基于重启Lanczos过程的模型降阶方法

计算机工程与科学

基于重启Lanczos过程的模型降阶方法

杨平1，徐康丽1，蒋耀林1,2

(1.新疆大学数学与系统科学学院，新疆乌鲁木齐 830046；2.西安交通大学数学与统计学院，陕西西安 710049)

收稿日期:2015-08-27 修回日期:2015-12-29 出版日期:2017-03-25 发布日期:2017-03-25
基金资助:
国家自然科学基金（11371287）；新疆自治区研究生科研创新项目（XJGRI2015007）

A model reduction method based

on restarted Lanczos process

YANG Ping1，XU Kang-li1，JIANG Yao-lin1,2

(1.College of Mathematics and System Sciences,Xinjiang University,Urumqi 830046;

2.College of Mathematics and Statistics,Xi’ an Jiaotong University,Xi’an 710049,China)

Received:2015-08-27 Revised:2015-12-29 Online:2017-03-25 Published:2017-03-25

摘要/Abstract

摘要：

针对大规模的线性时不变系统，提出了基于重启Lanczos过程的模型降阶方法。首先，通过重启Lanczos过程分别得到原始系统的可控Gram矩阵的近似矩阵及可观Gram矩阵的近似矩阵。然后，根据原始系统的可控Gram矩阵及可观Gram矩阵所满足的Lyapunov方程构造映射Sylvester方程并求解，对解进行双正交化，得到降阶所需的变换矩阵，从而得到降阶系统。运用此方法对大规模线性时不变系统进行降阶，能够得到具有较高近似精度的稳定的降阶系统。最后，数值算例验证了此方法是行之有效的。

关键词: 模型降阶, Gram矩阵, 重启Lanczos过程, 映射Sylvester方程

Abstract:

We propose a model reduction method based on the restarted Lanczos process for some large scale linear time-invariant systems. Firstly, we utilize the restarted Lanczos process to obtain the approximate matrices of both the reachability Gramian matrices and the observability Gramian matrices of the original system. Then, according to the Lyapunov equations satisfied by the reachability Gramian matrices and the observability Gramian matrices respectively, we can construct the projected Sylvester equations, solve them and carry out the biorthogonal process on the solution to get the required transformation matrices, thus obtaining the order-reduced system of the original system. Using the proposed algorithm to reduce the order of the large scale linear time-invariant system, we can obtain a stable reduced order system in higher precision. Numerical examples verify the effectiveness of the proposed algorithm.

Key words: model order reduction, gramian matrix, restarted Lanczos process, projected Sylvester equation

杨平1，徐康丽1，蒋耀林1,2. 基于重启Lanczos过程的模型降阶方法[J]. 计算机工程与科学.

YANG Ping1，XU Kang-li1，JIANG Yao-lin1,2.

A model reduction method based

on restarted Lanczos process

[J]. Computer Engineering & Science.