This paper studies the model reduction problem for negative imaginary (NI) systems. For a given high-order system that is stable and NI, our goal is to approximate it by a low-order NI system so that the norm of the approximation error system is minimised. By using the Galerkin projection, the model reduction problem is formulated as a minimisation problem over the Stiefel manifold. The first-order necessary condition is derived for the construction of a local optimal reduced-order system. A gradient descent algorithm is provided to solve the first-order necessary condition. The resulting reduced-order system preserves the stability and the NI structure of the original system. Finally, two examples are presented to demonstrate the effectiveness of the proposed model reduction method.

中文翻译：

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$H_{2}$ 负虚数系统的模型简化

本文研究了负虚数 (NI) 系统的模型约简问题。对于给定的稳定且 NI 的高阶系统，我们的目标是通过低阶 NI 系统对其进行近似，从而使近似误差系统的范数最小化。通过使用 Galerkin 投影，模型缩减问题被表述为 Stiefel 流形上的最小化问题。推导出构造局部最优降阶系统的一阶必要条件。提供梯度下降算法来求解一阶必要条件。由此产生的降阶系统保留了原始系统的稳定性和 NI 结构。最后，给出了两个例子来证明所提出的模型约简方法的有效性。