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New AFSMC Method for Nonlinear System with State-dependent Uncertainty: Application to Hexapod Robot Position Control

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Abstract

Conventional Adaptive Fuzzy Sliding Mode Control (AFSMC) method is extended for nonlinear affine systems with state-dependent upper bound of uncertainties. More general affine model of the system with state-dependent uncertainties is proposed where such a model is more applicable in robotics. Position control of a Stewart Manipulator (SM) is then considered as a challenging case study to experimentally verify the effectiveness of the proposed Extended AFSMC (E-AFSMC) method. The proposed method is encompassed of a fuzzy system for estimation of a nonlinear system, a robust controller for compensation of uncertainties and some appropriate adaptation laws for optimization of performance. The second Lyapunov theorem and Barbalat lemma are used to prove the closed-loop asymptotic stability. Furthermore, numerical simulations depict the robustness of the proposed controller and in particular, under the very critical situation of actuator saturation and unexpected uncertainties. The effectiveness of the proposed control method is validated through experimental results.

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References

  1. Kunquan Li, R.W.: Robust control of a walking robot system and controller design. Procedia Eng. 174, 947 (2017). https://doi.org/10.1016/J.PROENG.2017.01.246. http://www.sciencedirect.com/science/article/pii/S1877705817302461

    Article  Google Scholar 

  2. Sylvia Kohn-Rich, H.F.: Robust fuzzy logic tracking control of mechanical systems. J. Franklin Inst. 338(2-3), 353 (2001). https://doi.org/10.1016/S0016-0032(00)00093-4. http://www.sciencedirect.com/science/article/pii/S0016003200000934

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhengchao Li, J.Y., Zhao, X.: On robust control of continuous-time systems with state-dependent uncertainties and its application to mechanical systems. ISA Trans. 60, 12 (2016). https://doi.org/10.1016/J.ISATRA.2015.10.023. http://www.sciencedirect.com/science/article/pii/S0019057815002578

    Article  Google Scholar 

  4. Sun, W., Gao, H., Yao, B.: Adaptive robust vibration control of full-car active suspensions with electrohydraulic actuators. IEEE Trans. Control Syst. Technol. 21(6), 2417 (2013). https://doi.org/10.1109/TCST.2012.2237174. http://ieeexplore.ieee.org/document/6450065/

    Article  Google Scholar 

  5. Gao, H., Zhao, Y., Sun, W.: Input-delayed control of uncertain seat suspension systems with human-body model. IEEE Trans. Control Syst. Technol. 18(3), 591 (2010). https://doi.org/10.1109/TCST.2009.2024929. http://ieeexplore.ieee.org/document/5208208/

    Article  Google Scholar 

  6. Gao, H., Sun, W., Shi, P.: Robust sampled-data H∞ control for vehicle active suspension systems. IEEE Trans. Control Syst. Technol. 18(1), 238 (2010). https://doi.org/10.1109/TCST.2009.2015653. http://ieeexplore.ieee.org/document/5291703/

    Article  Google Scholar 

  7. Zhu, Y.F.S.J., Zheng, Y.F.: Analysis of non-linear dynamics of a two-degree-of-freedom vibration system with nonlinear damping and non-linear spring. J. Sound Vib. 271(1-2), 15 (2004). https://doi.org/10.1016/S0022-460X(03)00249-9. http://www.sciencedirect.com/science/article/pii/S0022460X03002499

    Article  Google Scholar 

  8. Zhao, X., Zhang, L., Shi, P., Karimi, H.R.: Robust control of continuous-time systems with state-dependent uncertainties and its application to electronic circuits. IEEE Trans. Ind. Electron. 61(8), 4161 (2014). https://doi.org/10.1109/TIE.2013.2286568. http://ieeexplore.ieee.org/document/6642110/

    Article  Google Scholar 

  9. Its’haki-Allerhand, L., Shaked, U.: H∞ control of a class of nonlinear systems and its application to control of electronic circuit with nonlinear elements. In: 2008 IEEE 25th Convention of Electrical and Electronics Engineers in Israel, pp. 457–460. IEEE (2008). https://doi.org/10.1109/EEEI.2008.4736570. http://ieeexplore.ieee.org/document/4736570/

