Automation and Remote Control

, Volume 76, Issue 5, pp 801–808 | Cite as

About the necessity of Popov criterion for a special Lyapunov function existence for the systems with multiple nonlinearities

  • M. M. Lipkovich
  • A. L. Fradkov
Topical Issue


Necessary and sufficient conditions for existence of Lyapunov function from the class “quadratic form plus integral of nonlinearity” (Lyapunov-Lurie function) for systems with several nonlinearities are considered. It is assumed that the nonlinearity graphs belong to the infinite sectors, i.e., belong to the union of the first and third quadrants in the plane. It is proven that Popov criterion is necessary and sufficient for existence of Lyapunov-Lurie function if the relative degree of the linear part is greater than one. The proof is based on the result concerning losslessness of the S-procedure for several respective quadratic constraints.


Remote Control Lyapunov Function Absolute Stability Domain Condition Nonlinear Control System 
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  1. 1.
    Letov, A.M., Stability in Nonlinear Control Systems, Princeton: Princeton Univ. Press, 1961.zbMATHGoogle Scholar
  2. 2.
    Lurie, A.I. and Postnikov, V.N., On the Stability Theory of Control Systems, Prikl. Mat. Mekh., 1944, vol. 8, no. 3, pp. 246–248.zbMATHGoogle Scholar
  3. 3.
    Malkin, I.G., Theory of Stability of Motion, Moscow: Nauka, 1966.zbMATHGoogle Scholar
  4. 4.
    Rozenvasser, E.N., The Absolute Stability of Nonlinear Systems, Autom. Remote Control, 1963, vol. 24, no. 3, pp. 283–291.zbMATHMathSciNetGoogle Scholar
  5. 5.
    Yakubovich, V.A., The Solution of Certain Matrix Inequalities in Automatic Control Theory, Sov. Math., 1962, vol. 3, no. 2, pp. 620–623.zbMATHGoogle Scholar
  6. 6.
    Brockett, R. and Lee, H., Frequency-Domain Instability Criteria for Time-Varying and Nonlinear Systems, Proc. IEEE, 1967, vol. 55, no. 5, pp. 604–619.CrossRefGoogle Scholar
  7. 7.
    Kalman, R.E., Lyapunov Functions for the Problem of Lur’e in Automatic Control, Proc. Natl. Acad. Sci. USA, 1963, vol. 49, no. 2, pp. 201–205.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Popov, V.M., Criterii de Stabilitate Pentru Sistemele Neliniare de Reglare Automata Bazate pe Utilizarea Transformatei Laplace, Stud. Cerc. Energetica, 1959, vol. 9, no. 1, pp. 119–135.Google Scholar
  9. 9.
    Yakubovich, V.A., The Solution of Certain Matrix Inequalities Encountered in Nonlinear Control Theory, Dokl. Akad. Nauk USSR, 1964, vol. 156, no. 2, pp. 278–281.Google Scholar
  10. 10.
    Popov, V.M., Hyperstability of Automatic Control Systems with Several Nonlinear Elements, Rev. Electrotech. Energ., 1964, vol. 9, no. 1, pp. 35–45.Google Scholar
  11. 11.
    Gantmacher, F.R and Yakubovich, V.A., Absolute Stability of Nonlinear Control Systems, Tr. II Vses. S”ezda teoret. prikl. mekh. (Proc. All-Union Congr. on Theor. and Appl. Mech.), Moscow: Nauka, 1966, pp. 30–63.Google Scholar
  12. 12.
    Aizerman, M.A. and Gantmacher, F.R., Absolute Stability of Regularized Systems, Moscow: Akad. Nauk USSR, 1963.Google Scholar
  13. 13.
    Gusev, S.V. and Likhtarnikov, A.L., Kalman-Popov-Yakubovich Lemma and the S-procedure: A Historical Essay, Autom. Remote Control, 2006, vol. 67, no. 11, pp. 1768–1810.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Yakubovich, V.A., The S-procedure in Nonlinear Control Theory, Vestn. Leningr. Univ., Math., 1977, vol. 4, pp. 73–93.Google Scholar
  15. 15.
    Fradkov, A.L., Duality Theorems for Certain Nonconvex Extremal Problems, Sib. Math. J., 1973, vol. 14, no. 2, pp. 247–264.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Polyak, B.T., Convexity of Quadratic Transformations and Its Use in Control and Optimization, J. Optim. Theory Appl., 1998, vol. 99, no. 3, pp. 553–583.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kamenetskii, V.A., Convolution Method for Matrix Inequalities and Absolute Stability Criteria for Stationary Control Systems, Autom. Remote Control, 1989, vol. 50, no. 5, part 1, pp. 598–607.MathSciNetGoogle Scholar
  18. 18.
    Rapoport, L.B., Frequency Criterion of Absolute Stability for Control Systems with Several Nonlinear Stationary Elements, Autom. Remote Control, 1989, vol. 50, no. 6, part 1, pp. 743–750.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Ionsian, U. and Susya, Chzhao, On the Problem of Absolute Stability of Control Systems with Several Nonlinear Stationary Elements in the case of an Infinite Sector, Autom. Remote Control, 1991, vol. 52, no. 1, part 1, pp. 26–34.zbMATHGoogle Scholar
  20. 20.
    Fradkov, A.L, Adaptivnoe upravlenie v slozhnykh sistemakh (Adaptive Control in Complex Systems), Moscow: Nauka, 1990.zbMATHGoogle Scholar
  21. 21.
    Yakubovich, V.A., Leonov, G.A. and Gelig, A.Kh., Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, Singapore: World Scientific, 2004.CrossRefzbMATHGoogle Scholar
  22. 22.
    Barbashin, E.A. and Krasovski, N.N., On the Stability of Motion as a Whole, Dokl. Akad. Nauk USSR, 1952, vol. 86, no. 3, pp. 453–456.zbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Saint Petersburg State UniversitySt. PetersburgRussia
  2. 2.Institute of Problems of Mechanical EngineeringSt. PetersburgRussia
  3. 3.ITMO UniversitySt. PetersburgRussia

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