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Mittag–Leffler stability is a property of fractional-order dynamical systems, also called fractional Lyapunov stability, requiring the evolution of the positive-definite functions to be Mittag–Leffler, rather than the exponential meaning in Lyapunov stability theory. Similarly, fractional Lyapunov function plays an important role in the study of Mittag–Leffler stability. The aim of this study is to create closed-loop systems for commensurate fractional-order non-linear systems (FONSs) with Mittag–Leffler stability. We extend the classical backstepping to fractional-order backstepping for stabilising (uncertain) FONSs. For this purpose, several conditions of control fractional Lyapunov functions for FONSs are investigated in terms of Mittag–Leffler stability. Within this framework, (uncertain) FONSs Mittag–Leffler stabilisation is solved via fractional-order backstepping and the global convergence of closed-loop systems is guaranteed. Finally, the efficiency and applicability of the proposed fractional-order backstepping are demonstrated in several examples.
References
-
-
1)
-
19. Farges, C., Moze, M., Sabatier, J.: ‘Pseudo-state feedback stabilization of commensurate fractional order systems’, Automatica, 2010, 46, (10), pp. 1730–1734 (doi: 10.1016/j.automatica.2010.06.038).
-
2)
-
33. Miroslav, K., Ioannis, K., Petar, V.K.: ‘Nonlinear and adaptive control design’ (Wiley, 1995).
-
3)
-
38. Yu, W., Luo, Y., Pi, Y.G.: ‘Fractional order modeling and control for permanent magnet synchronous motor velocity servo system’, Mechatronics, 2013, 23, (7), pp. 813–820 (doi: 10.1016/j.mechatronics.2013.03.012).
-
4)
-
6. Lu, J.-G., Chen, Y., Chen, W.: ‘Robust asymptotical stability of fractional-order linear systems with structured perturbations’, Comput. Math. Appl., 2013, 66, (5), pp. 873–882 (doi: 10.1016/j.camwa.2013.03.001).
-
5)
-
20. Zhang, X., Liu, L., Feng, G., Wang, Y.: ‘Asymptotical stabilization of fractional-order linear systems in triangular form’, Automatica, 2013, 49, (11), pp. 3315–3321 (doi: 10.1016/j.automatica.2013.08.002).
-
6)
-
27. Wang, J.R., Lv, L.L., Zhou, Y.: ‘New concepts and results in stability of fractional differential equations’, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, (6), pp. 2530–2538 (doi: 10.1016/j.cnsns.2011.09.030).
-
7)
-
4. Sabatier, J., Agrawal, O.P., Machado, J.A.T.: ‘Advances in fractional calculus’ (Springer, 2007).
-
8)
-
16. Li, C., Wang, J.C.: ‘Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0 < α < 1 Case’, J. Franklin Inst., 2012, 349, (7), pp. 2406–2419 (doi: 10.1016/j.jfranklin.2012.05.006).
-
9)
-
1. Baleanu, D., Machado, J.A.T., Luo, A.C.J.: ‘Fractional dynamics and control’ (Springer, 2012).
-
10)
-
32. Abooee, A., Haeri, M.: ‘Stabilisation of commensurate fractional-order polytopic non-linear differential inclusion subject to input non-linearity and unknown disturbances’, IET Control Theory Appl., 2013, 7, (12), pp. 1624–1633 (doi: 10.1049/iet-cta.2013.0038).
-
11)
-
21. Lakshmikantham, V., Leela, S., Sambandham, M.: ‘Lyapunov theory for fractional differential equations’, Commun. Appl. Anal., 2008, 12, (4), pp. 365–376.
-
12)
-
28. Zhou, X.-F., Hu, L.-G., Liu, S., Jiang, W.: ‘Stability criterion for a class of nonlinear fractional differential systems’, Appl. Math. Lett., 2014, 28, pp. 25–29 (doi: 10.1016/j.aml.2013.09.007).
-
13)
-
2. Cao, Y., Li, Y., Ren, W., Chen, Y.: ‘Distributed coordination of networked fractional order systems’, IEEE Trans. Syst. Man Cybern. B Cybern., 2010, 40, (2), pp. 362–370 (doi: 10.1109/TSMCB.2009.2024647).
-
14)
-
14. Dadras, S., Momeni, H.R.: ‘Passivity-based fractional-order integral sliding-mode control design for uncertain fractional-order nonlinear systems’, Mechatronics, 2013, 23, (7), pp. 880–887 (doi: 10.1016/j.mechatronics.2013.05.009).
-
15)
-
29. Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: ‘Lyapunov functions for fractional order systems’, Commun. Nonlinear Sci. Numer. Simul., 2014, 19, (9), pp. 2951–2957 (doi: 10.1016/j.cnsns.2014.01.022).
-
16)
-
X.J. Wen ,
Z.M. Wu ,
J.G. Lu
.
Stability analysis of a class of nonlinear fractional-order systems.
