Linear Controller Design: Limits of Performance by Stephen Boyd and Craig Barratt - HTML preview

CHAPTER 4 NORMS OF SIGNALS

as follows:

2

Z

1

rms =

(0) = 1

( )

(4.6)

kuk

R

S

!

d!

:

u

2

u

;1

We can interpret the last integral as follows: the average power in the signal is the

integral of the contribution at each frequency.

For stochastic signals, the analogs of the average-absolute or peak norm are less

often encountered than the RMS norm. For stationary, we de ne the average-

u

absolute norm as

aa =

( )

kuk

E

ju

t

j

which for ergodic signals agrees (with probability one) with our deterministic de -

nition of

aa. We interpret

aa as the expected or mean resource consumption.

kuk

kuk

The analog of the steady-state peak of is the essential sup norm,

u

ess sup = inf

( ( )

) = 0

kuk

fa

j

Prob

ju

t

j

a

g

or equivalently, the smallest number such that with probability one, ( )

.

a

ju

t

j

a

Under some mild technical assumptions about , this agrees with probability one

u

with the steady-state peak norm of de ned in section 4.2.1.

u

4.2.5

Amplitude Distributions

We can think of the steady-state peak norm, RMS norm, and average-absolute

norm as di ering in the relative weighting of large versus small signal values: the

steady-state peak norm is entirely dependent on the large values of a signal the

RMS norm is less dependent on the large values, and the average-absolute norm

less still.

This idea can be made precise by considering the notion of the amplitude distri-

bution ( ) of a signal , which is, roughly speaking, the fraction of the time the

F

a

u

u

signal exceeds the limit , or the probability that the signal exceeds the limit at

a

a

some particular time.

We rst consider stationary ergodic stochastic signals. The amplitude distribu-

tion is just the probability distribution of the absolute value of the signal:

( ) =

( ( )

)

F

a

Prob

ju

t

j

a

:

u

Since is stationary, this expression does not depend on .

u

t

We can also express ( ) in terms of the fraction of time the absolute value of

F

a

u

the signal exceeds the threshold . Consider the time interval 0 ]. Over this time

a

T

interval, the signal will spend some fraction of the total time with ( )

.

u

T

ju

t

j

a

( ) is the limit of this fraction as

:

F

a

T

!

1

u

( ) = lim

0

( )

ft

j

t

T

ju

t

j

ag

(4.7)

F

a

u

T

!1

T

4.2 COMMON NORMS OF SCALAR SIGNALS

77

where ( ) denotes the total length (Lebesgue measure) of a subset of the real line.

These two ideas are depicted in gure 4.9. The amplitude distribution of the signal

in gure 4.9 is shown in gure 4.10.

3

2

a = 1:5

1

()t

0

u

;1

H

Y

H

;2

a = 1:5

;

;3

0

1

2

3

4

5

6

7

8

9

10

t

Example of calculating F (1:5) for the signal u in gure 4.1.

Figure

4.9

u

For T = 10, t 0 t T u(t)

1:5 is the length of the shaded

f

j

j

j

g

intervals, and this length divided by T approximates F (1:5).

u

This last interpretation of the amplitude distribution in terms of the fraction of

time the signal exceeds any given threshold allows us to extend the notion of ampli-

tude distribution to some deterministic (non-stochastic) signals. For a deterministic

, we de ne ( ) to be the limit (4.7), provided this limit exists (it need not). All

u

F

a

u

of the results of this section hold for a suitably restricted set of deterministic signals,

if we use this de nition of amplitude distribution. There are many more technical

details in such a treatment of deterministic signals, however, so we continue under

the assumption that is a stationary ergodic stochastic process.

u

Clearly ( ) = 0 for

ss , and

( ) increases to one as decreases

F

a

a

>

kuk

F

a

a

u

1

u

to zero. Informally, we think of ( ) as spending a large fraction of time where

ju

t

j

the slope of ( ) is sharp if

decreases approximately linearly, we say ( ) is

F

a

F

ju

t

j

u

u

approximately uniformly distributed in amplitude. Figure 4.11 shows two signals

and their amplitude distribution functions.

We can compute the steady-state peak, RMS, and average-absolute norms of u

directly from its amplitude distribution. We have already seen that

ss = sup

( ) 0

(4.8)

kuk

fa

j

F

a

>

g:

1

u

The steady-state peak of a signal is therefore the value of at which the graph of

a

the amplitude distribution rst becomes zero.

78