Direct Transcription Solution of Optimal Control Problems with Delays

在现代技术的各种重要领域，从航空航天系统的控制到引导车辆，到确定药物治疗，再到生产工业产品，都需要找到执行不同任务的最佳或最佳方法。目标是提高效率和性能，降低成本。物理系统可以用数学方法来描述。该项目将开发数值算法、计算机软件和支持数学，使科学家和工程师能够利用这些数学描述来解决许多他们目前很难解决的最优控制问题。这将有助于一些国家需要的领域，包括更有效的工业过程。这项研究的最初应用领域将是化学过程、航空航天应用和生物医学问题，因为这些合作已经在进行中。长期影响甚至比这些领域更广泛，因为提高效率和更好的性能的需求是整个科学和技术的驱动需求。数学描述常常涉及微分方程。通常在物理过程或用于控制该过程的手段上存在物理或操作约束。一个比较成功的数值方法，经常用于复杂的最优控制问题是直接转录，其中问题被转换成一个离散问题，通过各种编程技术，然后迭代求解在特别选择的精细网格，直到一个高质量的解决方案被发现。直接转录在工业上是一种流行的技术。许多具有挑战性的问题在状态动力学或控制或两者中也有延迟。延迟可能由于多种原因发生，包括传输延迟、物理过程、执行器延迟或人为操作引起的延迟。例如，与状态相关的延迟在化学工业中很常见。到目前为止，这些延迟已经构成了挑战，阻碍了工业级直接转录算法的发展，这些算法用于计算具有延迟的状态和控制约束非线性系统的最优解。这个项目将解决这个困难。提出了发展的算法，有效的实现，并支持数学分析的直接转录解的状态和控制约束的最优控制问题的延迟。所提出的研究将产生新的算法和对数值方法的新的理论认识。这些类型的算法往往非常复杂。理论支持的指导方针将为这些类型的算法的用户和开发人员开发。拟议的项目将在最佳控制方面的一些重要问题上取得重大进展，训练下一代研究人员研究这些问题，并对国家需要的几个领域作出贡献。广泛的影响之一是能够更广泛地使用具有延迟的约束数学描述，这将允许更容易地模拟和优化各种物理系统，从而减少设计所需的时间并提高性能。将项目成果落实为工业级规范将大大加快并扩大其对许多行业的影响。

In a wide variety of important areas of modern technology from the control of aerospace systems to guiding vehicles to determining drug therapies to producing industrial products it is desired to find the optimal or best way to perform different tasks. The goal is improved efficiency and performance and lower cost. The physical system can be described mathematically. This project will develop numerical algorithms, computer software, and supporting mathematics that will enable scientists and engineers to take these mathematical descriptions and solve a number of optimal control problems that they currently have great difficulty in solving. This will contribute to a number of areas of national need including more efficient industrial processes. The initial areas of application of this research will be chemical processes, aerospace applications, and biomedical problems since those collaborations are already under way. The long term impact is even wider than these areas since the need for increased efficiency and better performance is a driving need throughout science and technology. Mathematical descriptions often involve differential equations. Frequently there are physical or operational constraints on either the physical process or the means being used to control this process. One of the more successful numerical approaches that is often used for complex optimal control problems is direct transcription where the problem is converted to a discrete problem by various programing techniques, and then iteratively solved on specially chosen refined grids until a high quality solution is found. Direct transcription is a popular technique in industry. Many challenging problems also have delays in either the state dynamics or the controls or both. Delays can occur for several reasons including transmission delays, physical processes, actuator delays, or human operator induced delays. State dependent delays are common in the chemical industry, for example. These delays, to date, have posed challenges that have hindered the development of industrial grade direct transcription algorithms for the computation of the optimal solution of state and control constrained nonlinear systems with delays. This project will address this difficulty. It is proposed to develop algorithms, efficient implementations, and supporting mathematical analysis for the direct transcription solution of state and control constrained optimal control problems with delays. The proposed research will result in new algorithms and new theoretical understanding of numerical methods. These types of algorithms tend to be very sophisticated. Theory backed guidelines will be developed for both users and developers of these types of algorithms. The proposed project will result in substantial advancement on a number of important problems in optimal control, the training of the next generation of researchers to work on these problems, and contributions to several areas of national need. Among the broad impacts are the capability to more widely use constrained mathematical descriptions with delays which will permit easier simulation and optimization of a wide variety of physical systems resulting in reduced time needed for design and improved performance. The implementation of the projects results into an industrial grade code will greatly hasten and broaden their impact across a number of industries.