# Approximation of complex µ

• Minyue Fu
Chapter
Part of the Communications and Control Engineering book series (CCE)

## Abstract

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:
$${\mu _\Delta }\left( M \right) = \inf \,\left\{ {\rho :\rho > 0,\det \left( {{I_n} - {\rho ^{ - 1}}\Delta M} \right) \ne 0,\forall \Delta \in B\left( \Delta \right)} \right\}$$
(22.1)
where
$$B\left( \Delta \right) = \,\left\{ {\Delta = diag\left. {\left\{ {{\Delta _1}, \cdots ,{\Delta _m}} \right\}} \right|\left. {{\Delta _i} \in {C^{{k_i} \times {k_i}}},} \right\|\left. {{\Delta _i}} \right\| \leqslant 1} \right\}$$
(22.2)
Let $$\hat{\mu }$$ be an approximation of µ. We call $$\hat{\mu }$$ an r-approximation, r > 0, if either
$$\mu \leqslant \hat \mu \leqslant \left( {1 + r} \right)\mu$$
(22.3)
or
$$\frac{\mu }{{1 + r}} \leqslant \hat \mu \leqslant \mu$$
(22.4)
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.

## Keywords

Computational Complexity Polynomial Algorithm Multidimensional System Interval Matrix Computational Complexity Analysis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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