# Output Tracking of Uncertain Fractional-order Systems via Robust Iterative Learning Sliding Mode Control

• Accepted : 2017.09.07
• Published : 2018.07.01
• 166 6

#### Abstract

This paper develops a novel controller called iterative learning sliding mode (ILSM) to control linear and nonlinear fractional-order systems. This control applies a combination structures of continuous and discontinuous controller, conducts the system output to the desired output and achieve better control performance. This controller is designed in the way to be robust against the external disturbance. It also estimates unknown parameters of fractional-order systems. The proposed controller unlike the conventional iterative learning control for fractional systems does not need to apply direct control input to output of the system. It is shown that the controller perform well in partial and complete observable conditions. Simulation results demonstrate very good performance of the iterative learning sliding mode controller for achieving the desired control objective by increasing the number of iterations in the control loop.

#### References

1. Alessandro Pisano, Milan Rade Rapaic, Zoran D. Jelicic and Elio Usai, "Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics," Int J Robust Nonlinear Control, vol. 20, pp. 2045-2056, Dec. 2010. https://doi.org/10.1002/rnc.1565
2. Christophe Tricaud and Yang Quan Chen, "An approximate method for numerically solving fractional order optimal control problems of general form," Computers and Mathematics with Applications, vol. 59, pp. 1644-1655, March 2010. https://doi.org/10.1016/j.camwa.2009.08.006
3. Zhen Wang, Xia Huang and Junwei Lu, "Sliding mode synchronization of chaotic and hyperchaotic systems with mismatched fractional derivatives," Transactions of the Institute of Measurement and Control, vol. 35, pp. 1713-1741, Dec. 2013.
4. Vineet Kumar, K.P.S. Rana and Puneet Mishra, "Robust speed control of hybrid electric vehicle using fractional order fuzzy PD and PI controllers in cascade control loop," Journal of the Franklin Institute, vol. 353, pp. 680-694, May 2016.
5. Sara Dadras and Hamid Reza momeni "Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty," Commun Nonlinear Sci Numer Simulat, vol. 17, pp. 367-377, June 2012. https://doi.org/10.1016/j.cnsns.2011.04.032
6. Yan Li, Yang Quan Chen, and Hyo-Sung Ahn, "A Generalized Fractional-Order Iterative Learning Control," in Proceedings of 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, USA, March 2011.
7. Yong-Hong Lan, "Iterative learning control with initial state learning for fractional order nonlinear systems," Computers and Mathematics with Applications, vol. 64, pp. 3210-3216, Nov. 2012. https://doi.org/10.1016/j.camwa.2012.03.086
8. Yong-Hong Lan and Yong Zhou, " $D^{\alpha}$ -type Iterative Learning Control for fractional-order linear time- delay systems," Asian Journal of Control, vol. 15, pp. 669-677, May 2013. https://doi.org/10.1002/asjc.623
9. Yan Li, Yang Quan Chen and Hyo-Sung Ahn, "A High-gain Adaptive Fractional-order Iterative Learning Control," in Proceedings of 11th IEEE International Conference on Control & Automation (ICCA), Taichung, Taiwan, June 2014.
10. Denis Matignon, "Stability results of fractional differential equations with applications to control processing," in Proceedings IEEE-SMC proceed Computational Engineering in Systems Applications multi conference (IMACS), Lille, France, March 1997.
11. Jian-Xin Xu and Jing Xu, "On Iterative Learning From Different Tracking Tasks in the Presence of Time-Varying Uncertainties," IEEE transactions on systems, man, and cybernetics, vol. 34, pp. 589-597, Feb. 2004. https://doi.org/10.1109/TSMCB.2003.818433
