This article presents a time-invariant extremum seeking controller (ESC) for nonlinear autonomous systems with limit cycles. For this time-invariant ESC, we propose a method to prove the closed-loop system has an asymptotically stable limit cycle. The method is based on a perturbation theorem for maps, and, unlike existing techniques that use averaging and singular perturbation tools, it is not limited to weakly nonlinear systems. We use a typical example system to show that our method does indeed establish asymptotic stability of the limit cycle with minimal amplitude. Utilizing the example, we provide a general guide for analytic computations that are required to apply our method. The corresponding Mathematica code is available as supplementary material.

中文翻译：

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极限环最小化时不变极值搜索控制的稳定性


本文提出了一种用于具有极限环的非线性自治系统的时不变极值搜索控制器 (ESC)。对于这种时不变的ESC，我们提出了一种证明闭环系统具有渐近稳定极限环的方法。该方法基于地图的微扰定理，并且与使用平均和奇异微扰工具的现有技术不同，它不限于弱非线性系统。我们使用一个典型的示例系统来表明我们的方法确实建立了具有最小振幅的极限环的渐近稳定性。利用该示例，我们为应用我们的方法所需的分析计算提供了一般指南。相应的 Mathematica 代码可作为补充材料提供。