Volume 14, Issue 2 (Journal of Control, V.14, N.2 Summer 2020)                   JoC 2020, 14(2): 89-99 | Back to browse issues page

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Jafari P, Teshnehlab M, Tavakoli-Kakhki M. Designing of a Control Approach for Uncertain Fractional Order Systems with Indirect Adaptive Fuzzy Controller and Frational Order Sliding Mode. JoC 2020; 14 (2) :89-99
URL: http://joc.kntu.ac.ir/article-1-584-en.html
1- Faculty of Electrical and Computer Engineering, Sistan and Baluchestan University
2- Faculty of Electrical Engineering, K.N. Toosi University of Technology
Abstract:   (6370 Views)
Todays according to the noticeable growth of the fractional order calculus in engineering sciences, this field has converted to a beloved context for researchers especially Control engineers. There have been designed various fractional order control methods accordingly. Also, it has been proved that adaptive fuzzy controllers are capable of controlling uncertain systems with disturbance if necessary, conditions have been provided. For this reason, in this paper, an indirect adaptive TSK fuzzy controller with fractional order sliding mode control is introduced to control a certain class of nonlinear fractional order systems. The fractional order stability of the closed-loop system is studied and based on a fractional order Lyapunov function candidate; fractional order adaptation laws are obtained. The fractional order adaptation law is proposed to adjust the free parameters in the consequence part of the adaptive TSK system. In addition, a robust adaptive law is proposed to reduce the influence of approximation error between true system functions and TSK fuzzy controller. Hence, using the fractional order Lyapunov theorem, the Mittag-Leffler stability of the closed-loop system is guaranteed. The numerical simulation shows validity and effectiveness of the introduced control strategy for fractional order nonlinear models that perturbed by disturbance and uncertainty.
Type of Article: Research paper | Subject: Special
Received: 2018/05/28 | Accepted: 2019/02/21 | Published: 2019/08/15

References
1. [1] R. Caponetto, Fractional order systems: modeling and control applications vol. 72: World Scientific, 2010. [DOI:10.1142/7709]
2. [2] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications vol. 198: Academic press, 1998.
3. [3] J.-D. Gabano and T. Poinot, "Fractional modelling and identification of thermal systems," Signal Processing, vol. 91, pp. 531-541, 2011. [DOI:10.1016/j.sigpro.2010.02.005]
4. [4] B. Vinagre and V. Feliu, "Modeling and control of dynamic system using fractional calculus: Application to electrochemical processes and flexible structures," in Proc. 41st IEEE Conf. Decision and Control, pp. 214-239, 2002.
5. [5] R. L. Magin, Fractional calculus in bioengineering: Begell House Redding, 2006.
6. [6] J. Sabatier, M. Aoun, A. Oustaloup, G. Grégoire, F. Ragot, and P. Roy, "Fractional system identification for lead acid battery state of charge estimation," Signal processing, vol. 86, pp. 2645-2657, 2006. [DOI:10.1016/j.sigpro.2006.02.030]
7. [7] R. Hilfer, Applications of fractional calculus in physics: World Scientific, 2000. [DOI:10.1142/3779]
8. [8] C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu-Batlle, Fractional-order systems and controls: fundamentals and applications: Springer Science & Business Media, 2010. [DOI:10.1007/978-1-84996-335-0]
9. [9] A. Oustaloup, X. Moreau, and M. Nouillant, "The CRONE suspension," Control Engineering Practice, vol. 4, pp. 1101-1108, 1996. [DOI:10.1016/0967-0661(96)00109-8]
10. [10] M. Zamani, M. Karimi-Ghartemani, N. Sadati, and M. Parniani, "Design of a fractional order PID controller for an AVR using particle swarm optimization," Control Engineering Practice, vol. 17, pp. 1380-1387, 2009. [DOI:10.1016/j.conengprac.2009.07.005]
11. [11] N. Aguila-Camacho and M. A. Duarte-Mermoud, "Fractional adaptive control for an automatic voltage regulator," ISA transactions, vol. 52, pp. 807-815, 2013. [DOI:10.1016/j.isatra.2013.06.005]
12. [12] Y. He and R. Gong, "Application of fractional-order model reference adaptive control on industry boiler burning system," in 2010 International Conference on Intelligent Computation Technology and Automation, pp. 750-753, 2010. [DOI:10.1109/ICICTA.2010.59]
