Improved closed-loop stability for fixed-point controller implementation using the delta operator

A stable discrete-time control system may achieve a lower than predicted performance or even become unstable when the discrete-time control law is implemented with a fixed-point digital control processor due to the finite word length (FWL) effects, which depend on the control law state-space realization and the discrete-time operator (e.g., the delta operator or the forward-shift operator) used to represent the control laws. To improve the closed-loop stability (and as a byproduct, performance) when the control law is implemented, a state-space approach that selects the control law realization to optimize a stability-related objective function is developed using the delta operator. Analytical and numerical comparison of the fixed-point performance of delta control laws with the performance of the corresponding forward-shift control laws quantifies the improved closed-loop stability of the delta realizations over those of the corresponding forward-shift realizations. It is also shown that there exists a simple mapping between the optimal FWL forward-shift control law realizations and the optimal delta control law realizations. The results are illustrated by the delta and forward-shift control law realizations of a discrete-time H_∞ control law designed for a teleoperation motion-scaling system