Collaborative Research: Control of Underactuated Mechanical Systems through Lyapunov Direct Methods

AbstractCollaborative Research：通过李亚普诺夫直接方法欠驱动机械系统的控制建议的项目涉及开发李亚普诺夫直接方法设计非线性欠驱动机械系统的稳定控制器，使系统成为闭环李亚普诺夫形式，而不是闭环哈密顿形式。 一个目标是表明，与闭环李雅普诺夫形式相关的控制器可以通过构成渐近稳定的某些线性（偏）微分方程的解析或数值解（在反馈中完成）来实现。 所提出的方法的优点是，在哈密顿形式的系统类是一个严格的子集的系统类的李雅普诺夫形式，从而产生一个更大的类的稳定控制器，可以提供系统性能的改善。 进一步预计，所提出的研究将提供一个基础，在此基础上处理的问题是棘手的使用其他方法，同时也导致一个统一的框架，利用受控拉格朗日和受控哈密顿的方法。 另一个重要的结果是，所获得的控制器是线性的速度，而现有的基于能量的非线性控制器的速度是二次的。 通过所提出的方法设计的控制律和通过控制拉格朗日，λ和IDA-PBC方法设计的控制律之间的比较将被制作。所提出的活动的主要推力是分析和进一步发展一种新的策略，李雅普诺夫直接方法，用于设计欠驱动机械系统的稳定控制器，以具有比自由度少的驱动器为特征的系统，例如其中一个或多个关节不被供电的机器人。 这类系统的例子有航天器、水下航行器、卫星、气垫船和集装箱船的货物起重机。 所提出的方法提供了对现有技术的改进，因为实现了更好的性能，同时没有遭受现有方法的一些常见问题。 预计存在可以通过所提出的技术处理的系统，而这些系统不能通过当前技术处理。 分析将包括比较的控制律产生的李雅普诺夫直接方法和目前公布的方法。 将鼓励数学和工程专业的本科生通过就业和课程学分参与该项目。 一些学生将帮助在KSU的REU计划的发展。 为了促进代表性不足的群体的参与，KSU少数民族工程计划和妇女在工程和科学计划办公室已承诺帮助学生招聘，这是一项与NSF资助的STEP计划相一致的活动。

AbstractCollaborative Research: Control of Underactuated Mechanical Systems through Lyapunov Direct MethodsThe proposed project involves developing Lyapunov direct methods for designing stabilizing controllers for nonlinear underactuated mechanical systems that render the system into closed loop Lyapunov form as opposed to closed loop Hamiltonian form. A goal is to show that the controller associated with the closed loop Lyapunov form can be implemented through either an analytical or a numerical solution (done in the feedback) of certain linear (partial) differential equations constituting asymptotic stability. The advantage of the proposed methodology is that the class of systems in Hamiltonian form is a strict subset of the class of systems in Lyapunov form thus yielding a larger class of stabilizing controllers that could offer improvements in system performance. It is further expected that the proposed research will provide a basis on which to treat problems that are intractable using other methods, while also leading to a unifying framework for methods utilizing controlled Lagrangians and controlled Hamiltonians. Another significant result is that the obtained controller is linear with respect to the velocities, whereas existing energy based nonlinear controllers are quadratic in the velocities. Comparisons between control laws designed through the proposed method and those designed through controlled Lagrangian, lambda, and IDA-PBC methods will be made.The primary thrust of the proposed activity is to analyze and further develop a new strategy, Lyapunov direct methods, for designing stabilizing controllers for underactuated mechanical systems, systems characterized by having fewer drives than degrees of freedom such as a robot where one or more joints are not powered. Examples of such systems are spacecraft, underwater vehicles, satellites, hovercraft and cargo cranes for container ships. The proposed method offers an improvement over the current art in that better performance is achieved while not suffering some common problems of existing methods. It is expected that there are systems that can be treated by the proposed techniques that cannot be treated by current techniques. The analysis will include comparisons of control laws produced by Lyapunov direct methods and current published methods. Undergraduate students in both mathematics and engineering will be encouraged to participate in the project through employment and course credit. Some students will help with development of an REU program at KSU. To facilitate participation of underrepresented groups, the KSU Minority Engineering Program and Women in Engineering and Science Program offices have committed to help in student recruitment, an activity that dovetails with their NSF funded STEP program.