The absence of periodic solutions for pendulum-like systems

The pendulum-like system is a special nonlinear system with multiple equilibria. Its periodic solutions are different from the universal nonlinear system´s. The frequency-domain criteria for the nonexistence of the second periodic solutions of the pendulum-like systems with MIMO can be converted to the time-domain criteria with the KYP lemma. By applying the frequency-domain criteria the method of the state-feedback controller design and the state-feedback controller design based on the state observer design is investigated. Examples are given to illustrate the results.