Skip to main content
SpringerLink
Log in

Frequency-domain stability conditions for hybrid systems

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

Abstract

Consideration was given to a special class of the hybrid systems with switchings of time-invariant linear right-hand sides. A narrower subclass of such systems, that of connected switched linear systems, was specified among them. The necessary and sufficient frequencydomain conditions (criteria) for the existence of a common quadratic Lyapunov function providing stability of the switched systems were proposed for them. The specified subclass includes control systems with several nonstationary nonlinearities from the finite sectors that are the matter at issue of the theory of absolute stability. For the connected switched linear systems of a special kind (triangular type systems), the separate necessary and separate sufficient existence conditions were obtained for such Lyapunov functions. The interrelations between these conditions were discussed in the example.

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Lin, H. and Antsaklis, P.J., Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results, IEEE Trans. Autom. Control, 2009, vol. 54, no. 2, pp. 308–322.

    Article  MathSciNet  MATH  Google Scholar 

  3. Vassilyev, S.N. and Kosov, A.A., Analysis of Hybrid Systems’ Dynamics Using the Common Lyapunov Functions and Multiple Homomorphisms, Autom. Remote Control, 2011, vol. 72, no. 6, pp. 1163–1183.

    Article  MathSciNet  MATH  Google Scholar 

  4. Blanchini, F., Colaneri, P., and Valcher, M.E., Is Stabilization of Switched Positive Linear Systems Equivalent to the Existence of an Hurwitz Convex Combination of the System Matrices?, in 50 IEEE Conf. Decision Control Eur. Control Conf. (CDC-ECC) Orlando, USA, 12–15 December, 2011, pp. 1834–1839.

    Chapter  Google Scholar 

  5. Aleksandrov, A.Yu., Kosov, A.A., and Platonov, A.V., On the Asymptotic Stability of Switched Homogeneous Systems, Syst. Control Lett., 2012, vol. 61, no. 1, pp. 127–133.

    Article  MathSciNet  MATH  Google Scholar 

  6. Sun, Y. and Wang, L., On Stability of a Class of Switched Nonlinear Systems, Automatica, 2013, vol. 49, no. 1, pp. 305–307.

    Article  MathSciNet  MATH  Google Scholar 

  7. Kouhia, Y., Bajcinca, N., Raisch, J., and Shorten, R., On the Quadratic Stability of Switched Linear Systems Associated with Symmetric Transfer Function Matrices, Automatica, 2014, vol. 50, no. 11, pp. 2872–2879.

    Article  MathSciNet  MATH  Google Scholar 

  8. Aleksandrov, A.Yu. and Platonov, A.V., On Stability of Solutions for a Class of Nonlinear Difference Systems with Switching, Autom. Remote Control, 2016, vol. 77, no. 5, pp. 779–788.

    Article  MathSciNet  MATH  Google Scholar 

  9. Shorten, R.N. and Narendra, K.S., Necessary and Sufficient Conditions for the Existence of a Common Quadratic Lyapunov Function for Two Stable Second Order Linear Time-invariant Systems, in Proc. Am. Control Conf., 1999, pp. 1410–1414.

    Google Scholar 

  10. Shorten, R.N. and Narendra, K.S., Necessary and Sufficient Conditions for the Existence of a Common Quadratic Lyapunov Function for a Finite Number of Stable Second Order Linear Time-invariant Systems, Int. J. Adapt. Control Signal Proc., 2003, vol. 16, no. 10, pp. 709–728.

    Article  MATH  Google Scholar 

  11. Pakshin, P.V. and Pozdyaev, V.V., Solvability Conditions for a Second-Order System of Linear Matrix Inequalities, J. Comp. Syst. Sci. Int., 2006, vol. 45, no. 5, pp. 681–689.

    Article  MATH  Google Scholar 

  12. Vassilyev, S.N. and Kosov, A.A., Common Lyapunov Functions and the Comparison Vector Functions in the Analysis of Hybrid Systems, in Analytical Mechanics, Stability, and Control: Proc. X Int. Chetaev Conf., vol. 2, section 2: Stability, Kazan, June 12–16, 2012 Kazan: Kazan. Gos. Tekh. Univ., 2012, pp. 162–176.

    Google Scholar 

  13. Aleksandrov, A.Yu. and Murzinov, I.E., On the Existence of a Common Lyapunov Function for a Family of Nonlinear Mechanical Systems with One Degree of Freedom, Nonlin. Dynam. Syst. Theory, 2012, vol. 12, no. 2, pp. 137–143.

