Simple Dynamics from Complex Data: Model Reduction for Control of Fluids

Clancy Rowley

Department of Mechanical and Aerospace Engineering, Princeton University

This talk addresses numerical techniques for extracting simplified, approximate dynamical models for complex systems, for instance governed by partial differential equations. Specifically, the focus is on low-dimensional models of fluid flows, with the goal of enabling tools from dynamical systems and control theory to be applied to these systems.

The ability to effectively control a fluid would enable many exciting technological advances, including the design of quieter, more efficient aircraft. Most of the flow control strategies tried so far have been largely ad hoc, and have not used many of the available tools from control theory, which can guide controller design as well as placement of sensors and actuators. These tools require knowledge of a model of the system in terms of a system of differential equations, and the equations governing a fluid, though known, are too complex for these tools to apply. We discuss recent developments in model reduction, which can be used to simplify an existing high-dimensional description, for instance in the form of a complex simulation, in order to obtain low-dimensional models tractable enough to be used for analysis and control, while retaining the essential physics. These techniques provide a bridge between complex problems and the mathematical tools useful for their analysis.

Specifically, the talk will focus on recent developments of two techniques, Proper Orthogonal Decomposition (POD) and balanced truncation. Each of these techniques has strengths and weaknesses, and we show how ideas from both techniques may be combined, to exploit their strengths. We illustrate the methods by obtaining reduced-order models for several fluid flows, including oscillations in a cavity flow, flow in a plane channel, and separating flow past an airfoil.