The existence and uniqueness of Volterra series for nonlinear systems

Given an input-output map described by a nonlinear control system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ẋ=f(x,u)</tex> and nonlinear output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y=h(x)</tex> , we present a simple straightforward means for obtaining a series representation of the output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t)</tex> in terms of the input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u(t)</tex> . When the control enters linearly, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ẋ =f(x)+ ug(x)</tex> , the method yields the existence of a Volterra series representation. The proof is constructive and explicitly exhibits the kernels. It depends on standard mathematical tools such as the Fundamental Theorem of Calculus and the Cauchy estimates for the Taylor series coefficients of analytic functions. In addition, the uniqueness of Volterra series representations is discussed.