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

CHAPTER 11 A PICTORIAL EXAMPLE

2

1:5

(b)

H

1

0:5

(a)

(b)

(c)

0:6H

+

0:3H

+

0:1H

(c)

(a)

H

H

0

(a)

(b)

(c)

;0:5

;0:2H

;

0:6H

+

1:8H

;1

;1

;0:5

0

0:5

1

1:5

2

Each point (

) corresponds to a closed-loop transfer matrix

Figure

11.1

that lies in the plane through

,

, and

.

(a)

(b)

(c)

H

H

H

H

Many of the gures in this chapter show level curves of functionals on slice,

H

,

i.e.

n

]

(a) + (b) + (1

) (c) = o

T

H

H

;

;

H

:

For quasiconvex functionals this set might not be a connected curve, so we de ne

the level curve to be the boundary of the convex set

n

]

(a)

(b)

(c)

o

T

+

+ (1

)

H

H

;

;

H

:

In most cases these two de nitions agree.

11.1

I/O Specifications

11.1.1

A Settling Time Limit

The step responses from the reference input to p for the three closed-loop systems

r

y

are shown in gure 11.2. Figure 11.3 shows the level curves of the reference to p

r

y

settling-time functional settle, ,

i.e.

settle

(a)

(b)

(c)

13 +

13 + (1

) 13

(11.1)

H

H

;

;

H

:

Recall from section 8.1.1 that settle, and therefore (11.1), are quasiconvex. From

gure 11.3 it can be seen that the level curves bound convex subsets.

11.1 I/O SPECIFICATIONS

251

1:6

(a)

s

13

1:4

;

;

(c)

s

13

1:2

;

;

1

@

I

@

(b)

0:8

s

13

0:6

0:4

0:2

0

;0:2

0

1

2

3

4

5

6

7

8

9

10

t

The step responses from the reference input, , to plant out-

Figure

11.2

r

put, , for the closed-loop transfer matrices

,

, and

.

(a)

(b)

(c)

y

H

H

H

p

2

1:5

9

1

8

0:5

7

0

5

6

7

8

9

;0:5

;1

;1

;0:5

0

0:5

1

1:5

2

Level curves of the step response settling time, from the

Figure

11.3

reference to , given by (11.1).

r

y

p

252