Given a matrix M ∈ С n×n and a set of positive integers χ = (k 1,⋯,k m ) with k 1+⋯+k m = n, the so-called complex structured singular value µχ(M), (complex µ for short), is defined as follows:
$$ {\mu _\Delta }\left( M \right) = \inf \,\left\{ {\rho :\rho > 0,\det \left( {{I_n} - {\rho ^{ - 1}}\Delta M} \right) \ne 0,\forall \Delta \in B\left( \Delta \right)} \right\} $$
$$ B\left( \Delta \right) = \,\left\{ {\Delta = diag\left. {\left\{ {{\Delta _1}, \cdots ,{\Delta _m}} \right\}} \right|\left. {{\Delta _i} \in {C^{{k_i} \times {k_i}}},} \right\|\left. {{\Delta _i}} \right\| \leqslant 1} \right\} $$
Let \(\hat{\mu }\) be an approximation of µ. We call \(\hat{\mu }\) an r-approximation, r > 0, if either
$$ \mu \leqslant \hat \mu \leqslant \left( {1 + r} \right)\mu $$
$$ \frac{\mu }{{1 + r}} \leqslant \hat \mu \leqslant \mu $$
Note that \(\hat{\mu }\) {a_o} \approx {c_o} \approx \sqrt {2{a_c}} ,{b_o} \approx {a_c} is an upper bound in the former case and an lower bound in the latter.




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  1. [1]
    J. C. Doyle, “Analysis of Feedback Systems with Structured Uncertainties,” Proc. IEE, pt. D, vol. 129, pp. 240–250, 1982.MathSciNetGoogle Scholar
  2. [2]
    Megretski, A., “On the gap between structured singular values and their upper bounds,” Proc. 32nd Conf. Decision and Control, San Antonio, TX, December, 1993.Google Scholar
  3. [3]
    G. E. Coxson and C. L. DeMarco, “The computational complexity of approximating the minimal perturbation scaling to achieve instability in an interval matrix,” Mathematics of Control, Signals, and Systems, 7 (No. 4), pp. 279–292, 1994.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    O. Toker and H. Ozbay, “On the Complexity of Purely Complex µ Computation and Related Problems in Multidimensional Systems,” IEEE Trans. Auto. Contr., vol. 43, no. 3, pp. 409–414, 1998.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    O. Toker, “On the conservatism of upper bound tests for structured singular value analysis,” Proc. 35th Conference on Decision and Control, Kobe, Japan, Dec. 1996, pp. 1295–1300.Google Scholar
  6. [6]
    M. Fu, “The real structured singular value is hardly approximate,” IEEE Trans. Auto. Contr., vol. 42, no. 9, pp. 1286–1288, 1997.MATHCrossRefGoogle Scholar
  7. [7]
    M. Fu and S. Dasgupta, “Computational complexity of real structured singular value in p-norm Setting,” to appear in IEEE Trans. Auto. Contr.Google Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Minyue Fu
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of NewcastleNewcastleAustralia

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