This paper presents a detailed response type and basin of attraction (BOA) analyses of a linear mass-spring-damper oscillator and a Duffing-Holmes (D-H) oscillator controlled by a class of position feedback controllers. First, the response-type comparison of both linear and D-H systems subjected to a, Negative Position Feedback (NPF), and Positive Position Feedback (PPF) controllers, and Hybrid Position Feedback (HPF) controller (which combines the previous two) is analyzed individually. Initially, the bistable system is expressed as two linear models around the stable equilibriums, and shows similar dynamic characteristics near the vicinity of stable equilibria. Three relevant response types are identified for the controlled D-H oscillator. These are the intra-well, single cross-well, and multiple cross-well response types describing all possible responses. With the BOA analyses, three response convergence types are defined. These are the convergence to state-1, convergence to state-2, and no convergence. The overall behavior of the bistable system under the hybrid controller is examined and described using these response- and convergence-type analyses. In this paper, it is shown that the HPF control concept provides desirable response for a wider range of systems and initial conditions when compared to the other simpler control schemes. The range of desirable controller parameters is identified.

中文翻译：

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混合位置反馈控制器下的 Duffing-Holmes 振荡器：响应类型和吸引力分析

本文介绍了由一类位置反馈控制器控制的线性质量-弹簧-阻尼振荡器和 Duffing-Holmes (DH) 振荡器的详细响应类型和吸引力盆 (BOA) 分析。首先，分析了采用负位置反馈 (NPF) 和正位置反馈 (PPF) 控制器以及混合位置反馈 (HPF) 控制器（结合了前两者）的线性和 DH 系统的响应类型比较分别。最初，双稳态系统表示为围绕稳定平衡的两个线性模型，并在稳定平衡附近显示出相似的动态特性。为受控的 DH 振荡器确定了三种相关的响应类型。这些是井内、单跨井、以及描述所有可能响应的多种跨井响应类​​型。通过 BOA 分析，定义了三种响应收敛类型。这些是收敛到状态 1、收敛到状态 2 和不收敛。使用这些响应型和收敛型分析来检查和描述混合控制器下双稳态系统的整体行为。本文表明，与其他更简单的控制方案相比，HPF 控制概念为更广泛的系统和初始条件提供了理想的响应。所需控制器参数的范围被识别。使用这些响应型和收敛型分析来检查和描述混合控制器下双稳态系统的整体行为。本文表明，与其他更简单的控制方案相比，HPF 控制概念为更广泛的系统和初始条件提供了理想的响应。所需控制器参数的范围被识别。使用这些响应型和收敛型分析来检查和描述混合控制器下双稳态系统的整体行为。本文表明，与其他更简单的控制方案相比，HPF 控制概念为更广泛的系统和初始条件提供了理想的响应。所需控制器参数的范围被识别。