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

CHAPTER 11 A PICTORIAL EXAMPLE

where

1 2

=

rms u( ) =

21 proc 2

22 sensor 22

(11.10)

H

kH

W

k2 + kH

W

k

:

Recall from section 8.2.2 that the RMS response to independent stochastic inputs

with known power spectral densities is a convex functional of

therefore rms u

H

is a convex function of , and (11.9) is a convex function of and .

H

2

1:75

1:5

1:5

1:25

1

1:0

0:75

0:5

0:5

0:25

0

0:25

;0:5

0:5

0:75

;1

;1

;0:5

0

0:5

1

1:5

2

Level curves of the RMS value of the actuator signal , with

Figure

11.11

u

sensor and actuator noises, given by (11.9).

11.3.3

RMS Gain Limit

Figure 11.12 shows the level curves of the worst case RMS actuator signal for any

u

reference input with RMS value bounded by 1, ,

r

i.e.

(a)

(b)

(c)

23 +

23 + (1

) 23

(11.11)

H

H

;

;

H

:

1

11.3 ACTUATOR EFFORT

259

1:5

275

225

1

175

125

0:5

0

25

75

;0:5

125

175

;1

225

;1:5

275

325

;2

;20

;15

;10

;5

0

5

10

15

20

Level curves of the worst case RMS actuator signal for

Figure

11.12

u

any reference input with RMS value bounded by 1, given by (11.11).

r

260