On a perturbed phase-locked loop system: A simple physical model

The stochastic Phase-Locked Loop (PLL) system, a non-linear dynamic circuit, is an appealing and non-trivial case in the general theory of dynamical systems. In this paper, the stochastically-influenced PLL system is the subject of investigation in lieu of the noise-free case. Significantly, Stochastic Differential Equation (SDE) descriptions refine estimation algorithms, stability conditions and control laws of dynamical systems in contrast to classical descriptions. More precisely, the stochastic version of the PLL system of this paper accounts non-linearity as well as the noise contribution at the output of the multiplier and noise in the resistor of a low pass filter. Subsequently, the PLL stochastic differential equation of this paper is recast by utilizing the Itô stochastic interpretation in contrast to the white noise interpretation. Importantly, the Itô interpretation is a formal stochastic interpretation, on the other hand, the white noise is an informal interpretation. The noise analysis of the PLL system is accomplished using the Kolmogorov forward and backward equations for the Markov process. As a result of this, we arrive at a set of two estimation evolution equations for the stochastic problem of concern here. This paper will be of interest to research communities in systems and control looking for noise analysis of dynamical systems in formal stochastic interpretations.