Volume 17, Issue 1 (Journal of Control, V.17, N.1 Spring 2023)
JoC 2023, 17(1): 35-49 |
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Dehghani H, Akbarzadeh Kalat A. Observer-based controller design for a nonlinear fractional order system. JoC 2023; 17 (1) :35-49
URL: http://joc.kntu.ac.ir/article-1-947-en.html
URL: http://joc.kntu.ac.ir/article-1-947-en.html
1- Faculty of Electrical Engineering, Sharood University of Technology
Abstract: (1211 Views)
In this paper, an observer-based controller is introduced for a class of nonlinear fractional order systems. First of all, considering a stable linear fractional order system known as the reference model, the controller is designed so that the closed loop system tracks the states of the reference system. In most systems, some states are unmeasurable or unreachable, so the controller must be designed based on the observer. The observer has been suggested in this research, using the well-known differential mean value theorem approach, converts the nonlinear error dynamics of the observer into linear and parameters-varying, so that the stability analysis is done simply using the Lyapunov quadratic function and linear matrix inequality. The stability analysis of the observer-based controller is also performed using the Lyapunov theorem in the following section. Finally, the efficiency and effectiveness of the proposed controller are shown through the simulation results of two nonlinear fractional order systems.
Keywords: Fractional-order system, Linear matrix inequality, Differential mean value theorem, Observer, Lyapunov theory.
Type of Article: Research paper |
Subject:
Special
Received: 2022/07/24 | Accepted: 2023/04/21 | ePublished ahead of print: 2023/06/11 | Published: 2023/06/22
Received: 2022/07/24 | Accepted: 2023/04/21 | ePublished ahead of print: 2023/06/11 | Published: 2023/06/22
References
1. [ ]I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
2. [ ]R. Hilfer, Application of Fractional Calculus in Physics, World Science Publishing, Singapore, 2000. [DOI:10.1142/3779]
3. [ ]A. Boulkroune, A. Bouzerbia and T. Bouden, 2016, "Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control", Neural Computing and Applications, vol. 27, no. 5, pp. 1349-1360. [DOI:10.1007/s00521-015-1938-4]
4. [ ]N. Laskin, 2000, "Fractional market dynamics", Physica A: Statistical Mechanics and its Applications, vol. 287, no. 3-4, pp. 482-492. [DOI:10.1016/S0378-4371(00)00387-3]
5. [ ]X. Yin, D. Yue and S. Hu, 2013, "Consensus of fractional Order heterogeneous multi-agent systems", IET Control Theory and Applications, vol.7, no. 2, pp. 314-322. [DOI:10.1049/iet-cta.2012.0511]
6. [ ]J.G. Lu, G. Chen, 2009, "Robust stability and stabilization of fractional-order interval systems: An LMI approach", IEEE Transaction on Automatic Control, vol. 54, no. 6, pp. 1294-1299. [DOI:10.1109/TAC.2009.2013056]
7. [ ]Y. Chen, Y. Wei, X. Zhou, Y. Wang, 2017, "Stability for nonlinear fractional order systems: an indirect approach", Nonlinear Dynamics. vol. 89, no. 2, pp. 1011-1018. [DOI:10.1007/s11071-017-3497-y]
8. [ ]Z. Song, K. Sun, S. Ling, 2017, "Stabilization and synchronization for a mechanical system via adaptive sliding mode control". ISA Transaction, vol. 68, pp. 353-366. [DOI:10.1016/j.isatra.2017.02.013]
9. [ ]N. Goléa, A. Goléa, K. Barra, T. Bouktir, 2008, "Observer-based adaptive control of robot manipulators: Fuzzy systems approach", Applied Soft Computing, vol. 8, no. 1, pp. 778-787. [DOI:10.1016/j.asoc.2007.05.011]