  10. Li, Z.: Robust control of PM spherical stepper motor based on neural networks. IEEE Trans. Ind. Electron. 56(8), 2945 (2009). https://doi.org/10.1109/TIE.2009.2023639. http://ieeexplore.ieee.org/document/5061562/

    Article  Google Scholar 

  11. Xia, Y., Zhu, Z., Fu, M., Wang, S.: Attitude tracking of rigid spacecraft with bounded disturbances. IEEE Trans. Ind. Electron. 58(2), 647 (2011). https://doi.org/10.1109/TIE.2010.2046611. http://ieeexplore.ieee.org/document/5439851/

    Article  Google Scholar 

  12. Liu, H., Lu, G., Zhong, Y.: Robust LQR attitude control of a 3-dof laboratory helicopter for aggressive maneuvers. IEEE Trans. Ind. Electron. 60(10), 4627 (2013). https://doi.org/10.1109/TIE.2012.2216233. http://ieeexplore.ieee.org/document/6290370/

    Article  Google Scholar 

  13. Zhang, Z., Park, J.H., Shao, H.: Adaptive synchronization of uncertain unified chaotic systems via novel feedback controls. Nonlinear Dyn. 81(1-2), 695 (2015). https://doi.org/10.1007/S11071-015-2020-6

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, Z., Zhao, X.: New results on robust control for a class of uncertain systems and its applications to Chua’s oscillator. Nonlinear Dynamics 84(4), 1929 (2016). https://doi.org/10.1007/s11071-016-2617-4

    Article  MathSciNet  MATH  Google Scholar 

  15. Xia, Y., Fu, M., Yang, H., Liu, G.-P.: Robust sliding-mode control for uncertain time-delay systems based on delta operator. IEEE Trans. Ind. Electron. 56(9), 3646 (2009). https://doi.org/10.1109/TIE.2008.2007987

    Article  Google Scholar 

  16. Hu, J., Wang, Z., Gao, H., Stergioulas, L.K.: Robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties, and randomly occurring nonlinearities. IEEE Trans. Ind. Electron. 59(7), 3008 (2012). https://doi.org/10.1109/TIE.2011.2168791

    Article  Google Scholar 

  17. Magni, R.S.L., De Nicolao, G., Magnani, L.: A stabilizing model-based predictive control algorithm for nonlinear systems. Automatica 37(9), 1351 (2001). https://doi.org/10.1016/S0005-1098(01)00083-8

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaloust, J., Qu, Z.: Attenuation of nonlinearly state-dependent uncertainties: Robust control design and its application to robotic manipulators. In: American Control Conference, pp. 3504–3505 (1994). https://doi.org/10.1109/ACC.1994.735231

  19. Pin, G., Raimondo, D., Magni, L., Parisini, T.: Robust model predictive control of nonlinear systems with bounded and state-dependent uncertainties. IEEE Trans. Autom. Control 54(7), 1681 (2009). https://doi.org/10.1109/TAC.2009.2020641. http://ieeexplore.ieee.org/document/5129691/

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhu, X.: Robust H∞ filtering for discrete-time systems with nonlinear uncertainties. In: 2006 1ST IEEE Conference on Industrial Electronics and Applications, pp. 1–6. IEEE (2006). https://doi.org/10.1109/ICIEA.2006.257251. http://ieeexplore.ieee.org/document/4025852/

  21. Liu, J., Wu, C., Wang, Z., Wu, L.: Reliable filter design for sensor networks using type-2 fuzzy framework. IEEE Trans. Ind. Inf. 13(4), 1742 (2017). https://doi.org/10.1109/TII.2017.2654323

    Article  Google Scholar 

  22. Wu, C., Liu, J., Xiong, Y., Wu, L.: Observer-based adaptive fault-tolerant tracking control of nonlinear nonstrict-feedback systems. IEEE Trans Neural Netw Learn Syst PP(99), 1 (2018). https://doi.org/10.1109/TNNLS.2017.2712619

