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

CHAPTER 8 PERFORMANCE SPECIFICATIONS

1:2

1:2

0:8

0:8

() 0:4

() 0:4

t

t

11

12

s

s

0

0

0:4

0:4

;

;

0

0:2

0:4

0:6

0:8

1

0

0:2

0:4

0:6

0:8

1

t

t

(a)

(b)

1:2

1:2

0:8

0:8

() 0:4

() 0:4

t

t

21

22

s

s

0

0

0:4

0:4

;

;

0

0:2

0:4

0:6

0:8

1

0

0:2

0:4

0:6

0:8

1

t

t

(c)

(d)

Design specications requiring the decoupling of responses to

Figure

8.7

step commands.

the inputs one at a time. Figure 8.8 shows the response of the commanded variables

to a particular command signal that has a set-point change in 1 at = 0 5 and

w

t

:

then a set-point change in 2 at = 2 0. The perturbation in 2 right after = 0 5

w

t

:

z

t

:

and the perturbation in 1 right after = 2 0 are due to command interaction.

z

t

:

The speci cation mimo env limits this perturbation, and guarantees that after the

H

set-point change for 2, for example, the e ect on 1 will fade away.

z

z

An extreme form of mimo env is to require that the o -diagonal step responses

H

be zero, or equivalently, that cc be diagonal. This is called exact or complete

H

decoupling:

dcpl =

cc is diagonal

H

fH

j

H

g

:

This speci cation forbids any command interaction at all, regardless of the com-

mand signals. dcpl is a ne.

H

8.1 INPUT/OUTPUT SPECIFICATIONS

181

2

1

w

@

@

R

1:5

@

I

@

1

1

z

0:5

0

2

z

;

;

;0:5

;

;

2

w

;1

0

0:5

1

1:5

2

2:5

3

3:5

4

t

An example of command interaction in a two-input, two-output

Figure

8.8

system. The individual step responses of the system are shown in gure 8.7.

Output 1 tracks a step change in the command signal 1. However, 1

z

w

z

is perturbed by a step change in the command signal 2. Similarly, 2 is

w

z

perturbed by a step change in 1.

w

Miscellaneous Step Response Specifications

Other speci cations often expressed in terms of the step response of cc include

H

monotonic step response and I/O slew-rate limits. The monotonic step response

constraint is

mono sr =

( ) is nondecreasing for all

0

H

f

H

j

s

t

t

g

=

( ) 0 for all

0

f

H

j

h

t

t

g

where ( ) is the impulse response of cc. Thus, mono sr requires that the com-

h

t

H

H

manded variable move only in the direction of the new set-point command in re-

sponse to a single abrupt change in set-point. mono sr is a stricter speci cation

H

than requiring that both the undershoot and overshoot be zero. mono sr is convex.

H

A related speci cation is a slew-rate limit on the step response:

slew sr =

d

( ) = ( )

slew for all

0

H

H

s

t

jh

t

j

M

t

:

dt

Response to Other Inputs

We close this section by noting that the response of c to any particular command

z

signal, and not just a unit step, can be substituted for the step response in all of

182