Application of the Volterra Series Paradigm to Memristive Systems

  • Alon Ascoli
  • Torsten Schmidt
  • Ronald Tetzlaff
  • Fernando Corinto


The advent of the memristor in the panorama of fundamental passive circuit elements and the recent development of physical nano-devices with memory resistance opens new opportunities in IC electronics. However, considerable progress in the design of novel memristor-based circuits and systems may not be achieved unless the nonlinear dynamics of these nano-devices is fully unfolded and modeled. Due to the strongly nonlinear behavior of the physical memristor, classical analysis and synthesis techniques from linear system theory may not be applied to memristor-based circuits. Within the framework of nonlinear system theory, the Volterra series paradigm offers a solid theoretical background which may support the modeling and analysis of these unique systems. After a brief pedagogical review of the Volterra series theory, this paper introduces a systematic technique enabling the accurate reproduction of the dynamical behavior of classes of memristor circuits.


Volterra Series Volterra System Volterra Kernel Impedance Matrix Kernel Equation 
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  1. 1.
    F. Corinto, A. Ascoli, A boundary condition-based approach to the modeling of memristor nanostructures. IEEE Trans. Circuits Syst. I 59(11), 2713–2726 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    F. Corinto, A. Ascoli, M. Gilli, Nonlinear dynamics of memristor oscillators. IEEE Trans. Circuits Syst. I 58(6), 1323–1336 (2011)Google Scholar
  3. 3.
    L.O. Chua, Memristor: the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)CrossRefGoogle Scholar
  4. 4.
    L.O. Chua, S.M. Kang, Memristive devices and systems. Proc. IEEE 64(2), 209–223 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    H.K. Khalil, Nonlinear Systems, 3rd edn. (Prentice Hall, Englewood Cliffs, 2002)MATHGoogle Scholar
  6. 6.
    M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Krieger Publishing Company, Malabar, FL, 2006)Google Scholar
  7. 7.
    L.O. Chua, C.Y. Ng, Frequency domain analysis of nonlinear systems. IEE J. Electron. Circuits Syst. 3(4), 165–185 (1979)CrossRefGoogle Scholar
  8. 8.
    S. Boyd, Y.S. Tang, L.O. Chua, Measuring Volterra kernels. IEEE Trans. Circuits Syst. CAS-30(8), 571–577 (1983)CrossRefGoogle Scholar
  9. 9.
    J.F. Barrett, The use of functionals in the analysis of non-linear physical systems. J. Electron. Control 15(6), 567–615 (1963)CrossRefGoogle Scholar
  10. 10.
    J. Waddington, F. Fallside, Analysis of non-linear differential equations by the Volterra series. Int. J. Control 3(1), 1–15 (1966)CrossRefGoogle Scholar
  11. 11.
    A.J. Krener, Linearization and bilinearization of control systems. Proceedings of Allerton Conference on Circuit and System Theory, pp. 834–843, University of Illinois, Urbana Champaign, Illinois, 1974Google Scholar
  12. 12.
    C. Lesiak, A. Krener, The existence und uniqueness of Volterra series for nonlinear systems. IEEE Trans. Automat. Control AC-23, 1090–1095 (1978)MathSciNetCrossRefGoogle Scholar
  13. 13.
    E.G. Gilbert, Functional expansions for the response nonlinear differential systems. IEEE Trans. Automat. Control AC-22(6), 909–921 (1977)CrossRefGoogle Scholar
  14. 14.
    W.J. Rugh, Nonlinear System Theory (The Johns Hopkins University Press, Baltimore, MD, 1981)MATHGoogle Scholar
  15. 15.
    I.W. Sandberg, A perspective on system theory. IEEE Trans. Circuits Syst., CAS-31(1), 88–103 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    S.P. Boyd, L.O. Chua, Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Circuits Syst. CAS-32(11), 1150–1161 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S.P. Boyd, L.O. Chua, Volterra series for nonlinear circuits. Proceedings of International Symposium on Circuits and Systems, p. 369, Kyoto, 1985Google Scholar
  18. 18.
    S.P. Boyd, L.O. Chua, C.A. Desoer, Analytical foundations of Volterra series. IMA J. Math. Control Info. 1(3), 243–282 (1984)CrossRefMATHGoogle Scholar
  19. 19.
    R.W. Brockett, Volterra series and geometric control theory. Automatica, 12, 167–176 (1976)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    I.W. Sandberg, Bounds for discrete-time Volterra series representations. IEEE Trans. Circuits Syst. I 46(1), 135–139 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    T. Siu, M. Schetzen, Convergence of Volterra series representation and BIBO stability of bilinear systems. Int. J. Syst. Sci. 22(12), 2679–2684 (1991)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    J.F. Barrett, The use of Volterra series to find region of stability of a non-linear differential equation. Int. J. Control 1(3), 209–216 (1965)CrossRefMATHGoogle Scholar
  23. 23.
    S.P. Boyd, Volterra Series: Engineering Fundamentals. PhD Dissertation Thesis, 1985Google Scholar
  24. 24.
    T. H\(\acute{e}\) lie, B. Laroche, Computation of convergence bounds for volterra series of linear-analytic single-input systems. IEEE Trans. Automat. Control 56(9), 2062–2072 (2011)Google Scholar
  25. 25.
    L.O. Chua, Resistance switching memories are memristors. Appl. Phys. A 102, 765–783 (2011)CrossRefGoogle Scholar
  26. 26.
    D.B. Strukov, G.S. Snider, D.R. Stewart, R.S. Williams, The missing memristor found. Nature 453, 80–83 (2008)CrossRefGoogle Scholar
  27. 27.
    F. Corinto, A. Ascoli, Memristive diode bridge with LCR filter. Electron. Lett. 48(14), 824–825 (2012)CrossRefGoogle Scholar
  28. 28.
    Z.-Q. Lang, S.A. Billings, Evaluation of output frequency responses of nonlinear systems under multiple inputs. IEEE Trans. Circuits Syst. II 47(1), 28–38 (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Alon Ascoli
    • 1
  • Torsten Schmidt
    • 2
  • Ronald Tetzlaff
    • 1
  • Fernando Corinto
    • 3
  1. 1.Technische Universität DresdenDresdenGermany
  2. 2.Hochschule AnsbachAnsbachGermany
  3. 3.Politecnico di TorinoTorinoItaly

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