On improving estimate of the region of attraction of a class of nonlinear time delay system

Villafuerte, R.; Mondié, Sabine

doi:10.1016/s1474-6670(17)69292-9

Cited by 7 publications

(6 citation statements)

References 2 publications

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“…(3) For instance, κ 1 is used in estimations of domains of attraction via linearization, Melchor-Aguilar and Niculescu (2006); Villafuerte and Mondié (2007); Alexandrova (2020); Scholl et al (2020). However, while for Lyapunov functions V (y) = y P y the largest lower bound in terms of y 2 is simply obtained by the minimum eigenvalue of P , λ min (P ) y 2 2 ≤ V y (y), nothing seems to be reported about the conservativity of known bounds on the LK functionals.…”

Section: Introductionmentioning

confidence: 99%

What ODE-Approximation Schemes of Time-Delay Systems Reveal about Lyapunov-Krasovskii Functionals

Scholl¹,

Hagenmeyer²,

Gröll³

2022

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

What ODE-Approximation Schemes of Time-Delay Systems Reveal about Lyapunov-Krasovskii Functionals

Scholl¹,

Hagenmeyer²,

Gröll³

2022

Preprint

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“…In this light, small initial disturbances can still result in departing trajectories. In relation to the results listed above, estimations on domains of attraction in time-delay systems are rare in the literature [11,15,27,58,83]. Time-delay systems require an initial function instead of the initial value, which would be sufficient in ODEs.…”

Section: Introductionmentioning

confidence: 99%

“…Of course, usually nothing is known about the conservativeness of such criteria. Results based on Lyapunov-Krasovskii functionals can be translated to such easier to handle norm criteria, which goes along with further conservativeness [58,83]. (vii) An overestimation of the radius of attraction (interior of the red dashed circle with radiusR X in Fig.…”

Section: Introductionmentioning

confidence: 99%

On norm-based estimations for domains of attraction in nonlinear time-delay systems

2020

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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 timeforward 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.

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“…In this light, small initial disturbances can still result in departing trajectories. In relation to the results listed above, estimations on domains of attraction in time delay systems are rare in literature [6,[18][19][20][21][22][23][24]. Time delay systems require an initial function instead of the initial value which would be sufficient in ODEs.…”

Section: Introductionmentioning

confidence: 99%

“…Of course, usually nothing is known about the conservativeness of such criteria. Lyapunov-Krasovskii results can be translated to such easier to handle norm criteria, which goes along with further conservativeness [18,19].…”

Section: Introductionmentioning

confidence: 99%

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

Scholl,

Hagenmeyer,

Gröll

2019

Preprint

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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.

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On improving estimate of the region of attraction of a class of nonlinear time delay system

Cited by 7 publications

References 2 publications

What ODE-Approximation Schemes of Time-Delay Systems Reveal about Lyapunov-Krasovskii Functionals

What ODE-Approximation Schemes of Time-Delay Systems Reveal about Lyapunov-Krasovskii Functionals

On norm-based estimations for domains of attraction in nonlinear time-delay systems

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

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