A Convex Computational Framework for Understanding and Controlling Nonlinear Systems

复杂的工程系统已经变得越来越普遍和自主，包括水下航行器、无人机和自动驾驶汽车，以及不太显眼的系统，如电源逆变器、电池存储设备、网络路由器和数据存储设备。长时间的自治不可避免地会导致一些意想不到的变化，如：系统配置的改变或降级；执行器或传感器故障；以及工作环境的演变。如果没有一种自动识别和适应这些变化的机制，我们社会的技术基础很可能会变得越来越不可靠。该项目的目标是提高复杂系统的自主性。具体来说，优化算法用于学习环境，分析这些信息，并设计控制策略。基于动态系统如何运作的基本理论，这些算法为认知和适应性提供了严格的基础。这项工作的影响将是在地球和空间上建立一个更加安全和可靠的技术基础设施。例如，提高自主系统的可靠性和持续时间将导致更快、更便宜的太空探索、更安全的交通网络和更可靠的通信网络。近年来，平方和算法（Sum-of-Squares， SOS）已成为研究非线性动力系统的有力工具。SOS的强大之处在于它对非二次Lyapunov函数的凸参数化。然而，不幸的是，这个凸公式假设动力学是已知的，并没有扩展到估计吸引区域（ROA），最小不变集（MIS）和前向可达性（FRS）。该项目的核心是新的观察，即ROA/MIS/FRS问题可以用Hamilton-Jacobi-Bellman （HJB）方程的解定义的值函数的子解或超解来表示。项目的第一部分利用这种等价性来表明ROA/MIS/FRS问题可以被提出为SOS优化问题，其中目标是子/超值函数的子水平集的体积最小化或最大化。项目的第二部分使用新的凸体积度量来解决这些SOS体积优化问题。项目的第三部分考虑了动力学未知的情况，并使用轨迹数据直接估计ROA和FRS，而不使用动态模型。该算法还应用于航天器姿态动力学控制。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Complicated engineered systems have become increasingly common and autonomous, with examples including underwater vehicles, UAVs, and self-driving cars, as well as less visible systems such as power inverters, battery storage devices, network routers, and data storage devices. Extended periods of autonomy inevitably lead to such unexpected changes as: altered or degraded system configuration; failures of actuators or sensors; and evolution of the working environment. Without an automated mechanism for identification of and adaptation to such changes, it is likely that the technological foundation of our society will become increasingly unreliable. The goal of the project is to improve autonomy in complicated systems. Specifically, optimization algorithms are used to learn the environment, analyze this information, and design control strategies. Based on fundamental theory of how dynamical systems operate, these algorithms provide a rigorous basis for cognizance and adaptability. The impact of this work will be a more safe and reliable technological infrastructure, both on earth and in space. Increasing the reliability and duration of autonomous systems will lead to, for example, faster and cheaper exploration of space, safer transportation networks, and more reliable communication networks.Recently, Sum-of-Squares (SOS) algorithms have become a powerful tool for understanding nonlinear dynamical systems. The power of SOS lies in its convex parametrization of non-quadratic Lyapunov functions. Unfortunately, however, this convex formulation assumes the dynamics are known and has not been extended to estimating the Region of Attraction (ROA), Minimum Invariant Set (MIS), and Forward Reachability (FRS). At the core of this project is the novel observation that the ROA/MIS/FRS problems can be expressed using sub- or super-solutions to a value function defined by the solution to a Hamilton-Jacobi-Bellman (HJB) equation. The first part of the project exploits this equivalence to show that the ROA/MIS/FRS problems can be posed as SOS optimization problems wherein the objective is volume minimization or maximization of sublevel sets of a sub/super-value function. The second part of the project uses new convex volume metrics to solve these SOS volume optimization problems. The third part of the project considers the case where the dynamics are unknown and uses trajectory data to directly estimate the ROA and FRS without use of a dynamic model. The algorithms are also applied to control of spacecraft attitude dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.