Asymptotic estimates for gradient-like distributed parameter systems with periodic nonlinearities

Many systems, arising in electrical and electronic engineering may be represented as an interconnection of LTI system and a periodic nonlinear block, as exemplified by phase-locked loops (PLL) and more general systems, based on controlled phase synchronization of several periodic processes (“phase synchronization” systems, or PSS). Typically such systems are featured by the gradient-like behavior, i.e. the system has infinite sequence of equilibria points, and any solution converges to one of them (which may be interpreted as the phase locking). This property however says nothing about the transient behavior of the system, whose important qualitative index is the maximal phase error. Before the phase is locked, the error may increase up to several periods, which phenomenon is known as cycle slipping and was introduced by J. Stoker for the model of mathematical pendulum. Since the cycle slipping is considered to be undesired behavior of PLLs, it is important to find efficient estimates for the number of slipped cycles. In the present paper, we address the problem of cycle-slipping for phase synchronization systems with infinite-dimensional linear part. New effective estimates for a number of slipped cycles are obtained by means of Popov´s method of “a priori integral indices”, which was originally developed for proving absolute stability of nonlinear systems.