Optimal Controller Realisations with the Smallest Dynamic Range

An approach is proposed to design optimal finite word length (FWL) realisations of digital controllers implemented in fixed-point arithmetic. A minimax-based search procedure is first used to obtain an optimal controller realisation that optimises 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 realisation may not have the smallest dynamic range. A measure is derived to indicate the dynamic range of a realisation. Based on an orthogonal transformation of this dynamic range measure for the optimal precision controller realisation, a numerical optimisation method is developed to make the controller realisation having the smallest dynamic range without sacrificing FWL closed-loop stability robustness.