Volterra级数理论研究进展与展望  

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

作　　者：彭志科[1] 程长明[1] 

机构地区：[1]上海交通大学机械与动力工程学院机械系统与振动国家重点实验室,上海200240

出　　处：《科学通报》2015年第20期1874-1888,共15页Chinese Science Bulletin

基　　金：国家杰出青年科学基金(11125209);国家自然科学基金(51121063;11472170)资助

摘　　要：非线性问题因其普遍性受到来自包括工程、物理和数学等众多领域学者的关注.针对非线性系统的建模、求解和分析等问题,人们发展出了多种数学理论和方法,Volterra级数就是其中之一.本文对Volterra级数的基本定义和由其发展而来的一些频域概念进行介绍,并分析它和Taylor级数、Wiener级数、NARMAX模型、Hammerstein模型、Wiener模型、Wiener-Hammerstein模型、谐波平衡法、摄动法和Adomian分解等非线性模型与求解方法之间的联系;探讨了其收敛性问题和核辨识问题研究中的挑战,总结了这方面的研究成果和进展.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.

关 键 词：VOLTERRA级数 广义频率响应函数 非线性输出频率响应函数 非线性系统辨识 收敛性 

分 类 号：O173[理学—数学]