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

摘要：微分代数方程(DAE)是状态变量和控制变量的隐式微分方程组。称为指数的整数值是衡量DAE与显式常微分方程式(ODE)的不同程度的指标之一。许多问题最自然地被初始建模为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，在优化过程中会发生DAE。由于DAE的重要性，近年来发展了各种数值技术来模拟和分析DAE，尽管这些方法大多只适用于特殊结构和低指数的特定类型的问题。遗憾的是，许多广泛使用的软件包无法接受DAE型号，或者如果它们可以接受，则型号必须是索引一或具有特殊结构。既要解决一般DAE积分器的问题，又要在当前软件中更广泛地使用DAE模型，一种方法是将DAE嵌入称为DAE完成式的常微分方程(ODE)中。将DAE解决方案嵌入到ODE中的想法已经存在一段时间了。然而，传统的方法要么让新的附加动力处于确定状态，要么只对特殊类别的系统起作用。一种更通用和自动化的嵌入类型，如拟议项目中要研究的嵌入类型，将极大地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一般DAE完井的分析、计算和应用。分析支持的算法将被构建为生成具有所需额外动态的完备性。所提出的研究代表了这一思想对一般非线性DAE的重大扩展和修正，以及在控制这些附加动力学性质方面的新结果的发展。所获得的结果将具有相当大的独立利益，但该提案将侧重于首先将新结果应用于模拟和控制。提出的研究将导致对微分代数方程的数值方法及其在控制和仿真中的应用的改进的算法和新的理论理解。在科学和工业中的许多问题中，越来越多地用所谓的微分方程来描述过程或机器。然后，这个数学模型被用来在计算机上模拟过程，并预测真实系统将会做什么，以设计更有效的过程，并提高性能。然而，当今科学和工程中感兴趣的问题正变得越来越复杂，以至于今天的数学模型往往是由几个不同的数学模型组合在一起形成的。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，有时并不是用当前的计算机软件就能解决的形式。然后，要将这些复杂的数学模型转换成易于使用的形式，可能需要花费大量的人力，有时还需要损失精度。转换过程不仅需要时间，还可能导致错误。该项目的研究人员将开发有理论支持的数学程序和算法，以促进和自动化将原始复杂数学模型转换为可与现有软件和算法一起使用的模型。这将减少从模型开发到能够使用模型的时间，提高模型的精度，减少引入误差，从而加快工业流程的设计，以及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社区的问题，以及开发更高效和更可靠的制造工艺。

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.