基于希尔伯特-施密特范数和交叉格莱姆的模型降阶方法

Approach to Model Reduction Based on Hilbert-Schmidt Norm and Cross-Gramian

为了克服基于交叉格莱姆矩阵的最小信息损失模型降阶方法的局限性——信息损失性能指标不满足非负性和范数意义导致其物理意义不直观,利用Hankel奇异值以及交叉格莱姆的信息属性推导出希尔伯特-施密特范数与交叉格莱姆的关系,进一步推理得出希尔伯特-施密特范数意义的信息损失性能指标,并给出了基于希尔伯特-施密特范数和交叉格莱姆的模型降阶方法.最后,通过数值算例表明,降阶效果大为改善,截断误差更小,验证了新的信息损失指标的合理性和有效性.

In order to overcome the limitation that the information loss performance index of the method for model reduction by minimizing information loss based on cross-Gramian matrix does not meet nonnegativity and any norm so that its physical meaning is not intuitive,the relationships between the Hilbert-Schmidt norm and the cross-Gramian are derived from the Hankel singular values of systems and the information property of the cross-Gramian.Via further theoretical reasoning,the information loss performance index based on Hilbert-Schmidt norm is obtained.Moreover,an approach to model reduction based on Hilbert-Schmidt norm and cross-Gramian is proposed.Finally,a numerical example is illustrated to verify that the performance of model reduction is greatly improved and the truncation error is smaller.Therefore,the rationality and the validity of the new performance index are proved.