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 as the approaches limit of infinity) are often used for this since they are easier to analyze. When the noise effects in the physical system are small, one is tempted to do an asymptotic analysis (noise intensity approaches limit of 0) on the diffusion approximation, and use this for the desired estimates on the original system. Such a procedure does not work in general: the double limit bandwidth approaches limit of infinity, intensity approaches limit of 0 is not always justified. Under quite broad conditions on the noise processes, it is justified for the systems studied here. 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. (Reprints)