13. [13] L. Jun-Guo, "Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems," Chinese Physics, vol. 14, p. 1517, 2005. [DOI:10.1088/1009-1963/14/8/007]
14. [14] C. Li and G. Chen, "Chaos and hyperchaos in the fractional-order Rössler equations," Physica A: Statistical Mechanics and its Applications, vol. 341, pp. 55-61, 2004. [DOI:10.1016/j.physa.2004.04.113]
15. [15] J. G. Lu, "Chaotic dynamics of the fractional-order Lü system and its synchronization," Physics Letters A, vol. 354, pp. 305-311, 2006. [DOI:10.1016/j.physleta.2006.01.068]
16. [16] Q. Yang and C. Zeng, "Chaos in fractional conjugate Lorenz system and its scaling attractors," Communications in Nonlinear Science and Numerical Simulation, vol. 15, pp. 4041-4051, 2010. [DOI:10.1016/j.cnsns.2010.02.005]
17. [17] N. Aguila-Camacho, M. A. Duarte-Mermoud, and J. A. Gallegos, "Lyapunov functions for fractional order systems," Communications in Nonlinear Science and Numerical Simulation, vol. 19, pp. 2951-2957, 2014. [DOI:10.1016/j.cnsns.2014.01.022]
18. [18] M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, and R. Castro-Linares, "Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems," Communications in Nonlinear Science and Numerical Simulation, vol. 22, pp. 650-659, 2015. [DOI:10.1016/j.cnsns.2014.10.008]
19. [19] Y. Li, Y. Chen, and I. Podlubny, "Mittag-Leffler stability of fractional order nonlinear dynamic systems," Automatica, vol. 45, pp. 1965-1969, 2009. [DOI:10.1016/j.automatica.2009.04.003]
20. [20] Y. Li, Y. Chen, and I. Podlubny, "Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability," Computers & Mathematics with Applications, vol. 59, pp. 1810-1821, 2010. [DOI:10.1016/j.camwa.2009.08.019]
21. [21] K. M. Passino, S. Yurkovich, and M. Reinfrank, Fuzzy control vol. 42: Citeseer, 1998. [DOI:10.1109/13.746327]
22. [22] L.-X. Wang, A course in fuzzy systems: Prentice-Hall press, USA, 1999.
23. [23] N. Golea, A. Golea, and K. Benmahammed, "Stable indirect fuzzy adaptive control," Fuzzy sets and Systems, vol. 137, pp. 353-366, 2003. [DOI:10.1016/S0165-0114(02)00279-8]
24. [24] S. Labiod and T. M. Guerra, "Adaptive fuzzy control of a class of SISO nonaffine nonlinear systems," Fuzzy Sets and Systems, vol. 158, pp. 1126-1137, 2007. [DOI:10.1016/j.fss.2006.11.013]
25. [25] C.-H. Wang, H.-L. Liu, and T.-C. Lin, "Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems," Fuzzy Systems, IEEE Transactions on, vol. 10, pp. 39-49, 2002. [DOI:10.1109/91.983277]
26. [26] M. Ö. Efe, "Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm," Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, vol. 38, pp. 1561-1570, 2008. [DOI:10.1109/TSMCB.2008.928227]
27. [27] T.-C. Lin, T.-Y. Lee, and V. E. Balas, "Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems," Chaos, Solitons & Fractals, vol. 44, pp. 791-801, 2011. [DOI:10.1016/j.chaos.2011.04.005]
28. [28] T.-C. Lin, C.-H. Kuo, T.-Y. Lee, and V. E. Balas, "Adaptive fuzzy H∞ tracking design of SISO uncertain nonlinear fractional order time-delay systems," Nonlinear Dynamics, vol. 69, pp. 1639-1650, 2012. [DOI:10.1007/s11071-012-0375-5]
29. [29] T.-C. Lin and T.-Y. Lee, "Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control," Fuzzy Systems, IEEE Transactions on, vol. 19, pp. 623-635, 2011. [DOI:10.1109/TFUZZ.2011.2127482]
30. [30] M. P. Aghababa, "Comments on "Adaptive fuzzy H∞ tracking design of SISO uncertain nonlinear fractional order time-delay systems"[Nonlinear Dyn. 69 (2012) 1639-1650]," Nonlinear Dynamics, vol. 70, pp. 2511-2513, 2012. [DOI:10.1007/s11071-012-0624-7]
31. [31] S. Tong, H.-X. Li, and W. Wei, "Comments on" Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems"," Fuzzy Systems, IEEE Transactions on, vol. 11, pp. 703-705, 2003. [DOI:10.1109/TFUZZ.2003.817843]
32. [32] T.-C. Lin, C.-H. Kuo, and V. E. Balas, "Uncertain fractional order chaotic systems tracking design via adaptive hybrid fuzzy sliding mode control," International Journal of Computers Communications & Control, vol. 6, pp. 418-427, 2011. [DOI:10.15837/ijccc.2011.3.2127]
33. [33] N. Ullah, S. Han, and M. Khattak, "Adaptive fuzzy fractional-order sliding mode controller for a class of dynamical systems with uncertainty," Transactions of the Institute of Measurement and Control, vol. 38, pp. 402-413, 2016. [DOI:10.1177/0142331215587042]
34. [34] N. Ullah, W. Shaoping, M. I. Khattak, and M. Shafi, "Fractional order adaptive fuzzy sliding mode controller for a position servo system subjected to aerodynamic loading and nonlinearities," Aerospace Science and Technology, vol. 43, pp. 381-387, 2015. [DOI:10.1016/j.ast.2015.03.020]
