Skip to main content
SpringerLink
Log in
Book cover

Self-Oscillations in Dynamic Systems pp 53–64Cite as

Birkhäuser

Self-Oscillation via Locus of a Perturbed Relay System Design (LPRS)

  • Chapter

  • 752 Accesses

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

Abstract

The Poincaré map considered above is a precise tool to find gains of the TRC . The drawback of this approach is in its complexity, which entails extensive computations. This chapter presents an alternative approach—based on the LPRS method, which in the solution of the analysis problem provides exact values of the parameters of self-excited oscillations and a precise solution of the input–output problem, when the plant is linear. Application of this method involved the use of specific computation formulas available within the LPRS method. Unlike other publications on the LPRS method that were focused on analysis, this chapter provides LPRS-based design of self-excited periodic motions. The experiments with inertia wheel pendulum are presented below to illustrate the results of this design.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

References

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

    Article  MathSciNet  Google Scholar 

  2. Albea, C., Canudas de Wit, C., Gordillo, F.: Adaptive control of the boost DC-AC converter. In: Proceedings of the IEEE International Conference on Control Applications, pp. 611–616. Singapore (2007)

    Google Scholar 

  3. Albea, C., Gordillo, F., Canudas de Wit, C.: Adaptive control design for a boost inverter. Control Eng. Pract. 19, 32–44 (2011)

    Google Scholar 

  4. Appleton, E.: Automatic synchronization of triode oscillators. Proc. Camb. Philos. Soc. 21(3), 231 (1923)

    Google Scholar 

  5. Arimoto, S.: Control Theory of Non-linear Mechanical Systems: A Passivity-Based and Circuit-Theoretic Approach. Oxford Engineering Science Series. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  6. Astashev, V.K., Korendyasev, G.K.: Thermomechanical model of cutter selfoscillation in perpendicular free cutting. J. Mach. Manuf. Reliab. 41(6), 3–10 (2012)

    Article  Google Scholar 

  7. Åström, K., Block, D., Spong, M.: The Reaction Wheel Pendulum. Lecture Notes for the Reaction Wheel Pendulum (Part of the Mechatronics Control Kit). Morgan & Claypool Publisher, San Rafael (2001)

    Google Scholar 

  8. Atherton, D.: Nonlinear Control Engineering–Describing Function Analysis and Design. Van Nostrand, Wokingham (1975)

    Google Scholar 

  9. Bejarano, F., Fridman, L.: High order sliding mode observer for linear systems with unbounded unknown inputs. Int. J. Robust Nonlinear Control 9, 1920–1929 (2010)

    MathSciNet  Google Scholar 

  10. Berkemeier, M., Fearing, R.: Tracking fast inverted trajectories of the underactuated acrobot. IEEE Trans. Robot. Autom. 15(4), 740–750 (1999)

    Article  Google Scholar 

  11. Best, R.: Phase-Locked Loops: Design, Simulation, and Application, 4th edn. McGraw-Hill, New York (1999)

    Google Scholar 

  12. Block, D., Astrom, K., Spong, M.: The Reaction Wheel Pendulum. Synthesis Lectures on Control and Mechatronics #1. Morgan & Claypool Publisher, San Rafael (2007)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Boiko, I.: Discontinuous Control Systems: Frequency-Domain Analysis and Design. Birkhäuser, Boston (2009)

    Google Scholar 

  15. Boiko, I., Fridman, L.: Analysis of chattering in continuous sliding-mode controllers. IEEE Trans. Autom. Control 50(9), 1442–1446 (2005)

    Article  MathSciNet  Google Scholar 

  16. Boiko, I., Fridman, L.: Frequency domain analysis of second order sliding modes. In: Advances in Variable Structure and Sliding Mode Control, pp. 125–142. Springer, Berlin (2006)

    Google Scholar 

  17. Boiko, I., Fridman, L., Castellanos, M.: Analysis of second-order sliding-mode algorithms in the frequency domain. IEEE Trans. Autom. Control 49(4), 946–950 (2004)

    Article  MathSciNet  Google Scholar 

  18. Boiko, I., Fridman, L., Pisano, A., Usai, E.: Analysis of chattering in systems with second-order sliding modes. IEEE Trans. Autom. Control 52(11), 2085–2102 (2007)

