Computing delay lyapunov matrices and H₂ norms for large-scale problems

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 H₂ optimal control. In this paper we present a general approach for computing delay Lyapunov matrices and H₂ norms for systems with multiple discrete delays, whose applicability extends toward problems where the matrices are large and sparse, and the associated positive semidefinite matrix has a low rank. The problems addressed are challenging, because the boundary value problem is matrix valued with a structure that is much harder to exploit than in the delay-free case, and its solution is in the generic situation nonsmooth. In contrast 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 are also given in terms of a projection method directly applied to a standard linear infinite-dimensional system, which is equivalent to the original time-delay system. Throughout the paper two didactic examples are used to illustrate the properties of the problem, the challenges and methodological choices, while numerical experiments are reported to illustrate the effectiveness of the algorithms.