Problem of cycle-slipping for infinite dimensional systems with MIMO nonlinearities

The concept of cycle slipping was introduced by J.J. Stocker for the mathematical pendulum with friction. After an impact, the pendulum make several revolutions around the suspension point before settling down in the lower stable equilibrium. The number of those revolutions is referred to as the number of cycles, slipped by the solution. In general, the number of slipped cycles may be defined for any system with periodic nonlinearity and gradient-like behavior, being an important characteristic of the transient process. In phase-locked loop (PLL) based systems, this number shows how large may be the phase error before the locking, which makes the problem of cycle slipping important for telecommunications and electronics. The problem addressed in the present paper is how to estimate the number of slipped cycles in infinite-dimensional systems, consisting of linear Volterra-type equation in the interconnection with a periodic MIMO nonlinearity. The techniques developed in the paper stem from the Popov´s approach of “a priori integral indices”, which was proposed originally as a tool for proving stability of nonlinear systems and was the prototype for the method of integral quadratic constraints (IQC). Employing novel types of Popov-type quadratic constraints, we obtain new frequency-domain estimates for the number of cycles slipped.