A Model for Optimum Control of Stochastic Sampled-Data Systems

A model is developed for a sampled-data feedback control system for processes having randomly-distributed outputs or involving measurement techniques that introduce a significant component of variance in the measured values of the controlled variable. The principal development concerns a simple proportional control system with fixed proportionality constant. An acceptable range for the constant is determined from stability criteria for the case of a normally-distributed output with constant process variance and is observed to coincide with well-known servo-mechanism design principles. The method of derivation involves successive operations on the normal probability density function. The effects on system response (time constant), oscillatory tendencies, and system variance of various values of the constant (within the acceptable range) are discussed. Finally, two classes of criterion functions are mentioned, from which optimum values of the proportionality constant can be determined. The first involves fixed tolerances on individual items of output, and the second treats costs that are functions of the deviation between desired and actual outputs.