Guaranteed robust stability of the closed-loop systems for digital controller implementations via orthogonal Hermitian transform

In this paper, an approach for robust stability analysis of a digital closed-loop system for digital controller implementations subject to finite word length (FWL) effects is proposed. Uncertainties caused by the roundoff and computational errors subject to FWL effects are expressed in function of mantissa bit number when the mode of floating-point arithmetic is used in the process. Then, based on the Small Gain Theorem and the Bellman-Grownwall Lemma, a sufficient stability criterion for the digital closed-loop system is derived. The eigenvalue sensitivity of the closed-loop system is developed in terms of mixed matrix-2/Frobenius norms. Then, by minimizing this eigenvalue sensitivity and using orthogonal Hermitian transform as well, an optimal similarity transformation can be obtained. By substituting this optimal transformation into the stability criterion, a minimum mantissa bit number used for implementing the stabilizing digital controllers can be determined. The main contributions are that this approach provides an analytical closed-form solution for obtaining the optimal transformation and, in addition to the stability criterion, leads to the implementation of the stabilizing controllers with a lower mantissa bit number when using this optimal one. Finally, detailed numerical design processes and simulation results are used to illustrate the effectiveness of the proposed scheme.