Global optimal realizations of finite precision digital controllers

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 all its 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. A computationally simple algorithm is presented.