Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4999774 | Automatica | 2017 | 10 Pages |
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)â¦u(t)+p, pâR) or scale-invariance (u(t)â¦pu(t), pâR>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 s0âR, 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.