SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1956-1992, January 2020.
This article proposes a framework which allows the study of stability robustness of equilibria of a nonlinear system in the face of parametric uncertainties from the point of view of bifurcation theory. In this context, a branch of equilibria is stable if bifurcations (i.e., qualitative changes of the steady-state solutions) do not occur as one or more bifurcation parameters are varied. The work focuses specifically on Hopf bifurcations, where a stable branch of equilibria meets a branch of periodic solutions. It is of practical interest to evaluate how the presence of uncertain parameters in the system alters the result of analyses performed with respect to a nominal vector field. Note that in this article bifurcation parameters have a different meaning than uncertain parameters. To answer the question, the concept of robust bifurcation margins is proposed based on the idea of describing the uncertain system in a Linear Fractional Transformation fashion. The robust bifurcation margins can be interpreted as nonlinear analogues of the structural singular value, or $\mu$, which provides robust stability margins for linear time invariant systems. Their computation is formulated as a nonlinear program aided by a continuation-based multistart strategy to mitigate the issue of local minima. Application of the framework is demonstrated on two case studies from the power system and aerospace literature.

中文翻译：

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基于鲁棒控制概念的分叉裕度计算

SIAM应用动力系统杂志，第19卷，第3期，第1956-1992页，2020年1月。
本文提出了一个框架，该框架允许从分叉理论的角度研究面对参数不确定性的非线性系统的平衡稳定性。在这种情况下，如果一个或多个分叉参数发生变化而不会发生分叉（即稳态解的质变），则平衡的分支是稳定的。这项工作专门针对Hopf分叉，其中平衡的稳定分支与周期解的分支相交。评估系统中不确定参数的存在如何改变相对于标称矢量场执行的分析结果具有实际意义。请注意，本文中的分叉参数与不确定参数具有不同的含义。为了回答这个问题，基于以线性分数阶变换方式描述不确定系统的思想，提出了鲁棒分支裕度的概念。鲁棒的分叉裕度可以解释为结构奇异值或\\ mu $的非线性类似物，它为线性时不变系统提供了鲁棒的稳定裕度。他们的计算公式化为非线性程序，并辅之以基于连续的多启动策略，以减轻局部极小值的问题。在电力系统和航空航天文献的两个案例研究中证明了该框架的应用。为线性时不变系统提供了强大的稳定性裕度。他们的计算公式化为非线性程序，并辅之以基于连续的多启动策略，以减轻局部极小值的问题。在电力系统和航空航天文献的两个案例研究中证明了该框架的应用。为线性时不变系统提供了强大的稳定性裕度。他们的计算公式化为非线性程序，并辅之以基于连续的多启动策略，以减轻局部极小值的问题。在电力系统和航空航天文献的两个案例研究中证明了该框架的应用。