Improved Closed-loop Stability with Fixed-point Control Law Implementation Using the Delta Operator

A stable discrete-time control system may become unstable when the discrete-time control law is implemented with a xed-point digital control processor (DCP) due to the nite word length (FWL) eeects which depend on the control law structures and the discrete-time representations (e.g., delta operator or forward-shift representations) used to represent the control laws. To improve closed-loop stability behavior when the control law is implemented, a state-space approach that optimizes a stability-related objective function is developed using the delta operator. Analytical and numerical comparison of the xed-point performance of delta operator control laws with the performance of the analogous forward-shift control laws quantiies the improved performance of the delta operator realizations over those of the corresponding forward-shift realizations. It is also shown that there exists a simple mapping between the optimal forward-shift control law realizations and the optimal delta operator control law realizations. The results are illustrated by the delta operator and forward-shift control law realizations of a discrete-time H 1 control law designed for a teleoperation motion-scaling system.