Papers by M. Lang and E.D. Sontag

This paper introduces two generalizations of systems invariant with respect to continuous sets of input transformations, that is, systems whose output dynamics remain invariant when applying a transformation to the input and simultaneously adjusting the initial conditions. These generalizations concern systems invariant with respect to time-dependent input transformations with exponentially increasing or decreasing ``strength'', and systems invariant with respect to transformations of the "nonlinear derivatives" of the input. Interestingly, these two generalizations of invariant systems encompass linear time-invariant (LTI) systems with real transfer function zeros of arbitrary multiplicity. Furthermore, the zero-dynamics of systems possessing our generalized invariances show properties analogous to those of LTI systems with transfer function zeros, generalizing concepts like pole-zero cancellation, the rejection of ramps by Hurwitz LTI systems with a zero at the origin with multiplicity two, and (to a certain extend) the superposition principle with respect to inputs zeroing the output.

In this article, we show that scale-invariant systems, as well as systems invariant with respect to other input transformations, can realize nonlinear differential operators: when excited by inputs obeying functional forms characteristic for a given class of invariant systems, the systems' outputs converge to constant values directly quantifying the speed of the input.