Bilinear Systems - A New Link to H2-norms, Relations to Stochastic Systems, and Further Properties
Bilinear Systems - A New Link to H2-norms, Relations to Stochastic Systems, and Further Properties
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双线性系统 - H2 范数的新链接、与随机系统的关系以及其他属性
DOI：
10.1137/19m1304106
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Martin Redmann
中科院分区：
文献类型：
--
作者：
Martin Redmann
在本文中，我们证明了几个新的结果，给新的见解双线性系统。我们证明了在什么条件下双线性系统是渐近稳定的。此外，我们提供了一个全球性的双线性系统的可达性的基础上一定的Gramian。到目前为止，使用相同的Gramian的可达性能量估计仅是局部的。本文的主要结果是两个双线性系统的输出误差和H_2误差之间的一个新的联系。这一结果在模型降阶领域有几个后果。它解释了为什么$\mathcal H_2$-最优模型降阶导致良好的近似的输出误差。此外，输出误差的基础上$\mathcalH_2 $-范数现在可以证明平衡相关的模型降阶计划在本文中。所有这些新结果都是基于这里建立的矩阵微分方程的Gronwall引理。
In this paper, we prove several new results that give new insights into bilinear systems. We show under which conditions a bilinear system is asymptotically stable. Moreover, we provide a global characterization of reachability in bilinear systems based on a certain Gramian. Reachability energy estimates using the same Gramian have only been local so far. The main result of this paper, however, is a new link between the output error and the $\mathcal H_2$-error of two bilinear systems. This result has several consequences in the field of model order reduction. It explains why $\mathcal H_2$-optimal model order reduction leads to good approximations in terms of the output error. Moreover, output errors based on the $\mathcal H_2$-norm can now be proved for balancing related model order reduction schemes in this paper. All these new results are based on a Gronwall lemma for matrix differential equations that is established here.