Local-global double algebras for slow H∞ adaptation. I. Inversion and stability

For Pt.II see ibid., vol.36, no.2, p.143-51 (1991). In this two-part work, a common algebraic framework is introduced for the frozen-time analysis of stability and H^∞ optimization in slowly time-varying systems, based on the notion of a normed algebra on which local and global products are defined. Relations between local stability, local (near) optimality, local coprime factorization, and global versions of these properties are sought. The framework is valid for time-domain disturbances in l^∞. H^∞ behavior is related to l^∞ input-output behavior via the device of an approximate isometry between frequency and time-domain norms. The authors presently elaborate on the double-algebra concept for Volterra operators which approximately commute with the shift, and summarize the main algebraic properties and norm inequalities. Local conditions for global invertibility are also obtained. Classical frozen-time stability conditions are incorporated in relations between local and global spectra