Robust analysis and synthesis for the nonexistence of periodic solutions in a class of nonlinear systems (Q2465464)

Periodic solutions or limit cycles exemplify the complex dynamic behavior of certain nonlinear systems. In most practical dynamical systems, such property should be suppressed because it may lead to instability. The Kalman-Yakubovich-Popov (KYP) lemma converts frequency domain inequalities (FDIs) into equivalent linear matrix inequalities (LMIs). The standard KYP lemma deals with FDIs in the whole frequency range, while practical control systems require often different properties in different frequency ranges. The generalized KYP (GKYP) lemma can treat FDIs in finite frequency range directly. In the presented paper, robust analysis and robust synthesis for the nonexistence of periodic solutions in the class of nonlinear systems are considered based on the GKYP lemma. First, LMI characterizations are established for such systems guaranteeing the nonexistence of periodic solutions in a certain frequency range. Then, the results are extended to robust analysis and robust synthesis with polytopic uncertainties. An output feedback controller is designed to guarantee the nonexistence of periodic solutions in such systems in virtue of parameter-dependent Lyapunov functions. A concrete application to the Chua circuit shows the applicability and validity of the proposed approach.