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

摘要：利用李雅普诺夫直接法控制欠驱动机械系统该项目涉及发展李雅普诺夫直接法，用于设计非线性欠驱动机械系统的稳定控制器，使系统变为闭环李雅普诺夫形式，而不是闭环哈密顿形式。目标是证明与闭环Lyapunov形式相关的控制器可以通过构成渐近稳定性的某些线性（偏）微分方程的解析解或数值解（在反馈中完成）来实现。所提出的方法的优点是，哈密顿形式的系统类是李雅普诺夫形式系统类的严格子集，从而产生更大的一类稳定控制器，可以提供系统性能的改进。进一步期望所提出的研究将为使用其他方法处理难以解决的问题提供基础，同时也为使用受控拉格朗日量和受控哈密顿量的方法提供统一的框架。另一个重要的结果是，得到的控制器相对于速度是线性的，而现有的基于能量的非线性控制器在速度上是二次的。将采用该方法设计的控制律与采用可控拉格朗日、lambda和IDA-PBC方法设计的控制律进行比较。所提出的活动的主要目的是分析并进一步开发一种新的策略，即Lyapunov直接方法，用于为欠驱动机械系统设计稳定控制器。欠驱动机械系统的特点是驱动器少于自由度，例如一个或多个关节没有动力的机器人。这种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当前技术的改进，在不遭受现有方法的一些常见问题的同时获得了更好的性能。预期存在可用所建议的技术处理而不能用当前技术处理的系统。分析将包括李亚普诺夫直接方法和目前发表的方法产生的控制律的比较。鼓励数学和工程专业的本科生通过就业和课程学分的方式参与该项目。一些学生将帮助KSU发展REU项目。为了促进代表性不足的群体的参与，KSU少数民族工程项目和女性参与工程和科学项目办公室承诺帮助招收学生，这一活动与他们的国家科学基金会资助的STEP项目相结合。

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.