EAGER: Revisiting Vibrational Control Theory

这个探索性研究（EAGER）项目旨在深化和扩展可用于振动控制的分析工具。振动控制为控制不稳定系统提供了独特的能力，而不需要反馈。考虑一个底部可移动的倒立摆。通过测量钟摆的位置，控制系统可以命令基座的左右运动，以保持它的直立。但如果无法测量位置，钟摆就会倒转。同样地，一个人可以通过观察扫帚的移动来保持手中扫帚的平衡。然而，如果那个人被要求戴上眼罩和厚手套，这项任务就不可能完成了。使用振动控制，通过以适当的频率和幅度上下移动基座，使倒立摆在没有反馈的情况下保持平衡。没有其他已知的方法可以做到这一点。许多重要的应用都可以从振动控制的系统应用中受益，包括化学反应、材料加工和柔性结构。然而，目前唯一开发的振动控制工具——一种称为一阶平均的数学技术——非常不适合工程设计。因此，本项目将探索三种有希望的替代方案。该项目的研究目标是探索和解决振动控制系统分析和设计的竞争方法。振动控制具有从根本上改变控制应用的潜力，这些应用的特点是潜在的不稳定性、有限的处理速度、高自由度、有限的驱动以及很少或没有传感。一阶平均是目前几乎专门用于分析和设计振动控制器的数学技术。该项目考虑了三种有希望的一阶平均替代方案，即使用幂级数的高阶平均，使用Volterra级数（“时间演算”）的高阶平均，以及基于Floquet理论的稳定性映射。本项目将解决这些方法之间的差异，并得出每种方法的基本优点和缺点。这个项目的结果将把振动控制从一个数学上的好奇变成一个强大而实用的工程工具。该项目包括对微型扑翼飞行器模型的验证，以及离子约束和过滤。

This EArly-concept Grant for Exploratory Research (EAGER) project seeks to deepen and extend the analytical tools available for vibrational control. Vibrational control offers unique capabilities for controlling unstable systems without the need for feedback. Consider an upside-down pendulum with a movable base. By measuring the pendulum position, a control system can command side-to-side movement of the base in order to keep it upright. But if position measurements are not available, the pendulum will topple over. Similarly, a person may balance a broomstick on their hand by watching it move. However, if that person were made to wear a blindfold and thick gloves, the task would be impossible. Using vibrational control allows the upside-down pendulum to be kept balanced with no feedback, by moving the base up and down at the right frequency and amplitude. There are no other known methods to accomplish this. Many important applications could benefit from a systematic application of vibrational control, including chemical reactions, material processing, and flexible structures. However the only currently developed tool for vibrational control -- a mathematical technique called first-order averaging -- is very poorly suited to engineering design. Therefore this project will explore three promising alternatives.The research objective of this project is to explore and resolve competing approaches to the analysis and design of vibrational control systems. Vibrational control has the potential to fundamentally transform control applications characterized by underlying instability, limited processing speed, high degrees of freedom, limited actuation, and little or no sensing. First-order averaging is the mathematical technique currently used almost exclusively to analyze and design vibrational controllers. This project considers three promising alternatives to first-order averaging, namely higher-order averaging using power series, higher-order averaging using Volterra series (the "chronological calculus"), and stability maps based on Floquet theory. This project will resolve discrepancies between these approaches, and derive the fundamental benefits and drawbacks of each. The results of this project will change vibrational control from a mathematical curiosity to a powerful and practical engineering tool. The project includes validation on models of flapping-wing micro air vehicles, and ion confinement and filtering.