Analysis of Nonlinear Oscillations via Lyapunov Approach

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本计划的目的是发展一个系统的和数值上易于处理的李雅普诺夫方法来评估非线性振荡的幅度。知识上的优点是李亚普诺夫方法提供了非线性振荡幅度的定量和可靠的表征。虽然非线性振荡已经得到了广泛的研究，但大多数努力都致力于对其复杂行为的定性描述。也存在一些近似但不可靠的方法，如描述函数法，来估计振荡的幅度。李亚普诺夫方法的主要工具包括描述一个带饱和函数的分段线性元和一个兼容的分段二次函数。这些工具将分析问题转化为数值上可处理的基于线性矩阵不等式的优化问题。该方法可适用于设计参数，以尽量减少不必要的振荡。这些方法将应用于某些电力电子系统的优化设计，以抑制不必要的振荡。该计划将对科学和工程的许多分支产生重大影响，因为非线性振荡存在于各种类型的系统中。研究成果将提交到跨学科期刊和会议上发表，从而促进李亚普诺夫理论等控制系统方法的应用。适当的研究成果将被纳入PI正在教授和更新的研究生课程“非线性系统”中。建议的工作将寻求研究生和本科生的广泛参与。特别是，将鼓励女性和少数民族学生参加该计划。在这个程序中，控制系统的一个基本工具，李雅普诺夫函数，将扩展到解决一个高度跨学科的主题-非线性振荡-这是在几乎所有的科学和工程分支研究。预计该计划将促进控制理论与其他研究领域的交叉施肥，并在非线性动力学方面产生革命性的发现。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Objective The objective of this program is to develop a systematic and numerically tractable Lyapunov approach for evaluation of the magnitude of nonlinear oscillations.Intellectual merit The intellectual merit is that the Lyapunov approach provides a quantitative and reliable characterization of the magnitude of nonlinear oscillation. Although nonlinear oscillations have been extensively studied, most efforts were devoted to qualitative description of the complex behavior. There also exist some approximate, but not reliable, methods, such as the describing function method, to estimate the magnitude of oscillations. The main tools of the Lyapunov approach include the description of a piecewise linear element with saturation functions and a compatible piecewise quadratic function. These tools convert the analysis problem into numerically tractable linear-matrix-inequality based optimization problem. The methods can be adapted for designing parameters for minimization of unwanted oscillations. The methods will be applied to optimal design of some power electronic systems for suppression of unwanted oscillations.Broader impacts This program will have significant impact on many branches of science and engineering since nonlinear oscillations exist in systems of various types. The research results will be submitted to interdisciplinary journals and conferences for publication, so as to promote the application of Lyapunov theory, and other methods for control systems. Appropriate research results will be incorporated into the graduate course ``Nonlinear Systems," which the PI has been teaching and renovating. The proposed work will seek broad participation from graduate and undergraduate students. In particular, female and minority students will be encouraged to participate in the program.In this program, a fundamental tool in control systems, the Lyapunov function, will be extended to address a highly interdisciplinary subject - nonlinear oscillation - which is studied in almost all branches of science and engineering. It is expected that this program will promote cross fertilization between control theory and other research fields, and yield revolutionary discovery in nonlinear dynamics.