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Figure 2-10.  Reduction to Separate Systems

In terms of the approximations to the margin discussed above, this 
reducibility will manifest itself as a problem such as divide-by-zero or 
nontermination. It really means that the minimum of the optimization 
problem is not achieved by any finite scaling.

A matrix M can be split into its reducible components using the following 
technique (refer to[BeP79]): 

1.

Form the matrix X = (

αI + M) – 1

 

for any

 

α larger than the spectral 

radius of M, for example 2

.

2.

Form Y = X + X

 where Y has a positive i,j entry if and only if 

δ

 and 

δ

 

are in the same reduced system; otherwise, the entries will be zero.

ssv( )

 checks for reducibility before invoking a scaling algorithm. The 

margins of each of the reduced systems then can be calculated separately, 
and the minimum taken.

In [DWS82], it is shown that a simple relation holds between the 
worst-case gain defined in Equation 2-3, and the stability margin. For 

γ > 0,

wcgain (jw) £ g if and only if m(Hred(jw) diag(g-1, 

l1(w),...,lk(w)) £ 1

where H

 is H with the rows and columns corresponding to all inputs and 

outputs deleted except the ones of interest (the qth input and the pth output). 
This can be interpreted as adding a fictitious uncertain transfer function 
from w

 to z

 with magnitude bound 

γ

–1

 at the given frequency. This 

additional uncertainty is called a performance loop as described in 
reference [BoB91].

δ

4

δ

2

δ

1