Robust stabilization—BIBO stability, distance notions and robustness optimization

Abstract

This paper studies robust stabilization of both linear shift-invariant causal systems in an l p setting and linear time-invariant causal continuous-time systems in an L p ( p = 1 or ∞) setting. Two key technical results in the paper establish the existence of l p and L p stable normalized coprime factorizations for discrete-time and continuous-time systems, respectively, which have coprime factorizations as l p and L p stable operators. Several distance measures for systems are then introduced including the graph metric, the ϱ function, the gap between the graphs of the systems, and the projection gap. It is shown that these distance measures lead to the weakest convergence notions for systems for which closed-loop stability is a robust property. The ϱ function can be computed using the Dahleh-Pearson theory for l 1 ( L 1) optimal control. Robustness optimization in a directed p function is shown to be closely related to robustness optimization for BIBO stable normalized coprime factor perturbations. This result connects the stability margin of Dahleh for coprime factor perturbations to the ϱ function. These considerations are further supported by a robustness result in terms of the projection gap.