A Semidefinite Programming Approach for Harmonic Balance Method

The harmonic balance method is broadly employed for analyzing and predicting the periodic steady-state solution. Most of the traditional methods in the literature do not guarantee global optimality. Due to its nonconvexity, it is a complex task to find a global solution to the harmonic balance problem in the frequency domain. A convex relaxation in the form of semidefinite programming has attracted attention since its introduction because it yields a global solution in most cases. This paper introduces a novel optimization-based approach to predict periodic solution by determining the Fourier series coefficients with high accuracy. Unlike the other commonly used methods, the proposed approach is completely independent of initial conditions. In our proposed method, the nonlinear constraints composed of time-dependent trigonometric functions are converted into nonlinear algebraic polynomial equations. Then, nonlinear unknowns are convexified through moment-sum of squares approach. However, computing a global solution costs a higher runtime. Our approach is validated through small examples which contain only polynomial nonlinearities. In all cases, the Mosek solver shows a better performance in comparison with SDPT3 and SeDuMi solvers. The proposed method shows high computational cost as a result of an increase in the positive semidefinite matrix size, which can depend on the number of harmonics and the degree of nonlinearity.