Advertisement

Stability Criteria on a Type of Differential Inclusions with Nonlinear Integral Delays

  • Shuli Guo
  • Lina Han
Chapter

Abstract

It is very difficult in directly analyzing the robust stability of uncertain differential inclusions.  If all points of the convex polyhedron are found, we can easily analyze the stability of convex points to obtain the stability of the uncertain differential inclusions. The new method of constructing polyhedron Lyapunov functional for given differential inclusions with nonlinear integral delays is presented. Algebraic criteria of asymptotical stable of the zero solution of a class of differential inclusion with nonlinear integral delays is outlined, too.  Moreover, the above conclusions are similarly generalized to show by the Ricatti matrix inequalities.  Finally, an interesting example is presented to illustrate the main result effectively.

Notes

Acknowledgements

This work is Supported by National Key Research and Development Program of China (2017YFF0207400).

References

  1. 1.
    Aubin JP, Cellina A. Differential inclusion. Berlin: Springer; 1984.CrossRefGoogle Scholar
  2. 2.
    Bacciotti A, Rosier L. Regularity of Lyapunov functions for stable systems. Syst Control Lett. 2000;41(2):265–70.CrossRefGoogle Scholar
  3. 3.
    Battilotti S. Robust stabilization of nonlinear systems with point wise norm-bounded uncertainties: a control Lyapunov function approach. IEEE Trans Autom Control. 1999;44(1):3–17.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Branicky MS. Multiple Lyapunov functions and other analysis tools for switched and hybird systems. IEEE Trans Autom Control. 1998;43(4):475–82.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boyd SP, Yang Q. Structured and simultanous Lyapunov functions for systems stability problem. Int J Control. 1995;49:2215–40.CrossRefGoogle Scholar
  6. 6.
    Dayawansa WP, Martin CF. A converse Lyapunov theorem for a class of dynamical systems which undergo switching. IEEE Trans Autom Control. 1999;44(4):751–60.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dugard L, Verrist EI. Stability and control of time-delay systems. Berlin: Springer; 1998. p. 303–17.CrossRefGoogle Scholar
  8. 8.
    Filippov AF. Differential equations with discontinous right-hand side. Moscow: Nauda; 1985.zbMATHGoogle Scholar
  9. 9.
    Guo S, Irene M, Si L, Han L. Several integral inequalities and their applications in nonlinear differential systems. Appl Math Comput. 2013;219:4266–77.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Guo S,  Yan S, Wen S. Absolute Stability of Nonlinear Systems with MIMO Bounded Time-Delays. American Society of Mechanical Engineers(ASME), Design Engineering Division, 2003,116(2).Google Scholar
  11. 11.
    Guo S,  Irene M,  Han L, Xin WF, Feng XJ.  Commuting matrices, equilibrium points for control systems with single saturated input.  Applied Mathematics and Computation. preprint.Google Scholar
  12. 12.
    Joaquain MC.  Stability of the convex hull of commuting matrixes. In: Proceedings of the 32nd conference on decision and control. San Antonio, Texas; 1993.Google Scholar
  13. 13.
    Huang L. Stability theory. Beijing: Peking University Press; 1992. p. 235–83.Google Scholar
  14. 14.
    Horn RA, Johnson CR. Topics in matrix analysis. Cambridge: Cambridge University Press; 1991.CrossRefGoogle Scholar
  15. 15.
    Kumpati SN, Jeyendran B. A common lyapunov function for stable LTI systems with commuting A-matrixes. IEEE Trans Autom Control. 1994;39(12):2469–71.CrossRefGoogle Scholar
  16. 16.
    Lee SH, Kim TH, Lim JT. A new stability analysis of switched systems. Automatica. 2000;36:917–22.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liberzon D, Hespanta JP, Morse AS. Stability of switched systems: a Lie-algebraic condition. Syst Control Lett North-Holland. 1999;37:117–22.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mareada KS, Balakrishan J. A common Lyapunov function for stable LTI systems with commuting A-martices. IEEE Trans Autom Control. 1994;39(12):2469–71.CrossRefGoogle Scholar
  19. 19.
    Molchanov AP, Pyatnitskiy YS. Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Syst Control Lett. 1989;13:59–64.MathSciNetCrossRefGoogle Scholar
  20. 20.
    La Salle J, Lefschetz S. Stability by Lyapunov’s direct method. New York: Academic Press; 1961.zbMATHGoogle Scholar
  21. 21.
    Polanski K. On absolute stability analysis by polyhydric Lypunov functions. Automatica. 2000;36:573–8.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schmitendorf WE, Barmish BR. Null controllability of linear systems with constrained controls. SIAM J Control Optim. 1980;18:327–45.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Shevitz D,  Paden B. Lyapunov stability theory of non-smooth systems. In: Proceeding of the 32nd conference on decision and control. San Antonio, Texas; 1993, pp. 416–421.Google Scholar
  24. 24.
    Tatsushi O, Yasuyuki F. Two conditions concerning common quadratic lyapunov functions for linear systems. IEEE Trans Autom Control. 1997;42(5):719–21.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Key Laboratory of Complex System Intelligent Control and Decision, School of AutomationBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Cardiovascular Internal Medicine of Nanlou Branch, National Clinical Research Center for Geriatric DiseasesChinese PLA General HospitalBeijingChina

Personalised recommendations