Volterra series theory: A state-of-the-art review

Abstract

Because many real-world systems are inherently nonlinear in nature, nonlinear problems have received considerable interest and attention from engineers, physicists, mathematicians, and other scientists. Many mathematical theories and methods have been developed to model, solve, and analyze nonlinear problems and systems. One such method uses the Volterra series, which describes the nonlinear relationship between system input and output. This is a powerful mathematical tool for the analysis of nonlinear systems, and is essentially an extension of the standard convolution description of linear systems. This paper introduces the basic definition of the Volterra series, together with some frequency domain concepts derived from the Volterra series. The connection between the Volterra series and other nonlinear system description models and nonlinear problem solving methods is discussed, including the Taylor series, Wiener series, NARMAX model, Hammerstein model, Wiener model, Wiener-Hammerstein model, harmonic balance method, perturbation method, and Adomian decomposition method. Challenging problems and the state-of-the-art in series convergence research and kernel identification studies are comprehensively introduced.