The objective is then simply ( ) with free. The solution of this free end-point optimal
z
T
T
control problem (which always occurs as
) yields
wc. This optimal control
T
!
1
kH
k
problem has linear dynamics and convex state and control constraints, so essentially all
numerical methods of solution will nd the value
wc (and not some local minimum).
kH
k
See, for example, the books by Pontryagin Pon62], Bryson and Ho BH75, ch.7], and
the survey article by Polak Pol73].
The Figures of Section 5.2.9
The input signals in gure 5.11 were computed by nely discretizing (with rst-order hold)
the optimization problem
10
Z
max
(10 ) ( )
1 0 h ; t w t dt
kw
k
1
_
1
kw
k
1
and solving the resulting linear program.
The input signals in gure 5.12 were computed from (5.6) with = 10.
T
The unit-energy input signals in gure 5.13 give the largest possible square root output en-
ergy for
5. These input signals were computed by nding the nite-time controllability
t
Gramian
5
Z
= 0 5]
5
5
contr =
T
T
At
T
A
t
= contr
A
contr A
W
W
0 e BB e
dt
W
;
e
W
e
:
where ( ) = (
) 1
;
H
s
C
sI
;
A
B
:
If is the largest eigenvalue of 1 2
1 2 and is the corresponding eigenvector,
=
obs
=
W
W
W
z
with
2 = 1, then the input signal
kz
k
( ) =
(5 )
1 2
T
for 0
5
T
A
;t
;
=
B
e
W
z
t
w
t
0
otherwise
has unit energy, and drives the system state to 1 2 at = 5. (It is actually the smallest
=
W
z
t
energy signal that drives the state from the origin to 1 2= in 5 seconds.) The output for
W
z
5,
t
( ) =
( 5) 1 2
A
t;
=
z
t
C
e
W
z
has square root energy
1 2
1 2
1 2
2
=
= ; max( = obs = ) 1=
W
W
W
(c.f. (5.39)).
