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

CHAPTER 4 NORMS OF SIGNALS

4.2.3

Average-Absolute Value

A measure that puts even less emphasis on large values of a signal (indeed, the

minimum emphasis possible to still be a norm) is its average-absolute value, de ned

by

1 Z T

aa = lim

( )

(4.2)

kuk

0 ju t j dt

T

!1

T

provided the limit exists (see the Notes and References). An example of a signal u

and its average-absolute norm

aa is shown in gure 4.6.

aa can be measured

kuk

kuk

with the circuit shown in gure 4.7 (c.f. the peak detector circuit in gure 4.2).

3

3

2

u aa

2 5

k

k

:

u aa

;

1

k

k

2

;

u(t) 0

u(t)j 1 5:

1

j

1

;

@

I

@

2

0 5

;

u aa

:

;k

k

3

0

;

0

2

4 t 6

8

10

0

2

4 t 6

8

10

(a)

(b)

A signal u and its average-absolute value u aa is shown in (a).

Figure

4.6

k

k

u aa is found by nding the average area under u , as shown in (b).

k

k

j

j

R

q

+

u

+

(t)

q

q

q

C

Vc

;

;

ro

q

q

If u varies much faster than the time constant RC then V will

Figure

4.7

c

be nearly proportional to the average of the peak of the input voltage u, so

that V = u aa. The resistor r

R ensures that the output impedance

k

k

c

o

of the bridge is low at all times.

The average-absolute norm

aa is useful in measuring average fuel or resource

kuk

use, when the fuel or resource consumption is proportional to ( ) . In contrast, the

ju

t

j

4.2 COMMON NORMS OF SCALAR SIGNALS

75

RMS norm

2rms is useful in measuring average power, which is often proportional

kuk

to ( )2. Examples of resource usage that might be measured with the average-

u

t

absolute norm are rocket fuel use, compressed air use, or power supply demand in

a conventional class B ampli er, as shown in gure 4.8.

Vcc

power

supply

q

q

+

u(t)

R

V

;

;

cc

Idealized version of a class B power amplier, with no bias

Figure

4.8

circuitry shown. Provided the amplier does not clip, i.e., u

V , the

k

k

1

cc

average power supplied by the power supply is proportional to u aa, the

k

k

average-absolute norm of u the average power dissipated in the load R is

proportional to u 2rms, the square of the RMS norm of u.

k

k

4.2.4

Norms of Stochastic Signals

For a signal modeled as a stationary stochastic process, the measure of its size most

often used is

2

rms = ;

( )2 1=

(4.3)

kuk

E

u

t

:

Because the process is stationary, the expression in (4.3) does not depend on . For

t

stochastic signals that approach stationarity as time goes on, we de ne

2

rms = lim

( )2 1=

kuk

E

u

t

:

t!1

If the signal is ergodic, then its RMS norm can be computed either by (4.3)

u

or (4.1): with probability one the deterministic and stochastic RMS norms are

equal.

The RMS norm can be expressed in terms of the autocorrelation of ,u

( ) = ( ) ( + )

(4.4)

R

E

u

t

u

t

u

or its power spectral density,

Z

( ) = 1 ( ) ;j!

(4.5)

S

!

R

e

d

u

u

;1

76