    MathSciNet  Google Scholar 

  23. Alvarez, J., Rosas, D., Pena, J.: Analog implementation of a robust control strategy for mechanical systems. IEEE Trans. Ind. Electron. 56(9), 3377 (2009). https://doi.org/10.1109/TIE.2009.2020706. http://ieeexplore.ieee.org/document/4838891/

    Article  Google Scholar 

  24. Magni, L., De Nicolao, G., Scattolini, R., Allgȯwer, F.: Robust model predictive control for nonlinear discrete-time systems. Int. J. Robust Nonlinear Control 13(3-4), 229 (2003). https://doi.org/10.1002/rnc.815

    Article  MathSciNet  MATH  Google Scholar 

  25. Magni, L., Scattolini, R.: Robustness and robust design of mpc for nonlinear discrete-time systems. In: Assessment and Future Directions of Nonlinear Model Predictive Control, pp. 239–254. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-72699-9_19

  26. Chisci, G.Z.L., Rossiter, J.A.: Systems with persistent disturbances: predictive control with restricted constraints. Automatica 37(7), 1019 (2001). https://doi.org/10.1016/S0005-1098(01)00051-6. http://www.sciencedirect.com/science/article/pii/S0005109801000516

    Article  MathSciNet  MATH  Google Scholar 

  27. Ojaghi, P., Bigdeli, N., Rahmani, M.: An LMI approach to robust model predictive control of nonlinear systems with state dependent uncertainties. J. Process. Control. 47, 1 (2016). https://doi.org/10.1016/j.jprocont.2016.08.012. http://www.sciencedirect.com/science/article/pii/S0959152416301135

    Article  Google Scholar 

  28. Wang, X., Yang, L., Sun, Y., Deng, K.: Adaptive model predictive control of nonlinear systems with state-dependent uncertainties. Int. J. Robust Nonlinear Control. https://doi.org/10.1002/rnc.3787 (2017)

  29. Soltanpour, M.R., Khooban, M.H., Soltani, M.: Robust fuzzy sliding mode control for tracking the robot manipulator in joint space and in presence of uncertainties. Robotica 32(03), 433 (2014). https://doi.org/10.1017/S0263574713000702. http://www.journals.cambridge.org/abstract_S0263574713000702

    Article  Google Scholar 

  30. Sadeghi, M.S., Vafamand, N., Khooban, M.H.: LMI-based stability analysis and robust controller design for a class of nonlinear chaotic power systems. J. Franklin Inst. 353(13), 2835 (2016). https://doi.org/10.1016/j.jfranklin.2016.04.021. http://linkinghub.elsevier.com/retrieve/pii/S0016003215301058

    Article  MathSciNet  MATH  Google Scholar 

  31. Veysi, M., Soltanpour, M.R., Khooban, M.H.: A novel self-adaptive modified bat fuzzy sliding mode control of robot manipulator in presence of uncertainties in task space. Robotica 33(10), 2045 (2015). https://doi.org/10.1017/S0263574714001258. http://www.journals.cambridge.org/abstract_S0263574714001258

    Article  Google Scholar 

  32. Gholami, A., Markazi, A.H.D.: A new adaptive fuzzy sliding mode observer for a class of MIMO nonlinear systems. Nonlinear Dyn. 70(3), 2095 (2012). https://doi.org/10.1007/s11071-012-0602-0

    Article  MathSciNet  Google Scholar 

  33. Ho, H.F., Wong, Y.K., Rad, A.B.: Simulation modelling practice and theory adaptive fuzzy sliding mode control with chattering elimination for nonlinear siso systems. Simul. Model. Pract. Theory 17(7), 1199 (2009). https://doi.org/10.1016/j.simpat.2009.04.004

    Article  Google Scholar 

  34. Poursamad, A., Davaie-Markazi, A.H.: Robust adaptive fuzzy control of unknown chaotic systems. Appl. Soft Comput. 9(3), 970 (2009). https://doi.org/10.1016/j.asoc.2008.11.014. http://www.sciencedirect.com/science/article/pii/S1568494608001774

