Exact and Parametric Dynamical System Representation of Nonlinear Functions

TL;DR

This paper introduces a fixed-initial-state constant-input dynamical system (FISCIDS) framework that provides exact parametric representations for a wide class of nonlinear functions, including many used in science and engineering.

Contribution

The paper proposes a novel FISCIDS method for representing nonlinear functions exactly, extending to functions beyond differentially algebraic ones.

Findings

Any differentially algebraic function has a quadratic FISCIDS representation.

Some analytic functions not differentially algebraic also have quadratic FISCIDS representations.

Most practical functions in science and engineering can be represented by quadratic FISCIDS.

Abstract

Parametric representations of various functions are fundamental tools in science and engineering. This paper introduces a fixed-initial-state constant-input dynamical system (FISCIDS) representation, which provides an exact and parametric model for a broad class of nonlinear functions. A FISCIDS representation of a given nonlinear function consists of an input-affine dynamical system with a fixed initial state and constant input. The argument of the function is applied as the constant input to the input-affine system, and the value of the function is the output of the input-affine system at a fixed terminal time. We show that any differentially algebraic function has a quadratic FISCIDS representation. We also show that there exists an analytic function that is not differentially algebraic but has a quadratic FISCIDS representation. Therefore, most functions in practical problems in science and engineering can be represented by a quadratic FISCIDS representation.