10. [ ]M.P.Aghababa, 2012, "Robust stabilization and synchronization of a class of fractional- order chaotic systems via a novel fractional sliding mode controller", Communications in Nonlinear Science and Numerical Simulation, vol.17, pp. 2670-2681. [DOI:10.1016/j.cnsns.2011.10.028]
11. [ ] Y. Li, Y. Chen, I. Podlubny, 2010, "Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability", Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1810-1821. [DOI:10.1016/j.camwa.2009.08.019]
12. [ ] J. Yu, H. Hu, S. Zhou, X. Lin, 2013, "Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems", Automatica, vol. 49, no. 6, pp. 1798-1803. [DOI:10.1016/j.automatica.2013.02.041]
13. [ ] C. Farges, M. Moze, J. Sabatier, 2010, "Pseudo-state feedback stabilization of commensurate fractional order systems", Automatica, vol. 46, pp. 1730-1734. [DOI:10.1016/j.automatica.2010.06.038]
14. [ ]J.C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, 2011, "A Lyapunov approach to the stability of fractional differentiel equations", Signl Processing, vol. 91 no. 3, pp. 437-445. [DOI:10.1016/j.sigpro.2010.04.024]
15. [ D.Y. Chen, R.F. Zhang, X.Z. Liu, X.Y. Ma, 2014, "Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks", Communications in Nonlinear Science and Numerical Simulation, vol. 19, pp. 4105-4121. [DOI:10.1016/j.cnsns.2014.05.005]
16. [ ] M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, R. Castro-Linares, 2015, "Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems", Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1-3, pp. 650-659. [DOI:10.1016/j.cnsns.2014.10.008]
17. [ ] S. Ibrir, M. Bettayeb, 2015, "New sufficient conditions for observer-based control of fractional-order uncertain systems", Automatica, vol. 59, pp. 216-223. [DOI:10.1016/j.automatica.2015.06.002]
18. [ ] A. Mohammadzadeh, S. Ghaemi, 2018, "Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay", Nonlinear Dynamics, vol. 93, pp. 1809-1821. [DOI:10.1007/s11071-018-4290-2]
19. [ ] T. Liu, F.Wang,W. Lu, X.Wang, 2019, "Global stabilization for a class of nonlinear fractional-order systems, International Journal of Modeling, "Simulation and Scientific Computing, vol. 10, pp. 1-10. [DOI:10.1142/S1793962319410095]
20. [ ] Y. H. Lan, Y. Zhou, 2011, "Lmi-based robust control of fractional-order uncertain linear systems", Computers and Mathematics with Applications, vol. 62, no.3, pp. 1460-1471. [DOI:10.1016/j.camwa.2011.03.028]
21. [ ] Y. H. Lan, H. X. Huang, Y.Zhou, 2012, "Observer-based robust control of α (1<α<2) fractional-order uncertain systems: a linear matrix inequality approach", IET Control Theory Application. vol. 6, pp. 229-234. [DOI:10.1049/iet-cta.2010.0484]
22. [ ] E. A. Boroujeni, H. R. Momeni, 2012, "Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems", Signal Processing.vol. 92, pp. 2365-2370,. [DOI:10.1016/j.sigpro.2012.02.009]
23. [ ] Z.S. Aghayan, A. Alfi, T. Machado, 2021, "Observer-based control approach for fractional-order delay systems of neutral type with saturating actuator". Mathematical Methods in Applied Science, vol. 44, no. 11, pp. 8554-8564. [DOI:10.1002/mma.7282]
24. [ ] H. F. Ghavidel, A. A Kalat, 2017, "Observer-based robust composite adaptive fuzzy control by uncertainty estimation for a class of nonlinear systems", Neurocomputing, vol. 230, pp.100-109. [DOI:10.1016/j.neucom.2016.12.001]