    Article  Google Scholar 

  35. Noroozi, N., Roopaei, M., Jahromi, M.Z.: Commun nonlinear sci numer simulat adaptive fuzzy sliding mode control scheme for uncertain systems. Commun. Nonlinear Sci. Numer. Simul. 14(11), 3978 (2009). https://doi.org/10.1016/j.cnsns.2009.02.015

    Article  MathSciNet  MATH  Google Scholar 

  36. Poursamad, A., Markazi, A.H.D.: Adaptive fuzzy sliding-mode control for multi-input multi-output chaotic systems. Chaos, Solitons & Fractals 42(5), 3100 (2009). https://doi.org/10.1016/j.chaos.2009.04.044

    Article  MathSciNet  MATH  Google Scholar 

  37. Nekoukar, V., Erfanian, A.: Adaptive fuzzy terminal sliding mode control for a class of MIMO uncertain nonlinear systems. Fuzzy Set. Syst. 179(1), 34 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Soltanpour, M.R., Khooban, M.H., Khalghani, M.R.: An optimal and intelligent control strategy for a class of nonlinear systems: adaptive fuzzy sliding mode. J. Vib. Control. 22(1), 159 (2016). https://doi.org/10.1177/1077546314526920

    Article  MathSciNet  MATH  Google Scholar 

  39. Niknam, T., Khooban, M.H., Kavousifard, A., Soltanpour, M.R.: An optimal type II fuzzy sliding mode control design for a class of nonlinear systems. Nonlinear Dyn. 75(1-2), 73 (2014). https://doi.org/10.1007/s11071-013-1050-1

    Article  MathSciNet  Google Scholar 

  40. Khooban, M.H., Niknam, T., Dehghani, M., Blaabjerg, F.: Free chattering hybrid sliding mode control for a class of non-linear systems: electric vehicles as a case study. IET Sci. Meas. Technol. 10(7), 776 (2016). https://doi.org/10.1049/iet-smt.2016.0091

    Article  Google Scholar 

  41. Craig, J.: Adaptive Control of Mechanical Manipulators, 1st edn. Addison-Wesley, New York (1988)

    Google Scholar 

  42. Wang, Z.: Lyapunov-Based Control Design for Uncertain MIMO Systems. Ph.D. Thesis. University of Central Florida, Orlando (2011)

    Google Scholar 

  43. Sastry, S.S., Bodson, M.: Adaptive Control: Stability, Convergence, and Robustness, 1St Edn. Courier Corporation, N. Chelmsford (1989)

    MATH  Google Scholar 

  44. Lebret, G., Liu, K., Lewis, F.L.: Dynamic analysis and control of a Stewart platform manipulator. J. Field Robot 10(5), 629 (1993). https://doi.org/10.1002/ROB.4620100506

    MATH  Google Scholar 

  45. Iqbal, S., Bhatti, A.I., Ahmed, Q.: Dynamic analysis and robust control design for stewart platform with moving payloads. In: Proceedings of the 17th World Congress, pp. 5324–5329 (2008). https://doi.org/10.3182/20080706-5-KR-1001.3036

  46. Chen, S.H., Fu, L.C.: Output feedback sliding mode control for a stewart platform with a nonlinear observer based forward kinematics solution. IEEE Trans. Control Syst. Technol. 21(1), 176 (2013). https://doi.org/10.1109/TCST.2011.2171964

    Article  Google Scholar 

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Authors and Affiliations

  1. Digital Control Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16844, Tehran, Iran

    Hamed Navvabi & A. H. D. Markazi

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  1. Hamed Navvabi
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  2. A. H. D. Markazi
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Correspondence to A. H. D. Markazi.

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Appendix

Appendix

In order to become clearer, list of main symbols of SM and controller are presented in Table 4.

Table 4 List of main symbols of SM and controller
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Navvabi, H., Markazi, A.H.D. New AFSMC Method for Nonlinear System with State-dependent Uncertainty: Application to Hexapod Robot Position Control. J Intell Robot Syst 95, 61–75 (2019). https://doi.org/10.1007/s10846-018-0850-4

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