Sensitivity minimization for controller implementations: fixed-point approach

A novel approach is proposed to analyze and minimize fixed-point errors for digital controller implementations based on sensitivity measures in magnitude and phase-supplement-angle of eigenvalues. First, uncertainties of the controller parameters caused by roundoff and computational errors using fixed-point computations are expressed in function of register length. Then, a stability criterion of the closed-loop system based on fixed-point statistical model is derived by means of small gain theorem and Bellman-Grownwall lemma. Thus, a measure that combines sensitivities of the magnitude and phase-supplement-angle of the closed-loop system eigenvalues with respect to controller parameters is constructed in the sense of mixed matrix-2/Frobenius norms. This measure is minimized by an optimal similarity transformation obtained from an analytically algebraic method. Based upon this transformation as well as the stability criterion, a least register length can be obtained. Finally, an example of the simplified model of a vertical take off and landing (VTOL) aircraft is performed to illustrate the effectiveness of the proposed scheme.