In this paper, the notions of pseudo Gramians are introduced for linear time-invariant semistable systems, which allow multiple semisimple poles at the origin. The proposed Gramians are the generalizations of standard Gramian matrices defined for asymptotically stable systems, and they can be computed by a set of Lyapunov equations. Furthermore, it is shown that the controllability and observability of a semistable system are indicated by the ranks of the pseudo Gramians, and the controllability and observability energy functions are also characterized using the pseudo Gramians. Additionally, the ${\mathcal{H}}_2$-norm and ${\mathcal{H}}_{\infty }$-norm of a semistable system are analyzed, and then the results are used for the model reduction of semistable systems. Finally, the effectiveness of the methods is illustrated by an example of a gene regulation network.

在本文中，针对线性时不变半稳定系统引入了伪Gramian概念，该系统允许在原点使用多个半简单极点。拟议的Gramian是为渐近稳定系统定义的标准Gramian矩阵的推广，可以通过一组Lyapunov方程进行计算。此外，还表明，半稳态系统的可控性和可观测性由拟革兰氏级数表示，并且可控性和可观测性能量函数也可通过拟革兰氏度来表征。此外，${\mathcal{H}}_2$-规范和 ${\mathcal{H}}_{\infty }$分析半稳定系统的-范数，然后将结果用于半稳定系统的模型简化。最后，通过基因调控网络的例子说明了该方法的有效性。