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

[72629]该项目的目标是开发一个可用于设计空间分布式控制系统的框架。具体来说，这个项目的结果是提供一种方法来优化放置传感器和执行器，并构建实用的降阶控制器。该方法基于计算数学和分布参数控制理论的独特结合。该框架是基于最优反馈控制的积分表示，产生于无限维控制问题的解决。积分核（反馈泛函增益）的计算可以通过求解无限维代数里卡蒂方程的数值近似来完成。然而，这种方法即使对于一个相当粗糙的网格上的二维热方程也是难以处理的。本文提出的方法是基于使用计算方法直接求解函数增益的钱德拉塞卡偏积分-微分方程。关于这些功能增益的平滑度和支持度的最新结果可以用来优化传感器和执行器的位置。此外，初步研究表明，增益也可以用于设计鲁棒低阶补偿器。这项工作的动机来自于对复杂物理系统有效、实用的控制器的需求。该提案的重点是开发一种适用于从材料加工到流动控制等广泛领域的方法。在这些领域的应用中，有几个至关重要的问题必须加以解决。其中包括在哪里放置传感器以获得控制器使用的精确测量，测量什么，在哪里放置执行器（实现控制），以及如何设计控制器使系统成为实时控制器。通常，有关传感器和执行器的问题是通过工程直觉来解决的。然而，我们提出的数学框架给出了这些问题的具体答案。此外，它为实时控制提供了几乎最优的动机（在某种意义上，设计来自数学优化问题的解决方案），并且具有鲁棒性，这意味着它在存在干扰或动态的情况下是有效的，这些可能在设计控制器时被忽略。这项工作的潜在好处是巨大的。它将适用于复杂的（和各种）问题，如稳定由空间站的航天飞机对接产生的振动，减少航天飞行器的阻力，以及半导体制造的最佳控制。它将为减少订单控制器的设计提供一种系统的方法，消除目前发生的许多猜测工作。

0072629KingThis 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.