Geometric homogeneity with applications to finite-time stability

Abstract

This paper studies properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution of this paper is a result relating regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result makes it possible to extend previous results on homogeneous systems to the geometric framework. As an application of our results, we consider finite-time stability of homogeneous systems. The main result that links homogeneity and finite-time stability is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity. We also show that the assumption of homogeneity leads to stronger properties for finite-time stable systems.

Adaptive continuous higher order sliding mode control

“…Plots: example 2 double-layer adaptiv From Proposition 8.1 presented in Bhat & Bernstein (2005), there exists an ϵ b ∈ (0, 1) such that for everyᾱ ∈ (1 − ϵ b , 1) the origin σ,σ . .…”

Fast Control Systems: Nonlinear Approach

“…The next theorem gives the most important result about scalability solutions to dhomogeneous evolution equations [49], [23], [41], [7], [28], [35].…”

Global finite-time observers for non linear systems

“…Nevertheless new conditions of stability and stabilization have been developped in the continuous finite-time domain, involving the settling time function, associated with the Lyapunov theory and homogeneous systems by Bhat and Bernstein in [11,12], and by Moulay and Perruquetti in [13,14]. There exists several methods achieving finite-time convergence, e.g.…”