Analysis, Computation, and Application of Completions of Constrained Dynamical Systems

翻译后摘要：微分代数方程（DAE）是一个隐式系统的状态和控制变量的微分方程。称为指数的整数量是DAE与显式常微分方程（ODE）之间差异的一种度量。许多问题最初最自然地建模为DAE，特别是那些使用计算机生成的数学模型进行分析和模拟的问题。当原始问题是DAE或由于激活了约束时，DAE会在优化中出现。由于DAE的重要性，近年来已经开发了各种数值技术来模拟和分析DAE，尽管这些方法大多数只适用于具有特殊结构和低指标的特定类别的问题。不幸的是，许多广泛使用的软件包不能接受DAE模型，或者如果它们可以接受DAE模型，则模型必须是索引1或具有特殊结构。解决通用DAE积分器问题和当前软件中DAE模型更广泛使用问题的一种方法是将DAE嵌入到常微分方程（ODE）中，称为DAE的完备化。将DAE解决方案嵌入到ODE中的想法已经存在了一段时间。然而，传统的方法要么留下了新的额外的动态下确定或再次只适用于特殊类别的系统。一种更普遍和自动化的嵌入方式，如拟议项目中将研究的嵌入方式，将大大扩展目前许多软件包的适用性。在这个项目中，研究人员和他的同事和学生将致力于分析，计算和应用一般DAE的完成。 分析支持的算法将被构造为生成具有所需额外动态的完成。拟议的研究代表了一个重大的扩展和修改这一想法，一般的非线性DAE和控制这些额外的动态性质的新成果的发展。所获得的结果将是相当独立的利益，但建议将集中在第一次应用新的结果模拟和控制。拟议的研究将导致改进的算法和新的理论理解的数值方法的微分代数方程及其使用的控制和simulation.Increasingly在许多问题在科学和工业的过程或机器被描述在什么是所谓的微分方程。 然后，这个数学模型被用来在计算机上模拟该过程，并预测真实的系统将做什么，以设计更有效的过程，并提高性能。 然而，今天在科学和工程中感兴趣的问题变得越来越复杂，使得今天的数学模型通常通过将几个不同的数学模型组合在一起来形成。 例如，生物技术系统的数学模型可能包括描述流体流动、化学反应和力学的方程。 这些复合数学模型通常非常复杂，并且有时不是以可用当前计算机软件解决的形式。 然后可能需要相当大的人力，有时甚至会损失准确性，才能将这些复杂的数学模型转换为易于使用的形式。 转换过程不仅需要时间，而且可能导致错误。 该项目的研究人员将开发理论支持的数学程序和算法，这些程序和算法将促进和自动化原始复杂数学模型到可与现有软件和算法一起使用的模型的转换。 这将减少模型开发和能够使用模型之间的时间，提高模型的准确性，减少错误的引入，从而加快工业过程的设计和许多物理系统的分析。 虽然本项目中研究的微分方程广泛存在，但在算法测试中，研究人员最初将主要利用机械工程和航空航天界的问题以及更有效和可靠的制造工艺的发展。

Abstract: A differential algebraic equation (DAE) is an implicit system of differential equations in state and control variables. An integer quantity called the index is one measure of how different a DAE is from being an explicit ordinary differential equation (ODE). Many problems are most naturally initially modeled as a DAE particularly those that are analyzed and simulated using computer generated mathematical models. DAEs occur in optimization when the original problem is a DAE or because of the activation of constraints. Because of DAEs importance, a variety of numerical techniques have been developed in recent years to simulate and analyze DAEs although most of these methods only work on specific classes of problems with special structure and low index. Unfortunately many widely used software packages cannot accept DAE models, or if they can accept them, the models must be index one or have special structure. One way to address both the problem of general DAE integrators and wider use of DAE models in current software, is to embed the DAE into an ordinary differential equation (ODE) called a completion of the DAE. The idea of embedding the DAE solutions into an ODE has been around for some time. However, traditional approaches either left the new additional dynamics under determined or again only worked for special classes of systems. A more general and automated type of embedding, such as that to be investigated in the proposed project, would greatly extend the applicability of many current software packages. In this project the investigators and his colleagues and students will work on the analysis, computation, and application of completions of general DAEs. Analysis backed algorithms will be constructed to generate completions with desired extra dynamics. The proposed research represents a major extension and modification of this idea to general nonlinear DAEs and the development of new results on controlling the nature of these additional dynamics. The obtained results will be of considerable independent interest but the proposal will focus on first applying the new results to simulation and control. The proposed research will result in improved algorithms and new theoretical understanding of numerical methods for differential algebraic equations and their use in control and simulation.Increasingly in many problems in science and industry the process or machine is described in terms of what are called differential equations. This mathematical model is then used to simulate the process on a computer and to predict what the real system would do, to design more efficient processes, and to improve performance. However, the problems of interest in science and engineering today are becoming increasingly complex so that today mathematical models are often formed by combining together several different mathematical models. A mathematical model of a biotechnological system, for example, might include equations describing fluid flow, chemical reactions, and mechanics. These composite mathematical models are often very complex and are sometimes not in a form that is ready to be solved with current computer software. It can then take considerable human effort, and sometimes a loss of accuracy, to convert these complex mathematical models to a form that can be readily used. The conversion process not only takes time but can lead to errors. The investigators of this project will develop theory backed mathematical procedures and algorithms that will facilitate and automate the conversion of the original complex mathematical models to models that can be used with existing software and algorithms. This will reduce the time between model development and being able to use the model, improve the accuracy of the models, reduce the introduction of errors, and thereby speed up the design of industrial processes, and the analysis of many physical systems. While the differential equations studied in this project occur widely, in the testing of the algorithms the investigators will initially draw primarily on problems from the mechanical engineering and aerospace communities and the development of more efficient and reliable manufacturing processes.