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

CHAPTER 9 DIFFERENTIAL SENSITIVITY SPECIFICATIONS

9.1.3

Example: Gain Variation

Our rst example concerns a change in gain, i.e.

0( ) =

0( )

(9.10)

P

s

P

s

so that

log 0( )

P

j

!

'

:

Hence from (9.8{9.9) we have

log ( )

( )

(9.11)

jT

j

!

j

'

<S

j

!

( )

( )

(9.12)

6

T

j

!

'

=S

j

!

:

It follows, for example, that the closed-loop convex speci cation

( 0) = 0

<S

j

!

guarantees that the magnitude of the I/O transfer function at the frequency 0 is

!

rst order insensitive to variations in .

To give a speci c example that compares the rst order deviation in ( ) to

jT

j

!

j

the real deviation, we take the standard example plant and controller (a) described

K

in section 2.4, and consider the e ect on ( ) of a gain perturbation of = 25%,

jT

j

!

j

which is about 2dB. Figure 9.1 shows:

std

(a)

( ) =

0 ( )

( )

P

j

!

K

j

!

jT

j

!

j

1 + std

0 ( ) (a)( )

P

j

!

K

j

!

which is the nominal magnitude of the I/O transfer function

pert

std

(a)

( ) = 1 25 0 ( )

( )

:

P

j

!

K

j

!

jT

j

!

j

1 + 1 25 std

0 ( ) (a)( )

:

P

j

!

K

j

!

which is the actual magnitude with the 25% gain increase in std

0

and

P

approx( ) = ( ) exp 0 25

1

jT

j

!

j

jT

j

!

j

:

<

1 + std

0 ( ) (a)( )

(9.13)

P

j

!

K

j

!

which is the magnitude of the perturbed I/O transfer function predicted by the rst

order perturbation formula (9.11). For this example, the rst order prediction gives

a good approximation of the perturbed magnitude of the I/O transfer function,

even for this 2dB gain change in std

0 .

P

9.1 BODE’S LOG SENSITIVITIES

199

10

5

0

;5

dB ;10

( )

p

ert

jT

j

!

j

;15

( )

appro

x

jT

j

!

j

;20

( )

jT

j

!

j

;25

;30

0:1

1

10

!

When

is replaced by 1 25

, the I/O transfer function's

std

std

Figure

9.1

P

:

P

0

0

magnitude changes from

to pert . approx is a rst order approxima-

jT

j

jT

j

jT

j

tion of

computed from (9.13). In this example the eect of a plant

p

ert

jT

j

gain change as large as 25% is well approximated using the dierential sen-

sitivity.

9.1.4

Example: Phase Variation

In this example we study the e ects on of a phase variation in 0, i.e.,

T

P

0( ) + 0( ) =

( )

)

j

!

0(

P

j

!

P

j

!

e

P

j

!

:

In this case we have, from (9.8{9.9),

log ( )

( ) ( )

(9.14)

jT

j

!

j

'

;

!

=S

j

!

( )

( ) ( )

(9.15)

6

T

s

'

!

<S

j

!

:

To guarantee that ( 0) is, for example, rst order insensitive to phase variations

jT

j

!

j

in 0( 0), we have the speci cation

P

j

!

( 0) = 0

=S

j

!

which is closed-loop a ne.

We now consider a speci c example that compares the actual e ect of a phase

variation in 0 to the e ect predicted by the rst order perturbational analy-

P

sis (9.14). As above, our plant is the standard example described in section 2.4,

together with the same controller (a). The speci c perturbed std

0 is

K

P

std

5

0 ( ) +

std

0 ( ) = 1 ; s

P

s

P

s

2 5 +

s

s

200