Zeros of nonlinear systems with input invariances Journal Article

Author(s): Lang, Moritz; Sontag, Eduardo
Article Title: Zeros of nonlinear systems with input invariances
Affiliation IST Austria
Abstract: 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.
Keywords: Nonlinear systems; Invariant systems; transmission zeros; differentiators; adaptive systems
Journal Title: Automatica
Volume: 81C
ISSN: 0005-1098
Publisher: IFAC, the International Federation of Automatic Control  
Date Published: 2017-04-20
Start Page: 46
End Page: 55
Copyright Statement: CC BY
Sponsor: FP7/2007-2013 no [291734]
DOI: 10.1016/j.automatica.2017.03.030
Notes: ML is grateful to the members of the Guet and Tkačik groups of IST Austria for valuable discussions, comments and support. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA Grant Agreement No. [291734]. Work was supported in part by grants AFOSR FA9550-14-1-0060 and NIH 1R01GM100473. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nathan Van De Wouw under the direction of Editor Andrew R. Teel.
Open access: yes (OA journal)
IST Austria Authors
  1. Moritz Lang
    4 Lang
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