Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning

Abstract

The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delayWe propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unknown is the value U (τ /2) ∈ R n×n , i.e., the delay Lyapunov matrix at time τ /2. This linear matrix equation with n 2 unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the n 2 × n 2 matrix associated to this linear system can be computed by solving a coupled matrix initial-value problem. A preconditioner for the iterative method is proposed based on solving a T-Sylvester equation M X +X T N = C, for which there are methods available in the literature. We prove that the preconditioner is effective under certain assumptions. The efficiency of the approach is illustrated by applying it to a time-delay system stemming from the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to n ≈ 1000, i.e., a linear system with n 2 ≈ 10 6 unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation.

Characterization and optimization of the smoothed spectral abscissa for time‐delay systems

“…Finally, in Section 4, we assume that the delays are commensurate, which led to a novel characterization of derivatives of the Lyapunov matrix allowing to compute matrix and derivatives simultaneously. For computing Lyapunov matrices of systems with incommensurate delays, we refer to the recent works …”

“…For computing Lyapunov matrices of systems with incommensurate delays, we refer to the recent works. 16,[27][28][29] How to cite this article: Gomez…”

Computing Delay Lyapunov Matrices and $\mathcal{H}_2$ Norms for Large-scale Problems

“…Some intuition behind this observation is given by Theorems 2.3 and 3.1: by construction precisely the matching moment between Υ and Υ N carry over to the projected transfer function Υ k . In experiments with very large n, we have kr << n for a realistic range for k values as in the second and third example, and the observed decay rate is slower, which is illustrated by a comparison between Inherent to the projection approach, the efficiency of the computational approach depends on whether or not accurate low rank approximations exist, whose determining factors are not well understood, and the projected system matrix G 2k must be stability preserving (this is the case for most problems and it was an important consideration in the methodological choices, but not always -a counter example is the 2nd example in [11] for n = 1023, where spurious roots are observed in the right half plane). The latter is not necessarily a strong limitation for the H 2 norm computation, since the L 2 norm of the low-order, projected transfer function Γ k can still be computed using other techniques different from solving the Lyapunov equation directly.…”

“…Then (1.4) can be reformulated as a standard boundary value problem for an ordinary differential equation of dimensions 2n 2 n m on the interval [0, h]. For the latter boundary value problem, the transition between starting and end time can be determined explicitly in the form of the action of a matrix exponential (the so-called semi-analytic approach [13,6]) or by a numerical time-integration scheme [11].…”

“…The common factor that leads to a poor scalability of the above vectorization based approaches with respect to the dimension n is that they rely on solving a system of equations in n 2 variables, possibly multiplied with a large factor, hence when using a direct solver the number of elementary operations amounts to O(n 6 ) operations. To the best of the authors' knowledge the only available method that allows to address large-scale problems is the one presented in [11] for the single delay case, which falls under the umbrella of shooting methods, with the transition map determined by time-integration. The key idea behind this approach, which has been shown to be effective for problems with n up to ≈ 1000, is to solve the linear system of equations arising from the shooting method using a preconditioned Krylov method, where the preconditioner is determined from the corresponding problem without delay.…”