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A novel Mittag–Leffler stable estimator for nonlinear fractional-order systems: a linear quadratic regulator approach

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Abstract

In this article, the estimation problem in a class of nonlinear fractional-order systems is solved by a stable estimator that generalizes the classical exponential observers, establishing a new class called Mittag–Leffler observers. The solution to the linear quadratic regulator problem is proposed to design optimal control laws for fractional-order linear systems and its applications. All the results are based on the Caputo derivative for commensurate fractional-order systems. Numerical simulations validate the proposed theory.

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References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  2. Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4(1), 75–89 (1977)

    Article  MathSciNet  Google Scholar 

  3. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  Google Scholar 

  4. Machado, J.A.T., Silva, M.F., Barbosa, R.S., Jesus, I.S., Reis, C.M., Marcos, M.G., Galhano, A.F.: Some applications of fractional calculus in engineering. Math. Probl. Eng. 2010, 1–34 (2010)

    Article  Google Scholar 

  5. Nutting, P.G.: A new general law of deformation. J. Frankl. Inst. 191(5), 679–685 (1921)

    Article  Google Scholar 

  6. Nutting, P.G.: A general stress–strain-time formula. J. Frankl. Inst. 235(5), 513–524 (1943)

    Article  Google Scholar 

  7. Scott Blair, G.W., Reiner, M.: The rheological law underlying the nutting equation. Appl. Sci. Res. 2(1), 225–234 (1951)

    Article  Google Scholar 

  8. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581–3592 (1997)

    Article  MathSciNet  Google Scholar 

  9. Gómez-Aguilar, J.F., Rosales-García, J.J., Bernal-Alvarado, J.J., Córdova-Fraga, T., Guzmán-Cabrera, R.: Fractional mechanical oscillators. Rev. Mex. Fís. 58(4), 348–352 (2012)

    Google Scholar 

  10. Gómez-Aguilar, J.F., Razo-Hernández, R., Granados-Lieberman, D.: A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response. Rev. Mex. Fís 60(1), 32–38 (2014)

    MathSciNet  Google Scholar 

  11. Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44(4), 554–566 (1996)

    Article  MathSciNet  Google Scholar 

  12. Yang, X.J., Machado, J.A.T., Cattani, C., Gao, F.: On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 47, 200–206 (2017)

    Article  Google Scholar 

  13. Sierociuk, D., Malesza, W., Macias, M.: Derivation, interpretation, and analog modelling of fractional variable order derivative definition. Appl. Math. Model. 39(13), 3876–3888 (2015)

    Article  MathSciNet  Google Scholar 

  14. Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5(4), 367–386 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Tarasov, V.E.: Geometric interpretation of fractional-order derivative. Fract. Calc. Appl. Anal. 19(5), 1200–1221 (2016)

    Article  MathSciNet  Google Scholar 

  16. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006)

    Article  Google Scholar 

  17. Sabatier, J., Merveillaut, M., Malti, R., Oustaloup, A.: How to impose physically coherent initial conditions to a fractional system? Commun. Nonlinear Sci. Numer. Simul. 15(5), 1318–1326 (2010)

    Article  MathSciNet  Google Scholar 

  18. Martínez-Fuentes, O., Martínez-Guerra, R.: Generalized synchronization in chaotic Liouvillian fractional systems. In: IEEE American Control Conference, pp. 2747–2752 (2016)

  19. Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A.: Fractional integral sliding modes for robust tracking of nonlinear systems. Nonlinear Dyn. 87(2), 895–901 (2017)

    Article  MathSciNet  Google Scholar 

  20. Maheri, M., Arifin, N.M.: Synchronization of two different fractional-order chaotic systems with unknown parameters using a robust adaptive nonlinear controller. Nonlinear Dyn. 85(2), 825–838 (2016)

