Stabilization via generalized homogeneous approximations

We introduce a notion of generalized homogeneous approximation at the origin and at infinity which extends the classical notions and captures a large class of nonlinear systems, including (lower and upper) triangular systems. Exploiting this extension and although this extension does not preserve the basic properties of the classical notion, we give basic results concerning stabilization and robustness of nonlinear systems, by designing a homogeneous (in the generalized sense) feedback controller which globally asymptotically stabilizes a chain of power integrators and makes it the dominant part at infinity and at the origin (in the generalized sense) of the dynamics. Stability against nonlinear perturbation follows from domination arguments.