Thus, the weight emphasizes the power spectral density of where ( ) is large,
u
jW
j
!
j
meaning it contributes more to the integral, and de-emphasizes the power spectral
density of where ( ) is small. We note from (4.12) that the weighted norm
u
jW
j
!
j
depends only on the magnitude of the weighting transfer function , and not its
W
phase. This is not true of all weighted norms.
We can view the e ect of the weight as changing the relative importance of
W
di erent frequencies in the total power integral (4.12). We can also interpret the
-weighted RMS norm in terms of the average power conceptual model shown in
W
gure 4.5, by changing the resistive load to a frequency-dependent load, i.e., a
R
more general passive admittance ( ). The load admittance is related to the
G
s
G
weight by the spectral factorization
W
( ) + ( )
G
s
G
;s
2
= ( ) ( )
(4.13)
W
s
W
;s
so that ( ) = ( ) 2. Since the average power dissipated in at a frequency
<G
j
!
jW
j
!
j
G
is proportional to the real part (resistive component) of ( ), we see that the
!
G
j
!
total average power dissipated in is given by the square of (4.12), or the square
G
of the -weighted RMS norm of .
W
u
For example, suppose that ( ) = (1 + p2 ) (1 + ), which gives up to 3dB
W
s
s
=
s
emphasis at frequencies above p2. The load admittance for this weight is ( ) =
G
s
(1+2 ) (1+ ), which we realize as the parallel connection of a 1 resistor and the
s
=
s
series connection of a 1 resistor and a 1 capacitor, as shown in gure 4.14.
F
q
+
u
1F
(t)
1
1
;
q
The average power dissipated in the termination admittance
Figure
4.14
G(s) = (1 + 2s)=(1 + s) is u 2 rms, the square of the W-weighted RMS
k
k
W
norm of the driving voltage u, where W(s) = (1 + p2s)=(1 + s).
Frequency domain weights are used less often with the other norms, mostly
because their e ect is harder to understand than the simple formula (4.12). One
common exception is the maximum slew rate, which is the peak norm used with the
weight ( ) = , a di erentiator:
W
s
s
slew rate = _
(4.14)
kuk
kuk
:
1
Maximum slew-rate speci cations occur frequently, especially on actuator signals.
For example, an actuator signal may represent the position of a large valve, which
4.2 COMMON NORMS OF SCALAR SIGNALS
83
is opened and closed by a motor that has a maximum speed. A graphical interpre-
tation of the slew-rate constraint
slew rate 1 is shown in gure 4.15.
kuk
The peak norm is sometimes used with higher order di erentiators:
2
acc = d u
kuk
2
dt
1
3
jerk = d u
kuk
3
:
dt
1
Weights that are successively higher order di erentiators yield the amusing snap,
crackle, and pop norms.
4
3
2
1
()t
0
u
;1
;2
;3
;4
0
1
2
3
4
5
6
7
8
9
10
t
An interpretation of the maximum slew-rate specication
Figure
4.15
_u
1: at every time t the graph of u must evolve within a cone, whose
k
k
sides have slopes of 1. Examples of these cones are shown for t = 0 3 6
u is slew-rate limited at t = 0 and t = 6, since _u(0) = 1 and _u(6) = 1.
;
4.2.9
Time Domain Weights
If the initial linear transformation consists of multiplying the signal by some given
function of time ( ), we refer to as a time domain weight. One example is the
w
t
w
ITAE (integral of time multiplied by absolute error) norm from classical control,
de ned as
Z
1
itae =
( )
kuk
0 tju t j dt:
84