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

CHAPTER 8 PERFORMANCE SPECIFICATIONS

1:4

1:2

1

0:8

()ts

0:6

0:4

0:2

rise

settle

;

;

;

;

0

0

1

2

3

4

5

6

7

8

9

10

t

The value of the rise-time functional, rise, is the earliest time

Figure

8.3

after which the step response always exceeds 0 8. The value of the settling-

:

time functional, settle, is the earliest time after which the step response is

always within 5% of 1 0.

:

settle =

settle( cc)

max

H

fH

j

H

T

g

are convex: if two step responses each settle to with 5% within some time limit

max, then so does their average. Thus, the rise-time and settling-time functionals

T

rise and settle are quasiconvex, i.e.,

rise( cc + (1

) ~cc) max rise( cc) rise( ~cc)

H

;

H

f

H

H

g

for all 0

1, but we do not generally have

rise( cc + (1

) ~cc)

rise( cc) + (1

) rise( ~cc)

H

;

H

H

;

H

so they are not convex. Figure 8.4 demonstrates two step responses for which this

inequality, with = 0 5, is violated.

:

We mention that there are other de nitions of rise-time functionals that are not

quasiconvex. One example of such a de nition is the time for the step response to

rise from the signal level 0.1 (10%) to 0.9 (90%):

10 90( cc) = inf

( ) 0 9 inf

( ) 0 1

H

ft

j

s

t

:

g

;

ft

j

s

t

:

g:

;

While this may be a useful functional of cc in some contexts, we doubt the utility

H

of the (nonconvex) speci cation 10 90( cc)

max, which can be satis ed by

H

T

;

a step response with a long initial delay or a step response with very large high

frequency oscillations.

8.1 INPUT/OUTPUT SPECIFICATIONS

177

1:2

1

@

I

@

0:8

~s

0:6

@

I

@

0:4

@

( + ~) 2

s

s

=

0:2

@

I

@

s

0

0

1

2

3

4

5

6

7

8

9

10

t

The rise times of the step responses and ~ are 5.0 and 0.8

Figure

8.4

s

s

seconds respectively. Their average, ( +~) 2, has a rise time of 3 38 5 8 2

s

s

=

:

>

:

=

seconds. This example shows that rise time is not a convex functional of ,

H

although it is quasiconvex.

General Step Response Envelope Specifications

Many of the step response speci cations considered so far are special cases of general

envelope constraints on the step response:

env =

min( )

( ) max( ) for all

0

(8.3)

H

f

H

j

s

t

s

t

s

t

t

g

where min( ) max( ) for all

0. An example of an envelope constraint is shown

s

t

s

t

t

in gure 8.5. The envelope constraint env is convex, since if two step responses lie

H

in the allowable region, then so does their average.

We mention one useful way that a general envelope constraint env can be

H

expressed as a functional inequality speci cation with a convex functional. We

de ne the maximum envelope violation,

max env viol( cc) = sup max ( )

max( ) min( )

( ) 0

H

t 0

fs

t

;

s

t

s

t

;

s

t

g

:

The envelope speci cation env can be expressed as

H

env =

max env viol( cc) 0

H

f

H

j

H

g

:

General Response-Time Functional

The quasiconvex functionals rise and settle are special cases of a simple, general

paradigm for measuring the response time of a unit step response. Suppose that we

178