A Novel Framework for the Efficient and Accurate Solutions of Complex Chance-Constrained Optimal Control Problems

这个项目将创建一个新的集成计算框架，用于在不确定性的存在下制定和解决最优控制问题。最优控制关注的是找到用户指定的输入动态系统，将产生最好的可能结果，在这个意义上，一些性能指标是尽可能小或尽可能大。通常，结果还必须满足其他约束，捕获系统不能或不得违反的物理限制或操作要求。在受到显著随机影响的不确定系统中，性能和约束都可以用概率来表征。其中一种提法涉及“机会限制”，要求某一特定的不良事件必须足够不可能发生-例如，两架飞机在彼此不安全的距离内通过的概率必须小于某一给定的阈值。不幸的是，机会约束往往导致公式计算上是棘手的。该项目旨在通过四个领域的创新来克服这一障碍，即1）以机会约束的形式表示不确定性，2）计算上易于处理的机会约束近似，3）连续最优控制问题的有效离散化，以及4）最优控制问题的结构化，以便仅使用局部信息就可以在许多不同的处理器之间进行分割。这些创新将被整合到一个统一的框架中，放大它们的优势，并最终实现复杂的不确定最优控制问题的准确和有效的解决方案。这项工作的结果将有利于快速多智能体轨迹规划的搜索，救援和侦察任务，以及涉及人体运动，空中交通管制，水下航行器控制和高超音速飞行器使命规划的应用。教育活动将包括通过佛罗里达大学学生科学培训方案和暑期科学研究所向高中学生和教师推广。目前，机会约束控制几乎完全由鲁棒模型预测控制主导，不可避免地涉及线性动力学和凸多面体机会约束，大多数由高斯随机参数组成。 与此相反，本计画则将弹道设计视为不确定环境下的非线性机会约束最佳控制问题。将研究以下关键方面：（a）不确定环境的建模及其对状态和控制变量的概率约束的贡献;（B）基于分裂伯恩斯坦近似和马尔可夫链蒙特卡罗的涉及非高斯概率测度的非线性、非凸和潜在高维机会约束的可扩展半解析近似;（c）用于离散化由机会约束最优控制问题引起的连续优化问题的高精度和低维变阶高斯求积方法;以及（d）一种新的大规模非线性规划问题求解器，用于快速和准确地解决由变量-阶高斯正交离散化在这方面的工作可以导致自主路径规划，可扩展到多智能体系统的重大贡献。 这将需要有效和准确的联合机会约束转换成计算上有吸引力的形式，可以被证明是一致的，并收敛到原来规定的机会约束。这项研究将奠定直接解决机会约束的最优轨迹设计的基础，通过离散化的转录问题，使用变阶正交配置方法，解决了使用非线性规划例程，采用了强大的反向通信架构，使并行处理连同一个国家的最先进的非线性规划算法。

This project will create a novel integrated computational framework for formulating and solving optimal control problems in the presence of uncertainty. Optimal control is concerned with finding the user-specified inputs to a dynamic system that will produce the best possible outcome, in the sense that some performance measure is made as small or as large as possible. Typically the outcome must also satisfy additional constraints, capturing physical limitations or operating requirements that the system cannot or must not violate. In uncertain systems subject to significant random influence, both performance and constraints may be characterized probabilistically. One such formulation involves "chance constraints," requiring that a specified undesirable event must be sufficiently unlikely -- for example, the probability that two aircraft will pass within an unsafe distance of each other must be less than a given threshold. Unfortunately chance constraints often lead to formulations that are computationally intractable. This project aims to overcome this obstacle through innovations in four areas, namely 1) the representation of uncertainty in the form of chance constraints, 2) the computationally tractable approximation of chance constraints, 3) the efficient discretization of continuous optimal control problems, and 4) the structuring of the optimal control problem so that it can be split among many different processors using only local information. These innovations will be integrated into a unified framework, amplifying their benefits and ultimately enabling accurate and efficient solution of complex uncertain optimal control problems. Results from this of this work will benefit rapid multi-agent trajectory planning for search, rescue and reconnaissance missions, as well as applications involving human motion, air-traffic control, underwater vehicle control, and hypersonic vehicle mission planning. Educational activities will include outreach to high school students and teachers through the University of Florida Student Science Training Program and Summer Science Institute.Presently, chance-constrained control is almost exclusively dominated by robust model predictive control, invariably involving linear dynamics and convex polyhedral chance constraints, mostly comprising Gaussian random parameters. In contrast, this project will pose trajectory design as a nonlinear chance-constrained optimal control problem in an uncertain environment. The following key aspects will be studied: (a) modeling of the uncertain environment and its contribution to probabilistic constraints on the state and control variables; (b) scalable semi-analytical approximation of nonlinear, nonconvex and potentially high dimensional chance constraints involving non-Gaussian probability measures based on split-Bernstein approximations and Markov chain Monte Carlo; (c) highly accurate and low-dimensional variable-order Gaussian quadrature methods for discretizing the continuous optimization problem arising from the chance-constrained optimal control problem; and (d) a novel large-scale nonlinear programming problem solver for rapidly and accurately solving problems arising from the variable-order Gaussian quadrature discretization. Work in this area can lead to significant contributions in autonomous path planning, extendable to multi-agent systems. This will require efficient and accurate conversion of the joint chance constraints into computationally attractive forms that can be shown to be consistent with and convergent to the originally prescribed chance constraints. This research will lay the foundation for the direct solution of chance-constrained optimal trajectory design by discretizing the transcribed problem using a variable order orthogonal collocation method, solved using an nonlinear programming routine that employs a powerful reverse communication architecture, enabling parallel processing together with a state-of-the-art nonlinear programming algorithm.