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Fractional Order Exponential Type Discrete-time Sliding Mode Control

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  • Control Theory and Applications
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Abstract

In this paper, a new fractional order exponential type reaching law (FOE-RL) is presented for the uncertain discrete-time system. The FOE-RL is constructed by adopting the exponential term and the Grünwald-Letnikov fractional order (FO) calculus of the switching function and the sign function. Compared to the integer order reaching laws (IO-RLs), the proposed reaching law is capable of regulating the system trajectory converging to a specified band whose width can be smaller than the upper bound of change rate of the disturbance and even approach zero. Hence, the proposed method has the ability to further suppress chattering and enhance the control accuracy compared to other reaching law strategies. The decrement band and the quasi-sliding mode domain (QSMD) of the uncertain system are analyzed. The sliding surface can be reached in finite steps, even though the system is suffering from uncertainties and disturbances. Numerical simulation examples are given to testify the validity of the presented method.

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References

  1. S. Mobayen, “Adaptive global terminal sliding mode control scheme with improved dynamic surface for uncertain nonlinear systems,” Int. J. Control Autom. Syst., vol. 16, no. 4, pp. 1692–1700, Aug. 2018.

    Google Scholar 

  2. C. Mnasri, D. Chorfi, and M. Gasmi, “Robust integral sliding mode control of uncertain networked control systems with multiple data packet losses,” Int. J. Control Autom. Syst., vol. 16, no. 5, pp. 2093–2102, Oct. 2018.

    Google Scholar 

  3. Y. Liu, “Sliding mode control for a class of uncertain discrete switched systems,” Int. J. Control Autom. Syst., vol. 16, no. 4, pp. 1716–1723, Aug. 2018.

    Google Scholar 

  4. Y. Wu, H. Shen, H. R. Karimi, and D. Duan, “Dissipativety-based fuzzy integral sliding mode control of continuous-time T-S fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 26, no. 3, pp. 1164–1176, Jun. 2018.

    Google Scholar 

  5. Y. Wu, H. R. Karimi, H. Shen, Z. Fang, and M. Liu, “Fuzzy-model-based sliding mode control of nonlinear descriptor systems,” IEEE Trans. Cybern., vol. 49, no. 9, pp. 3409–3419, Jun. 2018.

    Google Scholar 

  6. M. Pai, “Robust tracking and model following of uncertain dynamic systems via discrete-time integral sliding mode control,” Int. J. Control Autom. Syst., vol. 7, no. 3, pp. 381–387, Jun. 2009.

    Google Scholar 

  7. K. Furuta, “Sliding mode control of a discrete system,” Syst. Control Lett., vol. 14, pp. 144–152, Feb. 1990.

    MathSciNet  MATH  Google Scholar 

  8. W. Gao and J. C. Hung, “Variable structure control of nonlinear systems: A new approach,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 45–55, Feb. 1993.

    Google Scholar 

  9. W. Gao, Y. Wang, and A. Homaifa, “Discrete-time variable structure control systems,” IEEE Trans. Ind. Electron., vol. 42, no. 2, pp. 117–122, Apr. 1995.

    Google Scholar 

  10. X. Yu, B. Wang, and X. Li., “Computer-controlled variable structure systems: The state-of-the-art,” IEEE Trans. Ind. Inf., vol. 8, no. 2, pp. 197–205, May 2012.

    Google Scholar 

  11. P. Latosiski, “Sliding mode control based on the reaching law approach- a brief survey,” in Int. Conf. Methods Models Autom. Robot. (MMAR), Miedzyzdroje, Poland, pp. 519–524, 2017.

    Google Scholar 

  12. B. Jiang, H. R. Karimi, Y. Kao, and C. Gao, “A novel robust fuzzy integral sliding mode control for nonlinear semimarkovian jump T-S fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 26, no. 6, pp. 3594–3604, Dec. 2018.

    Google Scholar 

  13. A. Bartoszewicz, “Discrete-time quasi-sliding-mode control strategies,” IEEE Trans. Ind. Electron., vol. 45, no. 4, pp. 633–637, Apr. 1998.

    Google Scholar 

  14. A. Bartoszewicz and P. Leśniewski, “Reaching law approach to the sliding mode control of periodic review inventory systems,” IEEE Trans. Autom. Sci. Eng., vol. 11, no. 3, pp. 810–817, Jun. 2014.

    Google Scholar 

  15. A. Bartoszewicz and P. Leśniewski, “New switching and nonswitching type reaching laws for SMC of discrete time system,” IEEE Trans. Control Syst. Technol., vol. 24, no. 3, pp. 670–677, Mar. 2016.

    Google Scholar 

  16. A. Bartoszewicz and P. Latosiński, “Discrete time sliding mode control with reduced switching - a new reaching law approach,” Int. J. Robust Nonlinear Control, vol. 26, no. 1, pp. 47–68, Jan. 2016.

