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

CHAPTER 12 SOME ANALYTIC SOLUTIONS

i

x

i

x

i

x

i

x

i

i

i

i

1

3 027

6

3 677 11

4 479 16

9 641

;

:

;

:

:

:

2

7 227

7

9 641 12

9 641 17

1 660

:

;

:

;

:

;

:

3

9 374

8

3 018 13

9 641 18

9 641

;

:

;

:

;

:

;

:

4

1 836

9

9 641 14

5 682 19

4 398

:

:

;

:

:

5

9 641 10

9 641 15

9 641 20

0 343

:

:

:

;

:

The coecients in the parametrization (12.57) for the step

T

able

12.1

response in gure 12.4.

where = (10 11) . The undershoot and rise-time specications are

i

c

=

i

20

X

( )

0 7

for 0

1 0

(12.59)

x

s

t

;

:

t

:

i

i

i=1

20

X

( ) 0 8

for

1 0

(12.60)

x

s

t

:

t

:

i

i

i=1

where is the step response of ( 10 + 1) . By nely discretizing , (12.59) and (12.60)

;i

s

s=

t

i

yield (many) linear inequality constraints on , i.e.

x

T

= 1 ...

(12.61)

a

x

b

k

L:

k

k

(12.58) and (12.61) can be solved as a feasibility linear program. The particular coecients

that we used, shown in table 12.1, were found by minimizing

subject to (12.58)

kxk

1

and (12.61).

About the Examples in Sections 12.4 and 12.5

These two examples can be expressed as innite-dimensional linear programming problems.

The references for the next two chapters are relevant see also Luenberger

], Rock-

Lue69

afellar

], Reiland

], Anderson and Philpott

], and Anderson

R

oc74,

R

oc82

Rei80

AP84

and Nash

].

AN87

We solved the problem (12.47) (ignoring the third equality constraint) by rst solving its

dual problem, which is

max

(1 + 2 );1

(12.62)

+

1

T

:

2

trk

;t

;2t

k

e

e

k

1

2

1

This is a convex optimization problem in , which is readily solved. The mysterious ( )

2

R

t

that we used corresponds exactly to the optimum and for this dual problem.

1

2

This dual problem is sometimes called a semi-innite optimization problem since the con-

straint involves a \continuum" of inequalities (i.e.,

+

1 for each

0).

;t

;2t

j

e

e

j

t

1

2

Special algorithms have been developed for these problems see for example the surveys

by Polak

], Polak, Mayne, and Stimler

], and Hettich

].

Pol83

PMS84

Het78