Notes on Fading-Memory Conditions

AbstractIn some studies concerning the approximation of nonlinear systems, the concept of a weighting function w plays a central role. This paper, in which attention is focused on continuous-time cases, directs attention to some interesting properties of systems that have R+ fading memory or fading memory. In particular, we show that half-line input-output maps that have R+ fading memory with respect to some w in fact have R+ fading memory with respect to all such w’s. And we show that a similar proposition holds in a setting concerning Volterra-series approximations for continuous-time systems with inputs and outputs defined on all of R. We show also that, in that setting, fading memory is equivalent to uniform fading memory. Some related results are also described.

[1]  I. Sandberg A canonical form for discrete-time systems defined overZ+* , 1999 .

[2]  Bernard D. Coleman,et al.  Norms and semi-groups in the theory of fading memory , 1966 .

[3]  I. Sandberg Separation conditions and approximation of discrete-time and discrete-space systems , 1998 .

[4]  Irwin W. Sandberg,et al.  Structure theorems for nonlinear systems , 1991, Multidimens. Syst. Signal Process..

[5]  M. Stone The Generalized Weierstrass Approximation Theorem , 1948 .

[6]  Irwin W. Sandberg,et al.  Uniform approximation of discrete-time nonlinear systems , 1999, IJCNN'99. International Joint Conference on Neural Networks. Proceedings (Cat. No.99CH36339).

[7]  Irwin W. Sandberg,et al.  Z+ fading memory and extensions of input–output maps , 2001, Int. J. Circuit Theory Appl..

[8]  I. Sandberg The mathematical foundations of associated expansions for mildly nonlinear systems , 1983 .

[9]  Irwin W. Sandberg,et al.  A‐map representations and asymptotically almost periodic responses , 2001, Int. J. Circuit Theory Appl..

[10]  Irwin W. Sandberg,et al.  Uniform approximation of multidimensional myopic maps , 1997 .

[11]  I. Sandberg,et al.  Criteria for the approximation of nonlinear systems , 1992 .

[12]  B. D. Coleman,et al.  An approximation theorem for functionals, with applications in continuum mechanics , 1960 .

[13]  An adaptive nonlinear filter structure , 1991, 1991., IEEE International Sympoisum on Circuits and Systems.

[14]  Bernard D. Coleman,et al.  On the general theory of fading memory , 1968 .

[15]  I. P. Natanson,et al.  Theory of Functions of a Real Variable , 1955 .

[16]  Leon O. Chua,et al.  Fading memory and the problem of approximating nonlinear operators with volterra series , 1985 .

[17]  Approximately-finite memory and discrete-time input-output maps , 2000 .