Analysis of robustness of homogeneous systems with time delays using Lyapunov‐Krasovskii functionals

Abstract

The paper is devoted to stability analysis of homogeneous time-delay systems applying the Lyapunov-Krasovskii theory, and a generic structure of the functional is given that suits for any homogeneous system of non-zero degree (and can also be used for any dynamics admitting a homogeneous approximation). The obtained stability conditions are utilized to evaluate the domain of attraction for the delayed twisting control algorithm.

Stability analysis of switched homogeneous time-delay systems under synchronous and asynchronous commutation

“…W ≤ − From ( 9) and ( 10) it follows (see [17]) that˙ W ≤ −b 1 W 1+ ν µ for ν < 0 in the domain where…”

“…Using the properties of homogeneous functions the following result can be obtained [17]: Lemma 1. Let a, b ∈ R + and > 0, α > 0, β > 0, γ > 0, δ > 0 be given, then…”

“…Next, determine conditions under which there exist Lyapunov-Krasovskii functionals guaranteeing delay-independent asymptotic stability or uniform ultimate boundedness for the systems (1) and ( 2) under arbitrary switching. For this purpose we will use a recent result [17], which suggests a generic structure of Lyapunov-Krasovskii functional for a homogeneous system that is asymptotically stable in the delay-free case.…”

“…It has been shown that for any value of delay, the homogeneous systems, which are asymptotically stable in the delay-free case, are locally asymptotic stable for positive degree and globally asymptotic stable with respect to a compact set containing the origin for negative degree [16,9,45,15,5,6,23,59]. These results have been obtained using the Lyapunov-Razumikhin [23,59] and Lyapunov-Krasovskii [7,17] theories. Moreover, in [17] a canonical form of Lyapunov-Krasovskii functional has been proposed that can be used for any (locally) homogeneous systems.…”

“…These results have been obtained using the Lyapunov-Razumikhin [23,59] and Lyapunov-Krasovskii [7,17] theories. Moreover, in [17] a canonical form of Lyapunov-Krasovskii functional has been proposed that can be used for any (locally) homogeneous systems. This achievement allows the Lyapunov-Krasovskii approach to be effectively applied for this class of models.…”