A short survey on nonlinear models of the classic Costas loop: rigorous derivation and limitations of the classic analysis

Rigorous nonlinear analysis of the physical model of Costas loop --- a classic phase-locked loop (PLL) based circuit for carrier recovery, is a challenging task. Thus for its analysis, simplified mathematical models and numerical simulation are widely used. In this work a short survey on nonlinear models of the BPSK Costas loop, used for pre-design and post-design analysis, is presented. Their rigorous derivation and limitations of classic analysis are discussed. It is shown that the use of simplified mathematical models, and the application of non rigorous methods of analysis (e.g., simulation and linearization) may lead to wrong conclusions concerning the performance of the Costas loop physical model.Comment: Accepted to American Control Conference (ACC) 2015 (Chicago, USA

New results on cycle–slipping in pendulum–like systems

n this paper, we examine dynamics of multidimensional control systems obtained as feedback interconnections of stable linear blocks and periodic nonlinearities. The simplest of such systems is the model of mathematical pendulum (with viscous friction), so we call such systems pendulum-like. Other examples include, but are not limited to, coupled vibrating units, networks of oscillators, Josephson junction arrays and numerous synchronization circuits used in radio and telecommunication engineering. Typically, a pendulum-like system has infinite sequence of equilibria, and one of the central problems addressed in the theory of such systems is to find the conditions of "global stability", or gradient-like behavior ensuring that every solution converges to one of the equilibria points. If a system is gradient-like, another problem arises, being the main concern of this paper: can we find the terminal equilibrium, given the initial condition of the system? It is well known that solutions do not converge, in general, to the nearest equilibrium; this phenomenon is known as cycle-slipping. For a pendulum, cycle-slipping corresponds to multiple rotations of the pendulum about its suspension point. In this paper, we estimate the number of slipped cycles for general pendulum-like systems by means of periodic Lyapunov functions and the Kalman-Yakubovich-Popov lemma