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On the metric complexity of continuous‐time systems
Author(s) -
Yi Wang Le,
Lin Lin
Publication year - 1996
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/(sici)1099-1239(199604)6:3<221::aid-rnc148>3.0.co;2-f
Subject(s) - mathematics , sobolev space , sampling (signal processing) , metric (unit) , nonuniform sampling , interval (graph theory) , metric space , nyquist–shannon sampling theorem , discrete mathematics , sobolev inequality , pure mathematics , mathematical analysis , algorithm , combinatorics , computer science , operations management , filter (signal processing) , quantization (signal processing) , economics , computer vision
In this paper, metric complexities of certain classes of continuous‐time systems are studied, using the time‐domain sampling approach and the concepts of Kolmogorov, Gel'fand and sampling n ‐widths for certain classes of Sobolev space. A sampling theorem is obtained which extends Shannon's sampling theorem to systems with possibly non‐band‐limited spectra. The theorem demonstrates that continuous‐time systems in certain Sobolev spaces can be approximately reconstructed causally from their sampled systems. The Kolmogorov, Gel'fand and sampling n ‐widths of various uncertainty sets in the Sobolev spaces are derived. The results show that the sampling approach is in fact asymptotically optimal, when the sampling interval is selected to minimize the loss of information in the sampling process, for the modelling of systems in such Sobolev spaces.