2.4. µANALYSIS 49
Actually, the lowerb ound is always equal to µbut the implied optimization has local
maxima which are not global. For the upper bound Safonov and Doyle [72], have shown
that finding the infimum is a convex problem and hence more easily solved. However the
bound is equal to µonly in certain special cases. Here we use the infimum rather that
the miminum because Dmay have a element whichgo es to zero as the maximum
singular value decreases. So the limiting case (where an element of Dis zero) is not a
member of the set D.
The cases where the upper bound is equal to µare tabulated below.
q=0 q=1 q=2
m=0 equal less than or equal
m=1 equal equal less than or equal
m=2 equal less than or equal less than or equal
m=3 equal less than or equal less than or equal
m=4 less than or equal less than or equal less than or equal
Most practical applications of the µtheory involve models where q= 0. Here we see that
we have equalitywith the upp er bound for three or fewer blocks. Computational
experience has yet to produce an example where the bound differs by more than
15 percent. In practically motivated problems the gap is usually much less.
2.4.5 The Main Loop Theorem
We will introduce afunda mentaltheore min µanalysis: the main loop theorem. From
the previous discussion youwill see that there are several matrix properties that can be
expressed as µtests. The spectral radius and the maximum singular value are two such
quantities. The main loop theorem gives a way of testing such properties for perturbed
systems. The test is simply a larger µproblem. This is the theorem underlying the
extension from robust stability to robust performance.
Consider a partitioned matrix,
M=M11 M12
M21 M22 ,