---
_id: '1007'
abstract:
- lang: eng
  text: 'A nonlinear system possesses an invariance with 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 in R) or scale-invariance (u(t) -> pu(t), p in 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) systems with transfer functions
    having an irreducible zero at s0 in 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 second-
    and higher-order s0-invariances, 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 nth-order 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.'
article_processing_charge: Yes (in subscription journal)
author:
- first_name: Moritz
  full_name: Lang, Moritz
  id: 29E0800A-F248-11E8-B48F-1D18A9856A87
  last_name: Lang
- first_name: Eduardo
  full_name: Sontag, Eduardo
  last_name: Sontag
citation:
  ama: Lang M, Sontag E. Zeros of nonlinear systems with input invariances. <i>Automatica</i>.
    2017;81C:46-55. doi:<a href="https://doi.org/10.1016/j.automatica.2017.03.030">10.1016/j.automatica.2017.03.030</a>
  apa: Lang, M., &#38; Sontag, E. (2017). Zeros of nonlinear systems with input invariances.
    <i>Automatica</i>. International Federation of Automatic Control. <a href="https://doi.org/10.1016/j.automatica.2017.03.030">https://doi.org/10.1016/j.automatica.2017.03.030</a>
  chicago: Lang, Moritz, and Eduardo Sontag. “Zeros of Nonlinear Systems with Input
    Invariances.” <i>Automatica</i>. International Federation of Automatic Control,
    2017. <a href="https://doi.org/10.1016/j.automatica.2017.03.030">https://doi.org/10.1016/j.automatica.2017.03.030</a>.
  ieee: M. Lang and E. Sontag, “Zeros of nonlinear systems with input invariances,”
    <i>Automatica</i>, vol. 81C. International Federation of Automatic Control, pp.
    46–55, 2017.
  ista: Lang M, Sontag E. 2017. Zeros of nonlinear systems with input invariances.
    Automatica. 81C, 46–55.
  mla: Lang, Moritz, and Eduardo Sontag. “Zeros of Nonlinear Systems with Input Invariances.”
    <i>Automatica</i>, vol. 81C, International Federation of Automatic Control, 2017,
    pp. 46–55, doi:<a href="https://doi.org/10.1016/j.automatica.2017.03.030">10.1016/j.automatica.2017.03.030</a>.
  short: M. Lang, E. Sontag, Automatica 81C (2017) 46–55.
date_created: 2018-12-11T11:49:39Z
date_published: 2017-06-01T00:00:00Z
date_updated: 2023-10-17T08:51:18Z
day: '01'
ddc:
- '000'
department:
- _id: CaGu
- _id: GaTk
doi: 10.1016/j.automatica.2017.03.030
ec_funded: 1
external_id:
  isi:
  - '000403513900006'
file:
- access_level: open_access
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:11:29Z
  date_updated: 2018-12-12T10:11:29Z
  file_id: '4884'
  file_name: IST-2017-813-v1+1_ZerosOfNonlinearSystems.pdf
  file_size: 1401954
  relation: main_file
file_date_updated: 2018-12-12T10:11:29Z
has_accepted_license: '1'
isi: 1
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 46 - 55
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: Automatica
publication_identifier:
  issn:
  - 0005-1098
publication_status: published
publisher: International Federation of Automatic Control
publist_id: '6391'
pubrep_id: '813'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Zeros of nonlinear systems with input invariances
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 81C
year: '2017'
...