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

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CHAPTER 5 NORMS OF SYSTEMS

i.e., the square root of the -entropy of a transfer matrix is no smaller than its 2

H

norm. We also have the more complicated converse inequality

q

( ) 1 p log(1

2)

2

(5.26)

I

H

;

;

kH

k

where =

1. Thus, the relative increase in the square root of the

kH

k

=

<

1

-entropy over the 2 norm can be bounded by an expression that only depends

H

on how close the

norm is to the critical value . For example, if

2

H

kH

k

=

1

1

(

0 5), we have

2 p ( ) 1 073

2, i.e., if exceeds

by 6dB

:

kH

k

I

H

:

kH

k

kH

k

1

or more, then the square root of the -entropy and

2 cannot di er by more

kH

k

than about 0.6dB. Thus, the

norm of a transfer matrix must be near for the

H

1

square root of the -entropy to di er much from the 2 norm.

H

An important property of the entropy is that it is readily computed using state-

space methods, as we will see in section 5.6.5.

An Interpretation of the Entropy

Recall that the square of the 2 norm of a transfer function is the power of

H

H

its output when it is driven by a white noise. For the case of scalar (i.e., transfer

function) , we can give a similar interpretation of the -entropy of as the average

H

H

power of its output when it is driven by a white noise, and a certain random feedback

is connected around it, as shown in gure 5.19.

+

r

w

z

H

+

q

The -entropy of a transfer function is the average power of

Figure

5.19

H

its output when it is driven by a white noise and a random feedback

z

w

is connected around it. The values of the random feedback transfer function

are independent at dierent frequencies, and uniformly distributed on a

disk of radius 1 centered at the origin in .

=

C

The transfer function from to in gure 5.19, with a particular feedback

w

z

transfer function connected, is (1

), so the power of the output signal is

H

=

;

H

2

H

1

2 :

;

H

We now assume that is random, with ( ) and ( ) independent for = ,

j

!

j

!

6

and each ( ) uniformly distributed on the disk of radius 1 in the complex

j

!

=

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index-124_7.png

5.4 IMPORTANT PROPERTIES OF GAINS

115

plane. Then we have

2

H

= ( )

(5.27)

E

1

2 I H

;

H

where denotes expectation over the random feedback transfer functions .

E

Some feedback transfer functions decrease the power in the output, while other

feedback transfer functions increase it the inequality (5.25) shows that on average,

the power in the output is increased by the feedback. The limit (5.24) shows that

if the feedback is small, then it has little e ect on the output power (indeed, (5.26)

shows that the average e ect of the feedback on the output power is small unless

the

norm of the feedback is close to the inverse of the

norm of ).

H

H

H

1

1

5.3.6

Scaling and Weights

The discussion of section 4.3.5 concerning scalings and weights for norms of vector

signals has important implications for norms of MIMO systems gains especially are

a ected by the scaling used to measure the input and output vector signals. For

example, let us consider the e ect of scaling on the RMS gain of a (square) MIMO

system. Let be a scaling matrix (i.e., diagonal). The -scaled RMS value of

D

D

H

w

is

rms =

rms the -scaled RMS value of is

rms =

rms.

kH

w

k

kD

H

w

k

D

w

kw

k

kD

w

k

D

D

Thus, the -scaled RMS gain of is

D

H

1

sup

rms

~

;

rms

kD

H

w

k

= sup kDHD wk

rms=0

rms

~

=0

~ rms

kD

w

k

rms

kw

k

kw

k

6

kw

k

6

=

1

;

kD

H

D

k

1

the

norm of the diagonally pre- and post-scaled transfer matrix.

H

1

More general transfer matrix weights can be applied, e.g., post

pre .

kW

H

W

k

5.4

Important Properties of Gains

5.4.1

Gain of a Cascade Connection

An important property of gains is that the gain of the product of two transfer

matrices can be bounded in terms of the gains of the individual transfer matrices:

if

gn denotes any gain, then

k

k

2 1 gn

2 gn 1 gn

(5.28)

kH

H

k

kH

k

kH

k

:

This inequality is easily established it can be seen from gure 5.20. The prop-

erty (5.28) does not generally hold for norms of LTI systems that are not gains. For

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index-125_3.png

index-125_4.png

index-125_5.png

index-125_6.png

index-125_7.png

index-125_8.png

index-125_9.png

index-125_10.png

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116