Abstract
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.
Highlights
Nonlinear oscillators have been broadly studied in many areas of physics and engineering [1] and are of significant importance in mechanical [2], micro-electromechanical systems [3], power systems [4], circuits [5], and nonlinear structural dynamics [6] for the understanding and accurate prediction of motion
Based on the proposed framework, an efficient method is presented for predicting the periodic solutions of a nonlinear periodic dynamical system
The nonlinear dynamical equations are transformed into the frequency domain by Fourier expansion and alternating frequency time (AFT) technique is applied to handle the nonlinearity accurately
Summary
Nonlinear oscillators have been broadly studied in many areas of physics and engineering [1] and are of significant importance in mechanical [2], micro-electromechanical systems [3], power systems [4], circuits [5], and nonlinear structural dynamics [6] for the understanding and accurate prediction of motion. Efficient algorithms like interior point method have been extensively applied, which is able to compute an optimal solution of any given accuracy. These properties make SDP modeling adapted to many applications [25]–[27]. Recent works have focused on moment matrix relaxation for solving polynomial optimization problems [28], [29]. We propose a convexified HB optimization method for solving systems with polynomial nonlinearities. This approach aims at finding global solutions to polynomial optimization problems.
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