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
