Lang, MoritzIST Austria ; Sontag, Eduardo
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.
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Lang M, Sontag E. Zeros of nonlinear systems with input invariances. Automatica. 2017;81C:46-55. doi:10.1016/j.automatica.2017.03.030
Lang, M., & Sontag, E. (2017). Zeros of nonlinear systems with input invariances. Automatica, 81C, 46–55. https://doi.org/10.1016/j.automatica.2017.03.030
Lang, Moritz, and Eduardo Sontag. “Zeros of Nonlinear Systems with Input Invariances.” Automatica 81C (2017): 46–55. https://doi.org/10.1016/j.automatica.2017.03.030.
M. Lang and E. Sontag, “Zeros of nonlinear systems with input invariances,” Automatica, vol. 81C, pp. 46–55, 2017.
Lang M, Sontag E. 2017. Zeros of nonlinear systems with input invariances. Automatica. 81C, 46–55.
Lang, Moritz, and Eduardo Sontag. “Zeros of Nonlinear Systems with Input Invariances.” Automatica, vol. 81C, IFAC, the International Federation of Automatic Control, 2017, pp. 46–55, doi:10.1016/j.automatica.2017.03.030.