Periodic steady-state analysis of nonlinear oscillators based on multivariate polynomial roots finding

Abstract

Periodic steady-state analysis plays an important role in both theoretical topics and numerical simulations. It has been applied to numerous fields such as electronics, economics, biology, chemistry and so on. Particularly in electronics it is the basis of microwave and radio frequency (RF) circuit simulation. Although the topic has been studied for decades, periodic steady-state analysis still remains a difficulty in certain aspects including the analysis of the exact analytical formulas of limit cycles, as well as fast and accurate approximation of periodic steady states with unknown frequencies. In this thesis, two innovative methods are proposed in order to overcome two difficulties in the field of periodic steady-state analysis accordingly: on the one hand, a limit cycle identification method is developed to provide a robust method for computation of the exact analytical formulas of limit cycles. The method can be further extended to a wide range of nonlinear systems by the technique called state immersion. On the other hand, a method for highly accurate periodic steady-state approximation based on harmonic balancing is proposed. It combines the robustness of Macaulay matrix approach for small size polynomial root(s) finding, and the efficiency of a guided global optimization for higher order approximations. Thus, it is capable of computing approximations of periodic steady states with a high accuracy. Together, the two methods establish a reliable framework where highly accurate periodic steady-state analysis for a wide range of nonlinear systems can be performed.