    MathSciNet  MATH  Google Scholar 

  14. 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, vol. 48, no. 4, pp. 618–621.

    Article  MathSciNet  MATH  Google Scholar 

  15. King, C. and Nathanson, M., On the Existence of a Common Quadratic Lyapunov Function for a Rank One Difference, Linear Algebra Appl., 2006, vol. 419, nos. 2–3, pp. 400–416.

    Article  MathSciNet  MATH  Google Scholar 

  16. Laffey, T.J. and Smigoc, H., Common Lyapunov Solutions for Two Matrices Whose Difference has Rank One, Linear Algebra Appl., 2009, vol. 431, nos. 1–2, pp. 228–240.

    Article  MathSciNet  MATH  Google Scholar 

  17. Anan’evskii, I.M., Anokhin, N.V., and Ovseevich, A.I., Common Lyapunov Function in the Problem of Design of Control of Linear Dynamic Systems, in Problems of Stability and Control. Collected Papers for 80th Anniversary of Acad. V.M. Matrosov, Moscow: Fizmatlit, 2013, pp. 91–103.

    Google Scholar 

  18. 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  MATH  Google Scholar 

  19. Kamenetskiy, V.A. and Pyatnitskiy, Ye.S., An Iterative Method of Lyapunov Function Construction for Differential Inclusions, Syst. Control Lett., 1987, vol. 8, no. 5, pp. 445–451.

    Article  MathSciNet  MATH  Google Scholar 

  20. Filippov, A.F., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu, Moscow: Nauka, 1985. Translated under the title Differential Equations with Discontinuous Right-hand Sides, New York: Kluwer, 1988.

    Google Scholar 

  21. Alimov, Yu.I., On the Application of Lyapunov’s Direct Method to Differential Equations with Ambiguous Right Sides, Autom. Remote Control, 1961, vol. 22, no. 7, pp. 713–725.

    MathSciNet  MATH  Google Scholar 

  22. Boyd, S. and Yang, Q., Structured and Simultaneous Lyapunov Functions for System Stability Problems, Int. J. Control, 1989, vol. 49, no. 6, pp. 2215–2240.

    Article  MathSciNet  MATH  Google Scholar 

  23. 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, vol. 44, no. 11, pp. 1427–1437.

    MathSciNet  Google Scholar 

  24. Filippov, A.F., Stability Conditions in Homogeneous Systems with Arbitrary Regime Switching, Autom. Remote Control, 1980, vol. 41, no. 8, pp. 1078–1085.

    MathSciNet  MATH  Google Scholar 

  25. Gelig, A.X., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh sistem s needinstvennym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with Nonunique Equilibrium State), Moscow: Nauka, 1978.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. Barkin, A.I., Absolyutnaya ustoichivost’ sistem upravleniya (Absolute Stability of Control Systems), Moscow: Knizhnyi Dom “LIBROKOM,” 2012.

    Google Scholar 

  28. Molchanov, A.P. and Pyatnitskii, E.S., Absolute Instability of Nonlinear Nonstationary Systems. I, Autom. Remote Control, 1982, vol. 43, no. 1, pp. 13–20.

    MathSciNet  MATH  Google Scholar 

  29. Skorodinskii, V.I., Absolute Stability and Absolute Instability of Control Systems with Two Nonlinear Nonstationary Elements. I, II, Autom. Remote Control, 1981, vol. 42, no. 9, pp. 1149–1157; 1982, vol. 43, no. 6, pp. 775–781.

    MathSciNet  Google Scholar 

  30. Evtushenko, Yu.G., Metody resheniya ekstremal’nykh zadach i ikh primenenie v sistemakh optimizatsii (Methods to Solve Extremal Problems and Their Use in the Optimization Systems), Moscow: Nauka, 1982.

    MATH  Google Scholar 

Download references

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

Original Russian Text © V.A. Kamenetskiy, 2017, published in Avtomatika i Telemekhanika, 2017, No. 12, pp. 3–25.

This paper was recommended for publication by P.V. Pakshin, a member of the Editorial Board

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kamenetskiy, V.A. Frequency-domain stability conditions for hybrid systems. Autom Remote Control 78, 2101–2119 (2017). https://doi.org/10.1134/S0005117917120013

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117917120013

Keywords

Search

Navigation