Local-global double algebras for slow H{infty} adaptation; the case of l2 disturbances

In this paper, a common algebraic framework is described forthe frozen-time analysis of stability and H optimization inslowly time-varying systems, based on the notion of a normedalgebra on which local and global products are defined. Relationsbetween local stability, local (near) optimality. local coprimefactorization, and global versions of these properties are sought.The framework is valid for time-domain disturbances in l² (Z)^R. The first part of the paper elaborates the double-algebra conceptfor Volterra operators which approximately commute with theshift. The main algebraic properties and norm inequalities aresummarized. Local conditions for global inveribility are obtained.Classical frozen-time stability conditions are incorporatedin relations between local and global spectra. The paper continuesby establishing an explicit formula linking global and localsensitivity for systems with stable plants, where local sensitivityis a Lipschitz continuous function of data. Frequency-domainestimates of time-domain sensitivity norms. which become accurateas rates of time-variation approach zero, are obtained. Notionsof adaptive versus nonadaptive (robust) control are introduced.It is shown that adaptive control can achieve better sensitivitythan optimal nonadaptive control. It is demonstrated by an examplethat, in general, H-optimal interpolants do not depend Lipschitz-continuouslyon data. However, -suboptimal interpolants of the AAK central(maximal entropy) type are shown to satisfy a tractable Lipschitzcondition.