5 research outputs found
On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems
For nonlinear time-delay systems, domains of attraction are rarely studied
despite their importance for technological applications. The present paper
provides methodological hints for the determination of an upper bound on the
radius of attraction by numerical means. Thereby, the respective Banach space
for initial functions has to be selected and primary initial functions have to
be chosen. The latter are used in time-forward simulations to determine a first
upper bound on the radius of attraction. Thereafter, this upper bound is
refined by secondary initial functions, which result a posteriori from the
preceding simulations. Additionally, a bifurcation analysis should be
undertaken. This analysis results in a possible improvement of the previous
estimation. An example of a time-delayed swing equation demonstrates the
various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in
'Nonlinear Dynamics'. The final authenticated version is available online at
https://doi.org/10.1007/s11071-020-05620-8
On norm-based estimations for domains of attraction in nonlinear time-delay systems
For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects
What ODE-Approximation Schemes of Time-Delay Systems Reveal about Lyapunov-Krasovskii Functionals
Lyapunov-Krasovskii functionals are found to be related to Lyapunov functions
that prove partial stability in finite dimensional system approximations. These
approximations are ordinary differential equations, which, in the present
paper, originate from the Chebyshev (pseudospectral) collocation or the
Legendre tau method. Lyapunov functions that prove partial stability are simply
obtained by solving a Lyapunov equation. They approximate the
Lyapunov-Krasovskii functional. A formula for the partial positive definiteness
bound on the Lyapunov function is derived. The formula is also applied to a
numerical integration of the known Lyapunov-Krasovskii functional. An example
shows that both approaches converge to identical results, representing the
largest quadratic lower bound on complete-type or related functionals.Comment: 6 pages, 2 figures, "This work has been submitted to IFAC for
possible publication.