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

CHAPTER 5 NORMS OF SYSTEMS

( ) = 0 for

()

w

t

t

T

t

;

;

0

w

( ) rings for

z

t

t

T

;

;

()t 0

z

;10

;8

;6

;4

;2

0

2

4

6

8

10

t

;

T

The Hankel norm of a transfer function is the largest possible

Figure

5.7

square root energy in the output for

, given a unit-energy excitation

z

t

T

that stops at = .

w

t

T

responses of (a)

(b)

13 and 13 are shown in gure 5.8 and their frequency response

H

H

magnitudes in gure 5.9. The values of various norms of (a)

(b)

13 and 13 are shown

H

H

in table 5.1.

From the rst row of table 5.1 we see that the peak of the response of (a)

13 to

H

a step input is about the same as (b)

13 . Thus, in the sense of peak step response,

H

(a)

(b)

13 is about the same size as 13 .

H

H

If (a)

(b)

(a)

13 and 13 are driven by white noise, the RMS value of the output of 13

H

H

H

is less than half that of (b)

13 (second row). Figure 5.10 shows an example of the

H

Norm

(a)

(b)

13

13

gure

H

H

pk step 1 36 1 40

5.8

k

k

:

:

2

1 17 2 69

5.10

k

k

:

:

wc

1 60 1 68

5.11

k

k

:

:

pk gn

1 74 4 93

5.12

k

k

:

:

1 47 3 72

5.9

k

k

:

:

1

hankel

1 07 2 04

5.13

k

k

:

:

The values of six dierent norms of (a) and (b).

T

able

5.1

H

H

13

13

5.2 NORMS OF SISO LTI SYSTEMS

105

1:6

(b)

= 1 40

kH

k

:

pk

step

13

;

;

1:4

H

Y

(a)

= 1 36

H

kH

k

:

pk

step

13

1:2

1

@

I

@

(b)

0:8

( )

s

t

13

0:6

@

I

@

0:4

(a)

( )

s

t

13

0:2

0

;0:2

0

1

2

3

4

5

6

7

8

9

10

t

The step responses of the transfer functions in (5.17) and (5.18).

Figure

5.8

Note that (a)

= 1.36, and (b)

= 1.40.

kH

k

kH

k

pk

step

pk

step

13

13

10

(b)

= 3 72

(a)

= 1 47

kH

k

:

1

kH

k

:

13

1

13

;

;

?

1

j

H

j

(b)

( )

jH

j

!

j

13

0:1

;

;

;

;

(a)

( )

jH

j

!

j

13

0:01

0:1

1

10

100

!

The magnitudes of the transfer functions in (5.17) and (5.18).

Figure

5.9

Note that (a) = 1.47, and (b) = 3.72.

kH

k

kH

k

1

1

13

13

106