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

CHAPTER 11 A PICTORIAL EXAMPLE

11.6

Nonconvex Design Specifications

11.6.1

Controller Stability

Figure 11.22 shows the subset of slice that is achieved by an open-loop stable

H

controller, i.e.,

]T

H(a) + H(b) + (1

)H(c) is

;

;

achieved by a stable controller K

:

(11.27)

From gure 11.22, we see that (11.27) is a nonconvex subset of slice. We conclude

H

that a speci cation requiring open-loop controller stability,

P

1P

k stab = H

H = Pzw + PzuK(I yuK); yw

;

H

for some stable K that stabilizes P

is not in general convex.

3

2:5

2

1:5

1

0:5

0

;0:5

;1

;5

;4

;3

;2

;1

0

1

2

3

Region where the closed-loop transfer matrix is achieved

Figure

11.22

H

by a controller that is open-loop stable. It is not convex.

11.7

A Weighted-Max Functional

Consider the functional

'wt max( ) = max 'pk trk( ) 0:5'max sens( ) 15'rms yp(

)

f

g

11.7 A WEIGHTED-MAX FUNCTIONAL

269

where the functions 'pk trk, 'max sens, and 'rms yp are given by (11.2), (11.18),

and (11.6). The level curves of the function 'wt max(

) are shown in gure 11.23.

The function 'wt max will be used for several examples in chapter 14.

2

4:0

3:5

1:5

3:0

2:5

1

2:0

1:5

0:5

0

0:8

0:9

1:0

;0:5

1:1

1:2

1:3

1:4

;1

;1

;0:5

0

0:5

1

1:5

2

The level curves of wt max(

).

Figure

11.23

'

270