Limit Cycle Identification in Nonlinear Polynomial Systems

Shuqi Zhang, Haotian Liu, Kim Batselier, Ngai Wong

Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China.
Department of Electrical Engineering (ESAT)-SCD, KU Leuven/IBBT Future Health Department, Leuven, Belgium.

DOI: 10.4236/am.2013.49A004                

Abstract

We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.

Keywords

Limit Cycle Identification; Polynomial Representation; Roots Finding; Macaulay Matrix; Immersion

Conflicts of Interest

The authors declare no conflicts of interest.

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