Biomedical Engineering Reference
In-Depth Information
m
( · )
R + R
controlled variables u
:
that control the dynamical system. It can be
written formally as
T
f 0
g 0
min
u ( · ) , T
(
t
,
x
(
t
) ,
u
(
t
))
d t
+
(
T
,
x
(
T
))
0
x
(
t
)=
f
(
t
,
x
(
t
) ,
u
(
t
)) , ∀
t
[
0
,
T
]
u
(
t
)
U t , ∀
t
[
0
,
T
]
x
(
0
)=
x 0 ,
x
(
T
)
M 1
(16)
Here, f 0 is the cost function and g 0 is the final cost. The final time T can be either
fixed or be part of the control. The dynamical system is represented by function f ,
which gives the evolution of x and is controlled by u . At each time, the control may
be subject to constraints represented by the set U t and the set M 1 is a subset of
n
R
representing conditions on the final state. If we replace the constraints x
(
0
)=
x 0 and
x
,wehavea T -periodic optimal control problem (see [ 28 ]
for more precision on the consequences of this model).
A major tool of optimal control is Pontryagin's maximum principle [ 121 ]. It
gives necessary optimality conditions for the optimal trajectories. We denote the
Hamiltonian of the system by
(
T
)
M 1 by x
(
T
)=
x
(
0
)
n
i = 1 p i f i ( t , x , u ) + p 0 f 0
p 0
H
(
t
,
x
,
p
,
,
u
)=
(
t
,
x
,
u
) ,
p 1
p n
n
and p 0
where p
=(
,...,
) R
R
.If u
( · )
associated with the trajectory x
( · )
is an optimal control on
[
0
,
T
]
, then there exists a continuous application p
( · )
called
the adjoint vector and a nonpositive number p 0 such that for almost all t
[
0
,
T
]
,
)=
H
)=
H
p 0
p 0
x
(
t
p (
t
,
x
(
t
) ,
p
(
t
) ,
,
u
(
t
)) ,
p
(
t
x (
t
,
x
(
t
) ,
p
(
t
) ,
,
u
(
t
))
(17)
and we have the maximisation condition for almost all t
[
0
,
T
]
,
p 0
p 0
H
(
t
,
x
(
t
) ,
p
(
t
) ,
,
u
(
t
)) =
max
v
H
(
t
,
x
(
t
) ,
p
(
t
) ,
,
v
) .
(18)
U t
If in addition, the final time to reach the target M 1 is not fixed, we have the condition
g 0
p 0
p 0
max
v
H
(
T
,
x
(
T
) ,
p
(
T
) ,
,
v
)=
t (
T
,
x
(
T
))
U T
n
and if M 1 is manifold of
R
with a tangent space T x M 1 at x ,wehave
p 0
g
p
(
T
)
t (
T
,
x
(
T
))
T x ( T ) M 1 .
 
Search WWH ::




Custom Search