精确不稳定裕度分析与最小范数强镇定——相位变化率最大化——
Exact Instability Margin Analysis and Minimum-Norm Strong Stabilization -- phase change rate maximization --
AI总结：
本文针对单输入单输出线性时不变系统提出相位变化率最大化问题，通过将其转化为相位约束下的优化，证明最大值由常数或一阶全通函数取得，并给出精确鲁棒不稳定半径的表征条件，应用于鲁棒不稳定分析和最小范数强镇定。
摘要：
本文研究单输入单输出线性时不变系统的一个新优化问题，称为“相位变化率最大化”。该问题与两个控制问题相关，即针对稳定扰动的鲁棒不稳定分析和最小范数强镇定。我们定义了一个不稳定裕度指标，称为“鲁棒不稳定半径（RIR）”，它是使给定不稳定系统稳定的稳定扰动的最小$H_\infty$范数。本文有两个主要贡献。首先，我们证明通过小增益条件求解精确RIR的问题可以转化为在相位约束下最大化峰值频率处的相位变化率的问题。然后，我们证明该最大值由常数或一阶全通函数取得，并推导出RIR可以被精确表征的条件，这些条件以相位变化率表示。我们提供了两个实际应用来说明我们结果的实用性。
英文摘要：
This paper is concerned with a new optimization problem named "phase change rate maximization" for single-input-single-output linear time-invariant systems. The problem relates to two control problems, namely robust instability analysis against stable perturbations and minimum-norm strong stabilization. We define an index of the instability margin called "robust instability radius (RIR)" as the smallest $H_\infty$-norm of a stable perturbation that stabilizes a given unstable system. This paper has two main contributions. It is first shown that the problem of finding the exact RIR via the small-gain condition can be transformed into the problem of maximizing the phase change rate at the peak frequency with a phase constraint. Then, we show that the maximum is attained by a constant or a first-order all-pass function and derive conditions, under which the RIR can be exactly characterized, in terms of the phase change rate. Two practical applications are provided to illustrate the utility of our results.
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