arXiv subjects

Tessina H. Scholl

Publications and source records attributed to Tessina H. Scholl.

5 recordsLinked to original sources

arXiv2026

Lyapunov-Krasovskii Functionals of Robust Type for the Stability Analysis in Time-Delay Systems

Inspired by the widespread concept of Lyapunov-Krasovskii functionals of complete type, this article proposes an alternative class of functionals, termed Lyapunov-Krasovskii functionals of robust type. Their construction aims to improve resulting robustness bounds of linear systems with a constant delay. These refer to bounds on nonlinear or uncertain terms that can be added to the system without compromising the proof of stability. The article derives, via the classical Lyapunov-Krasovskii theorem, an $H_\infty$-norm-based small-gain robustness result established by a functional of known structure. While complete-type functionals are related to infinite-dimensional Lyapunov equations, the proposed functionals are related to infinite-dimensional algebraic Riccati equations. Existence of the functional relies on the operator-valued Kalman-Yakubovich-Popov lemma, which is made applicable due to a splitting approach. In particular, for any asymptotically stable nominal system, there exists a Lyapunov-Krasovskii functional of robust type that yields a nonzero bound on admissible perturbations. An example illustrates the reduction in conservatism compared with complete-type functionals.

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arXiv2026

A Lyapunov-Based Perspective on Absolute Stability

This article presents a unifying perspective on absolute stability concepts. In particular, it develops a Lyapunov-like explanatory framework for a nonscalar circle criterion with its small-gain and strict-passivity special cases. To this end, a general defining inequality for a Lyapunov-like function is proposed that avoids strict definiteness conditions, enabled by a strengthening of the sector constraint. We discuss different ways to derive a quadratic solution: via a linear matrix inequality (LMI), an algebraic Riccati equation, and a matrix equation. By exploiting the Kalman-Yakubovich-Popov (KYP) lemma, classical frequency-domain results are recovered. A passivity-index-based result is derived that simplifies the evaluation. Overall, the presented interrelations may be useful for both analysis and teaching.

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arXiv2023

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

The article proposes an approach to complete-type and related Lyapunov-Krasovskii functionals that neither requires knowledge of the delay-Lyapunov matrix function nor does it involve linear matrix inequalities. The approach is based on ordinary differential equations (ODEs) that approximate the time-delay system. The ODEs are derived via spectral methods, e.g., the Chebyshev collocation method (also called pseudospectral discretization) or the Legendre tau method. A core insight is that the Lyapunov-Krasovskii theorem resembles a theorem for Lyapunov-Rumyantsev partial stability in ODEs. For the linear approximating ODE, only a Lyapunov equation has to be solved to obtain a partial Lyapunov function. The latter approximates the Lyapunov-Krasovskii functional. Results are validated by applying Clenshaw-Curtis and Gauss quadrature to a semi-analytical result of the functional, yielding a comparable finite-dimensional approximation. In particular, the article provides a formula for a tight quadratic lower bound, which is important in applications. Examples confirm that this new bound is significantly less conservative than known results.

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arXiv2020

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.

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arXiv2019

Time Delay in the Swing Equation: A Variety of Bifurcations

The present paper addresses the swing equation with additional delayed damping as an example for pendulum-like systems. In this context, it is proved that recurring sub- and supercritical Hopf bifurcations occur if time delay is increased. To this end, a general formula for the first Lyapunov coefficient in second order systems with additional delayed damping and delay-free nonlinearity is given. In so far the paper extends results about stability switching of equilibria in linear time delay systems from Cooke and Grossman. In addition to the analytical results, periodic solutions are numerically dealt with. The numerical results demonstrate how a variety of qualitative behaviors is generated in the simple swing equation by only introducing time delay in a damping term.

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