Nonlinear step responses and how to find them

Abstract The unit step response is a standard tool for experimental identification; its shape is equivalent to a solution of an appropriate governing equation. In this study, we investigate two variants of nonlinear first-order governing equation, $${y'=k(1-y^\beta )/\beta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>β</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:math> , and mainly its rescaled version $${y'=k\left( 1-y^\beta \right) }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>β</mml:mi> </mml:msup> </mml:mfenced> </mml:mrow> </mml:math> . We explore a continuum of response shapes, ranging from constant zero to constant one, including standard shapes such as $$1-\exp (-t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $${\tanh (t)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>tanh</mml:mo> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , but also uncommon shapes such as a part of the logarithmic integral. Two methods for estimating the parameter $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> are proposed: one based on analyzing the curvature of a single response and the other based on analyzing initial slopes for multiple responses, while varying the amplitude of the step. Our investigation is conducted in the context of gas permeation through a homogeneous porous medium, where nonlinear response shapes have been observed. This context allows for a meaningful interpretation of the results, consistent with expectations for specific thermodynamic processes. However, neither method is constrained by this context, and either might be considered for any first-order system that fails to have amplitude-scale invariance or for which the shapes seem to exhibit an effect of saturation.