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Uncertain polytopic LPV modelling of robot manipulators and trajectory tracking

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  • Robot and Applications
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Abstract

This research work proposes a full state systematic feedback control design method for some classes of non-linear systems which are forced to follow a specific desired trajectory, such as robotic systems, using uncertain polytopic linear parameter-varying (LPV) modelling approach. An LPV representation of the system is generated from linearization of its usual Lagrangian equation about a desired state trajectory and is reduced to an uncertain polytopic one. A vector of scheduling signals from the desired trajectory information is produced to construct the LPV model. The control gain matrix is derived by solving a set of linear matrix inequalities (LMIs) that returns the sufficiently small value of the time derivative of the Lyapunov function. A sufficient condition is proposed to guarantee the asymptotic stability of the closed-loop LPV systems against the uncertainties on the vertices. The proposed scheme is applied to controller synthesis of a two-degree-of-freedom robotic manipulator trajectory tracking problem.

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References

  1. S. D. Ramos, A. C. J. Domingos, and E. Vazquez Silva, “An algorithm to verify asymptotic stability conditions of a certain family of systems of differential dquations,” Applied Mathematical Sciences, vol. 8, no. 31, pp. 1509–1520, 2014. [click]

    Article  MathSciNet  Google Scholar 

  2. X. Xuping and P. J. Antsaklis, “Stabilization of secondorder LTI switched systems,” International Journal of Control, vol. 73, no. 14, pp. 1261–1279, 2000. [click]

    Article  MathSciNet  MATH  Google Scholar 

  3. Z. Sun, “A robust stabilizing law for switched linear systems,” International Journal of Control, vol. 77, no. 4, pp. 389–398, 2004. [click]

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Amato, F. Garofalo, L. Glielmo, and A. Pironti, “Robust and quadratic stability via polytopic set covering,” International Journal of Robust and Nonlinear Control, vol. 5, no. 8, pp. 745–756, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  5. Y. Yanga, C. Xianga, and T. H. Leea, “Sufficient and necessary conditions for the stability of second-order switched linear systems under arbitrary switching,” International Journal of Control, vol. 85, no. 12, pp. 1977–1995, 2012. [click]

    Article  MathSciNet  Google Scholar 

  6. R. H. Ordóñez-Hurtado and M. A. Duarte-Mermoud, “Finding common quadratic Lyapunov functions for switched linear systems using particle swarm optimisation,” International Journal of Control, vol. 85, no. 1, pp. 12–25, 2012. [click]

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Tong, L. Zhang, P. Shi, and C. Wang, “A common linear copositive Lyapunov function for switched positive linear systems with commutable subsystems,” International Journal of Systems Science, vol. 44, no. 11, pp. 1994–2003, 2013. [click]

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Xiang and J. Xiao, “Finite-time stability and stabilisation for switched linear systems,” International Journal of Systems Science, vol. 44, no. 2, pp. 384–400, 2013. [click]

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear systems: A survey of recent results,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 308–322, 2009.

    Article  MathSciNet  Google Scholar 

  10. J. Xiong, J. Lam, Z. Shu, and X. Mao, “Stability analysis of continuous-time switched systems with a random switching signal,” IEEE Trans. Automat. Contr., vol. 59, no. 1, pp. 180–186, 2014.

    Article  MathSciNet  Google Scholar 

  11. Z. She and B. Xue, “Discovering multiple Lyapunov functions for switched hybrid systems,” SIAM Journal on Control and Optimization, vol. 52, no. 5, pp. 3312–3340, 2014. [click]

    Article  MathSciNet  MATH  Google Scholar 

  12. W. A. De Souza, M. C. M. Teixeira, M. P. A. Santim, R. Cardim, and E. Assuncao, “On switched control design of linear time-invariant systems with polytopic uncertainties,” Mathematical Problems in Engineering, vol. 2013, 2013.

    Google Scholar 

  13. R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton University Press, New Jersey, 2012.

    MATH  Google Scholar 

  14. H. Yang, J. Bin, and V. Cocquempot, Stabilization of Switched Nonlinear Systems with Unstable Modes, Springer, 2014.

    Book  MATH  Google Scholar 

  15. C. Hoffmann and H. Werner, “A survey of linear parametervarying control applications validated by experiments or high-fidelity simulations,” IEEE Transactions on Control Systems Technology, vol. 23, no. 2, pp. 416–433, 2014.

    Article  Google Scholar 

  16. J. S. Shamma, “An overview of LPV systems,” in Control of Linear Parameter Varying Systems with Applications, J. Mohammadpour and C. W. Scherer, Eds. Springer US, 2012, pp. 3–26.

    Chapter  Google Scholar 

  17. C. Hoffmann, S. M. Hashemi, H. S. Abbas, and H. Werner, “Benchmark problem–nonlinear control of a 3-DOF robotic manipulator,” Proc. of the 52nd IEEE Conference on Decision and Control (CDC), pp. 5534–5539, 2013.

    Chapter  Google Scholar 

  18. G. Cai, C. Hu, B. Yin, H. He, and X. Han, “Gain-scheduled H2 controller synthesis for continuous-time polytopic LPV systems,” Mathematical Problems in Engineering, vol. 2014, pp. 1–14, 2014.

