Skip to main content
SpringerLink
Log in

A Review on Self-oscillating Relay Feedback Systems and Its Application to Underactuated Systems with Degree of Underactuation One

  • Chapter
Advances in Sliding Mode Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 440))

Abstract

A tool for the design of a periodic motion in underactuated systems via generating a self-excited oscillation of a desired amplitude and frequency driven by a variable structure control is reviewed. In this chapter, we overview the capabilities of the two-relay controller to induce oscillations in dynamical systems. In this chapter, we will focus on underactuated mechanical systems with degree of underactuation one, that is, n degrees-of-freedom and n – 1 actuators only. Threemethods to set the frequency and amplitude of oscillation and its application to one-degree of underactuation systems are reviewed: describing function method, Locus of the perturbed relay system design (LPRS), and Poincaré map based design. Theoretical and practical open problems are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Aguilar, L., Boiko, I., Fridman, L., Freidovich, L.: Generating oscillations in inertia wheel pendulum via two relay controller. International Journal of Robust and Nonlinear Control 22, 318–330 (2012)

    Article  MathSciNet  Google Scholar 

  2. Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Generation of periodic motions for underactuated mechanical system via second-order sliding-modes. In: Proc. of the American Control Conference, Minnesota, USA, pp. 5396–5400 (2006)

    Google Scholar 

  3. Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Output excitation via continuous sliding-modes to generate periodic motion in underactuated systems. In: Proc. of the IEEE Conference on Decision and Control, San Diego, USA, pp. 1629–1634 (2006)

    Google Scholar 

  4. Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Output excitation via second-order sliding-modes to generate periodic motion for underactuated systems. In: Proc. of the 9th International Workshop on Variable Structure Systems, Alghero, Italy, pp. 359–364 (2006)

    Google Scholar 

  5. Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Generating self-excited oscillations for underactuated mechanical systems via two relay controller. International Journal of Control 82(9), 1678–1691 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Generating self-excited oscillations via two-relay controller. IEEE Trans. on Automatic Control 54(2), 416–420 (2009)

    Article  MathSciNet  Google Scholar 

  7. Aguilar, L., Boiko, I., Iriarte, R., Fridman, L.: Periodic motion of underactuated mechanical systems self-generated by variable structure controllers: design and experiments. In: 2007 European Control Conference, Kos, Greece, pp. 3796–3801 (2007)

    Google Scholar 

  8. Atherton, D.: Nonlinear control engineering–Describing Function Analysis and Design. Van Nostrand, Workingham (1975)

    Google Scholar 

  9. Berkemeier, M., Fearing, R.: Tracking fast inverted trajectories of the underactuated acrobot. IEEE Transactions on Robotics and Automation 15(4), 740–750 (1999)

    Article  Google Scholar 

  10. Boiko, I.: Oscillations and transfer properties of relay servo systems – the locus of a perturbed relay system approach. Automatica 41, 677–683 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boiko, I.: Discontinuous control systems: Frequency-domain analysis and design. Birkhäuser, Boston (2009)

    Google Scholar 

  12. Chevallereau, C., Abba, G., Aoustin, Y., Plestan, E., Canudas-de-Wit, C., Grizzle, J.: Rabbit: A testbed for advanced control theory. IEEE Control Systems Magazine 23(5), 57–79 (2003)

    Article  Google Scholar 

  13. Dranga, O., Navy, I.: Stability analysis of feedback controlled resonant DC-DC converter using Poincaré map function. In: IEEE Int. Symposium on Industrial Electronics, Pusan, Korea, pp. 2142–2147 (2001)

    Google Scholar 

  14. Estrada, A., Fridman, L.: Exact compensation of unmatched perturbation via quasi-continuous HOSM. In: 47th IEEE Conference on Decision and Control, Cancún, México, pp. 2202–2207 (2008)

    Google Scholar 

  15. Fendrich, O.: Describing functions in limit cycles. IEEE Transactions on Automatic Control 37(4), 486–488 (1992)

    Article  MathSciNet  Google Scholar 

  16. Fridman, L.: An averaging approach to chattering. IEEE Transactions on Automatic Control 46(8), 1260–1265 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fridman, L.: Slow periodic motion in variable structure systems. International Journal of Systems Science 33(14), 1145–1155 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grizzle, J., Moog, C., Chevallereau, C.: Nonlinear control of mechanical systems with an unactuated cyclic variable. IEEE Transactions on Automatic Control 50(5), 559–576 (2005)

    Article  MathSciNet  Google Scholar 

  19. Lai, J.: Power conditioning circuit topologies: Power conversion from low-voltage dc to high-voltage ac for single-phase grid-tie applications. IEEE Ind. Electronics Mag. 3(2), 24–34 (2009)

