On the absence of periodic solutions in interconnected phase systems

In this paper, a class of interconnected phase systems is discussed in terms of the absence of a specific kind of periodic solutions. Such systems can be viewed as an interconnection of several simpler subsystems with the interconnecting structure specified by a square matrix. Frequency-domain conditions as well as linear matrix inequalities conditions for nonexistence of limit cycles of the second kind 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 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 and the applicability and validity of the proposed method.