Abstract
This short note is about the conservatism of the standard upper bound, \(\bar \mu \)relative to the complex structured singular value, μ.This problem is first formulated by John Doyle. To follow the present note, only mathematical definitions of μ, and \(\bar \mu \) are necessary. For a tutorial introduction about complex μ and its importance in system analysis and design, see [10]. If M∈ ℂn ×n, then
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References
Boyd, S., L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia: Society for Industrial and Applied Mathematics, 1994.
Braatz, R. D., P. M. Young, J. C. Doyle, and M. Morari, “Computational Complexity of μ Calculation,” IEEE Transactions on Automatic Control, 39 (1994) pp. 1000–1002.
Coxson, G. E., and C. L. DeMarco, “The computational complexity of approximating the minimal perturbation scaling to achieve instability in an interval matrix,” Math, of Control, Signals, and Systems, 7 (1994) pp. 279–292.
Fu, M., “The real structured singular value is hard to approximate,” to appear in IEEE Transactions on Automatic Control
Garey, M., and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, San Francisco: W. H. Freeman, 1979.
Megretski, A., “On the gap between structured singular values and their upper bounds,” Proc. 32nd Conference on Decision and Control, San Antonio, TX, Dec. 1993, pp. 3461–3462.
Megretski, A., “Necessary and sufficient conditions of stability: a multiloop generalization of the circle criterion,” IEEE Transactions on Automatic Control, 38 (1993) pp. 753–756.
Megretski, A., and S. Treil, “Power distribution inequalities in optimization and robustness of uncertain systems,” Journal of Math. Systems, Estimation, Control, 3 (1993) pp. 301–319.
Nemirovskii, A., “Several NV-hard problems arising in robust stability analysis,” Math, of Control, Signals, and Systems, 6 (1993) pp. 99–105.
Packard, A., and J. C. Doyle, “The Complex Structured Singular Value,” Automatica, 29 (1993) pp. 71–109.
Poolla, K., and A. Tikku, “Robust Performance Against Time-Varying Structured Perturbations,” IEEE Transactions on Automatic Control, 40 (1995) pp. 1589–1602.
Poljak, S., and J. Rolin, “Checking robust nonsingularity is NP-hard,” Math, of Control, Signals, and Systems, 6 (1993) pp. 1–9.
Shamma, J., “Robust Stability with Time Varying Structured Uncertainly,” IEEE Transactions on Automatic Control, 39 (1994) pp. 714–724.
Toker, O., and H. Ozbay, “On the Complexity of Purely Complex fi Computation and Related Problems in Multidimensional Systems,” IEEE Transactions on Automatic Control, 43 (1998) pp. 409–414.
Toker, O., “On the conservatism of the upper bound tests for structured singular value analysis,” Proc. 35th Conference on Decision and Control, Kobe, Japan, Dec. 1996, pp. 1295–1300.
Toker, O., and H. Ozbay, “Complexity issues in robust stability of linear delay-differential systems,” Math. of Control, Signals, and Systems, 9 (1996) pp. 386–400.
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Toker, O., de Jager, B. (1999). Conservatism of the standard upper bound test: Is sup(\({{\bar \mu } \mathord{\left/ {\vphantom {{\bar \mu } \mu }} \right. \kern-\nulldelimiterspace} \mu }\)) finite? Is it bounded by 2?. In: Blondel, V., Sontag, E.D., Vidyasagar, M., Willems, J.C. (eds) Open Problems in Mathematical Systems and Control Theory. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0807-8_43
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