50 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
and two block structures, ∆1(compatible with M11)and∆
2(compatible with M22).
There are twop erturbed subsystems that wecan study here: Fu(M,∆1),where ∆1is
closed in a feedback loop around M11;andF
l
(M, ∆2), where ∆2is closed in a feedback
loop around M22.
We havealready seen that in the case of a dynamic system, the robust stability of
Fu(M,∆1) is analyzed by checking that µ1(M11)<1. Herewe have used µ1to indicate
that we are considering it with respect to the block structure ∆1. In the constant
matrix case, we say that the LFT, Fu(M,∆1) is well posed for all ∆1∈B∆1if and only
if µ1(M11)<1. This simply means that the inverse in the LFT equations is well defined
for all ∆1∈B∆1.
The well posedness discussion above applies equally well to Fl(M,∆2) and we will
denote the µtest for M22 by µ2(M22). However,instead of looking at µ2of M22 ,we
want to look at µ2(Fu(M,∆1)). Note that Fu(M,∆1) has the same dimensions as M22
and in fact Fu(M,∆1)=M
22 when ∆1= 0. In otherwords, what happens when we
apply the µ2test to the whole set of matrices generated by Fu(M,∆1).
Toanswer this question we need to introduce a larger block structure, denoted here
simply by ∆. This is simply the diagonal combination of the previous twostructures:
∆= diag(∆1,∆2).
Note that this has compatible dimensions with Mitself and the associated µtest will be
denoted by µ(M). Now we can answer the question about what happens to
µ2(Fu(M,∆1)) for all ∆1∈B∆1.
Theorem 9 (Main Loop Theorem)
µ(M)<1if and only if
µ1(M11)<1
and
max
∆1∈B∆1
µ2[Fu(M,∆1)] <1
This theorem underlies the fact that robust performance is a simple extension of robust
stability. It has a much more significant role in developing connections between the µ
theory and other theoretical aspects of control. The example in the following section is
an illustration of this point.