Design of nonlinear interconnections guaranteeing the absence of periodic solutions

In this paper, a novel methodology for scheming interconnection structure for a class of nonlinearly interconnected systems is proposed to guarantee the absence of a specific kind of periodic solutions. Those types of systems can be viewed as an interconnection of single-input single-output isolated subsystems with the interconnection structure specified by a square matrix. Frequency-domain inequalities conditions as well as linear matrix inequalities (LMIs) conditions for nonexistence of limit cycles of the second kind in the entire interconnected system are established. The main results also define the frequency range on which cycles of the second kind of the system cannot exist. Based on this LMI approach, an estimate of the largest bound of the frequency of cycles of the second kind can be explicitly computed by solving a generalized eigenvalue minimization problem. Numerical results show the effect of the nonlinear interaction on system behavior and the applicability and validity of the proposed method.