具有二次输出的双线性系统：$\mathcal{H}_2$ 分析、模型约简的最优性条件及算法解决方案
Bilinear Systems with Quadratic Outputs: $\mathcal{H}_2$ Analysis, Optimality Conditions for Model Reduction, and Algorithmic Solutions
AI总结：
研究具有二次输出的双线性系统，通过建立 $\mathcal{H}_2$ 内积、范数等构建 $\mathcal{H}_2$ 框架，推导输出界与最优性条件，提出算法计算满足条件的约简系统，推广现有最优方法并经数值例子验证有效性。
AI中文摘要：
具有二次输出的双线性系统（BQO）近来成为重要系统类别，在动力学和感兴趣量均非线性依赖状态的应用中自然出现。尽管对此类系统兴趣渐浓，但缺乏系统的 $\mathcal{H}_2$ 框架。本文通过为 BQO 系统建立 $\mathcal{H}_2$ 内积和范数，推导基于 $\mathcal{H}_2$ 范数的输出界，得到 $\mathcal{H}_2$ 最优模型约简的一阶最优性条件，建立了这样的框架。在此基础上提出算法，计算满足这些最优性条件的约简 BQO 系统，推广了现有双线性和线性二次输出系统的 $\mathcal{H}_2$ 最优方法。通过两个数值例子证明了所提框架的有效性。
英文摘要：
Bilinear systems with quadratic outputs (BQO) have recently emerged as an important system class, arising naturally in applications where both the dynamics and the quantities of interest depend nonlinearly on the state. Despite the growing interest in this class of systems, a systematic $\mathcal{H}_2$ framework for BQO systems has been lacking. In this paper, we develop such a framework by establishing an $\mathcal{H}_2$ inner product and norm for BQO systems, deriving output bounds in terms of the $\mathcal{H}_2$ norm, and obtaining first-order optimality conditions for $\mathcal{H}_2$ optimal model reduction. Building on these theoretical foundations, we propose an algorithm that computes a reduced BQO system satisfying these optimality conditions, and thus generalizing existing $\mathcal{H}_2$ optimal methods for bilinear and linear quadratic output systems. The effectiveness of the proposed framework is demonstrated on two numerical examples.
