2.4. µANALYSIS 43
descriptions are considered, where Bagain denotes the unit ball.
Power : BP =(wlim
T→∞
1
2TZT
−T
|w(t)|2dt ≤1)(2.12)
Energy: BL2=(wkwk2
2=Z∞
−∞
|w(t)|2dt ≤1)(2.13)
Magnitude : BL∞=(wkwk∞= ess sup
t|w(t)|≤1
)(2.14)
These norms are defined for scalar signalsfor clarity. The choice of wand eas the above
sets defines the performance criterion. The performance can be considered as a teston
the corresponding induced norm of the system. More formally,
Lemma 5 (Nominal Performance)
For al l winthe input set, eis in the output set
if and only if kG(s)k≤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∈BL2.Aswenotedin
Section 2.1.2, both of these cases lead to the following induced norm.
kG(s)k∞=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 kGk1as a performance measure where
kGk1=Z∞
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