Stochastic Systems with Small Noise, Analysis and Simulation: A Phase Locked Loop Example

摘要

Systems with wide bandwidth noise inputs are a common occurrence in stochastic control and communication theory and elsewhere; e.g., tracking or synchronization systems such as phase locked loops (PLL). One is often interested in calculating such quantities as the probability of escape from a desired error set, in some time interval, or the mean time for such escape. Diffusion approximations (the system obtained in the limit bandwidth approaches infinity) are often used for this, being easier to analyze. We study a particular form of the PLL owing to the great practical importance of the system and because it provides a useful vehicle for understanding the extent of validity of the asymptotic methods for such systems. The basic analytical techniques are from the theory of large deviations. One seeks information on the escape probabilities, mean times, and on the most likely exit paths and exit locations. Also, we seek information on the interactions between the signals to be tracked and the noise which are most likely to lead to exit. The large deviations technique is eminently suited to this job. Simulations are taken in order to understand the range of validity of the asymptotic method. Agreement between the predictions and sample estimates is good over noise intensity levels which seem to be ever larger than those typically occurring in practice.