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

CHAPTER 5 NORMS OF SYSTEMS

5.3.4

RMS Gain

The RMS gain of a MIMO system is important for several reasons, one being that it

is readily computed from state-space equations. The RMS gain of a MIMO transfer

matrix is

rms gn =

= sup

( ))

kH

k

kH

k

s

1

0 max(H

<s>

the

norm of a transfer matrix (c.f. (5.11), the analogous de nition for transfer

H

1

functions). Thus,

if and only if the transfer matrix is stable. For

kH

k

<

1

H

1

stable , we can express the

norm as the maximum of the maximum singular

H

H

1

value over all frequencies:

= sup max( ( ))

kH

k

H

j

!

1

!

as shown in gure 5.17. Note that the other singular values do not a ect

.

kH

k

1

10

9

@

I

8

@

1

kH

k

1

;

7

;

6

5

4

3

2

2

;

3

;

;

1

;

0

0:1

1

10

100

!

The

norm of a stable transfer matrix is the maxi-

Figure

5.17

H

H

1

mum over frequency of the maximum singular value, 1. The other singular

values, 2 ... , do not aect the

norm.

H

n

1

5.3.5

Entropy of a System

In this section we describe a measure of the size of a MIMO system, which is not

a norm, but is closely related to the 2 norm and the

norm. For

0 we

H

H

>

1

5.3 NORMS OF MIMO LTI SYSTEMS

113

de ne the -entropy of the system with transfer matrix as

H

8

2 Z 1

2

( ) = <

logdet;

;

( ) ( )

if

;

2

I

;

H

j

!

H

j

!

d!

kH

k

<

1

I

H

;1

:

if

1

kH

k

1

(c.f. (5.21)). The -entropy can also be expressed in terms of the singular values as

8

1 Z

n

1

X

>

2

( ) = < 2

log;1 ( ( ( )) )2

if

;

;

H

j

!

=

d!

kH

k

<

i

1

I

H

=1

;1

i

>

:

if

1

kH

k

1

(c.f. (5.22)). This last formula allows us to interpret the -entropy of as a measure

H

of its size, that puts a weight 2 log(1 ( )2) on a singular value , whereas

;

;

=

the 2 norm uses the weight 2. This weight function is shown in gure 5.18, with

H

2 shown for comparison.

2 log;1 ( )2

;

;

=

@

@

R

@

I

@

2

2

0

0

The -entropy of is a measure of its size that puts a weight

Figure

5.18

H

2 log(1 ( )2) on a singular value , whereas the 2 norm uses the

;

;

=

H

weight 2.

Since these two weight functions are close when is small compared to , we

i

see that

lim q ( ) =

2

(5.24)

I

H

kH

k

:

!1

From gure 5.18 we can see that

q

( )

2

(5.25)

I

H

kH

k

114