© 1991 by Institute of Mathematics and its Applications
Local–global double algebras for slow H
adaptation; the case of l2 disturbances
Department of Electrical and Computer Engineering, Wayne State University 5050 Anthony Wayne Drive, Detroit, Michigan 48202, U.S.A.
Department of Electrical Engineering, McGill University 3480 University street, Montreal, Quebec, Canada, H3A 2A7
In this paper, a common algebraic framework is described 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 l2 (Z)R.
The first part of the paper elaborates the double-algebra concept for Volterra operators which approximately commute with the shift. The main algebraic properties and norm inequalities are summarized. Local conditions for global inveribility are obtained. Classical frozen-time stability conditions are incorporated in relations between local and global spectra. The paper continues by establishing an explicit formula linking global and local sensitivity for systems with stable plants, where local sensitivity is a Lipschitz continuous function of data. Frequency-domain estimates of time-domain sensitivity norms. which become accurate as rates of time-variation approach zero, are obtained. Notions of adaptive versus nonadaptive (robust) control are introduced. It is shown that adaptive control can achieve better sensitivity than optimal nonadaptive control. It is demonstrated by an example that, in general, H-optimal interpolants do not depend Lipschitz-continuously on data. However, 
-suboptimal interpolants of the AAK central (maximal entropy) type are shown to satisfy a tractable Lipschitz condition.