    Article  Google Scholar 

  20. Levant, A.: High-order sliding modes: differentiation and output-feedback control. International Journal of Control 76, 924–941 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Martínez, S., Cortés, J., Bullo, F.: Motion Planning and Control Problems for Underactuated Robot. In: Bicchi, A., Christensen, H.I., Prattichizzo, D. (eds.) Control Problems in Robotics. STAR, vol. 4, pp. 59–74. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  22. Martínez-Salamero, L., Valderrama-Blavi, H., Giral, R., Alonso, C., Estibals, B., Cid-Pastor, A.: Self-oscillating DC-to-DC switching converters with transformer characteristics. IEEE Transactions on Aerospace and Electronic Systems 41(2), 710–716 (2005)

    Article  Google Scholar 

  23. Nakamura, Y., Suzuki, T., Koinuma, M.: Nonlinear behavior and control of a nonholonomic free-joint manipulator. IEEE Transactions on Robotics and Automation 13(6), 853–862 (1997)

    Article  Google Scholar 

  24. Orlov, Y., Aguilar, L., Acho, L., Ortiz, A.: Asymptotic harmonic generator and its application to finite time orbital stabilization of a friction pendulum with experimental verification. International Journal of Control 81(2), 227–234 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Orlov, Y., Riachy, S., Floquet, T., Richard, J.: Stabilization of the cart-pendulum system via quasi-homogeneous switched control. In: Proc. of the 2006 Int. Workshop on Variable Structure Systems, Alghero, Italy, pp. 139–142 (2006)

    Google Scholar 

  26. Robinett, I.R., Wilson, D.: What is a limit cycle? International Journal of Control 81(12), 1886–1900 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sanchis, P., Ursua, A., Gubia, E., Marroyo, L.: Buck-boost DC-AC inverter for a new control strategy. In: 35th Annual IEEE Power Electronics Specialist Conference, Aachen, Germany, pp. 3994–3998 (2004)

    Google Scholar 

  28. Santiesteban, R., Floquet, T., Orlov, Y., Riachy, S., Richard, J.: Second order sliding mode control for underactuated mechanical system II: orbital stabilization of an inverted pendulum with application to swing up/balancing control. International Journal of Robust Nonlinear Control 18(4–5), 544–556 (2008)

    Article  MathSciNet  Google Scholar 

  29. Shiriaev, A., Perram, J., Canudas-de-Wit, C.: Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual contraint approach. IEEE Transactions on Automatic Control 50(8), 1164–1176 (2005)

    Article  MathSciNet  Google Scholar 

  30. Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control in Electromechanical Systems. CRC Press, Boca Raton (1999)

    Google Scholar 

  31. Varigonda, S., Georgiou, T.: Dynamics of relay relaxation oscillators. IEEE Transactions on Automatic Control 46(1), 65–77 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Youssef, M., Jain, P.: A novel single stage AC–DC self-oscillating series-parallel resonant converter. IEEE Transactions on Power Electronics 21(6), 1735–1744 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto Politécnico Nacional, Ave. del parque 1310 Mesa de Otay, Tijuana, Baja California, Mexico, 22510

    Luis T. Aguilar & Rafael Iriarte

  2. Electrical Engineering Department, The Petroleum Institute, P.O. BOX 2533, Abu Dhabi, U.A.E.

    Igor Boiko

  3. Departamento de Ingeniería de Control, Universidad Nacional Autónoma de México, UNAM, Mexico City, México, D.F., 04510

    Leonid Fridman

Authors
  1. Luis T. Aguilar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Boiko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leonid Fridman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rafael Iriarte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis T. Aguilar .

Editor information

Editors and Affiliations

  1. , Interdisciplinary Programme in Systems, Indian Institute of Technology Bombay, ACRE Building 114 B, Mumbai, 400 076, Maharashtra, India

    B Bandyopadhyay

  2. , Dept. of Electrical Engineering, Indian Institute of Technology Delhi, Room No : II/335, New Delhi, 110 016, India

    S Janardhanan

  3. , Department of Electronics, University of Kent, Jennison Building - Room 1.54, Kent, CT2 7NT, United Kingdom

    Sarah K. Spurgeon

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Aguilar, L.T., Boiko, I., Fridman, L., Iriarte, R. (2013). A Review on Self-oscillating Relay Feedback Systems and Its Application to Underactuated Systems with Degree of Underactuation One. In: Bandyopadhyay, B., Janardhanan, S., Spurgeon, S. (eds) Advances in Sliding Mode Control. Lecture Notes in Control and Information Sciences, vol 440. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36986-5_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-36986-5_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36985-8

  • Online ISBN: 978-3-642-36986-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation