By Macri M., Nolasco M., Ricciardi T.
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Additional info for A shadowing lemma for abelian Higgs vortices
C, are matrices of suitable dimensions with entries in R and h ∈ R+ is a given delay. Introducing the delay operator δ, which is deﬁned for any time function f (t) by δf (t) = f (t − h), we can write Σd = x˙ (t) = y(t) = a i i=0 Ai δ x (t) + c i C δ i=0 i x (t). b i=0 Bi δ i u(t) (2) Formally, it is possible to substitute the delay operator δ with the algebraic indeterminate ∆, so that, denoting by A, B, C the matrices with entries in R = R[∆] given, respectively, by a b Ai ∆ i , A= i=0 c Bi ∆i , B= Ci ∆i , and C = i=0 i=0 we can associate to Σd the system Σ over the ring R = R[∆] of real polynomials in one indeterminate, deﬁned by the set of equations Σ= x(t + 1) = Ax(t) + Bu(t) y(t) = Cx(t).
Robustness had become a central issue in system and control theory, focusing the researchers’ attention from the study of a single model to the investigation of a set of models, described by a set of perturbations of a “nominal” model. This set, often indicated as the uncertainty model set, has to be suitably constructed to describe the inherent uncertainty about the system under consideration and to be used for analysis and design purposes. H∞ identiﬁcation methods deliver uncertainty model sets in a form suitable to be used by well-established robust design techniques, based on H∞ or µ optimization methods.
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