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

CHAPTER 9 DIFFERENTIAL SENSITIVITY SPECIFICATIONS

1:6

1:4

1:2

1

0:8

0:6

pert( )

s

t

0:4

approx( )

s

t

0:2

( )

s

t

0

;0:2

0

1

2

3

4

5

6

7

8

9

10

t

When std

0 is replaced by 0 8 std

0 , the step response changes

Figure

9.6

P

:

P

from to pert. The rst order approximation of pert is given by approx( ) =

s

s

s

s

t

( ) 0 2 ( ).

s

t

;

:

s

t

1:6

1:4

1:2

1

() 0:8

t

s

0:6

0:4

0:2

0

;0:2

0

1

2

3

4

5

6

7

8

9

10

t

The sensitivity of the step response to plant gain changes is

Figure

9.7

shown for the controller (a). The rst order approximation of the step

K

response falls in the shaded envelope when std

0 is replaced by

std

0 , for

P

P

0 8

1 2.

:

:

9.3 GENERAL DIFFERENTIAL SENSITIVITY

207

9.3.2

Some Convex Approximations

In many cases there are useful convex approximations to speci cations that limit

general di erential sensitivities of the closed-loop system.

Consider the speci cation

( ) 0 75

for

0

(9.27)

js

t

j

:

t

which limits the sensitivity of the step response to gain variations in 0. This

P

speci cation is equivalent to

pk step 0 75

kS

T

k

:

which is not closed-loop convex. We will describe two convex approximations for

the nonconvex speci cation (9.27).

Suppose

min( )

( ) max( ) for

0

(9.28)

s

t

s

t

s

t

t

is a design speci cation (see gure 8.5). A weak approximation of the sensitivity

speci cation (9.26) (along with the step response speci cation (9.28)) is that a

typical (and therefore xed) step response satis es the speci cation:

typ

75

kS

T

kpk step

0:

where typ is the transfer function that has unit step response

T

) +

)

typ( ) = min(

max(

s

t

s

t

s

t

2

:

A stronger approximation of (9.27) (along with the step response speci ca-

tion (9.28)) requires that the sensitivity speci cation be met for every step response

that satis es (9.28):

max

min( )

( ) max( ) for

0

0 75

f

kS

v

k

j

s

t

v

t

s

t

t

g

:

:

1

This is an inner approximation of (9.27), meaning that it is tighter than (9.27).

208