Analysis of Nonlinear Oscillations via Lyapunov Approach

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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.