On the small-μ theorem

✍ Scribed by AndréL. Tits; Michael K.H. Fan

• Book ID: 102640345
• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 304 KB
• Volume: 31
• Category: Article
• ISSN: 0005-1098
• DOI: 10.1016/0005-1098(95)00035-u

✦ Synopsis

respect to a block structure A is defined by p(M) = 0 if there is no A E A such that det (I + MA) = 0, and

otherwise (Doyle, 1982). A commonly quoted statement of the 'small-~ theorem' in the 'complex-p' case (i.e. R = 0) is that, given P E RIR"", (I + PA)-' E HZ"" for all realrational A E A such that . sup 5(A( jo)) % 1 wrrf

(1)

(2)

The small-p theorem (Doyle et al., 1982) is the foundation of the structured singular-value approach to robust control. It gives a necessary and sufficient condition on the 'p-norm' of a stable plant P(s) for the 'P-A' feedback loop to be stable for all real-rational stable proper structured uncertainty A of a given maximum size (K-norm). In this note we first show that a commonly quoted statement of this theorem is if and only if incorrect. Specifically, we exhibit plants P with unit 'p-norm' that cannot be destabilized by any real-rational stable proper sup &P(jw))< I. structured uncertainty of size one. We then provide a WE&! sup cL(P(jo)) = AWO)) = 1.