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

• 成果类型: Article
• 署名作者: Scholl, Tessina H.; Hagenmeyer, Veit; Groell, Lutz
• 署名单位: Helmholtz Association; Karlsruhe Institute of Technology
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3347497
• 发表日期: 2024
• 页码: 4614-4629
• 关键词: Delay systems; Gauss quadrature; Lyapunov-Krasovskii functional; operator-valued Lyapunov equation; pseudospectral discretization; Spectral methods

摘要

This 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 semianalytical 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.