Stability and Optimality Properties of Sequential Action Control for Nonlinear and Hybrid Systems

这个项目将极大地扩展机器人和车辆控制的一种强大的新方法，称为顺序动作控制（SAC）。控制复杂系统的一种广泛使用的方法是解决下几个控制脉冲的实时数值优化问题，其中所有脉冲都具有恒定的宽度。然而，即使使用强大的处理器，也很难足够快地计算这些值。SAC通过计算下一个控制脉冲的最佳宽度和相对开始时间来解决这一挑战。对于许多感兴趣的问题，这种控制策略的改变极大地简化了计算，以至于不可行的控制问题变得易于处理。SAC允许对一些问题进行分析解决，并将其他问题的计算速度提高了8个数量级。SAC与现代控制设计的共同特征自然兼容，包括在连续动态行为集合之间离散切换的混合系统；在量子化系统中，输入、状态和输出只能取一组有限的常数值；以及非线性动力学系统。在许多解析可解的情况下，SAC可以恢复全局最优控制信号。在其他代表性的测试用例中，计算的SAC输入提供的性能在数字上与最佳性能没有区别。如果微小的干扰导致系统迅速偏离预期行为，那么最优或接近最优的输入信号就没有价值。因此，实际的控制器还必须确保对被控系统的小干扰只会导致系统响应的小偏差——一种称为稳定性的特性。该项目旨在严格推导SAC性能保证，用于广泛的系统类别，以及显示SAC确保稳定性的条件。达尔文人形机器人将被用作这项研究的高维、非线性、混合试验台。达尔文机器人的控制可以在开源机器人操作系统（ROS）中实现，允许健壮且可验证的SAC分发用于传播。该项目的结果将使康复机器人、辅助设备、转子车辆和无人驾驶汽车等系统的控制得到极大改善和可验证，这些系统使用广泛可用的低成本计算平台，如移动电话。这个项目对社会的好处包括提高这些自动化基础设施系统的安全性和性能。该项目还包括课堂创新、国际合作、通过芝加哥科学与工业博物馆开展的外展活动，以及开源软件的传播。该项目的双重目的是将顺序动作控制（SAC）发展成为一种可操作的，近乎通用的方法，用于综合嵌入式实时控制，并提供最优性，稳定性和几何的基础结果。该方法计算效率高，适用于高维问题。此外，SAC自然地扩展到李群，在机器人和自动化等应用中很常见。该项目将解决三个基本问题。首先，它将确定SAC可以直接或迭代应用以实现最优控制的条件。其次，它将为稳定创造条件。第三，它将使SAC适应在李群上进化的系统，以实现多体机械系统的全局性能。对这项工作的更广泛影响包括外联、向康复转移技术、开发动态和分析方面的在线课程以及国际合作。PI目前正在与科学与工业博物馆合作，作为该项目的一部分，PI以及PI实验室的研究生和本科生将参加在博物馆主圆形大厅举行的国家机器人周展览，预计现场观众将超过10,000人。

This project will greatly extend a powerful new method for control of robots and vehicles, called sequential action control (SAC). One widely used approach to controlling complicated systems is to solve a real-time numerical optimization problem for the magnitude of the next few control pulses, where the all the pulses have a constant width. However even with powerful processors it can be difficult to compute these values fast enough. SAC addresses this challenge by instead computing the optimal width and relative start time of the next control pulse. For many problems of interest, this change in control strategy greatly simplifies computation, to the point that infeasible control problems become tractable. SAC allows analytical solutions to some problems, and speeds computations by up to eight orders of magnitude for others. SAC is naturally compatible with common features of modern control design, including hybrid systems that switch discretely between a collection of continuous dynamic behaviors; quantized systems where inputs, states, and outputs may take only a finite set of constant values; and systems with nonlinear dynamics. SAC can be shown to recover the globally optimal control signal in a number of analytically solvable cases. In other representative test cases, the computed SAC input provides performance that is numerically indistinguishable from the optimum. Optimal or near-optimal input signals are of no value if small disturbances cause the system to rapidly diverge from the desired behavior. Therefore practical controllers must also ensure that small disturbances to the controlled system cause only small deviations in the system response -- a property known as stability. This project seeks to rigorously derive SAC performance guarantees for a broad class of systems, as well as to show conditions under which SAC ensures stability. The Darwin humanoid robot will be used as a high-dimensional, nonlinear, hybrid testbed for this research. Control of the Darwin robot may be implemented in the open-source Robot Operating System (ROS), allowing a robust and verifiable SAC distribution for dissemination. The results of this project will enable greatly improved and verifiable control over systems such as rehabilitation robots, assistive devices, rotor vehicles, and driverless cars, using widely available and low-cost computing platforms such as mobile phones. Benefits to society from this project include enhanced safety and performance of these automated infrastructure systems. The project also includes classroom innovation, international collaboration, outreach activities through the Museum of Science and Industry in Chicago, and dissemination of open-source software.The twofold purpose of this project is to develop sequential action control (SAC) into an actionable, near-universal method for synthesizing embedded real-time control as well as to provide foundational results on optimality, stability, and geometry. The method is computationally efficient and scales to high dimensional problems. Moreover, SAC extends naturally to Lie groups, common in applications such as robotics and automation. The project will address three fundamental questions. First, it will identify conditions under which SAC can be applied directly or iteratively to achieve optimal control. Second, it will derive conditions for stability. Third, it will adapt SAC to systems evolving on Lie groups, to achieve global performance for multibody mechanical systems. The broader impacts for this work include outreach, technology transfer to rehabilitation, the development of online courses in dynamics and analysis, and international collaboration. The PI is currently working with the Museum of Science and Industry, and as part of the project the PI, and graduate and undergraduates involved in the PI's laboratory, will participate in a National Robotics Week exhibit in the main rotunda of the museum with an estimated viewership of over ten thousand on-site visitors.