(10), 3053–3058 (2012)MathSciNetCrossRef

15.

Jiao, Z., Chen, Y.Q., Podlubny, I.: Distributed-order dynamic systems: stability. Simulation, Applications and Perspectives, London (2012)

16.

Kamal, S., Chalanga, A., Moreno, J.A., Fridman, L., Bandyopadhyay, B.: Higher order super-twisting algorithm. In: 2014 13th International Workshop on Variable Structure Systems (VSS), IEEE, pp 1–5 (2014)

17.

Kochubei, A.N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340(1), 252–281 (2008)MathSciNetCrossRef

18.

Lazović, G., Vosika, Z., Lazarević, M., Simić-Krstić, J., Koruga, D.: Modeling of bioimpedance for human skin based on fractional distributed-order modified cole model. FME Transact. 42(1), 74–81 (2014)CrossRef

19.

Li, A., Liu, G., Luo, Y., Yang, X.: An indirect lyapunov approach to robust stabilization for a class of linear fractional-order system with positive real uncertainty. J. Appl. Math. Comput. 57(1), 39–55 (2018)MathSciNetCrossRef

20.

Li, Y., Chen, Y.: Theory and implementation of distributed-order element networks. Int. Des. Eng. Tech. Conf. Comput. Inf. Eng. Conf. 54808, 361–367 (2011)

21.

Li, Y., Chen, Y.Q.: Theory and implementation of weighted distributed order integrator. In: IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, IEEE, pp 119–124 (2012)

22.

Liu, L., Feng, L., Xu, Q., Zheng, L., Liu, F.: Flow and heat transfer of generalized maxwell fluid over a moving plate with distributed order time fractional constitutive models. Int. Commun. Heat Mass Transf. 116, 104679 (2020)CrossRef

23.

Mahmoud, G.M., Farghaly, A.A., Abed-Elhameed, T.M., Aly, S.A., Arafa, A.A.: Dynamics of distributed-order hyperchaotic complex van der pol oscillators and their synchronization and control. Eur. Phys. J. Plus 135(1), 1–16 (2020)CrossRef

24.

Mebarek-Oudina, F.: Numerical modeling of the hydrodynamic stability in vertical annulus with heat source of different lengths. Eng. Sci. Technol. Int. J. 20(4), 1324–1333 (2017)

25.

Morgado, M.L., Rebelo, M.: Numerical approximation of distributed order reaction-diffusion equations. J. Comput. Appl. Math. 275, 216–227 (2015)MathSciNetCrossRef

26.

Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: A novel continuous fractional sliding mode control. Int. J. Syst. Sci. 48(13), 2901–2908 (2017)MathSciNetCrossRef

27.

Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Non-smooth convex Lyapunov functions for stability analysis of fractional-order systems. Trans. Inst. Meas. Control. 41(6), 1627–1639 (2019)CrossRef

28.

Muñoz-Vázquez, A.J., Fernández-Anaya, G., Sánchez-Torres, J.D., Meléndez-Vázquez, F.: Predefined-time control of distributed-order systems. Nonlinear Dyn. 103(3), 2689–2700 (2021)CrossRef

29.

Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol. 87. Springer Science & Business Media, Berlin (2013)MATH

30.

Patnaik, S., Semperlotti, F.: Application of variable-and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators. Nonlinear Dyn. 100(1), 561–580 (2020)CrossRef

31.

Pérez-Ventura, U., Fridman, L.: Design of super-twisting control gains: a describing function based methodology. Automatica 99, 175–180 (2019)MathSciNetCrossRef

32.

Pisano, A., Rapaić, M.R., Jeličić, Z.D., Usai, E.: Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics. Int. J. Robust Nonlinear Control 20(18), 2045–2056 (2010)MathSciNetCrossRef

33.

Pisano, A., Rapaić, M.R., Usai, E., Jeličić, Z.D.: Continuous finite-time stabilization for some classes of fractional order dynamics. In: International Workshop on Variable Structure Systems, IEEE, pp 16–21 (2012)

34.

Podlubny, I.: Fractional Differential Equations. Elsevier, Amsterdam (1998)MATH

35.

Rayal, A., Verma, S.R.: An approximate wavelets solution to the class of variational problems with fractional order. J. Appl. Math. Comput. 65(1), 735–769 (2021)MathSciNetCrossRef

36.

Ross, B., Samko, S.G., Love, E.R.: Functions that have no first order derivative might have fractional derivatives of all orders less than one. Real Anal. Exch. 20(1), 140–157 (1994)MathSciNetCrossRef

37.

Taghavian, H., Tavazoei, M.S.: Stability analysis of distributed-order nonlinear dynamic systems. Int. J. Syst. Sci. 49(3), 523–536 (2018)MathSciNetCrossRef

38.

Toaldo, B.: Lévy mixing related to distributed order calculus, subordinators and slow diffusions. J. Math. Anal. Appl. 430(2), 1009–1036 (2015)MathSciNetCrossRef

39.

Wang, F., Liu, X.: Pseudo-state estimation for fractional order neural networks. Neural Process. Lett. pp 1–14 (2021)

40.

Wei, L., Liu, L., Sun, H.: Stability and convergence of a local discontinuous galerkin method for the fractional diffusion equation with distributed order. J. Appl. Math. Comput. 59(1), 323–341 (2019)MathSciNetCrossRef

41.

Yang, W., Chen, X., Zhang, X., Zheng, L., Liu, F.: Flow and heat transfer of viscoelastic fluid with a novel space distributed-order constitution relationship. Comput. Math. Appl. 94, 94–103 (2021)MathSciNetCrossRef

42.

Yang, Z., Zhang, J., Niu, Y.: Finite-time stability of fractional-order bidirectional associative memory neural networks with mixed time-varying delays. J. Appl. Math. Comput. 63(1), 501–522 (2020)MathSciNetCrossRef

43.

Zaky, M.A., Machado, J.T.: On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 52, 177–189 (2017)MathSciNetCrossRef

44.

Želi, V., Zorica, D.: Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law. Phys. A 492, 2316–2335 (2018)MathSciNetCrossRef

45.

Zhou, F., Zhao, Y., Li, Y., Chen, Y.: Design, implementation and application of distributed order pi control. ISA Trans. 52(3), 429–437 (2013)CrossRef

Titel: Fractional integro-differential sliding mode control of a class of distributed-order nonlinear systems
verfasst von: Aldo Jonathan Muñoz-Vázquez
Guillermo Fernández-Anaya
Juan Diego Sánchez-Torres
Publikationsdatum: 06.10.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Applied Mathematics and Computing / Ausgabe 4/2022
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-021-01632-8

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 4/2022

An efficient uniformly convergent numerical scheme for singularly perturbed semilinear parabolic problems with large delay in time

Randic index of bipolar fuzzy graphs and its application in network systems

Predator–prey system with multiple delays: prey’s countermeasures against juvenile predators in the predator–prey conflict

Positive solutions for fractional (p, q)-difference boundary value problems

Periodicity and stationary distribution of two novel stochastic epidemic models with infectivity in the latent period and household quarantine

Efficient regularized Newton-type algorithm for solving convex optimization problem

Premium Partner