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

CHAPTER 11 A PICTORIAL EXAMPLE

11.2.3

RMS Regulation

Suppose that proc and sensor are independent, zero-mean stochastic processes with

n

n

power spectral densities

proc( ) = 2proc

S

!

W

sensor( ) = 2sensor

S

!

W

where

proc = 0 04

W

:

sensor = 0 01

W

:

( , scaled white noises). Figure 11.9 shows the level curves of the RMS value of

i.e.

p due to these noises,

, the level curves of the function

y

i.e.

rms yp(

) = rms yp

(a) + (b) + (1

) (c)

(11.6)

'

H

H

;

;

H

where

1 2

=

rms yp( ) =

11 proc 2

12 sensor 22

(11.7)

H

k

H

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 yp

H

is a convex function of , and rms yp is a convex function of and .

H

'

11.3

Actuator Effort

11.3.1

A Particular Disturbance

We consider again the particular actuator-referred disturbance part( ) shown in

d

t

gure 11.7. Figure 11.10 shows the peak actuator signal due to the actuator-

u

referred disturbance part, ,

d

i.e.

(a)

(b)

(c)

21 +

21 + (1

) 21 part

(11.8)

H

H

;

;

H

d

1

which is a convex function on 2.

R

11.3.2

RMS Limit

Figure 11.11 shows the level curves of the RMS value of due to the noises described

u

in section 11.2.3, , the level curves of the function

i.e.

rms u

(a) + (b) + (1

) (c)

(11.9)

H

H

;

;

H

11.3 ACTUATOR EFFORT

257

2

0:07

0:06

1:5

0:05

0:04

1

0:5

0:04

0:05

0

0:06

0:07

;0:5

0:08

0:09

;1

;1

;0:5

0

0:5

1

1:5

2

Level curves of the RMS value of , with sensor and actuator

Figure

11.9

y

p

noises, given by (11.6).

2

0:6

0:55

0:5

1:5

0:45

0:4

0:35

1

0:3

0:25

0:5

0:2

0:2

0

0:25

0:3

;0:5

0:35

0:4

;1

;1

;0:5

0

0:5

1

1:5

2

Level curves of the peak actuator signal , due to the par-

Figure

11.10

u

ticular actuator-referred disturbance

( ) shown in gure 11.7, given

d

t