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

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.