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.


Computational Complexity Polynomial Algorithm Multidimensional System Interval Matrix Computational Complexity Analysis 
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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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