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of generality—so, roughly speaking, it can be solved. [SD83,SD84] 
discusses this optimization problem. 

Notice that:

so you have the following from Equation 2-5:

This inequality is thought to be nearly an equality, so that the left side is a 
good engineering approximation to the right side. No theory supports this 
generally held belief, but no example is known where the left side is more 
than 15% larger than the right side. Equality can be shown to hold provided 
k

 ≤ 3—for example, if there are three or fewer uncertain transfer functions 

[Doy82]. 

The approximation equation of 

μ(M) (Equation 2-5) is an upper bound. This means 

that the stability margin calculated using this approximation is conservative, that is, less 
than the actual stability margin. This optimization problem itself can be difficult. Osborne 
[Osb60] and Safonov [Saf82] provide two methods for finding good suboptimal scalings 
for Equation 2-5. 

Both Osborne’s and Safonov’s Perron-Frobenius scalings usually have 
been found to be close to the optimum for the optimization problem 
equation. The resulting approximations,

are thought to be good engineering approximations to 

μ. 

optscale( )

 

provides an iterative optimization function based on the ellipsoid 
algorithm.

σ

M

( )

σ DMD

1

–

(

)

=

1

=

σ

M

( ) μ M

( )

≥

uˆ

( ) σ

1

–

(

)

=

uˆ

( ) σ

1

–

(

)

=