Optimal controller and filter realizations using finite-precision, floating-point arithmetic

Abstract

The problem of reducing the fragility of digital controllers and filters implemented using finite-precision, floating-point arithmetic is considered. Floating-point arithmetic parameter uncertainty is multiplicative, unlike parameter uncertainty resulting from fixed-point arithmetic. Based on first-order eigenvalue sensitivity analysis, an upper bound on the eigenvalue perturbations is derived. Consequently, open-loop and closed-loop eigenvalue sensitivity measures are proposed. These measures are dependent upon the filter/controller realization. Problems of obtaining the optimal realization with respect to both the open-loop and the closed-loop eigenvalue sensitivity measures are posed. The problem for the open-loop case is completely solved. Solutions for the closed-loop case are obtained using non-linear programming. The problems are illustrated with a numerical example.

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

“…Denote the set of integers e e e as Z [e;e] . Assuming that no underflow or overflow occurs, which means that the exponent of x is within Z [e;e] , x is perturbed to [7] Q 2 (x) = x + x 2 j 2 j < 2 0( +1) :…”

“…First, 1 (w; ) can be used in different representation formats while the existing measures are only valid for a particular format. For example, the measures presented in [3]- [6] are fixed-point measures and the measure in [7] is a floating-point one. The measure 1(w; ) offers a unified framework to compare the FWL characteristics of a realization w in different formats.…”

“…For an illustrative purpose, we consider the case of = 1 (fixedpoint format) with F, G, J and M being nonzero matrices and H = 0 in (7). Denote the controller realization…”