    Article  MathSciNet  Google Scholar 

  19. Cáceres, R., Barbi, I.: A boost DC–AC converter: analysis, design, and experimentation. IEEE Trans. Power Electron. 14(1), 134–141 (1999)

    Article  Google Scholar 

  20. Canudas de Wit, C., Espiau, B., Urrea, C.: Orbital stabilization of underactuated mechanical systems. In: Proceedings of the 15th IFAC World Congress. Barcelona (2002)

    Google Scholar 

  21. Cardon, S., Iberall, A.: Oscillations in biological systems. Biosystems 3(3), 237–249 (1970)

    Article  Google Scholar 

  22. Chen, F., Liang, T., Lin, R., Chen, J.: A novel self-oscillating, boos-derived DC–DC converter with load regulation. IEEE Trans. Power Electron. 20(1), 65–74 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Choukchou-Braham, A., Cherki, B., Djemaï, M., Busawon, K.: Analysis and Control of Underactuated Mechanical Systems. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  25. Craig, J.: Introduction to Robotics: Mechanics and Control. Addison-Wesley Publishing, Massachusetts (1989)

    MATH  Google Scholar 

  26. di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Applied Mathematical Sciences, vol. 163. Springer, London (2008)

    Google Scholar 

  27. Epstein, I.R.: Nonlinear oscillations in chemical and biological systems. Physica D: Nonlinear Phenomena 51(1–3), 152–160 (1991)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  29. Estrada, A., Fridman, L.: Quasi-continuous HOSM control for systems with unmatched perturbations. Automatica 46(11), 1916–1919 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Fantoni, I., Lozano, R.: Nonlinear Control for Underactuated Mechanical Systems. Springer, London (2001)

    Google Scholar 

  31. Fendrich, O.: Describing functions in limit cycles. IEEE Trans. Autom. Control 37(4), 486–488 (1992)

    Article  MathSciNet  Google Scholar 

  32. Ferreira, A., Bejarano, F.J., Fridman, L.: Robust control with exact uncertainties compensation: with or without chattering? IEEE Trans. Control Syst. Technol. 19(5), 969–975 (2011)

    Article  Google Scholar 

  33. Ferreira, A., Rios, H., Rosales, A.: Robust regulation for a 3-DOF helicopter via sliding-mode observation and identification. J. Frankl. Inst. 349(2), 700–718 (2012)

    Article  MATH  Google Scholar 

  34. Filippov, A.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publisher, Dordrecht (1988)

    Book  MATH  Google Scholar 

  35. Fradkov, A., Pogromsky, A.: Introduction to Control of Oscillations and Chaos. Series on Nonlinear Science, vol. 35. World Scientific, Singapore (1998)

    Google Scholar 

  36. Fridman, L.: An averaging approach to chattering. IEEE Trans. Autom. Control 46(8), 1260–1265 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Fridman, L.: Slow periodic motion in variable structure systems. Int. J. Syst. Sci. 33(14), 1145–1155 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Fridman, L.: Slow periodic motions with internal sliding modes in the singularly perturbed relay systems. Int. J. Control. 75(7), 524–537 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Gelb, A., Velde, W.V.: Multiple-Input Describing Functions and Nonlinear Systems Design. McGraw-Hill, New York (1968)

    Google Scholar 

  40. Grizzle, J., Abba, G., Plestan, F.: Asymptotically stable walking for biped robots: analysis via systems with impulsive effects. IEEE Trans. Autom. Control 48(1), 51–64 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Hamel, B.: Contribution a l’etude mathematique des systemes de reglage par tout-ou-rien, C.E.M.V. Service Technique Aeronautique 17 (1949)

    Google Scholar 

  43. Hara, Y., Jahan, R.A.: Activation energy of aggregation-disaggregation self-oscillation of polymer chain. Int. J. Mol. Sci. 13, 16281–16290 (2012)

    Article  Google Scholar 

  44. Hsu, J., Meyer, A.: Modern Control Principles and Applications. McGraw Hill, New York (1968)

    MATH  Google Scholar 

  45. Hurmuzlu, Y., Génot, F., Brogliato, B.: Modeling, stability and control of biped robots—a general framework. Automatica 40, 1647–1664 (2004)

    Article  MATH  Google Scholar 

  46. Isidori, A.: Nonlinear Control Systems: An Introduction. Springer, Berlin (1989)

    Book  Google Scholar 

  47. Isidori, A.: Nonlinear Control Systems, 3rd edn. Springer, London (1995)

    Book  MATH  Google Scholar 

  48. Kelly, R., Llamas, J., Campa, R.: A measurement procedure for viscous and coulomb friction. IEEE Trans. Instrum. Meas. 49(4), 857–861 (2000)

