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.
