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2.4. µ ANALYSIS 43
descriptions are considered, where B again denotes the unit ball.
Power : BP =
w
lim
T→∞
1
2T
T
−T
| w(t) |
2
dt ≤ 1
(2.12)
Energy : BL
2
=
w
w
2
2
=
∞
−∞
| w(t) |
2
dt ≤ 1
(2.13)
Magnitude : BL
∞
=
w
w
∞
= ess sup
t
|w(t)|≤1
(2.14)
These norms are defined for scalar signals for clarity. The choice of w and e as the above
sets defines the performance criterion. The performance can be considered as a test on
the corresponding induced norm of the system. More formally,
Lemma 5 (Nominal Performance)
For all w in the input set, e is in the output set
if and only if G(s)≤1.
Only certain combinations of input and output sets lead to meaningful induced norms.
The H
∞
/µ approach is based on the cases w, e ∈ BP and w, e ∈ BL
2
.Aswenotedin
Section 2.1.2, both of these cases lead to the following induced norm.
G(s)
∞
=sup
ω
σ
max
[G(ω)].
The choice of other input and output sets can lead to meaningful norms with
engineering significance. For example w, e ∈ BL
∞
is arguably a more suitable choice for
some problems and leads to G
1
as a performance measure where
G
1
=
∞
0
|g(τ)| dτ.
and g(τ) is the convolution kernel (impulse response) of G(s). For a discussion on the
other possible selections of input and output sets, and the mathematical advantages of