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Exact and Parametric Dynamical System Representation of Nonlinear Functions

Toshiyuki Ohtsuka

TL;DR

The paper introduces FISCIDS, a fixed-initial-state, constant-input dynamical-system framework for exact, parametric nonlinear function representations. It defines R-/P-/Q-FISCIDS variants and proves that any function on a star-shaped domain has a FISCIDS representation, with R/P/Q forms being equivalent in expressiveness. It then shows that every nonsingular differentially algebraic (DA) function has a Q-FISCIDS representation, and provides a Katriel-style TT function that also admits a Q-FISCIDS representation, illustrating that Q-FISCIDS can capture a broader class than DA. The work thereby furnishes a rigorous basis for exact, parametric nonlinear function representations and suggests future avenues for applying these representations to mathematical problems and parameter identification. $ADE$s underpin the DA results, linking the framework to classical algebraic-differential structure, while the TT example hints at further generality beyond DA functions.$

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.

Exact and Parametric Dynamical System Representation of Nonlinear Functions

TL;DR

The paper introduces FISCIDS, a fixed-initial-state, constant-input dynamical-system framework for exact, parametric nonlinear function representations. It defines R-/P-/Q-FISCIDS variants and proves that any function on a star-shaped domain has a FISCIDS representation, with R/P/Q forms being equivalent in expressiveness. It then shows that every nonsingular differentially algebraic (DA) function has a Q-FISCIDS representation, and provides a Katriel-style TT function that also admits a Q-FISCIDS representation, illustrating that Q-FISCIDS can capture a broader class than DA. The work thereby furnishes a rigorous basis for exact, parametric nonlinear function representations and suggests future avenues for applying these representations to mathematical problems and parameter identification. s underpin the DA results, linking the framework to classical algebraic-differential structure, while the TT example hints at further generality beyond DA functions.$

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.

Paper Structure

This paper contains 5 sections, 6 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dynamical System Representation of Nonlinear Functions
  3. Q-FISCIDS Representation of DA Functions
  4. Q-FISCIDS Representation of a TT Function
  5. Conclusion

Key Result

Theorem 1

Any function $\phi: X \to \mathbb{R}^m$ on an open set $X$ that is star-shaped at the origin has a FISCIDS representation, which is expressed as follows:

Figures (3)

  • Figure 1: Flow-based representation and FISCIDS representation of a nonlinear function. In a flow-based representation (Fig. \ref{['fig:Flow']}), the argument of the nonlinear function is the initial state of an autonomous system. By contrast, in a FISCIDS representation (Fig. \ref{['fig:FISCIDS']}), the argument of the nonlinear function is a constant input of an input-affine system with a fixed initial state.
  • Figure 2: Snapshots of output $y(t;\xi)$ of the Q-FISCIDS representation for a function such that $y(1;\xi) = \exp \left({-\frac{1}{2}(\xi_1^2 + \xi_1 \xi_2 + \xi_2^2) + \xi_1 + \frac{1}{2}\xi_2} \right)$. The initial value $y(0;\xi) = 1$ is fixed for all $\xi$.
  • Figure 3: Inclusion relationships between classes of functions.

Theorems & Definitions (11)

  • Definition 1: FISCIDS Representation
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2: DA Function
  • Theorem 4
  • Example 1
  • Example 2
  • Theorem 5
  • ...and 1 more