Zeros of nonlinear systems with input invariances

A nonlinear system possesses an invariancewith respect to a set of transformations if its output dynamics remain invariant when transforming the input, and adjusting the initial condition accordingly. Most research has focused on invariances with respect to time-independent pointwise transformations like translational-invariance (u(t) → u(t) + p, p ∈ R) or scale-invariance (u(t) → pu(t), p ∈ R>0). In this article, we introduce the concept of s0-invariances with respect to continuous input transformations exponentially growing/decaying over time.We show that s0-invariant systems not only encompass linear time-invariant (LTI) systemswith transfer functions having an irreducible zero at s0 ∈ R, but also that the input/output relationship of nonlinear s0-invariant systems possesses properties well known from their linear counterparts. Furthermore,we extend the concept of s0-invariances to secondand higher-order s0invariances, corresponding to invariances with respect to transformations of the time-derivatives of the input, and encompassing LTI systems with zeros of multiplicity two or higher. Finally, we show that nthorder 0-invariant systems realize – under mild conditions – nth-order nonlinear differential operators: when excited by an input of a characteristic functional form, the system’s output converges to a constant value only depending on the nth (nonlinear) derivative of the input. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).