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

CHAPTER 8 PERFORMANCE SPECIFICATIONS

1:4

1:2

1

os

0:8

() 0:6

t

s

0:4

0:2

us

0

;0:2

;0:4

0

1

2

3

4

5

6

7

8

9

10

t

A typical step response and its overshoot ( os) and under-

Figure

8.1

s

shoot ( us). The asymptotic tracking specication asympt trk requires that

H

the step response converge to one as

.

t

!

1

The functionals os and us are usually used together with the asymptotic track-

ing constraint asympt trk otherwise we might use the relative overshoot and relative

H

undershoot de ned by

8

sup ( )

1

if

0

<

cc(0)

cc(0)

ros( cc) =

t 0 s t =H

;

H

>

H

:

+

if cc(0) 0

1

H

8

sup ( )

0

<

cc(0)

if cc(0)

rus( cc) =

t 0 ;s t =H

H

>

H

:

+

if cc(0) 0

1

H

:

It is less obvious that the speci cations

ros =

ros( cc)

(8.1)

H

fH

j

H

g

rus =

rus( cc)

(8.2)

H

fH

j

H

g

are convex. To see that the relative overshoot constraint ros is convex, we rewrite

H

it as

ros =

cc(0) 0 ( ) (1 + ) cc(0) 0 for all

0

H

fH

j

H

>

s

t

;

H

t

g

:

If

~

ros and 0

1, then

=

+ (1

) ~ satis es cc(0) 0,

H

H

2

H

H

H

;

H

H

>

and for each

0 we have ( ) (1 + ) cc(0) 0. Hence,

ros.

t

s

t

;

H

H

2

H

8.1 INPUT/OUTPUT SPECIFICATIONS

175

Since the functional inequality speci cations (8.1{8.2) are convex for each ,

the relative overshoot and relative undershoot functionals are quasiconvex they are

not, however, convex. If one step response, ( ), has a relative overshoot of 30%,

s

t

and another step response ~( ) has a relative overshoot of 10%, then their average

s

t

has a relative overshoot not exceeding 30% but it may exceed 20%, the average

of the two relative overshoots. An example of two such step responses is shown in

gure 8.2.

3

s

;

;

2:5

( + ~) 2

s

s

=

;

2

;

;

1:5

~s

;

;

1

0:5

0

0

1

2

3

4

5

6

7

8

9

10

t

The relative overshoot of the step responses and ~ are 30%

Figure

8.2

s

s

and 10% respectively. Their average, ( + ~) 2, has a relative overshoot of

s

s

=

23%. This example shows that relative overshoot is not a convex functional

of . It is, however, quasiconvex.

H

Rise Time and Settling Time

There are many de nitions of rise time and settling time in use we shall use

rise( cc) = inf

( ) 0 8 for

H

fT

j

s

t

>

:

t

T

g

settle( cc) = inf

( ) 1 0 05 for

H

fT

j

js

t

;

j

<

:

t

T

g

as illustrated in gure 8.3. The functional rise is usually used together with the

asymptotic tracking speci cation asympt trk we can also de ne relative or normal-

H

ized rise time.

The functional inequality speci cations

rise =

rise( cc)

max

H

fH

j

H

T

g

176