一种改进的线性系统的有限时间平衡截断

A Modified Time-limited Balanced Truncation for Linear Systems

本文提出一种改进的线性系统的有限时间平衡截断方法.该方法首先利用Shifted Legendre多项式对线性系统的有限时间可控Gram矩阵和可观Gram矩阵进行近似低秩分解,其中根据正交多项式与幂级数之间的关系,该近似低秩分解因子可以通过简单的递推公式得到,然后构造正交投影变换得到近似平衡系统,进而通过截断较小的Hankel奇异值对应的状态得到降阶系统.此外,本文还简要讨论了该降阶模型的稳定性.最后,通过数值算例验证了算法的有效性.

In this paper,a modified time-limited model order reduction(MOR) method for linear systems is presented.It uses the shifted Legendre polynomials expansion vectors to perform the approximate low-rank decomposition of the Gramians for the linear systems.According to the relationship between the orthogonal polynomials and power series,the low-rank decomposition factors are obtained by a simple recurrence formula.Then,the approximate balanced system can be generated by them.After that the reduced-order models(ROMs) are obtained by truncating the states corresponding to the small approximate Hankel singular values(HSVs).Furthermore,the stability of the ROMs is discussed briefly.In the end,a numerical experiment is given to illustrate the validity of the proposed methods.