Global optimal realizations of finite precision digital controllers

Abstract

The paper analyzes global solutions to the optimal digital controller realization problem based on maximizing a finite word length (FWL) closed-loop stability measure. For each closed-loop eigenvalue, a single-pole FWL stability function is first introduced, and a single-pole FWL stability measure is then defined as the maximum of the corresponding single-pole stability function over all the controller realizations. It is shown that the minimum of the single-pole stability measures for all the closed-loop eigenvalues is an upper bound of the optimal value for the optimal realization problem. An analytical method to compute a single-pole stability measure is developed, and an expression for all the realizations which achieve a given single-pole measure is derived. When a realization, which is a solution of the minimum single-pole measure, further satisfies the condition that the values of its all the single-pole stability functions are not less than the minimum single-pole measure, the minimum single-pole measure is the optimal value of the optimal realization problem and this realization is the solution for the optimal realization problem. An algorithm is presented to compute an optimal FWL controller realization. Unlike most of the existing methods relying on numerical optimization search algorithms, which can be computationally expensive and may easily be trapped at local optimal solutions, the proposed analytical approach guarantees to find a global optimal controller realization.

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