    Article  MathSciNet  Google Scholar 

  21. Muñiz-Montero, C., García-Jiménez, L.V., Sánchez-Gaspariano, L.A., Sánchez-López, C., González-Díaz, V.R., Tlelo-Cuautle, E.: New alternatives for analog implementation of fractional-order integrators, differentiators and PID controllers based on integer-order integrators. Nonlinear Dyn. 90(1), 241–256 (2017)

    Article  MathSciNet  Google Scholar 

  22. Lan, Y.H., Wang, L.L., Ding, L., Zhou, Y.: Full-order and reduced-order observer design for a class of fractional-order nonlinear systems. Asian J. Control 18(4), 1467–1477 (2016)

    Article  MathSciNet  Google Scholar 

  23. Li, C., Wang, J., Lu, J., Ge, Y.: Observer-based stabilisation of a class of fractional order non-linear systems for \(0< \alpha < 2 \) case. IET Control Theory Appl. 8(13), 1238–1246 (2014)

    Article  MathSciNet  Google Scholar 

  24. Cafagna, D., Grassi, G.: Observer-based synchronization for a class of fractional chaotic systems via a scalar signal: results involving the exact solution of the error dynamics. Int. J. Bifurc. Chaos 21(3), 955–962 (2011)

    Article  MathSciNet  Google Scholar 

  25. Matignon, D., d’Andrea–Novel, B.: Observer-based controllers for fractional differential systems. In: IEEE Conference on Decision and Control, pp. 4967–4972 (1997)

  26. Wei, Y.H., Sun, Z.Y., Hu, Y.S., Wang, Y.: On fractional order adaptive observer. Int. J. Autom. Comput. 12(6), 664–670 (2015)

    Article  Google Scholar 

  27. Djeghali, N., Djennoune, S., Bettayeb, M., Ghanes, M., Barbot, J.P.: Observation and sliding mode observer for nonlinear fractional-order system with unknown input. ISA Trans. 63, 1–10 (2016)

    Article  Google Scholar 

  28. Cruz-Victoria, J.C., Martínez-Guerra, R., Pérez-Pinacho, C.A., Gómez-Cortés, G.C.: Synchronization of nonlinear fractional order systems by means of PIr\(\alpha \) reduced order observer. Appl. Math. Comput. 262, 224–231 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Martínez-Guerra, R., Pérez-Pinacho, C.A., Gómez-Cortés, G.C., Cruz-Victoria, J.C.: Synchronization of incommensurate fractional order system. Appl. Math. Comput. 262, 260–266 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Meléndez-Vázquez, F., Martínez-Fuentes, O., Martínez-Guerra, R.: Fractional fault-tolerant dynamical controller for a class of commensurate-order fractional systems. Int. J. Syst. Sci. 49(1), 196–210 (2017)

    Article  MathSciNet  Google Scholar 

  31. Kou, S.R., Elliott, D.L., Tarn, T.J.: Exponential observers for nonlinear dynamic systems. Inf. Control 29(3), 204–216 (1975)

    Article  MathSciNet  Google Scholar 

  32. Xia, X.h, Gao, W.b: On exponential observers for nonlinear systems. Syst. Control Lett. 11(4), 319–325 (1988)

    Article  MathSciNet  Google Scholar 

  33. Hassan, M.F.: Observer-based controller for discrete-time systems: a state dependent Riccati equation approach. Nonlinear Dyn. 70(1), 693–707 (2012)

    Article  MathSciNet  Google Scholar 

  34. Chen, C.S.: Optimal nonlinear observers for chaotic synchronization with message embedded. Nonlinear Dyn. 61(4), 623–632 (2010)

    Article  MathSciNet  Google Scholar 

  35. Li, Y., Chen, Y.Q.: Fractional order linear quadratic regulator. In: IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, pp. 363–368 (2008)

  36. Mousa, M.E., Ebrahim, M.A., Moustafa Hassan, M.A.: Optimal fractional order proportional-integral-differential controller for inverted pendulum with reduced order linear quadratic regulator. In: Azar AT, Vaidyanathan S, Ouannas A (eds) Fractional Order Control and Synchronization of Chaotic Systems, vol. 688, pp. 225–252. Springer, Cham (2017)

