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

Scholl, Tessina H.; Hagenmeyer, Veit; Gröll, Lutz

doi:10.1007/s11071-020-05620-8

Cited by 11 publications

(4 citation statements)

References 75 publications

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“…However, starting the initial condition outside of the region of attraction (yellow trajectory), the solution converges to the centre manifold, but it is repelled on it by the unstable periodic orbit and continues to oscillate with evergrowing amplitude. Note that the approximation of the basin of attraction for time delay systems is a challenging problem due to the infinite-dimensional state space and out of the scope of this work, although there exist some promising techniques in the literature [67][68][69][70].…”

Section: Methods For Nonlinear Analysismentioning

confidence: 99%

Nonlinear effects of saturation in the car-following model

Martinovich

2022

Nonlinear Dyn

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The goal of this paper is to provide insight about the effect of acceleration saturation in the car-following model. In this contribution, we consider a heterogeneous, mixed-traffic scenario which contains both human-driven and autonomous vehicles subjected to time delays. Corresponding stability charts are provided from which one can tune the control parameters of the automated vehicles to achieve smooth traffic flow. By taking into account the acceleration saturation, it modifies the global behaviour of the system and reduces the range of the optimal technological parameters. On a demonstrative example, we highlight the complex dynamical phenomenon induced by the saturation and we attempt to connect these nonlinear investigations to the engineering practice and point out their relevance.

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Section: Methods For Nonlinear Analysismentioning

confidence: 99%

Nonlinear effects of saturation in the car-following model

Martinovich

2022

Nonlinear Dyn

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“…For autonomous time delay systems without control, 51 introduced the concept of safety functionals, which has been investigated further in Reference 52 by means of discretization. The relationship between discretization and functionals is discussed in Reference 53, while safe domains interpreted as basin of attraction were investigated for delayed dynamical systems in Reference 54. These works, however, do not address control systems with time delay.…”

Section: Introductionmentioning

confidence: 99%

Control barrier functionals: Safety‐critical control for time delay systems

Molnár

Ames

Orosz

2023

Intl J Robust & Nonlinear

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This work presents a theoretical framework for the safety-critical control of time delay systems. The theory of control barrier functions, that provides formal safety guarantees for delay-free systems, is extended to systems with state delay. The notion of control barrier functionals is introduced, to attain formal safety guarantees by enforcing the forward invariance of safe sets defined in the infinite dimensional state space. The proposed framework is able to handle multiple delays and distributed delays both in the dynamics and in the safety condition, and provides an affine constraint on the control input that yields provable safety. This constraint can be incorporated into optimization problems to synthesize pointwise optimal and provable safe controllers. The applicability of the proposed method is demonstrated by numerical simulation examples.

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

Cited by 11 publications

References 75 publications

Nonlinear effects of saturation in the car-following model

Nonlinear effects of saturation in the car-following model

Control barrier functionals: Safety‐critical control for time delay systems

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

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