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

CHAPTER 11 A PICTORIAL EXAMPLE

11.4

Sensitivity Specifications

11.4.1

A Log Sensitivity Specification

We consider the plant perturbation

std

0 ( ) =

std

0 ( )

P

s

P

s

, a gain variation in std

0 (see section 9.1.3). Figure 11.13 shows the level curves of

i.e.

P

the maximum logarithmic sensitivity of the magnitude of the I/O transfer function

13, over the frequency range 0

1, to these gain changes, ,

H

!

i.e.

sup @

log

) = max

( )

(11.12)

0

1

13(

=0

jH

j

!

j

0

1 j<S j! j

!

@

!

where

( ) = 1

(a)

(b)

(c)

13 ( ) +

13 ( ) + (1

) 13 ( )

S

j

!

;

H

j

!

H

j

!

;

;

H

j

!

:

As expected, the level curves in gure 11.13 bound convex subsets of slice.

H

2

0:8

0:7

0:6

0:5

0:4

1:5

0:3

0:2

0:1

1

0:5

0:4

0

0:5

0:6

;0:5

0:7

0:8

;1

;1

;0:5

0

0:5

1

1:5

2

Level curves of the logarithmic sensitivity of the magnitude

Figure

11.13

of the I/O transfer function

, over the frequency range 0

1, to

H

!

13

gain changes in the plant

, given by (11.12).

std

P

0

When the function (11.12) takes on the value 0.3, the maximum rst order

change in 13( ) , over 0

1, with a 25% plant gain change is exp(0 075),

jH

j

!

j

!

:

or 0 65dB. In gure 11.14 the actual maximum change in 13( ) is shown for

:

jH

j

!

j

points on the 0 3 contour of the function (11.12).

:

11.4 SENSITIVITY SPECIFICATIONS

261

2

q

0 85dB

:

0 63dB

:

q

1:5

0 58dB

:

q

0 55dB

:

q

1

0 53dB

:

q

0:5

0 63dB

:

q

0 51dB

:

0 54dB

q

0

0 35dB

:

q

:

q

0 39dB

:

q

;0:5

0 57dB

:

q

;1

;1

;0:5

0

0:5

1

1:5

2

To rst order, the peak change in 13( ) for 0

1

Figure

11.14

jH

j

!

j

!

along the 0.3 contour in gure 11.13, for a 25% gain change in std

0 , will be

P

0 65dB. The 0.3 contour from gure 11.13 is shown, together with the actual

:

peak change in 13( ) for 0

1 for several points on the contour.

jH

j

!

j

!

11.4.2

A Step Response Sensitivity Specification

In section 9.3 we considered the sensitivity of the I/O step response at t = 1 to

plant gain changes, i.e., Pstd

0 = Pstd

0 :

s (1) = @s(1)

@

:

=0

Figure 11.15 shows the subset of slice for which

H

s (1) 0:75:

j

j

This speci cation is equivalent to

1 Z 1 (1 T(j!))T(j!)

;

2

j!

ej! d! 0:75

(11.13)

;1

where

T(j!) = H(a)

13 (j!) + H(b)

13 (j!) + (1

)H(c)

13 (j!):

;

;

As we showed in section 9.3, and as is clear from gure 11.15, the step response

sensitivity speci cation (11.13) is not convex.

262