3.7. STRUCTURED SINGULAR VALUE(µ) ANALYSIS AND SYNTHESIS 107
3.7 Structured Singular Value (µ) Analysis andSynthesis
This section covers the functions used in the D-Kiterationprocedure. Theprimary
functions are the calculationof the controller (already discussed), the calculation of µ
and the fitting of rational Dscales. Severalsubroutines used for Dscale fitting are
useful in their own regard and are also discussed here. The discussion below assumes
that the reader if familiar with the definition and use of µ; only the implementation
issues are covered here. A simple application example is given here. To get a better idea
of the standard approaches to µand the D-Kiteration, the reader should refer to the
demos in Chapter 4. To find out more about the theoretical aspects of µ, including its
application to robust stability and robust performance problems, refer to Section 2.4.

3.7.1 Calculation of µ

The function mu calculates the structured singular value. µis is defined as a function of
a matrix and a specified block structure. The most common use involves calculating µat
each frequency of a system frequency response. For this reason the µfunction accepts
pdm input arguments as well as matrices. The block structure is specified to the
function in a coded form.
In general only upper and lower bounds for µare calculated. Theupper bound (for the
square ∆ block case) is given by,
µ(M)≤inf
D∈Dσmax DMD−1,
where Dis the set of all matrices which commutewith the p erturbation,∆. Recallthat
the lower bound is given by,
max
Q∈Qρ(MQ)≤µ(M),
where ρdenotes the spectral radius and Qis the set of all unitary perturbations of the
specified block structure. The matrix Qachieving this maximization is a destabilizing ∆.