Optimization and Simulation of Constrained Dynamical Systems

Stephen L.坎贝尔将进行优化，数值方法的约束微分方程和约束最优控制的研究。 这项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（DAE）和不等式约束的最优控制问题的直接转录方法的解决方案。 更具体地，DAE是隐式微分方程。 称为指数的整数量是DAE与显式常微分方程的区别的一种度量。 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时，由于约束的激活，以及当使用线方法数值求解偏微分方程时，DAE发生在优化中。 由于微分代数方程在实际应用中的重要性，近年来出现了多种数值方法来模拟和分析微分代数方程，尽管这些方法大多只适用于具有特殊结构和低指标的特定问题。 开发更通用的DAE集成器是本建议书的重点之一。 直接转录方法是一种流行的方法来解决最优控制问题。最近，它已被证明，通常的理论DAE积分器需要大大修改时，DAE发生在数值解不等式约束的最优控制问题的直接转录方法。 这个建议的第二个主要重点是不等式约束的最优控制问题的数值解使用直接转录方法。 在整个工业中，需要开发更有效的工艺和系统。 这对于提高性能、节约资源以及工业和军事竞争力至关重要。 这种提高的性能需要使用所讨论的系统的复杂数学模型。 在许多领域，利用物理过程或系统的更大、更复杂和更完整的模型变得越来越必要。 系统设计的两个重要部分是仿真和优化。 在应用中有许多限制。 这些限制从使命限制（例如在某些飞行走廊内飞行或拦截特定目标）到物理限制（例如机械臂与工具之间的接触）到财务或成本限制。 这些限制带来了理论和计算上的挑战，随着模型的复杂性而增加。 本项目旨在为具有约束的复杂系统的数值模拟和最优控制发展数学理论和数值算法。 提议者与包括航空航天工业在内的许多行业中使用的软件开发人员正在进行合作。 这项研究的结果将作为主要研究者合作的一部分纳入生产计算机代码，并与其他开发数值软件的人分享。 这些算法和软件将具有非常广泛的适用性。 目前的合作涉及航空航天工业，包括空间和飞机使命规划，机器人制造和化学过程的优化。 除了要完成的重要研究，研究生将在这项研究中发挥重要作用。 该项目的另一个重要贡献是培训这一领域的下一代研究人员。 与主要研究者的前学生一样，在这个项目上接受培训的学生将继续在工业，国家实验室和研究中心，武装部队和学术界工作。

Stephen L. Campbell will perform research on optimization, numerical methods for constrained differential equations, and constrained optimal control. This research will result in improved algorithms and new theoretical understanding of numerical methods for differential algebraic equations (DAEs) and the solution of inequality constrained optimal control problems by direct transcription approaches. More specifically, a DAE is an implicit differential equation. An integer quantity called the index is one measure of how different a DAE is from being an explicit ordinary differential equation. 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, because of the activation of constraints, and when partial differential equations are numerically solved using the method of lines. Because of the importance of DAEs in applications, 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. The development of more general DAE integrators is one focus of this proposal. Direct transcription methods are a popular approach for the solution of optimal control problems. Recently it has been shown that the usual theory of DAE integrators needs to be substantially modified when DAEs occur during the numerical solution by direct transcription methods of inequality constrained optimal control problems. A second primary focus of this proposal is the numerical solution of inequality constrained optimal control problems using direct transcription methods. Throughout industry there is a need for the development of more efficient processes and systems. This is essential for improved performance, conserving of resources, and industrial and military competitiveness. This increased performance requires working with complex mathematical models of the systems in question. In many areas it is becoming necessary to utilize larger, more complex, and more complete models of the physical process or system. Two important parts of systems design are simulation and optimization. In applications there are many constraints. These constraints range from mission constraints (such as flying within certain flight corridors or intercepting specific targets) to physical constraints (such as contact between a robotic arm and a tool) to financial or cost constraints. These constraints pose both theoretical and computational challenges which increase with the model's complexity. This project is to develop both mathematical theory and numerical algorithms for the numerical simulation and optimal control of complex systems with constraints. The proposer has ongoing collaborations with developers of software used in a number of industries including the aerospace industry. The results of this research will both be incorporated into production computer codes as part of collaborations of the Principal Investigator and also shared with others developing numerical software. These algorithms and software will have very wide applicability. The current collaborations involve the aerospace industry including space and aircraft mission planning, robotic manufacturing, and optimization of chemical processes. In addition to the significant research to be accomplished, graduate students will play a fundamental role in this research. The training of the next generation of researchers in this area is another important contribution of this project. As with prior students of the Principle Investigator, the students trained on this project will move on to work in Industry, National Laboratories and Research Centers, the Armed Forces, and Academia.