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

0072629KingThis项目的目标是开发一个框架，可用于设计空间分布式控制系统。 具体来说，这个项目的成果是提供一种方法来最佳地放置传感器和执行器，并构建实用的降阶控制器。 该方法是基于计算数学和分布参数控制理论的独特组合。该框架是基于一个积分表示的最优反馈控制，所产生的无限维控制问题的解决方案。积分的核（反馈函数增益）的计算可以通过求解无限维代数Riccati方程的数值近似来完成。然而，这种方法是棘手的，即使是二维热方程在一个相当粗糙的网格。本文提出的方法是基于使用的计算方法，以直接解决函数增益的Komrasekhar偏积分微分方程。 最近的结果，这些功能增益的平滑度和支持，然后可以利用最佳的位置传感器和执行器。 此外，初步的研究表明，如何增益也可以用来设计鲁棒低阶补偿器。这项工作的动机来自于需要有效率的，实用的控制器的复杂的物理系统。该提案的重点是开发一种适用于从材料加工到流量控制等各种领域的方法。 在这些地区的应用中，必须解决几个至关重要的问题。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值，测量什么，在哪里放置执行器（执行控制），以及如何设计控制器，以便系统实时控制。通常，关于传感器和执行器的问题通过工程直觉来解决。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动力，实时控制几乎是最优的（在这个意义上，设计来自数学优化问题的解决方案）并且是鲁棒的，这意味着它在存在干扰或在设计控制器时可能被忽略的动态时是有效的。 这项工作的潜在好处是巨大的。它将适用于复杂的（和不同的）问题，如稳定的振动所产生的航天飞机对接在空间站，减少航空航天器的阻力，和半导体制造的最佳控制。它将提供一个系统的方法来降低阶控制器的设计，消除了许多猜测，目前发生。

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.