Value of measurement frequency and accuracy in control of a stochastic bistable system

Abstract

A bistable system with a desirable and an undesirable state exhibits transitions between the states because of stochastic actions on the system. Selecting the operational point of a bistable system is a tradeoff between expected performance of the system and risk of transition to an undesired state. Hence control of a bistable system must consider the probability density function of the state. Measurement about state reduces state uncertainty according to the Bayesian rule. The author analyses how the measurement frequency and uncertainty affect the tradeoff between performance and risk, and thus seek for optimal measurement design. The author discretises the time by Euler integrating the stochastic differential equation describing the bistable system over an interval much shorter than any deterministic time scale, and discretise state to describe the system as a Markov chain. Measurements are made infrequently – every 50–100 time steps – and their uncertainty is Gaussian. The tradeoff problem is formulated by maximising system performance subject to constraint on the probability of undesired transition. The tradeoff is then obtained by varying the constraining probability. Similar tradeoffs are obtained either by frequent but uncertain and less frequent but more accurate measurements. Our analysis is motivated by the control of bioreactors.