An Optimization Approach for Nonlinear Optimal Feedback Control Design and Uncertainty Propagation

下一代自主系统的多分辨率关节能力，用于先进的机器人、康复、远程操作、制造和基础设施应用，通过独特的设计和复杂的机制成为可能。这些下一代机器人系统可以通过开发计算高效的工具来综合最优反馈控制律，以在存在模型和传感不确定性的情况下实现指定的输出信号统计，从而大大受益。为了支持包裹递送等商业应用，在不确定环境中运行的机器人系统必须以规定的容差协商特定的路点。这种在不确定环境中运行的车辆的路径选择的独特挑战涉及到不确定性传播和控制之间的强耦合。这一挑战的成功谈判取决于稳定的最优反馈控制律的发展，同时考虑到弥漫在动态系统中的不确定性。支撑这项研究的基本形式主义广泛适用于下一代机器人系统和无人驾驶车辆的控制。该项目包括将研究整合到涉及研究生和本科生的教育工作中，包括扩大接触科学和工程领域代表性不足人群的努力。推广活动包括与当地学校教师的技术交流会议和开设与机器人相关的教程课程以激励初中生。研究的重点是通过开发分别计算高效的汉密尔顿-雅可比-贝尔曼(HJB)和福克-普朗克-科尔莫戈洛夫(FPK)偏微分方程组(PDE)的计算高效解，研究稳定的最优反馈控制律和不确定性传播方法的统一方法。利用稀疏近似和非乘积求积规则的最新进展来求解HJB方程和FPK方程。提出了不确定性量化、最优控制理论和数值分析的知识状态，并将它们有效地结合在一起，实现了一个可扩展的动态系统研究框架。共轭无迹变换(CUT)技术与稀疏近似方法相结合，形成了驱动不确定性传播和最优反馈控制实现过程的使能工具。这项研究的结果是发展了不确定动态系统的非线性稳定反馈控制律，影响了工程中的各种估计和控制问题。拟议的研究将在几个基准问题上进行演示，以及两个关键应用，涉及陀螺系统控制中的最优动量传递和借助无人驾驶飞行器对地面区域的监视。

The multi-resolution articulation abilities of the next-generation autonomous systems, used in advanced robotics, rehabilitation, tele-operation, manufacturing and infrastructure applications are made possible by unique designs and complex mechanisms. These next generation robotic systems can benefit greatly by the development of computationally efficient tools to synthesize optimal feedback control laws to achieve specified output signal statistics in presence of model and sensing uncertainties. To support commercial applications such as package delivery, it is imperative that the robotic systems operating in uncertain environments negotiate specific waypoints with prescribed tolerance. This unique challenge of routing the vehicles operating in uncertain environments involves a strong coupling between the uncertainty propagation and control. Successful negotiation of this challenge is contingent on the development of stable optimal feedback control laws, while accounting for the uncertainties that pervade dynamical systems. The fundamental formalisms that underpin this research are widely applicable to the control of next generation robotic systems and unmanned vehicles. The project includes plans to integrate research into educational efforts involving graduate and undergraduate students, including expanding efforts to reach under-represented populations in science and engineering. Outreach activities include the technical interchange meetings with local school teachers and instituting robotics related tutorial courses to motivate middle and high school students.The focus of the research is to investigate a unified approach to stable optimal feedback control laws and uncertainty propagation methods by developing computationally efficient solutions to the Hamilton Jacobi Bellman (HJB) and Fokker-Planck-Kolmogorov (FPK) partial differential equations (PDEs), respectively. Recent advances in sparse approximation and non-product quadrature rules are exploited to solve the HJB and FPK equations. The intellectual merits are drawn from advancing the state of knowledge in uncertainty quantification, optimal control theory, and numerical analysis, and integrating them effectively to realize a scalable framework for study of dynamical systems. The Conjugate Unscented Transformation (CUT) technique in conjunction with sparse approximation methods forms an enabling tool to drive the uncertainty propagation and optimal feedback control realization processes. This research culminates in the development of nonlinear stable feedback control laws for uncertain dynamic systems, impacting various estimation and control problems in engineering. The proposed research will be demonstrated on several benchmark problems, along with two key applications involving optimal momentum transfer in control of gyroscopic systems and surveillance of a ground region with the help of unmanned vehicles.