Systematic experimental exploration of nonlinear structures with control-based continuation

在工程学和应用科学的众多研究领域中，存在着非线性结构或系统，对于这些结构或系统，存在着不充分的或多个相互竞争的数学模型。这通常是由于在适当的尺度上对物理学的理解不佳造成的。这方面的两个例子是研究飞机起落架中的摆振振荡，以及高速加工过程中颤振的开始。在这些领域，为了解决模型无法解决的细节，实验研究是至关重要的。然而，系统地研究实验中的非线性动力学是充满困难的，因为随着系统参数的变化，动力学可能发生突变；而且，这种变化可能是定性的，也可能是定量的。例如，对于特定的参数选择，一个系统可能只有一个稳定的稳态，而对于另一个选择，它可能变成双稳态。特别令人感兴趣的是不同类型的定性行为之间的边界(所谓的分叉)。这一建议旨在通过利用控制论和动力学系统的思想，使确定物理实验中的这些边界成为一项正常和例行公事的任务，从而揭示以前未见过的动力学现象。基于控制的延续是一种新的方法，于2008年在布里斯托尔发展，用于系统地表征物理非线性实验的定性行为。它的潜力已经得到了其他机构的实验者的认可，他们现在正将其应用到自己的系统中。其核心思想是，使用反馈控制和数字路径跟踪技术的组合，可以在物理实验中直接发现和跟踪系统的动力学特征，如稳定性变化或混合模式振荡的开始。因此，可以绘制出显示不同质量行为区域的“地图”或分叉图。要使以控制为基础的延续得到广泛采用，必须满足三个关键目标。(1)必须能够确定稳态的局部线性化，从而提供额外的动态信息和执行在线控制器设计/自适应的手段。(2)托换数值方法必须具有快速和稳健的特点。(3)必须证明基于控制的延拓方法对多自由度系统的可扩展性。这项提议试图解决所有三点。

In numerous research areas across engineering and the applied sciences there are nonlinear structures or systems for which there are inadequate or multiple competing mathematical models. This is often caused by a poor understanding of the physics at the appropriate scale. Two examples of this are the study of shimmy oscillations in aircraft landing gear, and the onset of chatter during high-speed machining. In these areas experimental investigations are of fundamental importance in order to resolve the details that the models cannot.However, systematically investigating nonlinear dynamics in an experiment is fraught with difficulties due to the potential for sudden changes in the dynamics as system parameters are varied; moreover, the changes may be qualitative as well as quantitative. For example, for particular choice of parameters a system may have a single stable steady-state whereas for another choice it may become bistable. Of particular interest are the boundaries between different types of qualitative behaviour (so-called bifurcations). This proposal seeks to make the determination of these boundaries in a physical experiment a normal and routine task by leveraging ideas from control theory and dynamical systems, and so reveal previously unseen dynamical phenomena.Control-based continuation is a new method, developed in Bristol in 2008, for systematically characterising the qualitative behaviour of a physical nonlinear experiment. Its potential has been recognised by experimenters in other institutions, who are now applying it to their own systems. The key idea is that dynamical features of the system, such as stability changes or the onset of mixed-mode oscillations, can be found and tracked directly in a physical experiment using a combination of feedback control and numerical path following techniques. Thus a `map' or bifurcation diagram showing different regions of qualitative behaviour can be traced out.To enable the widespread uptake of control-based continuation three key objectives must be satisfied. (1) It must be possible to determine the local linearisation of a steady-state, thus providing additional dynamical information and a means to perform on-line controller design/adaptation. (2) The underpinning numerical methods must be made fast and robust. (3) The scalability of control-based continuation to multi-degree-of-freedom systems must be demonstrated. This proposal seeks to address all three points.