Generalizations and new proof of the discrete-time positive reallemma and bounded real lemma
Chengshan Xiao
Hill, D.J.
Wireless Networks, Nortel, Nepean, Ont.;
Abstract
There are three different restatements claimed to be equivalent to
the definition of discrete-time positive realness (DTPR) in the
literature. These restatements were obtained by assuming that they are
similar to the results of continuous-time positive realness when the
transfer function has poles on the stability boundary. In this paper it
is shown that only one of them is equivalent to the DTPR lemma and
others are disproved by counter-examples. Furthermore, the DTPR lemma is
specialized for minimal systems which have all poles on the unit cycle,
the DTPR lemma is also generalized for nonminimal systems, the
discrete-time bounded real (DTBR) lemma is proven by a simple method,
and then the DTBR lemma is extended to the nonminimal case.
Continuous-time results are also briefly considered in the
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