12. Wajdi M. Ahmad, and Julien Clinton Sprott "Chaos in fractional-order autonomous nonlinear systems," Chaos, Solitons and Fractals, vol. 16, pp. 339-351, March 2003. https://doi.org/10.1016/S0960-0779(02)00438-1
13. Yan Li, Yang Quan Chen and Hyo-Sung Ahn, "Fractional-order iterative learning control for fractional-order linear systems," Asian Journal of Control, vol. 13, pp. 54-63, Jan. 2011. https://doi.org/10.1002/asjc.253
14. Seyed Hasan Hosseinnia, Reza Ghaderi, Abolfazl Ranjbar, Mohmad Mahmudian and Shaher Momani, "Sliding mode synchronization of an uncertain fractional order chaotic system," Computers & Mathematics with Applications, vol. 59, pp. 1637- 1643, March 2010. https://doi.org/10.1016/j.camwa.2009.08.021
15. Seyed Hasan Hosseinnia, Ines Tejado and Blas Vinagre, "Fractional-order reset control: Application to a servomotor," Mechatronics, vol. 23, pp. 781-788, Oct. 2013. https://doi.org/10.1016/j.mechatronics.2013.03.005
16. Seyed Hasan Hosseinnia, Ines Tejado, Vicente Milanes, Jorge Villagra and Blas Vinagre, "Experimental Application of Hybrid Fractional-Order Adaptive Cruise Control at Low Speed," IEEE transaction on control systems technology, vol. 22, pp. 2329-2336, Nov. 2014. https://doi.org/10.1109/TCST.2014.2308837
17. Ehsan Ghotb Razmju, Abolfazl Ranjbar, Zahra Rahmani and Reza Ghaderi, Robust synchronization and parameter identification of a unified fractional- order chaotic system: Springer-Verlag, 2012, pp. 173- 184.
18. Ehsan Ghotb Razmju, Abolfazl Ranjbar, Zahra Rahmani, Reza Ghaderi and Shaher Momani, Stabilization of fractional order unified chaotic systems via linear state feedback controller: Springer-Verlag, 2012, p. 85-94.
19. Igor Podlubny, Fractional differential equation: New York, CA: Academic Press, 1999.
20. Ricardo Enrique Gutierrez, Joao Mauricio Rosario and Jose Tenreiro Machado, "Fractional order calculus: basic concepts and engineering applications," Mathematical problems in engineering, doi:10.1155/ 2010/375858, March 2010. https://doi.org/10.1155/2010/375858
21. Ahmed M.A. El-Sayed, "Fractional order diffusion wave equation," International Journal of Theoretical Physics, vol. 35, pp. 311-322, Feb. 1996. https://doi.org/10.1007/BF02083817
22. Victor George Jenson and Gofrey Vaughan Jeffreys, Mathematical Methods in Chemical Engineering: New York, CA: Academic Press, 1997.
23. Dimitri Kusnezov, "Quantum levy processes and fractional kinetics," Phys. Rev. Lett, vol. 82, pp. 1136-1139, Feb. 1999. https://doi.org/10.1103/PhysRevLett.82.1136
24. Peter Torvik and Ronald Bagley, "On the appearance of the fractional derivative in the behavior of real materials," Trans. ASME, vol. 51, pp. 294-298, Jun 1984. https://doi.org/10.1115/1.3167615
25. Nooshin Bigdeli, "The design of a non-minimal state space fractional-order predictive functional controller for fractional systems of arbitrary order," Journal of Process Control, vol. 29, pp. 45-56, May 2015. https://doi.org/10.1016/j.jprocont.2015.03.004
26. Ehsan Ghotb Razmju, Abolfazl Ranjbar and Zahra Rahmani, "Using state feedback control to stabilize unstable equilibrium points of the unified fractional- order chaotic system," Scientific Research and Essays, vol. 6, pp. 5937-5950, Nov. 2010.
27. Chun Yin, Sara Dadras and Shou-ming Zhong, "Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional- order chaotic systems with fully unknown parameters," Journal of the Franklin Institute, vol. 349, pp. 3078- 3101, Dec. 2012. https://doi.org/10.1016/j.jfranklin.2012.09.009
28. Zheng-Ming Ge, and Chan-Yi Ou "Chaos synchronization of fractional order modified duffing systems with parameters excited by a chaotic signal," Chaos, Solitons and Fractals, vol. 35, pp. 705-717, Feb. 2008. https://doi.org/10.1016/j.chaos.2006.05.101