    MathSciNet  MATH  Google Scholar 

  17. G. Bartolini, A. Ferrara, and V. Utkin, “Adaptive sliding mode control in discrete-time systems,” Automatica, vol. 31, no. 5, pp. 769–773, Apr. 1995.

    MathSciNet  MATH  Google Scholar 

  18. H. Ma, J. Wu, and Z. Xiong, “A novel exponential reaching law of discrete-time sliding-mode control,” IEEE Trans. Ind. Electron., vol. 64, no. 5, pp. 3840–3850, May 2017.

    Google Scholar 

  19. Y. Niu, D. W. C. Ho, and Z. Wang, “Improved sliding mode control for discrete-time systems via reaching law,” IET Control Theory Appl., vol. 4, no. 11, pp. 2245–2251, Nov. 2010.

    MathSciNet  Google Scholar 

  20. H. Ma, J. Wu, and Z. Xiong, “Discrete-time sliding-mode control with improved quasi-sliding-mode domain,” IEEE Trans. Ind. Electron., vol. 63, no. 10, pp. 6292–6304, Oct. 2016.

    Google Scholar 

  21. H. Du, X. Yu, M. Chen, and S. Li, “Chattering-free discrete-time sliding mode control,” Automatica, vol. 68, no. 3, pp. 87–91, Jun. 2016.

    MathSciNet  MATH  Google Scholar 

  22. S. Chakrabarty and B. Bandyopadhyay, “A generalized reaching law for discrete time sliding mode control,” Automatica, vol. 52, no. 3, pp. 83–86, Feb. 2015.

    MathSciNet  MATH  Google Scholar 

  23. S. Chakrabarty and B. Bandyopadhyay, “A generalized reaching law with different convergence rates,” Automatica, vol. 63, no. 3, pp. 34–37, Jan. 2016.

    MathSciNet  MATH  Google Scholar 

  24. S. C. Qu, X. H. Xia, and J. F. Zhang, “Dynamics of discrete-time sliding-mode-control uncertain systems with a disturbance compensator,” IEEE Trans. Ind. Electron., vol. 61, no. 7, pp. 3502–3510, Jul. 2014.

    Google Scholar 

  25. Y. S. Eun, J. H. Kim, K. S. Kim, and D. Cho, “Discrete-time variable structure controller with a decoupled disturbance compensator and its application to a CNC servomechanism,” IEEE Trans. Control Syst. Technol., vol. 7, no. 4, pp. 414–423, Jul. 1999.

    Google Scholar 

  26. H. Ma and Y. Li, “Multi-power reaching law based discrete-time sliding mode control,” IEEE Access, vol. 7, pp. 49822–49829, Mar. 2019.

    Google Scholar 

  27. H. Ma, Y. Li, and Z. Xiong, “Discrete-time sliding-mode control with enhanced power reaching law,” IEEE Trans. Ind. Electron., vol. 66, no. 6, pp. 4629–4638, Aug. 2018.

    Google Scholar 

  28. C. Ma and H. Hori, “Fractional-order control: theory and application in motion control,” IEEE Ind. Electron. Mag., vol. 1, no. 4, pp. 6–16, Win. 2007.

    Google Scholar 

  29. Y. Wang, L. Gu, Y. Xu, and X. Cao, “Practical tracking control of robot manipulators with continuous fractional-order nonsingular terminal sliding mode,” IEEE Trans. Ind. Electron., vol. 63, no. 10, pp. 6194–6204, Oct. 2016.

    Google Scholar 

  30. Y. Chen, Y. Wei, and Y. Wang, “On 2 types of robust reaching laws,” Int. J. Robust Nonlinear Control, vol. 28, no. 6, pp. 2651–2667, Apr. 2018.

    MathSciNet  MATH  Google Scholar 

  31. P. Muthukumar, P. Balasubramaniam, and K. Ratnavelu, “Sliding mode control for generalized robust synchronization of mismatched fractional order dynamical systems and its application to secure transmission of voice,” ISA Trans., vol. 82, pp.51–61, Nov. 2018.

    Google Scholar 

  32. Q. Xu and Y. Li, “Micro-/nanopositioning using model predictive output integral discrete sliding mode control,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1161–1170, Feb. 2012.

    Google Scholar 

  33. K. Abidi, J. X. Xu, and X. Yu, “On the discrete-time integral sliding mode control,” IEEE Trans. Autom. Control, vol. 52, no. 4, pp. 709–715, Apr. 2007.

    MathSciNet  MATH  Google Scholar 

  34. S. Chakrabarty and B. Bandyopadhyay, “Minimum ultimate band design of discrete sliding mode control,” Asian J. Control, vol. 18, no. 1, pp. 1–9, Sep. 2015.

