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

CHAPTER 4 NORMS OF SIGNALS

2( )

u

t

1

A = 1:5 1:5

-

0:2 1:2

;

1

1( )

u

t

1

;

1

;

@

I

@

;

;

A = 1 0

0 1

A = 1:5 0

0 0:8

The constraint u

1 requires that the signal u(t) lie

Figure

4.20

k

k

A

1

inside a trapezoid at all times t. Three weighting matrices A are shown,

together with the corresponding limits on the signal u.

The rst inequality in (4.18) can be shown by replacing 2 by the upper bound

u

2 :

kuk

1

2

1 Z T

rms = lim

( )2

kuk

0 u t dt

T

!1

T

Z

lim 1 T

2

0 kuk dt

1

T

!1

T

=

2

kuk

:

1

The second inequality in (4.18) follows from the Cauchy-Schwarz inequality:

1 Z T

aa = lim

1 ( )

kuk

0 ju t j dt

T

!1

T

1 2

1 2

!

!

=

=

Z

Z

lim 1 T 12

1 T ( ) 2

0

dt

0 ju t j dt

T

!1

T

T

=

rms

kuk

:

It can also be shown that

ss

rms

aa

kuk

kuk

kuk

kuk

:

1

1

Another norm inequality, that gives a lower bound for

aa, is

kuk

2rms

aa

(4.20)

kuk

kuk

kuk

:

1

4.4 COMPARING NORMS

91

This inequality can be understood by considering the power ampli er shown in

gure 4.8, with a 1 load resistance. If we have

=

, then the ampli er

V

kuk

cc

1

does not saturate (clip), and the voltage on the load is the input voltage, ( ). The

u

t

average power delivered to the load is therefore

2rms. The average power supply

kuk

current is

aa, so the average power delivered by the power supply is

aa ,

kuk

kuk

V

cc

which is

aa

. Of course, the average power delivered to the load does not

kuk

kuk

1

exceed the average power drain on the power supply (the di erence is dissipated in

the transistors), so we have

2rms

aa

, which is (4.20).

kuk

kuk

kuk

1

If the signal is close to a switching signal, which means that it spends most of

u

its time near its peak value, then the values of the peak, steady-state peak, RMS

and average-absolute norms will be close. The crest factor of a signal gives an

indication of how much time a signal spends near its peak value. The crest factor

is de ned as the ratio of the steady-state peak value to the RMS value of a signal:

( ) =

ss

kuk

1

CF

u

rms :

kuk

Since

ss

rms, the crest factor of a signal is at least 1. The crest factor is

kuk

kuk

1

a measure of how rapidly the amplitude distribution of the signal increases below

=

ss

see gure 4.11. The two signals in gure 4.11 have crest factors of 1.15

a

kuk

1

and 5.66, respectively.

The crest factor can be combined with the upper bound in (4.20) to give a bound

on how much the RMS norm exceeds the average-absolute norm:

rms

kuk

( )

(4.21)

aa

CF

u

:

kuk

92