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

摘要：微分代数方程（DAE）是包含状态变量和控制变量的隐式微分方程组。称为指数的整数是衡量DAE与显式常微分方程（ODE）有多大不同的一种度量。许多问题最初都很自然地被建模为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时，或者由于激活了约束，在优化中就会出现DAE。由于DAEs的重要性，近年来发展了各种数值方法来模拟和分析DAEs，但这些方法大多只适用于具有特殊结构和低指数的特定类别的问题。不幸的是，许多广泛使用的软件包不能接受DAE模型，或者即使可以接受，模型也必须是索引1或具有特殊结构。解决一般DAE积分器和DAE模型在当前软件中广泛使用的问题的一种方法是将DAE嵌入到称为DAE补全的常微分方程（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，传统的方法要么不确定新的附加动态，要么只适用于特殊类型的系统。一种更一般和自动化的嵌入类型，比如在提议的项目中要研究的，将极大地扩展许多当前软件包的适用性。在这个项目中，研究者和他的同事和学生将对一般DAEs的分析、计算和应用进行研究。将构建分析支持算法，以生成具有所需额外动态的完井。所提出的研究代表了这一思想对一般非线性DAEs的主要扩展和修改，以及对控制这些附加动力学性质的新结果的发展。所获得的结果将具有相当大的独立意义，但建议将重点放在首先将新结果应用于仿真和控制上。所提出的研究将导致改进算法和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新的理论理解。在科学和工业的许多问题中，越来越多地用所谓的微分方程来描述过程或机器。然后用这个数学模型在计算机上模拟这个过程，并预测真实系统会做什么，设计更有效的过程，并提高性能。然而，当今科学和工程领域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种不同的数学模型组合而成的。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，有时不是用当前计算机软件可以解决的形式。然后，要将这些复杂的数学模型转换为易于使用的形式，可能需要大量的人力，有时甚至会失去准确性。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，以及对许多物理系统的分析。虽然本项目研究的微分方程广泛存在，但在算法的测试中，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，并开发更有效和可靠的制造工艺。

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.