    MathSciNet  MATH  Google Scholar 

  35. S. Kamal, A. Raman, and B. Bandyopadhyay, “Finite-time stabilization of fractional order uncertain chain of integrator: an integral sliding mode approach,” IEEE Trans. Autom. Control, vol. 58, no. 6, pp. 1597–1602, Jun. 2013.

    MathSciNet  MATH  Google Scholar 

  36. I. Podlubny, Fractional Differential Equations, Academic, New York, NY, USA, 1999.

    MATH  Google Scholar 

  37. G. Sun, L. Wu, Z. Kuang, Z. Ma, and J. Liu, “Practical tracking control of linear motor via fractional-order sliding mode,” Automatica, vol. 94, pp. 221–235, Aug. 2018.

    MathSciNet  MATH  Google Scholar 

  38. S. Fan, “A new extracting formula and a new distinguishing means on the one variable cubic equation,” Natural Science J. Hainan Teachers College, vol. 2, no. 2, pp. 91–98, 1989.

    MathSciNet  Google Scholar 

  39. X. Yan, J. Xu, Y. Zhu, J. Wang, Y. Yang, and C. Wang, “Downlink average rate and SINR distribution in cellular networks,” IEEE Trans. Commun., vol. 64, no. 2, pp. 847–862, Feb. 2016.

    Google Scholar 

  40. X. Wu, Y. Tang, and J. Cao, “Input-to-State stability of time varying switched systems with time-delays,” IEEE Trans. Autom. Control, vol. 64, no. 6, pp. 2537–2544, Jun. 2019.

    MathSciNet  MATH  Google Scholar 

  41. X. Wu, P. Shi, Y. Tang, and W. Zhang, “Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects,” Int. J. Robust Nonlinear Control, vol. 27, no. 10, pp. 1792–1809, Jul. 2017.

    MathSciNet  MATH  Google Scholar 

  42. X. Wu, Y. Tang, and W. Zhang, “Input-to-state stability of impulsive stochastic delayed systems under linear assumptions,” Automatica, vol. 66, pp. 195–204, Apr. 2016.

    MathSciNet  MATH  Google Scholar 

  43. C. Yin, Y. Chen, and S. Zhong, “Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems,” Automatica, vol. 50, no. 12, pp. 3173–3181, Dec. 2014.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of High Efficiency and Clean Mechanical Manufacture of MOE, School of Mechanical Engineering, Shandong University, Jinan, 250061, China

    Haifeng Ma

  2. The Hong Kong Polytechnic University, Department of Industrial and Systems Engineering, Kowloon, Hong Kong

    Haifeng Ma & Yangmin Li

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  1. Haifeng Ma
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Correspondence to Yangmin Li.

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recommended by Editor Hamid Reza Karimi. This work is supported by the National Science Foundation of China under Grant (51805327, 51575544), by the Hong Kong Scholars Program under Grant (XJ2017022), the research committee of the Hong Kong Polytechnic University under Grant (G-YZ1G, 1-ZE97), and in part by Tianjin Natural Science Foundation (16JCZDJC38000).

Haifeng Ma received his B.E. and M.E. degrees in mechanical engineering from Southwest Jiaotong University, Chengdu, China, in 2010 and 2013, respectively, and his Ph.D. degree in mechatronics from Shanghai Jiao Tong University, Shanghai, China, in 2017. He is currently working at the Key Laboratory of High Efficiency and Clean Mechanical Manufacture of MOE, School of Mechanical Engineering, Shandong University, Jinan, China. He is also a “Hong Kong Scholar” and a postdoctoral fellow in The Hong Kong Polytechnic University, Kowloon, Hong Kong, China. His research interests include sliding-mode control (SMC) theory and applications, vibration control and intelligent manufacturing.

Yangmin Li received his B.S. and M.S. degrees from Jilin University, Changchun, China, in 1985 and 1988, respectively, and his Ph.D. degree from Tianjin University, Tianjin, China, in 1994, all in mechanical engineering. He is currently a Full Professor of the Department of Industrial and Systems Engineering of The Hong Kong Polytechnic University. He has authored and coauthored 420 scientific papers in journals and conferences. His research interests include micro/nanomanipulation, compliant mechanism, precision engineering, robotics, multibody dynamics and control. Dr. Li is a Member of the ASME. He is an Associate Editor of the IEEE Trans. Auto. Sci. Eng., Associate Editor of Mechatronics, Associate Editor of the International Journal of Control, Automation, and Systems, and Associate Editor of IEEE Access.

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Ma, H., Li, Y. Fractional Order Exponential Type Discrete-time Sliding Mode Control. Int. J. Control Autom. Syst. 18, 374–383 (2020). https://doi.org/10.1007/s12555-018-0898-8

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