On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems

Abstract

For nonlinear time delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper introduces a novel method to determine an upper bound on the radius of attraction by a numerical approach. 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 is undertaken with the time delay as bifurcation parameter. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the overall methodology.

Time delay in the swing equation: A variety of bifurcations

“…e denotes the k-th derivative of h(x 1 ) in x 1 = 0. In order to evaluate (33), the characteristic matrix given by (11) with c := h e is considered. Due to (13), an evaluation at λ = iω can be simplified by a + ãe −iωτ = −iω + i c ω .…”

“…According to (33) normalization factors α p , α q ∈ C, have to be chosen, such that p := α p ic ω , q := α q 1 iω (39) fulfill a relative normalization p D∆(iω)q ! = 1, where the derivative of the characteristic matrix is given by D∆(λ ) = 1 0 0 1 − τ ãe −λ τ .…”

“…Such upper bounds are needed in order to prove that a minimum norm requirement on the domain of attraction is not fulfilled. We refer to Scholl et al 33 , Example VI.1.…”