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

CHAPTER 13 ELEMENTS OF CONVEX ANALYSIS

13.4.7

A Worst Case Norm

We consider the particular worst case norm described in section 5.1.3:

(H) = H wc = sup Hu

u

Mampl _u

Mslew :

k

k

fk

k

j

k

k

k

k

g

1

1

1

We rst rewrite as

Z

(H) = sup

1

v(t)h(t)dt v

Mampl _v

Mslew :

(13.5)

0

k

k

k

k

1

1

Now for each signal v we de ne the linear functional

v

Z

(H) = 1 v(t)h(t)dt:

0

We can express the worst case norm as a maximum of a set of these linear func-

tionals: (H)=sup v(H) v Mampl _v Mslew :

f

j

k

k

k

k

g

1

1

We proceed as follows to nd a subgradient of at the transfer matrix H0. Find

a signal v0 such that

v

Z

1

0

Mampl

_v0

Mslew

v0(t)h0(t)dt = (H0):

k

k

k

k

1

1

0

(It can be shown that in this case there always is such a v0 some methods for nding

v0 are described in the Notes and References for chapter 5.) Then a subgradient of

at H0 is given by

sg(H) = v0(H):

The same procedure works for any worst case norm: rst, nd a worst case

input signal u0 such that H0 wc = H0u0 output. This task must usually be done

k

k

k

k

to evaluate H0 wc anyway. Now nd any subgradient of the convex functional

k

k

u0(H) = Hu0 output it will be a subgradient of

wc at H0.

k

k

k

k

13.4.8

Subgradient for the Negative Dual Function

In this section we show how to nd a subgradient for

at , where is the dual

;

function introduced in section 3.6.2 and discussed in section 6.6. Recall from (6.8) of

section 6.6 that

can be expressed as the maximum of a family of linear functions

;

of we can therefore use the maximum tool to nd a subgradient.

We start by nding any Hach such that

( ) = 1 1(Hach) + + L L(Hach):

Then a subgradient of

at is given by

;

2

1(Hach) 3

g =

.

6

..

7

:

;

4

5

L(Hach)

13.5 SUBGRADIENTS ON A FINITE-DIMENSIONAL SUBSPACE

307

13.5

Subgradients on a Finite-Dimensional Subspace

In the previous section we determined subgradients for many of the convex func-

tionals we encountered in chapters 8{10. These subgradients are linear functionals

on the in nite-dimensional space of transfer matrices most numerical computation

will be done on nite-dimensional subspaces of transfer matrices (as we will see in

chapter 15). In this section we show how the subgradients computed above can be

used to calculate subgradients on nite-dimensional subspaces of transfer matrices.

Suppose that we have xed transfer matrices H0 H1 ... HN, and is some

convex functional on transfer matrices. We consider the convex function ' : N

R

!

given by

R

'(x) = (H0 + x1H1 + + xNHN):

To determine some g @'(~x), we nd a subgradient of at the transfer matrix

2

H0 + ~x1H1 + + ~xNHN, say, sg. Then

2

sg(H1) 3

g =

.

6

..

7

@'(~x):

2

4

5

sg(HN)

Let us give a speci c example using our standard plant of section 2.4. Consider

the weighted peak tracking error functional of section 11.1.2,

'pk trk(

) = W H(a)

13 + H(b)

13 + (1

)H(c)

13

1

;

;

;

pk gn

where

W = 0:5

s + 0:5

H(a)

44:1s3 + 334s2 + 1034s + 390

13 =

;

s6 + 20s5 + 155s4 + 586s3 + 1115s2 + 1034s + 390

H(b)

220s3 + 222s2 + 19015s + 7245

13 =

;

s6 + 29:1s5 + 297s4 + 1805s3 + 9882s2 + 19015s + 7245

H(c)

95:1s3 24:5s2 + 9505s + 2449

13 =

;

;

s6 + 33:9s5 + 425s4 + 2588s3 + 8224s2 + 9505s + 2449:

'pk trk has the form

'pk trk(

) = H0 + H1 + H2 pk gn

k

k

where H0=W H(c) 1;

H1 = W H(a) H(c)

;

H2 = W H(b) H(c) :

;

308