IEEE Trans. Circuit Syst. II: Express Briefs
,
1178 -
1183
-
17)
-
15. Matignon, D.: ‘Stability results for fractional differential equations with applications to control processing’. Computational Engineering in Systems Applications, 1996, 2, pp. 963–968.
-
18)
-
9. Padula, F., Alcántara, S., Vilanova, R., Visioli, A.: ‘ℋ∞ control of fractional linear systems’, Automatica, 2013, 49, (7), pp. 2276–2280 (doi: 10.1016/j.automatica.2013.04.012).
-
19)
-
J.G. Lu ,
G.R. Chen
.
Robust stability and stabilization of fractional-order interval systems: an LMI approach.
IEEE Trans. Autom Control
,
1294 -
1300
-
20)
-
37. Peng, J., Ye, X.: ‘Distributed adaptive controller for the output-synchronization of networked systems in semi-strict feedback form’, J. Frankl. Inst., Eng. Appl. Math., 2014, 351, (1), pp. 412–428 (doi: 10.1016/j.jfranklin.2013.09.004).
-
21)
-
24. Yu, J., Hu, H., Zhou, S., Lin, X.: ‘Generalized Mittag–Leffler stability of multi-variables fractional order nonlinear systems’, Automatica, 2013, 49, (6), pp. 1798–1803 (doi: 10.1016/j.automatica.2013.02.041).
-
22)
-
J. Sabatier ,
M. Moze ,
C. Farges
.
LMI stability conditions for fractional order systems.
Comput. Math. Appl.
,
1594 -
1609
-
23)
-
30. Trigeassou, J.C., Maamri, N., Oustaloup, A.: ‘Lyapunov stability of linear fractional systems. Part 1: definition of fractional energy’. ASME IDETC-CIE Conf., 2013.
-
24)
-
2. Petras, I.: ‘Fractional-order nonlinear systems: modeling, analysis and simulation’ (Springer, 2011).
-
25)
-
J. Trigeassou ,
N. Maamri ,
J. Sabatier ,
A. Oustaloup
.
A Lyapunov approach to the stability of fractional differential equations.
Signal Process.
,
3 ,
437 -
445
-
26)
-
Y. Li ,
Y.Q. Chen ,
I. Podlubny
.
Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability.
Comput. Math. Appl.
,
5 ,
1810 -
1821
-
27)
-
Y.H. Lan ,
Y. Zhou
.
LMI-based robust control of fractional-order uncertain linear systems.
Comput. Math. Appl.
,
1460 -
1471
-
28)
-
26. Burton, T.A.: ‘Fractional differential equations and Lyapunov functionals’, Theory Methods Appl., 2011, 74, (16), pp. 5648–5662 (doi: 10.1016/j.na.2011.05.050).
-
29)
-
11. Lan, Y.-H., Gu, H.B., Chen, C.-X., Zhou, Y., Luo, Y.-P.: ‘An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks’, Neurocomputing, 2014, 136, (20), pp. 235–242 (doi: 10.1016/j.neucom.2014.01.009).
-
30)
-
3. Uchaikin, V.V.: ‘Fractional derivatives for physicists and engineers’ (Springer, 2013).
-
31)
-
13. Shi, B., Yuan, J., Dong, C.: ‘Pseudo-state sliding mode control of fractional SISO nonlinear systems’, Adv. Math. Phys., 2013, .
-
32)
-
Y. Li ,
Y.Q. Chen ,
I. Podlubny
.
Mittag-Leffler stability of fractional order nonlinear dynamic systems.
Automatica
,
8 ,
1965 -
1969
-
33)
-
31. Trigeassou, J.C., Maamri, N., Oustaloup, A.: ‘Lyapunov stability of linear fractional systems. Part 2: derivation of a stability condition’. ASME IDETC-CIE Conf., 2013.
-
34)
-
34. Igor, P.: ‘Fractional differential equations’ (Academic Press, 1999).
-
35)
-
8. Farges, C., Fadiga, L., Sabatier, J.: ‘ℋ∞ analysis and control of commensurate fractional order systems’, Mechatronics, 2013, 23, (7), pp. 772–780 (doi: 10.1016/j.mechatronics.2013.06.005).
-
36)
-
7. Shen, J., Lam, J.: ‘State feedback ℋ∞ control of commensurate fractional-order systems’, Int. J. Syst. Sci., 2014, 45, (3), pp. 363–372 (doi: 10.1080/00207721.2012.723055).
-
37)
-
35. Tarasov, V.E.: ‘No violation of the Leibniz rule. No fractional derivative’, Commun. Nonlinear Sci. Numer. Simul., 2013, 18, (11), pp. 2945–2948 (doi: 10.1016/j.cnsns.2013.04.001).
-
38)
-
37. Aghababa, M.P.: ‘Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller’, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, pp. 2670–2681 (doi: 10.1016/j.cnsns.2011.10.028).
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