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

Abstract

A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential equation. This boundary value problem can be seen as a natural generalization of the classical Lyapunov matrix equation. Lyapunov matrices play an important role in constructing Lyapunov functionals and in H 2 optimal control. In this paper we present a general approach for computing delay Lyapunov matrices and H 2 norms for systems with multiple discrete delays, whose applicability extends towards problems where the matrices are large and sparse, and the associated positive semidefinite matrix (the "right-hand side" for the standard Lyapunov equation), has a low rank. The problems addressed are challenging, because besides that the boundary value problem is matrix valued with a structure that much harder to exploit than in the delay-free case, its solution is in the generic situation non-smooth. In contract to existing methods that are based on solving the boundary value problem directly, our method is grounded in solving standard Lyapunov equations of increased dimensions. It combines several ingredients: i) a spectral discretization of the system of delay equations, ii) a targeted similarity transformation which induces a desired structure and sparsity pattern and, at the same time, favors accurate low rank solutions of the corresponding Lyapunov equation, and iii) a Krylov method for large-scale matrix Lyapunov equations. The structure of the problem is exploited in such a way that the final algorithm does not involve a preliminary discretization step, and provides a fully dynamic construction of approximations of increasing rank. Interpretations in terms of a projection method directly applied to a standard linear infinite-dimensional system equivalent to the original timedelay system are also given. Throughout the paper two didactic examples are used to illustrate the properties of the problem, the challenges and methodological choices, while numerical experiments are presented at the end to illustrate the effectiveness of the algorithm.Key words. delay system, Lyapunov matrix equations, Krylov method   K (t) = A 0 K(t) + m i=1 A i K(t − τ i ), for almost all t ≥ 0, K(0) = I, K(t) = 0, for t < 0.

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…”

On the dual linear periodic time‐delay system: Spectrum and Lyapunov matrices, with application to analysis and balancing

“…A generally applicable but less efficient approach, which extends the results presented in section 2 of Reference to periodic systems, consists of inferring an approximation of U from the Lyapunov matrix associated with a spectral discretization of the delay equation. This approach requires solving a standard delay‐free periodic Lyapunov equation of increased dimensions.…”

Design of delay-based output-feedback controllers optimizing a quadratic cost function via the delay Lyapunov matrix

“…For the computation of Lyapunov matrices for systems with non-commensurate delays the reader is referred to [15] and [8], and for large scale systems to [22]. A useful property of the delay Lyapunov matrix of system ( 4) is that its elements are analytic functions of the system parameters.…”

“…In each of them, we consider that the initial condition of the system is unknown, but that satisfies x 0 1, and that the weighting matrices of the performance index (3) are given by Q = I n and R = 1. The objective function J wc (f) is computed from Theorem 9 by using equation (15), and its gradient is computed from Theorem 11 by employing equation (22). The first example is taken from [33].…”