35. [35] K. Khettab, S. Ladaci, and Y. Bensafia, "Fuzzy adaptive control of a fractional order chaotic system with unknown control gain sign using a fractional order Nussbaum gain," IEEE/CAA Journal of Automatica Sinica, 2016. [DOI:10.1109/STA.2015.7505141]
36. [36] K. Khettab, Y. Bensafia, and S. Ladaci, "Robust Adaptive Fuzzy Control for a Class of Uncertain Nonlinear Fractional Systems," in Recent Advances in Electrical Engineering and Control Applications, ed Cham: Springer pp. 276-294, 2017. [DOI:10.1007/978-3-319-48929-2_21]
37. [37] K. Khettab, Y. Bensafia, and S. Ladaci, "Robust Adaptive Interval Type-2 Fuzzy Synchronization for a Class of Fractional Order Chaotic Systems," in Fractional Order Control and Synchronization of Chaotic Systems, ed Cham: Springer, pp. 203-224, 2017. [DOI:10.1007/978-3-319-50249-6_7]
38. [38] A. Bouzeriba, A. Boulkroune, and T. Bouden, "Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems," International Journal of Machine Learning and Cybernetics, vol. 7, pp. 893-908, October 2016. [DOI:10.1007/s13042-015-0425-7]
39. [39] A. Boulkroune, A. Bouzeriba, and T. Bouden, "Fuzzy generalized projective synchronization of incommensurate fractional-order chaotic systems," Neurocomputing, vol. 173, pp. 606-614, 2016. [DOI:10.1016/j.neucom.2015.08.003]
40. [40] A. Bouzeriba, A. Boulkroune, T. Bouden, and S. Vaidyanathan, "Fuzzy adaptive synchronization of incommensurate fractional-order chaotic systems," in Advances and Applications in Chaotic Systems, ed: Springer, pp. 363-378, 2016. [DOI:10.1007/978-3-319-30279-9_15]
41. [41] M. Mohadeszadeh and H. Delavari, "Synchronization of uncertain fractional-order hyper-chaotic systems via a novel adaptive interval type-2 fuzzy active sliding mode controller," International Journal of Dynamics and Control, vol. 5, pp. 135-144, 2017. [DOI:10.1007/s40435-015-0207-9]
42. [42] P. Jafari, M. Teshnehlab, and M. Tavakoli-Kakhki, "Adaptive Type-2 Fuzzy System for Synchronization and Stabilization of Chaotic Nonlinear Fractional Order Systems," IET Control Theory & Applications, vol. 12, pp. 183-193, 2017. [DOI:10.1049/iet-cta.2017.0785]
43. [43] P. Jafari, M. Teshnehlab, and M. Tavakoli-Kakhki, "Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal," Nonlinear Dynamics, vol. 90, pp. 1037-1052, 2017. [DOI:10.1007/s11071-017-3709-5]
44. [44] محمد صالح تواضعی و مهسان توکلی کاخکی،سیستم‌ها و کنترل‌کننده‌های مرتبه کسری، انتشارات دانشگاه صنعتی خواجه نصیرالدین طوسی، 1394.
45. [45] J. A. Gallegos, M. A. Duarte-Mermoud, N. Aguila-Camacho, and R. Castro-Linares, "On fractional extensions of Barbalat Lemma," Systems & Control Letters, vol. 84, pp. 7-12, 2015. [DOI:10.1016/j.sysconle.2015.07.004]
46. [46] A. Sharifi, M. Aliyari Shoorehdeli, and M. Teshnehlab, "Semi-polynomial Takagi-Sugeno-Kang Type Fuzzy System for System Identification and Pattern Classification," Journal of Control, vol. 4, pp. 15-28, 2010.
47. [47] v. Bahrami, M. Mansouri, and M. Teshnehlab, "Designing Model Reference Fuzzy Controller Based on State Feedback Integral Control for Nonlinear Systems," Journal of Control, vol. 9, pp. 1-18, 2015.
48. [48] Y. Diao and K. M. Passino, "Adaptive neural/fuzzy control for interpolated nonlinear systems," IEEE Transactions on Fuzzy Systems, vol. 10, pp. 583-595, 2002. [DOI:10.1109/TFUZZ.2002.803493]
49. [49] Y. Yang, G. Feng, and J. Ren, "A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems," IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, vol. 34, pp. 406-420, 2004. [DOI:10.1109/TSMCA.2004.824870]
50. [50] M. P. Aghababa, "Design of hierarchical terminal sliding mode control scheme for fractional-order systems," IET Science, Measurement and Technology, vol. 9, pp. 122-133, 2014. [DOI:10.1049/iet-smt.2014.0039]
51. [51] M. P. Aghababa and H. P. Aghababa, "The rich dynamics of fractional-order gyros applying a fractional controller," Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 227, pp. 588-601, 2013. [DOI:10.1177/0959651813492326]
52. [52] S. H. Hosseinnia, R. Ghaderi, A. Ranjbar, J. Sadati, and S. Momani, "Synchronization of gyro systems via fractional-order adaptive controller," in New Trends in Nanotechnology and Fractional Calculus Applications, ed: Springer, pp. 495-502, 2010. [DOI:10.1007/978-90-481-3293-5_44]