Optimal realizations of fixed-point implemented digital controllers with the smallest dynamic range

Abstract

A novel approach is proposed to design optimal finite word length (FWL) realizations of digital controllers implemented in fixed-point arithmetic. A minimax-based search procedure is first formulated to obtain an optimal controller realization that optimizes an FWL closed-loop stability measure. Since this FWL closed-loop stability measure is solely linked to the fractional part or precision of fixed-point format, the resulting realization may not have the smallest dynamic range. A measure is then derived to indicate the dynamic range of fixed-point implemented realization. By choosing an appropriate orthogonal transformation of this dynamic range measure of the optimal precision controller realization, a numerical optimization method is developed to make the controller realization having the smallest dynamic range without sacrificing FWL closed-loop stability robustness. The proposed approach is more efficient than a direct optimization of some combined FWL closed-loop stability and dynamic range measure via a numerical means. The proposed approach is established within a unified framework that includes both the shift and delta operator parameterizations, which makes it possible to compare the closed-loop stability characteristics of the optimal FWL controller realizations using shift and delta operators, respectively. Through analysing the simulation results of a design example, some useful insights and understandings are obtained regarding the FWL controller realizations based on shift and delta operators.

Acknowledgements

J. Wu and J. Chu wish to thank the support of the National Natural Science Foundation of China (Grants No. 60374002 and No. 60421002), 973 program of China (Grant No. 2002CB312200) and program for New Century Excellent Talents in University (NCET-04-0547). S. Chen wish to thank the support of the United Kingdom Royal Academy of Engineering.