Shift and delta operator realisations for digital controllers with finite word length considerations

The implementation issues of digital controllers with finite word length (FWL) considerations are addressed. Both the shift and delta operator parametrisations of a general controller structure are considered. A unified formulation is adopted to derive a computationally tractable stability related measure that describes FWL closed-loop stability characteristics of different controller realisations. Within a given operator parametrisation, the optimal FWL controller realisation, which maximises the proposed stability related measure, is the solution of a nonlinear optimisation problem. The relationship between the z-operator and δ-operator controller parametrisations is analysed, and it is shown that the δ parametrisation has a better FWL closed-loop stability margin than the z-domain approach under a mild condition. A design example is included to verify the theoretical analysis and to illustrate the proposed optimisation procedure