发表机构
Technische Universität Ilmenau（埃尔朗根-纽伦堡工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于算子ADI迭代的延迟微分系统平衡截断方法，直接处理无限维历史状态，在Takenaka-Malmquist坐标中实现有限维计算，并证明H∞收敛性，数值实验验证了有效性。
AI 中文摘要
我们针对线性延迟微分系统提出了一种平衡截断方法，该方法基于对相关算子Lyapunov方程进行有限秩交替方向隐式（ADI）迭代。该方法直接作用于无限维历史状态实现，在Gramian近似或平衡过程中不离散化延迟区间。有限秩可达性和可观测性因子在Takenaka-Malmquist坐标中精确表示，将所需计算简化为有限维线性代数。这些因子产生一个有限维、无延迟的降阶模型。对于实数系统数据，实现保持实数运算。在ADI移位满足适当条件且Hankel奇异值存在间隙的情况下，我们证明了在$\mathcal H_\infty$范数下收敛到由无限维系统精确平衡截断所得的降阶传递函数。该构造可推广到有限多个离散延迟。数值实验验证了该方法与独立参考计算的一致性。实验中使用的完整MATLAB实现作为补充材料提供。
英文摘要
We develop a balanced truncation method for linear delay differential systems based on a finite-rank alternating-direction implicit (ADI) iteration for the associated operator Lyapunov equations. The method acts directly on the infinite-dimensional history-state realization and does not discretize the delay interval during Gramian approximation or balancing. The finite-rank reachability and observability factors are represented exactly in Takenaka--Malmquist coordinates, reducing the required computations to finite-dimensional linear algebra. These factors yield a finite-dimensional, delay-free reduced model. For real system data, the implementation remains in real arithmetic. Under suitable conditions on the ADI shifts and a gap in the Hankel singular values, we prove convergence in $\mathcal H_\infty$ to the reduced transfer function obtained by exact balanced truncation of the infinite-dimensional system. The construction extends to finitely many discrete delays. Numerical experiments validate the method against independent reference computations. The complete MATLAB implementation used in the experiments is provided as supplementary material.
