计算大规模问题中的延迟李雅普诺夫矩阵和H2范数
Computing delay Lyapunov matrices and H2 norms for large-scale problems
AI总结：
本文提出了一种计算具有多个离散延迟系统的延迟李雅普诺夫矩阵和H2范数的方法，通过谱离散化、相似变换和Krylov方法，避免了预处理离散化步骤，实现了动态构造高秩近似。
摘要：
一个指数稳定的线性时不变延迟微分方程组对应的延迟李雅普诺夫矩阵可以表征为涉及矩阵值延迟微分方程的边值问题的解。该边值问题可以视为经典李雅普诺夫矩阵方程的自然推广。本文提出了一种通用方法，用于计算具有多个离散延迟的系统中的延迟李雅普诺夫矩阵和H2范数，其适用性扩展到矩阵较大且稀疏，以及关联的半正定矩阵（标准李雅普诺夫方程的“右端项”）具有低秩的情况。与现有基于直接求解边值问题的方法不同，本文的方法基于求解增加维度的标准李雅普诺夫方程。它结合了几个成分：i）延迟方程系统的谱离散化，ii）针对性的相似变换，该变换诱导了期望的结构和稀疏性模式，同时有利于相应李雅普诺夫方程的准确低秩解，以及iii）用于大规模矩阵李雅普诺夫方程的Krylov方法。问题的结构被利用，使得最终算法不涉及预处理离散化步骤，并提供完全动态构造的递增秩近似。还给出了直接应用于等价于原始时滞系统的标准线性无限维系统的投影方法的解释。本文通过两个教学示例说明了问题的性质、挑战和方法论选择，最后通过数值实验展示了算法的有效性。
英文摘要：
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. We present a general approach for computing delay Lyapunov matrices and H2 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. 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 time-delay 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.
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