Convergence and computation of describing functions using a recursive Volterra series

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 ?(t)+b₁y(t)+b₂u²(t)y(t) = K(u?(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. Copyright © 2004 John Wiley & Sons, Ltd.