On validating closed-loop behaviour from noisy frequency-response measurements

Abstract

It is shown how noisy closed-loop frequency response measurements can be used to obtain pointwise in frequency bounds on the possible difference between an otherwise unknown closed-loop system and the closed-loop comprising a nominal model of the plant and a stabilising controller. To this end, the Vinnicombe's gap metric framework for robustness analysis plays a central role. Indeed, an optimisation problem and corresponding algorithm are proposed for estimating the chordal distance between the frequency responses of the nominal plant model and a plant that is consistent with the closed-loop data and a priori information, when projected onto the Riemann sphere. NotationLet C denote the field of complex numbers, C n×m the space of n × m matrices with complex entries, T := {z ∈ C : |z| = 1} the unit circle, and D ρ := {z ∈ C : |z| < ρ} the open disc of radius ρ > 0. The symbolD ρ is used to denote the closure of D ρ and for convenience, the sets D 1 andD 1 are denoted by D andD, respectively. Given ρ ≥ 1, let H ∞,ρ := {f : C →

System identification for achieving robust performance

“…Commonly, normalized coprime factorizations (McFarlane & Glover, 1990) are considered, since these have a close connection to robustness properties in the graph and (ν-) gap metric; see Georgiou and Smith (1990), de Callafon, Van den Hof and Bongers (1996) and Vinnicombe (2001). Specifically, related results that connect control performance and model validation based on these normalized coprime factorizations are reported in Steele and Vinnicombe (2001) and Date and Cantoni (2005). The dualYoula-Kučera structure (Anderson, 1998;de Callafon and Van den Hof, 1997;Douma & Van den Hof, 2005) further refines these coprime factorization-based model uncertainty structures by excluding candidate models that are not stabilized by the controller that is used during the identification experiment.…”