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

CHAPTER 5 NORMS OF SYSTEMS

all of its poles have negative real part. The power spectral density of the output z

of is then

H

( ) = ( ) ( ) 2

S

!

S

!

jH

j

!

j

z

w

and therefore

1 2

Z

=

1

rms =

1

( ) 2 ( )

(5.2)

kz

k

2

jH

j

!

j

S

!

d!

:

w

;1

Thus we assign to the norm

H

1 2

Z

=

1

rms =

1

( ) 2 ( )

(5.3)

kH

k

jH

j

!

j

S

!

d!

:

w

2

w

;1

The right-hand side of (5.3) has the same form as (4.12), with substituted

H

for . The interpretations are di erent, however: in (5.3), is some xed signal,

W

w

and we are measuring the size of the LTI system , whereas in (4.12), is a xed

H

W

weighting transfer function, and we are measuring the size of the signal .

w

5.2.3

Norm: RMS Response to White Noise

H

2

Consider the RMS response norm above. If ( ) 1 at those frequencies where

S

!

w

( ) is signi cant, then we have

jH

j

!

j

1

1 2

Z

=

1

rms

( ) 2

kH

k

jH

j

!

j

d!

:

w

2 ;1

It is convenient to think of such a signal as an approximation of a white noise signal,

a ctitious input signal with ( ) = 1 for all (and thus, in nite power, which

S

!

!

w

we conveniently overlook).

This important norm of a stable system is denoted

1 2

Z

=

1

2 =

1

( ) 2

kH

k

2

jH

j

!

j

d!

;1

(we assign

2 =

for unstable ), and referred to as the 2 norm of .

kH

k

1

H

H

H

Thus we have the important fact: the 2 norm of a transfer function measures

H

the RMS response of its output when it is driven by a white noise excitation.

The 2 norm can be given another interpretation. By the Parseval theorem,

H

1 2

Z

=

1

2 =

( )2

=

2

kH

k

0 h t dt

khk

the 2 norm of the impulse response of the LTI system. Thus we can interpret

L

h

the 2 norm of a system as the 2 norm of its response to the particular input

H

L

signal , a unit impulse.

5.2 NORMS OF SISO LTI SYSTEMS

97

5.2.4

A Worst Case Response Norm

Let us give an example of measuring the size of a transfer function using the worst

case response paradigm. Suppose that not much is known about except that

w

ampl and _

slew, i.e.,

is bounded by ampl and slew-rate

kw

k

M

kw

k

M

w

M

1

1

limited by slew. If the peak of the output is critical, a reasonable measure of

M

z

the size of is

H

wc = sup

ampl _

slew

kH

k

fkH

w

k

j

kw

k

M

kw

k

M

g

:

1

1

1

In other words,

wc is the worst case (largest) peak of the output, over all inputs

kH

k

bounded by ampl and slew-rate limited by slew.

M

M

5.2.5

Peak Gain

The peak gain of an LTI system is

pk gn = sup kHwk1

(5.4)

kH

k

=0

:

kw

k

1

kw

k

6

1

It can be shown that the peak gain of a transfer function is equal to the 1 norm

L

of its impulse response:

Z

1

pk gn =

( ) =

1

(5.5)

kH

k

0 jh t j dt khk :

The peak gain of a transfer function is nite if and only if the transfer function is

stable.

To establish (5.5) we consider the input signal

( ) = sgn( (

)) for 0

h

T

;

t

t

T

w

t

0

otherwise

(5.6)

which has

= 1 (the sign function, sgn( ), has the value 1 for positive argu-

kw

k

1

ments, and 1 for negative arguments). The output at time is

;

T

Z

( ) = T (

) ( )

z

T

0 w T ; t h t dt

Z

= T sgn( ( )) ( )

0

h

t

h

t

dt

Z

= T ( )

0 jh t j dt

which converges to

1 as

. So for large (and

stable, so that

khk

T

!

1

T

H

pk gn

), the signal (5.6) yields

near

1 it is also possible to

kH

k

<

1

kz

k

=kw

k

khk

1

1

show that there is a signal such that

=

1.

w

kz

k

=kw

k

khk

1

1

98