Automation and Remote Control

, Volume 72, Issue 1, pp 38–60 | Cite as

Stability investigation for linear periodic time-delayed systems using Fredholm theory

  • B. P. Lampe
  • E. N. Rosenwasser
Determinate Systems


A single-loop linear periodic system with time delay is considered. Using the mathematical tool of integral Fredholm equations of the second kind, a characteristic function is constructed, the roots of which are inverses of the multipliers of the system. Rigorous sufficient stability conditions are given, which are based on approximate representation of the characteristic function in the form of a polynomial.


Remote Control Unit Disk Integral Fredholm Equation Stability Investigation Closed Unit Disk 
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  1. 1.
    Michiels, W. and Niculescu, S.I., Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach, Philadelphia: SIAM, 2007.zbMATHGoogle Scholar
  2. 2.
    Halanay, A., Stability Theory of Linear Periodic Systems with Delay, Rev. Math. Pures Appl., 1961, no. 6(4), pp. 633–653.Google Scholar
  3. 3.
    Dolgii, Yu.F., Using Self-Conjugated Boundary Problems for Stability Investigation of Periodic Systems with Delay, Tr. Inst. Mat. Mekh., 2006, vol. 12, no. 2, pp. 78–87.MathSciNetGoogle Scholar
  4. 4.
    Dolgii, Yu.F. and Ulianov, E.V., Singular Values of Monodromy Operator and Sufficient Stability Conditions for Periodic System of Differential Equations with Constant Delay, Tr. Inst. Mat. Mekh., 2007, vol. 13, no. 2, pp. 66–79.Google Scholar
  5. 5.
    Insperger, T. and Stepan, G., Stability of the Damped Mathieu Equation with Time-Delay, J. Dynam. Syst., Meas. Control, 2003, vol. 125, no. 2, pp. 166–171.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ma, H., Butcher, E.A., and Bueler, E., Chebushev Expansion of Linear and Piecewise Linear Periodic Coefficients under Control Excitations, J. Dynam. Syst., tMeas. Control, 2003, vol. 125, no. 2, pp. 236–243.CrossRefGoogle Scholar
  7. 7.
    Butcher, E.A., Ma, H., Bueler, E., Averina, V., and Szabo, Z., Stability of Linear Time-Periodic Delay Differential Equations via Chebyshev Polynomials, Int. J. Numer. Meth. Eng., 2004, no. 59, pp. 859–922.Google Scholar
  8. 8.
    Rosenwasser, E.N., Kolebaniya nelineinykh sistem. Metod integral’nykh uravnenii (Oscillation of Nonlinear Systems. Integral Equation Method), Moscow: Nauka, 1969.Google Scholar
  9. 9.
    Rosenwasser, E.N. and Volovodov, S.K., Operatornye metody i kolebatel’nye processy (Operator Methods and Oscillation Processes), Moscow: Nauka, 1985.Google Scholar
  10. 10.
    Rosenwasser, E.N., Theory of Linear System with Stationary Delay and Periodically-Varying Parameter, Autom. Remote Control, 1964, no. 2, pp. 1067–1074.Google Scholar
  11. 11.
    Zverkin, A.M., Differentsial’no-raznostnye uravneniya s periodicheskimi koeffitsientami (Differentialdifference Equations with Periodic Coefficients). Appendix to Russian translation of R. Bellman and K.L. Cooke, Differential-Difference Equations, Moscow: Mir, 1967.Google Scholar
  12. 12.
    Stokes, A.A., Floquet Theory for Functional Differential Equations, Proc. Nat. Acad. Sci. USA, 1962, vol. 48, no. 8, pp. 1330–1334.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Privalov, I.I., Integral’nye uravneniya (Integral Equations), Moscow: ONTI, NKTP USSR, 1937.Google Scholar
  14. 14.
    Tricomi, F.G., Integral Equations, New York: Interscience, 1957. Translated under the title Integral’noe uravnenie, Moscow: Inostrannaya Literatura, 1960.Google Scholar
  15. 15.
    Mikhlin, S.G., Lektsii po lineinym integral’nym uravneniyam (Lections on Linear Integral Equations), Moscow: Fizmatgiz, 1959.Google Scholar
  16. 16.
    Smirnov, V.I., Kurs vysshei matematiki (A Course of Higher Mathematics), Moscow: Nauka, 1974, vol. 4, part 1.Google Scholar
  17. 17.
    Rosenwasser, E. and Lampe, B., Multivariable Computer-Controlled Systems. A Transfer-Function Approach, London: Springer, 2006.zbMATHGoogle Scholar
  18. 18.
    Gursa, E., Kurs matematicheskogo analiza (A Course of Mathematical Analysis), Moscow: ONTI, 1934, vol. 3, part 2.Google Scholar
  19. 19.
    Gohberg, I.C., Goldberg, S., and Krupnik, N., Traces and Determinants of Linear Operators, Berlin: Birkhauser, 2000.zbMATHGoogle Scholar
  20. 20.
    Rosenwasser, E. and Lampe, B., Computer Controlled Systems. Analysys and Design with Processorientated Models, London: Springer, 2000.Google Scholar
  21. 21.
    Chen, T. and Francis, B.A., Optimal Sampled-Data Control Systems, Berlin: Springer, 1995.zbMATHGoogle Scholar
  22. 22.
    Krasovskii, N.N., Nekotorye zadachi ustoichivosti dvizheniya (Some Problems of Motion Stability), Moscow: Fizmatgiz, 1959.zbMATHGoogle Scholar
  23. 23.
    Privalov, I.I., Vvedenie v teoriyu funktsii kompleksnogo peremennogo (Introduction to Theory of Functions of Complex Variable), Moscow: Gostekhizdat, 1948.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • B. P. Lampe
    • 1
  • E. N. Rosenwasser
    • 2
  1. 1.University of RostockRostockGermany
  2. 2.State University of Ocean TechnologySt. PetersburgRussia

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