Resumen de Zeros of nonlinear systems with input invariances
Moritz Lang, Eduardo D. Sontag
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 ∈ R ) or scale-invariance ( u ( t ) ↦ p u ( t ) , p ∈ R > 0 ). In this article, we introduce the concept of s 0 -invariances with respect to continuous input transformations exponentially growing/decaying over time. We show that s 0 -invariant systems not only encompass linear time-invariant (LTI) systems with transfer functions having an irreducible zero at s 0 ∈ R , but also that the input/output relationship of nonlinear s 0 -invariant systems possesses properties well known from their linear counterparts. Furthermore, we extend the concept of s 0 -invariances to second- and higher-order s 0 -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 n th-order 0 -invariant systems realize–under mild conditions– n th-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 n th (nonlinear) derivative of the input.