    Google Scholar 

  19. W. J. Rugh and J. S. Shamma, “Research on gain scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, 2000. [click]

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Xie, “Multi-objective H2/L2 performance controller synthesis for LPV systems,” Asian Journal of Control, vol. 14, no. 5, pp. 1273–1281, 2012. [click]

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Han and J. M. Lee, “Decentralized neural network control for guaranteed tracking error constraint of a robot manipulator,” International Journal of Control, Automation, and Systems, vol. 13, no. 4, pp. 906–915, 2015. [click]

    Article  Google Scholar 

  22. J. Yang, H. Su, Z. Li, D. Ao, and R. Song, “Adaptive control with a fuzzy tuner for cable-based rehabilitation robot,” International Journal of Control, Automation, and Systems, vol. 14, no. 3, pp. 865–875, 2016. [click]

    Article  Google Scholar 

  23. D. J. Stilwell and W. J. Rugh, “Interpolation of observer state feedback controllers for gain scheduling,” IEEE Transactions on Automatic Control, vol. 44, no. 6, pp. 1225–1229, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. J. Stilwell and W. J. Rugh, “Stability preserving interpolation methods for the synthesis of gain scheduled controllers,” Automatica, vol. 36, no. 5, pp. 665–671, 2000. [click]

    Article  MathSciNet  MATH  Google Scholar 

  25. Z. Yu, H. Chen, and P. Woo, “Gain scheduled LPV Hinfinity control based on LMI approach for a robotic manipulator,” Journal of Robotic Systems, vol. 19, no. 12, pp. 585–593, 2002. [click]

    Article  MATH  Google Scholar 

  26. C. Hua, Y. Yang, and P. X. Liu, “Output-feedback adaptive nontrol of networked teleoperation system with timevarying delay and bounded inputs,” IEEE/ASME Transactions on Mechatronics, vol. 20, no. 5, pp. 2009–2020, 2015.

    Article  Google Scholar 

  27. C. Hua, L. Zhang, and X. Guan, “Decentralized output feedback adaptive NN tracking control for timedelay stochastic nonlinear systems with prescribed performance,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 11, pp. 2749–2759, 2015.

    Article  MathSciNet  Google Scholar 

  28. C. Hoffmann and H. Werner, “Complexity of implementation and synthesis in linear parameter-varying control,” Proc. of the 19th IFAC World Congress, 2014, pp. 11749–11760. [click]

    Google Scholar 

  29. S. M. Hashemi, H. S. Abbas, and H. Werner, “LPV modelling and control of a 2-DOF robotic manipulator using PCA-based parameter set mapping,” Proc. of the 48th IEEE Conference on Decision and Control, pp. 7418–7423, 2009.

    Google Scholar 

  30. S. M. Hashemi, H. S. Abbas, and H. Werner, “Lowcomplexity linear parameter-varying modeling and control of a robotic manipulator,” Control Engineering Practice, vol. 20, no. 3, pp. 248–257, 2012. [click]

    Article  Google Scholar 

  31. M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control, John Wiley & Sons, INC, USA, 2006.

    Google Scholar 

  32. K. Kozlowski, Robot Motion and Control, Recent Developments, Springer-Verlag, London, UK, 2006.

    Book  MATH  Google Scholar 

  33. H. S. Ali, L. Boutat-Baddas, Y. Becis-Aubry, and M. Darouach, “H-infinity control of a SCARA robot using polytopic LPV approach,” Proc. of 14th Mediterranean Conference on Control and Automation, pp. 1–5, 2006.

    Google Scholar 

  34. R. Kelly, V. S. Davila, and A. Loria, Control of Robot Manipulators in Joint Space, Springer-Verlag, London, UK, 2005.

    Google Scholar 

  35. A. Kwiatkowski and H. Werner, “PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding,” IEEE Transactions on Control Systems Technology, vol. 16, no. 4, pp. 781–788, 2008.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Engineering, Shahed University, Khalij-Fars Highway, Tehran, Iran

    Mohammad Bagher Abolhasani Jabali & Mohammad Hosein Kazemi

Authors
  1. Mohammad Bagher Abolhasani Jabali
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  2. Mohammad Hosein Kazemi
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Corresponding author

Correspondence to Mohammad Hosein Kazemi.

Additional information

Recommended by Associate Editor Changchun Hua under the direction of Editor Myo Taeg Lim. This work is supported by Shahed University.

Mohammad Bagher Abolhasani Jabali received the B.Sc. and M.Sc. degrees from Amirkabir University, Tehran, Iran, in 2004 and 2007, respectively. He is currently pursuing a Ph.D. degree at Shahed University, Tehran, Iran. He has taught at Sadra Institute of Higher Education of Tehran from 2008. Currently, he is a Senior Expert with Iran Grid Management Company (IGMC). His current research interests include control and analysis of power system dynamics and stability.

Mohammad Hossein Kazemi received his B.Sc. degree in Electrical Engineering from the Khajeh Nasir University of Technology, Tehran, Iran. He received his M.Sc. and Ph.D. degrees in control engineering from the Sharif University of Technology, Tehran, Iran, and Amirkabir University, Tehran, Iran, in 1995 and 2001, respectively. He is currently an Assistant Professor in the Department of Electrical Engineering at the Shahed University, Tehran, Iran. His research interests include robotics, adaptive and robust control.

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Jabali, M.B.A., Kazemi, M.H. Uncertain polytopic LPV modelling of robot manipulators and trajectory tracking. Int. J. Control Autom. Syst. 15, 883–891 (2017). https://doi.org/10.1007/s12555-015-1432-1

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  • DOI: https://doi.org/10.1007/s12555-015-1432-1

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