The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited

Abstract

We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Padé approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the non-zero collocation points are chosen as the zeroes of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the $H^2$-norm, where, under mild conditions, super-geometric convergence is observed and, for a special case, super convergence is proved; both significantly faster than the algebraic convergence reported in previous work.

Figures (4)

• Figure 1: \newlabelfig:usual-conv0 Convergence of the ${H^2}$-norm of \ref{['eq:rdde']} with $A_0 = \spmqty{-2 1 \\ 3 -8}$, $A_1 = \spmqty{-1 -1 \\ -1 -1}$, $\tau = 1$, and $B = C = I_2$, for different discretizations. The solid line corresponds to the Lanczos tau method with Chebyshev polynomials of the second kind, the dashed line to pseudospectral collocation in Chebyshev extremal nodes.
• Figure 2: \newlabelfig:jacobi-conv0 Convergence of the ${H^2}$-norm of the system of \ref{['fig:usual-conv']}, for different discretizations. The solid line corresponds to the Lanczos tau method with Jacobi $\pqty{-\frac{1}{2},-\frac{3}{4}}$ polynomials, which do not satisfy \ref{['assum:oddeven']}, the dashed line to pseudospectral collocation in Chebyshev extremal nodes.
• Figure 3: \newlabelfig:super-conv0 Convergence of the ${H^2}$-norm of \ref{['eq:rdde']} with $A_0 = A_1 = -1$, $\tau = 1$, and $B = C = 1$, for different discretizations. The solid line corresponds to the Lanczos tau method with Chebyshev polynomials of the second kind, the dashed line to pseudospectral collocation in Chebyshev extremal nodes. For clarity of the figure, the relative error was lower bounded by $10^{-16}$.
• Figure 4: \newlabelfig:tf-approx0 Transfer function $G$ of the system of \ref{['fig:super-conv']}, and the approximation $G_1$ from a Lanczos tau method using Chebyshev polynomials of the second kind with $N=1$.

Theorems & Definitions (19)

• Proposition 2.1
• Proposition 3.1
• Proof 1
• Proposition 3.2
• Proof 2
• Theorem 4.1
• Proof 3
• Theorem 4.2
• Proof 4
• Corollary 4.3