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

CHAPTER 11 A PICTORIAL EXAMPLE

11.1.2

Some Worst Case Tracking Error Specifications

We now consider the tracking error. For our example, the tracking error transfer

function is given by

13 1

H

;

(we have not set up our example with the tracking error explicitly included in

the regulated variables). Figure 11.4 shows the level curves of the weighted peak

tracking error, ,

i.e.

pk trk(

) =

(a)

(b)

(c)

13 +

13 + (1

) 13 1

(11.2)

'

W

H

H

;

;

H

;

pk gn

where the weight is

( ) = 1

W

s

2 + 1:

s

(We use the symbol to denote the restriction of a functional to a nite-dimensional

'

domain.) Recall from section 8.1.2 that the weighted peak tracking error is a convex

functional of , and therefore pk trk is a convex function of and . From

H

'

gure 11.4 it can be seen that the level curves bound convex subsets.

Figure 11.5 shows the level curves of the peak tracking error, for reference inputs

bounded and slew-rate limited by 1, ,

i.e.

(a)

(b)

(c)

13 +

13 + (1

) 13 1

(11.3)

H

H

;

;

H

;

wc :

In section 6.3.2 we showed that a function of the form (11.3) is convex as expected,

the level curves in gure 11.5 bound convex subsets of slice.

H

In section 5.5.2 we showed that, for any transfer function ,

H

pk gn

wc 3

pk gn

kH

W

k

kH

k

kH

W

k

:

The reader should compare the level curves in gures 11.4 and 11.5 with this relation

in mind.

11.1 I/O SPECIFICATIONS

253

2

1:5

1

0:5

0:6

0:7

0:8

0

0:9

1:0

;0:5

1:1

1:2

1:3

;1

;1

;0:5

0

0:5

1

1:5

2

Level curves of the weighted peak tracking error, given

Figure

11.4

by (11.2).

2

1:5

1

0:5

1:0

0

1:2

1:4

1:6

1:8

;0:5

2:0

2:2

;1

;1

;0:5

0

0:5

1

1:5

2

Level curves of the peak tracking error, for reference inputs

Figure

11.5

bounded and slew-rate limited by 1, given by (11.3).

254