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综合计算框架，用于在存在不确定性的情况下制定和解决最优控制问题。最优控制关注的是为动态系统找到用户指定的输入，以产生可能的最佳结果，在某种意义上，某种性能指标被尽可能地小或大。通常，结果还必须满足其他约束，捕获系统不能或不能违反的物理限制或操作要求。在受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刻画。一种这样的表述涉及“机会约束”，即要求特定的不受欢迎的事件必须是完全不可能发生的--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给定的阈值。不幸的是，机会约束往往导致计算上难以处理的公式。该项目旨在通过四个方面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)机会约束的计算易于处理的近似，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，使得仅使用局部信息就可以将其分解到许多不同的处理器中。这些创新将被整合到一个统一的框架中，扩大它们的好处，最终能够准确有效地解决复杂的不确定最优控制问题。这项工作的结果将有助于搜索、救援和侦察任务的快速多智能体轨迹规划，以及涉及人体运动、空中交通管制、水下航行器控制和高超声速航行器任务规划的应用。教育活动将包括通过佛罗里达大学学生科学培训计划和暑期科学研究所向高中生和教师推广。目前，机会约束控制几乎完全由鲁棒模型预测控制主导，总是涉及线性动态和凸多面体机会约束，主要包括高斯随机参数。相比之下，该项目将把弹道设计问题归结为不确定环境下的非线性机会约束最优控制问题。(A)不确定环境的建模及其对状态变量和控制变量的概率约束的贡献；(B)基于分裂-Bernstein近似和马尔可夫链蒙特卡罗的涉及非高斯概率度量的非线性、非凸和潜在高维机会约束的可伸缩半解析近似；(C)用于离散机会约束最优控制问题的高精度和低维变阶高斯求积方法；以及(D)用于快速和准确地求解由变阶高斯求积离散化产生的问题的新型大规模非线性规划问题求解器。这一领域的工作可以在自主路径规划方面做出重大贡献，并可扩展到多智能体系统。这将需要有效和准确地将联合机会约束转换成计算上有吸引力的形式，这些形式可以被证明与最初规定的机会约束一致并收敛于该形式。这项研究将为机会约束最优弹道设计的直接解奠定基础，通过使用变阶正交配置法离散转录问题，使用采用强大的反向通信架构的非线性规划例程进行求解，使并行处理与最先进的非线性规划算法一起实现。

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.