Skip to main content
SpringerLink
Log in

Switched Systems, Lur’e Systems, Absolute Stability, Aizerman Problem

  • Nonlinear Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We distinguish a subclass of switched linear systems that we call pairwise connected. We show that the dynamics of such systems can be described by Lur’e systems. For pairwise connected systems, we obtain a sufficient frequency-domain condition for the existence of a quadratic Lyapunov function. The well-known Aizerman problem is reformulated for switched linear systems. We show an example of a system with switchings between three linear third order subsystems for which Aizerman’s problem has a positive solution.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lur’e, A.I. and Postnikov, V.N., On Stability Theory for Controllable Systems, Prikl. Mat. Mekh., 1944, vol. 8, no. 3, pp. 246–248.

    MATH  Google Scholar 

  2. Liberzon, M.R., Essays on the Absolute Stability Theory, Autom. Remote Control, 2006, vol. 67, no. 10, pp. 1610–1644.

    Article  MathSciNet  MATH  Google Scholar 

  3. Aizerman, M.A., On One Problem Related to Stability “At Large” for Dynamical Systems, Usp. Mat. Nauk, 1949, vol. 14, no. 4(32), pp. 187–188.

    Google Scholar 

  4. Pliss, V.A., Nekotorye problemy teorii ustoichivosti dvizheniya v tselom (Some Problems of Stability Theory for Motion as a Whole), Leningrad: Leningr. Gos. Univ., 1958.

    Google Scholar 

  5. Barabanov, N.E., On the Aizerman Problem for Nonstationary Third Order Systems, Differ. Uravn., 1993, vol. 29, no. 10, pp. 1659–1668.

    Google Scholar 

  6. Pyatnitskii, E.S., The Work of M.A. Aizerman on Control Theory, Motion Stability Theory, and Oscillation Theory, in Mark Aronovich Aizerman (1918–1992), Moscow: Fizmatlit, 2003, pp. 82–104.

    Google Scholar 

  7. Aizerman, M.A. and Gantmakher, F.R., Absolyutnaya ustoichivost’ nelineinykh reguliruemykh sistem (Absolute Stability of Nonlinear Controllable Systems), Moscow: Akad. Nauk, 1963.

    Google Scholar 

  8. Jayawardhana, B., Logemann, H., and Ryan, E.P., The Circle Criterion and Input-to-State Stability: New Perspectives on a Classical Result, IEEE Control Syst. Mag., 2011, vol. 31, no. 4, pp. 32–67.

    MathSciNet  MATH  Google Scholar 

  9. Shorten, R., Wirth, F., Mason, O., et al., Stability Sriteria for Switched and Hybrid Systems, SIAM Rev., 2007, no. 4, pp. 545–592.

    Article  MATH  Google Scholar 

  10. Duignan, R. and Curran, P.F., An Investigation on the Absolute Stability of Discrete and Continuous Time Lur’e Systems, 2007 Int. Symp. Nonlinear Theory Appl. NOLTA’07, Vancouver, Canada, September 16–19, 2007, pp. 349–352.

    Google Scholar 

  11. Valmorbida, G., Drummond, R., and Duncan, S.R., Regional Analysis of Slope-Restricted Lurie Systems, IEEE Trans. Autom. Control, 2019, vol. 64, no. 3, pp. 1201–1208.

    Article  MATH  Google Scholar 

  12. Yakubovich, V.A., Absolute Instability of Nonlinear Control Systems. II, Autom. Remote Control, 1971, vol. 32, no. 6, part 1, pp. 876–884.

    MATH  Google Scholar 

  13. Kamenetskii, V.A., Absolute Stability and Absolute Instability of Control Systems with Several Nonlinear Nonstationary Elements, Autom. Remote Control, 1983, vol. 44, no. 12, pp. 1543–1552.

    MathSciNet  Google Scholar 

  14. Liberzon, D., Switching in Systems and Control, Boston: Birkhäuser, 2003.

    Book  MATH  Google Scholar 

  15. Molchanov, A.P. and Pyatnitskii, E.S., Absolute Instability of Nonlinear Nonstationary Systems. I, II, Autom. Remote Control, 1982, vol. 43, no. 1, part 1, pp. 13–20; no. 2, part 1, pp. 147–157.

    Google Scholar 

  16. Molchanov, A.P. and Pyatnitskii, E.S., Lyapunov Functions that Specify Necessary and Sufficient Conditions of Absolute Stability of Nonlinear Nonstationary Control Systems. I, Autom. Remote Control, 1986, vol. 47, no. 3, part 1, pp. 344–354.

    MATH  Google Scholar 

  17. Kamenetskii, V.A. and Pyatnitskii, E.S., Gradient Method of Constructing Lyapunov Functions in Problems of Absolute Stability, Autom. Remote Control, 1987, vol. 48, no. 1, part 1, pp. 1–9.

    Google Scholar 

  18. Kamenetskiy, V.A., Frequency-Domain Stability Conditions for Hybrid Systems, Autom. Remote Control, 2017, 78, 12, pp. 2101–2119.

    Article  MathSciNet  MATH  Google Scholar 

  19. Shorten, R.S. and Narendra, K.S., On Common Quadratic Lyapunov Functions for Pairs of Stable LTI Systems whose System Matrices are in Companion Form, IEEE Trans. Autom. Control, 2003, 48, 4, pp. 618–621.

    Article  MATH  Google Scholar 

  20. Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh sistem s needinstvennym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with a Non-Unique Equilibrium State), Moscow: Nauka, 1978.

    Google Scholar 

  21. Kamenetskiy, V.A., Frequency-Domain Stability Conditions for Discrete-Time Switched Systems, Autom. Remote Control, 2018, 79, 8, pp. 13711–1389.

    Google Scholar 

  22. Anan’evskii, I.M., Anokhin, N.V., and Ovseevich, A.I., General Lyapunov Function in the Control Synthesis Problem for Linear Dynamical Systems, in Problemy ustoichivosti i upravleniya. Sb. nauch. st., posv. 80-letiyu akad. V.M. Matrosova (Problems of Stability and Control. CollectedWorks in Honor of the 80th Birthday of Acad. V.M. Matrosov), Moscow: Fizmatlit, 2013, pp. 91–103.

    Google Scholar 

  23. Filippov, A.F., On Some Problems in Optimal Control Theory, Vest. MGU, Ser. Matem., Mekh., Astron., Phiz. Khim., 1959, no. 2, pp. 25–32.

    Google Scholar 

  24. Pyatnitskii, E.S. and Skorodinskii, V.I., Numerical Method of Construction of Lyapunov Functions and Absolute Stability Criteria in the Form of Numerical Procedures, Autom. Remote Control, 1983, 44, 11, part 1, pp. 1427–1437.

    MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by the Program of the Presidium of the Russian Academy of Sciences “Modern Problems of Robotics,” project no. I.29.

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    V. A. Kamenetskiy

Authors
  1. V. A. Kamenetskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Kamenetskiy.

Additional information

Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 8, pp. 9–28.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kamenetskiy, V.A. Switched Systems, Lur’e Systems, Absolute Stability, Aizerman Problem. Autom Remote Control 80, 1375–1389 (2019). https://doi.org/10.1134/S0005117919080010

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117919080010

Keywords

Search

Navigation