A unified closed-loop stability measure for finite-precision digital controller realizations implemented in different representation schemes

Abstract

The closed-loop stability issue of finite-precision realizations is investigated for digital controllers implemented in three different arithmetic formats, namely fixed-point, floating-point and block-floating-point schemes. It is shown that the controller coefficient perturbations resulting from using different finite word length (FWL) representation schemes possess quite different properties. A unified FWL closed-loop stability measure is derived which is applicable to all the three arithmetic schemes. Unlike the existing works which only take into account the precision of a representation scheme with an assumption of an unlimited dynamic range, both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure. To facilitate the design of optimal finite-precision controller realizations, a computationally tractable FWL closed-loop stability measure is then introduced and the method of computing the value of this measure for a given controller realization is given. For each arithmetic scheme, the optimal controller realization is defined as the solution that maximizes the corresponding measure, and a numerical optimization approach is adopted to solve for the resulting optimal realization problem. The proposed design procedure provides a unified framework for true optimal controller implementation that requires a minimum bit length with maximum robustness to the FWL effect. Numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes.

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