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A class of uncertain pendulum-like systems with several nonlinearities

This paper considers the existence of cycles of the second kind in uncertain pendulum-like systems with several nonlinearities. Based on Kalman-Yakubovich-Popov (KYP) lemma, which establishes the equivalence between frequency domain inequalities (FDIs) and linear matrix inequalities (LMIs), LMI conditions guaranteeing the existence of cycles of the second kind for such nonlinear systems under parameter uncertainties are established. In virtue of the results, an interesting conclusion is reached that synthesis problem ensuring the existence of cycles of the second kind for such uncertain nonlinear system can be converted into synthesis problem for a system without uncertainties. A practical example about synchronous machine demonstrates the validity of the present approach.