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

CHAPTER 5 NORMS OF SYSTEMS

Thus we have for all with nonzero RMS value

w

rms

kH

w

k

rms

kH

k

1

kw

k

which shows that

rms gn

. By concentrating the power spectral density

kH

k

kH

k

1

of near a frequency max at which ( max)

, we have

w

!

jH

j

!

j

kH

k

1

rms

kH

w

k

rms

kH

k

:

1

kw

k

(Making this argument precise establishes (5.10).)

We can contrast the

and 2 norms by considering the associated inequality

H

H

1

speci cations. Equation (5.11) implies that the

norm-bound speci cation

H

1

(5.12)

kH

k

M

1

is equivalent to the speci cation:

rms

for all with

rms 1

(5.13)

kH

w

k

M

w

kw

k

:

In contrast, the 2 norm-bound speci cation

H

2

(5.14)

kH

k

M

is equivalent to the speci cation

rms

for a white noise.

(5.15)

kH

w

k

M

w

The

norm is often combined with a frequency domain weight:

H

1

=

kH

k

kW

H

k

:

W

1

1

If the weighting transfer function

and its inverse

1 are both stable, the

;

W

W

speci cation

1 can be expressed in the more classical form: is stable

kH

k

H

W

1

and

( )

( ) 1

;

for all

jH

j

!

j

jW

j

!

j

!

depicted in gure 5.4(b).

5.2.7

Shifted

Norm

H

1

A useful generalization of the

norm of a transfer function is its -exponentially

H

a

1

weighted 2 norm gain, de ned by

L

= sup ~ 2

kz

k

kH

k

1

a

~ 2=0 ~ 2

kw

k

kw

k

6

5.2 NORMS OF SISO LTI SYSTEMS

101

30

30

20

20

(dB) 10

(dB) 10

)

)j

j

0

0

(

(j!

j!

10

10

W

;

;

allowed region

j

W

j

1=

20

20

;

for ( )

jH

j

!

j

;

30

30

;

1

10

100 ; 1

10

100

!

!

(a)

(b)

An example of a frequency domain weight

that enhances

Figure

5.4

W

frequencies above = 10 is shown in (a). The specication

1

!

kW

H

k

1

requires that the magnitude of ( ) lie below the curve 1

( ) , as

H

j

!

=jW

j

!

j

shown in (b). In particular, ( ) must be below 20dB for

12.

jH

j

!

j

;

!

where ~( ) =

( ), ~( ) =

( ), and =

. It can be shown that

at

at

w

t

e

w

t

z

t

e

z

t

z

H

w

= sup

( ) =

(5.16)

kH

k

jH

s

j

kH

k

1

a

a

1

<s>;a

where ( ) = (

).

is called the -shifted transfer function formed from

H

s

H

s

;

a

H

a

a

a

, and the norm

is called the -shifted

norm of a transfer function.

H

k

k

a

H

1

a

1

This is shown in gure 5.5.

From (5.16), we see that the -shifted

norm of a transfer function is nite

a

H

1

if and only if the real parts of the poles of are less than . For

0, then,

H

;a

a

<

the shifted

norm can be used to measure the size of some unstable transfer

H

1

functions, for example,

1 ( 1)

2 = 1

k

=

s

;

k

1

;

(whereas 1 ( 1) = , meaning that its RMS gain is in nite). On the other

k

=

s

;

k

1

1

hand if

0, the -shifted

norm-bound speci cation

a

>

a

H

1

kH

k

M

1

a

(or even just

) guarantees that the poles of lie to the left of the line

kH

k

<

1

H

1

a

=

in the complex plane.

<s

;a

We can interpret the shifted transfer function (

) as follows. Given a block

H

s

;

a

diagram for that consists of integrators (transfer function 1 ), summing blocks,

H

=s

and scaling ampli ers, we replace each integrator with a transfer function 1 (

)

=

s

;

a

(called a \leaky integrator" when

0). The result is a block diagram of (

),

a

<

H

s

;

a

as shown in gure 5.6. In circuit theory, where is some network function, this is

H

called a uniform loading of .

H

102