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

CHAPTER 9 DIFFERENTIAL SENSITIVITY SPECIFICATIONS

so that the phase perturbation is

( ) = 2 tan 1

1

!

tan !

;

;

!

10 ;

5

which is plotted in gure 9.2. The maximum phase shift of 38 9 corresponds to

;

:

about 6dB of gain variation (see the discussion in section 9.1.2).

0

;5

;10

;15

;20

(degrees)

;25

;30

;35

;40

0:1

1

10

100

!

A specic phase shift in std.

Figure

9.2

P

0

Figure 9.3 shows the nominal magnitude of , the actual perturbed magnitude

T

caused by the phase shift in std

0 , and the perturbed magnitude predicted by the

P

rst order analysis,

approx( ) = ( ) exp

( )

1

(9.16)

jT

j

!

j

jT

j

!

j

;

!

=

1 + std

0 ( ) (a)( ) :

P

j

!

K

j

!

9.1.5

Other Log Sensitivities

We have seen that the logarithmic sensitivity of the I/O transfer function is given

T

by another closed-loop transfer function, the sensitivity . Several other important

S

closed-loop transfer functions have logarithmic sensitivities that are also closed-loop

transfer functions. Table 9.1 lists some of these.

From the top line of this table we see that a speci cation such as

log ( )

@

S

j

!

log 0( ) 2 for

bw

(9.17)

!

!

@

P

j

!

9.1 BODE’S LOG SENSITIVITIES

201

10

5

0

;5

dB ;10

pert( )

jT

j

!

j

;15

approx( )

jT

j

!

j

;20

( )

jT

j

!

j

;25

;30

0:1

1

10

!

When the phase factor (10 ) (10 + ) in std

0 is replaced

Figure

9.3

;

s

=

s

P

by (5 ) (5+ ), the magnitude of the I/O transfer function changes from

;

s

=

s

to pert . approx is a rst order approximation of pert computed

jT

j

jT

j

jT

j

jT

j

from (9.16).

log

@

H

H

log 0

@

P

1

0

;P

K

1 + 0

1 + 0

P

K

P

K

0

K

;P

K

1 + 0

1 + 0

P

K

P

K

0

1

P

1 + 0

1 + 0

P

K

P

K

0

1

P

K

1 + 0

1 + 0

P

K

P

K

The logarithmic sensitivity of some important closed-loop trans-

T

able

9.1

fer functions are also closed-loop transfer functions. In the general case,

however, the logarithmic sensitivity of a closed-loop transfer function need

not be another closed-loop transfer function.

202