Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study.

线性动力系统的平衡截断和奇异摄动近似产生满足涉及汉克尔奇异值的众所周知的误差界限的降阶模型。我们证明，如果对应于截断的 Hankel 奇异值的符号参数都相等，则对于单输入、单输出系统，这个界限同样成立。这些符号由相应线性模型的广义状态空间对称性决定。对于具有箭头实现的特殊类型的系统，可以直接从相应箭头矩阵的非对角项确定符号。我们描述了这种箭头系统如何在网络建模的某些应用中自然出现，并用激发本研究的电力系统模型来说明这些结果。