On the Pole Location of Reduced-order Model in Balanced Truncation Model Order Reduction

In this paper, we discuss model order reduction of linear time invariant systems based on balanced truncation technique. We introduce $\mathcal{D}$-stability theory, which is used to describe the pole region of the system. The transient response and steady-state response of a linear time invariant system are closely related to the position of its poles, therefore, if the reduced order model maintains the pole region of the original system, it can accurately approximate the input-output behavior of the original system within a time-limited interval. According to the $\mathcal{D}$-stability theory, we establish the pole region of system and we give a conjecture regarding whether the balanced truncation technique can preserve the pole region of the original system. Finally, we present a numerical example to demonstrate that the conjecture is not true in general.