    Chapter  Google Scholar 

  37. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1999)

    MATH  Google Scholar 

  38. Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)

    MATH  Google Scholar 

  39. Matignon, D., d’Andrea-Novel, B.: Some results on controllability and observability of finite-dimensional fractional differential systems. Comput. Eng. Syst. Appl. 2, 952–956 (1996)

    Google Scholar 

  40. Khalil, H.K.: Nonlinear Systems. Prentice-Hall, New Jersey (1996)

    Google Scholar 

  41. Li, Y., Chen, Y.Q., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)

    Article  MathSciNet  Google Scholar 

  42. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22(1), 650–659 (2015)

    Article  MathSciNet  Google Scholar 

  43. Kleinman, D.: On an iterative technique for Riccati equation computations. IEEE Trans. Autom. Control 13(1), 114–115 (1968)

    Article  Google Scholar 

  44. Raghavan, S., Hedrick, J.K.: Observer design for a class of nonlinear systems. Int. J. Control 59(2), 515–528 (1994)

    Article  MathSciNet  Google Scholar 

  45. Aqeel, M., Azam, A., Ahmad, S.: Control of chaos: Lie algebraic exact linearization approach for the Lü system. Eur. Phys. J. Plus 132, 426 (2017)

    Article  Google Scholar 

  46. Dias, F.S., Mello, L.F.: Hopf bifurcations and small amplitude limit cycles in Rucklidge systems. Electron. J. Differ. Equ. 2013(48), 1–9 (2013)

    MathSciNet  MATH  Google Scholar 

  47. Aqeel, M., Ahmad, S.: Analytical and numerical study of Hopf bifurcation scenario for a three-dimensional chaotic system. Nonlinear Dyn. 84(2), 755–765 (2016)

    Article  MathSciNet  Google Scholar 

  48. Messias, M., De Carvalho Braga, D., Mello, L.F.: Degenerate Hopf bifurcations in Chua’s system. Int. J. Bifurc. Chaos 19(2), 497–515 (2009)

    Article  MathSciNet  Google Scholar 

  49. Azam, A., Aqeel, M., Ahmad, S., Ahmad, F.: Chaotic behavior of modified stretch-twist-fold (STF) flow with fractal property. Nonlinear Dyn. 90(1), 1–12 (2017)

    Article  MathSciNet  Google Scholar 

  50. Dias, F.S., Mello, L.F., Zhang, J.G.: Nonlinear analysis in a Lorenz-like system. Nonlinear Anal. Real World Appl. 11(5), 3491–3500 (2010)

    Article  MathSciNet  Google Scholar 

  51. Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55–61 (2004)

    Article  MathSciNet  Google Scholar 

  52. Zhang, W., Zhou, S., Li, H., Zhu, H.: Chaos in a fractional-order Rössler system. Chaos Solitons Fractals 42(3), 1684–1691 (2009)

    Article  Google Scholar 

  53. Lee, S.M.: The double-simple pendulum problem. Am. J. Phys. 38(4), 536–537 (1970)

    Article  Google Scholar 

  54. Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading, MA (1980)

    MATH  Google Scholar 

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Acknowledgements

The first author thanks the National Council for Science and Technology (CONACyT), Mexico, for the financial support of Ph.D. Grant 295538.

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  1. Departamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508, Col. San Pedro Zacatenco, 07360, Ciudad de México, Mexico

    Oscar Martínez-Fuentes & Rafael Martínez-Guerra

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  1. Oscar Martínez-Fuentes
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  2. Rafael Martínez-Guerra
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Correspondence to Oscar Martínez-Fuentes.

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Martínez-Fuentes, O., Martínez-Guerra, R. A novel Mittag–Leffler stable estimator for nonlinear fractional-order systems: a linear quadratic regulator approach. Nonlinear Dyn 94, 1973–1986 (2018). https://doi.org/10.1007/s11071-018-4469-6

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