Notes on \(\mu \) and \(\ell _1\) robustness tests. (English) Zbl 1274.93218

Summary: An upper bound for the complex structured singular value related to a linear time-invariant system over all frequencies is given. It is in the form of the spectral radius of the \(\mathcal {H}_{\infty }\)-norm matrix of SISO input-output channels of the system when uncertainty blocks are SISO. In the case of MIMO uncertainty blocks; the upper bound is the \(\infty \)-norm of a special non-negative matrix derived from \(\mathcal {H}_{\infty }\)-norms of SISO channels of the system. The upper bound is fit into the inequality relation between the results of \(\mu \) and \(\ell _1\) robustness tests.


93D09 Robust stability
93C35 Multivariable systems, multidimensional control systems
Full Text: Link


[1] Dahleh M. A., Khammash M.: Controller design for plants with structured uncertainty. Automatica 29 (1993), 37-56 · Zbl 0772.93028 · doi:10.1016/0005-1098(93)90173-Q
[2] Dahleh M. A., Diaz-Bobillo I. J.: Control of Uncertain Systems. A Linear Programming Approach. Prentice Hall, NJ 1995 · Zbl 0838.93007
[3] Doyle J. C.: Structured uncertainty in control system design. Proc. of 24th Conference on Decision and Control, Ft. Lauderdale FL 1985, pp. 260-265
[4] Fiedler M.: Special Matrices and Their Application in Numerical Mathematics. SNTL - Nakladatelství technické literatury, Prague 1981 · Zbl 0531.65008
[5] Khammash M. H., Pearson J. B.: Performance robustness of discrete-time systems with structured uncertainty. IEEE Trans. Automat. Control 36 (1991), 398-412 · Zbl 0754.93063 · doi:10.1109/9.75099
[6] Khammash M. H., Pearson J. B.: Analysis and design for robust performance with structured uncertainty. Systems Control Lett. 20 (1993), 179-187 · Zbl 0768.93065 · doi:10.1016/0167-6911(93)90059-F
[7] Khammash M. H.: Necessary and sufficient conditions for the robustness of time-varying systems with applications to sampled-data systems. IEEE Trans. Automat. Control 38 (1993), 49-57 · Zbl 0777.93018 · doi:10.1109/9.186311
[8] Packard A., Doyle J. C.: The complex structured singular value. Automatica 29 (1993), 71-109 · Zbl 0772.93023 · doi:10.1016/0005-1098(93)90175-S
[9] Tits A. L., Fan M. K. H.: On the small-\(\mu \) theorem. Automatica 31 (1995), 1199-1201 · Zbl 0831.93021 · doi:10.1016/0005-1098(95)00035-U
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.