A Computational Framework for Reduced Feedback Control Design for Spatially Distributed Systems

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

0072629 King This goal of this project is to develop a framework that can be used to design spatially distributed control systems. Specifically, the outcome of this project is to provide a methodology to optimally place sensors and actuators and to construct practical reduced order controllers. The approach is based on a unique combination of computational mathematics and distributed parameter control theory. The framework is based on an integral representation of the optimal feedback control that arises from the solution of the infinite dimensional control problem. Computation of the kernel of the integral (feedback functional gain) may be accomplished by solving a numerical approximation of the infinite dimensional algebraic Riccati equation. However, this approach is intractable for even the two dimensional heat equation on a fairly coarse grid. The approach proposed herein is based on the use of computational methods to directly solve Chandrasekhar partial integro-differential equations for functional gains. Recent results on the smoothness and support of these functional gains can then be exploited to optimally place sensors and actuators. Moreover, initial studies show how the gains can also be used to design robust low- order compensators. The motivation for this work comes from the need to have efficient, practical controllers for complicated physical systems. The focus of the proposal is the development of an approach that is applicable to a wide variety of areas ranging from materials processing to flow control. In applications from such areas, there are several issues of primary importance that must be addressed. Among these are where to place sensors to obtain accurate measurements to be used by the controller, what to measure, where to place actuators (which implement the control), and how to design the controller so that the system is controller in real-time. Typically, the issues with respect to sensors and actuators are solved through engineering intuition. However, the mathematical framework that we propose gives specific answers to these questions. Moreover, it provides motivation for real-time control which is nearly optimal (in the sense that the design comes from the solution to a mathematical optimization problem) and is robust, which means that it is effective in the presence of disturbances, or dynamics which may have been neglected in designing the controller. The potential benefits of this work are enormous. It would be applicable to complicated (and varied) problems such as stabilizing vibrations arising from shuttle docking at the space station, reduction of drag in aerospace vehicles, and optimal control of semiconductor manufacturing. It would provide a systematic approach to reduced order controller design, eliminating much of the guesswork that currently takes place.