52 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Therefore µ1(A)<1 is equivalentto our system being stable. Furthermore,the
maximum modulus theorem for a stable system tells us that,
kP(z)k∞=sup
|z|≥1
σmax(P(z))
=sup
|z
−1
|≤1
σmax(Fu(Pss,z−1I)
=sup
∆
1
∈B∆1
µ2(Fu(Pss,∆1),
if ∆2is defined as a single full block of dimensions matching the input-output
dimensions of P(z). The main loop theorem (Theorem 9) immediately suggests the
following result.
Lemma 10
µ(Pss)<1 if and only if
P(z) is stable
and
kP(z)k∞<1.
Note that this tests whether or not the ∞norm is less than one. It doesn’t actually
calculate the ∞norm. To do this we have toset up a scaled system and search over the
scaling with makes µ(Pss)=1.
Toa pply this to a robust performance problem, consider the configuration shown in
Figure 2.10. Thisis a state-space representationof a perturbed system.It would
typically model an uncertain closed-loop system where the performance objective is
kek≤kwk, for all w∈BL2and all ∆ ∈B∆.
The real-valued matrix, Gss,is,
G
ss =
AB
1B
2
C
1D
11 D12
C2D21 D22
.