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.


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


[1] Balas, G. J.; Doyle, J. C.; Glover, K.; Packard, A.; Smith, R., (μ-Analysis and Synthesis Toolbox (1991), MUSYN, Inc. and Math Works, Inc)
[2] Boyd, S.; Desoer, C. A., Subharmonic functions and performance bounds on linear time-invariant feedback systems, IMA J. Math. Control Inf., 2, 153-170 (1985)
[3] Doyle, J. C., Analysis of feedback systems with structured uncertainties, (Proc. IEE, PtD, 129 (1982)), 242-250
[4] Doyle, J. C.; Wall, J. E.; Stein, G., Performance and robustness analysis for structured uncertainty, (Proc. 21st IEEE Conf. on Decision and Control. Proc. 21st IEEE Conf. on Decision and Control, Orlando, FL (1982)), 629-636
[5] Khargonekar, P. P.; Georgiou, T. T.; Pascoal, A. M., On the roubst stabilization of time-invariant plants with unstructured uncertainty, IEEE Trans. Autom. Control, AC-32, 201-207 (1987) · Zbl 0616.93062
[6] Kishore, A. P.; Pearson, J. B., Uniform stability and performance in \(H_∞\), (Proc. 31st IEEE Conf. on Decision and Control. Proc. 31st IEEE Conf. on Decision and Control, Tuscon, AZ (1992)), 1991-1996 · Zbl 1101.05040
[7] Parkard, A.; Doyle, J. C., The complex structured singular value, Automatica, 29, 71-109 (1993) · Zbl 0772.93023
[8] Packard, A.; Pandey, P., Continuity properties of the real/complex structured singular value, IEEE Trans. Autom. Control, AC-38, 415-428 (1993) · Zbl 0791.93023
[9] Tits, A. L., The small μ theorem without real-rational assumption, (Proc. 1995 American Control Conf.. Proc. 1995 American Control Conf., Seattle, WA (1995)), to appear
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.