@article{1007,
  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.},
  author       = {Lang, Moritz and Sontag, Eduardo},
  issn         = {0005-1098},
  journal      = {Automatica},
  pages        = {46 -- 55},
  publisher    = {International Federation of Automatic Control},
  title        = {{Zeros of nonlinear systems with input invariances}},
  doi          = {10.1016/j.automatica.2017.03.030},
  volume       = {81C},
  year         = {2017},
}