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

CHAPTER 13 ELEMENTS OF CONVEX ANALYSIS

The level curves of 'pk trk are shown in gure 11.4. A subgradient g @'pk trk( )

2

is given by

2

3

Z

1

sgn(h(t))h1(t)dt

g =

sg(H1)

6

7

sg

0

(H

6

7

2) =

Z

6

7

1

4

sgn(h(t))h2(t)dt 5

0

where h is the impulse response of H0 + H1 + H2, h1 is the impulse response of

H1, and h2 is the impulse response of H2 (see section 13.4.6).

Consider the point = 1, = 0, where 'pk trk(1 0) = 0:837. A subgradient at

this point is

g =

0:168

0:309 :

(13.6)

;

In gure 13.6 the level curve 'pk trk( ) = 0:837 is shown, together with the

subgradient (13.6) at the point 1 0]T. As expected, the subgradient determines a

half-space that contains the convex set

]T 'pk trk(

) 'pk trk(1 0) = 0:837 :

2

0 837

:

1:5

1

0:5

x

q

?

0

g

;0:5

;1

;1

;0:5

0

0:5

1

1:5

2

The level curve pk trk(

) = 0 837 is shown, together with

Figure

13.6

'

:

the subgradient (13.6) at the point 1 0] .

T

NOTES AND REFERENCES

309

Notes and References

Convex Analysis

Rockafellar's book

] covers convex analysis in detail. Other texts covering this

R

oc70

material are Stoer and Witzgall

], Barbu and Precupanu

], Aubin and Vin-

SW70

BP78

ter

], and Demyanov and Vasilev

]. The last three consider the innite-

A

V82

D

V85

dimensional (Banach space) case, carefully distinguishing between the many important dif-

ferent types of continuity, compactness, and so on, which we have not considered. Daniel's