On Existence of Invariant Sets for the Internal Dynamics of Homogeneous Differentiators

In this paper, we consider the internal error dynamics of a set of arbitrary order homogeneous systems, known as homogeneous differentiators, for which we provide sufficient conditions for the existence of positively invariant sets. Such conditions consist in verifying the existence of real roots of a polynomial whose coefficients depend linearly on the gains of the system. For the dynamics restricted to the invariant sets, we prove that the asymptotic stability of the origin depends on the sign of the real roots of the mentioned polynomial. Moreover, for the case of nonpositive homogeneity degree, we establish the existence of real roots such that the asymptotic stability of the origin of the restricted dynamics implies the asymptotic stability of the origin of the unrestricted dynamics.