Local–global double algebras for slow H∞ adaptation; the case of l2 disturbances

In this two-part paper, 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, lotal coprime factorization, and global versions of these properties are sought. The framework is valid for time-domain disturbances in 1". H"-be- havior is related to 1" input-output behavior via the device of an approximate isometry between frequency and time-domain norms. Part I elaborates the double-algebra concept for Volterra operators which approximately commute with the shift. The main algebraic prop- erties and norm inequalities are summarized. Local conditions for global invertibility are obtained. Clalssical frozen-time stability conditions are incorporated in relations between local and global spectra.

[1]  P. Khargonekar,et al.  Polynomial matrix-fraction representations for linear time-varying systems , 1986 .

[2]  Tryphon T. Georgiou,et al.  On the robust stabilizability of uncertain linear time-invariant plants using nonlinear time-varying controllers , 1987, Autom..

[3]  George Zames,et al.  Logarithmic Variation Criteria for the Stability of Systems with Time-Varying Gains , 1968 .

[4]  P. Duren.  Theory of H[p] spaces , 1970 .

[5]  C. Desoer.  Slowly varying discrete system xi+1=Aixi , 1970 .

[6]  W. Arveson.  Interpolation problems in nest algebras , 1975 .

[7]  B. S. Razumikhin.  Optimal Control Problems , 1984 .

[8]  D.L. Elliott,et al.  Feedback systems: Input-output properties , 1976, Proceedings of the IEEE.

[9]  Kumpati S. Narendra,et al.  Stability of nonlinear time-varying feedback systems , 1968, Autom..

[10]  Hans Schneider,et al.  Algebraic and topological aspects of feedback stabilization , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[11]  Malcolm C. Smith.  Well-posedness of H ∞ optimal control problems , 1990 .

[12]  M. Kreĭn,et al.  In nite Hankel block matrices and related extension problems , 1978 .

[13]  George Zames,et al.  H∞ optimization and slowly time varying systems , 1987, 26th IEEE Conference on Decision and Control.

[14]  李幼升,et al.  Ph , 1989 .

[15]  C. Desoer,et al.  Feedback system design: The fractional representation approach to analysis and synthesis , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[16]  Bruce A. Francis,et al.  Uniformly optimal control of linear feedback systems , 1985, Autom..

[17]  G. Zames.  Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses , 1981 .

[18]  Munther A. Dahleh,et al.  On slowly time-varying systems , 1991, Autom..