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Stability analysis of a family of continuous state feedback synthesis: Theory and experiments

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  • Control Theory and Applications
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Abstract

A family of continuous state feedback synthesis based on sliding mode control is analyzed using a Lyapunov approach, such that the compensation of growing perturbations together with state variables is shown. Robustness properties of a family of controllers, varying from the well known twisting sliding mode control law to the PD controller, are studied. A non-smooth Lyapunov function is proposed such that global finite-time stability of the origin is demonstrated and tuning rules for the control gains are obtained. Moreover, since the Lyapunov function is strict, an upper bound for the convergence time of the closed loop system can be estimated, in spite of the growing perturbations with respect to the state. To illustrate the performance and robustness properties of the feedback synthesis, experimental results are presented, using a one-link pendulum as a test bed.

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References

  1. Y. Orlov, Y. Aoustin, and C. Chevallereau, “Finite time stabilization of a perturbed double integrator-part I: Continuous sliding mode-based output feedback synthesis,” IEEE Transactions on Automatic Control, vol. 56, no. 3, pp. 614–618, 2011.

    Article  MathSciNet  Google Scholar 

  2. S. V. Emelyanov, S. K. Korovin, and L. V. Levantovsky, “Second order sliding modes in controlling uncertain systems,” Soviet Journal of Compute and System Science, vol. 24, no. 4, pp. 63–68, 1986.

    MathSciNet  Google Scholar 

  3. Y. Orlov, J. Álvarez, L. Acho and L. Aguilar, “Global position regulation of friction manipulators via switched chattering control,” International Journal of Control, vol. 76, no. 14, pp. 1446–1452, 2003. [click]

    Article  MathSciNet  MATH  Google Scholar 

  4. Y. Orlov, “Discontinuous systems: Lyapunov analysis and robust synthesis under uncertainty conditions,” Communications and Control Engineering, Springer, London, UK, 2009.

    Google Scholar 

  5. A. Polyakov, D. Efimov, and W. Perruquetti, “Sliding mode control design for MIMO systems: implicit Lyapunov function approach,” Proc. of the European Control Conference, pp. 2612–2617, 2014.

    Google Scholar 

  6. G. Qin and Z. Duan, “Output chattering attenuation between two tracking controllers,” International Journal of Control, Automation and Systems, vol. 10, no. 3, pp 651–658, 2012. [click]

    Article  Google Scholar 

  7. S. P. Bhat and D. S. Bernstein, “Continuous finite-time stabilization of the translational and rotational double integrators,” IEEE Transactions on Automatic Control, vol. 43, no. 5, pp. 678–682, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Santiesteban, L. Fridman, and J. A. Moreno, “Finite time convergence analysis for twisting controller via a strict lyapunov function,” Proc. of the 11th Workshop on Variable Structure Systems, Mexico City, pp. 1–6, 2010.

    Google Scholar 

  9. R. Santiesteban, “Time convergence estimation of a perturbed double integrator: Family of continuous sliding mode based output feedback synthesis,” Proc. of the European Control Congress, Zurich, pp. 3764–3769, 2013.

    Google Scholar 

  10. R. Santiesteban, A. Gárate-García, and R. Bautista-Quintero, “A family of continuous state feedback synthesis: Lyapunov approach,” Proc. of the Congreso Nacional de Control Automático, Mexico, pp. 448–453, 2013.

    Google Scholar 

  11. A. F. Filippov, Differential Equations with Discontinuous Right-hand Sides, Kluwer Academic Publisher, 1988.

    Book  MATH  Google Scholar 

  12. Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, Sliding Mode Control and Observation, Springer, New York, 2014.

    Book  Google Scholar 

  13. M. T. Angulo, J. A. Moreno, and L. Fridman, “Robust exact uniformly convergent arbitrary order differentiator,” Automatica, vol. 4, no. 8, pp. 2489–1495, 2013.

    Article  MathSciNet  Google Scholar 

  14. J. A. Moreno, “On discontinuous observers for second order systems: properties, analysis and design,” Advances in Sliding Mode Control, Springer Berlin Heidelberg, pp. 243–265, 2013. [click]

    Chapter  Google Scholar 

  15. J. Dávila, L. Fridman, and A. Levant, “Second-order sliding-mode observer for mechanical systems,” IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1785–1789, 2005.

    Article  MathSciNet  Google Scholar 

  16. K. Lee, J. Back, and I. Choy, “Nonlinear disturbance observer based robust attitude tracking controller for quadrotor UAVs,” International Journal of Control, Automation and Systems, vol. 12, no. 6, pp. 1266–1275, 2014. [click]

    Article  Google Scholar 

  17. M. C. Pai, “Observer-based adaptive sliding mode control for robust tracking and model following,” International Journal of Control, Automation and Systems, vol. 11, no. 2, pp. 225–232, 2013. [click]

    Article  MathSciNet  Google Scholar 

  18. Y. Orlov, “Finite-time stability and robust control synthesis of uncertain switched systems,” SIAM Journal on Control and Optimization, vol. 43, pp. 1253–1271, 2005. [click]

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Kelly, J. Llamas, and R. Campa, “A measurement procedure for viscous and coulomb friction,” IEEE Transactions on Instrumentation and Measurement, vol. 49, no. 4, pp. 857–861, 2000.

    Article  Google Scholar 

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Correspondence to Araceli Gárate-García.

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Recommended by Associate Editor Huaping Liu under the direction of Editor Fuchun Sun. This work was partially supported by IDSCEA, CONACyT and UPA under project FOMIX AGS-2012-01-207544. The authors would like to thank CICESE research center for providing the facilities and the equipment for the experimental setup.

Raúl Santiesteban-Cos was born in Mexico City on May 13, 1977. He received his Master’s degree in Applied Mathematics from Instituto Potosino de Investigación Científica y Tecnológica (IPICYT) and the Ph.D. degree in Electronics and Telecommunications, with orientation in instrumentation and control from Centro de Investigación Científica y de Educación Superior de Ensenada (CICESE), Baja California, Mexico, in 2004 and 2008, respectively. He is currently a Professor in the Department of Metal-Mechanics from Tecnológico Nacional de México since 2010. His research interests include nonlinear dynamics, mechanical systems, friction, sliding modes, and ordinary differential equation with inclusions.

Araceli Gárate-García was born in Mexicali, Mexico. She received her B.Eng. degree in electronics in the Mexicali Institute of Technology in 2003, her master’s degree in electronics and telecommunications from CICESE research center, Mexico, in 2006 and a Ph.D. Dual Degree from CICESE and the University of Nantes, France in 2011. Since 2011 she has been with the Polytechnic University of Aguascalientes (UPA) in the Department of Research and Postgraduate Studies. She is the head of the research group “Control, Automation and Instrumentation of Systems” at UPA. Her research focuses on algebraic methods of control theory, analysis and control of time-delay systems, symbolic computation and systems modelling.

Oscar Montaño-Godinez obtained his Master’s degree in Electronics and Telecommunications from CICESE, Mexico, in 2012, and received his Ph.D. degree in Robotics and Automation from l’Université de Nantes, France in 2016. He joined 2J Antennas USA, Corp in 2017. His research interests include analysis of nonlinear systems, applications to electromechanical systems, modeling and control of robotic manipulators and bipeds.

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Santiesteban-Cos, R., Gárate-García, A. & Montaño-Godinez, O. Stability analysis of a family of continuous state feedback synthesis: Theory and experiments. Int. J. Control Autom. Syst. 15, 1011–1019 (2017). https://doi.org/10.1007/s12555-014-0513-6

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  • DOI: https://doi.org/10.1007/s12555-014-0513-6

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