    Article  Google Scholar 

  49. Khalil, H.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002)

    MATH  Google Scholar 

  50. Koenig, D.R., Weig, E.M.: Voltage-sustained self-oscillation of a nano-mechanical electron shuttle. Appl. Phys. Lett. 101(213111), 1–5 (2012)

    Google Scholar 

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

    Article  Google Scholar 

  52. Lee, H., Kim, Y., Jeon, H.: On the linearization via restricted class of dynamic feedback. IEEE Trans. Autom. Control 45(7), 1385–1391 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  53. Levant, A.: High-order sliding modes: differentiation and output feedback control. Int. J. Control 76(11), 924–941 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  54. Levant, A.: High-order sliding mode controllers. IEEE Trans. Autom. Control 50(11), 1812–1816 (2005)

    Article  MathSciNet  Google Scholar 

  55. Levant, A.: Chattering analysis. IEEE Trans. Autom. Control 55(6), 1380–1389 (2010)

    Article  MathSciNet  Google Scholar 

  56. Lin, A.T., Lin, C.: Peniotron forward wave self-oscillations. Appl. Phys. Lett. 64(9), 1088–1090 (1994)

    Article  Google Scholar 

  57. Loeb, J.: Frequency response. In: Advances in Nonlinear Servo Theory, pp. 260–268. The Macmillan Company, New York (1956)

    Google Scholar 

  58. Mancini, R.: Op Amps For Everyone: Design Reference. Texas Instruments, Dallas (2002)

    Google Scholar 

  59. 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 Trans. Aerosp. Electron. Syst. 41(2), 710–716 (2005)

    Article  Google Scholar 

  60. Melkikh, A.V., Dolgirev, Y.E.: Self-oscillations in oscillating heat pipes. High Temp. 44(4), 542–547 (2006)

    Article  Google Scholar 

  61. Meza, M., Aguilar, L., Shiriaev, A., Freidovich, L., Orlov, Y.: Periodic motion planning and nonlinear H ∞ tracking control of a 3-DOF underactuated helicopter. Int. J. Syst. Sci. 42(5), 829–838 (2011)

    Article  MATH  Google Scholar 

  62. Molinari, B.P.: A strong controllability and observability in linear multivariable control. IEEE Trans. Autom. Control 21, 761–764 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  63. Morris, K., Rebarber, R.: Invariant zeros of siso infinite-dimensional systems. Int. J. Control 83(12), 2573–2579 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  64. Neǐmark, Y.: The Method of Point Transformations in the Theory of Nonlinear Oscillations. Nauka, Moscow (1972)

    Google Scholar 

  65. Oh, S., Pathak, K., Agrawal, S., Pota, H., Garratt, M.: Approaches for a tether-guided landing of an autonomous helicopter. IEEE Trans. Robot. 22(3), 536–544 (2006)

    Article  Google Scholar 

  66. Olivier, J.C., Le Claire, J.C., Loron, L.: An efficient switching frequency limitation process applied to high dynamic voltage supply. IEEE Trans. Power Electron. 23(1), 153–162 (2008)

    Article  Google Scholar 

  67. Orlov, Y.: Discontinuous Systems: Lyapunov Analysis and Robust Synthesis Under Uncertain Conditions. Springer, London (2009)

    Google Scholar 

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

    Google Scholar 

  69. 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. Int. J. Control 81(2), 227–234 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  70. Plestan, F., Grizzle, J., Westervelt, E., Abba, G.: Stable walking of a 7-DOF biped robot. IEEE Trans. Robot. Autom. 19(4), 653–668 (2003)

    Article  Google Scholar 

  71. Quanser: 3D helicopter system with active disturbance. Techical report (2004)

    Google Scholar 

  72. Rabinovich, M.I.: Self-oscillations of distributed systems. Radiophys. Quantum Electron. 17(4), 361–385 (1974)

    Article  Google Scholar 

  73. Raptis, I., Valavanis, K., Vachtsevanos, G.: Linear tracking control for small-scale unmanned helicopters. IEEE Trans. Control Syst. Technol. 20(4), 995–1009 (2012)

