This paper is a comparative analysis of two prominent iterative algorithms for model order reduction of linear time-varying (LTV) periodic systems where the system's matrices are singular. Our proposed method is based on a reformulation of the LTV model to an equivalent linear time-invariant (LTI) model using a suitable discretization procedure. The resulting LTI model is reduced in two ways, once by applying a balanced truncation method and once by applying a Krylov-based method known as iterative rational Krylov algorithm (IRKA). During the application of balanced truncation, the low-rank Cholesky factorized alternating directions implicit (LRCF-ADI) method is used to estimate the solutions of the corresponding LTI form of Lyapunov equations. Since the system's matrices are singular, the concept of pseudo-inverse is adopted to compute the shift parameters needed in the LRCF-ADI iterations. For the Krylov-based IRKA, our work is twofold. We solve the time-invariant Lyapunov equation for the observability Gramian and apply a moment-matching Krylov technique. The accuracy and effectiveness of the two proposed techniques are demonstrated with the help of frequency response graphs, bode plots, and eigenstructure of the main and reduced models.

中文翻译：

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基于电路的周期控制问题的模型约简策略比较分析

本文是对线性时变 (LTV) 周期系统的模型降阶的两种主要迭代算法的比较分析，其中系统矩阵是奇异的。我们提出的方法基于使用合适的离散化程序将 LTV 模型重新表述为等效线性时不变 (LTI) 模型。生成的 LTI 模型以两种方式减少，一种是通过应用平衡截断方法，另一种是通过应用称为迭代有理 Krylov 算法 (IRKA) 的基于 Krylov 的方法。在平衡截断的应用过程中，低秩Cholesky分解交替方向隐式（LRCF-ADI）方法用于估计Lyapunov方程的相应LTI形式的解。由于系统的矩阵是奇异的，采用伪逆的概念来计算 LRCF-ADI 迭代所需的移位参数。对于基于 Krylov 的 IRKA，我们的工作是双重的。我们求解了可观测性 Gramian 的时不变 Lyapunov 方程，并应用了矩匹配 Krylov 技术。借助主模型和简化模型的频率响应图、波特图和特征结构，证明了这两种技术的准确性和有效性。