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On the small-\(\mu\) theorem. (English) Zbl 0831.93021

Summary: Counterexamples to a commonly quoted statement of the small-\(\mu\) theorem are exhibited. We exhibit plants \(P\) with unit ‘\(\mu\)-norm’ that cannot be destabilized by any real-rational stable proper structured uncertainty of size one. A detailed proof of a correct statement is provided. It includes the case where the uncertainty is not constrained to be real- rational.

MSC:

93B35 Sensitivity (robustness)
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93B36 \(H^\infty\)-control
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