Convergence and computation of describing functions using a recursive Volterra series

J. W. Glass, Matthew Franchek

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Presented in this paper is a comparison of algorithms for computing an approximation to the sinusoidal input describing function (SIDF) for the nonlinear differential equation ẏ(t) + b1y(t) + b 2u2(t)y(t) = K(u̇(t) + b3u(t)) The importance of this nonlinear differential equation comes from the context of nonlinear feedback controller design. Specifically, this equation is either a linear lead or lag controller (depending on the coefficient values) augmented with a nonlinear, polynomial type term. Consequently, obtaining a SIDF representation of this nonlinear differential equation or creating a process to obtain SIDFs for other similar differential equations, will facilitate nonlinear controller design using classical loop shaping tools. The two SIDF approximations studied include the well-established harmonic balance method and a Volterra series based algorithm. In applying the Volterra series, several theoretical issues were addressed including the development of a recursive solution that calculates high order Volterra transfer functions, and the guarantee of convergence to an arbitrary accuracy. Throughout the paper, case studies are presented.

Original languageEnglish (US)
Pages (from-to)1469-1488
Number of pages20
JournalInternational Journal of Robust and Nonlinear Control
Issue number18
StatePublished - Dec 2004


  • Nonlinear control
  • Nonlinear system approximation
  • Sinusoidal input describing functions
  • Volterra series

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering
  • Applied Mathematics


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