    Article  Google Scholar 

  74. Robinett, III, R., Wilson, D.: What is a limit cycle? Int. J. Control 81(12), 1886–1900 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  75. Romanov, Y.A., Romanova, Y.Y.: Self-oscillations in semiconductor superlattices. J. Exp. Theor. Phys. 91(5), 1033–1045 (2000)

    Article  Google Scholar 

  76. 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, pp. 3994–3998. Aachen (2004)

    Google Scholar 

  77. Sanchis, P., Ursæa, A., Gubia, E., Marroyo, L.: boost DC–AC inverter: a new control strategy. IEEE Trans. Power Electron. 20(2), 343–353 (2005)

    Google Scholar 

  78. 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. Int. J. Robust Nonlinear Control 18(4–5), 544–556 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  79. Sastry, S.: Nonlinear Systems: Analysis, Stability, and Control. Springer, New York (1999)

    Book  MATH  Google Scholar 

  80. Shiriaev, A., Perram, J., Canudas-de-Wit, C.: Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraint approach. IEEE Trans. Autom. Control 50(8), 1164–1176 (2005)

    Article  MathSciNet  Google Scholar 

  81. Shiriaev, A., Freidovich, L., Robertsson, A., Sandberg, A.: Virtual-holonomic-constraints-based design stable oscillations of furuta pendulum: theory and experiments. IEEE Trans. Robot. 23(4), 827–832 (2007)

    Article  Google Scholar 

  82. Shiriaev, A., Freidovich, L., Manchester, I.: Can we make a robot ballerina perform a pirouette? orbital stabilization of periodic motions of underactuated mechanical systems. Annu. Rev. Control. 32(2), 200–211 (2008)

    Article  Google Scholar 

  83. Shiriaev, A., Freidovich, L., Gusev, S.: Transverse linearization for controlled mechanical systems with several passive degrees of freedom. IEEE Trans. Autom. Control 55(4), 893–906 (2010)

    Article  MathSciNet  Google Scholar 

  84. Shtessel, Y., Edwards, C., Fridman, L., Levant, A.: Sliding Mode Control and Observation. Birkhäuser, Boston (2013)

    Google Scholar 

  85. Spong, M., Vidyasagar, M.: Robot Dynamics and Control. Wiley, New York (1989)

    Google Scholar 

  86. Trentelman, H., Stoorvogel, A., Hautus, M.: Control Theory for Linear Systems. Springer, London (2001)

    Book  MATH  Google Scholar 

  87. Tsypkin, Y.: Relay Control Systems. Cambridge University Press, Cambridge (1984)

    MATH  Google Scholar 

  88. Utkin, V.: Sliding Modes in Control Optimization. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

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

    Google Scholar 

  90. Varigonda, S., Georgiou, T.: Dynamics of relay relaxation oscillators. IEEE Trans. Autom. Control 46(1), 65–77 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  91. Weldon, J., Alemán, B., Sussman, A., Gannett, W., Zettl, A.: Sustained mechanical self-oscillations in carbon nanotubes. Nano Lett. 10, 1728–1733 (2010)

    Article  Google Scholar 

  92. Westerberg, S., Mettin, U., Shiriaev, A., Freidovich, L., Orlov, Y.: Motion planning and control of a simplified helicopter model based on virtual holonomic constraints. In: Proceedings of the 14th International Conference on Advanced Robotics, pp. 1–6. Munich (2009)

    Google Scholar 

  93. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Text in Applied Mathematics, 2nd edn. Springer, New York (2000)

    Google Scholar 

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

    Article  Google Scholar 

  95. Zheng, B., Zhong, Y.: Robust attitude regulation of a 3-DOF helicopter benchmark: theory and experiments. IEEE Trans. Ind. Electron. 58(2), 660–670 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CITEDI, Instituto Politecnico Nacional, Tijuana, BC, Mexico

    Luis T. Aguilar

  2. Department of Electrical Engineering, The Petroleum Institute, Abu Dhabi, UAE

    Igor Boiko

  3. Division de Ingenieria Electrica, Universidad Nacional Autonoma de Mexico, Mexico City, DF, Mexico

    Leonid Fridman & Rafael Iriarte

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

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Aguilar, L.T., Boiko, I., Fridman, L., Iriarte, R. (2015). Self-Oscillation via Locus of a Perturbed Relay System Design (LPRS) . In: Self-Oscillations in Dynamic Systems. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-23303-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation