Skip to main content
SpringerLink
Log in

On Lyapunov stability of interconnected nonlinear systems: recursive integration methodology

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper evolved from an endeavor to construct a Lyapunov function of interconnected nonlinear systems described by ordinary differential equations. This construction is an intractable problem and crucial part of the design when applied to problems of high dimension with intricate structure. The design obtained is applicable if the system under investigation is a generalization of the strict feedback form and whose dependency graphs satisfies the decomposition into lower triangular form. This design is mainly developed for moderated nonlinear continuous time dynamical systems. However, as we show in the example, it is emphasized that the present results can be extended to discontinuous systems provided they can be approximated by smooth modifications. A key step in our approach which based on back integrating procedure is to decompose the interconnected system into subsystems using graph theoretic decomposition. In the proposed methodology, the postulated Lyapunov function is obtained by backward integration the composite system trajectory. This novel approach leads straightforward and by one shot methods to both Lyapunov function of the composite system and its time derivative. A practical example is presented to illustrate the effectiveness of the results.

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. Chiang, H.D., Fekich-Ahmed, L.: A constructive methodology for estimating the stability region of interconnected nonlinear system. IEEE Trans. Circuits Syst. 37(5), 577–588 (1990)

    Article  MATH  Google Scholar 

  2. Vidyasagar, M.: Decomposition techniques for large scale systems with nonadditive interactions: stability and stabilizability. IEEE Trans. Automat. Contr. AC-25(4), 773–779 (1980)

    Article  MathSciNet  Google Scholar 

  3. Siljak, D.D.: Decentralized Control of Complex Systems. Academic Press, Inc. Harcourt Brace Jovanovich, San Diego (1991)

    Google Scholar 

  4. Jamshidi, M.: Large-Scale Systems: Modeling, Control, and Fuzzy Logic. Prentice-Hall, Englewood Cliffs (1997)

    MATH  Google Scholar 

  5. Michel, A.N.: Qualitative Analysis of Dynamical Systems, 2nd edn. Dekker, New York (2001)

    Google Scholar 

  6. Chin, S.M.: A general method to derive Lyapunov functions for nonlinear systems. Int. J. Control 44(2), 381–393 (1986)

    Article  MathSciNet  Google Scholar 

  7. Hu, T., Lin, Z.: Properties of the composite quadratic Lyapunov functions. IEEE Trans. Automat. Contr. 49(7), 1162–1167 (2004)

    Article  MathSciNet  Google Scholar 

  8. Mazenc, F., Nestic, D.: Strong Lyapunov functions for systems satisfying the conditions of La Salle. IEEE Trans. Automat. Contr. 49(6), 1026–1030 (2004)

    Article  Google Scholar 

  9. Papachristodoulou, A., Prajma, S.: On the construction of Lyapunov functions using the sum of squares decomposition. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, pp. 3482–3486 (2002)

  10. Ito, H.: State-dependent scaling problems and stability of interconnected IBSS and ISS systems. IEEE Trans. Automat. Contr. 51, 1626–1643 (2006)

    Article  Google Scholar 

  11. Nersesov, S.G., Haddad, W.M.: On the stability and control of nonlinear dynamical systems via vector Lyapunov functions. IEEE Trans. Automat. Contr. 51(2), 203–215 (2006)

    Article  MathSciNet  Google Scholar 

  12. Xiang, J., Chen, G.: On the V-stability of complex dynamical network. Automatica 43, 1049–1057 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kidouche, M., Charef, A.: A constructive methodology of Lyapunov functions of composite systems with nonlinear interconnection term. Int. J. Robot. Autom. 21(1), 19–24 (2006)

    Google Scholar 

  14. Kidouche, M.: The variable gradient method for generating Lyapunov functions of nonlinear composite system. In: 11th IEEE International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, pp. 233–237 (2005)

  15. Tarjan, R.: Depth first research and linear graph algorithm. SIAM J. Comput. 1(2), 146–160 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  16. Deo, N.: Graph Theory with Application to Engineering and Computer Science. Prentice-Hall, Englewood Cliffs (1974)

    Google Scholar 

  17. Reinhard, D.: Graph Theory, 2nd edn. Springer, New York (2000)

    Google Scholar 

  18. Callier, F.M., Chan, W.S., Desoer, C.A.: Input-output stability theory of interconnected systems using decomposition techniques. IEEE Trans. Circuits Syst. CAS-23(21) (1976)

  19. Knuth, D.E.: The Art of Computer Programming. Fundamental Algorithms, vol. 1. Addison-Wesley, Reading (1973)

    Google Scholar 

  20. Freeman, R.A., Kokotovic, P.V.: Robust Nonlinear Control Design: State Space and Lyapunov Techniques. Springer, New York (1996)

    MATH  Google Scholar 

  21. Lakshmikantham, V., Matrosov, V.M., Sivasundaram, S.: Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. Kluwer Academic, Dordrecht (1991)

    MATH  Google Scholar 

  22. Sastry, S.: Nonlinear Systems: Analysis, Stability and Control. Springer, Berlin (1999)

    MATH  Google Scholar 

  23. Baccioto, A.: Lyapunov Functions and Stability in Control Theory, 2nd edn. Springer, New York (2005)

    Google Scholar 

  24. Dubljevic, S., Kazantzis, N.: A new Lyapunov design approach for nonlinear systems based on Zubov’s method. Automatica 38, 1999–2007 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mazenc, F., Malisoff, M., Queiroz, M.S.: Further results on strict Lyapunov functions for rapidly time varying nonlinear systems. Automatica 42, 1663–1671 (2006)

    Article  MATH  Google Scholar 

  26. Cai, C., Tell, A.R., Goebel, R.: Smooth Lyapunov functions for hybrid systems—Part I: existence is equivalent to robustness. IEEE Trans. Automat. Contr. 52(7), 1264–1277 (2007)

    Article  Google Scholar 

  27. Sepulchre, R., Jankovic, M., Kokotovic, P.V.: Constructive Nonlinear Control. Springer, London (1997)

    MATH  Google Scholar 

  28. Krishnamurthy, P., Khorrami, F.: Global robust state-feedback for nonlinear systems via dynamic high gain scaling. In: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, pp. 6139–6144 (2003)

  29. Merritt, H.E.: Hydraulic Control System. Wiley, New York (1967)

    Google Scholar 

  30. Hong, Y., Zheng-Jin, F., Xu-Yong, W.: Nonlinear control for a class of hydraulic servo system. J. Zhejiang Univ. Sci. 5(11), 1413–1417 (2004)

    Article  Google Scholar 

  31. Yao, B., Chiu, F.P.: Nonlinear adaptive robust control of electro hydraulic servo system with discontinuous projection. In: Proceedings of the 37th IEEE Conference on Decision and Control, Tempa, FL, pp. 2265–2270 (1998)

  32. Polyakov, A., Poznyak, A.: Lyapunov function for finite time convergence analysis: Twisting controller for second order sliding mode realization. Automatica 45, 444–448 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Fitz Simens, P.M., Palazzolo, J.J.: Modelling of a one degree of freedom active hydraulic mount. ASME J. Dyn. Syst., Meas. Control 118(3), 439–448 (1996)

    Article  Google Scholar 

  34. Fitz Simens, P.M., Palazzolo, J.J.: Friction compensation for an industrial hydraulic robot. IEEE Control Syst. 19(1), 25–32 (1999)

    Article  Google Scholar 

  35. Sohl, G.A., Bobrow, B.J.: Experiments and simulations on the nonlinear control of hydraulic servo system. IEEE Trans. Control Syst. Technol. 7(2), 238–247 (1999)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Automatica, F.H.C, University of Boumerdes, Boumerdes, Algeria

    M. Kidouche & H. Habbi

Authors
  1. M. Kidouche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Habbi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Kidouche.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kidouche, M., Habbi, H. On Lyapunov stability of interconnected nonlinear systems: recursive integration methodology. Nonlinear Dyn 60, 183–191 (2010). https://doi.org/10.1007/s11071-009-9588-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-009-9588-7

Keywords

Search

Navigation