25. [ ] H. F. Ghavidel, A. A Kalat, 2018, "Observer-based hybrid adaptive fuzzy control for affine and nonaffine uncertain nonlinear systems", Neural Computing and Applications, vol. 30, pp. 1187-1202. [DOI:10.1007/s00521-016-2732-7]
26. [ ] D. Valério, J.J. Trujillo, M. Rivero, J.T. Machado, D. Baleanu, 2013, "Fractional calculus: a survey of useful formulas". The European Physical Journal Special Topics, vol. 222, no. 8, pp. 1827-46. [DOI:10.1140/epjst/e2013-01967-y]
27. [ ] D. Chen, R. Zhang, X. Liu, X. Ma, 2014, "Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks", Communications in Nonlinear Science and Numerical Simulation, vol. 19, pp. 4105-4121. [DOI:10.1016/j.cnsns.2014.05.005]
28. [ ] N. Aguila-Camacho, M.A. Duarte-Mermoud, J. Gallegos, 2014, "Lyapunov functions for fractional order systems", Communications in Nonlinear Science and Numerical Simulation, vol. 19, pp. 2951-7. [DOI:10.1016/j.cnsns.2014.01.022]
29. [ ] M-A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, R. Castro-Linares, 2014, "Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems", Communications in Nonlinear Science and Numerical Simulation vol. 22, no. 1-3, pp. 650-659. [DOI:10.1016/j.cnsns.2014.10.008]
30. [ ] V. Sharma, V. Agrawal, B. Sharma, R. Nath, 2016, "Unknown input nonlinear observer design for continuous and discrete time systems with input recovery scheme". Nonlinear Dynamics vol. 85, no. 1, pp. 645-658. [DOI:10.1007/s11071-016-2713-5]
31. [ ]V. Sharma, M. Shukla, B.B. Sharma, 2018, "Unknown input observer design for a class of fractional order nonlinear systems", Chaos, Soliton and Fractal, vol. 115, pp. 96-107. [DOI:10.1016/j.chaos.2018.08.017]
32. [ ] A. Zemouche, M. Boutayeb, 2009, "A unified H∞ adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities", System and Control Letter, vol. 58, pp. 282- 288. [DOI:10.1016/j.sysconle.2008.11.007]
33. [ ] A. Zemouche, M. Boutayeb, G.I. Bara, 2005, "Observer design for nonlinear systems. An approach based on the differential mean value theorem". Proc CDC-ECC'05, 44th IEEE Conference on IEEE, pp.6353-6358.
34. [ ] S. Boyd, L. Vandenberghe, 2001, "Convex optimization with engineering applications", in: Lecture Notes, Stanford University, Stanford.
35. [ ] M. M. Polycarpous, P. A. Ioannouq, 1996, " A Robust Adaptive Nonlinear Control Design", Automarica, vol. 32, pp. 423-427. [DOI:10.1016/0005-1098(95)00147-6]
36. [ ] I. Petras, D. Bednarova, 2011 , "Control of fractional-order nonlinear systems: a review", Acta Mechanica et Automatica, vol.5, no.2, pp. 96-100.
37. [ ] S. Dadras, H.R. Momeni, 2010, "Control of a fractional-order economical systems via sliding mode" , Physica A: Statistical Mechanics and its Applications, vol. 389, pp. 2434-2442. [DOI:10.1016/j.physa.2010.02.025]
38. [ ] M. Bettayeb, S. Djennoune, 2016, "Design of sliding mode controllers for nonlinear fractional-order systems via diffusive representation", Nonlinear Dynamics. vol. 84, pp. 593-605. [DOI:10.1007/s11071-015-2509-z]
39. [ ] A. Zemouche and M. Boutayeb, 2013, "On LMI conditions to design observers for Lipschitz nonlinear systems," Automatica, vol. 49, pp. 585-591. [DOI:10.1016/j.automatica.2012.11.029]
40. [ ] E. A. Boroujeni, H. R. Momeni, 2012, "Observer Based Control of a Class of NonlinearFractional Order Systems using LMI", World Academy of Science, Engineering and Technology, vol. 6 (1), pp. 81-84.
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